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(-x^2/b^2 + y^2/a^2 = 1);Expression(H) = (y = x^2/4);Focus(H) = F;Directrix(H) = l;L1:Line;L2:Line;Asymptote(G)={L1,L2};Intersection(l, L1) = A;Intersection(l,L2)=B;Abs(LineSegmentOf(A, B)) = 4*Abs(LineSegmentOf(O, F))", "coords": {} }, { "index": 16, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0016.png", "answer": "{4, 4*sqrt(2)}", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (y^2/4 + x^2/m = 1);Eccentricity(G) = sqrt(2)/2", "coords": {} }, { "index": 17, "success": false, "conic_type": "parabola", "error": "validation: parabola_point_off_curve", "params": { "p": 14.875267028808594, "direction": "right" }, "optimization": { "final_loss": 7.913284778594971, "converged": false, "iterations": 500 }, "output_path": "results/train/parabola/problem_0017.png", "answer": "5", "fact_expr": "G: Parabola;a: Number;P: Point;F: Point;Expression(G) = (y^2 = a*x);Negation(a=0);Coordinate(P) = (-1, 4);PointOnCurve(P, G);Focus(G) = F;Distance(P, F) = Abs(LineSegmentOf(P, F))", "coords": { "P": [ -1.0, 4.0 ] }, "validation_reasons": [ "parabola_point_off_curve" ] }, { "index": 18, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0018.png", "answer": "1", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);M: Point;PointOnCurve(M, G);Distance(M, Focus(G)) = 2", "coords": {} }, { "index": 19, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 20, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { 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"error": null, "params": { "a": 1.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0023.png", "answer": "y = 4*x - 7", "fact_expr": "G: Hyperbola;H: Line;P1: Point;P2: Point;P: Point;Coordinate(P) = (2, 1);Expression(G) = (x^2 - y^2/2 = 1);PointOnCurve(P, H);Intersection(H, G) = {P1, P2};MidPoint(LineSegmentOf(P1, P2)) = P", "coords": { "P": [ 2.0, 1.0 ] } }, { "index": 24, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0024.png", "answer": "y^2=4*x", "fact_expr": "C: Parabola;O: Origin;Vertex(C) = O;PointOnCurve(Focus(C), xAxis) = True;G: Line;Expression(G) = (y = x);Intersection(G, C) = {A, B};A: Point;B: Point;P: Point;Coordinate(P) = (2, 2);MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ 2.0, 2.0 ] } }, { "index": 25, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0025.png", "answer": "6", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);P: Point;PointOnCurve(P, G) = True;Distance(P, yAxis) = 4", "coords": {} }, { "index": 26, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0026.png", "answer": "5", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;A: Point;F: Point;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;PointOnCurve(O, G);Intersection(G, C) = {A, B};Abs(LineSegmentOf(A, F)) = 6;Abs(LineSegmentOf(B, F)) = 8;AngleOf(A, F, B) = pi/2;O: Origin", "coords": {} }, { "index": 27, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0027.png", "answer": "b^2", "fact_expr": "G: Hyperbola;b: Number;a: Number;F1: Point;P: Point;F2: Point;M: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, G);Negation(P=Vertex(G));LeftFocus(G) = F1;RightFocus(G) = F2;Intersection(InscribedCircle(TriangleOf(F1, P, F2)), RealAxis(G)) = M", "coords": {} }, { "index": 28, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.23606797749979, "b": 2.0, "major_axis": "x", "x_coef": 5.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0028.png", "answer": "8-4*sqrt(3)", 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"results/train/hyperbola/problem_0032.png", "answer": "2*sqrt(3)", "fact_expr": "G: Hyperbola;H: Circle;F: Point;O: Origin;A: Point;B: Point;Expression(G) = (x^2 - y^2/3 = 1);RightFocus(G) = F;Center(H)=F;Radius(H)=LineSegmentOf(F,O);L1:Line;L2:Line;Asymptote(G)={L1,L2};Intersection(L1,H)=A;Intersection(L2,H)=B;Negation(A=O);Negation(B=O)", "coords": {} }, { "index": 33, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 34, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0034.png", "answer": "[\\sqrt{2}/2, 1)", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;c: Number;F1: Point;F2: Point;Focus(G) = {F1, F2};Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);M: Point;PointOnCurve(M, G);DotProduct(VectorOf(F1, M), VectorOf(F2, M)) = 0;e: Number;Eccentricity(G) = e", "coords": {} }, { "index": 35, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0035.png", "answer": "34/3", "fact_expr": "G: Hyperbola;P: Point;Expression(G) = (x^2/9 - y^2/16 = 1);PointOnCurve(P, G);Distance(P, LeftDirectrix(G)) = 16/5", "coords": {} }, { "index": 36, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0036.png", "answer": "sqrt(5)/2", "fact_expr": "H: Line;Expression(H) = (m + x - 3*y = 0);m: Number;Negation(m=0);G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;A: Point;B: Point;L1: Line;L2: Line;Asymptote(G) = {L1, L2};Intersection(H, L1) = A;Intersection(H, L2) = B;P: Point;Coordinate(P) = (m, 0);Abs(LineSegmentOf(P, A)) = Abs(LineSegmentOf(P, B))", "coords": {} }, { "index": 37, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0037.png", "answer": "12*sqrt(2)/7", "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2/3 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;l: Line;PointOnCurve(F2, l);Inclination(l) = pi/4;A: Point;B: Point;Intersection(l, C) = {A, B}", "coords": {} }, { "index": 38, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 39, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.06247315928339958, "direction": "up" }, "optimization": { "final_loss": 0.0014067133888602257, "converged": true, "iterations": 338 }, "output_path": "results/train/parabola/problem_0039.png", "answer": "x^2=(1/4)*y", "fact_expr": "G: Parabola;O:Origin;H: Point;Coordinate(H) = (1, 4);Vertex(G)=O;PointOnCurve(Focus(G),yAxis);PointOnCurve(H,G)", "coords": { "H": [ 1.0, 4.0 ] } }, { "index": 40, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0040.png", "answer": "(2,2)", "fact_expr": "G: Parabola;M: Point;P: Point;F: Point;Expression(G) = (y^2 = 2*x);Coordinate(M) = (4, 2);Focus(G) = F;PointOnCurve(P, G);WhenMin(Abs(LineSegmentOf(P,M))+Abs(LineSegmentOf(P,F)))", "coords": { "M": [ 4.0, 2.0 ] } }, { "index": 41, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0041.png", "answer": "sqrt(5)/2", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;A: Point;LeftVertex(C) = A;L1: Line;L2: Line;PointOnCurve(A, L1) = True;PointOnCurve(b, L2) = True;J1: Line;J2: Line;Asymptote(C) = {J1, J2};IsPerpendicular(L1, J1) = True;IsPerpendicular(L2, J2) = True;FootPoint(L1, J1) = M;FootPoint(L2, J2) = N;M: Point;N: Point;O: Origin;Abs(LineSegmentOf(M, N)) = (4/5)*Abs(LineSegmentOf(O, A))", "coords": {} }, { "index": 42, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 43, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0043.png", "answer": "{(sqrt(3),+oo),(-oo,-sqrt(3))}", "fact_expr": "G: Hyperbola;H: Line;m: Real;Expression(G) = (x^2/4 - y^2 = 1);Expression(H) = (y = m + x);NumIntersection(H, G)=2", "coords": {} }, { "index": 44, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0044.png", "answer": "-1", "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;l1:Line;l2:Line;Expression(G) = (y^2 = 4*x);PointOnCurve(Focus(G), l);Intersection(l, G) = {A, B};TangentOfPoint(A,G)=l1;TangentOfPoint(B,G)=l2", "coords": {} }, { "index": 45, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0045.png", "answer": "{ApplyUnit(60,degree),ApplyUnit(120,degree)}", "fact_expr": "l: Line;G: Parabola;p: Number;F: Point;A: Point;B: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;PointOnCurve(F, l);Intersection(l,G)={A,B};Abs(LineSegmentOf(F, A)) = 3*Abs(LineSegmentOf(F, B))", "coords": {} }, { "index": 46, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 47, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0047.png", "answer": "1/16", "fact_expr": "G: Parabola;p: Number;H: Circle;p>0;Expression(G) = (y^2 = 2*p*x);Expression(H) = (-6*x + x^2 + y^2 - 7 = 0);IsTangent(H,Directrix(G))", "coords": {} }, { "index": 48, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.708203932499369, "b": 4.47213595499958, "major_axis": "x", "x_coef": 45.0, "y_coef": 20.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0048.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Ellipse;Expression(G) = (x^2/45 + y^2/20 = 1);F1: Point;F2: Point;Focus(G) = {F1,F2};O: Origin;H: Line;PointOnCurve(O,H) = True;Intersection(H,G) = {A,B};A: Point;B: Point;Area(TriangleOf(A,B,F2)) = 20", "coords": {} }, { "index": 49, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 5.0, "b": 4.898979485566356, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0049.png", "answer": "x^2/56+y^2/7=1", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/25 - y^2/24 = 1);H: Ellipse;Focus(G) = Focus(H);Distance(Directrix(H),Center(H)) = 8", "coords": {} }, { "index": 50, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0050.png", "answer": "x^2/4+y^2/3=1", "fact_expr": "C: Hyperbola;G: Ellipse;Expression(C) = (x^2 - y^2/3 = 1);InterReciprocal(Eccentricity(C),Eccentricity(G))", "coords": {} }, { "index": 51, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0051.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, Directrix(G));IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2));LineSegmentOf(P, F1)*LineSegmentOf(P, F2) = 4*(a*b)", "coords": {} }, { "index": 52, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 53, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0053.png", "answer": "2", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Focus(C) = {F1, F2};PointOnCurve(P, C);IsPerpendicular(VectorOf(P,F1),VectorOf(P,F2));Area(TriangleOf(P, F1, F2)) = 4", "coords": {} }, { "index": 54, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0054.png", "answer": "sqrt(2)", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;H: Line;A: Point;B: Point;F1: Point;O:Origin;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;PointOnCurve(F1, H);Intersection(H, Asymptote(C)) = {A, B};IsDiameter(LineSegmentOf(A,B),G);PointOnCurve(O,G)", "coords": {} }, { "index": 55, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0055.png", "answer": "2*x+y-2=0", "fact_expr": "G: Parabola;A: Point;B: Point;O: Origin;M: Point;Expression(G) = (y^2 = 2*x);Coordinate(M) = (1, 0);PointOnCurve(M,LineOf(A,B));Intersection(LineOf(A,B), G) = {A, B};Slope(LineSegmentOf(O, A))+Slope(LineSegmentOf(O, B)) = 1", "coords": { "M": [ 1.0, 0.0 ] } }, { "index": 56, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 57, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0057.png", "answer": "18\n20", "fact_expr": "G: Ellipse;H: Line;A: Point;B: Point;F2: Point;F1: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(G) = {F1, F2};PointOnCurve(F1,H);Intersection(H,G)={A,B}", "coords": {} }, { "index": 58, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0058.png", "answer": "1\n2", "fact_expr": "C1: Hyperbola;C2:Hyperbola;b: Number;a: Number;F: Point;Expression(C1) = (-y^2/b^2 + x^2/a^2 = 1);Expression(C2) = (-y^2/16 + x^2/4 = 1);Coordinate(F) = (sqrt(5), 0);Asymptote(C1)=Asymptote(C2);RightFocus(C1) = F", "coords": { "F": [ 2.23606797749979, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 59, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0059.png", "answer": "x = -2", "fact_expr": "G: Parabola;p: Number;H: Circle;p>0;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (8*y - 4*x + x^2 + y^2 + 19 = 0);PointOnCurve(Center(H), G)", "coords": {} }, { "index": 60, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0060.png", "answer": "20", "fact_expr": "F1: Point;F2: Point;Focus(G) = {F1, F2};G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);H: Line;A: Point;B: Point;PointOnCurve(F1, H);Intersection(H, G) = {A, B}", "coords": {} }, { "index": 61, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 62, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0062.png", "answer": "x^2=4*y", "fact_expr": "C: Parabola;F: Point;Coordinate(F) = (0, 1);Focus(C) = F", "coords": { "F": [ 0.0, 1.0 ] } }, { "index": 63, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0063.png", "answer": "2/3", "fact_expr": "G: Ellipse;Expression(G) = (y^2/5 + x^2/a^2 = 1);a: Number;a>sqrt(5);F: Point;LeftFocus(G) = F;H: Line;Expression(H) = (x = m);m: Number;A: Point;B: Point;Intersection(G,H) = {A,B}\t;Max(Perimeter(TriangleOf(F,A,B))) = 12", "coords": {} }, { "index": 64, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0064.png", "answer": "y=pm*(4/3)*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1)", "coords": {} }, { "index": 65, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0065.png", "answer": "x^2/4 + y^2/2 = 1", "fact_expr": "C: Ellipse;b: Number;a: Number;D: Point;A: Point;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftVertex(C)=A;UpperVertex(C)=D;LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P,LineSegmentOf(A,D));Max(DotProduct(VectorOf(P, F1),VectorOf(P, F2)))=2;Min(DotProduct(VectorOf(P,F1),VectorOf(P,F2)))=-2/3", "coords": {} }, { "index": 66, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0066.png", "answer": "{sqrt(2),sqrt(3)}", "fact_expr": "H: Line;Expression(H) = (y = k*x + 1);k: Number;k > 0;G: Hyperbola;Expression(G) = (x^2 - y^2/2 = 1);NumIntersection(H, G) = 1", "coords": {} }, { "index": 67, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 68, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0068.png", "answer": "(0,12)", "fact_expr": "G: Hyperbola;k: Real;Expression(G) = (x^2/4 - y^2/k = 1);Eccentricity(G) =e;e:Number;In(e,(1,2))", "coords": {} }, { "index": 69, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0069.png", "answer": "5", "fact_expr": "E: Ellipse;b: Number;a: Number;G: Line;A: Point;B: Point;F2: Point;F1: Point;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(E) = F1;RightFocus(E) = F2;PointOnCurve(F1, G);Intersection(G, E) = {A, B};Abs(LineSegmentOf(A, F1)) = 3*Abs(LineSegmentOf(F1, B));Abs(LineSegmentOf(A, B)) = 4;Perimeter(TriangleOf(A, B, F2)) = 16", "coords": {} }, { "index": 70, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0070.png", "answer": "sqrt(5)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);C:Curve;Expression(C)=(x^2/a^2+y^2=1);a>0;a:Number;Focus(G)=RightFocus(C)", "coords": {} }, { "index": 71, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0071.png", "answer": "[4, +oo)", "fact_expr": "l: Line;C: Parabola;P: Point;Q: Point;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {P, Q}", "coords": {} }, { "index": 72, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 73, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0073.png", "answer": "9", "fact_expr": "l: Line;C: Hyperbola;A: Point;F: Point;B: Point;O:Origin;Expression(C) = (-y^2/9 + x^2/a^2 = 1);PointOnCurve(O, l);Intersection(l, C) = {A, B};LeftFocus(C) = F;IsPerpendicular(LineSegmentOf(A, F), LineSegmentOf(B, F));a:Number;a>0", "coords": {} }, { "index": 74, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0074.png", "answer": "sqrt(6)/3", "fact_expr": "C: Ellipse;b: Number;a: Number;A: Point;B: Point;M: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(B) = (0, a);Intersection(RightDirectrix(C), xAxis) = A;VectorOf(A,B)=3*VectorOf(A,M);PointOnCurve(M, C)", "coords": {} }, { "index": 75, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0075.png", "answer": "3, 25/3", "fact_expr": "G: Ellipse;m: Real;Expression(G) = (x^2/5 + y^2/m = 1);Eccentricity(G) = sqrt(10)/5", "coords": {} }, { "index": 76, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0076.png", "answer": "2", "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;P: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(120, degree);Area(TriangleOf(F1, P, F2)) = sqrt(3)", "coords": {} }, { "index": 77, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0077.png", "answer": "27/4", "fact_expr": "G: Parabola;Expression(G) = (y = x^2);A: Point;B: Point;Coordinate(A) = (1, 1);Coordinate(B) = (-2, 4);M: Point;Intersection(TangentOnPoint(A,G),TangentOnPoint(B,G))=M;PointOnCurve(A,G);PointOnCurve(B,G)", "coords": { "A": [ 1.0, 1.0 ], "B": [ -2.0, 4.0 ] } }, { "index": 78, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0078.png", "answer": "3", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2/5 + y^2/m = 1);PointOnCurve(Focus(G), xAxis);e:Number;Eccentricity(G)=e;e = sqrt(10)/5", "coords": {} }, { "index": 79, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0079.png", "answer": "6", "fact_expr": "P: Point;PointOnCurve(P, G);G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);H1: Circle;Expression(H1) = (y^2 + (x + 1)^2 = 1);PointOnCurve(Q, H1) = True;Q: Point;R: Point;H2:Circle;Expression(H2) = (y^2 + (x - 1)^2 = 1);PointOnCurve(R, H2) = True", "coords": {} }, { "index": 80, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0080.png", "answer": "(\\sqrt{2} - 1, 1)", "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;F2: Point;c: Number;P: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);RightFocus(G) = F2;LeftFocus(G) = F1;PointOnCurve(P, G);a/Sin(AngleOf(P, F1, F2)) = c/Sin(AngleOf(P, F2, F1))", "coords": {} }, { "index": 81, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0081.png", "answer": "-1", "fact_expr": "G: Parabola;H: Line;t: Number;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (y = t + x);Intersection(H, G) = {A, B};IsChordOf(LineSegmentOf(A,B),G);XCoordinate(MidPoint(LineSegmentOf(A, B))) = 3;Negation(A=B)", "coords": {} }, { "index": 82, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0082.png", "answer": "3", "fact_expr": "G: Hyperbola;a: Number;b: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;A:Point;LeftVertex(G)=A;PointOnCurve(P,G);3*VectorOf(P,A)=2*VectorOf(P,F1)+VectorOf(P,F2)", "coords": {} }, { "index": 83, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 2.0, "major_axis": "y", "x_coef": 4.0, "y_coef": 8.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0083.png", "answer": "x+y-3=0", "fact_expr": "P: Point;Coordinate(P) = (1, 2);l: Line;G: Ellipse;Expression(G) = (x^2/4 + y^2/8 = 1);MidPoint(InterceptChord(l, G)) = P", "coords": { "P": [ 1.0, 2.0 ] } }, { "index": 84, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 85, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_0085.png", "answer": "1/5", "fact_expr": "G: Ellipse;k: Real;H: Circle;Expression(G) = (k*y^2 + x^2 = 1);Expression(H) = (x^2 + y^2 = 4);PointOnCurve(Focus(G), H)", "coords": {} }, { "index": 86, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0086.png", "answer": "4", "fact_expr": "C: Ellipse;b: Number;a: Number;A: Point;F1: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (sqrt(5), 0);LeftFocus(C) = F1;RightFocus(C) = F2;RightVertex(C) = A;Eccentricity(C) = sqrt(5)/5", "coords": { "A": [ 2.23606797749979, 0.0 ] } }, { "index": 87, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0087.png", "answer": "3*sqrt(5)/5", "fact_expr": "l: Line;G: Hyperbola;b: Number;a: Number;A: Point;B: Point;C: Point;D: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(RightFocus(G),l);IsPerpendicular(l,xAxis);Intersection(l, G) = {A, B};Intersection(l,Asymptote(G))={C,D};3*Abs(LineSegmentOf(A, B)) = 2*Abs(LineSegmentOf(C, D))", "coords": {} }, { "index": 88, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0088.png", "answer": "(1, 2)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P: Point;PointOnCurve(P, G);F: Point;Focus(G) = F;M: Point;Coordinate(M) = (3, 2);WhenMin(LineSegmentOf(P, M) + LineSegmentOf(P, F))", "coords": { "M": [ 3.0, 2.0 ] } }, { "index": 89, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 90, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 91, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0091.png", "answer": "8/3", "fact_expr": "C: Parabola;G: Line;A: Point;B: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C)=F;Inclination(G)=pi/3;PointOnCurve(F,G);Intersection(G, C) = {A, B};IsChordOf(LineSegmentOf(A, B), C)", "coords": {} }, { "index": 92, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 93, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0093.png", "answer": "6", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2 = 1)", "coords": {} }, { "index": 94, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 95, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 4.47213595499958, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0095.png", "answer": "17", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/16 - y^2/20 = 1);Focus(G) = {F1,F2};PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 9", "coords": {} }, { "index": 96, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0096.png", "answer": "-9/4", "fact_expr": "O: Origin;T: Ellipse;Expression(T) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;Eccentricity(T) = sqrt(2)/2;B: Point;Coordinate(B)=(0,1);OneOf(Vertex(T)) = B;P: Point;PointOnCurve(P, T);PointOnCurve(P, LineOf(P, A));PointOnCurve(P, LineOf(P, C));A: Point;Intersection(LineOf(P, A), T) = A;C: Point;Intersection(LineOf(P, C), T) = C;D: Point;MidPoint(LineSegmentOf(P, A)) = D;E: Point;MidPoint(LineSegmentOf(P, C)) = E;k1: Number;k2: Number;Slope(LineOf(P, A)) = k1;Slope(LineOf(P, C)) = k2;k1<0;k2<0;Slope(LineOf(O, D))+Slope(LineOf(O, E)) = 2", "coords": { "B": [ 0.0, 1.0 ] } }, { "index": 97, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0097.png", "answer": "-1", "fact_expr": "M: Point;Coordinate(M) = (0, 2);G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;PointOnCurve(F, LineOf(A,B)) = True;Intersection(LineOf(A,B),G) = {A,B};A: Point;B: Point;DotProduct(VectorOf(A, M), VectorOf(F, M)) = 0", "coords": { "M": [ 0.0, 2.0 ] } }, { "index": 98, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0098.png", "answer": "4", "fact_expr": "A: Point;B: Point;Abs(LineSegmentOf(A, B)) = 10;C:Curve;Expression(C)=(y^2=4*x);IsChordOf(LineSegmentOf(A,B),C);PointOnCurve(Focus(C),LineSegmentOf(A,B))", "coords": {} }, { "index": 99, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0099.png", "answer": "y=x-4/3", "fact_expr": "G: Ellipse;Expression(G) = (x^2/2 + y^2 = 1);M: Point;UpperVertex(G) = M;l: Line;P: Point;Q: Point;Intersection(l, G) = {P, Q};H: Point;Coordinate(H) = (1, 0);Orthocenter(TriangleOf(P, Q, M)) = H", "coords": { "H": [ 1.0, 0.0 ] } }, { "index": 100, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0100.png", "answer": "sqrt(26)-2*sqrt(3)", "fact_expr": "G: Hyperbola;A: Point;P: Point;F: Point;Expression(G) = (x^2/3 - y^2 = 1);Coordinate(A) = (3, 1);PointOnCurve(P, RightPart(G));RightFocus(G) = F", "coords": { "A": [ 3.0, 1.0 ] } }, { "index": 101, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0101.png", "answer": "4", "fact_expr": "C: Hyperbola;F1: Point;A: Point;M: Point;F2: Point;Expression(C) = (x^2 - y^2/3 = 1);Coordinate(M) = (2/3, 0);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(A, C);Distance(M, LineOf(A, F1)) = Distance(M, LineOf(A, F2))", "coords": { "M": [ 0.6666666666666666, 0.0 ] } }, { "index": 102, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0102.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(G) = F;H: Circle;Center(H) = F;Radius(H) = a;IsTangent(H,Asymptote(G)) = True", "coords": {} }, { "index": 103, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0103.png", "answer": "y=pm*sqrt(2)*x", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;H: Line;PointOnCurve(F1, H);Inclination(H) = ApplyUnit(30, degree);P: Point;Intersection(H, RightPart(G)) = P;IsPerpendicular(LineSegmentOf(P, F2), LineSegmentOf(F1, F2))", "coords": {} }, { "index": 104, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0104.png", "answer": "7", "fact_expr": "PointOnCurve(Focus(G),LineOf(A,B)) = True;G: Parabola;Expression(G) = (y^2 = 4*x);Intersection(LineOf(A,B),G) = {A,B};A: Point;x1: Number;y1: Number;B: Point;x2: Number;y2: Number;Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);x1 + x2 = 5", "coords": {} }, { "index": 105, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0105.png", "answer": "2*sqrt(3)/3", "fact_expr": "G: Hyperbola;M: Point;F1: Point;F2: Point;Expression(G) = (x^2 - y^2/2 = 1);Focus(G)={F1,F2};PointOnCurve(M, G);LineSegmentOf(M, F1)*LineSegmentOf(M, F2) = 0", "coords": {} }, { "index": 106, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 107, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0107.png", "answer": "y^2=4*x", "fact_expr": "l: Line;Expression(l) = (y = m*x - 4*m);m: Number;G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;A: Point;B: Point;Intersection(l, G) = {A, B};H: Circle;IsDiameter(LineSegmentOf(A, B), H);O: Origin;PointOnCurve(O, H)", "coords": {} }, { "index": 108, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0108.png", "answer": "{-4/5,-1}", "fact_expr": "H: Line;Expression(H) = (y = x*(a + 1) - 1);G: Parabola;Expression(G) = (y^2 = a*x);a: Number;Negation(a=0);NumIntersection(H, G)=1", "coords": {} }, { "index": 109, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 110, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0110.png", "answer": "(6,4*sqrt(3))", "fact_expr": "G: Parabola;A: Point;F: Point;P: Point;l: Line;Expression(G) = (y^2 = 8*x);Focus(G) = F;Directrix(G)=l;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P,A),l);FootPoint(LineSegmentOf(P,A),l)=A;Slope(LineOf(A,F)) = -sqrt(3)", "coords": {} }, { "index": 111, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0111.png", "answer": "6", "fact_expr": "m: Line;G: Parabola;p: Number;l: Line;P: Point;M: Point;F: Point;p>0;y0:Number;Expression(G) = (y^2 = 2*(p*x));Coordinate(P) = (2, y0);Focus(G) = F;TangentOfPoint(P,G)=l;PointOnCurve(P,m);IsParallel(m,xAxis);L:Line;PointOnCurve(F,L);IsParallel(L,l);Intersection(L,m)=M;Abs(LineSegmentOf(P, M)) = 5", "coords": {} }, { "index": 112, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_ecc_mismatch", "params": { "a": 1.9257268905639648, "b": 1.7993911322594118, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 2.5576271215184017e-15, "converged": true, "iterations": 2 }, "output_path": "results/train/ellipse/problem_0112.png", "answer": "2*sqrt(3)", "fact_expr": "C: Ellipse;b: Number;a: Number;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Eccentricity(C) = 1/2;FocalLength(C) = 2", "coords": {}, "validation_reasons": [ "ellipse_ecc_mismatch" ] }, { "index": 113, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0113.png", "answer": "13", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P,G) = True;Abs(LineSegmentOf(P, F1)) = 7", "coords": {} }, { "index": 114, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 0.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0114.png", "answer": "sqrt(5)/2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - 4*y^2 = 1)", "coords": {} }, { "index": 115, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 116, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 117, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -2.0, 0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_0117.png", "answer": "y^2=8*x", "fact_expr": "G: Circle;H: Line;C:Circle;Expression(G) = (y^2 + (x - 2)^2 = 1);Expression(H) = (x + 1 = 0);IsOutTangent(G,C);IsTangent(H,C)", "coords": {} }, { "index": 118, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 119, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0119.png", "answer": "(x-5)^2+y^2=16", "fact_expr": "G: Hyperbola;C: Circle;Expression(G) = (x^2/9 - y^2/16 = 1);Center(C) = RightFocus(G);IsTangent(Asymptote(G), C)", "coords": {} }, { "index": 120, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0120.png", "answer": "1+sqrt(2)", "fact_expr": "F2: Point;F1: Point;Focus(G) = {F1, F2};G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;P: Point;Q: Point;IsChordOf(LineSegmentOf(P, Q), G);PointOnCurve(F1, LineSegmentOf(P, Q));IsPerpendicular(LineSegmentOf(P, Q), xAxis);AngleOf(P, F2, Q) = ApplyUnit(90, degree);e: Number;Eccentricity(G) = e", "coords": {} }, { "index": 121, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0121.png", "answer": "(3, pm*2*sqrt(3))", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;H: Line;PointOnCurve(F, H);A: Point;B: Point;Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, F)) = 4", "coords": {} }, { "index": 122, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 123, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 124, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0124.png", "answer": "1/3", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(C) = F;Intersection(Directrix(C), xAxis) = A;A: Point;G: Circle;Center(G) = F;Radius(G) = LineSegmentOf(A,F);OneOf(Intersection(G,C)) = B;B: Point;O: Origin;AngleOf(A,B,O) = theta;theta: Number", "coords": {} }, { "index": 125, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0125.png", "answer": "2", "fact_expr": "l: Line;C: Parabola;p: Number;A: Point;B: Point;O: Origin;F: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;k:Number;Slope(l)=k;PointOnCurve(F,l);Intersection(l,C)={A,B};Abs(LineSegmentOf(A, B)) = 6*Abs(LineSegmentOf(O, F))", "coords": {} }, { "index": 126, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0126.png", "answer": "y^2=4*x", "fact_expr": "l: Line;C: Parabola;p: Number;A: Point;B: Point;F: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};VectorOf(A, F) = 2*VectorOf(F, B);YCoordinate(MidPoint(LineSegmentOf(A, B))) = sqrt(2)/2;IsChordOf(LineSegmentOf(A, B),C)", "coords": {} }, { "index": 127, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0127.png", "answer": "2*sqrt(3)/3", "fact_expr": "E: Parabola;F: Point;M: Point;Expression(E) = (y^2 = 2*x);Focus(E) = F;PointOnCurve(M, E);O: Origin", "coords": {} }, { "index": 128, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0128.png", "answer": "y^2/4 + x^2/3 = 1", "fact_expr": "G: Ellipse;b: Number;a: Number;B: Point;F2: Point;F1: Point;a > b;b > 0;Expression(G) = (x^2/b^2 + y^2/a^2 = 1);UpperFocus(G) = F1;LowerFocus(G) = F2;RightVertex(G) = B;Abs(LineSegmentOf(B,F2)) = Abs(LineSegmentOf(F1, F2));Abs(LineSegmentOf(F1, F2)) = 2", "coords": {} }, { "index": 129, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 5.196152422706632, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0129.png", "answer": "13", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 - y^2/27 = 1);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2;Abs(LineSegmentOf(P, F1)) = 7", "coords": {} }, { "index": 130, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 131, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0131.png", "answer": "3", "fact_expr": "G: Line;Expression(G) = (y = 2*x - 3);C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;PointOnCurve(Focus(C), G)", "coords": {} }, { "index": 132, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0132.png", "answer": "sqrt(3)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l: Line;Slope(l) > 0;Intersection(l, C) = {M, N};M: Point;N: Point;Abs(LineSegmentOf(M, F)) = 3*Abs(LineSegmentOf(N, F));PointOnCurve(F, l)", "coords": {} }, { "index": 133, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 134, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 135, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0135.png", "answer": "2*sqrt(3)/3", "fact_expr": "G: Hyperbola;m: Number;H: Parabola;Expression(G) = (-y^2 + x^2/m = 1);Expression(H) = (y^2 = 8*x);Focus(H) = RightFocus(G)", "coords": {} }, { "index": 136, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0136.png", "answer": "4", "fact_expr": "G: Parabola;p: Number;H: Ellipse;p>0;Expression(G) = (y^2 = -2*p*x);Expression(H) = (x^2/9 + y^2/5 = 1);PointOnCurve(RightFocus(H), Directrix(G))", "coords": {} }, { "index": 137, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 138, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 3.0, 0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_0138.png", "answer": "6", "fact_expr": "l: Line;C1: Parabola;C2: Circle;A: Point;M: Point;N: Point;B: Point;O: Origin;Expression(Directrix(C1))=(x=-3);Expression(C2)=((x-3)^2+y^2=1);Vertex(C1) = O;Center(C2)=C;C:Point;PointOnCurve(C,l);Intersection(l,C1)={A,B};Intersection(l,C2) = {M,N};Abs(LineSegmentOf(A, M)) < Abs(LineSegmentOf(A, N))", "coords": {} }, { "index": 139, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0139.png", "answer": "3", "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/9 + y^2/m = 1);m: Real;FocalLength(G) = 4*sqrt(3)", "coords": {} }, { "index": 140, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0140.png", "answer": "{sqrt(5), sqrt(5)/2}", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;L1: Line;L2: Line;d1: Number;d2: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, G);Asymptote(G) = {L1, L2};Distance(P, L1) = d1;Distance(P, L2) = d2;d1*d2 = (2/5)*a*b", "coords": {} }, { "index": 141, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 142, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0142.png", "answer": "(pm*2*sqrt(5), 5)", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);P: Point;PointOnCurve(P, G);Distance(P, Focus(G)) = 6", "coords": {} }, { "index": 143, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0143.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Circle;F1: Point;P: Point;F2: Point;O:Origin;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P,G);Abs(LineSegmentOf(P,F2))=Abs(LineSegmentOf(F1,F2));Center(H)=O;Radius(H)=b;IsIntersect(LineOf(P,F1),H);Eccentricity(G)=e;e:Number", "coords": {} }, { "index": 144, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0144.png", "answer": "y=pm*(sqrt(6)/2)*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/b^2 = 1);b: Number;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, RightPart(G));Abs(VectorOf(P, F1)) = 6;DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0", "coords": {} }, { "index": 145, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0145.png", "answer": "10", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);A: Point;PointOnCurve(A, G);Z: Line;PointOnCurve(A, Z);IsPerpendicular(Z, Directrix(G));B: Point;FootPoint(Z, Directrix(G)) = B;Abs(LineSegmentOf(A, B)) = 5;F: Point;Focus(G) = F", "coords": {} }, { "index": 146, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 147, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 148, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0148.png", "answer": "6", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/5 = 1)", "coords": {} }, { "index": 149, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0149.png", "answer": "{-3,-25/3}", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2/5 - y^2/m = 1);Eccentricity(G) = e;e=sqrt(10)/5;e:Number", "coords": {} }, { "index": 150, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0150.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/4 = 1)", "coords": {} }, { "index": 151, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0151.png", "answer": "5/2", "fact_expr": "C: Circle;Expression(C) = (-10*y + x^2 + y^2 + 16 = 0);A: Point;B: Point;D: Point;PointOnCurve(A, C);PointOnCurve(B, C);PointOnCurve(D, C);Distance(A, OneOf(Asymptote(G))) = 1;Distance(B, OneOf(Asymptote(G))) = 1;Distance(D, OneOf(Asymptote(G))) = 1;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number", "coords": {} }, { "index": 152, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0152.png", "answer": "(9 - sqrt(17))/8", "fact_expr": "G: Parabola;H: Circle;P: Point;Q: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (x^2 + (y - 4)^2 = 1);PointOnCurve(P, G);PointOnCurve(Q, H);WhenMin(Distance(P, Q) + Distance(P, Directrix(G)))", "coords": {} }, { "index": 153, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0153.png", "answer": "y=pm*(3/4)*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1)", "coords": {} }, { "index": 154, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0154.png", "answer": "9", "fact_expr": "G: Hyperbola;A: Point;P: Point;F: Point;Expression(G) = (x^2/4 - y^2/12 = 1);Coordinate(A) = (1, 4);LeftFocus(G) = F;PointOnCurve(P, RightPart(G))", "coords": { "A": [ 1.0, 4.0 ] } }, { "index": 155, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0155.png", "answer": "4*sqrt(3)-4", "fact_expr": "A: Point;Coordinate(A) = (3, 0);P: Point;PointOnCurve(P, G) = True;G: Parabola;Expression(G) = (y^2 = 8*x);H: Circle;Expression(H) = (y^2 + (x - 2)^2 = 1);Q: Point;PointOnCurve(Q, H) = True", "coords": { "A": [ 3.0, 0.0 ] } }, { "index": 156, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0156.png", "answer": "3", "fact_expr": "G: Hyperbola;b: Number;b>0;Expression(G) = (x^2/4 - y^2/b^2 = 1);Expression(OneOf(Asymptote(G))) = (3*x + 2*y = 0)", "coords": {} }, { "index": 157, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0157.png", "answer": "2*sqrt(2-1)", "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (4, 0);PointOnCurve(P, G);Focus(G)=F;Abs(LineSegmentOf(P,A))=2*Abs(LineSegmentOf(P,F))", "coords": { "A": [ 4.0, 0.0 ] } }, { "index": 158, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0158.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (x^2/9 - y^2/m = 1);Expression(Asymptote(G)) = (y = pm*x*(sqrt(5)/3));F: Point;Focus(G) = F", "coords": {} }, { "index": 159, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0159.png", "answer": "2*sqrt(3)/3", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2", "coords": {} }, { "index": 160, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 161, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0161.png", "answer": "sqrt(2)", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Eccentricity(C) = sqrt(3)/2;F: Point;RightFocus(C) = F;G: Line;PointOnCurve(F,G) = True;Slope(G) = k;k: Number;k>0;Intersection(G,C) = {A,B};A: Point;B: Point;VectorOf(A, F) = 3*VectorOf(F, B)", "coords": {} }, { "index": 162, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 163, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 164, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0164.png", "answer": "6", "fact_expr": "C: Parabola;p: Number;p>0;P:Point;Expression(C) = (x^2 = 2*(p*y));PointOnCurve(P,C);Distance(P, xAxis) = Distance(P,Focus(C)) -3", "coords": {} }, { "index": 165, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0165.png", "answer": "(1, sqrt(13)/3) + (sqrt(5), +oo)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;LeftFocus(C) = F1;RightFocus(C) = F2;F1: Point;F2: Point;Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);c: Number;N: Point;Coordinate(N) = (-c, (3*b^2)/(2*a));M: Point;PointOnCurve(M, LeftPart(C)) = True;Abs(LineSegmentOf(M, F2)) + Abs(LineSegmentOf(M, N)) > 4*b", "coords": {} }, { "index": 166, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0166.png", "answer": "a/2 - p/2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;A: Point;B: Point;M: Point;a:Number;a>2*p;IsChordOf(LineSegmentOf(A,B),G);Abs(LineSegmentOf(A, B)) = a;MidPoint(LineSegmentOf(A, B)) = M", "coords": {} }, { "index": 167, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0167.png", "answer": "4/5", "fact_expr": "C: Ellipse;a: Number;b: Number;G: Line;O: Origin;P: Point;M: Point;N: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Expression(G) = (3*x + 5*y - 1 = 0);Intersection(G, C) = {M, N};MidPoint(LineSegmentOf(M, N)) = P;Slope(LineOf(O, P)) = 3/5", "coords": {} }, { "index": 168, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 169, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 170, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0170.png", "answer": "2*sqrt(7)\n(pm*2*sqrt(7),0)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/9 = 1)", "coords": {} }, { "index": 171, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.47213595499958, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0171.png", "answer": "10", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/20 - y^2/5 = 1)", "coords": {} }, { "index": 172, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 173, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 0.9220545291900635, "b": 2.765991687774658, "orientation": "horizontal" }, "optimization": { "final_loss": 4.6528161874448415e-06, "converged": true, "iterations": 186 }, "output_path": "results/train/hyperbola/problem_0173.png", "answer": "3", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/81 + x^2/a^2 = 1);a: Number;a>0;Expression(OneOf(Asymptote(G))) = (y = 3*x)", "coords": {} }, { "index": 174, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0174.png", "answer": "8", "fact_expr": "G: Ellipse;A: Point;B: Point;F2: Point;F1: Point;Expression(G) = (x^2/25 + y^2/16 = 1);LeftFocus(G) = F1;PointOnCurve(F1, LineSegmentOf(A, B));Abs(LineSegmentOf(A, F2)) + Abs(LineSegmentOf(B, F2)) = 12;RightFocus(G) = F2;IsChordOf(LineSegmentOf(A,B),G)", "coords": {} }, { "index": 175, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 176, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 177, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 178, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0178.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (x^2 = -4*y);K: Point;PointOnCurve(K, G);F: Point;Coordinate(F) = (0, -1);E: Point;Coordinate(E) = (1, -3)", "coords": { "F": [ 0.0, -1.0 ], "E": [ 1.0, -3.0 ] } }, { "index": 179, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0179.png", "answer": "2", "fact_expr": "G: Parabola;p: Number;A: Point;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;PointOnCurve(A, G);XCoordinate(MidPoint(LineSegmentOf(A,F))) = 1", "coords": {} }, { "index": 180, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0180.png", "answer": "20", "fact_expr": "l: Line;G: Ellipse;F2: Point;A: Point;B: Point;F1: Point;Expression(G) = (x^2/25 + y^2/9 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;Inclination(l) = pi/3;PointOnCurve(F1, l);Intersection(l, G) = {A, B}", "coords": {} }, { "index": 181, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.4641016151377544, "major_axis": "x", "x_coef": 16.0, "y_coef": 12.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0181.png", "answer": "[1,4]", "fact_expr": "M: Point;Coordinate(M) = (m, 0);m: Real;PointOnCurve(M, MajorAxis(G)) = True;G: Ellipse;Expression(G) = (x^2/16 + y^2/12 = 1);P: Point;PointOnCurve(P, G) = True;WhenMin(Abs(LineSegmentOf(M, P)));RightVertex(G) = P", "coords": {} }, { "index": 182, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 2.0, "major_axis": "x", "x_coef": 8.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0182.png", "answer": "x^2 - y^2 / 3 = 1", "fact_expr": "C: Hyperbola;G: Ellipse;H: Line;Expression(G) = (x^2/8 + y^2/4 = 1);Expression(H) = (y = sqrt(3)*x);Focus(C) = Focus(G);OneOf(Asymptote(C))=H", "coords": {} }, { "index": 183, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0183.png", "answer": "1", "fact_expr": "H: Parabola;G: Hyperbola;n: Number;F: Point;Expression(H) = (y^2 = 8*x);Expression(G) = (x^2/3 - y^2/n = 1);Focus(H) = F;OneOf(Focus(G))=F", "coords": {} }, { "index": 184, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 185, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 186, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0186.png", "answer": "(-1/2,0)", "fact_expr": "E: Parabola;Expression(E) = (y^2 = -2*x)", "coords": {} }, { "index": 187, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0187.png", "answer": "1/(Cos(alpha)-Sin(alpha))", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;F2: Point;F1: Point;P: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;IsDiameter(LineSegmentOf(F1,F2),H);Intersection(H,RightPart(G))=P;AngleOf(P, F1, F2) = alpha;alpha:Number", "coords": {} }, { "index": 188, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0188.png", "answer": "sqrt(15)", "fact_expr": "G: Ellipse;F: Point;P: Point;O: Origin;M: Point;Expression(G) = (x^2/9 + y^2/5 = 1);LeftFocus(G) = F;PointOnCurve(P,G);Quadrant(P)=2;MidPoint(LineSegmentOf(P,F))=M;Abs(LineSegmentOf(O,F))=Abs(LineSegmentOf(O,M))", "coords": {} }, { "index": 189, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0189.png", "answer": "y=pm*sqrt(3)*x", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;B: Point;F: Point;A: Point;LeftVertex(C)=A;RightFocus(C)=F;e:Number;Eccentricity(C)=e;PointOnCurve(B,RightPart(C));Negation(B=RightVertex(C));AngleOf(B,F,A)/AngleOf(B,A,F)=e", "coords": {} }, { "index": 190, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0190.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;a: Number;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G)=F2;L:Line;OneOf(Asymptote(G))=L;Expression(L)=(y = x*(b/a));Distance(F2, L) = 2*a", "coords": {} }, { "index": 191, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0191.png", "answer": "1", "fact_expr": "G: Parabola;Expression(G) = (x^2 = -2*y)", "coords": {} }, { "index": 192, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0192.png", "answer": "16", "fact_expr": "G: Hyperbola;P: Point;Q: Point;F1:Point;F2: Point;Expression(G) = (x^2/16 - y^2/9 = 1);Focus(G)={F1,F2};PointOnCurve(F1, LineSegmentOf(P, Q));IsChordOf(LineSegmentOf(P, Q), G);Inclination(LineSegmentOf(P, Q)) = ApplyUnit(60, degree)", "coords": {} }, { "index": 193, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0193.png", "answer": "sqrt(6)/2", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;O: Origin;F: Point;RightFocus(C) = F;G: Line;PointOnCurve(F, G);Z1: Line;Z2: Line;Asymptote(C) = {Z1, Z2};P: Point;Intersection(G, Z1) = P;Q: Point;Intersection(G, Z2) = Q;IsPerpendicular(LineSegmentOf(O, P), LineSegmentOf(P, Q));VectorOf(Q, F) = 3*VectorOf(F, P)", "coords": {} }, { "index": 194, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0194.png", "answer": "sqrt(6)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/2 = 1);P: Point;PointOnCurve(P, OneOf(Asymptote(G)));Distance(P, Center(G)) = 3", "coords": {} }, { "index": 195, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0195.png", "answer": "2*sqrt(3)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(C) = F;l: Line;Directrix(C) = l;M: Point;PointOnCurve(M, l);N: Point;Intersection(LineSegmentOf(M, F), C) = N;VectorOf(M, N) = 2*VectorOf(N, F);O: Origin;S: Number;Area(TriangleOf(O, F, N)) = S;S = sqrt(3)", "coords": {} }, { "index": 196, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0196.png", "answer": "pi/2", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;M: Point;Intersection(Directrix(C), xAxis) = M;Z1: Line;Z2: Line;TangentOfPoint(M, C) = {Z1, Z2};P: Point;Q: Point;TangentPoint(Z1, C) = P;TangentPoint(Z2, C) = Q", "coords": {} }, { "index": 197, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 198, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0198.png", "answer": "10", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x);H: Line;PointOnCurve(Focus(G), H);A: Point;B: Point;Intersection(H, G) = {A, B};M: Point;MidPoint(LineSegmentOf(A, B)) = M;Distance(M, Directrix(G)) = 5", "coords": {} }, { "index": 199, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0199.png", "answer": "x + pm*y = 0", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 1)", "coords": {} }, { "index": 200, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0200.png", "answer": "sqrt(5)", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Distance(Focus(G), Asymptote(C)) = sqrt(5)/5", "coords": {} }, { "index": 201, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0201.png", "answer": "x^2/16+y^2/25=1", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(G) = (x^2/5 - y^2/4 = -1);Eccentricity(H)=3/5;Focus(H) = Focus(G)", "coords": {} }, { "index": 202, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.8284271247461903, "b": 2.8284271247461903, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0202.png", "answer": "14 + 8*sqrt(2)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 8);F2: Point;RightFocus(G) = F2;P: Point;Q: Point;PointOnCurve(F2, LineSegmentOf(P, Q)) ;IsChordOf(LineSegmentOf(P, Q), G) ;LineSegmentOf(P, Q) = 7;F1: Point;LeftFocus(G) = F1", "coords": {} }, { "index": 203, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0203.png", "answer": "26", "fact_expr": "l: Line;G: Hyperbola;A: Point;B: Point;F1: Point;F2: Point;Expression(G) = (x^2/16 - y^2/9 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(F2, l);Intersection(l,RightPart(G))={A, B};Abs(LineSegmentOf(A, B))=5", "coords": {} }, { "index": 204, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 205, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0205.png", "answer": "sqrt(2)/2", "fact_expr": "G: Parabola;H: Line;P: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (y = x + 2);PointOnCurve(P, G)", "coords": {} }, { "index": 206, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0206.png", "answer": "14", "fact_expr": "C: Circle;G: Parabola;Expression(G) = (y^2 = 4*x);PointOnCurve(Focus(G), C);PointOnCurve(Center(C), Directrix(G));Negation(PointOnCurve(Center(C), xAxis));H: Line;Expression(H) = (x + sqrt(3)*y - 3 = 0);IsTangent(H, C)", "coords": {} }, { "index": 207, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 208, "success": false, "conic_type": "parabola", "error": "validation: parabola_point_off_curve", "params": { "p": 0.9927864074707031, "direction": "right" }, "optimization": { "final_loss": 0.00015326803259085864, "converged": true, "iterations": 44 }, "output_path": "results/train/parabola/problem_0208.png", "answer": "(1, 0)\n5", "fact_expr": "G: Parabola;p: Number;P: Point;Expression(G) = (y^2 = 2*p*x);Coordinate(P) = (4, 4);PointOnCurve(P, G)", "coords": { "P": [ 4.0, 4.0 ] }, "validation_reasons": [ "parabola_point_off_curve" ] }, { "index": 209, "success": false, "conic_type": "parabola", "error": "validation: parabola_point_off_curve", "params": { "p": 8.218917846679688, "direction": "right" }, "optimization": { "final_loss": 5.290376663208008, "converged": true, "iterations": 487 }, "output_path": "results/train/parabola/problem_0209.png", "answer": "(-1/4, 0)", "fact_expr": "G: Parabola;m: Number;H: Point;Expression(G) = (y^2 = m*x);Coordinate(H) = (-1, 1);PointOnCurve(H,G)", "coords": { "H": [ -1.0, 1.0 ] }, "validation_reasons": [ "parabola_point_off_curve" ] }, { "index": 210, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0210.png", "answer": "x^2=sqrt(6)*y", "fact_expr": "G: Parabola;p: Number;A: Point;F: Point;B: Point;C: Point;E: Point;l: Line;L: Line;p>0;Expression(G) = (x^2 = 2*(p*y));Coordinate(C) = (0, (9/2)*p);Focus(G) = F;Directrix(G) = l;PointOnCurve(A,L);IsPerpendicular(l,L);FootPoint(l,L)=B;Intersection(LineSegmentOf(A, F), LineSegmentOf(B, C)) = E;Abs(LineSegmentOf(C, F)) = 2*Abs(LineSegmentOf(A, F));Area(TriangleOf(A, C, E)) = sqrt(3)", "coords": {} }, { "index": 211, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 212, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_ecc_mismatch", "params": { "a": 1.9257268905639648, "b": 1.7993911322594118, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 2.5576271215184017e-15, "converged": true, "iterations": 2 }, "output_path": "results/train/ellipse/problem_0212.png", "answer": "12", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2/16 + y^2/m = 1);PointOnCurve(Focus(G), xAxis);Eccentricity(G) = 1/2", "coords": {}, "validation_reasons": [ "ellipse_ecc_mismatch" ] }, { "index": 213, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0213.png", "answer": "8", "fact_expr": "P:Point;x1:Number;y1:Number;G:Ellipse;Expression(G)=(x^2/4+y^2=1);Coordinate(P)=(x1,y1);PointOnCurve(P,G)", "coords": {} }, { "index": 214, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0214.png", "answer": "y=2*x-3", "fact_expr": "G: Parabola;H: LineSegment;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(P) = (2, 1);IsChordOf(H, G);MidPoint(H)=P", "coords": { "P": [ 2.0, 1.0 ] } }, { "index": 215, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0215.png", "answer": "4*sqrt(5)/5", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;P: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = ((x + 1)^2 + (y - 2)^2 = 4);Coordinate(P) = (1, 2);L:Line;OneOf(Asymptote(G))=L;PointOnCurve(P,L)", "coords": { "P": [ 1.0, 2.0 ] } }, { "index": 216, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0216.png", "answer": "sqrt(2)/2", "fact_expr": "G: Parabola;O: Origin;P: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G)=F;PointOnCurve(P,G);Abs(LineSegmentOf(O, P)) = Abs(LineSegmentOf(P, F))", "coords": {} }, { "index": 217, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.3300262689590454, "b": 2.30381441116333, "orientation": "horizontal" }, "optimization": { "final_loss": 6.658357051492203e-06, "converged": true, "iterations": 122 }, "output_path": "results/train/hyperbola/problem_0217.png", "answer": "x^2/16 - y^2/48 = 1", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x);H: Parabola;Expression(H) = (y^2 = 32*x);OneOf(Focus(G)) = Focus(H)", "coords": {} }, { "index": 218, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0218.png", "answer": "9", "fact_expr": "F: Point;LeftFocus(G) = F;G: Hyperbola;Expression(G) = (x^2/4 - y^2/12 = 1);Coordinate(A) = (1, 4);A: Point;P: Point;PointOnCurve(P, RightPart(G)) = True", "coords": { "A": [ 1.0, 4.0 ] } }, { "index": 219, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0219.png", "answer": "3*sqrt(7)", "fact_expr": "C: Hyperbola;Expression(C) = (-x^2/3 + y^2/4 = 1);F1: Point;F2: Point;UpperFocus(C) = F1;LowerFocus(C) = F2;Z: Line;PointOnCurve(F2, Z);IsPerpendicular(Z, yAxis);P: Point;Q: Point;Intersection(Z, C) = {P, Q}", "coords": {} }, { "index": 220, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 221, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.3530144097625203, "b": 3.0920111302279403, "major_axis": "x", "x_coef": 12.000000000000005, "y_coef": 8.999999999999996 }, "optimization": { "final_loss": 0.0017180106211009743, "converged": true, "iterations": 38 }, "output_path": "results/train/ellipse/problem_0221.png", "answer": "x^2/12+y^2/9=1", "fact_expr": "G: Ellipse;H: Point;Coordinate(H) = (2, -sqrt(6));I: Point;Coordinate(I) = (2*sqrt(2), sqrt(3));PointOnCurve(H, G);PointOnCurve(I, G)", "coords": { "H": [ 2.0, -2.449489742783178 ], "I": [ 2.8284271247461903, 1.7320508075688772 ] } }, { "index": 222, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0222.png", "answer": "(\\sqrt{5} - 1)/2", "fact_expr": "C: Ellipse;b: Number;a: Number;B: Point;F: Point;A: Point;e: Number;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;RightVertex(C)=A;UpperVertex(C)=B;IsPerpendicular(LineSegmentOf(B, F), LineSegmentOf(B, A));Eccentricity(C) = e", "coords": {} }, { "index": 223, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 224, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 225, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0225.png", "answer": "2*sqrt(2)", "fact_expr": "C: Parabola;P: Point;x0:Number;y0:Number;Expression(C) = (x^2 = 4*y);Coordinate(P) = (x0, y0);PointOnCurve(P, C);Distance(P, Focus(C)) = 3", "coords": {} }, { "index": 226, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_ecc_mismatch", "params": { "a": 1.9257268905639648, "b": 1.7993911322594118, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 2.5576271215184017e-15, "converged": true, "iterations": 2 }, "output_path": "results/train/ellipse/problem_0226.png", "answer": "3", "fact_expr": "C1: Ellipse;b:Number;00;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-1, 0);Coordinate(F2)=(1, 0);LeftFocus(C) =F1;RightFocus(C)=F2;PointOnCurve(P, C);Range(Eccentricity(C))=[2,4]", "coords": { "F1": [ -1.0, 0.0 ], "F2": [ 1.0, 0.0 ] } }, { "index": 231, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0231.png", "answer": "4*sqrt(3)", "fact_expr": "G: Ellipse;Expression(G) = (x^2 + 4*y^2 = 16)", "coords": {} }, { "index": 232, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 233, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 234, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 235, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0235.png", "answer": "9", "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/25 + y^2/9 = 1);Focus(C) = {F1, F2};PointOnCurve(P, C);AngleOf(F1, P, F2) = ApplyUnit(90, degree)", "coords": {} }, { "index": 236, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 237, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0237.png", "answer": "8", "fact_expr": "G: Hyperbola;H: Line;M: Point;F2: Point;N: Point;F: Point;Expression(G) = (x^2/4 - y^2/3 = 1);LeftFocus(G) = F;PointOnCurve(F, H);Intersection(H, LeftPart(G)) = {M, N};RightFocus(G) = F2", "coords": {} }, { "index": 238, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0238.png", "answer": "3 + 2*sqrt(2)", "fact_expr": "G: Hyperbola;H: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/2 - y^2 = 1);Expression(H) = (x^2/4 + y^2 = 1);LeftFocus(G) = F1;LeftFocus(H) = F1;RightFocus(H) = F2;RightFocus(G) = F2;Quadrant(P) = 1;Intersection(H, G) = P", "coords": {} }, { "index": 239, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0239.png", "answer": "3*sqrt(10)/10", "fact_expr": "G: Hyperbola;M: Point;Expression(G) = (x^2 - y^2/9 = 1);Coordinate(M) = (-1, 0)", "coords": { "M": [ -1.0, 0.0 ] } }, { "index": 240, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0240.png", "answer": "2*y^2 = x - 1", "fact_expr": "G: Parabola;A: Point;Expression(G) = (y^2 = x);Coordinate(A) = (2, 0);P: Point;PointOnCurve(P, G);M: Point;MidPoint(LineSegmentOf(A, P)) = M", "coords": { "A": [ 2.0, 0.0 ] } }, { "index": 241, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 242, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.1622776601683795, "b": 2.449489742783178, "major_axis": "x", "x_coef": 10.0, "y_coef": 6.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0242.png", "answer": "16*sqrt(30)/9", "fact_expr": "G:Ellipse;Expression(G)=(x^2/10+y^2/6=1);l:Line;Inclination(l)=ApplyUnit(60,degree);A:Point;B:Point;Intersection(l,G)={A,B};Abs(LineSegmentOf(A,B))=8*sqrt(30)/9;PointOnCurve(C,G);C:Point;Negation(C=A);Negation(C=B)", "coords": {} }, { "index": 243, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0243.png", "answer": "5*sqrt(2)/4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 10*x);P: Point;PointOnCurve(P, G);H: Line;Expression(H) = (x + y + 5 = 0)", "coords": {} }, { "index": 244, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0244.png", "answer": "{y^2=-16*x,x^2=16*y}", "fact_expr": "G: Parabola;H: Line;O: Origin;Expression(H) = (x - y + 4 = 0);Vertex(G) = O;SymmetryAxis(G)=axis;PointOnCurve(Focus(G), H)", "coords": {} }, { "index": 245, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0245.png", "answer": "1", "fact_expr": "C: Parabola;G: Line;F: Point;P: Point;Q: Point;k:Number;Expression(C) = (y^2 = 4*x);Expression(G) = (y = k*(x - 1));Focus(C) = F;Intersection(G, C) = {P, Q}", "coords": {} }, { "index": 246, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 247, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0247.png", "answer": "sqrt(3)-1", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;A: Point;B: Point;F: Point;a > b;b > 0;L:Line;O:Origin;RightFocus(G) = F;PointOnCurve(O,L);Intersection(L,G) = {A, B};Abs(LineSegmentOf(A, B)) = 4;Abs(LineSegmentOf(B, F)) = 2*sqrt(3);AngleOf(A, B, F) = ApplyUnit(30, degree)", "coords": {} }, { "index": 248, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0248.png", "answer": "pm*1", "fact_expr": "l: Line;E: Parabola;p: Number;G: Circle;A: Point;B: Point;C: Point;D: Point;F: Point;p>0;Expression(E) = (y^2 = 2*p*x);Expression(G) = (x^2-p*x+y^2-(3/4)*p^2=0);Focus(E) = F;PointOnCurve(F, l);Intersection(l, E) = {A, B};Intersection(l,G)={C,D};Abs(LineSegmentOf(A, B)) = 2*Abs(LineSegmentOf(C, D))", "coords": {} }, { "index": 249, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/train/hyperbola/problem_0249.png", "answer": "x^2/1 - y^2/3 = 1", "fact_expr": "P: Point;Coordinate(P) = (2, -3);G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;PointOnCurve(P, G) = True;F1: Point;F2: Point;Focus(G) = {F1, F2} ;Distance(F1, F2) = 4", "coords": { "P": [ 2.0, -3.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 250, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 1.0, 2.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_0250.png", "answer": "{-3/4, 3/4}", "fact_expr": "H: Circle;Expression(H) = (-4*y - 2*x + x^2 + y^2 + 4 = 0);G: Hyperbola;Expression(G) = (x^2 - y^2/b^2 = 1);b: Real;IsTangent(H,Asymptote(G)) = True", "coords": {} }, { "index": 251, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 252, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0252.png", "answer": "9/4", "fact_expr": "G: Parabola;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(A,G);PointOnCurve(B,G);VectorOf(A, F) = 2*VectorOf(F, B);IsChordOf(LineSegmentOf(A, B), G)", "coords": {} }, { "index": 253, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0253.png", "answer": "4*sqrt(2)/3", "fact_expr": "m: Line;C: Ellipse;G: Point;Expression(C) = (x^2/2 + y^2 = 1);Coordinate(G) = (-1, 0);PointOnCurve(G, m);Slope(m) = 1", "coords": { "G": [ -1.0, 0.0 ] } }, { "index": 254, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 255, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0255.png", "answer": "[-b^2,+\\infty)", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G)", "coords": {} }, { "index": 256, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.02777777798473835, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0256.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Parabola;Expression(G) = (y = 9*x^2)", "coords": {} }, { "index": 257, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 2.0, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0257.png", "answer": "{(-oo, -sqrt(3)/2), (sqrt(3)/2, +oo)}", "fact_expr": "G: Hyperbola;H: Line;Expression(G) = (x^2/4 - y^2/3 = -1);O: Origin;PointOnCurve(O, H);NumIntersection(H, G) = 2", "coords": {} }, { "index": 258, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 259, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2631799883090584, "b": 2.5275303393962245, "orientation": "horizontal" }, "optimization": { "final_loss": 4.481942266905428e-06, "converged": true, "iterations": 95 }, "output_path": "results/train/hyperbola/problem_0259.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;b>0;Expression(G) = (x^2 - y^2/b^2 = 1);OneOf(Asymptote(G)) = Z;Expression(Z) = (y = 2*x);Z: Line", "coords": {} }, { "index": 260, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 261, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0261.png", "answer": "3/4", "fact_expr": "C: Parabola;Expression(C) = (y^2 = x);l: Line;PointOnCurve(Focus(C), l);A: Point;B: Point;Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, B)) = 2;IsChordOf(LineSegmentOf(A, B), C);G: Point;MidPoint(LineSegmentOf(A, B)) = G", "coords": {} }, { "index": 262, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0262.png", "answer": "sqrt(2)", "fact_expr": "l: Line;C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Intersection(l, C) = {A, B};A: Point;B: Point;M: Point;MidPoint(LineSegmentOf(A, B)) = M;Slope(l)*Slope(LineSegmentOf(O, M)) = 1;O: Origin", "coords": {} }, { "index": 263, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0263.png", "answer": "sqrt(3)", "fact_expr": "H: Line;Expression(H) = (y = x*(b/(2*a)));G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;A: Point;B: Point;Intersection(H, LeftPart(G)) = A;Intersection(H, RightPart(G)) = B;F: Point;RightFocus(G) = F;IsPerpendicular(LineSegmentOf(A, B), LineSegmentOf(B, F))", "coords": {} }, { "index": 264, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0264.png", "answer": "2*sqrt(6) + sqrt(3)", "fact_expr": "C: Hyperbola;D: Circle;A: Point;B: Point;F: Point;Expression(C) = (x^2/3 - y^2/3 = 1);Expression(D) = (x^2 + (y + 3*sqrt(2))^2 = 3);RightFocus(C) = F;PointOnCurve(A, LeftPart(C));PointOnCurve(B, D)", "coords": {} }, { "index": 265, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0265.png", "answer": "1", "fact_expr": "C1: Hyperbola;Expression(C1) = (x^2/a^2 - y^2/b^2 = 1);C2: Hyperbola;Expression(C2) = (x^2/a^2 - y^2/b^2 = -1);a: Number;b: Number;e1: Number;e2:Number;Eccentricity(C1) = e1;Eccentricity(C2) = e2", "coords": {} }, { "index": 266, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0266.png", "answer": "5/3", "fact_expr": "F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;P: Point;PointOnCurve(P, G) = True;Abs(LineSegmentOf(P, F1)) + Abs(LineSegmentOf(P, F2)) = 3*b;Abs(LineSegmentOf(P, F1))*Abs(LineSegmentOf(P, F2)) = (9/4)*a*b", "coords": {} }, { "index": 267, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0267.png", "answer": "-1", "fact_expr": "l: Line;G: Parabola;Expression(G) = (y^2 = 2*x);M: Point;N: Point;x1:Number;x2:Number;y1:Number;y2:Number;Coordinate(M) = (x1, y1);Coordinate(N) = (x2, y2);Intersection(l, G) = {M, N};PointOnCurve(Focus(G), l)", "coords": {} }, { "index": 268, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0268.png", "answer": "sqrt(2)/2", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;RightFocus(G) = F;O: Origin;H: Line;PointOnCurve(O, H);A: Point;P: Point;Intersection(H, G) = {A, P};IsPerpendicular(LineSegmentOf(P, F), xAxis) ;Intersection(LineOf(A, F), G) = B;B: Point;IsPerpendicular(LineSegmentOf(P, B), LineSegmentOf(P, A));e: Number;Eccentricity(G) = e", "coords": {} }, { "index": 269, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0269.png", "answer": "y^2/2 - x^2/4 = 1", "fact_expr": "G: Hyperbola;A: Point;H:Hyperbola;Expression(G) = (x^2/2 - y^2 = 1);Coordinate(A) = (2, -2);PointOnCurve(A, H);Asymptote(G)=Asymptote(H)", "coords": { "A": [ 2.0, -2.0 ] } }, { "index": 270, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 271, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0271.png", "answer": "-1/4", "fact_expr": "m: Line;G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);P1: Point;P2: Point;Intersection(m, G) = {P1, P2};P: Point;MidPoint(LineSegmentOf(P1, P2)) = P;k1: Number;k2: Number;Slope(m) = k1;Negation(k1=0);O: Origin;Slope(LineOf(O, P)) = k2", "coords": {} }, { "index": 272, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0272.png", "answer": "7", "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;P: Point;F: Point;Expression(G) = (x^2 = 2*y);Coordinate(P) = (1, 3);Focus(G) = F;PointOnCurve(P, l);Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ 1.0, 3.0 ] } }, { "index": 273, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0273.png", "answer": "sqrt(26)-2*sqrt(3)", "fact_expr": "G: Hyperbola;A: Point;P: Point;F: Point;Expression(G) = (x^2/3 - y^2 = 1);Coordinate(A) = (3, 1);PointOnCurve(P, RightPart(G));RightFocus(G)=F", "coords": { "A": [ 3.0, 1.0 ] } }, { "index": 274, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0274.png", "answer": "1/2", "fact_expr": "G: Parabola;M: Point;A: Point;B: Point;O: Origin;F: Point;Expression(G) = (y^2 = 12*x);Coordinate(M) = (-1, 0);Focus(G) = F;PointOnCurve(A,G);PointOnCurve(B,G);VectorOf(M,A)=lambda*VectorOf(M,B);lambda:Number;lambda>0;WhenMin(Area(TriangleOf(A,B,O))+Area(TriangleOf(A,F,O)))", "coords": { "M": [ -1.0, 0.0 ] } }, { "index": 275, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0275.png", "answer": "4*sqrt(3)", "fact_expr": "C: Parabola;P: Point;F: Point;Expression(C) = (y^2 = 8*x);Focus(C) = F;PointOnCurve(P, C);Abs(LineSegmentOf(P, F)) = 8", "coords": {} }, { "index": 276, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 277, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 278, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0278.png", "answer": "4/5", "fact_expr": "C: Ellipse;b: Number;a: Number;G: Line;A: Point;F: Point;B: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x = a/2);RightFocus(C) = F;Intersection(G, C) = {A, B};AngleOf(A, F, B) = ApplyUnit(120, degree)", "coords": {} }, { "index": 279, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0279.png", "answer": "y^2=2*sqrt(2)*x", "fact_expr": "G: Parabola;p: Number;H: Line;F: Point;N: Point;M: Point;D: Point;l: Line;p>0;Expression(G) = (y^2 = 2*p*x);Focus(G) = F;Directrix(G) = l;PointOnCurve(F, H);Slope(H) = sqrt(3);Intersection(H, G)=M;Quadrant(M)=1;IsPerpendicular(LineSegmentOf(M,N),l);FootPoint(LineSegmentOf(M,N),l)=N;Intersection(LineOf(N,F),yAxis)=D;Abs(LineSegmentOf(M, D)) = sqrt(6)", "coords": {} }, { "index": 280, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0280.png", "answer": "x=3/2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2 = 1)", "coords": {} }, { "index": 281, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 282, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 283, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0283.png", "answer": "(-oo,-3/2)+(3*sqrt(3)/8,3/2)", "fact_expr": "F: Point;G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);LeftFocus(G) = F;P: Point;PointOnCurve(P, G) = True;Slope(LineOf(F, P)) > sqrt(3);O: Origin", "coords": {} }, { "index": 284, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 285, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 286, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0286.png", "answer": "sqrt(3)", "fact_expr": "C: Hyperbola;b: Number;a: Number;F1: Point;F2: Point;P: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C)=F1;RightFocus(C)=F2;Abs(LineSegmentOf(F1, F2)) = 2*sqrt(3);PointOnCurve(P, C);Min(DotProduct(VectorOf(P, F1), VectorOf(P, F2))) = -2", "coords": {} }, { "index": 287, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0287.png", "answer": "sqrt(2)/2", "fact_expr": "A: Point;F: Point;G: Ellipse;a: Number;b: Number;B: Point;M: Point;e: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);UpperVertex(G)=A;LowerVertex(G)=B;RightFocus(G)=F;Intersection(LineOf(A, F),RightDirectrix(G)) = M;IsParallel(LineOf(M,B),xAxis);Eccentricity(G) = e", "coords": {} }, { "index": 288, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0288.png", "answer": "sqrt(3)-1", "fact_expr": "C: Ellipse;b: Number;a: Number;G: Line;k: Number;M: Point;N: Point;F: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);k>0;Expression(G) = (y = k*x);LeftFocus(C) = F;Intersection(G,C)={M,N};Quadrant(M)=1;Abs(LineSegmentOf(M, N)) = 2*sqrt(a^2 - b^2);Abs(LineSegmentOf(F, M)) <= sqrt(3)*Abs(LineSegmentOf(F, N))", "coords": {} }, { "index": 289, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 3.753767905018077, "b": 2.2935134786480793, "major_axis": "x", "x_coef": 14.999999999999998, "y_coef": 5.000000000000001 }, "optimization": { "final_loss": 0.00019830568017831433, "converged": true, "iterations": 62 }, "output_path": "results/train/ellipse/problem_0289.png", "answer": "x^2/15 + y^2/5 = 1", "fact_expr": "G: Ellipse;P: Point;Q:Point;Coordinate(P) = (sqrt(3), -2);Coordinate(Q)=(-2*sqrt(3),1);PointOnCurve(P, G);PointOnCurve(Q, G)", "coords": { "P": [ 1.7320508075688772, -2.0 ], "Q": [ -3.4641016151377544, 1.0 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 290, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0290.png", "answer": "(x-2*\\sqrt{5})^2+y^2=25", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/16 = 1);RightFocus(G) = Center(H);H:Circle;Length(InterceptChord(OneOf(Asymptote(G)),H))=6", "coords": {} }, { "index": 291, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0291.png", "answer": "y = 0", "fact_expr": "G: Parabola;H: Circle;I: Line;A: Point;F: Point;Expression(G) = (x^2 = 4*y);Coordinate(F) = (0, 1);Focus(G) = F;PointOnCurve(A, G);IsDiameter(LineSegmentOf(A, F), H);IsTangent(H, I)", "coords": { "F": [ 0.0, 1.0 ] } }, { "index": 292, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0292.png", "answer": "y^2=8*x", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(C) = F;l: Line;Directrix(C) = l;O: Origin;P: Point;PointOnCurve(P, C);Abs(LineSegmentOf(O, P)) = Abs(LineSegmentOf(P, F));Q: Point;Intersection(OverlappingLine(LineSegmentOf(F, P)), l) = Q;Area(TriangleOf(O, F, Q)) = 8*sqrt(2)", "coords": {} }, { "index": 293, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 294, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0294.png", "answer": "(1, 2)", "fact_expr": "G: Parabola;B: Point;A: Point;Expression(G) = (y^2 = 4*x);Coordinate(B) = (3, 2);PointOnCurve(A, G);WhenMin(Distance(A, B) + Distance(A, Focus(G)))", "coords": { "B": [ 3.0, 2.0 ] } }, { "index": 295, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0295.png", "answer": "y^2 - x^2/3 = 1", "fact_expr": "G: Hyperbola;m: Number;n: Number;H: Parabola;Expression(G) = (y^2/n + x^2/m = 1);Expression(H) = (x^2 = 8*y);Eccentricity(G) = 2;OneOf(Focus(G))=Focus(H)", "coords": {} }, { "index": 296, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0296.png", "answer": "pm*sqrt(3)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 8*x);F: Point;Focus(C) = F;l: Line;PointOnCurve(F, l) ;Intersection(l, C) = {A, B};A: Point;B: Point;Intersection(l, Directrix(C)) = P;P: Point;VectorOf(A, P) = 3*VectorOf(B, P)", "coords": {} }, { "index": 297, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0297.png", "answer": "16/5", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;M: Point;m: Number;Coordinate(M) = (1, m);PointOnCurve(M, G);F: Point;Focus(G) = F;Distance(M, F) = 5;d: Number;Distance(Vertex(G), LineOf(M, F)) = d", "coords": {} }, { "index": 298, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0298.png", "answer": "10", "fact_expr": "G: Parabola;P: Point;F: Point;M: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(P,G);l1:Line;PointOnCurve(P,l1);IsPerpendicular(Directrix(G),l1);FootPoint(Directrix(G),l1)=M;Focus(G)=F;LineSegmentOf(P, F) = 5", "coords": {} }, { "index": 299, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0299.png", "answer": "8", "fact_expr": "l: Line;G: Ellipse;A: Point;B: Point;F2: Point;F1: Point;Expression(G) = (x^2/4 + y^2/3 = 1);LeftFocus(G) = F1;RightFocus(G)=F2;PointOnCurve(F1, l);Intersection(l, G) = {A, B}", "coords": {} }, { "index": 300, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0300.png", "answer": "2", "fact_expr": "G: Parabola;M: Point;O: Origin;Expression(G) = (y^2 = 4*x);PointOnCurve(M, G);Distance(M, O) = sqrt(5)", "coords": {} }, { "index": 301, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.1503682136535645, "b": 2.5736796855926514, "orientation": "horizontal" }, "optimization": { "final_loss": 5.191216132516274e-06, "converged": true, "iterations": 118 }, "output_path": "results/train/hyperbola/problem_0301.png", "answer": "x^2/6 - y^2/3 = 1", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Focus(C) = {F1, F2};Eccentricity(C) = sqrt(6)/2;PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) - Abs(LineSegmentOf(P, F2)) = 2*sqrt(6)", "coords": {} }, { "index": 302, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 4.0, "major_axis": "x", "x_coef": 36.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0302.png", "answer": "5", "fact_expr": "G: Ellipse;F1: Point;M: Point;O: Origin;N: Point;Expression(G) = (x^2/36 + y^2/16 = 1);PointOnCurve(M, G);LeftFocus(G) = F1;Distance(M, F1) = 2;MidPoint(LineSegmentOf(M, F1)) = N", "coords": {} }, { "index": 303, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0303.png", "answer": "1/4", "fact_expr": "G: Parabola;p: Number;p>0;Expression(G) = (y^2 = 2*p*x);Coordinate(Focus(G))=(1/8,0)", "coords": {} }, { "index": 304, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 305, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.1503682136535645, "b": 2.5736796855926514, "orientation": "horizontal" }, "optimization": { "final_loss": 5.191216132516274e-06, "converged": true, "iterations": 118 }, "output_path": "results/train/hyperbola/problem_0305.png", "answer": "sqrt(2)/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Ellipse;a > b;b > 0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = sqrt(6)/2", "coords": {} }, { "index": 306, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0306.png", "answer": "8", "fact_expr": "G: Ellipse;H: Line;F2: Point;A: Point;B: Point;F1: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(G) = {F1, F2};PointOnCurve(F1, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(F2, A)) + Abs(LineSegmentOf(F2, B)) = 12", "coords": {} }, { "index": 307, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 308, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0308.png", "answer": "sqrt(2)/2", "fact_expr": "l: Line;G: Ellipse;a: Number;b: Number;F: Point;P: Point;M: Point;A: Point;B: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(M) = (1, 1/2);LowerVertex(G)=P;LeftFocus(G) = F;IsParallel(LineSegmentOf(F, P),l);MidPoint(LineSegmentOf(A,B))=M;Intersection(l, G) = {A, B}", "coords": { "M": [ 1.0, 0.5 ] } }, { "index": 309, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0309.png", "answer": "1", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/b^2 = 1);b: Number;b>0;Expression(Asymptote(G)) = (y = pm*1/2*x)", "coords": {} }, { "index": 310, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0310.png", "answer": "18", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2 - y^2/3 = 1);PointOnCurve(P, RightPart(G));LeftFocus(G) = F1;RightFocus(G) = F2;Abs(LineSegmentOf(P, F1)) + Abs(LineSegmentOf(P, F2)) = 10", "coords": {} }, { "index": 311, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0311.png", "answer": "4/3", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;M: Point;A: Point;F: Point;N: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;LeftVertex(C)=A;Center(G)=F;Radius(G)=Abs(LineSegmentOf(F,A));Intersection(G,RightPart(C))={M, N};PointOnCurve(N,PerpendicularBisector(LineSegmentOf(A,M)))", "coords": {} }, { "index": 312, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0312.png", "answer": "{sqrt(34)/3, sqrt(34)/5}", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/n + x^2/m = 1);m: Number;n: Number;Expression(Asymptote(G)) = (pm*3*y + 5*x = 0)", "coords": {} }, { "index": 313, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0313.png", "answer": "3/2", "fact_expr": "C: Ellipse;Expression(C) = (x^2/25 + y^2/16 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C)", "coords": {} }, { "index": 314, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0314.png", "answer": "1", "fact_expr": "M: Point;Coordinate(M) = (-1, 2);C: Parabola;Expression(C) = (y^2 = 4*x);G: Line;PointOnCurve(Focus(C), G);k: Number;Slope(G) = k;A: Point;B: Point;Intersection(G, C) = {A, B};AngleOf(A, M, B) = ApplyUnit(90, degree)", "coords": { "M": [ -1.0, 2.0 ] } }, { "index": 315, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 316, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.4641016151377544, "major_axis": "y", "x_coef": 12.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0316.png", "answer": "9", "fact_expr": "G: Ellipse;A: Point;B: Point;P: Point;Expression(G) = (x^2/12 + y^2/16 = 1);Coordinate(A) = (0,-2);Coordinate(B) = (0, 2);PointOnCurve(P, G);Abs(VectorOf(A, P)) - Abs(VectorOf(B, P)) = 2", "coords": { "A": [ 0.0, -2.0 ], "B": [ 0.0, 2.0 ] } }, { "index": 317, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 318, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0318.png", "answer": "2", "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/25 + y^2/9 = 1);PointOnCurve(P, G);Distance(P, LeftDirectrix(G)) = 5/2", "coords": {} }, { "index": 319, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0319.png", "answer": "sqrt(5)", "fact_expr": "C: Hyperbola;b: Number;a: Number;F: Point;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);OneOf(Focus(C))=F;OneOf(Endpoint(ImageinaryAxis(C)))=B;PointOnCurve(MidPoint(LineSegmentOf(F,B)), C)", "coords": {} }, { "index": 320, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.449489742783178, "major_axis": "x", "x_coef": 9.0, "y_coef": 6.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0320.png", "answer": "16*pi/3", "fact_expr": "G: Ellipse;A: Point;B: Point;F: Point;Expression(G) = (x^2/9 + y^2/6 = 1);RightFocus(G) = F;IsChordOf(LineSegmentOf(A, B), G);IsPerpendicular(LineSegmentOf(A, B), MajorAxis(G));WhenMax(Perimeter(TriangleOf(A, B, F)))", "coords": {} }, { "index": 321, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0321.png", "answer": "2*sqrt(5)", "fact_expr": "l: Line;C: Ellipse;A: Point;B: Point;Expression(C) = (x^2 + 4*y^2 = 16);Expression(l) = (x + 2*y - 4 = 0);Intersection(l, C) = {A, B};IsChordOf(LineSegmentOf(A, B), C)", "coords": {} }, { "index": 322, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0322.png", "answer": "y=pm*(sqrt(2)/2)*x", "fact_expr": "F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Z: Line;PointOnCurve(F2, Z);IsPerpendicular(Z, OneOf(Asymptote(G)));M: Point;FootPoint(Z, OneOf(Asymptote(G))) = M;Abs(VectorOf(M, F1)) = 3*Abs(VectorOf(M, F2))", "coords": {} }, { "index": 323, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0323.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;b: Number;l1: Line;l2: Line;Expression(G) = (x^2/144 - y^2/b^2 = 1);Asymptote(G) = {l1, l2};IsPerpendicular(l1, l2)", "coords": {} }, { "index": 324, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0324.png", "answer": "sqrt(2)/2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);A: Point;B: Point;F: Point;Focus(G) = F;M: Point;N: Point;PointOnCurve(A, G);PointOnCurve(B, G);AngleOf(A, F, B) = pi/2;M = MidPoint(LineSegmentOf(A, B));Projection(M, Directrix(G)) = N", "coords": {} }, { "index": 325, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 326, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0326.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;A: Point;F2: Point;B: Point;F1: Point;l1: Line;l2: Line;a>0;b>0;L:Line;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;OneOf(Asymptote(G))=l1;OneOf(Asymptote(G))=l2;Negation(l1=l2);PointOnCurve(F2,L);IsPerpendicular(L,l1);FootPoint(L,l1)=A;Intersection(L,l2)=B;VectorOf(A, F2) = -VectorOf(F2, B)/2", "coords": {} }, { "index": 327, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 3.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0327.png", "answer": "(pm*1,0)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/3 + y^2/2 = 1)", "coords": {} }, { "index": 328, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 5.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0328.png", "answer": "{8,28}", "fact_expr": "G: Hyperbola;M: Point;F1: Point;F2:Point;Expression(G) = (x^2/25 - y^2/9 = 1);PointOnCurve(M, G);LeftFocus(G) = F1;Distance(M, F1) = 18;RightFocus(G)=F2", "coords": {} }, { "index": 329, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 330, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0330.png", "answer": "9", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/16 + y^2/9 = 1);PointOnCurve(P, G);Focus(G) = {F1,F2};AngleOf(F1,P,F2)=ApplyUnit(90,degree)", "coords": {} }, { "index": 331, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 332, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0332.png", "answer": "{3,5}", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (y^2/4 + x^2/m = 1);FocalLength(G) = 2", "coords": {} }, { "index": 333, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 334, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0334.png", "answer": "x^2=4*y", "fact_expr": "G: Parabola;Expression(G) = (x^2 = m*y);m: Number;m>0;P: Point;PointOnCurve(P, G);F: Point;Focus(G) = F;M: Point;Coordinate(M) = (0, 7);Distance(P, F) = 5;Distance(P, M) = 5", "coords": { "M": [ 0.0, 7.0 ] } }, { "index": 335, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0335.png", "answer": "8", "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2/m = 1);m: Number;0 < m;m < 4;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;l: Line;PointOnCurve(F2, l);A: Point;B: Point;Intersection(l, C) = {A, B}", "coords": {} }, { "index": 336, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0336.png", "answer": "sqrt(2)/2", "fact_expr": "A: Point;Coordinate(A) = (1, e);B: Point;Coordinate(B) = (e, sqrt(3)/2);G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;PointOnCurve(A, G) = True;PointOnCurve(B, G) = True;e: Number;Eccentricity(G) = e", "coords": {} }, { "index": 337, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0337.png", "answer": "{[-1-sqrt(3),0) \u222a (0,1+sqrt(3)]}", "fact_expr": "l: Line;G: Ellipse;M: Point;Q: Point;N: Point;x0:Number;Expression(G) = (x^2/2 + y^2 = 1);Intersection(l,yAxis)=M;PointOnCurve(Q,l);Negation(Q=M);XCoordinate(Q) = x0;PointOnCurve(N, G);AngleOf(M, Q, N) = ApplyUnit(45, degree);Expression(l)=(y=1);Negation(x0=0)", "coords": {} }, { "index": 338, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0338.png", "answer": "(\\sqrt{2}-1, 1)", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);c: Number;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);a/Sin(AngleOf(P, F1, F2)) = c/Sin(AngleOf(P, F2, F1))", "coords": {} }, { "index": 339, "success": false, "conic_type": "parabola", "error": "validation: parabola_point_off_curve", "params": { "p": 3.9731175899505615, "direction": "right" }, "optimization": { "final_loss": 0.00018243117665406317, "converged": true, "iterations": 127 }, "output_path": "results/train/parabola/problem_0339.png", "answer": "4*x+y-20=0", "fact_expr": "G: Parabola;p: Number;L:Line;A: Point;F: Point;B: Point;C: Point;Expression(G) = (y^2 = 2*p*x);Coordinate(A) = (1, 4);Focus(G) = F;PointOnCurve(B,G);PointOnCurve(C,G);VectorOf(F, A) + VectorOf(F, B) + VectorOf(F, C) = 0;PointOnCurve(B,L);PointOnCurve(C,L);PointOnCurve(A,G)", "coords": { "A": [ 1.0, 4.0 ] }, "validation_reasons": [ "parabola_point_off_curve" ] }, { "index": 340, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.463192105293274, "b": 2.070901870727539, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4661259228887502e-06, "converged": true, "iterations": 106 }, "output_path": "results/train/hyperbola/problem_0340.png", "answer": "y = pm*sqrt(2)*x", "fact_expr": "C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(C) = sqrt(3)", "coords": {} }, { "index": 341, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0341.png", "answer": "{1/2, sqrt(3)/2}", "fact_expr": "A: Point;F: Point;G: Ellipse;a: Number;b: Number;P: Point;B: Point;e: Number;c: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (c, 0);RightFocus(G) = F;UpperVertex(G) = A;LowerVertex(G) = B;Intersection(LineOf(A, F), G) = P;Slope(LineSegmentOf(P, B)) = sqrt(3)/4;Eccentricity(G) = e", "coords": {} }, { "index": 342, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0342.png", "answer": "[\\sqrt{3}/2,1)", "fact_expr": "C: Ellipse;b: Number;a: Number;F1: Point;Q: Point;F2: Point;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Focus(C)={F1,F2};PointOnCurve(Q, C);AngleOf(F1, Q, F2) = ApplyUnit(120, degree);Eccentricity(C)=e;e:Number;a > b;b > 0", "coords": {} }, { "index": 343, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 344, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0344.png", "answer": "x^2/8+y^2/2=1", "fact_expr": "T: Ellipse;Expression(T) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Length(MajorAxis(T)) = 2 * Length(MinorAxis(T));F: Point;LeftFocus(T) = F;PointOnCurve(F, G) = True;Inclination(G) = ApplyUnit(45, degree);G: Line;Intersection(G, T) = {A, B};A: Point;B: Point;Abs(LineSegmentOf(A, B)) = (8*sqrt(2))/5", "coords": {} }, { "index": 345, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0345.png", "answer": "3", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);P: Point;PointOnCurve(P, C);l: Line;Directrix(C) = l;d1: Number;Distance(P, l) = d1;G: Circle;Expression(G) = ((x + 2)^2 + (y + 4)^4 = 4);d2: Number;F: Point;PointOnCurve(F, G);Distance(P, F) = d2", "coords": {} }, { "index": 346, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.8425120115280151, "b": 1.6167607307434082, "orientation": "horizontal" }, "optimization": { "final_loss": 1.3765338735538535e-05, "converged": true, "iterations": 54 }, "output_path": "results/train/hyperbola/problem_0346.png", "answer": "4*sqrt(7)/7", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/n^2 + x^2/m^2 = 1);m: Number;n: Number;Eccentricity(G) = 4/3;H: Hyperbola;Expression(H) = (y^2/n^2 - x^2/m^2 = 1)", "coords": {} }, { "index": 347, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 348, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 349, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0349.png", "answer": "5/3", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, RightPart(G));Abs(LineSegmentOf(P, F1)) = 4*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 350, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0350.png", "answer": "24", "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;Expression(G) = (y^2 = -12*x);PointOnCurve(Focus(G),l);Intersection(l, G) = {A, B};XCoordinate(MidPoint(LineSegmentOf(A,B))) = -9", "coords": {} }, { "index": 351, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0351.png", "answer": "3*sqrt(5)", "fact_expr": "l: Line;C: Hyperbola;a: Number;b: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, C);Expression(l)=(x-2*y=0);RightFocus(C)=F2;LeftFocus(C)=F1;IsPerpendicular(OneOf(Asymptote(C)),l);Abs(LineSegmentOf(P,F1))-Abs(LineSegmentOf(P,F2))=3", "coords": {} }, { "index": 352, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 5.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0352.png", "answer": "pm*4*y+5*x=0", "fact_expr": "G: Hyperbola;Expression(G) = (25*x^2 - 16*y^2 = 400)", "coords": {} }, { "index": 353, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 354, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 1.0, -0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_0354.png", "answer": "6\n16/25", "fact_expr": "G: Hyperbola;H: Circle;m: Number;Expression(G) = (9*x^2 - 16*y^2 = -144);Expression(H) = (m - 2*x + x^2 + y^2 = 0);IsTangent(Asymptote(G),H)", "coords": {} }, { "index": 355, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 356, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0356.png", "answer": "8", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;PointOnCurve(F, H) ;Inclination(H) = pi/4;H: Line;Intersection(G, H) = {A, B};A: Point;B: Point", "coords": {} }, { "index": 357, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 358, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0358.png", "answer": "x^2+y^2=1", "fact_expr": "G: Ellipse;P: Point;D: Point;E:Point;Expression(G) = (x^2 + y^2/4 = 1);PointOnCurve(P, G);PointOnCurve(P,LineSegmentOf(P,D));IsPerpendicular(LineSegmentOf(P,D),xAxis);FootPoint(LineSegmentOf(P,D),xAxis)=D;MidPoint(LineSegmentOf(P,D))=E", "coords": {} }, { "index": 359, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0359.png", "answer": "2", "fact_expr": "C: Parabola;P: Point;Q: Point;F: Point;M: Point;N: Point;R: Point;l: Line;Expression(C) = (y^2 = 4*x);Focus(C) = F;Directrix(C) = l;PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P,Q),l);FootPoint(LineSegmentOf(P,Q),l)=Q;MidPoint(LineSegmentOf(P,Q))=M;MidPoint(LineSegmentOf(P,F))=N;Intersection(LineSegmentOf(M,N),xAxis)=R;AngleOf(N, R, F) = ApplyUnit(60, degree)", "coords": {} }, { "index": 360, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0360.png", "answer": "2", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(l) = (x + sqrt(3)*y = 0);IsPerpendicular(OneOf(Asymptote(C)), l);Distance(OneOf(Focus(C)), l) = 1", "coords": {} }, { "index": 361, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0361.png", "answer": "(4, \\pm 4\\sqrt{2})", "fact_expr": "l: Line;E: Parabola;A: Point;F: Point;B: Point;Expression(E) = (y^2 = 8*x);Focus(E) = F;PointOnCurve(F, l);Intersection(l, E) = {A, B};Abs(LineSegmentOf(A, F)) = 6", "coords": {} }, { "index": 362, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0362.png", "answer": "sqrt(2)/2", "fact_expr": "G: Ellipse;b: Number;a: Number;B: Point;F2: Point;F1: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Focus(G)={F1,F2};OneOf(Endpoint(MinorAxis(G)))=B;AngleOf(F1,B,F2)=ApplyUnit(90,degree)", "coords": {} }, { "index": 363, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0363.png", "answer": "y = x - 1", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;P: Point;Expression(G) = (y^2 = 2*x);Coordinate(P) = (2, 1);PointOnCurve(P, H);Intersection(H, G) = {A, B};IsChordOf(LineSegmentOf(A, B), G);MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ 2.0, 1.0 ] } }, { "index": 364, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0364.png", "answer": "3/4", "fact_expr": "G: Parabola;M: Point;Expression(G) = (y^2 = x);PointOnCurve(M, G);Distance(M, Focus(G)) = 1", "coords": {} }, { "index": 365, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0365.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;A: Point;O: Origin;B: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 4*x);l1: Line;l2: Line;Asymptote(G) = {l1, l2};Intersection(l1, Directrix(H)) = A;Intersection(l2, Directrix(H)) = B;Area(TriangleOf(A, O, B)) = sqrt(3)", "coords": {} }, { "index": 366, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0366.png", "answer": "x^2/4 - y^2/3 = 1", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;I: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 4*(sqrt(7)*x));Coordinate(I) = (2, sqrt(3));PointOnCurve(I,OneOf(Asymptote(G)));PointOnCurve(OneOf(Focus(G)), Directrix(H))", "coords": { "I": [ 2.0, 1.7320508075688772 ] } }, { "index": 367, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0367.png", "answer": "sqrt(2)/2", "fact_expr": "C: Ellipse;b: Number;G: Line;F: Point;A: Point;O: Origin;B: Point;b>0;c:Number;Expression(C) = (x^2/4 + y^2/b^2 = 1);Expression(G) = (7*x - sqrt(14)*y = 0);Coordinate(F) = (c, 0);RightFocus(C) = F;Intersection(G, C) = {A, B};Abs(VectorOf(A, O)) = Abs(VectorOf(A, F))", "coords": {} }, { "index": 368, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 3.4641016151377544, "major_axis": "y", "x_coef": 12.0, "y_coef": 36.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0368.png", "answer": "x+y-4=0", "fact_expr": "G: Ellipse;B: Point;A: Point;N: Point;Expression(G) = (3*x^2 + y^2 = 36);Coordinate(N) = (1, 3);PointOnCurve(A, G);PointOnCurve(B, G);MidPoint(LineSegmentOf(A,B))=N", "coords": { "N": [ 1.0, 3.0 ] } }, { "index": 369, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0369.png", "answer": "sqrt(m)", "fact_expr": "l1:Line;l2:Line;l:Line;C: Parabola;A: Point;B: Point;P: Point;F: Point;m:Number;Expression(C) = (x^2 = 4*y);Focus(C) = F;PointOnCurve(F,l);Intersection(l,C)={A,B};Negation(A=B);TangentOnPoint(A,C)=l1;TangentOnPoint(B,C)=l2;Intersection(l1,l2)=P;Abs(LineSegmentOf(A, B)) = m", "coords": {} }, { "index": 370, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0370.png", "answer": "7", "fact_expr": "G: Parabola;M: Point;Expression(G) = (x^2 = -4*y);PointOnCurve(M, G);Distance(M, Focus(G)) = 7", "coords": {} }, { "index": 371, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.8284271247461903, "major_axis": "y", "x_coef": 8.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0371.png", "answer": "(0, pm*1)\n1/3", "fact_expr": "G: Ellipse;Expression(G) = (x^2/8 + y^2/9 = 1)", "coords": {} }, { "index": 372, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 6.0, "b": 8.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0372.png", "answer": "{28, 4}", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/36 - y^2/64 = 1);P: Point;PointOnCurve(P, C);A: Point;Coordinate(A) = (-10, 0);B: Point;Coordinate(B) = (10, 0);Abs(LineSegmentOf(P, A)) = 16", "coords": { "A": [ -10.0, 0.0 ], "B": [ 10.0, 0.0 ] } }, { "index": 373, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0373.png", "answer": "x^2 - y^2/2 = 1", "fact_expr": "G: Hyperbola;a: Number;H: Parabola;Expression(G) = (-y^2/2 + x^2/a^2 = 1);Expression(H) = (y^2 = 4*(sqrt(3)*x));RightFocus(G) = Focus(H)", "coords": {} }, { "index": 374, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0374.png", "answer": "y=pm*sqrt(3)*x", "fact_expr": "C: Hyperbola;b: Number;a: Number;O: Origin;F: Point;A: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;RightVertex(C) = A;MidPoint(LineSegmentOf(O,F)) = A", "coords": {} }, { "index": 375, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.75, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0375.png", "answer": "[sqrt(2)/2,1]", "fact_expr": "C: Parabola;M: Point;F: Point;A: Point;Expression(C) = (x^2 = 3*y);Focus(C)=F;Intersection(Directrix(C),yAxis)=A;PointOnCurve(M, C)", "coords": {} }, { "index": 376, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0376.png", "answer": "5/7", "fact_expr": "C: Ellipse;b: Number;a: Number;G: Line;A: Point;F: Point;B: Point;e: Number;a > b;b > 0;O: Origin;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;PointOnCurve(O, G);Intersection(C, G) = {A, B};Abs(LineSegmentOf(A, B)) = 10;Abs(LineSegmentOf(A, F)) = 6;Cos(AngleOf(A, B, F)) = 4/5;Eccentricity(C) = e", "coords": {} }, { "index": 377, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0377.png", "answer": "y^2 - x^2 = 2", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2 + y^2/2 = 1);Endpoint(MajorAxis(H)) = Vertex(G);Eccentricity(G) * Eccentricity(H) = 1", "coords": {} }, { "index": 378, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.4641016151377544, "major_axis": "x", "x_coef": 16.0, "y_coef": 12.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0378.png", "answer": "(2, 3)", "fact_expr": "P: Point;Quadrant(P) = 1;G: Ellipse;Expression(G) = (x^2/16 + y^2/12 = 1);PointOnCurve(P, G);F1: Point;F2: Point;Focus(G) = {F1, F2};Distance(P, F1) - Distance(P, F2) = 2", "coords": {} }, { "index": 379, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0379.png", "answer": "y=pm*(sqrt(2)/2)*x", "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/24 + y^2/12 = 1)", "coords": {} }, { "index": 380, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.4149149626832767, "b": 2.450642358201144, "orientation": "horizontal" }, "optimization": { "final_loss": 1.0319236343469396e-05, "converged": true, "iterations": 94 }, "output_path": "results/train/hyperbola/problem_0380.png", "answer": "x^2/(2/3)-y^2/2=1", "fact_expr": "C: Hyperbola;G: Point;Coordinate(G) = (1, 1);PointOnCurve(G, C);Expression(OneOf(Asymptote(C))) = (y = sqrt(3)*x)", "coords": { "G": [ 1.0, 1.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 381, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0381.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);P: Point;PointOnCurve(P, G);Distance(P, Focus(G)) = 6", "coords": {} }, { "index": 382, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0382.png", "answer": "8\n12", "fact_expr": "G: Ellipse;H: Circle;C:Circle;P: Point;M: Point;N: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Expression(H) = (y^2 + (x + 4)^2 = 1);Expression(C) = ((x-4)^2+y^2=1);PointOnCurve(P, G);PointOnCurve(M, H);PointOnCurve(N, C)", "coords": {} }, { "index": 383, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0383.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "C: Hyperbola;a: Number;G: Parabola;m: Real;a>0;Expression(C) = (-y^2/4 + x^2/a^2 = 1);Expression(G) = (y^2 = 2*(m*x));Eccentricity(C) = 3/5;LeftFocus(C) = Focus(G)", "coords": {} }, { "index": 384, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 385, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.463192105293274, "b": 2.070901870727539, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4661259228887502e-06, "converged": true, "iterations": 106 }, "output_path": "results/train/hyperbola/problem_0385.png", "answer": "y=pm*sqrt(2)*x/2", "fact_expr": "C1: Ellipse;Expression(C1) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;C2: Hyperbola;Expression(C2)=(-y^2/b^2+x^2/a^2=1);Eccentricity(C1)*Eccentricity(C2) = sqrt(3)/2", "coords": {} }, { "index": 386, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0386.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;PointOnCurve(H, OneOf(Asymptote(G)));H: Point;Coordinate(H) = (4, 4*sqrt(3))", "coords": { "H": [ 4.0, 6.928203230275509 ] } }, { "index": 387, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.23606797749979, "b": 2.0, "major_axis": "x", "x_coef": 5.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0387.png", "answer": "{(sqrt(15)/2,1),(-sqrt(15)/2,1),(sqrt(15)/2,-1),(-sqrt(15)/2,-1)}", "fact_expr": "P: Point;G: Ellipse;Expression(G) = (x^2/5 + y^2/4 = 1);PointOnCurve(P, G) = True;Focus(G) = {F1, F2};F1: Point;F2: Point;Area(TriangleOf(P, F1, F2)) = 1", "coords": {} }, { "index": 388, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/train/hyperbola/problem_0388.png", "answer": "2*sqrt(3)/3", "fact_expr": "F1: Point;F2: Point;C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;LeftFocus(C) = F1;RightFocus(C) = F2;M: Point;Coordinate(M) = (3, sqrt(2));PointOnCurve(M, C) = True;Distance(F2, LineOf(M, F1)) = 4*sqrt(6)/9", "coords": { "M": [ 3.0, 1.4142135623730951 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 389, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0389.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;c: Number;FocalLength(G) = 2*c;A: Point;RightVertex(G) = A;H: Parabola;Expression(H) = (x^2 = 2*(p*y));p: Number;p>0;F: Point;Focus(H) = F;Length(InterceptChord(Directrix(H), G)) = 2*c;Abs(LineSegmentOf(F, A)) = c", "coords": {} }, { "index": 390, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0390.png", "answer": "2", "fact_expr": "C: Parabola;G: Line;M: Point;A: Point;B: Point;Expression(C) = (y^2 = 8*x);Coordinate(M) = (-2, 2);k:Number;PointOnCurve(Focus(C), G);Slope(G) = k;Intersection(G, C) = {A, B};DotProduct(VectorOf(M, A), VectorOf(M, B)) = 0", "coords": { "M": [ -2.0, 2.0 ] } }, { "index": 391, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 392, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 393, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 394, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0394.png", "answer": "(\\sqrt{2}/2,1)", "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;O: Origin;A1: Point;A2: Point;e: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftVertex(G) = A1;RightVertex(G) = A2;PointOnCurve(P,G);Negation(A1=P);Negation(A2=P);DotProduct(VectorOf(P, O), VectorOf(P, A1)) = 0;Eccentricity(G) = e", "coords": {} }, { "index": 395, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0395.png", "answer": "2", "fact_expr": "Q: Point;Coordinate(Q) = (2*sqrt(2), 0);G: Parabola;Expression(G) = (y = x^2/4);P: Point;x1: Number;y1: Number;Coordinate(P) = (x1, y1);PointOnCurve(P, G)", "coords": { "Q": [ 2.8284271247461903, 0.0 ] } }, { "index": 396, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0396.png", "answer": "sqrt(3)", "fact_expr": "l: Line;G: Ellipse;b: Number;B: Point;F2: Point;A: Point;F1: Point;0 < b;b < 2;Expression(G) = (x^2/4 + y^2/b^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, l);Intersection(l, G) = {A, B};Max(Abs(LineSegmentOf(A, F2)) + Abs(LineSegmentOf(B, F2))) = 5", "coords": {} }, { "index": 397, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0397.png", "answer": "2", "fact_expr": "C: Ellipse;Expression(C) = (x^2 + 2*y^2 = 4);F1:Point;F2:Point;P:Point;Focus(C)={F1,F2};OneOf(Endpoint(MinorAxis(C)))=P", "coords": {} }, { "index": 398, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_focus_mismatch;ellipse_point_off_curve", "params": { "a": 2.5848636627197266, "b": 2.4273612433297647, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 1.9079626818174794e-06, "converged": true, "iterations": 111 }, "output_path": "results/train/ellipse/problem_0398.png", "answer": "x^2/18+y^2/9^2=1", "fact_expr": "C: Ellipse;b: Number;a: Number;G: Point;F: Point;a>0;b>0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(G) = (-3, (3*sqrt(2))/2);Coordinate(F) = (3, 0);RightFocus(C) = F;PointOnCurve(G, C)", "coords": { "G": [ -3.0, 2.121320343559643 ], "F": [ 3.0, 0.0 ] }, "validation_reasons": [ "ellipse_focus_mismatch", "ellipse_point_off_curve" ] }, { "index": 399, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_0399.png", "answer": "2*x-3*y+25=0", "fact_expr": "l: Line;C: Circle;M: Point;A: Point;B: Point;Q:Point;Expression(C) = (x^2 + y^2 = 25);Coordinate(M) = (-2, 3);PointOnCurve(M, l);Intersection(l, C) = {A, B};L1:Line;L2:Line;TangentOfPoint(A,C)=L1;TangentOfPoint(B,C)=L2;Intersection(L1,L2)=Q", "coords": {} }, { "index": 400, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0400.png", "answer": "pm*1", "fact_expr": "l: Line;G: Hyperbola;H: Circle;A: Point;B: Point;m: Number;Expression(G) = (x^2 - y^2/2 = 1);Expression(H) = (x^2 + y^2 = 5);Expression(l) = (m + x - y = 0);Intersection(l, G) = {A, B};Negation(A=B);PointOnCurve(MidPoint(LineSegmentOf(A,B)), H)", "coords": {} }, { "index": 401, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0401.png", "answer": "16", "fact_expr": "C: Hyperbola;b: Number;P: Point;F: Point;M: Point;b>0;Expression(C) = (x^2/4 - y^2/b^2 = 1);Coordinate(P) = (0, b);LeftFocus(C)=F;Distance(F,OneOf(Asymptote(C)))=4;PointOnCurve(M, RightPart(C))", "coords": {} }, { "index": 402, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0402.png", "answer": "(0, pm*2)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/4 = 1);P: Point;PointOnCurve(P, G);F1: Point;F2: Point;Focus(G) = {F1, F2};Area(TriangleOf(P, F1, F2)) = 2*sqrt(5)", "coords": {} }, { "index": 403, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0403.png", "answer": "(6-4*sqrt(2),6+4*sqrt(2))", "fact_expr": "C: Parabola;A: Point;F: Point;M: Point;N: Point;a: Real;l:Line;Expression(C) = (y^2 = 8*x);Focus(C)=F;Coordinate(A) = (a, 0);PointOnCurve(A,l);Intersection(l,C)={M,N};Negation(M=N);DotProduct(VectorOf(F,M),VectorOf(F,N))<0", "coords": {} }, { "index": 404, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 405, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0009275674819946, "direction": "right" }, "optimization": { "final_loss": 1.0271783139614854e-06, "converged": true, "iterations": 82 }, "output_path": "results/train/parabola/problem_0405.png", "answer": "2*sqrt(2)", "fact_expr": "G: Parabola;p: Number;M: Point;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(M) = (2, 2*sqrt(2));PointOnCurve(M, G);Focus(G) = F", "coords": { "M": [ 2.0, 2.8284271247461903 ] } }, { "index": 406, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.0, "b": 0.5, "major_axis": "y", "x_coef": 0.25, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0406.png", "answer": "sqrt(3)", "fact_expr": "G: Parabola;H: Ellipse;O: Origin;Expression(H) = (4*x^2 + y^2 = 1);Vertex(G) = O;Focus(G) = OneOf(Focus(H))", "coords": {} }, { "index": 407, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0407.png", "answer": "8/3", "fact_expr": "G: Parabola;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(A, G);PointOnCurve(B, G);VectorOf(A, F) = 3*VectorOf(F, B);IsChordOf(LineSegmentOf(A, B), G)", "coords": {} }, { "index": 408, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 409, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0409.png", "answer": "3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P: Point;PointOnCurve(P, G);XCoordinate(P) = 2", "coords": {} }, { "index": 410, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.2631799883090584, "b": 2.5275303393962245, "orientation": "horizontal" }, "optimization": { "final_loss": 4.481942266905428e-06, "converged": true, "iterations": 95 }, "output_path": "results/train/hyperbola/problem_0410.png", "answer": "y^2/4-x^2=1", "fact_expr": "G: Hyperbola;H: Point;Coordinate(H) = (1, 2*sqrt(2));PointOnCurve(H, G);Expression(OneOf(Asymptote(G))) = (y = 2*x)", "coords": { "H": [ 1.0, 2.8284271247461903 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 411, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0411.png", "answer": "8", "fact_expr": "l: Line;G: Parabola;a: Number;A: Point;B: Point;F:Point;a>0;Expression(G) = (y = a*x^2);Coordinate(F)=(0, 2) ;Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B}", "coords": { "F": [ 0.0, 2.0 ] } }, { "index": 412, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 413, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.6457513110645907, "major_axis": "x", "x_coef": 16.0, "y_coef": 7.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0413.png", "answer": "7", "fact_expr": "C: Ellipse;Expression(C) = (x^2/16 + y^2/7 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;O: Origin;P: Point;PointOnCurve(P, C) = True;IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2)) = True", "coords": {} }, { "index": 414, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0414.png", "answer": "2", "fact_expr": "G: Hyperbola;a: Number;b: Number;H: Ellipse;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2/6 + y^2/4 = 1);RightFocus(G)=RightVertex(H);RightFocus(H)=RightVertex(G)", "coords": {} }, { "index": 415, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.4641016151377544, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0415.png", "answer": "[-sqrt(3)/3,sqrt(3)/3]", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/12 - y^2/4 = 1);H: Line;F: Point;RightFocus(C) = F;PointOnCurve(F,H);NumIntersection(H,RightPart(C))=1", "coords": {} }, { "index": 416, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 417, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0417.png", "answer": "3", "fact_expr": "G: Parabola;M: Point;F: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(M, G);Focus(G) = F;Distance(M, F) = 4", "coords": {} }, { "index": 418, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 419, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0419.png", "answer": "4*sqrt(5)/5", "fact_expr": "C: Circle;Expression(C) = (-2*y - 8*x + x^2 + y^2 + 16 = 0);A: Point;B: Point;PointOnCurve(A, C) = True;PointOnCurve(B, C) = True;P: Point;G: Parabola;Expression(G) = (x^2 = 2*y);PointOnCurve(P, G) = True;WhenMax(AngleOf(A, P, B)) = True", "coords": {} }, { "index": 420, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0420.png", "answer": "10", "fact_expr": "G: Parabola;M: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(M, G);Distance(M, Focus(G)) = 11", "coords": {} }, { "index": 421, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 422, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0422.png", "answer": "sqrt(5)", "fact_expr": "l: Line;G: Parabola;H: Circle;A: Point;B: Point;P: Point;Expression(G) = (x^2 = 4*y);Slope(l) = 1/2;PointOnCurve(Focus(G), l);Intersection(l, G) = {A, B};IsDiameter(LineSegmentOf(A, B), H);TangentPoint(H, Directrix(G)) = P", "coords": {} }, { "index": 423, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0423.png", "answer": "5-sqrt(2)", "fact_expr": "C: Parabola;G: Line;m: Number;P: Point;Q: Point;F: Point;l: Line;Expression(C) = (y^2 = 8*x);Expression(G) = (-4*m + x*(m + 1) + y - 6 = 0);Focus(C) = F;PointOnCurve(P, C);PointOnCurve(F, l);IsPerpendicular(l,G);FootPoint(l,G)=Q", "coords": {} }, { "index": 424, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0424.png", "answer": "2", "fact_expr": "l: Line;G: Parabola;M: Point;A: Point;B: Point;Expression(G) = (y = x^2);Coordinate(M) = (1, 2);Expression(l)=(x-y+1=0);Intersection(l, G) = {A, B}", "coords": { "M": [ 1.0, 2.0 ] } }, { "index": 425, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0425.png", "answer": "2", "fact_expr": "G: Parabola;p: Number;A: Point;B: Point;F: Point;x0:Number;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(A) = (x0, 2);Focus(G) = F;Intersection(Directrix(G), xAxis) = B;PointOnCurve(A,G);Abs(LineSegmentOf(A,B))=sqrt(2)*Abs(LineSegmentOf(A,F))", "coords": {} }, { "index": 426, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 427, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 428, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0428.png", "answer": "2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;H: Circle;Expression(H) = (-6*x + x^2 + y^2 - 7 = 0);IsTangent(Directrix(G),H) = True", "coords": {} }, { "index": 429, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0429.png", "answer": "(0,\\sqrt{2}/2]", "fact_expr": "C: Ellipse;b: Number;a: Number;P: Point;B1: Point;B2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Endpoint(MinorAxis(C))={B1,B2};PointOnCurve(P, C);Abs(LineSegmentOf(P, B1)) <= Abs(LineSegmentOf(B1, B2))", "coords": {} }, { "index": 430, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 2.0, "major_axis": "x", "x_coef": 8.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0430.png", "answer": "(2,0)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/8 + y^2/4 = 1)", "coords": {} }, { "index": 431, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0431.png", "answer": "2", "fact_expr": "C: Hyperbola;P: Point;F1: Point;M: Point;x0:Number;y0:Number;x0>0;y0>0;F2: Point;Expression(C) = (x^2/4 - y^2/5 = 1);Coordinate(M) = (2, 1);Coordinate(P) = (x0, y0);LeftFocus(C) =F1;RightFocus(C)=F2;DotProduct(VectorOf(P, F1), VectorOf(M, F1))/Abs(VectorOf(P, F2)) = DotProduct(VectorOf(F2, F1), VectorOf(M, F1))/Abs(VectorOf(F2, F1))", "coords": { "M": [ 2.0, 1.0 ] } }, { "index": 432, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0432.png", "answer": "(1,\\sqrt{2}]", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (-4*x + x^2 + y^2 + 2 = 0);IsIntersect(Asymptote(G),H)", "coords": {} }, { "index": 433, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 434, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0434.png", "answer": "4", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);F1: Point;OneOf(Focus(G))=F1;M: Point;PointOnCurve(M,G);Abs(LineSegmentOf(M, F1)) = 2;O: Origin;N: Point;MidPoint(LineSegmentOf(M,F1)) = N", "coords": {} }, { "index": 435, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0435.png", "answer": "sqrt(2)/2", "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F: Point;A:Point;B:Point;a>0;b>0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P,F),xAxis);IsParallel(LineSegmentOf(O,P),LineSegmentOf(A,B));RightVertex(G)=A;UpperVertex(G)=B;O:Origin", "coords": {} }, { "index": 436, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_ecc_mismatch", "params": { "a": 1.9257268905639648, "b": 1.7993911322594118, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 2.5576271215184017e-15, "converged": true, "iterations": 2 }, "output_path": "results/train/ellipse/problem_0436.png", "answer": "9, 16", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2/12 + y^2/m = 1);Eccentricity(G) = 1/2", "coords": {}, "validation_reasons": [ "ellipse_ecc_mismatch" ] }, { "index": 437, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 438, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0438.png", "answer": "x^2/4+y^2/3=1", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;B1: Point;B2: Point;UpperVertex(G) = B1;LowerVertex(G) = B2;C: Point;MidPoint(LineSegmentOf(B1, F2)) = C;DotProduct(VectorOf(B1, F1), VectorOf(B1, F2)) = 2;IsPerpendicular(LineSegmentOf(C, F1), LineSegmentOf(B1, F2))", "coords": {} }, { "index": 439, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.242640687119285, "b": 3.0, "major_axis": "x", "x_coef": 18.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0439.png", "answer": "[1/5, 5]", "fact_expr": "P: Point;Coordinate(P) = (0, 9/2);G: Ellipse;Expression(G) = (x^2/18 + y^2/9 = 1);A: Point;B: Point;PointOnCurve(A, G);PointOnCurve(B, G);lambda: Number;VectorOf(P, A) = lambda*VectorOf(P, B)", "coords": { "P": [ 0.0, 4.5 ] } }, { "index": 440, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "major_axis": "x", "x_coef": 3.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0440.png", "answer": "2*sqrt(2)", "fact_expr": "G: Ellipse;F1: Point;F2: Point;Focus(G) = {F1, F2};Expression(G) = (x^2/3 + y^2 = 1)", "coords": {} }, { "index": 441, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0441.png", "answer": "3/2", "fact_expr": "M: Point;Coordinate(M) = (1/2, 0);l: Line;PointOnCurve(M, l);G: Parabola;Expression(G) = (y^2 = 2*x);A: Point;B: Point;Intersection(l, G) = {A, B};C: Point;Coordinate(C) = (2, 0)", "coords": { "M": [ 0.5, 0.0 ], "C": [ 2.0, 0.0 ] } }, { "index": 442, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0442.png", "answer": "{3, 15}", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 - y^2/16 = 1);PointOnCurve(P, G);OneOf(Focus(G))=F1;OneOf(Focus(G))=F2;Negation(F1=F2);Distance(P, F1) = 9", "coords": {} }, { "index": 443, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0443.png", "answer": "x^2/8+y^2/4=1", "fact_expr": "G: Parabola;H: Ellipse;O: Origin;Expression(G) = (y^2 = 8*x);Center(H) = O;OneOf(Focus(H)) = Focus(G);Coordinate(OneOf(Vertex(H)))=(0,2)", "coords": {} }, { "index": 444, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.4149149626832767, "b": 2.450642358201144, "orientation": "horizontal" }, "optimization": { "final_loss": 1.0319236343469396e-05, "converged": true, "iterations": 94 }, "output_path": "results/train/hyperbola/problem_0444.png", "answer": "x^2 - y^2 / 3 = 1", "fact_expr": "G: Hyperbola;H: Point;Coordinate(H) = (2, 3);PointOnCurve(H, G);Expression(Asymptote(G)) = (y = pm*(sqrt(3)*x))", "coords": { "H": [ 2.0, 3.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 445, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 446, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0446.png", "answer": "4", "fact_expr": "l: Line;C: Ellipse;b: Number;A: Point;B: Point;F1: Point;F2: Point;0 < b;b < 1;Expression(C) = (x^2 + y^2/b^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F2, l);Intersection(l, C) = {A, B}", "coords": {} }, { "index": 447, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0447.png", "answer": "(0,5/2)", "fact_expr": "C: Parabola;p: Number;A: Point;B: Point;p>0;x0:Number;Expression(C) = (x^2 = 2*p*y);Coordinate(B) = (0, -p/2);Coordinate(A) = (x0, p/2);PointOnCurve(A, C);Abs(LineSegmentOf(A, B)) = 5*sqrt(2)", "coords": {} }, { "index": 448, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 449, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0449.png", "answer": "pm*4", "fact_expr": "C: Parabola;p: Number;A: Point;p>0;Expression(C) = (x^2 = 2*p*y);Coordinate(A) = (m, 4);PointOnCurve(A, C);Distance(A, Focus(C)) = 5;m:Number", "coords": {} }, { "index": 450, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0450.png", "answer": "-4/5", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2:Point;I: Point;lambda:Number;Expression(G) = (x^2/16 - y^2/9 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, RightPart(G));Incenter(TriangleOf(P,F1,F2)) = I;Area(TriangleOf(I,P,F1)) =Area(TriangleOf(I,P,F2))- lambda*Area(TriangleOf(I,F1,F2))", "coords": {} }, { "index": 451, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 452, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.8087351322174072, "b": 1.3174467086791992, "orientation": "horizontal" }, "optimization": { "final_loss": 6.625325568165863e-06, "converged": true, "iterations": 78 }, "output_path": "results/train/hyperbola/problem_0452.png", "answer": "16", "fact_expr": "C: Hyperbola;b: Number;a: Number;F1: Point;F2: Point;P: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-sqrt(5), 0);Coordinate(F2) = (sqrt(5), 0);Expression(Asymptote(C)) = (y = pm*(x/2));LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, RightPart(C))", "coords": { "F1": [ -2.23606797749979, 0.0 ], "F2": [ 2.23606797749979, 0.0 ] } }, { "index": 453, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0453.png", "answer": "sqrt(6)/3", "fact_expr": "l: Line;G: Ellipse;a: Number;b: Number;M: Point;A: Point;B: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(M) = (1, 1);PointOnCurve(M, l);Slope(l) = -1/3;Intersection(l, G) = {A, B};VectorOf(A, M) = VectorOf(M, B)", "coords": { "M": [ 1.0, 1.0 ] } }, { "index": 454, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 455, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 456, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.449489742783178, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0456.png", "answer": "(3,0)", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/6 - y^2/3 = 1)", "coords": {} }, { "index": 457, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 458, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0458.png", "answer": "{8/3, 8}", "fact_expr": "G: Hyperbola;P: Point;Expression(G) = (x^2/4 - y^2/5 = 1);PointOnCurve(P, G) ;Distance(P, RightFocus(G)) = 8", "coords": {} }, { "index": 459, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 460, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0460.png", "answer": "16/5", "fact_expr": "l1: Line;Expression(l1) = (4*x - 3*y + 12 = 0);l2: Line;Expression(l2) = (x = -1);G: Parabola;Expression(G) = (y^2 = 4*x);P: Point;PointOnCurve(P, G) = True", "coords": {} }, { "index": 461, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.23606797749979, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0461.png", "answer": "(pm*3, 0)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/5 - y^2/4 = 1)", "coords": {} }, { "index": 462, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 463, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 464, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.5030499696731567, "b": 2.00545072555542, "orientation": "horizontal" }, "optimization": { "final_loss": 1.3176339734854992e-06, "converged": true, "iterations": 109 }, "output_path": "results/train/hyperbola/problem_0464.png", "answer": "y=pm*(4/3)*x", "fact_expr": "C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(C) = 5/3", "coords": {} }, { "index": 465, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0465.png", "answer": "(1, \\sqrt{2}]", "fact_expr": "H: Line;Expression(H) = (y = x);G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;NumIntersection(H, G) = 0", "coords": {} }, { "index": 466, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0466.png", "answer": "-5", "fact_expr": "G: Parabola;p: Number;H: Line;A: Point;F: Point;B: Point;P: Point;l: Line;lambda:Number;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;Directrix(G) = l;PointOnCurve(F, H);Intersection(H, G) = {A,B};Quadrant(B)=1;Intersection(H, l) = P;VectorOf(A, F) = (2/3)*VectorOf(F, B);VectorOf(A, P) = lambda*VectorOf(A, F)", "coords": {} }, { "index": 467, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0467.png", "answer": "2", "fact_expr": "G: Ellipse;b: Number;A: Point;B: Point;F: Point;0 < b;b < 2;Expression(G) = (x^2/4 + y^2/b^2 = 1);Intersection(G, yAxis) = {A, B};OneOf(Focus(G)) = F", "coords": {} }, { "index": 468, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0468.png", "answer": "3*sqrt(5)/5", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;c: Number;Coordinate(F) = (c, 0);RightFocus(C) = F;G: Circle;Center(G) = F;Radius(G) = a;A: Point;B: Point;Intersection(G, OneOf(Asymptote(C))) = {A, B};Abs(LineSegmentOf(A, B)) = (2/3)*c", "coords": {} }, { "index": 469, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0469.png", "answer": "3", "fact_expr": "l: Line;G: Parabola;p: Number;A: Point;F: Point;B: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Slope(l)=sqrt(3);Focus(G)=F;PointOnCurve(F,l);Intersection(l,G)={A,B};Quadrant(A)=1", "coords": {} }, { "index": 470, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0470.png", "answer": "5", "fact_expr": "G: Hyperbola;H: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 - y^2 = 1);Expression(H) = (x^2/9 + y^2/4 = 1);Focus(H)={F1,F2};Focus(G)={F1,F2};OneOf(Intersection(H, G)) =P", "coords": {} }, { "index": 471, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0471.png", "answer": "3", "fact_expr": "G: Hyperbola;m: Number;H: Parabola;Expression(G) = (-x^2 + y^2/m = 1);Expression(H) = (x^2 = 8*y);Focus(H) = OneOf(Focus(G))", "coords": {} }, { "index": 472, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0472.png", "answer": "y^2/4 - x^2/16 = 1", "fact_expr": "G: Hyperbola;H: Point;C:Hyperbola;Expression(G) = (x^2 - 4*y^2 = 4);Coordinate(H) = (2, sqrt(5));Asymptote(C) = Asymptote(G);PointOnCurve(H,C)", "coords": { "H": [ 2.0, 2.23606797749979 ] } }, { "index": 473, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 474, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 475, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0475.png", "answer": "3", "fact_expr": "G: Parabola;H: Circle;P: Point;Q: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (y^2 + (x - 5)^2 = 1);PointOnCurve(P, G);PointOnCurve(Q, H)", "coords": {} }, { "index": 476, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0476.png", "answer": "-1/2", "fact_expr": "l: Line;G: Ellipse;O: Origin;P: Point;P1: Point;P2: Point;M: Point;k1:Number;k2:Number;Negation(k1 = 0);Expression(G) = (x^2 + 2*y^2 = 2);Coordinate(M) = (-2, 0);PointOnCurve(M, l);Intersection(l, G) = {P1, P2};MidPoint(LineSegmentOf(P1, P2)) = P;Slope(l) = k1;Slope(LineOf(O, P)) = k2", "coords": { "M": [ -2.0, 0.0 ] } }, { "index": 477, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0477.png", "answer": "sqrt(2)", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F2: Point;F1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P,F2),xAxis);Sin(AngleOf(P,F1,F2))=1/3", "coords": {} }, { "index": 478, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0478.png", "answer": "2", "fact_expr": "G: Parabola;p: Number;Q: Point;p>0;Expression(G) = (y^2 = 2*(p*x));PointOnCurve(Q, G);Min(Distance(Q, Focus(G))) = 1", "coords": {} }, { "index": 479, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0479.png", "answer": "5/2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/12 = 1);M: Point;PointOnCurve(M, G);Distance(M, RightFocus(G)) = 3", "coords": {} }, { "index": 480, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0480.png", "answer": "1", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(90, degree)", "coords": {} }, { "index": 481, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0481.png", "answer": "x^2/(9/4)-y^2/4=1", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);Z: Hyperbola;Asymptote(Z) = Asymptote(G);H: Point;Coordinate(H) = (-3,2*sqrt(3));PointOnCurve(H, Z)", "coords": { "H": [ -3.0, 3.4641016151377544 ] } }, { "index": 482, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.0625, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0482.png", "answer": "31/16", "fact_expr": "G: Parabola;P: Point;F: Point;Expression(G) = (y = 4*x^2);Coordinate(P) = (x0, y0);Focus(G) = F;PointOnCurve(P, G);Abs(LineSegmentOf(P, F)) = 2;x0:Number;y0:Number", "coords": {} }, { "index": 483, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0483.png", "answer": "5/3", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(l) = (4*x-5*y = 20);PointOnCurve(OneOf(Focus(C)),l);PointOnCurve(OneOf(Endpoint(ImageinaryAxis(C))),l)", "coords": {} }, { "index": 484, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0484.png", "answer": "y=pm*x", "fact_expr": "C: Hyperbola;b: Number;a: Number;F1: Point;H: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;L:Line;PointOnCurve(F2,L);IsPerpendicular(OneOf(Asymptote(C)),L);FootPoint(L,OneOf(Asymptote(C)))=H;Area(TriangleOf(F1, H, F2)) = b^2", "coords": {} }, { "index": 485, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 486, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0486.png", "answer": "8", "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Slope(l)=1;PointOnCurve(F,l);Focus(G)=F;F: Point;Intersection(l,G)={A,B}", "coords": {} }, { "index": 487, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 488, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0488.png", "answer": "sqrt(2)", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y^2 + (x - 2)^2 = 2);IsTangent(Asymptote(C),G)", "coords": {} }, { "index": 489, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0489.png", "answer": "9", "fact_expr": "C: Ellipse;F1: Point;P: Point;F2: Point;Expression(C) = (x^2/25 + y^2/9 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(F1, P), LineSegmentOf(F2, P))", "coords": {} }, { "index": 490, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_ecc_mismatch", "params": { "a": 1.9257268905639648, "b": 1.7993911322594118, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 2.5576271215184017e-15, "converged": true, "iterations": 2 }, "output_path": "results/train/ellipse/problem_0490.png", "answer": "3/2", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2/2 + y^2/m = 1);PointOnCurve(Focus(G), xAxis);Eccentricity(G) = 1/2", "coords": {}, "validation_reasons": [ "ellipse_ecc_mismatch" ] }, { "index": 491, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2631799883090584, "b": 2.5275303393962245, "orientation": "horizontal" }, "optimization": { "final_loss": 4.481942266905428e-06, "converged": true, "iterations": 95 }, "output_path": "results/train/hyperbola/problem_0491.png", "answer": "3/5", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/(3 - n) + x^2/n = 1);n: Number;Expression(Asymptote(G)) = (y=pm*2*x)", "coords": {} }, { "index": 492, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0492.png", "answer": "9", "fact_expr": "G: Hyperbola;A: Point;P: Point;F: Point;Expression(G) = (x^2/4 - y^2/12 = 1);Coordinate(A) = (1, 4);LeftFocus(G) = F;PointOnCurve(P, RightPart(G))", "coords": { "A": [ 1.0, 4.0 ] } }, { "index": 493, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 494, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0010501146316528, "direction": "right" }, "optimization": { "final_loss": 3.455598402979376e-07, "converged": true, "iterations": 81 }, "output_path": "results/train/parabola/problem_0494.png", "answer": "2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;H: Point;Coordinate(H) = (1, 2);PointOnCurve(H, G)", "coords": { "H": [ 1.0, 2.0 ] } }, { "index": 495, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0495.png", "answer": "1/2", "fact_expr": "O: Origin;C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;LeftFocus(C) = F;A: Point;PointOnCurve(A, C) ;IsPerpendicular(LineSegmentOf(A, F), xAxis) ;Area(TriangleOf(F, A, O)) = b^2/4", "coords": {} }, { "index": 496, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0496.png", "answer": "sqrt(5)", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;A: Point;F: Point;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y = (4/3)*x);RightFocus(C) = F;Intersection(G, C) = {A, B};IsPerpendicular(LineSegmentOf(A, F), LineSegmentOf(B, F))", "coords": {} }, { "index": 497, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 498, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0498.png", "answer": "2*sqrt(7)/3", "fact_expr": "C: Parabola;G: Line;P: Point;A: Point;B: Point;Expression(C) = (y^2 = 4*x);Coordinate(P) = (-2, 0);PointOnCurve(P, G);Intersection(G, C) = {A, B};Abs(LineSegmentOf(P, A)) = Abs(LineSegmentOf(A, B))/2;O:Origin", "coords": { "P": [ -2.0, 0.0 ] } }, { "index": 499, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0499.png", "answer": "18/5", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};H: Line;PointOnCurve(F2, H);A: Point;B: Point;Intersection(H, G) = {A, B}", "coords": {} }, { "index": 500, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 501, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0501.png", "answer": "{pi/3,(2*pi)/3}", "fact_expr": "l: Line;C: Parabola;A: Point;F: Point;B: Point;Expression(C) = (y^2 = 8*x);Expression(l) = (y = k*(x - 2));Intersection(l, C) = {A, B};Focus(C) = F;Abs(LineSegmentOf(A, F)) = 3*Abs(LineSegmentOf(B, F));k:Number", "coords": {} }, { "index": 502, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0502.png", "answer": "10", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;Expression(G) = (x^2 = 8*y);Slope(H)=1/2;PointOnCurve(Focus(G),H);Intersection(H,G)={A,B}", "coords": {} }, { "index": 503, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0503.png", "answer": "1", "fact_expr": "G: Hyperbola;a: Number;H: Ellipse;Expression(G) = (-y^2/2 + x^2/a = 1);Expression(H) = (x^2/4 + y^2/a^2 = 1);Focus(G) = Focus(H)", "coords": {} }, { "index": 504, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.2631799883090584, "b": 2.5275303393962245, "orientation": "horizontal" }, "optimization": { "final_loss": 4.481942266905428e-06, "converged": true, "iterations": 95 }, "output_path": "results/train/hyperbola/problem_0504.png", "answer": "sqrt(5)/2", "fact_expr": "G: Hyperbola;P:Point;O:Origin;Coordinate(P) = (1, 3);Expression(OneOf(Asymptote(G))) = (y = 2*x);PointOnCurve(Focus(G),axis);Center(G)=O;PointOnCurve(P,G)", "coords": { "P": [ 1.0, 3.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 505, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0505.png", "answer": "16/3", "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B};XCoordinate(A) + XCoordinate(B) = 10/3", "coords": {} }, { "index": 506, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0506.png", "answer": "sqrt(3)/3", "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;P: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G)=F1;L:Line;PointOnCurve(F1, L);IsPerpendicular(L,xAxis);Intersection(L, G) = P;RightFocus(G)=F2;AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 507, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0507.png", "answer": "sqrt(2)/2", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;M: Point;Coordinate(M) = (a*(sqrt(2)/2), b*(sqrt(2)/2));PointOnCurve(M, C);F: Point;RightFocus(C) = F;B: Point;UpperVertex(C) = B;O: Origin;Area(TriangleOf(B, F, O)) = sqrt(2)*Area(TriangleOf(B, F, M))", "coords": {} }, { "index": 508, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0508.png", "answer": "sqrt(3)", "fact_expr": "l: Line;C: Hyperbola;b: Number;F1: Point;M: Point;N: Point;b>0;Expression(C) = (x^2/4 - y^2/b^2 = 1);LeftFocus(C)=F1;PointOnCurve(F1, l);Intersection(l,LeftPart(C)) = {M, N};IsPerpendicular(l,xAxis);Abs(LineSegmentOf(M,N))=3;F2:Point;RightFocus(C)=F2", "coords": {} }, { "index": 509, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 510, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.23606797749979, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0510.png", "answer": "6", "fact_expr": "G: Hyperbola;H: Parabola;p: Number;p>0;Expression(G) = (x^2/5 - y^2/4 = 1);Expression(H) = (y^2 = 2*p*x);PointOnCurve(LeftFocus(G),Directrix(H))", "coords": {} }, { "index": 511, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0511.png", "answer": "5/3", "fact_expr": "G: Hyperbola;b: Number;a: Number;e: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G)))=(4*x-3*y=0);Eccentricity(G) = e", "coords": {} }, { "index": 512, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 513, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0513.png", "answer": "2*sqrt(3)/3", "fact_expr": "H: Parabola;Expression(H) = (y^2 = 4*x);G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Distance(Focus(H),Asymptote(G)) = 1/2", "coords": {} }, { "index": 514, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_focus_mismatch", "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0514.png", "answer": "6", "fact_expr": "G: Ellipse;Expression(G) = (y^2/4 + x^2/a^2 = 1);a: Number;a>0;F: Point;OneOf(Focus(G)) = F;Coordinate(F) = (-sqrt(5), 0)", "coords": { "F": [ -2.23606797749979, 0.0 ] }, "validation_reasons": [ "ellipse_focus_mismatch" ] }, { "index": 515, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.23606797749979, "b": 2.0, "major_axis": "x", "x_coef": 5.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0515.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "P: Point;PointOnCurve(P, G);G: Ellipse;Expression(G) = (x^2/5 + y^2/4 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};AngleOf(F1, P, F2) = ApplyUnit(30, degree)", "coords": {} }, { "index": 516, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0516.png", "answer": "8*sqrt(2)/3", "fact_expr": "l: Line;C: Ellipse;F1: Point;A: Point;B: Point;Expression(C) = (x^2/2 + y^2 = 1);Expression(l) = (y = x - 1);LeftFocus(C) = F1;Intersection(l, C) = {A, B}", "coords": {} }, { "index": 517, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0517.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(H) = (2*sqrt(2), 0);OneOf(Focus(G))=H;Expression(Asymptote(G)) = (y = pm*x)", "coords": { "H": [ 2.8284271247461903, 0.0 ] } }, { "index": 518, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0518.png", "answer": "y=pm*3*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/9 = 1)", "coords": {} }, { "index": 519, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0519.png", "answer": "2", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(C) = F;A: Point;RightVertex(C) = A;Z: Line;PointOnCurve(F, Z);IsPerpendicular(Z, xAxis);P: Point;Intersection(Z, C) = P;Slope(LineOf(A, P)) = sqrt(3)*Slope(OneOf(Asymptote(C)))", "coords": {} }, { "index": 520, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 521, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 522, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0522.png", "answer": "2", "fact_expr": "G: Hyperbola;H: Parabola;p: Number;Expression(G) = (x^2 - y^2 = 1);p>0;Expression(H) = (y^2 = 2*p*x);PointOnCurve(LeftVertex(G),Directrix(H))", "coords": {} }, { "index": 523, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0523.png", "answer": "{18, 2}", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/16 - y^2/9 = 1);PointOnCurve(P, G);Distance(P, F1) = 10;OneOf(Focus(G))=F1;OneOf(Focus(G))=F2;Negation(F1=F2)", "coords": {} }, { "index": 524, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "y", "x_coef": 16.0, "y_coef": 25.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0524.png", "answer": "20", "fact_expr": "F1: Point;F2: Point;Focus(G) = {F1, F2};G: Ellipse;Expression(G) = (x^2/16 + y^2/25 = 1);L: Line;PointOnCurve(F1, L);M: Point;N: Point;Intersection(L, G) = {M, N}", "coords": {} }, { "index": 525, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0525.png", "answer": "(2, 1)", "fact_expr": "l: Line;G: Hyperbola;A: Point;Expression(G) = (x^2/2 - y^2 = 1);Slope(l) = 1;TangentPoint(l, G) = A;Quadrant(A) = 1", "coords": {} }, { "index": 526, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0526.png", "answer": "y^2=-4*x", "fact_expr": "G: Parabola;F: Point;Coordinate(F) = (-1, 0);Focus(G)=F", "coords": { "F": [ -1.0, 0.0 ] } }, { "index": 527, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0527.png", "answer": "4/5", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);M: Point;N: Point;PointOnCurve(M, G);PointOnCurve(N, G);IsPerpendicular(LineSegmentOf(O, M), LineSegmentOf(O, N));O: Origin", "coords": {} }, { "index": 528, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0528.png", "answer": "1", "fact_expr": "C: Parabola;l: Line;K: Point;P: Point;F: Point;x1:Number;y1:Number;y1>0;Expression(C) = (y^2 = 4*x);Coordinate(P) = (x1, y1);Focus(C) = F;Directrix(C)=l;Intersection(l, xAxis) = K;PointOnCurve(P, C);WhenMax(Abs(LineSegmentOf(P,K))/Abs(LineSegmentOf(P,F)))", "coords": {} }, { "index": 529, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0529.png", "answer": "(2*sqrt(7))/7", "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2 = 1);B: Point;UpperVertex(C) = B;O1: Circle;r: Number;Expression(O1) = (x^2 + y^2 = r^2);0 < r;r < 6/7;l: Line;TangentOfPoint(B, O1) = l;Q: Point;OneOf(Intersection(l, C)) = Q;O: Origin;LineSegmentOf(O, Q) = sqrt(13)/2", "coords": {} }, { "index": 530, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0530.png", "answer": "2*sqrt(2)-5", "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2/3 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;M: Point;PointOnCurve(M,C) = True;E: Circle;Expression(E) = ((x - 3)^2 + (y - 2)^2 = 1);N: Point;PointOnCurve(N,E) = True", "coords": {} }, { "index": 531, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0531.png", "answer": "5/3", "fact_expr": "G: Hyperbola;a: Number;b: Number;a>0;b>0;P: Point;F1: Point;F2: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, RightPart(G));Abs(LineSegmentOf(P,F1))=4*Abs(LineSegmentOf(P,F2))", "coords": {} }, { "index": 532, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0532.png", "answer": "2*sqrt(3)/3", "fact_expr": "C: Hyperbola;a: Number;b: Number;c: Number;F: Point;O: Origin;d: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (c, 0);OneOf(Focus(C)) = F;Distance(F, Asymptote(C)) = d;Abs(LineSegmentOf(O, F)) = 2*d", "coords": {} }, { "index": 533, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_0533.png", "answer": "y=pm*sqrt(3)*x", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Eccentricity(C) = 2;G: Line;Expression(G) = (x + y + 2 = 0);PointOnCurve(Focus(C), G)", "coords": {} }, { "index": 534, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0534.png", "answer": "1", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2 = 1)", "coords": {} }, { "index": 535, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0535.png", "answer": "9", "fact_expr": "E: Hyperbola;F1: Point;F2: Point;P: Point;LeftFocus(E) = F1;RightFocus(E) = F2;PointOnCurve(P, E);Expression(E) = (x^2/4 - y^2/9 = 1);Abs(LineSegmentOf(P, F1)) = 5", "coords": {} }, { "index": 536, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0536.png", "answer": "(\\sqrt{3} + 1)/2", "fact_expr": "C: Hyperbola;b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;A: Point;B: Point;LeftFocus(C) = F1;RightFocus(C) = F2;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(A, RightPart(C));PointOnCurve(B, RightPart(C));Circumcenter(TriangleOf(F1, A, B)) = F2;DotProduct((VectorOf(B, A) + VectorOf(B, F1)), VectorOf(A, F1)) = 0", "coords": {} }, { "index": 537, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.6457513110645907, "major_axis": "x", "x_coef": 16.0, "y_coef": 7.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0537.png", "answer": "4/3", "fact_expr": "G: Ellipse;A: Point;B: Point;F2: Point;F1: Point;x1: Number;y1: Number;x2: Number;y2: Number;Expression(G) = (x^2/16 + y^2/7 = 1);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);LeftFocus(G) = F1;RightFocus(G) = F2;IsChordOf(LineSegmentOf(A, B),G);PointOnCurve(F1, LineSegmentOf(A, B));Perimeter(InscribedCircle(TriangleOf(A, B, F2))) = pi", "coords": {} }, { "index": 538, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0538.png", "answer": "sqrt(5)+1", "fact_expr": "G: Hyperbola;b: Number;a: Number;F1: Point;P: Point;F2: Point;Q: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;Quadrant(P)=1;PointOnCurve(P,Asymptote(G));AngleOf(F1, P, F2) = pi/2;Intersection(LineOf(P,F1), LeftPart(G)) = Q;MidPoint(LineSegmentOf(P,F1)) = Q", "coords": {} }, { "index": 539, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 540, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0540.png", "answer": "8", "fact_expr": "E: Parabola;Expression(E) = (x^2 = 4*y);G: Line;Expression(G) = (y = x + 1);P: Point;Q: Point;Intersection(G, E) = {P, Q};IsChordOf(LineSegmentOf(P, Q), E)", "coords": {} }, { "index": 541, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0541.png", "answer": "1/2", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;RightFocus(G) = F1;l1: Line;RightDirectrix(G) = l1;L: LineSegment;PointOnCurve(F1, L);IsPerpendicular(L, xAxis);IsChordOf(L, G);Length(L) = Distance(F1, l1)", "coords": {} }, { "index": 542, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0542.png", "answer": "y^2/12 - x^2/27 = 1", "fact_expr": "G: Hyperbola;C:Hyperbola;H: Point;Expression(G) = (x^2/9 - y^2/4 = 1);Coordinate(H) = (-3, 4);Asymptote(C) = Asymptote(G);PointOnCurve(H,C)", "coords": { "H": [ -3.0, 4.0 ] } }, { "index": 543, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "y", "x_coef": 3.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0543.png", "answer": "2+2*sqrt(5)", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 2*(p*y));p: Number;p>0;H: Ellipse;Expression(H) = (x^2/3 + y^2/4 = 1);C: Circle;M: Point;B: Point;A: Point;F: Point;l:Line;L:Line;Focus(G)=F;OneOf(Focus(H))=F;Directrix(G)=l;Intersection(l,yAxis)=M;PointOnCurve(F,L);Intersection(L, G) = {A, B};IsDiameter(LineSegmentOf(B,M),C);PointOnCurve(A, C)", "coords": {} }, { "index": 544, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0544.png", "answer": "x^2+(y-1)^2=1/5", "fact_expr": "G: Parabola;H: Circle;Expression(G) = (x^2 = 4*y);Focus(G) = Center(H);C:Curve;Expression(C)=(y^2/4-x^2=1);IsTangent(H,Asymptote(C))", "coords": {} }, { "index": 545, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0545.png", "answer": "9", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2))", "coords": {} }, { "index": 546, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0546.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (-4*x + x^2 + y^2 + 2 = 0);IsTangent(Asymptote(G),H)", "coords": {} }, { "index": 547, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.2631799883090584, "b": 2.5275303393962245, "orientation": "horizontal" }, "optimization": { "final_loss": 4.481942266905428e-06, "converged": true, "iterations": 95 }, "output_path": "results/train/hyperbola/problem_0547.png", "answer": "x^2 - y^2/4 = 1", "fact_expr": "G: Hyperbola;H: Point;Coordinate(H) = (sqrt(2), 2);Expression(Asymptote(G))=(y=pm*2*x);PointOnCurve(H,G)", "coords": { "H": [ 1.4142135623730951, 2.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 548, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0548.png", "answer": "y=pm*(sqrt(5)/2)*x", "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/4 + y^2/5 = 1)", "coords": {} }, { "index": 549, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0549.png", "answer": "(3*sqrt(2)+sqrt(10))/2", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;F1: Point;A: Point;F2: Point;O: Origin;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(O, G);Slope(G)=sqrt(3);Intersection(G, RightPart(C)) = A;AngleOf(F1, A, F2) = (2*pi)/3", "coords": {} }, { "index": 550, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 551, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0551.png", "answer": "9/2", "fact_expr": "G: Parabola;H: Line;A: Point;F: Point;B: Point;Expression(G) = (y^2 = 2*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B}", "coords": {} }, { "index": 552, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 553, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0553.png", "answer": "16*sqrt(3)/3", "fact_expr": "C: Parabola;G: Line;O: Origin;A: Point;B: Point;F: Point;Expression(C) = (y^2 = 8*x);Focus(C) = F;PointOnCurve(F, G);Inclination(G) = ApplyUnit(60, degree);Intersection(G, C) = {A, B}", "coords": {} }, { "index": 554, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 555, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0555.png", "answer": "1", "fact_expr": "G: Hyperbola;a: Real;H: Ellipse;Expression(G) = (-y^2/2 + x^2/a = 1);Expression(H) = (x^2/4 + y^2/a^2 = 1);Focus(H) = Focus(G)", "coords": {} }, { "index": 556, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0556.png", "answer": "y=pm*((1+(sqrt(3)/3))*x)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;H: Circle;Expression(H) = (x^2 + y^2 = a^2);L: Line;TangentOfPoint(F2, H) = L;M: Point;Intersection(L, LeftPart(G)) = M;AngleOf(F1, M, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 557, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0557.png", "answer": "8", "fact_expr": "C: Ellipse;P: Point;F1: Point;Q: Point;F2: Point;Expression(C) = (x^2/4 + y^2/3 = 1);Focus(C) = {F1, F2};PointOnCurve(P, C);Intersection(OverlappingLine(LineSegmentOf(P, F1)), C) = Q", "coords": {} }, { "index": 558, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0558.png", "answer": "64/3", "fact_expr": "l: Line;C: Parabola;G: Line;A: Point;B: Point;F: Point;Expression(C) = (y^2 = 16*x);Expression(G) = (x + sqrt(3)*y - 1 = 0);Focus(C) = F;PointOnCurve(F,l);IsPerpendicular(l,G);Intersection(l, C) = {A, B}", "coords": {} }, { "index": 559, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0559.png", "answer": "8*sqrt(2)", "fact_expr": "l: Line;C: Parabola;O: Origin;M: Point;N: Point;Expression(C) = (y^2 = 4*x);Intersection(l, C) = {M, N};Slope(LineSegmentOf(O, M))*Slope(LineSegmentOf(O, N)) = -1/2", "coords": {} }, { "index": 560, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.6457513110645907, "major_axis": "x", "x_coef": 16.0, "y_coef": 7.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0560.png", "answer": "120*sqrt(3)/11", "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/7 = 1);F: Point;RightFocus(G) = F;P: Point;PointOnCurve(P, G);A: Point;Coordinate(A) = (0, 3*sqrt(3));WhenMax(Perimeter(TriangleOf(A, P, F)))", "coords": { "A": [ 0.0, 5.196152422706632 ] } }, { "index": 561, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0561.png", "answer": "10+sqrt(26)", "fact_expr": "G: Ellipse;P: Point;A: Point;F1:Point;F2: Point;Coordinate(A) = (2, 1);Expression(G) = (x^2/25 + y^2/16 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(P,G)", "coords": { "A": [ 2.0, 1.0 ] } }, { "index": 562, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.449489742783178, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 6.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0562.png", "answer": "1/2", "fact_expr": "C: Ellipse;L: Line;B: Point;A: Point;M: Point;Expression(C) = (x^2/6 + y^2/3 = 1);Coordinate(M) = (-1, 1);Slope(L)=k;k:Number;Intersection(L, C) = {A, B};MidPoint(LineSegmentOf(A,B)) = M", "coords": { "M": [ -1.0, 1.0 ] } }, { "index": 563, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 564, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0564.png", "answer": "5", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);H: Line;Expression(H) = (sqrt(3)*x - y - sqrt(3) = 0);A: Point;B: Point;Intersection(H, G) = {A, B};O: Origin;F: Point;Intersection(H,xAxis) = F;lambda: Number;mu: Number;VectorOf(O, F) = lambda*VectorOf(O, A) + mu*VectorOf(O, B)", "coords": {} }, { "index": 565, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 566, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0566.png", "answer": "10", "fact_expr": "G: Parabola;M: Point;P: Point;F: Point;l: Line;Expression(G) = (y^2 = 4*x);PointOnCurve(P, G);PointOnCurve(P, l);IsPerpendicular(l, Directrix(G));FootPoint(l, Directrix(G)) = M;Abs(LineSegmentOf(P, M)) = 5;Focus(G) = F", "coords": {} }, { "index": 567, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0567.png", "answer": "x=-1/8", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;F: Point;Focus(G) = F;H: Line;PointOnCurve(F, H);Intersection(H, G) = {P, Q};P: Point;Q: Point;1/Abs(LineSegmentOf(P, F)) + 1/Abs(LineSegmentOf(F, Q)) = 8", "coords": {} }, { "index": 568, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0568.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, G) = {A, B};E: Point;Intersection(l, Directrix(G)) = E;MidPoint(LineSegmentOf(A, E)) = F;Abs(LineSegmentOf(B, F)) = 2", "coords": {} }, { "index": 569, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 570, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0570.png", "answer": "2", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;M: Point;N: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Vertex(C)={M,N};PointOnCurve(P,C);Slope(LineOf(P, N))*Slope(LineOf(P,M)) = 3", "coords": {} }, { "index": 571, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0571.png", "answer": "9", "fact_expr": "C: Parabola;F: Point;P: Point;M: Point;N: Point;l: Line;Expression(C) = (y^2=8*x);Focus(C) = F;Directrix(C)=l;PointOnCurve(P, l);Intersection(LineOf(P,F),C)={M,N};VectorOf(P,M)=3*VectorOf(M,F)", "coords": {} }, { "index": 572, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0572.png", "answer": "[1/2, sqrt(3)/3]", "fact_expr": "F1: Point;Coordinate(F1) = (-c, 0);c: Number;F2: Point;Coordinate(F2) = (c, 0);C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;Focus(C) = {F1, F2};P: Point;PointOnCurve(P, C) = True;DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 2*c^2", "coords": {} }, { "index": 573, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0573.png", "answer": "1", "fact_expr": "G: Parabola;A: Point;B: Point;C: Point;D: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(Focus(G),LineSegmentOf(A,B));PointOnCurve(Focus(G),LineSegmentOf(C,D));IsChordOf(LineSegmentOf(A,B),G);IsChordOf(LineSegmentOf(C,D),G);IsPerpendicular(LineSegmentOf(A,B),LineSegmentOf(C,D))", "coords": {} }, { "index": 574, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0574.png", "answer": "2*sqrt(2)", "fact_expr": "G: Parabola;A: Point;P: Point;Expression(G) = (x^2 = 8*y);Coordinate(A) = (2, 0);PointOnCurve(P, G)", "coords": { "A": [ 2.0, 0.0 ] } }, { "index": 575, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0575.png", "answer": "6-2*sqrt(5)", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/2 - y^2 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;l: Line;PointOnCurve(F1, l);A: Point;Intersection(l, LeftPart(C)) = A;B: Point;Intersection(l, RightPart(C)) = B;IsPerpendicular(l, LineSegmentOf(F2, B))", "coords": {} }, { "index": 576, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 577, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 578, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 579, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0579.png", "answer": "2*sqrt(3)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1)", "coords": {} }, { "index": 580, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0580.png", "answer": "8/5", "fact_expr": "H: Point;Coordinate(H) = (0, 2);G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1)", "coords": { "H": [ 0.0, 2.0 ] } }, { "index": 581, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 582, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 583, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0583.png", "answer": "1", "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/4 + y^2/3 = 1);PointOnCurve(P, G)", "coords": {} }, { "index": 584, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0584.png", "answer": "(6*sqrt(5)/5)-1", "fact_expr": "G: Parabola;Expression(G) = (y^2 = -4*x);l: Line;Expression(l) = (2*x + y - 4 = 0);A: Point;PointOnCurve(A, G);Distance(A, yAxis) = m;Distance(A, l) = n;m: Number;n: Number", "coords": {} }, { "index": 585, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 586, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.125, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0586.png", "answer": "2", "fact_expr": "G: Parabola;A: Point;B: Point;M: Point;F: Point;Expression(G) = (x^2 = y/2);Coordinate(M) = (x0, y0);x0:Number;y0:Number;PointOnCurve(F,LineSegmentOf(A, B));IsChordOf(LineSegmentOf(A, B),G);Focus(G) = F;MidPoint(LineSegmentOf(A, B)) = M;y0 = 7/8", "coords": {} }, { "index": 587, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0587.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);e: Number;Abs(LineSegmentOf(P, F1)) = 2*a;AngleOf(F1, P, F2) = pi/3;Eccentricity(G) = e", "coords": {} }, { "index": 588, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0588.png", "answer": "[\\sqrt{2}, +\\infty)", "fact_expr": "G: Ellipse;Expression(G) = (y^2 + x^2/a^2 = 1);a: Real;a>1;P: Point;PointOnCurve(P, G);O: Origin;l: Line;PointOnCurve(P, l);PointOnCurve(O, l);H: Circle;Expression(H) = (x^2 + y^2 = a^2 + 1);A: Point;B: Point;Intersection(l, H) = {A, B};a^2 = Abs(LineSegmentOf(P, A))*Abs(LineSegmentOf(P, B)) + 1", "coords": {} }, { "index": 589, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0589.png", "answer": "3", "fact_expr": "M: Point;Coordinate(M) = (1, t);t: Number;t>0;H: Parabola;Expression(H) = (y^2 = 2*p*x);p: Number;p>0;PointOnCurve(M, H);Distance(M, Focus(H)) = 5;G: Hyperbola;Expression(G) = (-y^2/9 + x^2/a^2 = 1);a: Real;a>0;A: Point;LeftVertex(G) = A;IsParallel(OneOf(Asymptote(G)), LineOf(A, M)) = True", "coords": {} }, { "index": 590, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.4641016151377544, "major_axis": "y", "x_coef": 12.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0590.png", "answer": "16", "fact_expr": "G: Ellipse;Expression(G) = (x^2/12 + y^2/16 = 1);F1: Point;UpperFocus(G) = F1;L: Line;PointOnCurve(F1, L) = True;Intersection(L, G) = {A, B};A: Point;B: Point;LowerFocus(G) = F2;F2: Point", "coords": {} }, { "index": 591, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0591.png", "answer": "16", "fact_expr": "G: Hyperbola;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/9 - y^2/16 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;AngleOf(F1,P,F2)=ApplyUnit(90,degree);PointOnCurve(P,G)", "coords": {} }, { "index": 592, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.6457513110645907, "major_axis": "x", "x_coef": 9.0, "y_coef": 7.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0592.png", "answer": "-sqrt(7)/3", "fact_expr": "l: Line;C: Ellipse;F: Point;A: Point;B: Point;O: Origin;Expression(C) = (x^2/9 + y^2/7 = 1);LeftFocus(C) = F;PointOnCurve(O, l);Intersection(l, C) = {A, B};Quadrant(A)=2;AngleOf(F, A, B) = AngleOf(B, F, O)", "coords": {} }, { "index": 593, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0593.png", "answer": "x^2=8*y", "fact_expr": "l: Line;G: Parabola;Expression(G) = (x^2 = 2*p*y);p: Number;p>0;PointOnCurve(Focus(G), l) ;Intersection(l, G) = {A, B};B: Point;A: Point;Length(LineSegmentOf(A, B)) = 6;Distance(MidPoint(LineSegmentOf(A, B)), xAxis) = 1", "coords": {} }, { "index": 594, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 595, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 596, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 5.196152422706632, "major_axis": "x", "x_coef": 36.0, "y_coef": 27.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0596.png", "answer": "6", "fact_expr": "F1: Point;F2: Point;G: Ellipse;Expression(G) = (x^2/36 + y^2/27 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;A: Point;PointOnCurve(A, G) = True;O: Origin;B: Point;VectorOf(O, B) = (VectorOf(O, A) + VectorOf(O, F1))/2;C: Point;VectorOf(O, C) = (VectorOf(O, A) + VectorOf(O, F2))/2", "coords": {} }, { "index": 597, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0597.png", "answer": "4", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, H);Inclination(H) = pi/2;Intersection(H, G) = {A, B}", "coords": {} }, { "index": 598, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0598.png", "answer": "2*sqrt(3)", "fact_expr": "G: Parabola;p: Number;A: Point;p>0;Expression(G) = (x^2 = 2*p*y);Coordinate(A) = (m, 1);PointOnCurve(A, G);Distance(A, Focus(G)) = 4;m:Number", "coords": {} }, { "index": 599, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0599.png", "answer": "y=pm*((sqrt(2)/2)*x)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;Length(ImageinaryAxis(G)) = 2;FocalLength(G) = 2*sqrt(3)", "coords": {} }, { "index": 600, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0600.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1)", "coords": {} }, { "index": 601, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.23606797749979, "b": 2.0, "major_axis": "x", "x_coef": 5.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0601.png", "answer": "5/3", "fact_expr": "G: Ellipse;H: Line;O: Origin;A: Point;B: Point;Expression(G) = (x^2/5 + y^2/4 = 1);PointOnCurve(RightFocus(G),H);Slope(H)=2;Intersection(H,G)={A,B}", "coords": {} }, { "index": 602, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0602.png", "answer": "2", "fact_expr": "G: Parabola;P: Point;Expression(G) = (x^2 = -4*y);Coordinate(P) = (2, -1);PointOnCurve(P,G)", "coords": { "P": [ 2.0, -1.0 ] } }, { "index": 603, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0603.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (2, 0);RightFocus(G) = F;Distance(F, OneOf(Asymptote(G))) = sqrt(3)", "coords": { "F": [ 2.0, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 604, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 605, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.2352789464429823, "b": 2.2352789464429823, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4312591569844774e-06, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_0605.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Hyperbola;F1: Point;F2: Point;Coordinate(F1) = (-sqrt(10), 0);Coordinate(F2) = (sqrt(10), 0);Focus(G) = {F1, F2};Expression(Asymptote(G)) = (y = pm*(x/2))", "coords": { "F1": [ -3.1622776601683795, 0.0 ], "F2": [ 3.1622776601683795, 0.0 ] } }, { "index": 606, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 607, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 608, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 609, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0609.png", "answer": "sqrt(2)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;L: Line;PointOnCurve(F, L) = True;Intersection(L, G) = {A, B};A: Point;B: Point;O: Origin;Abs(LineSegmentOf(A, F)) = 3", "coords": {} }, { "index": 610, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0610.png", "answer": "y=pm*(3/4)*x", "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (x^2/16 - y^2/m = 1);FocalLength(G) = 10", "coords": {} }, { "index": 611, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0611.png", "answer": "4", "fact_expr": "G: Parabola;D: Point;l1:Line;l2:Line;F: Point;A: Point;B: Point;C: Point;Expression(G) = (y^2 = 2*x);Focus(G)=F;PointOnCurve(F,l1);PointOnCurve(F,l2);IsPerpendicular(l1,l2);Intersection(l1,G)={A,B};Intersection(l2,G)={C,D}", "coords": {} }, { "index": 612, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0612.png", "answer": "9", "fact_expr": "G: Ellipse;P: Point;Z1: Circle;Z2: Circle;Expression(G) = (x^2/25 + y^2/9 = 1);Expression(Z1) = (y^2 + (x + 4)^2 = 1/4);Expression(Z2) = (y^2 + (x - 4)^2 = 1/4);Q: Point;R: Point;PointOnCurve(P, G);PointOnCurve(Q, Z1);PointOnCurve(R, Z2)", "coords": {} }, { "index": 613, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0613.png", "answer": "sqrt(5)/2", "fact_expr": "C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(OneOf(Asymptote(C))) = (y = x/2)", "coords": {} }, { "index": 614, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0614.png", "answer": "8", "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;P: Point;F: Point;Expression(G)=(x^2=12*y);Coordinate(P)=(2,1);Focus(G)=F;PointOnCurve(P,l);Intersection(l,G)={A,B};MidPoint(LineSegmentOf(A,B)) = P", "coords": { "P": [ 2.0, 1.0 ] } }, { "index": 615, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0615.png", "answer": "sqrt(2)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/2 + y^2 = 1)", "coords": {} }, { "index": 616, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.4641016151377544, "major_axis": "x", "x_coef": 16.0, "y_coef": 12.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0616.png", "answer": "[2,6]", "fact_expr": "C: Ellipse;Expression(C) = (x^2/16 + y^2/12 = 1);F: Point;LeftFocus(C) = F;M: Point;PointOnCurve(M, C) = True", "coords": {} }, { "index": 617, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0617.png", "answer": "4", "fact_expr": "G: Ellipse;F1: Point;P: Point;O: Origin;M: Point;F2: Point;Expression(G) = (x^2/25 + y^2/16 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);MidPoint(LineSegmentOf(F1, P)) = M;Abs(LineSegmentOf(O, M)) = 3", "coords": {} }, { "index": 618, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 619, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0619.png", "answer": "9", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G) = True;AngleOf(P, F1, F2) = ApplyUnit(90, degree)", "coords": {} }, { "index": 620, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 621, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0621.png", "answer": "4", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (y^2/3 + x^2/m = 1);Coordinate(OneOf(Focus(G))) = (-1, 0)", "coords": {} }, { "index": 622, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0622.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);F1: Point;LeftFocus(G) = F1;P: Point;PointOnCurve(P, G);A: Point;Coordinate(A) = (1, 1)", "coords": { "A": [ 1.0, 1.0 ] } }, { "index": 623, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0623.png", "answer": "[sqrt(5)/2,sqrt(5)]", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;P:Point;D:Point;e:Number;c:Number;a>1;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (-y/b + x/a = 1);Coordinate(P) = (-1, 0);Coordinate(D) = (1, 0);FocalLength(G)=2*c;Eccentricity(G)=e;Distance(P,H)+Distance(D,H)=s;s:Number;s>=(4/5)*c", "coords": { "P": [ -1.0, 0.0 ], "D": [ 1.0, 0.0 ] } }, { "index": 624, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0624.png", "answer": "3", "fact_expr": "G: Parabola;A: Point;B: Point;F: Point;Expression(G) = (y^2 = x);Focus(G) = F;PointOnCurve(A, G);PointOnCurve(B, G);Distance(MidPoint(LineSegmentOf(A, B)), yAxis) = 5/4", "coords": {} }, { "index": 625, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0625.png", "answer": "4", "fact_expr": "G: Parabola;H: Line;P: Point;A: Point;d:Number;Expression(G) = (y^2 = 8*x);Expression(H) = (x = -2);Coordinate(A) = (1, sqrt(15));PointOnCurve(P, G);Distance(P, H) = d", "coords": { "A": [ 1.0, 3.872983346207417 ] } }, { "index": 626, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0626.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;b: Number;a: Number;A: Point;P: Point;B: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftVertex(G) = A;RightVertex(G) = B;PointOnCurve(P, G);Negation(P = A);Negation(P = B);Slope(LineOf(P,A))*Slope(LineOf(P,B))=2", "coords": {} }, { "index": 627, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 628, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0628.png", "answer": "{pi/3, ApplyUnit(60, degree)}", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;M: Point;N: Point;x1: Number;x2: Number;y1: Number;y2: Number;Coordinate(M) = (x1, y1);Coordinate(N) = (x2, y2);PointOnCurve(M, C);PointOnCurve(N, C);x1 + x2 + 2 = 2*LineSegmentOf(M, N)", "coords": {} }, { "index": 629, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0629.png", "answer": "2", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);G: Line;PointOnCurve(Focus(C), G);Slope(G) = 1;Intersection(G, C) = {A, B};A: Point;B: Point;M: Point;Coordinate(M) = (-1, m);m: Number;AngleOf(A, M, B) = ApplyUnit(90, degree)", "coords": {} }, { "index": 630, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0630.png", "answer": "8", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);L: Line;Expression(L) = (y = x - 1);Intersection(L, C) = {M, N};M: Point;N: Point", "coords": {} }, { "index": 631, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 632, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0632.png", "answer": "(-4, \\pm 4\\sqrt{2})", "fact_expr": "G: Parabola;Expression(G) = (y^2 = -8*x);P:Point;Distance(P,Focus(G))=6;PointOnCurve(P,G)", "coords": {} }, { "index": 633, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0633.png", "answer": "2/5", "fact_expr": "C: Ellipse;A: Point;B: Point;D: Point;F: Point;l: Line;Expression(C) = (x^2/25 + y^2/9 = 1);RightFocus(C) = F;PointOnCurve(F,l);IsParallel(l, yAxis) = False ;Intersection(l,C)={A,B};PerpendicularBisector(LineSegmentOf(A,B))=l1;Intersection(l1,MajorAxis(C))=D;l1:Line", "coords": {} }, { "index": 634, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0634.png", "answer": "6/5", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;RightFocus(C) = F;G: Line;Slope(G) = sqrt(3);PointOnCurve(F, G);A: Point;B: Point;Intersection(G, C) = {A, B};VectorOf(A, F) = 4*VectorOf(F, B)", "coords": {} }, { "index": 635, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_0635.png", "answer": "(x^2/4+y^2=1)&Negation(y=0)", "fact_expr": "G: Circle;C1:Line;C2:Line;C3:Line;C4:Line;D: Point;C: Point;A: Point;B: Point;M:Point;Expression(G) = (x^2 + y^2 = 4);PointOnCurve(A,C1);Expression(C2)=(x=2);Intersection(C1,C2)=C;PointOnCurve(B,C3);Expression(C4)=(x=-2);Intersection(C3,C4)=D;Coordinate(A) = (-2, 0);Coordinate(B) = (2, 0);IsTangent(LineOf(C,D),G);Intersection(LineOf(A,C),LineOf(B,D))=M", "coords": {} }, { "index": 636, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0636.png", "answer": "y=pm*(sqrt(5)/2)*x", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;P: Point;PointOnCurve(P, C);A: Point;LeftVertex(C) = A;F: Point;RightFocus(C) = F;IsPerpendicular(LineSegmentOf(P, F), LineSegmentOf(A, F));2*Abs(LineSegmentOf(P, F)) = Abs(LineSegmentOf(A, F))", "coords": {} }, { "index": 637, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0637.png", "answer": "-5", "fact_expr": "F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);P: Point;PointOnCurve(P, G);M: Point;Coordinate(M) = (6, 4)", "coords": { "M": [ 6.0, 4.0 ] } }, { "index": 638, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0638.png", "answer": "sqrt(3)+1", "fact_expr": "G: Hyperbola;b: Number;a: Number;O: Origin;P: Point;F2: Point;F1: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, RightPart(G));LeftFocus(G) = F1;RightFocus(G) = F2;DotProduct((VectorOf(O,P)+VectorOf(O,F2)),VectorOf(F2,P))=0;Abs(LineSegmentOf(P,F1))=sqrt(3)*Abs(LineSegmentOf(P,F2))", "coords": {} }, { "index": 639, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.0, "b": 0.5, "major_axis": "x", "x_coef": 1.0, "y_coef": 0.25 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0639.png", "answer": "y^2=pm*2*sqrt(3)*x", "fact_expr": "G: Parabola;H: Ellipse;Expression(H) = (x^2 + 4*y^2 = 1);Vertex(G)=O;O:Origin;Focus(G)=OneOf(Focus(H))", "coords": {} }, { "index": 640, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0640.png", "answer": "3", "fact_expr": "G: Parabola;l1:Line;l2:Line;P: Point;Expression(G) = (y^2 = 4*x);Expression(l1)=(4*x-3*y+11=0);Expression(l2)=(x+1=0);PointOnCurve(P, G)", "coords": {} }, { "index": 641, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 642, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0642.png", "answer": "{18, 2}", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/9 + x^2/a^2 = 1);a: Number;P: Point;PointOnCurve(P, G);Expression(OneOf(Asymptote(G))) = (3*x + 4*y = 0);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;Abs(LineSegmentOf(P, F1)) = 10", "coords": {} }, { "index": 643, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0643.png", "answer": "1/3", "fact_expr": "C: Ellipse;b: Number;a: Number;B: Point;F: Point;A: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;UpperVertex(C) = B;OneOf(Intersection(LineOf(B,F),C)) = A;LineSegmentOf(A, F)/LineSegmentOf(B, F) = 4/5", "coords": {} }, { "index": 644, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0644.png", "answer": "(pm*2*sqrt(2),0)\ny=pm*(sqrt(15)/3)*x", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Ellipse;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2/8 + y^2/5 = 1);Eccentricity(G) = (2*sqrt(6))/3;Vertex(G)=Focus(H)", "coords": {} }, { "index": 645, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0645.png", "answer": "[\\sqrt{2}/2,1)", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;F1: Point;F2: Point;P: Point;a > b;b > 0;Focus(G)={F1,F2};PointOnCurve(P,G);Abs(LineSegmentOf(P,F1)+LineSegmentOf(P,F2))=Abs(LineSegmentOf(F1,F2))", "coords": {} }, { "index": 646, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0646.png", "answer": "6", "fact_expr": "C: Parabola;p: Number;G: Line;A: Point;F: Point;B: Point;O: Origin;p>0;Expression(C) = (y^2 = 2*p*x);Focus(C) = F;PointOnCurve(F, G);Intersection(G, C) = {A, B};Abs(LineSegmentOf(A, F)) = 5*Abs(LineSegmentOf(B, F))", "coords": {} }, { "index": 647, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0647.png", "answer": "(1,\\sqrt{10}]", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = 3*x);NumIntersection(G, H)=0;Eccentricity(G)=e;e:Number", "coords": {} }, { "index": 648, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0648.png", "answer": "(\\pi/6, \\pi/2)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, RightPart(G));Abs(LineSegmentOf(P, F2)) = Abs(LineSegmentOf(F1, F2));Q: Point;MidPoint(LineSegmentOf(P, F1)) = Q", "coords": {} }, { "index": 649, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0649.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 4)", "coords": {} }, { "index": 650, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 651, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 652, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0652.png", "answer": "7/2", "fact_expr": "G: Parabola;H: Line;B: Point;A: Point;P:Point;D:Point;F: Point;Expression(G) = (x^2 = y);Coordinate(P) = (1, 0);Coordinate(D) = (0, 2);PointOnCurve(P,H);Intersection(H, G) = {A, B};Focus(G)=F;PointOnCurve(D,PerpendicularBisector(LineSegmentOf(A,B)))", "coords": { "P": [ 1.0, 0.0 ], "D": [ 0.0, 2.0 ] } }, { "index": 653, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 654, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0654.png", "answer": "16", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;Expression(G) = (y^2 = 8*x);Expression(H) = (x + y - 2 = 0);Intersection(H, G) = {A, B}", "coords": {} }, { "index": 655, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 656, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 657, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 658, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0658.png", "answer": "sqrt(3)-1", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;H: Circle;Expression(H) = (x^2 + y^2 = c^2);c: Number;P: Point;Quadrant(P) = 2;Intersection(G, H) = P;F1: Point;Coordinate(F1) = (-c, 0);LeftFocus(G) = F1;O: Origin;Distance(O, LineOf(P, F1)) = sqrt(3)/2*c", "coords": {} }, { "index": 659, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0659.png", "answer": "66/5", "fact_expr": "G: Hyperbola;H: Point;P: Point;Expression(G) = (x^2/16 - y^2/9 = 1);Coordinate(H) = (5, 0);PointOnCurve(P, G);Distance(P, H) = 8.5", "coords": { "H": [ 5.0, 0.0 ] } }, { "index": 660, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0660.png", "answer": "(-sqrt(2)/2, -1/2) + (-1/2, 1/2) + (1/2,sqrt(2)/2)", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/4 - y^2 = 1);l: Line;Expression(l) = (y = k*x - 1);k: Real;NumIntersection(l, C) = 2", "coords": {} }, { "index": 661, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 662, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0662.png", "answer": "(5, 0)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1)", "coords": {} }, { "index": 663, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0663.png", "answer": "sqrt(10)/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;F: Point;E: Point;O: Origin;P: Point;c:Number;c>0;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2 + y^2 = a^2/4);Coordinate(F) = (-c, 0);LeftFocus(G) = F;l:Line;TangentOfPoint(F,H)=l;TangentPoint(l,H)=E;Intersection(OverlappingLine(LineSegmentOf(F,E)),RightPart(G))=P;VectorOf(O, P) = 2*VectorOf(O, E) - VectorOf(O, F)", "coords": {} }, { "index": 664, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.4641016151377544, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0664.png", "answer": "8", "fact_expr": "G: Hyperbola;H: Parabola;p: Number;Expression(G) = (x^2/12 - y^2/4 = 1);p>0;Expression(H) = (y^2 = 2*p*x);Focus(H) = RightFocus(G)", "coords": {} }, { "index": 665, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 666, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0666.png", "answer": "sqrt(10)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;PointOnCurve(G,OneOf(Asymptote(C))) =True;G: Point;Coordinate(G) = (1, -3)", "coords": { "G": [ 1.0, -3.0 ] } }, { "index": 667, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0667.png", "answer": "sqrt(10)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;b > a;a > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;l: Line;PointOnCurve(F1, l);Slope(l) = 1;A: Point;B: Point;L1: Line;L2: Line;Asymptote(G) = {L1, L2};Intersection(l, L1) = A;Intersection(l, L2) = B;Abs(LineSegmentOf(A, F1)) = Abs(LineSegmentOf(A, B))", "coords": {} }, { "index": 668, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0668.png", "answer": "8", "fact_expr": "C: Parabola;Expression(C) = (x^2 = 4*y);F: Point;Focus(C) = F;l: Line;PointOnCurve(F, l);Inclination(l) = pi/4;A: Point;B: Point;Intersection(l, C) = {A, B};Negation(A=B)", "coords": {} }, { "index": 669, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0669.png", "answer": "4", "fact_expr": "G: Hyperbola;p: Number;H: Parabola;p>0;Expression(G) = (x^2/3 - 16*y^2/p^2 = 1);Expression(H) = (y^2 = 2*p*x);PointOnCurve(LeftFocus(G),Directrix(H))", "coords": {} }, { "index": 670, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0670.png", "answer": "16/3", "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;F: Point;Inclination(l) = ApplyUnit(60, degree);Focus(G) = F;PointOnCurve(F, l);Expression(G) = (y^2 = 4*x);Intersection(l, G) = {A, B};IsChordOf(LineSegmentOf(A, B), G)", "coords": {} }, { "index": 671, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 4.47213595499958, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0671.png", "answer": "17", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/20 = 1);P: Point;PointOnCurve(P, G) = True;H: Point;Coordinate(H) = (6, 0);Distance(P,H) = 9;I: Point;Coordinate(I) = (-6,0)", "coords": { "H": [ 6.0, 0.0 ], "I": [ -6.0, 0.0 ] } }, { "index": 672, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0672.png", "answer": "x^2/9 - y^2/3 = 1", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2 + 4*y^2 = 16);Focus(G) = Focus(H);Expression(OneOf(Asymptote(G))) = (x + sqrt(3)*y = 0)", "coords": {} }, { "index": 673, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0673.png", "answer": "{1, -1}", "fact_expr": "G: Parabola;B: Point;A: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, LineOf(A, B));Intersection(LineOf(A, B), G) = {A, B};XCoordinate(MidPoint(LineSegmentOf(A, B))) = 3", "coords": {} }, { "index": 674, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0674.png", "answer": "4", "fact_expr": "G: Parabola;H: Circle;P: Point;Expression(G) = (y^2 = 4*x);Expression(H) = ((x + 2)^2 + (y - 4)^2 = 1);PointOnCurve(P, G);PointOnCurve(Q, H);Q:Point", "coords": {} }, { "index": 675, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 676, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0676.png", "answer": "32", "fact_expr": "G: Parabola;H: Line;Q: Point;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Coordinate(Q) = (4, 0);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);x1:Number;y1:Number;x2:Number;y2:Number;PointOnCurve(Q, H);Intersection(H, G) = {A,B}", "coords": { "Q": [ 4.0, 0.0 ] } }, { "index": 677, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 2.449489742783178, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0677.png", "answer": "7*sqrt(3)", "fact_expr": "G: Hyperbola;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/3 - y^2/6 = 1);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2;DotProduct((VectorOf(F1, P) - VectorOf(F2, P)),(VectorOf(F1, P) + VectorOf(F2, P))) = 72", "coords": {} }, { "index": 678, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0678.png", "answer": "y^2/4 - x^2/12 = 1", "fact_expr": "G: Hyperbola;m: Number;n: Number;H: Parabola;Expression(G) = (y^2/n + x^2/m = 1);Expression(H) = (x^2 = 16*y);Eccentricity(G) = 2;OneOf(Focus(G)) = Focus(H)", "coords": {} }, { "index": 679, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2135674953460693, "b": 2.4270472526550293, "orientation": "horizontal" }, "optimization": { "final_loss": 4.453341261978494e-06, "converged": true, "iterations": 125 }, "output_path": "results/train/hyperbola/problem_0679.png", "answer": "x^2-4*y^2=1", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;p: Number;M: Point;A: Point;m:Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);p>0;Expression(H) = (y^2 = 2*p*x);Coordinate(M) = (1, m);PointOnCurve(M, H);Distance(M, Focus(H)) = 5;LeftVertex(G)=A;Eccentricity(G)=sqrt(5)/2;IsPerpendicular(OneOf(Asymptote(G)),LineOf(A,M))", "coords": {} }, { "index": 680, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_0680.png", "answer": "2", "fact_expr": "G: Hyperbola;a: Number;b: Number;H: Parabola;p: Number;A: Point;O: Origin;B: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);p>0;Expression(H) = (y^2 = 2*(p*x));l1:Line;l2:Line;Asymptote(G)={l1,l2};Intersection(l1, Directrix(H)) = A;Intersection(l2, Directrix(H)) = B;Eccentricity(G) = 2;Area(TriangleOf(A, O, B)) = sqrt(3)", "coords": {} }, { "index": 681, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0681.png", "answer": "sqrt(2) - 1", "fact_expr": "G: Hyperbola;H: Curve;P: Point;Q: Point;M: Point;O: Origin;Expression(G) = (x^2/2 - y^2 = 1);Expression(H) = (x^2 + (y - 3)^2 = 4);PointOnCurve(P, G);PointOnCurve(Q, H);MidPoint(LineSegmentOf(P, Q)) = M", "coords": {} }, { "index": 682, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0682.png", "answer": "(\\sqrt{2}-1,1)", "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;P: Point;F2: Point;a > b;b > 0;c:Number;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(P,G);a/Sin(AngleOf(P,F1,F2))=c/Sin(AngleOf(P,F2,F1))", "coords": {} }, { "index": 683, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 684, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0684.png", "answer": "1/2", "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;l: Line;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;LeftDirectrix(G) = l;H:LineSegment;PointOnCurve(F1,H);IsPerpendicular(H,xAxis);Length(H)=Distance(F1,l);IsChordOf(H,G)", "coords": {} }, { "index": 685, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0685.png", "answer": "sqrt(3)/3", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;LeftFocus(C) = F;G: Line;PointOnCurve(F,G) = True;Slope(G) = sqrt(3)/3;A: Point;B: Point;Intersection(G, C) = {A, B};VectorOf(A, F) = 3*VectorOf(F, B)", "coords": {} }, { "index": 686, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0686.png", "answer": "[4+2*sqrt(3),8)", "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2/3 = 1);F: Point;RightFocus(C) = F;O: Origin;PointOnCurve(O, l);l: Line;Intersection(l, C) = {A, B};A: Point;B: Point", "coords": {} }, { "index": 687, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0687.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p>0;p: Number;P: Point;Coordinate(P) = (2, m);m: Number;PointOnCurve(P, G) = True;F: Point;Focus(G) = F;Distance(P, F) = 4", "coords": {} }, { "index": 688, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_0688.png", "answer": "3*x^2-y^2=1", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = 2;Distance(Focus(G),Asymptote(G))=1", "coords": {} }, { "index": 689, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0689.png", "answer": "y^2=4*sqrt(3)*x", "fact_expr": "G: Parabola;p: Number;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 2*(p*x));Focus(G)=F;Negation(A=Vertex(G));PointOnCurve(A,LineSegmentOf(A,B));IsPerpendicular(LineSegmentOf(A,B),Directrix(G));FootPoint(LineSegmentOf(A,B),Directrix(G))=B;Abs(LineSegmentOf(B, F)) = Abs(LineSegmentOf(A, F));Area(TriangleOf(A, B, F)) = 12*sqrt(3);PointOnCurve(A,G)", "coords": {} }, { "index": 690, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0690.png", "answer": "x+2*y-3=0", "fact_expr": "C: Parabola;B: Point;A: Point;M: Point;Expression(C) = (x^2 = 4*y);Coordinate(M) = (-1, 2);PointOnCurve(A, C);PointOnCurve(B, C);VectorOf(A, M) = VectorOf(M, B)", "coords": { "M": [ -1.0, 2.0 ] } }, { "index": 691, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0691.png", "answer": "1", "fact_expr": "G: Ellipse;A: Point;B: Point;O:Origin;P: Point;H:Line;Expression(G) = (x^2/4 + y^2/3 = 1);IsChordOf(LineSegmentOf(A,B),G);Expression(H) = (y = -1/4);Coordinate(P) = (0, 1);2*VectorOf(O, A) + VectorOf(O, B) - 3*VectorOf(O, P) = 0", "coords": { "P": [ 0.0, 1.0 ] } }, { "index": 692, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0692.png", "answer": "sqrt(5)/2", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Length(RealAxis(C)) = 16;F: Point;LeftFocus(C) = F;M: Point;PointOnCurve(M, OneOf(Asymptote(C)));O: Origin;IsPerpendicular(LineSegmentOf(O, M), LineSegmentOf(M, F));Area(TriangleOf(O, M, F)) = 16", "coords": {} }, { "index": 693, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0693.png", "answer": "sqrt(3)/2", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;c: Number;Coordinate(F) = (c, 0);RightFocus(C) = F;G: Line;Expression(G) = (x - 2*sqrt(2)*y = 0);A: Point;B: Point;Intersection(G, C) = {A, B};DotProduct(VectorOf(A, F), VectorOf(B, F)) = 0", "coords": {} }, { "index": 694, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0694.png", "answer": "5/3", "fact_expr": "E: Hyperbola;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F2: Point;F1: Point;LeftFocus(E) = F1;RightFocus(E) = F2;G: Line;PointOnCurve(F1, G);A: Point;B: Point;Intersection(G, E) = {A, B};Abs(LineSegmentOf(A, F2)) = Abs(LineSegmentOf(F1, F2));e: Number;VectorOf(B, F1) = 2*VectorOf(F1, A);Eccentricity(E) = e", "coords": {} }, { "index": 695, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_0695.png", "answer": "x^2 - y^2 / 3 = 1", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;Distance(OneOf(Focus(C)), OneOf(Asymptote(C))) = sqrt(3);Eccentricity(C) = 2", "coords": {} }, { "index": 696, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0696.png", "answer": "pm*8", "fact_expr": "G: Parabola;p: Number;H: Circle;Expression(G) = (x^2 = p*y);Expression(H) = (-4*x + x^2 + y^2 = 0);IsTangent(Directrix(G),H)", "coords": {} }, { "index": 697, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0697.png", "answer": "pm*sqrt(3)/2", "fact_expr": "G: Parabola;M: Point;Expression(G) = (y = x^2);PointOnCurve(M, G);Distance(M, Focus(G)) = 1", "coords": {} }, { "index": 698, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0698.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-x^2/b^2 + y^2/a^2 = 1);Inclination(OneOf(Asymptote(G)))=pi/6", "coords": {} }, { "index": 699, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0699.png", "answer": "3", "fact_expr": "F: Point;Coordinate(F) = (1, 0);E: Ellipse;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;RightFocus(E) = F;G: Line;B: Point;LowerVertex(E) = B;PointOnCurve(B, G) ;PointOnCurve(F, G) ;A: Point;Intersection(G, E) = {B, A};4*VectorOf(B, F) = 5*VectorOf(F, A)", "coords": { "F": [ 1.0, 0.0 ] } }, { "index": 700, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0700.png", "answer": "2*sqrt(5)", "fact_expr": "G: Parabola;H: Line;A: Point;P: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (x = -1);Coordinate(A) = (3, 4);PointOnCurve(P, G);Distance(P, H) = d;d:Number", "coords": { "A": [ 3.0, 4.0 ] } }, { "index": 701, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 702, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0702.png", "answer": "sqrt(2)/2", "fact_expr": "G: Parabola;p: Number;A: Point;B: Point;M: Point;N: Point;F: Point;l:Line;p>0;Expression(G) = (y^2 = 2*p*x);Directrix(G)=l;Focus(G) = F;PointOnCurve(A,G);PointOnCurve(B,G);AngleOf(A, F, B) = ApplyUnit(90, degree);IsChordOf(LineSegmentOf(A,B),G);MidPoint(LineSegmentOf(A,B))=M;PointOnCurve(M,LineSegmentOf(M,N));IsPerpendicular(LineSegmentOf(M,N),l);FootPoint(LineSegmentOf(M,N),l)=N", "coords": {} }, { "index": 703, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0703.png", "answer": "7", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;lambda:Number;Expression(G) = (x^2/4 + y^2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);PointOnCurve(MidPoint(LineSegmentOf(P,F1)), yAxis);Abs(LineSegmentOf(P, F1)) = lambda*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 704, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0704.png", "answer": "2*(sqrt(2)+1)", "fact_expr": "Q: Point;Coordinate(Q) = (-2, 0);G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;H: Line;PointOnCurve(Focus(G), H);A: Point;B: Point;Intersection(H, G) = {A, B};P: Point;Intersection(H, yAxis) = P;VectorOf(A, B) = 3*VectorOf(B, P);Slope(LineOf(Q, A)) = 1", "coords": { "Q": [ -2.0, 0.0 ] } }, { "index": 705, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0705.png", "answer": "9/2", "fact_expr": "C: Parabola;G: Line;A: Point;O: Origin;B: Point;F: Point;Expression(C) = (y^2 = 6*x);Focus(C) = F;PointOnCurve(F, G);Intersection(G, C) = {A, B}", "coords": {} }, { "index": 706, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 707, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 708, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.23606797749979, "b": 2.0, "major_axis": "x", "x_coef": 5.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0708.png", "answer": "8-4*sqrt(3)", "fact_expr": "P: Point;Q: Curve;Expression(Q) = (x^2/5 + y^2/4 = 1);PointOnCurve(P, Q) = True;F1: Point;F2: Point;Focus(Q) = {F1, F2};AngleOf(F1, P, F2) = ApplyUnit(30, degree)", "coords": {} }, { "index": 709, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0709.png", "answer": "(x-1)^2 + (y-2)^2 = 4", "fact_expr": "C: Circle;Quadrant(Center(C)) = 1;H: Line;Expression(H) = (y = 2*x);PointOnCurve(Center(C), H);G: Parabola;Expression(G) = (y^2 = 4*x);IsTangent(C, Directrix(G));IsTangent(C, xAxis)", "coords": {} }, { "index": 710, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 711, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 712, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0712.png", "answer": "2", "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2 = 1);A1: Point;A2: Point;LeftVertex(C) = A1;RightVertex(C) = A2;P: Point;PointOnCurve(P, C);Slope(LineOf(A1, P)) = 1/2", "coords": {} }, { "index": 713, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0713.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2 + 2*y^2 = 2);PointOnCurve(P, G);Focus(G)={F1,F2};AngleOf(F1, P, F2) = ApplyUnit(90, degree)", "coords": {} }, { "index": 714, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 2.8284271247461903, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0714.png", "answer": "4", "fact_expr": "l: Line;G: Hyperbola;A: Point;B: Point;Expression(G) = (x^2/16 - y^2/8 = 1);PointOnCurve(OneOf(Focus(G)),l);IsPerpendicular(l,RealAxis(G));Intersection(l, G) = {A, B}", "coords": {} }, { "index": 715, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0715.png", "answer": "1/2", "fact_expr": "l: Line;C: Ellipse;b: Number;a: Number;A: Point;B: Point;M: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(M) = (-2, 1);Eccentricity(C) = sqrt(3)/2;Intersection(l, C) = {A, B};MidPoint(LineSegmentOf(A,B)) = M", "coords": { "M": [ -2.0, 1.0 ] } }, { "index": 716, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0716.png", "answer": "12", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;x1: Number;x2: Number;y2: Number;Expression(G) = (y^2 = 8*x);Coordinate(A) = (x1, x2);Coordinate(B) = (x2, y2);PointOnCurve(Focus(G), H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 16", "coords": {} }, { "index": 717, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0717.png", "answer": "(\\sqrt{5} - 1)/2", "fact_expr": "C: Ellipse;b: Number;a: Number;G: Circle;P: Point;A: Point;F: Point;O: Origin;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftVertex(C) = A;LeftFocus(C) = F;TangentPoint(LineOf(A, P), G) = P;Center(G) = O;Radius(G) = b;IsPerpendicular(LineSegmentOf(P, F), xAxis)", "coords": {} }, { "index": 718, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0718.png", "answer": "8", "fact_expr": "l: Line;Focus(G) = F;F: Point;PointOnCurve(F,l) = True;G: Parabola;Expression(G) = (y^2 = 8*x);Intersection(l,G) = {A,B};A: Point;B: Point;Distance(MidPoint(LineSegmentOf(A,B)),yAxis) = 2", "coords": {} }, { "index": 719, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0719.png", "answer": "6", "fact_expr": "G: Parabola;p: Number;H: Line;B: Point;C: Point;F: Point;A: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G)=F;PointOnCurve(F,H);Intersection(H,G) = {B, A};Intersection(H,Directrix(G))=C;Between(B,F,C)=True;Abs(LineSegmentOf(B, C)) = 2*Abs(LineSegmentOf(B, F));Abs(LineSegmentOf(A, F)) = 12", "coords": {} }, { "index": 720, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0720.png", "answer": "sqrt(6)/3", "fact_expr": "C: Ellipse;b: Number;a: Number;A: Point;P: Point;B: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Endpoint(MajorAxis(C))={A,B};PointOnCurve(P, C);AngleOf(A, P, B) = ApplyUnit(120, degree)", "coords": {} }, { "index": 721, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0721.png", "answer": "12", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);F: Point;Focus(G) = F;A: Point;Coordinate(A) = (5, 4);P: Point;PointOnCurve(P, G);Negation(PointOnCurve(P, LineOf(A, F)))", "coords": { "A": [ 5.0, 4.0 ] } }, { "index": 722, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 4.0, -0.0 ], "radius": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_0722.png", "answer": "sqrt(15)", "fact_expr": "G: Hyperbola;a: Number;H: Circle;a>0;Expression(G) = (-y^2 + x^2/a^2 = 1);Expression(H) = (-8*x + x^2 + y^2 = 0);RightFocus(G)=Center(H)", "coords": {} }, { "index": 723, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0723.png", "answer": "x^2/12 - y^2/8 = 1", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/4 = 1);H: Point;Coordinate(H) = (3*sqrt(2), 2);Z: Hyperbola;Focus(G) = Focus(Z);PointOnCurve(H, Z)", "coords": { "H": [ 4.242640687119286, 2.0 ] } }, { "index": 724, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0724.png", "answer": "(1,3)+(3,+oo)", "fact_expr": "G: Ellipse;m: Number;H: Line;Expression(G) = (y^2/3 + x^2/m = 1);Expression(H) = (y = x + 2);NumIntersection(H, G) = 2", "coords": {} }, { "index": 725, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0725.png", "answer": "[2, +oo)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F;PointOnCurve(F, H);Inclination(H) = (2*pi)/3;NumIntersection(H, LeftPart(G)) = 1", "coords": {} }, { "index": 726, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 9.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0726.png", "answer": "2", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/2 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 4", "coords": {} }, { "index": 727, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0727.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;l: Line;A: Point;B: Point;O:Origin;c:Number;b > a;a > 0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (a, 0);Coordinate(B) = (0, b);HalfFocalLength(G)=c;Distance(O,l)=(sqrt(3)/4)*c;PointOnCurve(A,l);PointOnCurve(B,l)", "coords": {} }, { "index": 728, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0728.png", "answer": "10*sqrt(2)", "fact_expr": "l: Line;G: Parabola;C: Point;D: Point;A: Point;B: Point;Expression(G) = (y^2 = 16*x);Expression(l) = (2*x - y + 1 = 0);Intersection(l, G) = {A, B};L1:Line;L2:Line;PointOnCurve(A,L1);PointOnCurve(B,L2);IsPerpendicular(l,L1);IsPerpendicular(l,L2);Intersection(L1, xAxis) = C ;Intersection(L2, xAxis) = D", "coords": {} }, { "index": 729, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0729.png", "answer": "(1/4,+\\infty)", "fact_expr": "G: Ellipse;m: Number;H: Line;Expression(G) = (x^2/3 + y^2/m = 1);Expression(H) = (x + 2*y - 2 = 0);NumIntersection(G, H) = 2", "coords": {} }, { "index": 730, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.125, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0730.png", "answer": "1/4", "fact_expr": "G: Parabola;Expression(G) = (y = 2*x^2)", "coords": {} }, { "index": 731, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 732, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 733, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0733.png", "answer": "4*sqrt(2)", "fact_expr": "G: Parabola;P: Point;F: Point;O: Origin;Expression(G) = (x^2 = 4*y);PointOnCurve(P, G);Focus(G) = F;Abs(LineSegmentOf(P, F)) = 5", "coords": {} }, { "index": 734, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0734.png", "answer": "pm*sqrt(2)", "fact_expr": "G: Parabola;A: Point;B: Point;O: Origin;M: Point;Expression(G) = (y^2=2*x);PointOnCurve(A,G);PointOnCurve(B,G);DotProduct(VectorOf(O,A),VectorOf(O,B))=0;Intersection(LineOf(A,B),xAxis)=M;LineSegmentOf(A,M)=2*LineSegmentOf(B,M)", "coords": {} }, { "index": 735, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 736, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 737, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 738, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0738.png", "answer": "(-\\infty,-4] \\cup [12,+\\infty)", "fact_expr": "G: Parabola;A: Point;B: Point;C: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (4, 4);PointOnCurve(B,G);PointOnCurve(C,G);IsPerpendicular(LineSegmentOf(A, B), LineSegmentOf(B, C))", "coords": { "A": [ 4.0, 4.0 ] } }, { "index": 739, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0739.png", "answer": "3*sqrt(3)/5", "fact_expr": "F1: Point;F2: Point;Focus(G) = {F1, F2};G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);P: Point;PointOnCurve(P, G);AngleOf(P, F1, F2) = ApplyUnit(120, degree)", "coords": {} }, { "index": 740, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 741, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 742, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 743, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0743.png", "answer": "10/3", "fact_expr": "G: Ellipse;A: Point;B: Point;F2: Point;F1: Point;x1:Number;x2:Number;y1:Number;y2:Number;Expression(G) = (x^2/25 + y^2/16 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;IsChordOf(LineSegmentOf(A,B),G);Perimeter(InscribedCircle(TriangleOf(A,B,F2)))=2*pi;Coordinate(A)=(x1,y1);Coordinate(B)=(x2,y2);PointOnCurve(F1, LineSegmentOf(A, B))", "coords": {} }, { "index": 744, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0744.png", "answer": "40", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 8*y);H: Line;PointOnCurve(Focus(G), H);Slope(H) = 2;A: Point;B: Point;Intersection(H, G) = {A, B};IsChordOf(LineSegmentOf(A,B),G)", "coords": {} }, { "index": 745, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0745.png", "answer": "[-2*sqrt(3), 2*sqrt(3)]", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);P: Point;PointOnCurve(P, C);F: Point;Focus(C) = F;A: Point;m: Real;Coordinate(A) = (3, m);Min(Abs(LineSegmentOf(P, A)) + Abs(LineSegmentOf(P, F))) = 4", "coords": {} }, { "index": 746, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0746.png", "answer": "17/4", "fact_expr": "G: Ellipse;A: Point;M: Point;F: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Coordinate(A) = (2, 2);RightFocus(G) = F;PointOnCurve(M, G)", "coords": { "A": [ 2.0, 2.0 ] } }, { "index": 747, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0747.png", "answer": "2", "fact_expr": "G: Parabola;A: Point;B: Point;P: Point;Expression(G) = (x^2 = 4*y);PointOnCurve(A, G);PointOnCurve(B, G);Negation(A=B);MidPoint(LineSegmentOf(A,B)) = P;Abs(LineSegmentOf(A, B)) = 6", "coords": {} }, { "index": 748, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0748.png", "answer": "2,8", "fact_expr": "G: Parabola;p: Number;M: Point;p>0;Expression(G) = (y^2 = 2*(p*x));PointOnCurve(M, G);Distance(M, xAxis) = 4;Distance(M,Focus(G))=5", "coords": {} }, { "index": 749, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.463192105293274, "b": 2.070901870727539, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4661259228887502e-06, "converged": true, "iterations": 106 }, "output_path": "results/train/hyperbola/problem_0749.png", "answer": "sqrt(6)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Eccentricity(C) = sqrt(3);G: Point;Coordinate(G) = (3, 0)", "coords": { "G": [ 3.0, 0.0 ] } }, { "index": 750, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0750.png", "answer": "(1, \\sqrt{21}/3]", "fact_expr": "C: Hyperbola;b: Number;a: Number;A2: Point;P: Point;A1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftVertex(C) = A1;RightVertex(C) = A2;PointOnCurve(A2, LineSegmentOf(A2, P));IsPerpendicular(Asymptote(C), LineSegmentOf(A2, P));FootPoint(Asymptote(C), LineSegmentOf(A2, P)) = P;Abs(LineSegmentOf(A1, P))/Abs(LineSegmentOf(A2, P)) >= 2", "coords": {} }, { "index": 751, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0751.png", "answer": "(-1, 2)", "fact_expr": "A: Point;Coordinate(A) = (-3, 2);G: Parabola;Expression(G) = (y^2 = -4*x);F: Point;Focus(G) = F;P: Point;PointOnCurve(P, G);WhenMin(Abs(LineSegmentOf(P, A)) + Abs(LineSegmentOf(P, F)))", "coords": { "A": [ -3.0, 2.0 ] } }, { "index": 752, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.1503682136535645, "b": 2.5736796855926514, "orientation": "horizontal" }, "optimization": { "final_loss": 5.191216132516274e-06, "converged": true, "iterations": 118 }, "output_path": "results/train/hyperbola/problem_0752.png", "answer": "y=pm*sqrt(5)*x", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Eccentricity(G) = sqrt(6)", "coords": {} }, { "index": 753, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0753.png", "answer": "sqrt(a^2+b^2-a*b)", "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;A: Point;B: Point;O: Origin;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (x0, y0);x0:Number;y0:Number;l:Line;TangentOfPoint(P,G)=l;Expression(l)=(x0*x/a^2+y0*y/b^2=1);Intersection(l,xAxis)=A;Intersection(l,yAxis)=B;WhenMin(Abs(LineSegmentOf(A,B)));PointOnCurve(P,G)", "coords": {} }, { "index": 754, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0754.png", "answer": "sqrt(2)/4", "fact_expr": "G: Hyperbola;a: Number;b: Number;P: Point;F1: Point;F2: Point;M: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, RightPart(G));LeftFocus(G) = F1;RightFocus(G) = F2;Slope(OneOf(Asymptote(G))) = sqrt(7);Incenter(TriangleOf(P, F1, F2)) = M;lambda: Number;Area(TriangleOf(P, M, F1)) = Area(TriangleOf(P, M, F2)) + lambda*Area(TriangleOf(M, F1, F2))", "coords": {} }, { "index": 755, "success": false, "conic_type": "parabola", "error": "validation: parabola_focus_mismatch", "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0755.png", "answer": "y^2 = 4*x", "fact_expr": "C: Parabola;G: Line;A: Point;B: Point;F: Point;O: Origin;Expression(G) = (y = x);Coordinate(F) = (2, 2);PointOnCurve(Focus(C), xAxis);Intersection(G, C) = {A,B};MidPoint(LineSegmentOf(A, B)) = F;Vertex(C) = O", "coords": { "F": [ 2.0, 2.0 ] }, "validation_reasons": [ "parabola_focus_mismatch" ] }, { "index": 756, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 1.0, 1.0 ], "radius": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_0756.png", "answer": "(x-3/2)^2+(y-2)^2=5/4", "fact_expr": "C: Circle;Expression(C) = ((x - 1)^2 + (y - 1)^2 = 9);A: Point;Coordinate(A) = (2, 3);L: LineSegment;PointOnCurve(A, L) = True;IsChordOf(L, C) = True;MidPoint(L) = P;P: Point", "coords": {} }, { "index": 757, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0757.png", "answer": "{9, 7}", "fact_expr": "G: Ellipse;Expression(G) = (y^2/8 + x^2/m = 1);m: Number;FocalLength(G) = 2", "coords": {} }, { "index": 758, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0758.png", "answer": "sqrt(2)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;M: Point;LeftVertex(C) = M;G: Line;IsParallel(G, xAxis);A: Point;B: Point;Intersection(G, C) = {A, B};DotProduct(VectorOf(M, A), VectorOf(M, B)) = 0", "coords": {} }, { "index": 759, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7195594310760498, "b": 1.7224669456481934, "orientation": "horizontal" }, "optimization": { "final_loss": 2.391175712546101e-06, "converged": true, "iterations": 83 }, "output_path": "results/train/hyperbola/problem_0759.png", "answer": "y=pm*x", "fact_expr": "G: Hyperbola;a: Number;b: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = sqrt(2)", "coords": {} }, { "index": 760, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 761, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0761.png", "answer": "3*x+4*y-7=0", "fact_expr": "Gamma: Ellipse;Expression(Gamma) = (x^2/4 + y^2/3 = 1);P: Point;Coordinate(P) = (1, 1);l: Line;PointOnCurve(P, l);A: Point;B: Point;Intersection(l, Gamma) = {A, B};IsChordOf(LineSegmentOf(A, B), Gamma);MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ 1.0, 1.0 ] } }, { "index": 762, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 2.0, -0.0 ], "radius": 1.4142135623730951 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_0762.png", "answer": "{2*sqrt(3)/3, 2}", "fact_expr": "G: Hyperbola;H: Circle;Expression(H) = (-4*x + x^2 + y^2 + 2 = 0);Length(InterceptChord(OneOf(Asymptote(G)), H)) = 2", "coords": {} }, { "index": 763, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 2.6457513110645907, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0763.png", "answer": "5+2*sqrt(5)", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/9 - y^2/7 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;E: Circle;Expression(E) = (-4*y + x^2 + y^2 + 3 = 0);P: Point;PointOnCurve(P, RightPart(C)) = True;Q: Point;PointOnCurve(Q, E) = True", "coords": {} }, { "index": 764, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 765, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0765.png", "answer": "[(3/4)*sqrt(5),9/4]", "fact_expr": "G: Ellipse;P: Point;A: Point;Coordinate(A) = (1/4, 0);Expression(G)= (x^2/4 + y^2/3 = 1);PointOnCurve(P,G)", "coords": { "A": [ 0.25, 0.0 ] } }, { "index": 766, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 767, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0767.png", "answer": "sqrt(3)/3", "fact_expr": "G: Ellipse;b: Number;a: Number;F2: Point;B2: Point;C: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F2;LowerVertex(G) = B2;PointOnCurve(C, G);VectorOf(F2, B2) = 2*VectorOf(C, F2)", "coords": {} }, { "index": 768, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0768.png", "answer": "(2*sqrt(3))/3", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;H: Circle;Expression(H) = (y^2 + (x - 2)^2 = 1);IsTangent(Asymptote(G), H)", "coords": {} }, { "index": 769, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0769.png", "answer": "10", "fact_expr": "l: Line;G: Parabola;A: Point;O: Origin;B: Point;M: Point;P: Point;Q: Point;H: Circle;Expression(G) = (y^2 = 4*x);Coordinate(M) = (6, 0);Intersection(l, G) = {A, B};Slope(LineOf(O, A))*Slope(LineOf(O, B)) = -1;Center(H) = MidPoint(LineSegmentOf(A, B));Radius(H) = sqrt(2);Intersection(H, l) = {P, Q}", "coords": { "M": [ 6.0, 0.0 ] } }, { "index": 770, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.196152422706632, "b": 4.0, "major_axis": "x", "x_coef": 27.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0770.png", "answer": "1/5", "fact_expr": "G: Ellipse;l: Line;P: Point;Expression(G) = (x^2/27 + y^2/16 = 1);Expression(l) = (4*x + 3*y - 25 = 0);PointOnCurve(P, G)", "coords": {} }, { "index": 771, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0771.png", "answer": "2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;H: Circle;Expression(H) = (-2*y + x^2 + y^2 - 1 = 0);Length(InterceptChord(Directrix(G), H)) = 2", "coords": {} }, { "index": 772, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 773, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 774, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0774.png", "answer": "2*sqrt(3)/3", "fact_expr": "F: Point;E: Hyperbola;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;RightFocus(E) = F;L: Line;PointOnCurve(F, L);L1: Line;L2: Line;OneOf(Asymptote(E))=L1;OneOf(Asymptote(E))=L2;Negation(L1=L2);IsPerpendicular(L, L1);FootPoint(L, L1) = A;A: Point;Intersection(L, L2) = B;B: Point;Abs(LineSegmentOf(O, F)) = Abs(LineSegmentOf(F, B));O: Origin", "coords": {} }, { "index": 775, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0775.png", "answer": "sqrt(2)/2", "fact_expr": "G: Ellipse;H: Line;F:Point;m:Number;m>0;M:Point;Expression(H) = (y = x*(sqrt(2)/2));Expression(G) = (x^2/16 + y^2/m^2 = 1);OneOf(Intersection(H,G))=M;Projection(M,xAxis)=F;RightFocus(G)=F", "coords": {} }, { "index": 776, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0776.png", "answer": "2", "fact_expr": "G: Parabola;P: Point;F: Point;O: Origin;Expression(G) = (y^2 = 4*x);PointOnCurve(P, G);Focus(G) = F;Distance(P, F) = 5", "coords": {} }, { "index": 777, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0777.png", "answer": "4", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;c:Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) =F1;RightFocus(G)=F2;Eccentricity(G) = (2*sqrt(3))/3;FocalLength(G) = 2*c;2*a^2 = 3*c;PointOnCurve(P,G);DotProduct(VectorOf(P,F1),VectorOf(P,F2))=2", "coords": {} }, { "index": 778, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5002657771110535, "direction": "right" }, "optimization": { "final_loss": 8.410525879298802e-06, "converged": true, "iterations": 102 }, "output_path": "results/train/parabola/problem_0778.png", "answer": "5/2", "fact_expr": "G: Parabola;p: Number;M: Point;Expression(G) = (y^2 = 2*(p*x));Coordinate(M) = (2, 2);PointOnCurve(M, G)", "coords": { "M": [ 2.0, 2.0 ] } }, { "index": 779, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0779.png", "answer": "pm*1", "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;F: Point;M: Point;Expression(G) = (y^2 = 8*x);Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A, B)) = M;Distance(M, Directrix(G)) = 8", "coords": {} }, { "index": 780, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0780.png", "answer": "20", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2/5 + y^2/m = 1);PointOnCurve(Focus(G), yAxis);Eccentricity(G)=e;e:Number;e=sqrt(3)/2", "coords": {} }, { "index": 781, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0781.png", "answer": "[\\sqrt{3}/2, 1)", "fact_expr": "C1: Ellipse;Expression(C1) = (x^2/a^2 + y^2/b^2 = 1);b: Number;a: Number;a > b;b > 0;C2: Circle;Expression(C2) = (x^2 + y^2 = b^2);P: Point;PointOnCurve(P, C1);A: Point;B: Point;TangentOfPoint(P, C2) = {LineOf(P, A), LineOf(P, B)};TangentPoint(LineOf(P, A), C2) = A;TangentPoint(LineOf(P, B), C2) = B;AngleOf(B, P, A) = pi/3", "coords": {} }, { "index": 782, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0782.png", "answer": "6", "fact_expr": "C: Parabola;F: Point;M: Point;N: Point;Expression(C) = (y^2 = 12*x);Focus(C) = F;PointOnCurve(M, C);Intersection(OverlappingLine(LineSegmentOf(F, M)), yAxis) = N;VectorOf(F, M) = 2*VectorOf(M, N)", "coords": {} }, { "index": 783, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 784, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0784.png", "answer": "1/2", "fact_expr": "G: Ellipse;H: Line;m: Number;A: Point;P: Point;B: Point;Q: Point;k1: Number;k2: Number;Expression(G) = (x^2/4 + y^2 = 1);Expression(H) = (x = m*y + 6/5);LeftVertex(G) = A;RightVertex(G) = B;Intersection(H, G) = {P, Q};Slope(LineOf(A, P)) = k1;Slope(LineOf(B, Q)) = k2", "coords": {} }, { "index": 785, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 786, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0786.png", "answer": "(2,2*sqrt(2))", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P:Point;Quadrant(P)=1;PointOnCurve(P,G);Distance(P,Focus(G))=3", "coords": {} }, { "index": 787, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0787.png", "answer": "y^2=20*x", "fact_expr": "G: Hyperbola;H: Parabola;Expression(G) = (x^2/16 - y^2/9 = 1);Vertex(H) = Center(G);Focus(H) = RightFocus(G)", "coords": {} }, { "index": 788, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 789, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 2.449489742783178, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0789.png", "answer": "16*sqrt(3)/5", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2/6 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;H: Line;PointOnCurve(F2, H);Inclination(H) = ApplyUnit(30, degree);A: Point;B: Point;Intersection(H, G) = {A, B};O: Origin", "coords": {} }, { "index": 790, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0790.png", "answer": "x^2/25+y^2/9=1", "fact_expr": "C: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Focus(C) = {F1, F2};PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2));Area(TriangleOf(P, F1, F2)) = 9;Perimeter(TriangleOf(P, F1, F2))=18", "coords": {} }, { "index": 791, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0791.png", "answer": "3", "fact_expr": "G: Ellipse;p: Number;H: Point;Expression(G) = (y^2/4 + x^2/p = 1);Coordinate(H) = (0, -1);OneOf(Focus(G)) = H", "coords": { "H": [ 0.0, -1.0 ] } }, { "index": 792, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0792.png", "answer": "5/4", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1);e: Number;Eccentricity(G) = e", "coords": {} }, { "index": 793, "success": false, "error": "Circle from diameter: cannot parse diameter pattern" }, { "index": 794, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5002657771110535, "direction": "right" }, "optimization": { "final_loss": 8.410525879298802e-06, "converged": true, "iterations": 102 }, "output_path": "results/train/parabola/problem_0794.png", "answer": "5/2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;M: Point;Coordinate(M) = (2, 2);PointOnCurve(M, G)", "coords": { "M": [ 2.0, 2.0 ] } }, { "index": 795, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 796, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0796.png", "answer": "5/4", "fact_expr": "A: Point;B: Point;G: Parabola;F: Point;Expression(G) = (y^2 = x);Focus(G) = F;PointOnCurve(A, G);PointOnCurve(B, G);Abs(LineSegmentOf(A, F)) + Abs(LineSegmentOf(B, F)) = 3", "coords": {} }, { "index": 797, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0797.png", "answer": "3", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);A: Point;PointOnCurve(A, G);F: Point;LeftFocus(G) = F;Distance(A, F) = 4;B: Point;MidPoint(LineSegmentOf(A, F)) = B;O: Origin", "coords": {} }, { "index": 798, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 799, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 800, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0800.png", "answer": "5/3", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, RightPart(G));Abs(LineSegmentOf(P, F2)) = Abs(LineSegmentOf(F1, F2));H: Circle;Expression(H) = (x^2 + y^2 = a^2);IsIntersect(LineOf(P, F1), H)", "coords": {} }, { "index": 801, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.6259966743174588, "b": 2.298015331604694, "orientation": "horizontal" }, "optimization": { "final_loss": 1.93011531933813e-06, "converged": true, "iterations": 91 }, "output_path": "results/train/hyperbola/problem_0801.png", "answer": "2*sqrt(3)", "fact_expr": "G: Hyperbola;H: Point;Coordinate(H) = (1, 2);PointOnCurve(H, G);Expression(Asymptote(G)) = (y = pm*(sqrt(2)*x))", "coords": { "H": [ 1.0, 2.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 802, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0802.png", "answer": "y^2=18*x", "fact_expr": "H: Ellipse;Expression(H) = (x^2/9 + y^2/5 = 1);G: Parabola;Center(H) = Vertex(G);LeftDirectrix(H) = Directrix(G)", "coords": {} }, { "index": 803, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 804, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 805, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0805.png", "answer": "sqrt(10)", "fact_expr": "G: Parabola;H: Circle;P: Point;A: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (y^2 + (x - 4)^2 = 2);PointOnCurve(P, G);l: Line;TangentOfPoint(P, H) = l;TangentPoint(l, H) = A", "coords": {} }, { "index": 806, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0806.png", "answer": "2*sqrt(3)/3", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2/b^2 = 1);b: Number;Length(ImageinaryAxis(G)) = 2", "coords": {} }, { "index": 807, "success": false, "error": "Circle from diameter: missing point coordinates" }, { "index": 808, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0808.png", "answer": "2", "fact_expr": "H: Line;Expression(H) = (m - m*y + x = 0);m: Number;G: Parabola;Expression(G) = (x^2 = 2*(p*y));p: Number;p>0;PointOnCurve(Focus(G), H)", "coords": {} }, { "index": 809, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 810, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0810.png", "answer": "7/2", "fact_expr": "G: Parabola;P: Point;A: Point;M: Point;d: Number;Expression(G) = (y^2 = 2*x);Coordinate(A) = (7/2, 4);PointOnCurve(P, G);Distance(P, Directrix(G)) = d;Projection(P, yAxis) = M", "coords": { "A": [ 3.5, 4.0 ] } }, { "index": 811, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 812, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 813, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0813.png", "answer": "1/2", "fact_expr": "G: Ellipse;a:Number;b:Number;F1:Point;F2:Point;P:Point;a>b;b>0;PointOnCurve(Focus(G), xAxis);Expression(G) = (y^2/b^2 + x^2/a^2 = 1);OneOf(Endpoint(MinorAxis(G)))=P;Focus(G)={F1,F2};Radius(InscribedCircle(TriangleOf(P,F1,F2)))=b/3", "coords": {} }, { "index": 814, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0814.png", "answer": "3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;H: Line;PointOnCurve(F, H);A: Point;B: Point;Intersection(H, G) = {A, B};VectorOf(A, F) = 2*VectorOf(F, B)", "coords": {} }, { "index": 815, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 2.8503371627061314, "b": 1.8676572177554456, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 5.51866980271156e-07, "converged": true, "iterations": 103 }, "output_path": "results/train/ellipse/problem_0815.png", "answer": "x^2/9+y^2/4=1", "fact_expr": "P: Point;Coordinate(P) = (-3, 0);Q: Point;Coordinate(Q) = (0, -2);G: Ellipse;PointOnCurve(P, G);PointOnCurve(Q, G)", "coords": { "P": [ -3.0, 0.0 ], "Q": [ 0.0, -2.0 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 816, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0816.png", "answer": "(\\sqrt{3}+1)/2", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;G: Circle;Center(G) = F2;Radius(G) = Abs(LineSegmentOf(F1, F2));A: Point;B: Point;Intersection(G, RightPart(C)) = {A, B};Abs(LineSegmentOf(A, B)) = sqrt(3)*Abs(LineSegmentOf(F1, F2))", "coords": {} }, { "index": 817, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0817.png", "answer": "16/3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, G) = {A, B};K: Point;Intersection(Directrix(G), xAxis) = K;Abs(LineSegmentOf(A, F)) = 2*Abs(LineSegmentOf(F, K))", "coords": {} }, { "index": 818, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0818.png", "answer": "(1, 2)", "fact_expr": "G: Hyperbola;b: Number;a: Number;A: Point;B: Point;P: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (1, 2);Coordinate(B) = (-1, 2);IsPerpendicular(VectorOf(B, P),VectorOf(A, P));NumIntersection(Asymptote(G),Locus(P))=0", "coords": { "A": [ 1.0, 2.0 ], "B": [ -1.0, 2.0 ] } }, { "index": 819, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.23606797749979, "b": 2.0, "major_axis": "x", "x_coef": 5.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0819.png", "answer": "(0,pm*2)", "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/5 + y^2/4 = 1);PointOnCurve(P, G);Area(TriangleOf(P,F1,F2))=2;F1:Point;F2:Point;Focus(G)={F1,F2}", "coords": {} }, { "index": 820, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0820.png", "answer": "3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);H: Point;x0: Number;y0: Number;Coordinate(H) = (x0, y0);PointOnCurve(H, G);z: Number;z=x0^2+(1/2)*y0^2+3", "coords": {} }, { "index": 821, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0821.png", "answer": "16", "fact_expr": "G: Ellipse;H: Line;A: Point;B: Point;F2: Point;F1: Point;Expression(G) = (x^2/16 + y^2/9 = 1);OneOf(Focus(G)) = F1;PointOnCurve(F1, H);Intersection(H, G) = {A, B};OneOf(Focus(G)) = F2;Negation(F1 = F2)", "coords": {} }, { "index": 822, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0822.png", "answer": "y^2/2 - x^2/2 = 1", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2 + y^2/2 = 1);Vertex(G) = Endpoint(MajorAxis(H));Eccentricity(G)*Eccentricity(H) = 1", "coords": {} }, { "index": 823, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0823.png", "answer": "4*x+pm*3*y=0", "fact_expr": "C: Hyperbola;G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);Focus(C)=Endpoint(MajorAxis(G));Vertex(C)=Focus(G)", "coords": {} }, { "index": 824, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0824.png", "answer": "4*sqrt(3)/3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);l: Line;F: Point;Focus(G) = F;PointOnCurve(F, l) = True;Intersection(l, G) = {A, B};A: Point;B: Point;VectorOf(B, A) = 4*VectorOf(B, F);O: Origin", "coords": {} }, { "index": 825, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 5.0, "major_axis": "x", "x_coef": 36.0, "y_coef": 25.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0825.png", "answer": "9", "fact_expr": "G: Ellipse;P: Point;F1:Point;F2:Point;Expression(G) = (x^2/36 + y^2/25 = 1);PointOnCurve(P, G);Distance(P,F1) = 3;OneOf(Focus(G))=F1;OneOf(Focus(G))=F2;Negation(F1=F2)", "coords": {} }, { "index": 826, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0826.png", "answer": "3", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;A: Point;F: Point;B: Point;O: Origin;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(O, l);Intersection(l, C) = {A, B};Negation(A = B);LeftFocus(C) = F;Abs(LineSegmentOf(A, F)) = (5/3)*Abs(LineSegmentOf(B, F));Abs(LineSegmentOf(O, A)) = b", "coords": {} }, { "index": 827, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0010501146316528, "direction": "right" }, "optimization": { "final_loss": 3.455598402979376e-07, "converged": true, "iterations": 81 }, "output_path": "results/train/parabola/problem_0827.png", "answer": "(-\\infty,-1)", "fact_expr": "C: Parabola;p: Number;A: Point;B: Point;x1:Number;y1:Number;y1>-6;y1<-2;p>0;Expression(C) = (y^2 = 2*p*x);Coordinate(A) = (1, 2);Coordinate(B) = (x1, y1);PointOnCurve(A,C);PointOnCurve(B, C)", "coords": { "A": [ 1.0, 2.0 ] } }, { "index": 828, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0828.png", "answer": "{ApplyUnit(45,degree),ApplyUnit(135,degree)}", "fact_expr": "l: Line;G: Parabola;H: Circle;A: Point;B: Point;Expression(G) = (y^2 = 8*x);PointOnCurve(Focus(G), l);Intersection(l, G) = {A, B};IsDiameter(LineSegmentOf(A,B),H);Radius(H)=8", "coords": {} }, { "index": 829, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 7.0, "b": 4.898979485566356, "major_axis": "x", "x_coef": 49.0, "y_coef": 24.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0829.png", "answer": "24", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2:Point;Expression(G) = (x^2/49 + y^2/24 = 1);PointOnCurve(P,G);Focus(G)={F1,F2};IsPerpendicular(LineSegmentOf(P,F1),LineSegmentOf(P,F2))", "coords": {} }, { "index": 830, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_0830.png", "answer": "x^2 - y^2 / 3 = 1", "fact_expr": "H: Parabola;Expression(H) = (y^2 = 8*x);G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;PointOnCurve(OneOf(Focus(G)), Directrix(H)) = True;Eccentricity(G) = 2", "coords": {} }, { "index": 831, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2135674953460693, "b": 2.4270472526550293, "orientation": "horizontal" }, "optimization": { "final_loss": 4.453341261978494e-06, "converged": true, "iterations": 125 }, "output_path": "results/train/hyperbola/problem_0831.png", "answer": "5*x^2-(5/4)*y^2=1", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x = y^2/4);OneOf(Focus(G))=Focus(H);Eccentricity(G) = sqrt(5)", "coords": {} }, { "index": 832, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.449489742783178, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0832.png", "answer": "sqrt(3)", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/6 - y^2/3 = 1)", "coords": {} }, { "index": 833, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 834, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0834.png", "answer": "sqrt(3)", "fact_expr": "l: Line;E: Hyperbola;b: Number;a: Number;G: Circle;P: Point;Q: Point;F1: Point;a>0;b>0;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2 + y^2 = a^2);LeftFocus(E) = F1;PointOnCurve(F1, l);Intersection(l, Asymptote(E)) = P;Quadrant(P) = 1;TangentPoint(l, G) = Q;Abs(LineSegmentOf(P, Q)) = 2*Abs(LineSegmentOf(F1, Q));e: Number;Eccentricity(E) = e", "coords": {} }, { "index": 835, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 836, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 837, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 838, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 839, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 840, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0840.png", "answer": "15", "fact_expr": "G: Ellipse;M: Point;P: Point;F1: Point;Expression(G) = (x^2/25 + y^2/16 = 1);Coordinate(M) = (6, 4);PointOnCurve(P, G);LeftFocus(G) = F1", "coords": { "M": [ 6.0, 4.0 ] } }, { "index": 841, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 16.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0841.png", "answer": "x^2/4 - y^2/9 = 1", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/16 + y^2/3 = 1);Focus(G) = Focus(H);Expression(OneOf(Asymptote(G))) = (y = (3/2)*x)", "coords": {} }, { "index": 842, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 843, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0843.png", "answer": "4", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/5 = 1)", "coords": {} }, { "index": 844, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0844.png", "answer": "9/4", "fact_expr": "H: Line;Expression(H) = (-k + 4*(k*x) - 4*y = 0);k: Number;G: Parabola;Expression(G) = (y^2 = x);A: Point;B: Point;Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 4;L: Line;Expression(L) = (x + 1/2 = 0);IsChordOf(LineSegmentOf(A, B), G)", "coords": {} }, { "index": 845, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0845.png", "answer": "-14", "fact_expr": "l: Line;C: Parabola;G: Line;M: Point;N: Point;P: Point;F: Point;Expression(C) = (x^2 = 4*y);Focus(C) = F;PointOnCurve(F, G);Slope(G) = 1;Intersection(G, C) = {M, N};IsTangent(l, C);IsParallel(l,LineSegmentOf(M,N));PointOnCurve(P, l)", "coords": {} }, { "index": 846, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0846.png", "answer": "(2,+oo)", "fact_expr": "G: Line;Expression(G) = (y = sqrt(3)*x);C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;NumIntersection(G, C) = 2", "coords": {} }, { "index": 847, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0847.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Slope(OneOf(Asymptote(G))) = -2", "coords": {} }, { "index": 848, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0848.png", "answer": "16/3", "fact_expr": "G: Hyperbola;Q:Circle;Expression(G) = (x^2/9 - y^2/16 = 1);PointOnCurve(Center(Q), G);PointOnCurve(OneOf(Vertex(G)),Q);PointOnCurve(OneOf(Focus(G)),Q)", "coords": {} }, { "index": 849, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0849.png", "answer": "8", "fact_expr": "G: Hyperbola;a: Number;Expression(G) = (-y^2/12 + x^2/a^2 = 1);Eccentricity(G) = e;e:Number;e=2", "coords": {} }, { "index": 850, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 851, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0851.png", "answer": "p/y0", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;P: Point;Q: Point;IsChordOf(LineSegmentOf(P, Q), G);x0: Number;y0: Number;Negation(y0=0);M: Point;Coordinate(M) = (x0, y0);MidPoint(LineSegmentOf(P, Q)) = M", "coords": {} }, { "index": 852, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0852.png", "answer": "{2,-2}", "fact_expr": "G: Parabola;O:Origin;p:Point;m:Number;Vertex(G)=O;PointOnCurve(Focus(G),xAxis);PointOnCurve(p,G);Coordinate(p)=(1,m);Distance(p,Focus(G))=2", "coords": {} }, { "index": 853, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 4.7896847246870875, "b": 2.5151959567993787, "major_axis": "x", "x_coef": 25.0, "y_coef": 7.000000000000001 }, "optimization": { "final_loss": 1.3352682552042877e-05, "converged": true, "iterations": 90 }, "output_path": "results/train/ellipse/problem_0853.png", "answer": "x^2/25+y^2/7=1", "fact_expr": "G: Ellipse;H: Point;I: Point;Coordinate(H) = (-5, 0);Coordinate(I) = (0, sqrt(7));PointOnCurve(H,G);PointOnCurve(I,G)", "coords": { "H": [ -5.0, 0.0 ], "I": [ 0.0, 2.6457513110645907 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 854, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0854.png", "answer": "sqrt(3)+1", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;Center(G) = F1;HalfFocalLength(C) = c;c: Number;Radius(G) = c;G: Circle;Intersection(G, C) = {P, Q};P: Point;Q: Point;IsTangent(LineSegmentOf(P, F2), G) = True", "coords": {} }, { "index": 855, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 856, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 857, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 2.4813292026519775, "b": 2.2365526058027783, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 6.31275318580922e-05, "converged": true, "iterations": 91 }, "output_path": "results/train/ellipse/problem_0857.png", "answer": "x^2/9+y^2/3=1", "fact_expr": "G: Ellipse;b: Number;a: Number;P1: Point;P2: Point;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(P1) = (sqrt(6), 1);Coordinate(P2) = (-sqrt(3), -sqrt(2));PointOnCurve(P1,G);PointOnCurve(P2,G)", "coords": { "P1": [ 2.449489742783178, 1.0 ], "P2": [ -1.7320508075688772, -1.4142135623730951 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 858, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.1622776601683795, "b": 2.23606797749979, "major_axis": "x", "x_coef": 10.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0858.png", "answer": "y=x+3", "fact_expr": "C: Ellipse;M: Point;N: Point;D: Point;Expression(C) = (x^2/10 + y^2/5 = 1);Coordinate(D) = (-2, 1);IsChordOf(LineSegmentOf(M,N),C);MidPoint(LineSegmentOf(M,N))=D", "coords": { "D": [ -2.0, 1.0 ] } }, { "index": 859, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0859.png", "answer": "8", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;A: Point;m: Number;Coordinate(A) = (2, m);PointOnCurve(A, G);Distance(A, Focus(G)) = 6", "coords": {} }, { "index": 860, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 861, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0861.png", "answer": "{1/2, 1/3}", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;H: Parabola;Expression(H) = (y^2 = 2*(p*x));p: Number;Focus(H) = F2;P: Point;OneOf(Intersection(C, H)) = P;Cos(AngleOf(P, F1, F2)) = 5/7", "coords": {} }, { "index": 862, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0862.png", "answer": "(0,3/4]", "fact_expr": "l: Line;G: Ellipse;A: Point;B: Point;F2: Point;F1: Point;Expression(G) = (x^2/4 + y^2/3 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, l);Intersection(l, G) = {A, B};r:Number;Radius(InscribedCircle(TriangleOf(A,B,F2)))=r", "coords": {} }, { "index": 863, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0863.png", "answer": "12", "fact_expr": "l: Line;C: Ellipse;A: Point;B: Point;F1: Point;Expression(C) = (x^2/9 + y^2/5 = 1);Intersection(l, C) = {A, B};LeftFocus(C) = F1;PointOnCurve(RightFocus(C),l)", "coords": {} }, { "index": 864, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0864.png", "answer": "x^2/16+y^2/12=1", "fact_expr": "G: Hyperbola;C: Ellipse;Expression(G) = (x^2 - y^2/3 = 1);Focus(G)=Focus(C);Eccentricity(C)=1/2", "coords": {} }, { "index": 865, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0865.png", "answer": "(x+pm*2)^2+(y-1)^2=4", "fact_expr": "G: Parabola;H: Circle;Expression(G) = (x^2 = 4*y);PointOnCurve(Center(H), G);IsTangent(Directrix(G),H);IsTangent(yAxis,H)", "coords": {} }, { "index": 866, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.3300262689590454, "b": 2.30381441116333, "orientation": "horizontal" }, "optimization": { "final_loss": 6.658357051492203e-06, "converged": true, "iterations": 122 }, "output_path": "results/train/hyperbola/problem_0866.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(Asymptote(G))=(y=pm*(sqrt(3)*x))", "coords": {} }, { "index": 867, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 868, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0868.png", "answer": "1+sqrt(2)", "fact_expr": "C: Hyperbola;b: Number;a: Number;c:Number;B: Point;A: Point;F2: Point;F1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(B) = (3*c, 0);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(A, C);IsPerpendicular(LineSegmentOf(A,F2),xAxis);AngleOf(F1,A,B)=ApplyUnit(90,degree);HalfFocalLength(C)=c", "coords": {} }, { "index": 869, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 870, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0870.png", "answer": "1/2", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;A: Point;B: Point;UpperVertex(G) = A;LowerVertex(G) = B;M: Point;Intersection(LineOf(A, F2), G) = {A, M};AngleOf(F1, A, F2) = ApplyUnit(90, degree)", "coords": {} }, { "index": 871, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0871.png", "answer": "(1,\\sqrt{2})", "fact_expr": "m: Line;Expression(m) = (y = k*x + 1);k: Number;G: Hyperbola;Expression(G) = (x^2 - y^2 = 1);NumIntersection(m,LeftPart(G)) = 2", "coords": {} }, { "index": 872, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 873, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0873.png", "answer": "{pm*1, pm*sqrt(2)}", "fact_expr": "l: Line;Expression(l) = (y = k*x + 1);k: Number;C: Hyperbola;Expression(C) = (x^2 - y^2 = 1);NumIntersection(l, C) = 1", "coords": {} }, { "index": 874, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0874.png", "answer": "sqrt(3)/3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;F: Point;Focus(G) = F;A: Point;B: Point;PointOnCurve(A, G);PointOnCurve(B, G);AngleOf(A, F, B) = ApplyUnit(120, degree);IsChordOf(LineSegmentOf(A, B), G);M: Point;MidPoint(LineSegmentOf(A, B)) = M;l: Line;Directrix(G) = l;M1: Point;Projection(M, l) = M1", "coords": {} }, { "index": 875, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 876, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 2.0, -0.0 ], "radius": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_0876.png", "answer": "2*sqrt(2)", "fact_expr": "G: Ellipse;O: Circle;Expression(O) = (-4*x + x^2 + y^2 - 12 = 0);PointOnCurve(Focus(G), xAxis);Eccentricity(G) = (sqrt(2)/2);Length(MajorAxis(G))=Radius(O)", "coords": {} }, { "index": 877, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0877.png", "answer": "1", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;F:Point;x1:Number;x2:Number;y1:Number;y2:Number;Expression(G) = (y^2 = 4*x);Coordinate(A) = (x1,y1);Coordinate(B) = (x2,y2);PointOnCurve(F,H);Intersection(H,G) = {A,B};Focus(G)=F", "coords": {} }, { "index": 878, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0878.png", "answer": "sqrt(5)", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;H:Line;k: Number;A: Point;B: Point;F: Point;a>0;b>0;Negation(k=0);Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = k*x);Intersection(H, C) = {A, B};IsDiameter(LineSegmentOf(A,B),G);RightFocus(C)=F;PointOnCurve(F,G);Area(TriangleOf(A, B, F)) = 4*a^2", "coords": {} }, { "index": 879, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0879.png", "answer": "4", "fact_expr": "G: Parabola;p: Number;H: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(H) = (-2, 3);Distance(H, Focus(G)) = 5", "coords": { "H": [ -2.0, 3.0 ] } }, { "index": 880, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0880.png", "answer": "sqrt(15)", "fact_expr": "C: Parabola;H:Point;l1:Line;l2:Line;M: Point;N: Point;Expression(C) = (y^2 = 4*x);Coordinate(H) = (0, 1);Expression(l1) = (x - 2*y - 3 = 0);PointOnCurve(H,l2);IsPerpendicular(l1,l2);Intersection(l2, C) = {M, N}", "coords": { "H": [ 0.0, 1.0 ] } }, { "index": 881, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0881.png", "answer": "2/9", "fact_expr": "P: Point;C: Hyperbola;Expression(C) = (x^2/16 - y^2/9 = 1);PointOnCurve(P, LeftPart(C)) = True;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 882, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0882.png", "answer": "(0,2-sqrt(3))+(2+sqrt(3),+oo)", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;F: Point;A: Point;B: Point;t: Real;c:Number;a>0;b>0;t>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(l)=(y=sqrt(2)*(x-c));Coordinate(F) = (c, 0);RightFocus(C) = F;Intersection(l,C)={A,B};VectorOf(A, F) = t*VectorOf(F, B)", "coords": {} }, { "index": 883, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0883.png", "answer": "y^2=4*x", "fact_expr": "C: Parabola;O: Origin;Vertex(C) = O;PointOnCurve(Focus(C), xAxis);G: Line;Expression(G) = (y = x);A: Point;B: Point;Intersection(G, C) = {A, B};P: Point;Coordinate(P) = (2, 2);MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ 2.0, 2.0 ] } }, { "index": 884, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0884.png", "answer": "2*sqrt(2)", "fact_expr": "l: Line;C:Curve;G: Circle;r: Number;A: Point;B: Point;F: Point;r>0;Expression(G) = (y^2 + (x - 5)^2 = r^2);Expression(C)=(y^2 = 2*(p*x));Focus(C) = F;PointOnCurve(F, l);Inclination(l) = ApplyUnit(45, degree);Intersection(l, C) = {A, B};TangentPoint(l,G)=MidPoint(LineSegmentOf(A,B));p: Number;p>0", "coords": {} }, { "index": 885, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 0.9980636721375151, "b": 0.9980636721375151, "orientation": "horizontal" }, "optimization": { "final_loss": 1.454111102587757e-05, "converged": true, "iterations": 69 }, "output_path": "results/train/hyperbola/problem_0885.png", "answer": "x^2 - y^2 = 1", "fact_expr": "C: Hyperbola;F1: Point;F2: Point;Coordinate(F1) = (-sqrt(2), 0);Coordinate(F2) = (sqrt(2), 0);Focus(C) = {F1, F2};I: Point;OneOf(Vertex(C)) = I;Coordinate(I) = (1, 0)", "coords": { "F1": [ -1.4142135623730951, 0.0 ], "F2": [ 1.4142135623730951, 0.0 ], "I": [ 1.0, 0.0 ] } }, { "index": 886, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 6.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0886.png", "answer": "y=pm*(3/4)*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/36 - y^2/9 = 1)", "coords": {} }, { "index": 887, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 888, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0888.png", "answer": "3*sqrt(5)", "fact_expr": "G: Parabola;p: Number;M: Point;F: Point;O:Origin;p>0;t:Number;Expression(G) = (y^2 = 2*(p*x));Coordinate(M) = (3, t);PointOnCurve(M, G);Focus(G) = F;Distance(M, F) = Abs(LineSegmentOf(M, F)) ;Abs(LineSegmentOf(M, F)) = p", "coords": {} }, { "index": 889, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 890, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0890.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/3 = 1)", "coords": {} }, { "index": 891, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0891.png", "answer": "5/2", "fact_expr": "P: Point;G: Parabola;Expression(G) = (y^2 = 2*x);PointOnCurve(P, G) = True;Q: Point;Coordinate(Q) = (2, 1)", "coords": { "Q": [ 2.0, 1.0 ] } }, { "index": 892, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0892.png", "answer": "sqrt(3)-1", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;c: Number;FocalLength(G) = 2*c;H: Line;Expression(H) = (y = -sqrt(3)*(-c + x));M: Point;OneOf(Intersection(H, G)) = M;AngleOf(M, F2, F1) = 2*AngleOf(M, F1, F2)", "coords": {} }, { "index": 893, "success": false, "conic_type": "parabola", "error": "validation: parabola_point_off_curve", "params": { "p": 10.4933443069458, "direction": "up" }, "optimization": { "final_loss": 80.80186462402344, "converged": true, "iterations": 470 }, "output_path": "results/train/parabola/problem_0893.png", "answer": "x^2 = -y/4", "fact_expr": "G: Parabola;O: Origin;Vertex(G) = O;PointOnCurve(Focus(G),yAxis) = True;H: Point;Coordinate(H) = (1, -4);PointOnCurve(H,G) = True", "coords": { "H": [ 1.0, -4.0 ] }, "validation_reasons": [ "parabola_point_off_curve" ] }, { "index": 894, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0894.png", "answer": "4", "fact_expr": "G: Parabola;P: Point;A: Point;F: Point;Expression(G) = (x^2 = 4*y);Coordinate(A) = (2, 3);PointOnCurve(P, G);Focus(G) = F", "coords": { "A": [ 2.0, 3.0 ] } }, { "index": 895, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 896, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0896.png", "answer": "{2,8}", "fact_expr": "G: Ellipse;m: Real;Expression(G) = (x^2/4 + y^2/m = 1);Eccentricity(G) = sqrt(2)/2", "coords": {} }, { "index": 897, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0897.png", "answer": "7/2", "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (y^2 = 2*x);Coordinate(A) = (3, 2);Focus(G) = F;PointOnCurve(P, G)", "coords": { "A": [ 3.0, 2.0 ] } }, { "index": 898, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0898.png", "answer": "(\\sqrt{2},+\\infty)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F;PointOnCurve(F, H);Inclination(H) = ApplyUnit(45, degree);NumIntersection(H, RightPart(G)) = 1", "coords": {} }, { "index": 899, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 900, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.75, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0900.png", "answer": "(9/4, pm*(3*sqrt(3)/2))", "fact_expr": "G: Parabola;B: Point;A: Point;F: Point;Expression(G) = (y^2 = 3*x);PointOnCurve(A, G);Focus(G) = F;H:Line;TangentOnPoint(A,G)=H;Intersection(H,xAxis)=B;AngleOf(B,A,F)=ApplyUnit(30,degree)", "coords": {} }, { "index": 901, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0901.png", "answer": "5", "fact_expr": "H: Line;Slope(H) = 2;G: Parabola;Expression(G) = (y^2 = 4*x);PointOnCurve(Focus(G), H);A: Point;B: Point;Intersection(H, G) = {A, B}", "coords": {} }, { "index": 902, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0902.png", "answer": "y^2=2*x", "fact_expr": "G: Parabola;p: Number;p>0;Expression(G) = (y^2 = 2*p*x);Distance(Focus(G),Directrix(G))=1", "coords": {} }, { "index": 903, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0903.png", "answer": "(1/2,1)", "fact_expr": "G: Ellipse;a: Number;b: Number;M: Point;F2: Point;P: Point;F1: Point;O: Origin;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);RightFocus(G)=F2;LeftFocus(G)=F1;PointOnCurve(P,G);IsPerpendicular(LineOf(F2,M),LineOf(O,P));Intersection(LineOf(F2,M),LineSegmentOf(F1,P))=M;Abs(LineSegmentOf(F1, M)) = 2*Abs(LineSegmentOf(M, P))", "coords": {} }, { "index": 904, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0904.png", "answer": "2*sqrt(2)/3", "fact_expr": "G: Parabola;p: Number;H: Line;M: Point;N: Point;F: Point;l: Line;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;Directrix(G) = l;PointOnCurve(F, H);Intersection(H, G) = M;Intersection(H, l) = N;VectorOf(M, F) = VectorOf(M, N)/3", "coords": {} }, { "index": 905, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0905.png", "answer": "x^2/3 - y^2/6 = 1", "fact_expr": "G: Hyperbola;Center(G) = O;O: Origin;Eccentricity(G) = sqrt(3);H: Parabola;Expression(H) = (y^2 = 4*x);OneOf(Directrix(G)) = Directrix(H)", "coords": {} }, { "index": 906, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0906.png", "answer": "sqrt(13)", "fact_expr": "l: Line;E: Ellipse;A: Point;F1: Point;B: Point;F2: Point;O:Origin;Expression(E) = (x^2/4 + y^2/3 = 1);LeftFocus(E) = F1;RightFocus(E) = F2;PointOnCurve(O,l);Intersection(l, E) = {A, B};IsPerpendicular(LineSegmentOf(A,F1),xAxis);IsPerpendicular(LineSegmentOf(B,F2),xAxis)", "coords": {} }, { "index": 907, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 908, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0908.png", "answer": "8", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;x1:Number;x2:Number;y1:Number;y2:Number;Expression(G)=(y^2 = 4*x);Coordinate(A)=(x1,y1);Coordinate(B)=(x2,y2);PointOnCurve(Focus(G),H);Intersection(H, G) ={A,B};x1+x2=6", "coords": {} }, { "index": 909, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0909.png", "answer": "24", "fact_expr": "C: Parabola;Expression(C) = (x^2 = 4*y);F: Point;Focus(C) = F;l1: Line;l2: Line;k1: Number;k2: Number;Slope(l1) = k1;Slope(l2) = k2;PointOnCurve(F, l1);PointOnCurve(F, l2);A: Point;B: Point;Intersection(l1, C) = {A, B};D: Point;E: Point;Intersection(l2, C) = {D, E};k1*k2 = -2", "coords": {} }, { "index": 910, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0910.png", "answer": "sqrt(2)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;c: Number;FocalLength(C) = 2*c;A: Point;RightVertex(C) = A;M: Point;N: Point;PointOnCurve(M,l) = True;PointOnCurve(N,l) = True;Abs(LineSegmentOf(A, M)) = Abs(LineSegmentOf(A, N));AngleOf(M, A, N) = ApplyUnit(120, degree);Abs(LineSegmentOf(A, N)) = c;l: Line;OneOf(Asymptote(C)) = l", "coords": {} }, { "index": 911, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0911.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x);P: Point;PointOnCurve(P, G);H: Line;Expression(H) = (y = x + 2)", "coords": {} }, { "index": 912, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 913, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0913.png", "answer": "sqrt(5)+2*sqrt(2)", "fact_expr": "G: Ellipse;A: Point;P: Point;F: Point;Expression(G) = (x^2/2 + y^2 = 1);Coordinate(A) = (0, 1/2);RightFocus(G) = F;PointOnCurve(P, G)", "coords": { "A": [ 0.0, 0.5 ] } }, { "index": 914, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0914.png", "answer": "2", "fact_expr": "C: Parabola;G: Circle;H: Line;A: Point;B: Point;P: Point;k:Number;k>0;Expression(C) = (y^2 = 8*x);Coordinate(P) = (1, 0);PointOnCurve(P, H);Slope(H) = k;Intersection(H, C) = {A, B};IsDiameter(LineSegmentOf(A,B),G);Length(InterceptChord(xAxis,G))=Length(InterceptChord(yAxis,G))", "coords": { "P": [ 1.0, 0.0 ] } }, { "index": 915, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0915.png", "answer": "20", "fact_expr": "G: Ellipse;H: Line;P: Point;Q: Point;F2: Point;F1: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(G) ={F1,F2};PointOnCurve(F1, H);Intersection(H, G) = {P, Q}", "coords": {} }, { "index": 916, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0916.png", "answer": "(2-sqrt(2))/2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);l: Line;Slope(l)<0;A: Point;Coordinate(A) = (1, 2);B: Point;Intersection(l, G) = {A, B};Q: Point;MidPoint(LineSegmentOf(A, B)) = Q;l1: Line;PointOnCurve(Q, l1);IsPerpendicular(l1, yAxis);C: Point;Intersection(l1, G) = C;P: Point;2*VectorOf(Q, C) = VectorOf(C, P);O: Origin", "coords": { "A": [ 1.0, 2.0 ] } }, { "index": 917, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0917.png", "answer": "3", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Focus(C) = {F1, F2};PointOnCurve(P, C);IsPerpendicular(VectorOf(P, F1), VectorOf(P, F2));Area(TriangleOf(P, F1, F2)) = 9", "coords": {} }, { "index": 918, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0918.png", "answer": "5", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);P: Point;PointOnCurve(P, G);XCoordinate(P) = 3", "coords": {} }, { "index": 919, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0919.png", "answer": "1/3 + 2*sqrt(21)/3", "fact_expr": "G: Ellipse;C: Circle;P: Point;Q: Point;Expression(G) = (x^2 + 4*y^2 = 4);Expression(C) = (x^2 + (y - 2)^2 = 1/9);PointOnCurve(P, C);PointOnCurve(Q, G)", "coords": {} }, { "index": 920, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_0920.png", "answer": "-12", "fact_expr": "C: Hyperbola;a: Number;G: Parabola;m: Real;a>0;Expression(C) = (-y^2/4 + x^2/a^2 = 1);Expression(G) = (y^2 = m*x);Eccentricity(C)=2*sqrt(5)/5;LeftFocus(C)=Focus(G)", "coords": {} }, { "index": 921, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 922, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0922.png", "answer": "2*sqrt(2)", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);P: Point;PointOnCurve(P, G);Distance(P, Focus(G)) = 3", "coords": {} }, { "index": 923, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0923.png", "answer": "x-4*y+3=0", "fact_expr": "G: Parabola;L: Line;A: Point;B: Point;M: Point;Expression(G) = (x^2 = 8*y);Coordinate(M) = (1, 1);Intersection(L,G)={A,B};MidPoint(LineSegmentOf(A,B))=M", "coords": { "M": [ 1.0, 1.0 ] } }, { "index": 924, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0924.png", "answer": "12", "fact_expr": "C: Parabola;Expression(C) = (x^2 = 4*y);F: Point;Focus(C) = F;PointOnCurve(F, G);G: Line;Intersection(G, C) = {A, B};A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), C) ;Distance(MidPoint(LineSegmentOf(A, B)), xAxis) = 5", "coords": {} }, { "index": 925, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 926, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.4149149626832767, "b": 2.450642358201144, "orientation": "horizontal" }, "optimization": { "final_loss": 1.0319236343469396e-05, "converged": true, "iterations": 94 }, "output_path": "results/train/hyperbola/problem_0926.png", "answer": "x^2 - y^2 / 3 = 1", "fact_expr": "G: Hyperbola;H: Point;Coordinate(H) = (2, 3);PointOnCurve(H, G);Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x)", "coords": { "H": [ 2.0, 3.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 927, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0927.png", "answer": "2", "fact_expr": "G: Hyperbola;F1: Point;F2: Point;P: Point;Expression(G) = (x^2/4 - y^2 = 1);Focus(G)={F1,F2};PointOnCurve(P, G);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0", "coords": {} }, { "index": 928, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 929, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0929.png", "answer": "x^2 - y^2/3 = 1", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;l: Line;OneOf(Asymptote(C)) = l;Inclination(l) = pi/3;Distance(OneOf(Focus(C)), l) = sqrt(3)", "coords": {} }, { "index": 930, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0930.png", "answer": "(1+sqrt(5))/2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;A: Point;B: Point;F1: Point;F2: Point;a>0;b>0;Focus(G) = {F1, F2};L:Line;IsPerpendicular(L,xAxis);Intersection(L,G)={A,B};PointOnCurve(F2, L);Radius(InscribedCircle(TriangleOf(A,B,F1)))=a", "coords": {} }, { "index": 931, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0931.png", "answer": "12", "fact_expr": "G:Parabola;Expression(G) = (y^2 = 4*x);F:Point;Focus(G)=F;l:Line;A: Point;B: Point;Intersection(l,G)={A,B};M: Point;PointOnCurve(F, l);MidPoint(LineSegmentOf(A,B))=M;Distance(M,Directrix(G))=6", "coords": {} }, { "index": 932, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 9.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0932.png", "answer": "2\n(2/3)*pi", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 + y^2/2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 4", "coords": {} }, { "index": 933, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0933.png", "answer": "{sqrt(6)/2,sqrt(6)/3}", "fact_expr": "G: Ellipse;Expression(G) = (y^2 + x^2/a^2 = 1);a: Real;a>0;Eccentricity(G) = sqrt(3)/3", "coords": {} }, { "index": 934, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0934.png", "answer": "y=pm*sqrt(3)*(x-1/2)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*x);F: Point;Focus(C)= F;l: Line;PointOnCurve(F, l) = True;Intersection(l, C) = {A, B};A: Point;B: Point;Abs(LineSegmentOf(A, F)) = 3*Abs(LineSegmentOf(B, F))", "coords": {} }, { "index": 935, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0935.png", "answer": "32", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);H: Line;P: Point;Coordinate(P) = (4, 0);PointOnCurve(P, H);A: Point;x1: Number;y1: Number;Coordinate(A) = (x1, y1);B: Point;x2: Number;y2: Number;Coordinate(B) = (x2, y2);Intersection(H, G) = {A, B}", "coords": { "P": [ 4.0, 0.0 ] } }, { "index": 936, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0936.png", "answer": "8", "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/25 + y^2/9 = 1);PointOnCurve(P, G);F1:Point;F2:Point;OneOf(Focus(G))=F1;OneOf(Focus(G))=F2;Negation(F1=F2);Distance(P, F1) = 2", "coords": {} }, { "index": 937, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0937.png", "answer": "(4, 2)", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (x - y = 2);Intersection(H, G) = {A, B}", "coords": {} }, { "index": 938, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 939, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 940, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0940.png", "answer": "2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;H: Line;Inclination(H) = pi/4;PointOnCurve(F, H);A: Point;B: Point;Intersection(H, G) = {A, B};M: Point;Intersection(PerpendicularBisector(LineSegmentOf(A, B)), xAxis) = M", "coords": {} }, { "index": 941, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0941.png", "answer": "(0,sqrt(6)/3]", "fact_expr": "C: Ellipse;a: Number;b: Number;A: Point;P: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (0, 2*b);PointOnCurve(P,C);Abs(LineSegmentOf(P,A))<=3*b", "coords": {} }, { "index": 942, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0942.png", "answer": "8/5", "fact_expr": "l: Line;G: Ellipse;A: Point;B: Point;Expression(G) = (x^2/4 + y^2 = 1);PointOnCurve(RightFocus(G), l);Slope(l)=1;Intersection(l, G) = {A, B};IsChordOf(LineSegmentOf(A,B),G)", "coords": {} }, { "index": 943, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0943.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x + 2*y - 1 = 0);IsPerpendicular(OneOf(Asymptote(G)), H)", "coords": {} }, { "index": 944, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0944.png", "answer": "4", "fact_expr": "G: Parabola;M: Point;Expression(G) = (y^2 = 8*x);Coordinate(M) = (x0, y0);PointOnCurve(M, G);Distance(M, Focus(G)) = 6;x0:Number;y0:Number", "coords": {} }, { "index": 945, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.0625, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0945.png", "answer": "1/8", "fact_expr": "G: Parabola;F: Point;l: Line;Expression(G) = (y = 4*x^2);Focus(G) = F;Directrix(G) = l", "coords": {} }, { "index": 946, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.4641016151377544, "major_axis": "x", "x_coef": 16.0, "y_coef": 12.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0946.png", "answer": "16", "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/12 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G);PointOnCurve(F1, LineSegmentOf(A, B))", "coords": {} }, { "index": 947, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 948, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 949, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0949.png", "answer": "x^2 - y^2/6 = 1", "fact_expr": "G: Hyperbola;b: Number;a: Number;F1: Point;M: Point;F2: Point;O: Origin;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Focus(G)={F1,F2};PointOnCurve(M, G);AngleOf(F1, M, F2) = ApplyUnit(120, degree);Abs(LineSegmentOf(O,M))=sqrt(3)*a;Area(TriangleOf(F1,M,F2))=2*sqrt(3)", "coords": {} }, { "index": 950, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0950.png", "answer": "1", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G)", "coords": {} }, { "index": 951, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0951.png", "answer": "9/4", "fact_expr": "O: Origin;F: Point;Focus(C) = F;C: Parabola;Expression(C) = (y^2 = 2*x);l: Line;Expression(l) = (y = m*(2*x - 1));A: Point;B: Point;Intersection(l, C) = {A, B};Quadrant(A) = 1;Abs(LineSegmentOf(A, F)) = 2*Abs(LineSegmentOf(B, F));m: Number", "coords": {} }, { "index": 952, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0952.png", "answer": "2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;H: Line;PointOnCurve(Focus(G), H);IsPerpendicular(H, SymmetryAxis(G));A: Point;B: Point;Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 4", "coords": {} }, { "index": 953, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0953.png", "answer": "(9/4,3)", "fact_expr": "G: Parabola;P: Point;A: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (4, 3);Focus(G) = F;PointOnCurve(P, G);WhenMin(Abs(LineSegmentOf(P,A))+Abs(LineSegmentOf(P,F)))", "coords": { "A": [ 4.0, 3.0 ] } }, { "index": 954, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0954.png", "answer": "{sqrt(2), sqrt(10)/3}", "fact_expr": "G: Line;Expression(G) = (y = a);C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;P: Point;Intersection(G, OneOf(Asymptote(C))) = P;A2: Point;A1: Point;LeftVertex(C) = A1;RightVertex(C) = A2;Abs(LineSegmentOf(P, A2)) = (sqrt(5)/2)*Abs(LineSegmentOf(A1, A2))", "coords": {} }, { "index": 955, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 956, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0956.png", "answer": "[\\sqrt{2}/2, 1)", "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;P: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(90, degree)", "coords": {} }, { "index": 957, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 958, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0958.png", "answer": "9", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);F: Point;Focus(G) = F;A: Point;Coordinate(A) = (-1, 8);P: Point;PointOnCurve(P, G) = True", "coords": { "A": [ -1.0, 8.0 ] } }, { "index": 959, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 960, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0960.png", "answer": "pi/3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;M: Point;Intersection(Directrix(G), xAxis) = M;N: Point;PointOnCurve(N, G);Abs(LineSegmentOf(M, N)) = 2*Abs(LineSegmentOf(N, F))", "coords": {} }, { "index": 961, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0961.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;b: Number;a: Number;F: Point;H: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;PointOnCurve(F,H);Slope(H) = 1;NumIntersection(H,G) = 1", "coords": {} }, { "index": 962, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0962.png", "answer": "-sqrt(15)/sqrt(15)", "fact_expr": "Slope(l) = k;k: Number;k<0;l: Line;F: Point;Coordinate(F) = (0, 1);PointOnCurve(F, l) ;H: Curve;Expression(H) = (y = x^2/4)&(x>=0);G: Line;Expression(G) = (y = -1);Intersection(l, H) = A;Intersection(l, G) = B;B: Point;A: Point;Abs(LineSegmentOf(F, B)) = 5*Abs(LineSegmentOf(F, A))", "coords": { "F": [ 0.0, 1.0 ] } }, { "index": 963, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0963.png", "answer": "2*sqrt(6)", "fact_expr": "G: Parabola;H: Line;A: Point;M: Point;N: Point;Expression(G) = (y^2 = 2*x);Coordinate(A) = (1, 0);PointOnCurve(A, H);Inclination(H)=pi/4;Intersection(H,G)={M,N}", "coords": { "A": [ 1.0, 0.0 ] } }, { "index": 964, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0964.png", "answer": "2", "fact_expr": "H: Parabola;Expression(H) = (y^2 = -8*x);G: Hyperbola;Expression(G) = (-y^2/3 + x^2/m = 1);m: Number;PointOnCurve(RightFocus(G), Directrix(H))", "coords": {} }, { "index": 965, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 966, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0966.png", "answer": "16", "fact_expr": "C: Parabola;l1:Line;l2:Line;A: Point;B: Point;D: Point;E: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F,l1);PointOnCurve(F,l2);IsPerpendicular(l1,l2);Intersection(l1,C)={A,B};Intersection(l2,C)={D,E}", "coords": {} }, { "index": 967, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 968, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0968.png", "answer": "2*x+pm*3*y=0", "fact_expr": "G: Hyperbola;a: Number;Expression(G) = (x^2/9 - y^2/a = 1);Coordinate(RightFocus(G)) = (sqrt(13), 0)", "coords": {} }, { "index": 969, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0969.png", "answer": "7", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;l: Line;PointOnCurve(F, l) = True;Intersection(l, G) = {A, B};A: Point;B: Point;XCoordinate(A) + XCoordinate(B) = 5", "coords": {} }, { "index": 970, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0970.png", "answer": "10", "fact_expr": "C: Parabola;p: Number;P: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(P) = (3, y0);y0: Number;PointOnCurve(P, C);Distance(P, Directrix(C)) = 8", "coords": {} }, { "index": 971, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0971.png", "answer": "sqrt(3)/3", "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;P: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;L:Line;PointOnCurve(F1, L);IsPerpendicular(L,xAxis);Intersection(L, G) = P;AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 972, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 973, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0973.png", "answer": "sqrt(5)", "fact_expr": "G: Parabola;H: Circle;M: Point;l:Line;Expression(G) = (y^2 = 4*x);PointOnCurve(M, G);Center(H)=M;IsTangent(xAxis,H);Directrix(G)=l;IsTangent(l,H);O:Point;Vertex(G)=O", "coords": {} }, { "index": 974, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0974.png", "answer": "-3", "fact_expr": "l: Line;M: Point;Coordinate(M) = (0, 3);PointOnCurve(M, l);G: Parabola;Expression(G) = (y = x^2);E: Point;F: Point;Intersection(l, G) = {E, F};Negation(IsPerpendicular(l, yAxis));P: Point;t: Real;Coordinate(P) = (0, t);PointOnCurve(P, yAxis);PointOnCurve(Incenter(TriangleOf(P, E, F)), yAxis)", "coords": { "M": [ 0.0, 3.0 ] } }, { "index": 975, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0975.png", "answer": "6", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;P: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};MidPoint(LineSegmentOf(A,B))=M;L:Line;PointOnCurve(M,L);IsPerpendicular(L,Directrix(G));Intersection(L,G)=P;IsChordOf(LineSegmentOf(A,B),G);Abs(LineSegmentOf(P, F)) = 3/2;M:Point", "coords": {} }, { "index": 976, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 977, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0977.png", "answer": "-2/9", "fact_expr": "A: Point;B: Point;G: Ellipse;Expression(G) = (x^2/9+y^2/4 = 1);IsChordOf(LineSegmentOf(A, B), G);l: Line;Expression(l) = (x+2*y-3 = 0);IsPerpendicular(LineSegmentOf(A, B), l);P: Point;MidPoint(LineSegmentOf(A, B)) = P;O: Point;Center(G) = O", "coords": {} }, { "index": 978, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0978.png", "answer": "sqrt(2) +1", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, H) = True;IsPerpendicular(H, xAxis) = True;H: Line;Intersection(H, G) = {A, B};B: Point;A: Point;DotProduct(VectorOf(F1, A), VectorOf(F1, B)) = 0", "coords": {} }, { "index": 979, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 980, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_0980.png", "answer": "(x - \\sqrt{3}/2)^2 + (y - 1/2)^2 = 1", "fact_expr": "O: Circle;G: Line;M: Point;N: Point;P: Point;Expression(O) = (x^2 + y^2 = 16);Coordinate(P) = (sqrt(3), 1);PointOnCurve(P, G);Intersection(G, O) = {M, N};IsChordOf(LineSegmentOf(M, N), O);Q: Point;MidPoint(LineSegmentOf(M, N)) = Q", "coords": {} }, { "index": 981, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.0654510707333538, "b": 2.6632060182000887, "orientation": "horizontal" }, "optimization": { "final_loss": 3.6995812463854516e-05, "converged": true, "iterations": 75 }, "output_path": "results/train/hyperbola/problem_0981.png", "answer": "sqrt(29)/5", "fact_expr": "C: Hyperbola;G: Point;Coordinate(G) = (3, 8);Expression(Asymptote(C)) = (y = pm*(5/2)*x);PointOnCurve(G, C)", "coords": { "G": [ 3.0, 8.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 982, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0982.png", "answer": "9", "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;F: Point;Expression(C) = (y^2 = 8*x);Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};Distance(A, Directrix(C)) = 6", "coords": {} }, { "index": 983, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0983.png", "answer": "2", "fact_expr": "G: Parabola;M: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(M, G);Distance(M, Focus(G)) = 3", "coords": {} }, { "index": 984, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 985, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0985.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/b^2 - y^2/a^2 = 1);b: Number;a: Number;l1: Line;l2: Line;Asymptote(G) = {l1, l2};IsPerpendicular(l1, l2)", "coords": {} }, { "index": 986, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0986.png", "answer": "sqrt(5)/2", "fact_expr": "l: Line;C: ConicSection;a: Number;P: Point;Expression(C) = (y^2/16 + x^2/a = 1);Coordinate(P) = (-1, 6);Expression(l)=(y=a*x+2);PointOnCurve(P,l)", "coords": { "P": [ -1.0, 6.0 ] } }, { "index": 987, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0987.png", "answer": "8", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 + y^2/4 = 1);PointOnCurve(P,G);Focus(G)={F1, F2};DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0", "coords": {} }, { "index": 988, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0988.png", "answer": "sqrt(30)/3", "fact_expr": "G: Ellipse;Expression(G) = (x^2/2 + y^2 = 1);A: Point;B: Point;LeftVertex(G) = A;RightVertex(G) = B;P: Point;PointOnCurve(P, G);Quadrant(P) = 1;lambda: Number;Abs(LineSegmentOf(P, A)) = lambda*Abs(LineSegmentOf(P, B));AngleOf(P, B, A) = 2*AngleOf(P, A, B)", "coords": {} }, { "index": 989, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "major_axis": "x", "x_coef": 3.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0989.png", "answer": "(0,pm*1)", "fact_expr": "G: Ellipse;F1: Point;A: Point;F2: Point;B: Point;Expression(G) = (x^2/3 + y^2 = 1);Focus(G) = {F1, F2};PointOnCurve(A, G);PointOnCurve(B, G);VectorOf(F1, A) = 5*VectorOf(F2, B)", "coords": {} }, { "index": 990, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0990.png", "answer": "sqrt(2)/2", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;p>0;F: Point;Focus(C) = F;l: Line;Directrix(C) = l;A: Point;PointOnCurve(A, C) = True;Center(G) = F;Radius(G) = LineSegmentOf(F, A);G: Circle;Intersection(G, l) = {B, D};B: Point;D: Point;AngleOf(B, F, D) = ApplyUnit(120, degree);Area(TriangleOf(A, B, D)) = sqrt(3)", "coords": {} }, { "index": 991, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_0991.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;M: Point;m: Number;Coordinate(M) = (4, m);PointOnCurve(M, G);Distance(M, Directrix(G)) = 6", "coords": {} }, { "index": 992, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0992.png", "answer": "4*sqrt(7)", "fact_expr": "C: Hyperbola;M: Point;N: Point;O: Origin;Expression(C) = (x^2/9-y^2/4 = 1);Expression(LineOf(M,N))=(y=x/3+2);Intersection(LineOf(M,N),C)={M,N}", "coords": {} }, { "index": 993, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0993.png", "answer": "3/2", "fact_expr": "A: Point;F: Point;Focus(Gamma) = F;Gamma: Parabola;Expression(Gamma) = (y^2 = 4*x);PointOnCurve(A, Gamma) = True;Negation(A = O);O: Origin;Intersection(PerpendicularBisector(LineSegmentOf(O, A)), xAxis) = B;B: Point;VectorOf(A, F) = 2*VectorOf(C, F)", "coords": {} }, { "index": 994, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0994.png", "answer": "(1, -4)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 16*x);F: Point;Focus(C) = F;P: Point;PointOnCurve(P, C);B: Point;Coordinate(B) = (-4, 6);WhenMax(Abs(LineSegmentOf(P, B))/Abs(LineSegmentOf(P, F)))", "coords": { "B": [ -4.0, 6.0 ] } }, { "index": 995, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_0995.png", "answer": "8*sqrt(3)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);M: Point;PointOnCurve(M, G);F: Point;Focus(G) = F;AngleOf(M, F, O) = ApplyUnit(120, degree);N: Point;Coordinate(N) = (-2, 0);O: Origin", "coords": { "N": [ -2.0, 0.0 ] } }, { "index": 996, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 8.0, "b": 6.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_0996.png", "answer": "96/5", "fact_expr": "G: Hyperbola;P: Point;Expression(G) = (x^2/64 - y^2/36 = 1);PointOnCurve(P, G);Distance(P,LeftFocus(G)) = 8", "coords": {} }, { "index": 997, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0997.png", "answer": "16", "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/25 + y^2/16 = 1);Focus(C)={F1,F2};PointOnCurve(P,C)", "coords": {} }, { "index": 998, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 2.0, "major_axis": "x", "x_coef": 8.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_0998.png", "answer": "(pm*sqrt(6),-1)", "fact_expr": "G: Ellipse;Q: Point;P: Point;Expression(G) = (x^2/8 + y^2/4 = 1);Coordinate(Q) = (0, 1);PointOnCurve(P, G);WhenMax(Distance(P,Q))", "coords": { "Q": [ 0.0, 1.0 ] } }, { "index": 999, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1000, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1001, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1001.png", "answer": "4", "fact_expr": "l: Line;C: Parabola;Expression(C) = (y^2 = 4*x);A: Point;O: Origin;B: Point;PointOnCurve(F, l);Intersection(l, C) = {A, B};P: Point;Q: Point;F: Point;Focus(C) = F;Intersection(LineOf(O,A),Directrix(C))=P;Intersection(LineOf(O,B),Directrix(C))=Q", "coords": {} }, { "index": 1002, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1003, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1003.png", "answer": "2*sqrt(3)/3", "fact_expr": "l: Line;C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;OneOf(Focus(C)) = F;PointOnCurve(F, l) = True;IsPerpendicular(l, RealAxis(C)) = True;A: Point;B:Point;Vertex(C) = {A, B};P: Point;PointOnCurve(P, l) = True;AngleOf(A, P, B) = ApplyUnit(60, degree)", "coords": {} }, { "index": 1004, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1004.png", "answer": "16*sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1))*Abs(LineSegmentOf(P, F2)) = 64", "coords": {} }, { "index": 1005, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1005.png", "answer": "(1/4,1)", "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (1, 1);Focus(G) = F;PointOnCurve(P, G);WhenMin(Abs(LineSegmentOf(P, A)) + Abs(LineSegmentOf(P, F)))", "coords": { "A": [ 1.0, 1.0 ] } }, { "index": 1006, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1006.png", "answer": "(8*sqrt(34))/17", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;M: Point;m: Number;Coordinate(M) = (1, m);PointOnCurve(M, G);F: Point;Focus(G) = F;Distance(M, F) = 5;d: Number;Distance(Vertex(G), LineOf(M, F)) = d", "coords": {} }, { "index": 1007, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1007.png", "answer": "sqrt(41)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l: Line;Directrix(C) = l;M: Point;Intersection(l, xAxis) = M;P: Point;PointOnCurve(P, C);Abs(LineSegmentOf(P, F)) = 5", "coords": {} }, { "index": 1008, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1008.png", "answer": "4*a", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};H: Line;PointOnCurve(F1, H);P: Point;Q: Point;Intersection(H, G) = {P, Q}", "coords": {} }, { "index": 1009, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1009.png", "answer": "5*sqrt(2)/3", "fact_expr": "G: Ellipse;H: Line;Expression(G) = (x^2 + y^2/2 = 1);Expression(H) = (y = 2*x - 1)", "coords": {} }, { "index": 1010, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.4641016151377544, "major_axis": "x", "x_coef": 16.0, "y_coef": 12.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1010.png", "answer": "12/5", "fact_expr": "G: Ellipse;F2: Point;M: Point;F1: Point;Expression(G) = (x^2/16 + y^2/12 = 1);Focus(G)={F1,F2};PointOnCurve(M, G);IsPerpendicular(LineSegmentOf(M,F1),xAxis)", "coords": {} }, { "index": 1011, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1012, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1013, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1014, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1014.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;H: Line;Expression(G) = (x^2/4 - y^2/5 = 1);Expression(H) = (x + 2*y - 8 = 0)", "coords": {} }, { "index": 1015, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1015.png", "answer": "[-1,4]", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/4 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G)", "coords": {} }, { "index": 1016, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1016.png", "answer": "(x - 5)^2 + y^2 = 9", "fact_expr": "C: Hyperbola;E: Circle;Expression(C) = (x^2/16 - y^2/9 = 1);Center(E) = RightFocus(C);IsTangent(Asymptote(C), E)", "coords": {} }, { "index": 1017, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1018, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1018.png", "answer": "pm*sqrt(2)", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (y^2 + x^2/m^2 = 1);FocalLength(G) = 2", "coords": {} }, { "index": 1019, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1019.png", "answer": "x=1", "fact_expr": "G: Parabola;Expression(G) = (y^2 = -4*x)", "coords": {} }, { "index": 1020, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25031906366348267, "direction": "up" }, "optimization": { "final_loss": 9.454704922973178e-06, "converged": true, "iterations": 108 }, "output_path": "results/train/parabola/problem_1020.png", "answer": "x^2 = y", "fact_expr": "G: Parabola;H: Point;O: Origin;Coordinate(H) = (2, 4);PointOnCurve(H, G);Vertex(G) = O;PointOnCurve(Focus(G), yAxis)", "coords": { "H": [ 2.0, 4.0 ] } }, { "index": 1021, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1021.png", "answer": "sqrt(5)", "fact_expr": "l: Line;Expression(l) = (y=2*(x-sqrt(5)));C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;RightFocus(C) = F;F: Point;PointOnCurve(F,l) = True;NumIntersection(l,C) = 1", "coords": {} }, { "index": 1022, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_1022.png", "answer": "x^2 - y^2 / 3 = 1", "fact_expr": "C1: Hyperbola;C2: Parabola;M: Point;F: Point;Expression(C1)=(x^2/a^2 - y^2/b^2 = 1);Eccentricity(C1)=2;RightFocus(C1)=F;Focus(C2)=F;Center(C1)=Vertex(C2);Intersection(C1,C2)=M;Abs(LineSegmentOf(M, F)) = 5;a:Number;b:Number;a>0;b>0", "coords": {} }, { "index": 1023, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1024, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1025, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1025.png", "answer": "3", "fact_expr": "G: Hyperbola;P: Point;F2: Point;F1: Point;e:Number;Expression(G) = (x^2 - y^2/3 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;Eccentricity(G)=e;PointOnCurve(P,G);AngleOf(P,F2,F1)=ApplyUnit(60,degree)", "coords": {} }, { "index": 1026, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1027, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1027.png", "answer": "x^2/4+y^2/3=1", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Length(MajorAxis(C)) = 4;FocalLength(C) = 2", "coords": {} }, { "index": 1028, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1029, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1029.png", "answer": "7", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 - y^2/12 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 3", "coords": {} }, { "index": 1030, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.23606797749979, "b": 2.0, "major_axis": "x", "x_coef": 5.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1030.png", "answer": "5/3", "fact_expr": "G: Ellipse;H: Line;O: Origin;A: Point;B: Point;Expression(G) = (x^2/5 + y^2/4 = 1);PointOnCurve(RightFocus(G), H);Slope(H) = 2;Intersection(H, G) = {A, B}", "coords": {} }, { "index": 1031, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1031.png", "answer": "(\\sqrt{7}-2,1)", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;D: Point;XCoordinate(D) = a/3;PointOnCurve(D, G) = True;Distance(D, LeftFocus(G)) > Distance(D, RightDirectrix(G))", "coords": {} }, { "index": 1032, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1032.png", "answer": "sqrt(30)/3", "fact_expr": "G: Ellipse;Expression(G) = (x^2 + 2*y^2 = 4);H:LineSegment;IsChordOf(H,G);Coordinate(MidPoint(H))=(1,1)", "coords": {} }, { "index": 1033, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1034, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0707900524139404, "b": 1.5409371596115455, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 3.1717604031648706e-06, "converged": true, "iterations": 78 }, "output_path": "results/train/ellipse/problem_1034.png", "answer": "pm*sqrt(3)", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Eccentricity(G) = 2/3;F: Point;RightFocus(G) = F;l: Line;PointOnCurve(F,l) = True;Slope(l) = k;k: Number;Intersection(l,G) = {A,B};A: Point;B: Point;VectorOf(A, F) = 2*VectorOf(F, B)", "coords": {} }, { "index": 1035, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 3.0, 4.0 ], "radius": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_1035.png", "answer": "6*x-8*y-21=0", "fact_expr": "M: Circle;P: Point;O: Origin;T: Point;L:Line;Expression(M)=((x - 3)^2 + (y - 4)^2 = 4);PointOnCurve(P, L);TangentOfPoint(P, M)=L;TangentPoint(L,M)=T;LineSegmentOf(P, O) = LineSegmentOf(P, T)", "coords": {} }, { "index": 1036, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1036.png", "answer": "3*sqrt(5)/5", "fact_expr": "G: Hyperbola;b: Number;a: Number;C: Circle;a>0;b>0;l1: Line;l2: Line;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(C) = (-6*x + x^2 + y^2 + 5 = 0);Asymptote(G) = {l1, l2};IsTangent(l1, C);IsTangent(l2, C)", "coords": {} }, { "index": 1037, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 5.196152422706632, "major_axis": "y", "x_coef": 27.0, "y_coef": 36.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1037.png", "answer": "y^2/4 - x^2/5 = 1", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/27 + y^2/36 = 1);Focus(G) = Focus(H);IsIntersect(G,H) = True;Coordinate(OneOf(Intersection(G,H))) = (sqrt(15),4)", "coords": {} }, { "index": 1038, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.681220143106833, "b": 2.2412689930799665, "orientation": "horizontal" }, "optimization": { "final_loss": 4.985250663519028e-06, "converged": true, "iterations": 95 }, "output_path": "results/train/hyperbola/problem_1038.png", "answer": "{x^2/36 - y^2/64 = 1, y^2/64 - x^2/36 = 1}", "fact_expr": "G: Hyperbola;Expression(Asymptote(G)) = (y = pm*(4/3)*x);FocalLength(G) = 20", "coords": {} }, { "index": 1039, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1039.png", "answer": "2", "fact_expr": "C: Hyperbola;b: Number;a: Number;G:Line;O: Origin;D: Point;E: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x = a);l1:Line;l2:Line;Asymptote(C)={l1,l2};Intersection(l1,G)=D;Intersection(l2,G)=E;FocalLength(C) = 4", "coords": {} }, { "index": 1040, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1040.png", "answer": "y=pm*sqrt(2)*x", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;Z: Line;PointOnCurve(F2, Z);IsPerpendicular(Z, xAxis);P: Point;OneOf(Intersection(Z, G)) = P;AngleOf(P, F1, F2) = pi/6", "coords": {} }, { "index": 1041, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1041.png", "answer": "x^2 = sqrt(3)*y", "fact_expr": "G: Parabola;p: Number;H: Line;A:Point;B:Point;p>0;Expression(G) = (x^2 = 2*p*y);Inclination(H)=ApplyUnit(60,degree);Intersection(H,G)={A,B};XCoordinate(A)+XCoordinate(B)=3", "coords": {} }, { "index": 1042, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1042.png", "answer": "x^2/2 - y^2/2 = 1", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/4 + y^2/2 = 1);Focus(H) = Vertex(G);Vertex(H)=Focus(G)", "coords": {} }, { "index": 1043, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1043.png", "answer": "sqrt(3)/2", "fact_expr": "G: Parabola;H: Ellipse;k: Number;Expression(G) = (y^2 = 12*x);k>0;Expression(H) = (k*y^2 + x^2 = 3*k);OneOf(Focus(H)) = Focus(G)", "coords": {} }, { "index": 1044, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1044.png", "answer": "(0,1/2)", "fact_expr": "G: Ellipse;b: Number;a: Number;c: Number;e: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);HalfFocalLength(G) = c;a*c - b^2 + c^2 < 0;Eccentricity(G) = e", "coords": {} }, { "index": 1045, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1045.png", "answer": "10", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/5 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);Negation(P=LeftVertex(G));Negation(P=RightVertex(G))", "coords": {} }, { "index": 1046, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1046.png", "answer": "ApplyUnit(90, degree)", "fact_expr": "G: Parabola;p: Number;H: Line;F: Point;B: Point;A: Point;A1: Point;B1: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};L1: Line;L2: Line;PointOnCurve(A, L1);PointOnCurve(B, L2);IsPerpendicular(L1, Directrix(G));IsPerpendicular(L2, Directrix(G));FootPoint(L1, Directrix(G)) = A1;FootPoint(L2, Directrix(G)) = B1", "coords": {} }, { "index": 1047, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1047.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;L: Line;PointOnCurve(F1, L);IsPerpendicular(L, Asymptote(G));M: Point;FootPoint(L, Asymptote(G)) = M;O: Origin;Tan(AngleOf(M, F1, O)) = 1/2", "coords": {} }, { "index": 1048, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1048.png", "answer": "pi/3", "fact_expr": "G: Parabola;p: Number;P: Point;Q: Point;M: Point;N: Point;F: Point;L: Line;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G)=F;PointOnCurve(P,G);PointOnCurve(Q,G);MidPoint(LineSegmentOf(P,Q)) = M;PointOnCurve(M, L);IsPerpendicular(L, Directrix(G));FootPoint(L, Directrix(G))=N;Abs(LineSegmentOf(M, N)) = Abs(LineSegmentOf(P, Q))", "coords": {} }, { "index": 1049, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1049.png", "answer": "sqrt(6)/3", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;A: Point;RightVertex(C) = A;G: Line;Expression(G) = (x = a/2);M: Point;N: Point;Intersection(G, C) = {M, N};DotProduct(VectorOf(A, M), VectorOf(A, N)) = 0", "coords": {} }, { "index": 1050, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1050.png", "answer": "[2*sqrt(2), 4)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);M: Point;PointOnCurve(M, C);C1: Circle;A: Point;B: Point;D: Point;Coordinate(A) = (5,0);Coordinate(B) = (2,sqrt(3));Coordinate(D) = (3,-2);PointOnCurve(A,C1);PointOnCurve(B,C1);PointOnCurve(D,C1);l1: Line;l2: Line;PointOnCurve(M,l1);PointOnCurve(M,l2);IsTangent(l1,C1);IsTangent(l2,C1);TangentPoint(l1,C1) = P;TangentPoint(l2,C1) = Q;P: Point;Q: Point", "coords": { "A": [ 5.0, 0.0 ], "B": [ 2.0, 1.7320508075688772 ], "D": [ 3.0, -2.0 ] } }, { "index": 1051, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1051.png", "answer": "x^2 - y^2 = 1", "fact_expr": "C: Hyperbola;G: Parabola;a:Number;b:Number;a>0;b>0;P: Point;l: Line;l1: Line;l2: Line;Expression(G) = (y^2 = 4*x);Coordinate(P) = (0, b);Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(Focus(G),l);PointOnCurve(P,l);IsParallel(l,l1);IsPerpendicular(l,l2);OneOf(Asymptote(C)) = l1;OneOf(Asymptote(C)) = l2;Negation(l1=l2)", "coords": {} }, { "index": 1052, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1053, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1053.png", "answer": "12/5", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/16 - y^2/9 = 1)", "coords": {} }, { "index": 1054, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1054.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;P: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 4*x);Coordinate(P) = (x0, -2);Intersection(Directrix(H),OneOf(Asymptote(G)))=P;x0:Number", "coords": {} }, { "index": 1055, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.4149149626832767, "b": 2.450642358201144, "orientation": "horizontal" }, "optimization": { "final_loss": 1.0319236343469396e-05, "converged": true, "iterations": 94 }, "output_path": "results/train/hyperbola/problem_1055.png", "answer": "3*x^2/4-y^2/4=1", "fact_expr": "C: Hyperbola;Expression(Asymptote(C)) = (y = pm*(sqrt(3)*x));G: Point;Coordinate(G) = (2, 2*sqrt(2));PointOnCurve(G, C)", "coords": { "G": [ 2.0, 2.8284271247461903 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 1056, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1056.png", "answer": "7/6", "fact_expr": "C: Parabola;p: Number;M: Point;F: Point;A: Point;B: Point;N: Point;Expression(C) = (x^2 = 2*p*y);Coordinate(M) = (2*sqrt(3), 0);Focus(C) = F;Intersection(LineOf(F,M),C)={A,B};Between(B,F,M);Intersection(LineOf(F,M),Directrix(C))=N;Abs(LineSegmentOf(B, N)) = 7*Abs(LineSegmentOf(B, F))", "coords": { "M": [ 3.4641016151377544, 0.0 ] } }, { "index": 1057, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1057.png", "answer": "sqrt(2)", "fact_expr": "G: Parabola;C: Circle;r: Number;P: Point;Expression(G) = (x^2 = 4*y);r>0;Expression(C) = ((x - 1)^2 + (y - 2)^2 = r^2);Intersection(G,C)=P;IsTangent(TangentOnPoint(P,G),C)", "coords": {} }, { "index": 1058, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1058.png", "answer": "4/3", "fact_expr": "G: Parabola;A: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(A, G);Quadrant(A)=1;Distance(A, Directrix(G)) = 5", "coords": {} }, { "index": 1059, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1060, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1060.png", "answer": "3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P: Point;y1: Number;Coordinate(P) = (2, y1);PointOnCurve(P, G);F: Point;Focus(G) = F", "coords": {} }, { "index": 1061, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1061.png", "answer": "97+56*sqrt(3)", "fact_expr": "G: Parabola;p: Number;l: Line;O: Origin;F: Point;A: Point;B: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Expression(l) = (-p/2 + x - sqrt(3)*y = 0);Focus(G) = F;Intersection(G, l) = {A, B};Quadrant(A)=1;Quadrant(B)=4", "coords": {} }, { "index": 1062, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1062.png", "answer": "8", "fact_expr": "G: Ellipse;H: Line;M: Point;A: Point;B: Point;k:Number;Expression(G) = (x^2/4 + y^2 = 1);Expression(H) = (y = k*(x + sqrt(3)));Coordinate(M) = (sqrt(3), 0);Intersection(G, H) = {A, B}", "coords": { "M": [ 1.7320508075688772, 0.0 ] } }, { "index": 1063, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1064, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1065, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1065.png", "answer": "1/2", "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;H: Point;x1: Number;y1: Number;x2: Number;y2: Number;Expression(G) = (y^2 = 4*x);Coordinate(H) = (0, 2);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);PointOnCurve(H, l);Intersection(l, G) = {A, B}", "coords": { "H": [ 0.0, 2.0 ] } }, { "index": 1066, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1066.png", "answer": "1", "fact_expr": "G: Hyperbola;b: Number;b>0;Expression(G) = (x^2/4 - y^2/b^2 = 1);Expression(Asymptote(G)) = (y = pm*(x/2))", "coords": {} }, { "index": 1067, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 9.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1067.png", "answer": "{x-y-2=0, x+y+2=0}", "fact_expr": "C: Ellipse;Expression(C) = (x^2/9 + y^2/3 = 1);l: Line;k:Number;Expression(l) = (y = k*x - 2);A: Point;B: Point;Intersection(l, C) = {A, B};P: Point;Coordinate(P) = (0, 1);Abs(LineSegmentOf(P, A)) = Abs(LineSegmentOf(P, B))", "coords": { "P": [ 0.0, 1.0 ] } }, { "index": 1068, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1068.png", "answer": "4*sqrt(3)", "fact_expr": "C: Parabola;G: Line;A: Point;B:Point;F: Point;M: Point;l: Line;Expression(C) = (y^2 = 4*x);Focus(C) = F;Directrix(C) = l;PointOnCurve(F, G);Inclination(G)=ApplyUnit(60,degree);Intersection(G, C) = {A,B};Quadrant(A)=1;L:Line;PointOnCurve(A,L);IsPerpendicular(l,L);FootPoint(l,L)=M", "coords": {} }, { "index": 1069, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_ecc_mismatch;ellipse_focus_mismatch", "params": { "a": 2.8909608023808606, "b": 2.375632646735382, "major_axis": "x", "x_coef": 8.000000000000002, "y_coef": 6.000000000000001 }, "optimization": { "final_loss": 2.1275904167232744e-15, "converged": true, "iterations": 2 }, "output_path": "results/train/ellipse/problem_1069.png", "answer": "x^2/8+y^2/6=1", "fact_expr": "G: Ellipse;F1: Point;F2: Point;O: Origin;Coordinate(F1) = (-sqrt(2), 0);Coordinate(F2) = (sqrt(2), 0);Center(G) = O;Focus(G)={F1,F2};Eccentricity(G) = 1/2", "coords": { "F1": [ -1.4142135623730951, 0.0 ], "F2": [ 1.4142135623730951, 0.0 ] }, "validation_reasons": [ "ellipse_ecc_mismatch", "ellipse_focus_mismatch" ] }, { "index": 1070, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1070.png", "answer": "(6*sqrt(5)-10)/5", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/5 = 1);M: Point;PointOnCurve(M, G);F: Point;LeftFocus(G) = F;A: Point;Coordinate(A) = (-2, 1);WhenMin(Abs(VectorOf(A, M)) + (3/2)*Abs(VectorOf(M, F)))", "coords": { "A": [ -2.0, 1.0 ] } }, { "index": 1071, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1071.png", "answer": "5", "fact_expr": "C: Parabola;P: Point;Q: Point;F: Point;O: Origin;T: Point;l: Line;Expression(C) = (y^2 = 4*x);Focus(C) = F;Directrix(C) = l;PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P, Q), l) ;FootPoint(LineSegmentOf(P, Q), l)= Q;Intersection(LineSegmentOf(Q, F), yAxis) = T;Abs(LineSegmentOf(O, T)) = 2", "coords": {} }, { "index": 1072, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1073, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 2.2117717266082764, "b": 1.940546499547044, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 9.025489068839395e-07, "converged": true, "iterations": 109 }, "output_path": "results/train/ellipse/problem_1073.png", "answer": "4*sqrt(3)", "fact_expr": "G: Ellipse;Expression(G) = (y^2/4 + x^2/m^2 = 1);m: Number;H: Point;Coordinate(H) = (-2, sqrt(3));PointOnCurve(H, G)", "coords": { "H": [ -2.0, 1.7320508075688772 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 1074, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1075, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1075.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(90, degree)", "coords": {} }, { "index": 1076, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.125, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1076.png", "answer": "sqrt(26)/2", "fact_expr": "G: Line;Expression(G) = (y = 5*x - 3);H: Curve;Expression(H) = (y = 2*x^2)", "coords": {} }, { "index": 1077, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7195594310760498, "b": 1.7224669456481934, "orientation": "horizontal" }, "optimization": { "final_loss": 2.391175712546101e-06, "converged": true, "iterations": 83 }, "output_path": "results/train/hyperbola/problem_1077.png", "answer": "2*sqrt(6)", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Parabola;A: Point;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y^2 = 8*x);Eccentricity(C) = sqrt(2);Intersection(C,Directrix(G)) = {A, B};Abs(LineSegmentOf(A, B)) = 2", "coords": {} }, { "index": 1078, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1078.png", "answer": "sqrt(2)/2", "fact_expr": "M: Point;Coordinate(M) = (1, 1);G: Line;PointOnCurve(M, G) = True;Slope(G) = -1/2;C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Intersection(G, C) = {A, B};A: Point;B: Point;MidPoint(LineSegmentOf(A, B)) = M", "coords": { "M": [ 1.0, 1.0 ] } }, { "index": 1079, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.3300262689590454, "b": 2.30381441116333, "orientation": "horizontal" }, "optimization": { "final_loss": 6.658357051492203e-06, "converged": true, "iterations": 122 }, "output_path": "results/train/hyperbola/problem_1079.png", "answer": "8", "fact_expr": "C: Hyperbola;b: Number;a: Number;O: Origin;P: Point;F2: Point;F1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(Asymptote(C)) = (y = pm*(sqrt(3)*x));PointOnCurve(P, RightPart(C));LeftFocus(C) = F1;RightFocus(C) = F2;Min(DotProduct(VectorOf(O,P),VectorOf(O,F2)))=2*a", "coords": {} }, { "index": 1080, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1080.png", "answer": "7*sqrt(5)/10", "fact_expr": "G: Parabola;H: Line;M: Point;N: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (2*x - y + 4 = 0);PointOnCurve(M, G);PointOnCurve(N, H)", "coords": {} }, { "index": 1081, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1081.png", "answer": "(1, (1+sqrt(5))/2)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(G) = F;L: Line;PointOnCurve(F, L);IsPerpendicular(L, Asymptote(G));P: Point;FootPoint(L, Asymptote(G)) = P;Q: Point;Intersection(L, yAxis) = Q;O: Origin;Abs(LineSegmentOf(F, P)) < Abs(LineSegmentOf(O, Q))", "coords": {} }, { "index": 1082, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1082.png", "answer": "{0, -1/2}", "fact_expr": "G: Line;Expression(G) = (y = a*x - 1);a: Real;H: Curve;Expression(H) = (y^2 = 2*x);NumIntersection(G, H) = 1", "coords": {} }, { "index": 1083, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1083.png", "answer": "2*sqrt(2) - 3", "fact_expr": "G: Ellipse;C: Circle;P: Point;A: Point;B: Point;l1: Line;l2: Line;Expression(G) = (x^2/4 + y^2/3 = 1);Expression(C) = (y^2 + (x - 1)^2 = 1);PointOnCurve(P, G);TangentOfPoint(P, C) = {l1, l2};TangentPoint(l1, C) = A;TangentPoint(l2, C) = B", "coords": {} }, { "index": 1084, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1084.png", "answer": "2*sqrt(3)", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2 - y^2 = 1);Focus(G)={F1,F2};PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2))", "coords": {} }, { "index": 1085, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1085.png", "answer": "30/13", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/5 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P,F2),xAxis) = True", "coords": {} }, { "index": 1086, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1087, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1087.png", "answer": "1", "fact_expr": "G: Hyperbola;a: Number;H: Ellipse;F1: Point;F2:Point;P: Point;Expression(G) = (-y^2 + x^2/a^2 = 1);Expression(H) = (x^2/4 + y^2 = 1);Focus(G)={F1,F2};Focus(H)={F1,F2};OneOf(Intersection(G,H))=P", "coords": {} }, { "index": 1088, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1089, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1089.png", "answer": "8", "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;M: Point;F: Point;Expression(G) = (y = x^2/8);Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B};Intersection(Directrix(G), yAxis) = M;WhenMax(Abs(LineSegmentOf(A,M))/Abs(LineSegmentOf(A,F)));IsChordOf(LineSegmentOf(A,B),G)", "coords": {} }, { "index": 1090, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1090.png", "answer": "{3,5}", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (y^2/4 + x^2/m = 1);FocalLength(G) = 2", "coords": {} }, { "index": 1091, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1092, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1093, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1093.png", "answer": "2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);P:Point;PointOnCurve(P,G)", "coords": {} }, { "index": 1094, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1094.png", "answer": "2", "fact_expr": "G: Parabola;A: Point;B: Point;Expression(G) = (y^2 = 2*x);PointOnCurve(A, G);PointOnCurve(B, G);Distance(A, Focus(G))+Distance(B,Focus(G)) = 5", "coords": {} }, { "index": 1095, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1096, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1097, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1097.png", "answer": "{4,2*sqrt(2)}", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (y^2/2 + x^2/m = 1);Eccentricity(G) = sqrt(2)/2", "coords": {} }, { "index": 1098, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1098.png", "answer": "x^2/4 + y^2/2 = 1", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;Focus(C) = {F1, F2};Abs(LineSegmentOf(F1, F2)) = 2*sqrt(2);P: Point;PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) - Abs(LineSegmentOf(P, F2)) = a;IsPerpendicular(LineSegmentOf(P, F2), LineSegmentOf(F1, F2))", "coords": {} }, { "index": 1099, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1099.png", "answer": "1", "fact_expr": "G: Hyperbola;m: Real;H: Ellipse;Expression(G) = (-y^2/2 + x^2/m = 1);Expression(H) = (x^2/4 + y^2/m^2 = 1);Focus(H) = Focus(G)", "coords": {} }, { "index": 1100, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 2.0, -0.0 ], "radius": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_1100.png", "answer": "x=-2", "fact_expr": "G: Parabola;H: Circle;Expression(H) = (-4*x + x^2 + y^2 = 0);O:Origin;Vertex(G)=O;Focus(G)=Center(H)", "coords": {} }, { "index": 1101, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1101.png", "answer": "sqrt(10)", "fact_expr": "C: Hyperbola;a: Number;b: Number;G: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(G) = (1, 3);PointOnCurve(G, OneOf(Asymptote(C)))", "coords": { "G": [ 1.0, 3.0 ] } }, { "index": 1102, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1102.png", "answer": "4", "fact_expr": "G: Parabola;p: Number;H: Line;A: Point;F: Point;B: Point;p>0;Expression(G) = (y^2 = 2*(p*x));PointOnCurve(F,H);Intersection(H,G) = {B, A};Focus(G)=F;Abs(LineSegmentOf(A,F))=2*Abs(LineSegmentOf(B,F));2*Abs(LineSegmentOf(B,F))=6", "coords": {} }, { "index": 1103, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 3.0, "major_axis": "x", "x_coef": 36.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1103.png", "answer": "x+2*y-8=0", "fact_expr": "G: Ellipse;H: LineSegment;P: Point;Expression(G) = (x^2/36 + y^2/9 = 1);Coordinate(P) = (4, 2);MidPoint(H)=P;IsChordOf(H,G)", "coords": { "P": [ 4.0, 2.0 ] } }, { "index": 1104, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1104.png", "answer": "2", "fact_expr": "G: Hyperbola;a: Number;Expression(G) = (-y^2/9 + x^2/a = 1);Inclination(OneOf(Asymptote(G))) = ApplyUnit(60, degree)", "coords": {} }, { "index": 1105, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1105.png", "answer": "x^2/4 - y^2/12 = 1", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F: Point;O: Origin;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (1, sqrt(3));PointOnCurve(P, Asymptote(C));RightFocus(C) = F;AngleOf(F, P, O) = ApplyUnit(90, degree)", "coords": { "P": [ 1.0, 1.7320508075688772 ] } }, { "index": 1106, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1106.png", "answer": "[-5,5]", "fact_expr": "P: Point;x1: Number;y1: Number;Coordinate(P) = (x1, y1);C: Curve;Expression(C) = (x^2/4 + y^2/3 = 1);PointOnCurve(P, C)", "coords": {} }, { "index": 1107, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1107.png", "answer": "1", "fact_expr": "G: Parabola;p: Number;H: Point;x0: Number;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(H) = (x0, 2);x0 > p/2;PointOnCurve(H, G);Distance(H, Focus(G)) = 5/2", "coords": {} }, { "index": 1108, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1109, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1110, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.6495774984359741, "b": 1.117531657218933, "orientation": "horizontal" }, "optimization": { "final_loss": 9.364227298647165e-05, "converged": true, "iterations": 59 }, "output_path": "results/train/hyperbola/problem_1110.png", "answer": "sqrt(2)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;Coordinate(F1) = (-2, 0);Coordinate(F2) = (2, 0);LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C);Min(DotProduct(VectorOf(P, F1), VectorOf(P, F2))) = -2", "coords": { "F1": [ -2.0, 0.0 ], "F2": [ 2.0, 0.0 ] } }, { "index": 1111, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1111.png", "answer": "x + 2*y - 3 = 0", "fact_expr": "G: LineSegment;H: Curve;Expression(H) = (x^2 + 2*y^2 = 4);Coordinate(MidPoint(G)) = (1, 1);IsChordOf(G, H)", "coords": {} }, { "index": 1112, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1112.png", "answer": "3*x+4*y-5=0", "fact_expr": "l: Line;G: Hyperbola;B: Point;A: Point;P: Point;Expression(G) = (x^2/4 - y^2 = 1);Coordinate(P) = (3, -1);PointOnCurve(P, l);Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A,B)) = P", "coords": { "P": [ 3.0, -1.0 ] } }, { "index": 1113, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1113.png", "answer": "[\\pi/6, 5\\pi/6]", "fact_expr": "O: Origin;theta: Number;H: Line;PointOnCurve(O, H);Inclination(H) = theta;G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;A: Point;B: Point;Intersection(H, G) = {A, B};F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;AngleOf(F1, A, F2) = pi/2;e: Number;Eccentricity(G) = e;In(e, [sqrt(2)/2, sqrt(6)/3])", "coords": {} }, { "index": 1114, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1114.png", "answer": "-2", "fact_expr": "G: Hyperbola;P: Point;A1: Point;F2: Point;Expression(G) = (x^2 - y^2/3 = 1);LeftVertex(G)=A1;RightFocus(G)=F2;PointOnCurve(P,RightPart(G))", "coords": {} }, { "index": 1115, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1115.png", "answer": "sqrt(15)/8", "fact_expr": "P: Point;C1: Curve;Expression(C1) = (y^2 = 16*x);PointOnCurve(P, C1);Q: Point;C2: Curve;Expression(C2) = (y^2 + (x - 4)^2 = 9);PointOnCurve(Q, C2);O: Origin", "coords": {} }, { "index": 1116, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1116.png", "answer": "28", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;Expression(G) = (y^2 = 16*x);PointOnCurve(Focus(G), H);Intersection(H, G) = {A, B};XCoordinate(MidPoint(LineSegmentOf(A, B))) = 10", "coords": {} }, { "index": 1117, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1118, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1119, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1119.png", "answer": "8", "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;F: Point;Expression(C) = (y^2 = 6*x);Focus(C) = F;PointOnCurve(F, l);Slope(l) = sqrt(3);Intersection(l, C) = {A, B};IsChordOf(LineSegmentOf(A,B),C)", "coords": {} }, { "index": 1120, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1120.png", "answer": "8", "fact_expr": "G: Ellipse;P: Point;F1: Point;x0: Number;y0: Number;Coordinate(P) = (x0, y0);Expression(G) = (x^2/25 + y^2/16 = 1);LeftFocus(G) = F1;PointOnCurve(P, G)", "coords": {} }, { "index": 1121, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1121.png", "answer": "2*sqrt(6)", "fact_expr": "C: Parabola;M: Point;Expression(C) = (y = x^2/8);PointOnCurve(M, C);Distance(M, Focus(C)) = 5", "coords": {} }, { "index": 1122, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.581065485469099, "b": 1.581065485469099, "orientation": "horizontal" }, "optimization": { "final_loss": 1.6812139285133018e-06, "converged": true, "iterations": 95 }, "output_path": "results/train/hyperbola/problem_1122.png", "answer": "x^2/2 - y^2/3 = 1", "fact_expr": "C: Hyperbola;F1: Point;Coordinate(F1) = (-sqrt(5), 0);F2: Point;Coordinate(F2) = (sqrt(5), 0);Focus(C) = {F1, F2};PointOnCurve(F2, G);G: Line;Intersection(G, RightPart(C)) = {A, B};A: Point;B: Point;VectorOf(A, F2) = 3*VectorOf(F2, B);3*Abs(LineSegmentOf(A, B)) = 4*Abs(LineSegmentOf(B, F1))", "coords": { "F1": [ -2.23606797749979, 0.0 ], "F2": [ 2.23606797749979, 0.0 ] } }, { "index": 1123, "success": false, "error": "\n= 1 $ $ (a > b > 0) $, the coordinates of its right focus are $ (c, 0)\n ^\nParseException: Expected end of text, found '$' (at char 4), (line:1, col:5)" }, { "index": 1124, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.3228756555322954, "b": 2.2912878474779204, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1124.png", "answer": "y = pm*sqrt(3)*x", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Ellipse;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2/16 + y^2/9 = 1);Eccentricity(G) = 2;Focus(G) = Focus(H)", "coords": {} }, { "index": 1125, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1126, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1127, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1127.png", "answer": "3/2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1), xAxis);Cos(AngleOf(P, F2, F1)) = 12/13", "coords": {} }, { "index": 1128, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1128.png", "answer": "3", "fact_expr": "G: Parabola;p: Number;m:Number;H: Circle;I: Point;P: Point;Q: Point;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (y^2 + (x - 3)^2 = 1);Coordinate(I) = (5, m);PointOnCurve(I,G);Focus(G)=F;Distance(I,F)=6;PointOnCurve(P,G);PointOnCurve(Q,H)", "coords": {} }, { "index": 1129, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1129.png", "answer": "[sqrt(5)/5, sqrt(3)/3]", "fact_expr": "E: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;c:Number;e: Number;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(E) =F1;RightFocus(E)=F2;PointOnCurve(P,E);Range(Max(DotProduct(VectorOf(P,F1),VectorOf(P,F2))))=[2*c^2,4*c^2];c=sqrt(a^2-b^2);Eccentricity(E) = e", "coords": {} }, { "index": 1130, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1131, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1131.png", "answer": "sqrt(2)", "fact_expr": "M: Point;F: Point;G: Parabola;N: Point;l: Line;Expression(G) = (y = x^2/2);Focus(G) = F;Directrix(G) = l;PointOnCurve(M, l);Intersection(LineSegmentOf(M, F), G) = N;Abs(LineSegmentOf(M, N)) = sqrt(2)*Abs(LineSegmentOf(N, F))", "coords": {} }, { "index": 1132, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1132.png", "answer": "7", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;e: Number;Eccentricity(G) = e;e = 1/2;A: Point;B: Point;LeftVertex(G) = A;RightVertex(G) = B;P: Point;PointOnCurve(P, G);Negation(P=A);Negation(P=B);alpha: Number;beta: Number;Inclination(LineOf(P, A)) = alpha;Inclination(LineOf(P, B)) = beta", "coords": {} }, { "index": 1133, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1133.png", "answer": "(2/3, -1/3)", "fact_expr": "H: Line;Expression(H) = (y = x - 1);G: Ellipse;Expression(G) = (x^2/2 + y^2 = 1);A: Point;B: Point;Intersection(H, G) = {A, B}", "coords": {} }, { "index": 1134, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1134.png", "answer": "-2", "fact_expr": "G: Hyperbola;H: Line;P: Point;O: Origin;A: Point;B: Point;Expression(G) = (x^2 - y^2 = 1);Expression(H) = (x + 2*y + 3 = 0);Intersection(G, H) = {A, B};MidPoint(LineSegmentOf(A,B))=P", "coords": {} }, { "index": 1135, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.449489742783178, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1135.png", "answer": "6/5", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/6 - y^2/3 = 1);F1: Point;F2: Point;Focus(G) = {F1,F2};M: Point;PointOnCurve(M, G) = True;IsPerpendicular(LineSegmentOf(M, F1), xAxis) = True", "coords": {} }, { "index": 1136, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1136.png", "answer": "y^2=8*x", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;p>0;G: Circle;Vertex(C) = Center(G);Intersection(C, G) = {A, B};A: Point;B: Point;Intersection(G, Directrix(C)) = {D, E};D: Point;E: Point;Abs(LineSegmentOf(A, B)) = 4*sqrt(2);Abs(LineSegmentOf(D, E)) = 2*sqrt(5)", "coords": {} }, { "index": 1137, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1137.png", "answer": "y=pm*sqrt(2)*x", "fact_expr": "G: Hyperbola;b: Number;a: Number;A: Point;F1:Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(A,G);IsPerpendicular(LineSegmentOf(A,F2),xAxis);Radius(InscribedCircle(TriangleOf(A,F1,F2)))=(sqrt(3)-1)*a", "coords": {} }, { "index": 1138, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1138.png", "answer": "25", "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/25 + y^2/9 = 1);PointOnCurve(P, G);F1:Point;F2:Point;Focus(G)={F1,F2};Distance(P, F1)*Distance(P,F2) = m;m:Number", "coords": {} }, { "index": 1139, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1139.png", "answer": "x^2/2 - y^2/2 = 1", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Distance(OneOf(Focus(G)),OneOf(Asymptote(G)))=sqrt(2);Expression(OneOf(Asymptote(G))) = (y = x)", "coords": {} }, { "index": 1140, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 0.8176298651812968, "b": 2.452642014841466, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4992967561765154e-06, "converged": true, "iterations": 283 }, "output_path": "results/train/hyperbola/problem_1140.png", "answer": "x^2 - y^2/9 = 1", "fact_expr": "G: Hyperbola;Coordinate(OneOf(Focus(G))) = (sqrt(10), 0);Expression(Asymptote(G)) = (y = pm*3*x)", "coords": {} }, { "index": 1141, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1141.png", "answer": "-2", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G)", "coords": {} }, { "index": 1142, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1142.png", "answer": "5/2", "fact_expr": "G: Parabola;P: Point;Expression(G) = (y^2 = 8*x);Coordinate(P) = (1/2, 2);PointOnCurve(P,G)", "coords": { "P": [ 0.5, 2.0 ] } }, { "index": 1143, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1143.png", "answer": "(\\sqrt{5}-1)/2", "fact_expr": "G: Ellipse;b: Number;a: Number;F: Point;A: Point;B: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (-c, 0);c:Number;Coordinate(A) = (a, 0);Coordinate(B) = (0, b);LeftFocus(G)=F;RightVertex(G)=A;UpperVertex(G)=B;DotProduct(VectorOf(F, B), VectorOf(A, B)) = 0", "coords": {} }, { "index": 1144, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1144.png", "answer": "[sqrt(3)/3, sqrt(10)/5]", "fact_expr": "H: Ellipse;Expression(H) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;G: Hyperbola;Expression(G) = (-y^2/n^2 + x^2/m^2 = 1);m: Number;n: Number;m>0;n>0;F1: Point;F2: Point;Focus(H) = {F1, F2};Focus(G) = {F1, F2};P: Point;OneOf(Intersection(H, G)) = P;e1: Number;e2: Number;Eccentricity(H) = e1;Eccentricity(G) = e2;AngleOf(F1, P, F2) = pi/3;In(e2, [sqrt(2), sqrt(3)])", "coords": {} }, { "index": 1145, "success": false, "conic_type": "parabola", "error": "validation: parabola_focus_mismatch", "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1145.png", "answer": "y^2=4*x", "fact_expr": "C: Parabola;O: Origin;Vertex(C) = O;PointOnCurve(Focus(C), xAxis);G: Line;Expression(G) = (y = x);A: Point;B: Point;Intersection(G, C) = {A, B};F: Point;Coordinate(F) = (2, 2);MidPoint(LineSegmentOf(A, B)) = F", "coords": { "F": [ 2.0, 2.0 ] }, "validation_reasons": [ "parabola_focus_mismatch" ] }, { "index": 1146, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.75, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1146.png", "answer": "sqrt(5)/6", "fact_expr": "C: Parabola;l: Line;O: Origin;M: Point;N: Point;Expression(C) = (y^2 = 3*x);Expression(l) = (3*x - y - 1 = 0);Intersection(l, C) = {M, N}", "coords": {} }, { "index": 1147, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1147.png", "answer": "1/3", "fact_expr": "G: Ellipse;H: Line;A: Point;B: Point;N: Point;F: Point;Expression(G) = (x^2/9 + y^2/5 = 1);LeftFocus(G) = F;PointOnCurve(F, H);Negation(IsPerpendicular(xAxis, H));Intersection(H, G) = {A, B};Intersection(PerpendicularBisector(LineSegmentOf(A, B)), xAxis) = N", "coords": {} }, { "index": 1148, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 2.0, "major_axis": "x", "x_coef": 8.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1148.png", "answer": "ApplyUnit(90, degree)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/8 + y^2/4 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);Area(TriangleOf(F1, P, F2)) = 4", "coords": {} }, { "index": 1149, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.33732376966822, "b": 1.7454107534785184, "orientation": "horizontal" }, "optimization": { "final_loss": 1.8920117601792354e-05, "converged": true, "iterations": 51 }, "output_path": "results/train/hyperbola/problem_1149.png", "answer": "5/4", "fact_expr": "G: Hyperbola;PointOnCurve(Focus(G), xAxis);Expression(Asymptote(G)) = (y = pm*(3/4)*x)", "coords": {} }, { "index": 1150, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1150.png", "answer": "sqrt(3)", "fact_expr": "G: Ellipse;H: Line;B: Point;A: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, H);Intersection(H, G) = {A, B};MidPoint(LineSegmentOf(A,B)) = F2", "coords": {} }, { "index": 1151, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1151.png", "answer": "4", "fact_expr": "C: Parabola;A: Point;P: Point;F: Point;Expression(C) = (y^2 = 4*x);Coordinate(A) = (3, 1);PointOnCurve(P, C);Focus(C) = F", "coords": { "A": [ 3.0, 1.0 ] } }, { "index": 1152, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/train/hyperbola/problem_1152.png", "answer": "7", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Expression(Asymptote(G)) = (y = pm*x*2*sqrt(6)/5);H: Point;Coordinate(H) = (5, 3*sqrt(2));PointOnCurve(H,G) = True", "coords": { "H": [ 5.0, 4.242640687119286 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 1153, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1153.png", "answer": "(1, \\sqrt{2})", "fact_expr": "E: Hyperbola;b: Number;a: Number;P: Point;a>0;b>0;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (2, 0);Distance(P, Asymptote(E))0;b>0;c: Number;HalfFocalLength(G) = c;-4*a*c + b^2 < 0", "coords": {} }, { "index": 1155, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.23606797749979, "b": 2.0, "major_axis": "y", "x_coef": 4.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1155.png", "answer": "8 - 4*sqrt(3)", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 + y^2/5 = 1);PointOnCurve(P, G);Focus(G) = {F1, F2};AngleOf(F1, P, F2) = ApplyUnit(30, degree)", "coords": {} }, { "index": 1156, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 10.0, "b": 6.0, "major_axis": "x", "x_coef": 100.0, "y_coef": 36.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1156.png", "answer": "4", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2:Point;Expression(G) = (x^2/100 + y^2/36 = 1);PointOnCurve(P, G);OneOf(Focus(G)) =F1;OneOf(Focus(G)) = F2;Negation(F1= F2);Distance(P, F1) = 6", "coords": {} }, { "index": 1157, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1157.png", "answer": "{(b^2/4+4)&(0,4],(2*b)&(b>4)}", "fact_expr": "G: Curve;b: Number;H: Point;b>0;Expression(G) = (x^2/4 + y^2/b^2 = 1);Coordinate(H) = (x1, y1);x1:Number;y1:Number;PointOnCurve(H,G)", "coords": {} }, { "index": 1158, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1158.png", "answer": "[a-sqrt(a^2-b^2),a+sqrt(a^2-b^2)]", "fact_expr": "P: Point;G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;PointOnCurve(P, G);F1: Point;OneOf(Focus(G)) = F1", "coords": {} }, { "index": 1159, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1159.png", "answer": "1/2", "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;l1: Line;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F1;RightDirectrix(G) = l1;f: LineSegment;PointOnCurve(F1, f);IsPerpendicular(f, xAxis);IsChordOf(f, G);Length(f) = Distance(F1, l1)", "coords": {} }, { "index": 1160, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1161, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1161.png", "answer": "sqrt(3)-1", "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Focus(G)={F1,F2};PointOnCurve(P, G);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;Tan(AngleOf(P,F1,F2))=sqrt(3)/3", "coords": {} }, { "index": 1162, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1162.png", "answer": "32", "fact_expr": "C:Parabola;Expression(C)=(y^2=4*x);P:Point;Coordinate(P)=(4,0);L:Line;PointOnCurve(P,L);A:Point;B:Point;x1:Number;x2:Number;y1:Number;y2:Number;Coordinate(A)=(x1,y1);Coordinate(B)=(x2,y2);Intersection(L,C)={A,B}", "coords": { "P": [ 4.0, 0.0 ] } }, { "index": 1163, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1163.png", "answer": "[-1,1]", "fact_expr": "l: Line;G: Parabola;Q: Point;Expression(G) = (y^2 = 8*x);Intersection(Directrix(G), xAxis) = Q;PointOnCurve(Q, l);IsIntersect(l,G)", "coords": {} }, { "index": 1164, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1164.png", "answer": "2*sqrt(5)", "fact_expr": "C: Ellipse;b: Number;a: Number;P: Point;F: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, C);OneOf(Focus(C))=F;Max(Abs(LineSegmentOf(P, F))) = 5;Min(Abs(LineSegmentOf(P,F)))=1", "coords": {} }, { "index": 1165, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1165.png", "answer": "1/2", "fact_expr": "G: Hyperbola;H: Point;Expression(G) = (x^2 - y^2/3 = 1);Coordinate(H) = (0, 1)", "coords": { "H": [ 0.0, 1.0 ] } }, { "index": 1166, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1167, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1167.png", "answer": "3*sqrt(5)/5", "fact_expr": "A: Point;Coordinate(A) = (0, -4);B: Point;Coordinate(B) = (3, 2);G: Parabola;Expression(G) = (y^2 = x);D: Point;PointOnCurve(D, G) = True", "coords": { "A": [ 0.0, -4.0 ], "B": [ 3.0, 2.0 ] } }, { "index": 1168, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 3.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1168.png", "answer": "-2/3", "fact_expr": "l: Line;C: Ellipse;D: Point;A: Point;B: Point;k1: Number;k2: Number;Expression(C) = (x^2/3 + y^2/2 = 1);O: Origin;PointOnCurve(O, l);Intersection(l, C) = {A, B};UpperVertex(C) = D;Slope(LineOf(A, D)) = k1;Slope(LineOf(B, D)) = k2", "coords": {} }, { "index": 1169, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1169.png", "answer": "8", "fact_expr": "G: Parabola;A: Point;B: Point;O: Origin;Expression(G) = (y^2 = 8*x);PointOnCurve(A,G);PointOnCurve(B,G);IsPerpendicular(LineOf(A,B),xAxis);Area(TriangleOf(O, A, B)) = 8", "coords": {} }, { "index": 1170, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1171, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1172, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1172.png", "answer": "4*x + 9*y = 13", "fact_expr": "l: Line;G: Ellipse;P: Point;Expression(G) = (x^2/9 + y^2/4 = 1);Coordinate(P) = (1, 1);PointOnCurve(P, l) ;MidPoint(InterceptChord(l, G)) = P", "coords": { "P": [ 1.0, 1.0 ] } }, { "index": 1173, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_ecc_mismatch", "params": { "a": 1.9257268905639648, "b": 1.7993911322594118, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 2.5576271215184017e-15, "converged": true, "iterations": 2 }, "output_path": "results/train/ellipse/problem_1173.png", "answer": "x^2/4+y^2/3=1", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;Eccentricity(C) = 1/2;L: Line;PointOnCurve(F2,L) = True;Intersection(L,C) = {A,B};A: Point;B: Point;Perimeter(TriangleOf(F1,A,B)) = 8", "coords": {}, "validation_reasons": [ "ellipse_ecc_mismatch" ] }, { "index": 1174, "success": false, "error": "Circle from diameter: missing point coordinates" }, { "index": 1175, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1175.png", "answer": "2", "fact_expr": "C: Parabola;Expression(C) = (x^2 = 2*y);P: Point;Q: Point;PointOnCurve(P, C);PointOnCurve(Q, C);Abs(LineSegmentOf(P, Q)) = 5", "coords": {} }, { "index": 1176, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1176.png", "answer": "sqrt(10)/2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;LeftFocus(G) = F1;RightFocus(G) = F2;F1: Point;F2: Point;H: Line;PointOnCurve(F2, H) = True;Intersection(H, RightPart(G)) = {A, B};B: Point;A: Point;DotProduct(VectorOf(B, F1), VectorOf(B, F2)) = 0;Sin(AngleOf(F1, A, F2)) = 3/5", "coords": {} }, { "index": 1177, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1177.png", "answer": "1", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(90, degree)", "coords": {} }, { "index": 1178, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1178.png", "answer": "3", "fact_expr": "G: Parabola;P: Point;Q: Point;F: Point;M: Point;Expression(G) = (y^2 = 4*x);Focus(G)=F;PointOnCurve(P,G);PointOnCurve(Q,G);Negation(P=Q);lambda:Number;lambda>1;VectorOf(P, F) = lambda*VectorOf(F, Q);MidPoint(LineSegmentOf(P,Q))=M;Distance(M, Directrix(G)) = 8/3", "coords": {} }, { "index": 1179, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1180, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1180.png", "answer": "2*sqrt(3)/3", "fact_expr": "l1: Line;l2:Line;l:Line;G: Hyperbola;b: Number;a: Number;F: Point;B: Point;A: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Asymptote(G)={l1,l2};RightFocus(G)=F;PointOnCurve(F,l);IsPerpendicular(l,l1);Intersection(l1,l)=A;Intersection(l2,l)=B;VectorOf(F, B) = 2*VectorOf(A, F)", "coords": {} }, { "index": 1181, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1182, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1182.png", "answer": "(3/2, 2*sqrt(3))", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);F: Point;Focus(G) = F;A: Point;Coordinate(A) = (2, 2*sqrt(3));P: Point;PointOnCurve(P, G);WhenMin(Abs(LineSegmentOf(P, A)) + Abs(LineSegmentOf(P, F)))", "coords": { "A": [ 2.0, 3.4641016151377544 ] } }, { "index": 1183, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1184, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 5.0, -0.0 ], "radius": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_1184.png", "answer": "{(y^2 = 20*x)&(x > 0), (y = 0)&(x < 0)}", "fact_expr": "C: Circle;Expression(C) = (-10*x + x^2 + y^2 = 0);H: Circle;P: Point;IsTangent(H, yAxis);IsOutTangent(C, H);Center(H) = P", "coords": {} }, { "index": 1185, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1186, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1186.png", "answer": "2", "fact_expr": "E: Hyperbola;Expression(E) = (-x^2/b^2 + y^2/a^2 = 1);a: Number;b: Number;a>0;b>0;C: Circle;Expression(C) = (y^2 + (x - 2)^2 = 1);IsTangent(Asymptote(E),C) = True", "coords": {} }, { "index": 1187, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1187.png", "answer": "2", "fact_expr": "C: Parabola;Expression(C) = (x^2 = 4*y);F: Point;Focus(C) = F;M: Point;Intersection(Directrix(C), yAxis) = M;A: Point;B: Point;l: Line;PointOnCurve(F, l);Intersection(l, C) = {A, B};AngleOf(A, M, B) = ApplyUnit(90, degree)", "coords": {} }, { "index": 1188, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1188.png", "answer": "(x-2)^2+y^2=16", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);F: Point;Focus(G) = F;l: Line;Directrix(G) = l;H: Circle;Center(H) = F;IsTangent(l, H)", "coords": {} }, { "index": 1189, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "y", "x_coef": 9.0, "y_coef": 25.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1189.png", "answer": "25", "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/9 + y^2/25 = 1);PointOnCurve(P, G);F1: Point;F2: Point;Focus(G) = {F1, F2};m: Number;Distance(P, F1)*Distance(P, F2) = m", "coords": {} }, { "index": 1190, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1190.png", "answer": "6*sqrt(2)", "fact_expr": "G: Parabola;H: Line;O: Origin;A: Point;B: Point;F: Point;Expression(G) = (x^2 = 8*y);PointOnCurve(F, H);Focus(G) = F;Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, F)) = 6", "coords": {} }, { "index": 1191, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1192, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1192.png", "answer": "16", "fact_expr": "G: Hyperbola;m: Real;Expression(G) = (x^2/9 - y^2/m = 1);FocalLength(G) = 10", "coords": {} }, { "index": 1193, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1193.png", "answer": "3", "fact_expr": "G: Hyperbola;m: Number;F1: Point;A: Point;l:Line;Expression(G) = (-y^2/16 + x^2/m = 1);Coordinate(F1) = (5, 0);OneOf(Focus(G)) = F1;OneOf(Asymptote(G))=l;PointOnCurve(F1, LineSegmentOf(F1, A));IsPerpendicular(LineSegmentOf(F1, A),l);Intersection(LineSegmentOf(F1, A), l) = A;O:Origin", "coords": { "F1": [ 5.0, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 1194, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1194.png", "answer": "pm*1", "fact_expr": "G: Hyperbola;a: Number;H: Ellipse;Expression(G) = (-y^2/2 + x^2/a^2 = 1);Expression(H) = (x^2/4 + y^2/a^2 = 1);Focus(G)=Focus(H)", "coords": {} }, { "index": 1195, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1195.png", "answer": "(-1/4,1)", "fact_expr": "C: Curve;Expression(C) = (y^2 = -4*x);P: Point;PointOnCurve(P, C);F: Point;Focus(C) = F;A: Point;Coordinate(A) = (-2, 1);WhenMin(Distance(P, F) + Distance(P, A))", "coords": { "A": [ -2.0, 1.0 ] } }, { "index": 1196, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1196.png", "answer": "3*sqrt(15)", "fact_expr": "C: Ellipse;Expression(C) = (x^2/25 + y^2/9 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) = 4", "coords": {} }, { "index": 1197, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 3.872983346207417, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1197.png", "answer": "13", "fact_expr": "G: Hyperbola;C1: Circle;C2: Circle;P: Point;M: Point;N: Point;l1:Line;l2:Line;Expression(G) = (x^2 - y^2/15 = 1);Expression(C1) = ((x + 4)^2 + y^2 = 4);Expression(C2) = ((x - 4)^2 + y^2 = 1);PointOnCurve(P, RightPart(G));TangentOfPoint(P,C1)=l1;TangentOfPoint(P,C2)=l2;TangentPoint(l1,C1)=M;TangentPoint(l2,C2)=N;PointOnCurve(P,l1);PointOnCurve(P,l2)", "coords": {} }, { "index": 1198, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1199, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/train/hyperbola/problem_1199.png", "answer": "2*sqrt(3)/3", "fact_expr": "C: Hyperbola;b: Number;a: Number;A: Point;F1: Point;F2: Point;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (sqrt(15)/2, 1/2);PointOnCurve(A, C);LeftFocus(C) = F1;RightFocus(C) = F2;DotProduct(VectorOf(A, F1), VectorOf(A, F2)) = 0", "coords": { "A": [ 1.9364916731037085, 0.5 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 1200, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1201, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1202, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1203, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1203.png", "answer": "1/4", "fact_expr": "G: Hyperbola;H: Parabola;a: Number;A: Point;B: Point;l:Line;Expression(G) = (x^2/4 - y^2 = 1);a>0;Expression(H) = (y = a*x^2);Directrix(H)=l;L1:Line;L2:Line;Asymptote(G)={L1,L2};Intersection(l,L1)=A;Intersection(l,L2)=B;Abs(LineSegmentOf(A, B)) = 4", "coords": {} }, { "index": 1204, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1204.png", "answer": "2", "fact_expr": "C: Hyperbola;b: Number;a: Number;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b = sqrt(3)*a", "coords": {} }, { "index": 1205, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1205.png", "answer": "x^2/4 + y^2/2 = 1", "fact_expr": "E: Ellipse;b: Number;a: Number;G: Line;P: Point;F2: Point;Q: Point;F1: Point;O:Origin;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(E) = F1;RightFocus(E) = F2;PointOnCurve(O,G);Intersection(G, E) = {P, Q};IsPerpendicular(LineSegmentOf(P, F2), LineSegmentOf(F2, Q));Area(TriangleOf(P,F,Q2))=a^2/2;Abs(LineSegmentOf(F2, Q)) + Abs(LineSegmentOf(P, F2)) = 4", "coords": {} }, { "index": 1206, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1206.png", "answer": "3", "fact_expr": "E: Parabola;A: Point;F: Point;m: Number;Expression(E) = (y^2 = 4*x);Coordinate(A) = (2, m);Focus(E) = F;PointOnCurve(A, E)", "coords": {} }, { "index": 1207, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1208, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1208.png", "answer": "1/2", "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;F2: Point;M: Point;F1: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;UpperVertex(G) = A;MidPoint(LineSegmentOf(A, F2)) = M;IsPerpendicular(LineSegmentOf(M, F1), LineSegmentOf(A, F2))", "coords": {} }, { "index": 1209, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1209.png", "answer": "16/3", "fact_expr": "E: Parabola;Expression(E) = (y^2 = 4*x);F: Point;Focus(E) = F;l: Line;Directrix(E) = l;m: Line;PointOnCurve(F, m);A: Point;B: Point;Intersection(m, E) = {A, B};P: Point;Q: Point;Intersection(PerpendicularBisector(LineSegmentOf(A, F)), l) = P;Intersection(PerpendicularBisector(LineSegmentOf(A, F)), xAxis) = Q;AngleOf(A, F, P) = AngleOf(A, F, Q)", "coords": {} }, { "index": 1210, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1211, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1211.png", "answer": "sqrt(5)/5", "fact_expr": "F1: Point;F2: Point;Focus(C) = {F1, F2};FocalLength(C) = 2;C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;P: Point;PointOnCurve(P, C);l: Line;TangentOnPoint(P, C) = l;Distance(F1, l) * Distance(F2, l) = 4;PointOnCurve(P, l)", "coords": {} }, { "index": 1212, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1213, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1213.png", "answer": "2", "fact_expr": "C: Parabola;p: Number;G: Line;M: Point;A: Point;B: Point;l: Line;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(M) = (1, 0);Directrix(C) = l;PointOnCurve(M, G);Slope(G)=sqrt(3);Intersection(l, G) = A;OneOf(Intersection(G,C))=B;VectorOf(A, M) = VectorOf(M, B)", "coords": { "M": [ 1.0, 0.0 ] } }, { "index": 1214, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_1214.png", "answer": "x^2 - y^2/3 = 1", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Point;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(G) = (sqrt(2), sqrt(3));PointOnCurve(G, C);Eccentricity(C) = 2", "coords": { "G": [ 1.4142135623730951, 1.7320508075688772 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 1215, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1215.png", "answer": "2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;H: Line;PointOnCurve(F, H);Inclination(H) = ApplyUnit(45, degree);A: Point;B: Point;Intersection(H, G) = {A, B};Length(LineSegmentOf(A, B)) = 8", "coords": {} }, { "index": 1216, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1217, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1217.png", "answer": "2*sqrt(3)", "fact_expr": "M: Hyperbola;Expression(M) = (x^2/3 - y^2 = 1);F: Point;RightFocus(M) = F;C: Circle;Expression(C) = (x^2 + (y + 1)^2 = 1/2);A: Point;B: Point;L:Line;TangentOfPoint(F,C)=L;Intersection(L,RightPart(M))={A,B}", "coords": {} }, { "index": 1218, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "y", "x_coef": 9.0, "y_coef": 25.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1218.png", "answer": "(3, 0), (-3, 0)", "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/9 + y^2/25 = 1);PointOnCurve(P, G);F1:Point;F2:Point;Focus(G)={F1,F2};m:Number;Distance(P, F1)*Distance(P,F2) = m;WhenMax(m)", "coords": {} }, { "index": 1219, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1219.png", "answer": "-2", "fact_expr": "l: Line;C: Parabola;G: Circle;A: Point;B: Point;M: Point;k: Number;p:Number;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(M) = ((-1)*2, (-1)*2);Slope(l) = k;PointOnCurve(Focus(C),l);Intersection(l,C)={A,B};IsDiameter(LineSegmentOf(A,B),G);Intersection(Directrix(C),G)=M", "coords": { "M": [ -2.0, -2.0 ] } }, { "index": 1220, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1220.png", "answer": "{(6,6*sqrt(2)),(6,-6*sqrt(2))}", "fact_expr": "C:Parabola;P:Point;Expression(C) = (y^2 = 12*x);PointOnCurve(P,C);Distance(P,Focus(C))=9", "coords": {} }, { "index": 1221, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1222, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1223, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1223.png", "answer": "2/3", "fact_expr": "C: Ellipse;Expression(C) = (x^2/a^2+y^2/b^2 = 1);a: Number;b: Number;a>b;b>0;F: Point;LeftFocus(C) = F;H: Line;PointOnCurve(F, H);Intersection(H, C) = {A, B};l: Line;Inclination(l) = ApplyUnit(60, degree);LineSegmentOf(A, F) = 2*LineSegmentOf(F, B);A: Point;B: Point", "coords": {} }, { "index": 1224, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1224.png", "answer": "4*x+9*y-13=0", "fact_expr": "G: Ellipse;A: Point;B: Point;C: Point;Expression(G) = (x^2/9 + y^2/4 = 1);Coordinate(C) = (1, 1);IsChordOf(LineSegmentOf(A,B), G) = True;MidPoint(LineSegmentOf(A, B)) = C", "coords": { "C": [ 1.0, 1.0 ] } }, { "index": 1225, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1225.png", "answer": "y^2 = 4*x", "fact_expr": "C: Parabola;p: Number;Expression(C) = (y^2 = 2*p*x);p > 0;F: Point;Focus(C) = F;M: Point;x0: Number;x0 > p/2;Coordinate(M) = (x0, 2*sqrt(2));PointOnCurve(M, C);Z: Circle;Center(Z) = M;H: Line;Expression(H) = (x = p/2);E: Point;G: Point;Intersection(Z, H) = {E, G};Sin(AngleOf(M, F, G)) = 1/3", "coords": {} }, { "index": 1226, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1226.png", "answer": "x^2/5 - y^2/20 = 1", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;H: Parabola;Expression(H) = (y^2 = 2*(p*x));p: Number;p>0;Coordinate(Intersection(OneOf(Asymptote(G)), Directrix(H))) = (-4/3, 8/3);M: Point;x0: Number;Coordinate(M) = (x0, 4);OneOf(Intersection(G, H)) = M", "coords": {} }, { "index": 1227, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 6.928203230275509, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1227.png", "answer": "25", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/48 = 1);P: Point;PointOnCurve(P, RightPart(G));C1:Circle;Expression(C1) = ((x + 7)^2 + y^2 = 4);C2:Circle;Expression(C2) = ((x - 7)^2 + y^2 = 1);M: Point;N: Point;l1:Line;l2:Line;TangentPoint(l1,C1)=M;TangentPoint(l2,C2)=N;TangentOfPoint(P,C1)=l1;TangentOfPoint(P,C2)=l2", "coords": {} }, { "index": 1228, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1228.png", "answer": "4*sqrt(2)/3", "fact_expr": "C: Ellipse;Expression(C) = (x^2/2 + y^2 = 1);l: Line;Expression(l) = (x - y + 1 = 0);A: Point;B: Point;Intersection(l, C) = {A, B}", "coords": {} }, { "index": 1229, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1229.png", "answer": "sqrt(2)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;H: Curve;Expression(H) = (y = ln(x) + 1);IsTangent(OneOf(Asymptote(C)),H) = True", "coords": {} }, { "index": 1230, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1230.png", "answer": "2", "fact_expr": "G: Parabola;p: Number;H: Circle;p>0;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (-6*x + x^2 + y^2 - 7 = 0);IsTangent(H, Directrix(G))", "coords": {} }, { "index": 1231, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1231.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, LeftPart(G));LeftFocus(G) = F1;RightFocus(G) = F2;Cos(AngleOf(P,F1,F2))=Sin(AngleOf(P,F2,F1));Sin(AngleOf(P,F2,F1))=sqrt(5)/5", "coords": {} }, { "index": 1232, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1232.png", "answer": "1/3", "fact_expr": "C: Ellipse;b: Number;a: Number;O: Origin;P: Point;F1: Point;F2: Point;G: Point;I: Point;a > b;b > 0;lambda:Number;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);PointOnCurve(G, LineSegmentOf(O, P));Abs(LineSegmentOf(O, P)) = 4*Abs(LineSegmentOf(O, G));Incenter(TriangleOf(P,F1,F2))=I;VectorOf(I, G) = lambda*VectorOf(F1, F2)", "coords": {} }, { "index": 1233, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "y", "x_coef": 2.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1233.png", "answer": "2*sqrt(2)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/2 + y^2/4 = 1)", "coords": {} }, { "index": 1234, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1235, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1235.png", "answer": "2*sqrt(2)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 2)", "coords": {} }, { "index": 1236, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 7.937253933193772, "b": 6.164414002968976, "major_axis": "x", "x_coef": 63.0, "y_coef": 38.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1236.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Ellipse;Expression(G)=(x^2/63+y^2/38=1);H: Circle;Expression(H)=(x^2+y^2-6*x-8*y=0);C:Ellipse;FocalLength(G) = FocalLength(C);PointOnCurve(Center(H),C)", "coords": {} }, { "index": 1237, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1238, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1238.png", "answer": "4", "fact_expr": "C: Hyperbola;F: Point;Expression(C) = (x^2/9 - y^2/16 = 1);RightFocus(C) = F", "coords": {} }, { "index": 1239, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1239.png", "answer": "(x^2/4 + 5*y^2/4 = 1) & Negation(y=0)", "fact_expr": "E: Ellipse;Expression(E) = (x^2/9 + y^2/5 = 1);M: Point;PointOnCurve(M, E);F1: Point;F2: Point;Focus(E) = {F1, F2}", "coords": {} }, { "index": 1240, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1241, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1241.png", "answer": "4", "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;P: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G)=F;PointOnCurve(F, l);Intersection(l, G) = {A, B};l1:Line;l2:Line;TangentOnPoint(A,G)=l1;TangentOnPoint(B,G)=l2;Intersection(l1,l2)=P", "coords": {} }, { "index": 1242, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_1242.png", "answer": "(-2, 0)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Length(RealAxis(C)) = 2;Eccentricity(C) = 2", "coords": {} }, { "index": 1243, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1244, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1244.png", "answer": "(1/8, pm*(sqrt(2)/4))", "fact_expr": "G: Parabola;Expression(G) = (y^2 = x);P:Point;Distance(P,Directrix(G))=Distance(P,Vertex(G));PointOnCurve(P,G)", "coords": {} }, { "index": 1245, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1245.png", "answer": "3", "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;M: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(Focus(G),l);Intersection(l,G) = {A,B};MidPoint(LineSegmentOf(A, B)) = M;Abs(LineSegmentOf(A, B)) = 8", "coords": {} }, { "index": 1246, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1246.png", "answer": "4", "fact_expr": "G: Parabola;C: Circle;M: Point;A: Point;F: Point;Expression(G) = (y^2 = 4*x);Expression(C) = ((x - 4)^2 + (y - 1)^2 = 1);PointOnCurve(M, G);Focus(G) = F;PointOnCurve(A, C)", "coords": {} }, { "index": 1247, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1247.png", "answer": "sqrt(3)", "fact_expr": "C: Curve;P: Point;Q: Point;Expression(C) = (y^2 = 2*x);Coordinate(P) = (2, 0);PointOnCurve(Q, C)", "coords": { "P": [ 2.0, 0.0 ] } }, { "index": 1248, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1248.png", "answer": "8*sqrt(2)/3", "fact_expr": "G: Hyperbola;H: Line;Expression(G) = (x^2 - y^2/4 = 1);Expression(H) = (y = x + 1)", "coords": {} }, { "index": 1249, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1249.png", "answer": "1/2*sqrt(a^2-1)", "fact_expr": "G: Ellipse;Expression(G) = (y^2 + x^2/a^2 = 1);a: Number;a>1;F: Point;OneOf(Focus(G)) = F;P: Point;PointOnCurve(P, G);O: Origin;Abs(LineSegmentOf(O, P)) = Abs(LineSegmentOf(O, F));S: Number;Area((TriangleOf(O, P, F))) = S", "coords": {} }, { "index": 1250, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1251, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1251.png", "answer": "8/3", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 8*y);F: Point;Focus(G) = F;l: Line;Directrix(G) = l;P: Point;PointOnCurve(P, G);A: Point;IsPerpendicular(LineSegmentOf(P, A), l);FootPoint(LineSegmentOf(P, A), l) = A;Inclination(LineOf(A, F)) = ApplyUnit(60, degree)", "coords": {} }, { "index": 1252, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1253, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1253.png", "answer": "2*sqrt(2)/3", "fact_expr": "G: Parabola;A: Point;B: Point;F: Point;P: Point;k: Number;k>0;Expression(G) = (y^2 = 4*x);Intersection(Directrix(G), xAxis) = P;Slope(LineOf(A,B)) = k;Intersection(H,G)={A,B};Focus(G) = F;Abs(LineSegmentOf(F, A)) = 2*Abs(LineSegmentOf(F, B));H:Line;PointOnCurve(P,H);Slope(H)=k", "coords": {} }, { "index": 1254, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1254.png", "answer": "3/5", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P: Point;PointOnCurve(P, G);L: Line;PointOnCurve(P, L);IsPerpendicular(L, Directrix(G));M: Point;FootPoint(L, Directrix(G)) = M;Abs(LineSegmentOf(P, M)) = 5;F: Point;Focus(G) = F", "coords": {} }, { "index": 1255, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1255.png", "answer": "x^2/3 - y^2/9 = 1", "fact_expr": "C: Hyperbola;H:Hyperbola;G: Point;Expression(C) = (x^2/4 - y^2/12 = 1);Coordinate(G) = (3, 3*sqrt(2));Asymptote(C)=Asymptote(H);PointOnCurve(G, H)", "coords": { "G": [ 3.0, 4.242640687119286 ] } }, { "index": 1256, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1256.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/9 + x^2/a^2 = 1);a: Number;a>0;Expression(OneOf(Asymptote(G))) = (3*x - 2*y = 0)", "coords": {} }, { "index": 1257, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1257.png", "answer": "x^2 - y^2 / 3 = 1", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;Inclination(OneOf(Asymptote(C))) = ApplyUnit(60, degree);G: Ellipse;Expression(G) = (x^2/5 + y^2 = 1);FocalLength(C) = FocalLength(G)", "coords": {} }, { "index": 1258, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1258.png", "answer": "2", "fact_expr": "G: Ellipse;Expression(G) = (x^2 + 2*y^2 = 2);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G)", "coords": {} }, { "index": 1259, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1259.png", "answer": "(\\sqrt{3} + \\sqrt{7})/2", "fact_expr": "C: Hyperbola;b: Number;a: Number;O: Origin;O1: Circle;P: Point;A: Point;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Center(O1) = O;Radius(O1) = FocalLength(C);Intersection(O1, xAxis) = {A, B};OneOf(Intersection(O1, C)) = P;Abs(LineSegmentOf(P, A)) = sqrt(3)*Abs(LineSegmentOf(P, B))", "coords": {} }, { "index": 1260, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2135674953460693, "b": 2.4270472526550293, "orientation": "horizontal" }, "optimization": { "final_loss": 4.453341261978494e-06, "converged": true, "iterations": 125 }, "output_path": "results/train/hyperbola/problem_1260.png", "answer": "16", "fact_expr": "C: Hyperbola;b: Number;a: Number;O: Origin;M: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(C)=sqrt(5)/2;LeftFocus(C)=F1;RightFocus(C)=F2;PointOnCurve(M,OneOf(Asymptote(C)));IsPerpendicular(LineSegmentOf(O,M),LineSegmentOf(M,F2));Area(TriangleOf(O,M,F2))=16;F1:Point", "coords": {} }, { "index": 1261, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1262, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1263, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1263.png", "answer": "3/2", "fact_expr": "G: Ellipse;m: Number;m>0;Expression(G) = (y^2 + x^2/m = 1);PointOnCurve(Focus(G), xAxis);FocalLength(G) = sqrt(2)", "coords": {} }, { "index": 1264, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1265, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1265.png", "answer": "-4", "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;F:Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (x1,y1);Coordinate(B) = (x2,y2);Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B};x1:Number;x2:Number;y1:Number;y2:Number", "coords": {} }, { "index": 1266, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1266.png", "answer": "3+2*sqrt(2)", "fact_expr": "P: Point;Coordinate(P) = (-1, 2);G: Parabola;Expression(G) = (y^2 = 4*x);TangentPoint(LineSegmentOf(P, A), G) = A;TangentPoint(LineSegmentOf(P, B), G) = B;A: Point;B: Point;Coordinate(A) = (x1, y1);x1: Number;y1: Number;Coordinate(B) = (x2, y2);x2: Number;y2: Number;y1 >y2;Intersection(LineOf(A,B), xAxis) = C;C: Point;TangentOfPoint(P,G)={LineSegmentOf(P,A),LineSegmentOf(P,B)}", "coords": { "P": [ -1.0, 2.0 ] } }, { "index": 1267, "success": false, "error": "could not convert string to float: 'a'" }, { "index": 1268, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "y", "x_coef": 2.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1268.png", "answer": "sqrt(2)/2", "fact_expr": "G: Ellipse;Expression(G) = (2*x^2 + y^2 = 4)", "coords": {} }, { "index": 1269, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1270, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1270.png", "answer": "-4/9", "fact_expr": "O: Origin;A: Point;M: Point;N: Point;l: Line;C: Ellipse;k: Number;k1:Number;Expression(C) = (x^2/9 + y^2/4 = 1);Slope(l)=k;Intersection(l, C) = {M, N};MidPoint(LineSegmentOf(M, N)) = A;Slope(LineOf(O,A))=k1", "coords": {} }, { "index": 1271, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1271.png", "answer": "2*sqrt(2)", "fact_expr": "E: Ellipse;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(E) = F1;RightFocus(E) = F2;O: Origin;PointOnCurve(O, G);G: Line;Intersection(G, E) = {P, Q};P: Point;Q: Point;IsPerpendicular(LineSegmentOf(P, F2), LineSegmentOf(F2, Q));Area(TriangleOf(P, F2, Q)) = a^2/2;Abs(LineSegmentOf(F2, Q)) + Abs(LineSegmentOf(P, F2)) = 4", "coords": {} }, { "index": 1272, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1272.png", "answer": "1+sqrt(2)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;A: Point;B: Point;F1: Point;F2: Point;l:Line;a>0;b>0;Expression(G) = (-x^2/b^2 + y^2/a^2 = 1);LowerFocus(G) = F1;PointOnCurve(F1,l);IsPerpendicular(l,yAxis);Intersection(l,G)={A,B};IsDiameter(LineSegmentOf(A,B),H);PointOnCurve(F2,H);UpperFocus(G)=F2", "coords": {} }, { "index": 1273, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1274, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1274.png", "answer": "[-3/4,-3/8]", "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2/3 = 1);A1: Point;A2: Point;Endpoint(MajorAxis(C)) = {A1, A2};P: Point;PointOnCurve(P, C) = True;Range(Slope(LineOf(P, A1))) = [1, 2]", "coords": {} }, { "index": 1275, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1275.png", "answer": "1", "fact_expr": "C: Parabola;p: Number;A: Point;B: Point;F: Point;D: Point;d: Number;p>0;Expression(C) = (y^2 = 2*(p*x));PointOnCurve(A, C);PointOnCurve(B, C);Focus(C) = F;AngleOf(A, F, B) = ApplyUnit(60, degree);MidPoint(LineSegmentOf(A, B)) = D;Distance(D, Directrix(C)) = d", "coords": {} }, { "index": 1276, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1276.png", "answer": "2", "fact_expr": "G: Parabola;p: Number;Expression(G) = (y^2 = 2*(p*x));P: Point;XCoordinate(P) = 4;PointOnCurve(P, G);Distance(P, Focus(G)) = 5", "coords": {} }, { "index": 1277, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1277.png", "answer": "sqrt(25)-2*sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2 = 1);P: Point;PointOnCurve(P, RightPart(G));F: Point;RightFocus(G) = F;A: Point;Coordinate(A) = (3, 1)", "coords": { "A": [ 3.0, 1.0 ] } }, { "index": 1278, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1278.png", "answer": "[1/3,1)", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C);PointOnCurve(F2, PerpendicularBisector(LineSegmentOf(P, F1))) = True", "coords": {} }, { "index": 1279, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1279.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;H: Line;Expression(G) = (x^2/4 - y^2/5 = 1);Expression(H) = (sqrt(5)*x - 2*y = 0)", "coords": {} }, { "index": 1280, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1280.png", "answer": "2*sqrt(3)/3", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;M: Point;PointOnCurve(M,Asymptote(C)) ;O: Origin;Abs(LineSegmentOf(O, F2)) = Abs(LineSegmentOf(O, M));Abs(LineSegmentOf(M, F1)) - Abs(LineSegmentOf(M, F2)) = 2*sqrt(2)*a", "coords": {} }, { "index": 1281, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1281.png", "answer": "x^2 - y^2 / 3 = 1", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;P: Point;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 8*x);OneOf(Focus(G))=F;Focus(H)=F;OneOf(Intersection(G,H))=P;Abs(LineSegmentOf(P, F)) = 5", "coords": {} }, { "index": 1282, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1282.png", "answer": "10", "fact_expr": "E: Hyperbola;P: Point;F1: Point;F2: Point;Expression(E) = (x^2/9 - y^2/16 = 1);LeftFocus(E) = F1;RightFocus(E) = F2;PointOnCurve(P, E);Abs(LineSegmentOf(P, F1)) = 4", "coords": {} }, { "index": 1283, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1284, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1285, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1286, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1286.png", "answer": "4\n-16", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);l: Line;A: Point;B: Point;F: Point;Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B};x1: Number;x2: Number;y1: Number;y2: Number", "coords": {} }, { "index": 1287, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1287.png", "answer": "2*sqrt(6)", "fact_expr": "l: Line;G: Parabola;H: Circle;m:Real;A: Point;B: Point;Expression(G) = (y^2 = 8*x);Expression(H) = ((x - 1)^2 + (y - 1)^2 = 6);Expression(l) = (-m + m*x - y + 1 = 0);PointOnCurve(Focus(G),l);Length(InterceptChord(l,H))=LineSegmentOf(A,B)", "coords": {} }, { "index": 1288, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1288.png", "answer": "2*a", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>b;b>0;M: Point;PointOnCurve(M, G);Negation(M=Endpoint(MinorAxis(G)));B1: Point;B2: Point;Endpoint(MinorAxis(G)) = {B1, B2};N: Point;K: Point;Intersection(LineOf(M, B1), xAxis) = N;Intersection(LineOf(M, B2), xAxis) = K;O: Origin", "coords": {} }, { "index": 1289, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1289.png", "answer": "ApplyUnit(30,degree)", "fact_expr": "G: Parabola;N: Point;F: Point;M: Point;Expression(G) = (x^2 = 4*y);Focus(G) = F;Intersection(Directrix(G), yAxis) = M;PointOnCurve(N, G);Abs(LineSegmentOf(N, F)) = (sqrt(3)/2)*Abs(LineSegmentOf(M, N))", "coords": {} }, { "index": 1290, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1291, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1291.png", "answer": "0", "fact_expr": "P: Point;PointOnCurve(P,C1) = True;C1: Ellipse;Expression(C1) = (x^2/4 + y^2/3 = 1);F: Point;LeftFocus(C1) = F;C2: Circle;Expression(C2) = ((x + 2)^2 + (y - 4)^2 = 1);Q: Point;PointOnCurve(Q,C2) = True", "coords": {} }, { "index": 1292, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 2.309742261840055, "b": 1.632524068785969, "orientation": "horizontal" }, "optimization": { "final_loss": 4.273270740250148e-06, "converged": true, "iterations": 84 }, "output_path": "results/train/hyperbola/problem_1292.png", "answer": "x^2/2-y^2=1", "fact_expr": "G: Hyperbola;O: Origin;P: Point;SymmetryAxis(G)=axis;Expression(Asymptote(G)) = (y =pm*(sqrt(2)/2)*x);PointOnCurve(P, G);Coordinate(P)=(2,1);Center(G)=O", "coords": { "P": [ 2.0, 1.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 1293, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1294, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1294.png", "answer": "-3/4", "fact_expr": "C: Parabola;p: Number;A: Point;F: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(A) = (-2, 3);PointOnCurve(A, Directrix(C));Focus(C) = F", "coords": { "A": [ -2.0, 3.0 ] } }, { "index": 1295, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1295.png", "answer": "8/3", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 8*x);F: Point;Focus(C) = F;l: Line;Directrix(C) = l;M: Point;PointOnCurve(M, l);Q: Point;OneOf(Intersection(LineOf(M, F), C)) = Q;VectorOf(F, M) = 3*VectorOf(F, Q)", "coords": {} }, { "index": 1296, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1296.png", "answer": "sqrt(5)/5", "fact_expr": "l: Line;C: Parabola;A: Point;F: Point;B: Point;P: Point;M: Point;N: Point;k:Number;k>0;Expression(C) = (y^2 = 4*x);Coordinate(P) = (-1, 0);Focus(C) = F;PointOnCurve(P, l);Slope(l) = k;Intersection(l, C) = {A, B};Intersection(LineOf(A,F), C) = M;Intersection(LineOf(B,F), C) = N;Abs(LineSegmentOf(A, F))/Abs(LineSegmentOf(F, M)) + Abs(LineSegmentOf(B, F))/Abs(LineSegmentOf(F, N)) = 18", "coords": { "P": [ -1.0, 0.0 ] } }, { "index": 1297, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1298, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1299, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1300, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1300.png", "answer": "3/2", "fact_expr": "G: Parabola;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Length(LineSegmentOf(A,B)) = 5;PointOnCurve(Endpoint(LineSegmentOf(A, B)), G)", "coords": {} }, { "index": 1301, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1301.png", "answer": "pm*1", "fact_expr": "G: Ellipse;H: Line;m: Real;A: Point;B: Point;Expression(G) = (x^2 + 2*y^2 = 2);Expression(H) = (y = m + x);Intersection(G, H) = {A, B};Abs(LineSegmentOf(A, B)) = (4*sqrt(2))/3", "coords": {} }, { "index": 1302, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.1622776601683795, "b": 1.0, "major_axis": "x", "x_coef": 10.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1302.png", "answer": "6*sqrt(2)", "fact_expr": "H: Circle;Expression(H) = (x^2 + (y - 6)^2 = 2);G: Ellipse;Expression(G) = (x^2/10 + y^2 = 1);P: Point;Q: Point;PointOnCurve(P, H);PointOnCurve(Q, G)", "coords": {} }, { "index": 1303, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1303.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;c: Number;Coordinate(F) = (c, 0);Focus(G) = F;A: Point;RightVertex(G) = A;Distance(A,OneOf(Asymptote(G))) = c/2", "coords": {} }, { "index": 1304, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1304.png", "answer": "4*sqrt(3)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l: Line;Directrix(C) = l;P: Point;PointOnCurve(P, C);Quadrant(P) = 1;Z: Line;PointOnCurve(P, Z);IsPerpendicular(Z, l);Q: Point;FootPoint(Z, l) = Q;Perimeter(TriangleOf(P, F, Q)) = 12", "coords": {} }, { "index": 1305, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1306, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1306.png", "answer": "(1,\\sqrt{2})", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Parabola;e: Number;x0:Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y^2 = x);XCoordinate(OneOf(Intersection(OneOf(Asymptote(C)),G)))=x0;Eccentricity(C) = e;x0>1", "coords": {} }, { "index": 1307, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1307.png", "answer": "3", "fact_expr": "P: Point;m: Number;n: Number;Coordinate(P) = (m, n);PointOnCurve(P, G);G: Parabola;Expression(G) = (x^2 = -8*y)", "coords": {} }, { "index": 1308, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1309, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1309.png", "answer": "4", "fact_expr": "E: Parabola;Expression(E) = (y^2 = 4*x);F: Point;Focus(E) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;x1: Number;x2: Number;y1: Number;y2: Number;Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);Intersection(l, E) = {A, B}", "coords": {} }, { "index": 1310, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1311, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1311.png", "answer": "2*sqrt(2-lambda)", "fact_expr": "G: Hyperbola;Expression(Z) = (x^2/2 + y^2/lambda = 1);Z: Curve;Z = G;lambda: Number", "coords": {} }, { "index": 1312, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1312.png", "answer": "sqrt(5)/3", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;A: Point;B: Point;PointOnCurve(A, C);PointOnCurve(B, C);VectorOf(A, F1) = 2*VectorOf(F1, B);DotProduct(VectorOf(A, F2), VectorOf(A,F1)) = 0", "coords": {} }, { "index": 1313, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1313.png", "answer": "x^2/5 - y^2/4 = 1", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;C: Circle;Expression(C) = (-6*x + x^2 + y^2 + 5 = 0);IsTangent(Asymptote(G),C) = True;Center(C) = RightFocus(G)", "coords": {} }, { "index": 1314, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1314.png", "answer": "4", "fact_expr": "G: Ellipse;M: Point;F1: Point;O: Origin;N: Point;Expression(G) = (x^2/25 + y^2/9 = 1);LeftFocus(G)=F1;PointOnCurve(M,G);MidPoint(LineSegmentOf(M, F1)) = N;Abs(LineSegmentOf(O, N)) = 3", "coords": {} }, { "index": 1315, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1315.png", "answer": "9", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 - y^2/9 = 1);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2;Abs(LineSegmentOf(P, F1)) = 5", "coords": {} }, { "index": 1316, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1316.png", "answer": "sqrt(2)/2", "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;P: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(90, degree)", "coords": {} }, { "index": 1317, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1317.png", "answer": "sqrt(2)/2", "fact_expr": "C: Circle;Gamma:Ellipse;Expression(Gamma) = (x^2/a^2+y^2/b^2=1);Expression(C) = ((x - 1)^2 + (y - 1)^2 = 2);a:Number;b:Number;a>b;b>0;F:Point;B:Point;RightFocus(Gamma)=F;UpperVertex(Gamma)=B;PointOnCurve(F,C);PointOnCurve(B,C)", "coords": {} }, { "index": 1318, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1319, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1320, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 3.872983346207417, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1320.png", "answer": "6", "fact_expr": "G: Hyperbola;H: Circle;C:Circle;P: Point;M: Point;N: Point;Expression(G) = (x^2 - y^2/15 = 1);Expression(H) = (y^2 + (x + 4)^2 = 9);Expression(C)=(y^2 + (x - 4)^2 = 1);PointOnCurve(P, RightPart(G));PointOnCurve(M, H);PointOnCurve(N, C)", "coords": {} }, { "index": 1321, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1321.png", "answer": "(pm*2,1)", "fact_expr": "G: Parabola;P: Point;Expression(G) = (x^2 = 4*y);PointOnCurve(P, G);Distance(P, Focus(G)) = 2", "coords": {} }, { "index": 1322, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1322.png", "answer": "x^2/4+y^2/3=1", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;RightFocus(C)=F;Coordinate(F)=(1,0);B: Point;UpperVertex(C)=B;Slope(LineOf(F,B)) = -sqrt(3)", "coords": { "F": [ 1.0, 0.0 ] } }, { "index": 1323, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1323.png", "answer": "x^2/9+y^2/7=1", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;Focus(C) = {F1, F2};H: Line;PointOnCurve(F1, H);A: Point;B: Point;Intersection(H, C) = {A, B};Perimeter(TriangleOf(A, B, F2)) = 12;Eccentricity(C) = sqrt(2)/3", "coords": {} }, { "index": 1324, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 0.9220545291900635, "b": 2.765991687774658, "orientation": "horizontal" }, "optimization": { "final_loss": 4.6528161874448415e-06, "converged": true, "iterations": 186 }, "output_path": "results/train/hyperbola/problem_1324.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;Expression(OneOf(Asymptote(G))) = (y = 3*x)", "coords": {} }, { "index": 1325, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1326, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1326.png", "answer": "2", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);Intersection(Directrix(C),xAxis)=M;A: Point;M: Point;B: Point;F:Point;Focus(C)=F;G:Line;Intersection(G, C) = {A, B};PointOnCurve(F, G);Slope(LineOf(M,A))=k1;Slope(LineOf(M,B))=k2;k1:Number;k2:Number", "coords": {} }, { "index": 1327, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1328, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1328.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;b: Number;a: Number;F: Point;A: Point;B: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;IsPerpendicular(LineSegmentOf(F, A), LineSegmentOf(F, B));L1:Line;L2:Line;Asymptote(G)={L1,L2};Intersection(RightDirectrix(G),L1)=A;Intersection(RightDirectrix(G),L2)=B", "coords": {} }, { "index": 1329, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1329.png", "answer": "2", "fact_expr": "G: Parabola;p: Number;P: Point;K: Point;F: Point;l: Line;p>0;y0:Number;Expression(G) = (x^2 = 2*p*y);Coordinate(P) = (4,y0);Focus(G) = F;Directrix(G) = l;PointOnCurve(P, G);Intersection(l,yAxis) = K;Abs(LineSegmentOf(P, K)) = sqrt(2)*Abs(LineSegmentOf(P, F))", "coords": {} }, { "index": 1330, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1330.png", "answer": "2", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;A: Point;F2: Point;B: Point;F1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, G);Intersection(G,LeftPart(C))={A,B};Sin(AngleOf(A, B, F2)/2) = 1/4;Abs(LineSegmentOf(A, B)) = Abs(LineSegmentOf(B, F2))", "coords": {} }, { "index": 1331, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1332, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1332.png", "answer": "7/5", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;P: Point;F2: Point;Q: Point;F1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, G);Intersection(G, LeftPart(C)) = {P, Q};Abs(LineSegmentOf(P, F2)) = 2*c;c:Number;c=sqrt(a^2+b^2);Area(TriangleOf(P,F1,F2))/Area(TriangleOf(Q,F1,F2)) = 2/3", "coords": {} }, { "index": 1333, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1334, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 3.872983346207417, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1334.png", "answer": "6", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/15 = 1);P: Point;PointOnCurve(P, G);F1: Point;F2: Point;OneOf(Focus(G)) = F1;OneOf(Focus(G)) = F2;Negation(F1=F2);Distance(P, F1) = 4", "coords": {} }, { "index": 1335, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 8.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1335.png", "answer": "1/4", "fact_expr": "H: Ellipse;Expression(H) = (x^2/8 + y^2/2 = 1);G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;A: Point;B: Point;Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 2", "coords": {} }, { "index": 1336, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1336.png", "answer": "[1,+oo)&\\Negation(a=2)", "fact_expr": "G: Ellipse;a: Number;H: Line;m: Number;Expression(G) = (y^2/2 + x^2/a = 1);Expression(H) = (x = m*y + 1);IsIntersect(H, G)", "coords": {} }, { "index": 1337, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1337.png", "answer": "pi/2", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(P,G);Abs(VectorOf(P, F1) + VectorOf(P, F2)) = 2*sqrt(3)", "coords": {} }, { "index": 1338, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1338.png", "answer": "5/2", "fact_expr": "C: Hyperbola;O: Origin;P: Point;F: Point;Expression(C) = (x^2/4 - y^2/5 = 1);OneOf(Focus(C)) = F;PointOnCurve(P, C);Abs(LineSegmentOf(O, P)) = Abs(LineSegmentOf(O, F))", "coords": {} }, { "index": 1339, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.23606797749979, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1339.png", "answer": "3", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/5 - y^2/4 = 1);F: Point;OneOf(Focus(C)) = F;P: Point;PointOnCurve(P, Asymptote(C));O: Origin;Abs(LineSegmentOf(O, P)) = Abs(LineSegmentOf(O, F))", "coords": {} }, { "index": 1340, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1341, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1341.png", "answer": "sqrt(6)", "fact_expr": "G: Parabola;B: Point;A: Point;F: Point;Expression(G) = (y^2 = 2*x);Coordinate(B) = (-1, 0);Focus(G) = F;PointOnCurve(A, G);WhenMax(2*Abs(LineSegmentOf(A, B))/(2*Abs(LineSegmentOf(A, F)) + 1))", "coords": { "B": [ -1.0, 0.0 ] } }, { "index": 1342, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/train/hyperbola/problem_1342.png", "answer": "x^2/4-y^2=1", "fact_expr": "C1: Hyperbola;H: Point;Expression(C1)=(-y^2/b^2 + x^2/a^2 = 1);Coordinate(H) = (4, sqrt(3));Expression(Asymptote(C1)) = (y = pm*(x/2));PointOnCurve(H,C1);a: Number;b: Number;a>0;b>0", "coords": { "H": [ 4.0, 1.7320508075688772 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 1343, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1343.png", "answer": "sqrt(2)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(C) = F;A1: Point;A2: Point;LeftVertex(C) = A1;RightVertex(C) = A2;PointOnCurve(F, l) = True;L1: Line;L2: Line;OneOf(Asymptote(C)) = L1;OneOf(Asymptote(C)) = L2;Negation(L1 = L2);IsParallel(l, L1) = True;l: Line;Intersection(l, L2) = P;P: Point;G: Circle;PointOnCurve(P, G) = True;IsDiameter(LineSegmentOf(A1, A2), G) = True", "coords": {} }, { "index": 1344, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1344.png", "answer": "y=pm*sqrt(3)*x", "fact_expr": "G: Hyperbola;b: Number;a: Number;O: Circle;A: Point;C: Point;B: Point;F: Point;l1: Line;l2: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(O) = (x^2 + y^2 = a^2);LeftFocus(G) = F;TangentOfPoint(F, O) = {l1, l2};TangentPoint(l1, O) = A;TangentPoint(l2, O) = B;LeftVertex(G) = C;AngleOf(A, C, B) = ApplyUnit(120, degree)", "coords": {} }, { "index": 1345, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1345.png", "answer": "9", "fact_expr": "F1: Point;F2: Point;C: Ellipse;Expression(C) = (x^2/25 + y^2/9 = 1);Focus(C) = {F1, F2};P: Point;PointOnCurve(P, C);AngleOf(F1, P, F2) = ApplyUnit(90, degree)", "coords": {} }, { "index": 1346, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.449489742783178, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1346.png", "answer": "5 + sqrt(6)", "fact_expr": "G: Hyperbola;H: Ellipse;P: Point;Expression(G) = (x^2/6 - y^2/3 = 1);Expression(H) = (x^2/25 + y^2/16 = 1);Quadrant(P) = 1;Intersection(G, H) = P", "coords": {} }, { "index": 1347, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1347.png", "answer": "{2, 8}", "fact_expr": "G: Ellipse;m: Number;e: Number;Expression(G) = (y^2/4 + x^2/m = 1);Eccentricity(G) = e ;e = sqrt(2)/2", "coords": {} }, { "index": 1348, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 2.0, "major_axis": "x", "x_coef": 8.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1348.png", "answer": "x = pm*4", "fact_expr": "G: Ellipse;Expression(G) = (x^2/8 + y^2/4 = 1)", "coords": {} }, { "index": 1349, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 0.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1349.png", "answer": "pm*1/2", "fact_expr": "E: Hyperbola;F: Line;k: Number;Expression(E) = (x^2 - 4*y^2 = 1);Expression(F) = (y = k*x);F = OneOf(Asymptote(E))", "coords": {} }, { "index": 1350, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1350.png", "answer": "16", "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (x^2/4 - y^2/m = 1);Length(ImageinaryAxis(G)) = 2*Length(RealAxis(G))", "coords": {} }, { "index": 1351, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1351.png", "answer": "8/3", "fact_expr": "G: Ellipse;a: Number;M: Point;A: Point;B: Point;N: Point;F1: Point;F2: Point;a>1;Expression(G) = (y^2 + x^2/a^2 = 1);LeftFocus(G) =F1;RightFocus(G)=F2;LeftVertex(G)=A;UpperVertex(G)=B;PointOnCurve(M, G);PointOnCurve(N, G);Max(Area(TriangleOf(M,A,B)))=2*(sqrt(2)+1)", "coords": {} }, { "index": 1352, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 4.898979485566356, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1352.png", "answer": "24", "fact_expr": "F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;G: Hyperbola;Expression(G) = (x^2 - y^2/24 = 1);P: Point;H: Ellipse;Expression(H) = (x^2/49 + y^2/24 = 1);OneOf(Intersection(G, H)) = P", "coords": {} }, { "index": 1353, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1353.png", "answer": "(0, -1/2)", "fact_expr": "G: Parabola;Expression(G) = (x^2 = -2*y)", "coords": {} }, { "index": 1354, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1354.png", "answer": "x^2/4 - y^2/5 = 1", "fact_expr": "P: Point;F1: Point;G: Hyperbola;b: Number;a: Number;M: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(M) = (3/2, 0);PointOnCurve(P,G);Focus(G)={F1,F2};Distance(P,F1)=6;Distance(P,F2)=2;Distance(M,LineOf(P,F1))=Distance(M,LineOf(P,F2))", "coords": { "M": [ 1.5, 0.0 ] } }, { "index": 1355, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.449489742783178, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 6.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1355.png", "answer": "8", "fact_expr": "G: Parabola;m: Real;H: Ellipse;Expression(G) = (y^2 = m*x);Expression(H) = (x^2/6 + y^2/2 = 1);Focus(G) = RightFocus(H)", "coords": {} }, { "index": 1356, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1356.png", "answer": "1", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;O: Origin;Expression(G) = (x^2/4 + y^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);DotProduct((VectorOf(O,P)+VectorOf(O,F2)),VectorOf(P,F2))=0", "coords": {} }, { "index": 1357, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1357.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, LeftPart(G));M: Point;MidPoint(LineSegmentOf(P, F1)) = M;O: Origin;IsPerpendicular(LineSegmentOf(O, M), LineSegmentOf(P, F1)) = True;2*Abs(LineSegmentOf(P, F1)) = Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 1358, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1358.png", "answer": "(1,2*sqrt(3)/3)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 + (x - 2)^2 = 1);IsIntersect(Asymptote(G),H)", "coords": {} }, { "index": 1359, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1359.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1);P: Point;PointOnCurve(P, G);F1: Point;F2: Point;Focus(G) = {F1, F2};AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 1360, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1360.png", "answer": "(2,0)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x)", "coords": {} }, { "index": 1361, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1361.png", "answer": "(3, 9)", "fact_expr": "G: Parabola;Expression(G) = (y = x^2);Expression(OneOf(TangentOnPoint(P, G))) = (6*x - y - 9 = 0);P: Point", "coords": {} }, { "index": 1362, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1362.png", "answer": "-2", "fact_expr": "G: Ellipse;A: Point;P: Point;F: Point;Expression(G) = (x^2/4 + y^2/3 = 1);Coordinate(A) = (1, 2);LeftFocus(G) = F;PointOnCurve(P, G)", "coords": { "A": [ 1.0, 2.0 ] } }, { "index": 1363, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1363.png", "answer": "y^2/8 - x^2/2 = 1", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;a>0;b>0;Expression(G) = (-x^2/b^2 + y^2/a^2 = 1);Expression(H) = (x^2 = 4*(sqrt(10)*y));Expression(OneOf(Asymptote(G))) = (y = -2*x);OneOf(Focus(G)) = Focus(H)", "coords": {} }, { "index": 1364, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1364.png", "answer": "3/2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/5 = 1)", "coords": {} }, { "index": 1365, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 2.0, 0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_1365.png", "answer": "y^2 = -8*x", "fact_expr": "l: Line;F: Circle;E: Circle;Expression(F) = (y^2 + (x + 2)^2 = 1);IsOutTangent(E,F);IsTangent(E, l);M: Point;Center(E) = M;Expression(l)=(x=1)", "coords": {} }, { "index": 1366, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1366.png", "answer": "y=-2*x", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);l: Line;A: Point;B: Point;Intersection(l, C) = {A, B};Coordinate(MidPoint(LineSegmentOf(A, B))) = (1/2, -1)", "coords": {} }, { "index": 1367, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1367.png", "answer": "2-sqrt(6)/2", "fact_expr": "G: Ellipse;H: Line;P: Point;Q: Point;x1:Number;x2:Number;y1:Number;y2:Number;Expression(G) = (x^2/2 + y^2 = 1);Expression(H) = (x + 4*y - 4 = 0);Coordinate(P) = (x1, y1);Coordinate(Q) = (x2, y2);PointOnCurve(P, G);PointOnCurve(Q, H)", "coords": {} }, { "index": 1368, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1369, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1369.png", "answer": "8/5", "fact_expr": "G: Hyperbola;P: Point;Expression(G) = (x^2/9 - y^2/16 = 1);Coordinate(P) = (2, 0)", "coords": { "P": [ 2.0, 0.0 ] } }, { "index": 1370, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1371, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1371.png", "answer": "-3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;H: Line;PointOnCurve(F, H);A: Point;B: Point;Intersection(H, G) = {A, B};O: Origin", "coords": {} }, { "index": 1372, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1372.png", "answer": "2", "fact_expr": "C: Parabola;Expression(C) = (x^2 = 4*y);l: Line;Directrix(C) = l;P: Point;PointOnCurve(P, l);A: Point;B: Point;TangentOfPoint(P,C)={LineOf(P,A),LineOf(P,B)};TangentPoint(LineOf(P,A),C)=A;TangentPoint(LineOf(P,B),C)=B;Q: Point;IsChordOf(LineSegmentOf(A, B), C);MidPoint(LineSegmentOf(A, B)) = Q", "coords": {} }, { "index": 1373, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1374, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1374.png", "answer": "2/3", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/4 = 1);M: Point;PointOnCurve(M, G);H: Circle;Expression(H) = (x^2 + y^2 = 2);L1: Line;L2: Line;TangentOfPoint(M, H) = {L1, L2};A: Point;B: Point;TangentPoint(L1, H) = A;TangentPoint(L2, H) = B;L: Line;PointOnCurve(A, L);PointOnCurve(B, L);P: Point;Q: Point;Intersection(L, xAxis) = P;Intersection(L, yAxis) = Q;O: Origin", "coords": {} }, { "index": 1375, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1375.png", "answer": "2", "fact_expr": "C: Parabola;G:Circle;H: Line;P: Point;M: Point;Q: Point;N: Point;F: Point;Expression(C) = (y^2 = 4*x);Expression(G) = (-2*x + x^2 + y^2 = 0);Focus(C) = F;PointOnCurve(F, H);Intersection(H, C) = {P, Q};Quadrant(P)=1;Quadrant(M)=1;Intersection(H,G)={M,N}", "coords": {} }, { "index": 1376, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1376.png", "answer": "sqrt(2)-1", "fact_expr": "G: Ellipse;b: Number;a: Number;M: Point;F1: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Focus(G) = {F1, F2};PointOnCurve(M,G);IsPerpendicular(LineSegmentOf(M,F1),xAxis);AngleOf(F1,M,F2)=ApplyUnit(45,degree)", "coords": {} }, { "index": 1377, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1377.png", "answer": "sqrt(3)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*x);F: Point;Focus(C) = F;l: Line;PointOnCurve(F,l);IsPerpendicular(l,xAxis);B: Point;A: Point;Intersection(l, C) = {A, B};G: Circle;IsDiameter(LineSegmentOf(A,B),G)", "coords": {} }, { "index": 1378, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1378.png", "answer": "y=pm*(3/2)*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/9 = 1)", "coords": {} }, { "index": 1379, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1380, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1380.png", "answer": "y^2 = -36*(x - 4)", "fact_expr": "G: Hyperbola;H: Parabola;Expression(G) = (x^2/16 - y^2/9 = 1);Vertex(H) = RightVertex(G);Focus(H) = LeftFocus(G)", "coords": {} }, { "index": 1381, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1381.png", "answer": "sqrt(3)/2", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1)", "coords": {} }, { "index": 1382, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 2.0987446308135986, "b": 1.8757982470995966, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 7.841294542546938e-06, "converged": true, "iterations": 109 }, "output_path": "results/train/ellipse/problem_1382.png", "answer": "9/2", "fact_expr": "C: Ellipse;b: Number;a: Number;Q: Point;F1: Point;F2: Point;P: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(Q) = (2, sqrt(2));LeftFocus(C) = F1;RightFocus(C) = F2;Abs(LineSegmentOf(F1, F2)) = 4;PointOnCurve(Q, C);PointOnCurve(P, C)", "coords": { "Q": [ 2.0, 1.4142135623730951 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 1383, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1383.png", "answer": "sqrt(3)/3", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P, F2), LineSegmentOf(F1, F2));AngleOf(P, F1, F2) = ApplyUnit(30, degree)", "coords": {} }, { "index": 1384, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1384.png", "answer": "sqrt(3)/3", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;L: Line;PointOnCurve(F2, L) = True;IsPerpendicular(L, xAxis) = True;Intersection(L, C) = {A, B};A: Point;B: Point;Intersection(LineSegmentOf(F1, A), yAxis) = D;D: Point;IsPerpendicular(LineSegmentOf(B, D), LineSegmentOf(F1, A)) = True", "coords": {} }, { "index": 1385, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1385.png", "answer": "x^2/3 - y^2/12 = 1", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/4 = 1);C: Hyperbola;Asymptote(C) = Asymptote(G);M: Point;Coordinate(M) = (2, 2);PointOnCurve(M, C)", "coords": { "M": [ 2.0, 2.0 ] } }, { "index": 1386, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1386.png", "answer": "(\\sqrt{2}, 2)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;G: Line;Expression(G) = (y = x);IsIntersect(C, G);P: Point;PointOnCurve(P, C);Negation(P=Vertex(C));AngleOf(P, F2, F1) = 3*AngleOf(P, F1, F2)", "coords": {} }, { "index": 1387, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1387.png", "answer": "{0, -1/2}", "fact_expr": "G: Parabola;H: Line;k: Real;Expression(G) = (y^2 = 8*x);Expression(H) = (y = k*x - 4);NumIntersection(H, G) = 1", "coords": {} }, { "index": 1388, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1388.png", "answer": "2*sqrt(3)/3", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/2 - y^2/4 = 1)", "coords": {} }, { "index": 1389, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1389.png", "answer": "1/2", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;l: Line;O: Circle;Expression(O) = (x^2 + y^2 = b^2);IsTangent(l, O);Inclination(l) = pi/3;PointOnCurve(RightVertex(C), l)", "coords": {} }, { "index": 1390, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1390.png", "answer": "16/3", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/9 + y^2/4 = 1);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2;AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 1391, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1391.png", "answer": "3", "fact_expr": "C: Parabola;p: Number;B: Point;P: Point;A: Point;E: Point;F: Point;l: Line;l1:Line;p>0;Expression(C) = (y^2 = 2*p*x);Focus(C) = F;Directrix(C) = l;PointOnCurve(A,C);PointOnCurve(A,l1);IsPerpendicular(l,l1);FootPoint(l,l1)=B;Coordinate(P)=(9*p/2,0);Intersection(LineSegmentOf(A,F),LineSegmentOf(B,P))=E;Abs(LineSegmentOf(P, F)) = 2*Abs(LineSegmentOf(A, F));Area(TriangleOf(P, A, E)) = 6*sqrt(3)", "coords": {} }, { "index": 1392, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1392.png", "answer": "5", "fact_expr": "G: Parabola;C: Circle;M: Point;A: Point;F: Point;Expression(G) = (x^2 = 4*y);Expression(C) = ((x + 1)^2 + (y - 5)^2 = 1);PointOnCurve(M, G);PointOnCurve(A, C);Focus(G)=F", "coords": {} }, { "index": 1393, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1393.png", "answer": "2*sqrt(3)", "fact_expr": "G: Parabola;M: Point;N: Point;F: Point;Expression(G) = (y^2 = 8*x);Focus(G) = F;Intersection(Directrix(G),xAxis)=M;PointOnCurve(N,G);2*Abs(LineSegmentOf(N, F)) = Abs(LineSegmentOf(M, N))", "coords": {} }, { "index": 1394, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1394.png", "answer": "9/4", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 + y^2/3 = 1);PointOnCurve(P, G);Focus(G) = {F1, F2};Radius(InscribedCircle(TriangleOf(F1, P, F2))) = 1/2", "coords": {} }, { "index": 1395, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1395.png", "answer": "12", "fact_expr": "l: Line;G: Parabola;p: Number;H: Circle;p>0;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (y^2 + (x - 2)^2 = 4);Slope(l) = sqrt(3)/3;PointOnCurve(Focus(G), l);IsTangent(l, H)", "coords": {} }, { "index": 1396, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1396.png", "answer": "pm*4", "fact_expr": "G: Parabola;p: Number;Q: Point;m: Real;Expression(G) = (y^2 = 2*p*x);Coordinate(Q) = (1, m);PointOnCurve(Q, G);Distance(Q,Focus(G))=5", "coords": {} }, { "index": 1397, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1397.png", "answer": "x=-1", "fact_expr": "l: Line;C: Parabola;Expression(C) = (y^2 = 4*x);PointOnCurve(Focus(C), l);A: Point;B: Point;Intersection(l, C) = {A, B};Z1: Line;Z2: Line;TangentOfPoint(A, C) = Z1;TangentOfPoint(B, C) = Z2;P: Point;Intersection(Z1, Z2) = P", "coords": {} }, { "index": 1398, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1399, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1399.png", "answer": "x^2/9 - y^2/4 = 1", "fact_expr": "E: Hyperbola;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;LeftFocus(E) = F;L: Line;H: Point;PointOnCurve(F, L);IsPerpendicular(OneOf(Asymptote(E)), L);FootPoint(OneOf(Asymptote(E)), L) = H;O: Origin;Abs(LineSegmentOf(F, H)) = 2;Area(TriangleOf(F, O, H)) = 3", "coords": {} }, { "index": 1400, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1401, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.9959803819656372, "direction": "up" }, "optimization": { "final_loss": 3.6780886603082763e-07, "converged": true, "iterations": 56 }, "output_path": "results/train/parabola/problem_1401.png", "answer": "5/4", "fact_expr": "G: Parabola;a: Number;A: Point;Expression(G) = (x^2 = a*y);Coordinate(A) = (1, 1/4);PointOnCurve(A, G)", "coords": { "A": [ 1.0, 0.25 ] } }, { "index": 1402, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1403, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "major_axis": "x", "x_coef": 3.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1403.png", "answer": "5*sqrt(2)/2", "fact_expr": "C: Ellipse;G: Line;P: Point;Expression(C) = (x^2/3 + y^2 = 1);Expression(G) = (x + y - 3 = 0);PointOnCurve(P, C)", "coords": {} }, { "index": 1404, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0009275674819946, "direction": "right" }, "optimization": { "final_loss": 1.0271783139614854e-06, "converged": true, "iterations": 82 }, "output_path": "results/train/parabola/problem_1404.png", "answer": "3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;P: Point;Coordinate(P) = (2, 2*sqrt(2));PointOnCurve(P, G)", "coords": { "P": [ 2.0, 2.8284271247461903 ] } }, { "index": 1405, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0010501146316528, "direction": "right" }, "optimization": { "final_loss": 3.455598402979376e-07, "converged": true, "iterations": 81 }, "output_path": "results/train/parabola/problem_1405.png", "answer": "2+2*sqrt(2)", "fact_expr": "H:Ellipse;G:Parabola;M:Point;Coordinate(M)=(1,2);PointOnCurve(M,H);PointOnCurve(M,G);Focus(G)=Focus(H);PointOnCurve(Focus(G),xAxis);PointOnCurve(Focus(H),xAxis);SymmetryAxis(H)=axis;O:Origin;Vertex(G)=O", "coords": { "M": [ 1.0, 2.0 ] } }, { "index": 1406, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_1406.png", "answer": "x^2/4+y^2=1", "fact_expr": "O: Origin;O1: Circle;Expression(O1) = (x^2 + y^2 = 4);O2: Circle;Expression(O2) = (x^2 + y^2 = 1);A: Point;PointOnCurve(A, O1);B: Point;Intersection(LineSegmentOf(O, A), O2) = B;L: Line;IsPerpendicular(L, xAxis);PointOnCurve(A, L);C: Point;Intersection(L, xAxis) = C;D: Point;L1: Line;IsPerpendicular(L1, LineSegmentOf(A, C));PointOnCurve(B, L1);Intersection(L1, LineSegmentOf(A, C)) = D", "coords": {} }, { "index": 1407, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1408, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1408.png", "answer": "x^2/9+y^2/5=1", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(G) = (x^2/4 - y^2/5 = 1);Focus(G)=Vertex(H);Vertex(G)=Focus(H)", "coords": {} }, { "index": 1409, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1409.png", "answer": "4", "fact_expr": "C: Parabola;p: Number;A: Point;D: Point;B: Point;F: Point;l: Line;x0:Number;p>0;x0>0;Expression(C) = (y^2 = 2*p*x);Coordinate(A) = (1, 2);Coordinate(D) = (x0, 0);Focus(C)=F;Directrix(C)=l;Distance(F, l) = 2;PointOnCurve(B, C);AngleOf(A,B,D)=AngleOf(A,D,B);AngleOf(A,D,B)=ApplyUnit(45,degree)", "coords": { "A": [ 1.0, 2.0 ] } }, { "index": 1410, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1410.png", "answer": "sqrt(3)/2", "fact_expr": "P: Point;B1: Point;G: Ellipse;b: Number;a: Number;B2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LowerVertex(G)=B1;UpperVertex(G)=B2;PointOnCurve(P, G);Slope(LineOf(P,B1))=1/4;Slope(LineOf(P,B2))=-1", "coords": {} }, { "index": 1411, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1411.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 2)", "coords": {} }, { "index": 1412, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1412.png", "answer": "1/2", "fact_expr": "C: Ellipse;a: Number;b: Number;B: Point;F1: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C)=F1;RightFocus(C)=F2;UpperVertex(C)=B;Radius(CircumCircle(TriangleOf(B,F1,F2)))=(2*b)/3", "coords": {} }, { "index": 1413, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1413.png", "answer": "-1/4", "fact_expr": "E: Ellipse;b: Number;a: Number;A: Point;P: Point;B: Point;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (-a, 0);Coordinate(B) = (a, 0);PointOnCurve(P, E);Negation(P=A);Negation(P=B);Eccentricity(E) = sqrt(3)/2", "coords": {} }, { "index": 1414, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1414.png", "answer": "pm*1/2", "fact_expr": "P: Point;C: Ellipse;Expression(C) = (x^2/4 + y^2 = 1);UpperVertex(C) = P;O: Origin;PointOnCurve(O, l) = True;l: Line;Intersection(l, C) = {A, B};A: Point;B: Point;Area(TriangleOf(P, A, B)) = sqrt(2)", "coords": {} }, { "index": 1415, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1415.png", "answer": "5/3", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;e: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, RightPart(G));Abs(LineSegmentOf(P, F1)) = 4*Abs(LineSegmentOf(P, F2));Eccentricity(G) = e", "coords": {} }, { "index": 1416, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1416.png", "answer": "20", "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;Expression(G) = (x^2 = -4*y);Expression(l) = (2*x - y - 1 = 0);Intersection(l, G) = {A, B}", "coords": {} }, { "index": 1417, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1418, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1419, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.3484079837799072, "b": 1.8638464212417603, "orientation": "horizontal" }, "optimization": { "final_loss": 4.620347226591548e-06, "converged": true, "iterations": 88 }, "output_path": "results/train/hyperbola/problem_1419.png", "answer": "12/5", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;Coordinate(F1) = (-3, 0);Coordinate(F2) = (3, 0);LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, Asymptote(C));Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))", "coords": { "F1": [ -3.0, 0.0 ], "F2": [ 3.0, 0.0 ] } }, { "index": 1420, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1420.png", "answer": "sqrt(2)/4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x);M: Point;PointOnCurve(M, G) = True;F: Point;Focus(G) = F;Distance(M, F) = 3/2;O: Origin", "coords": {} }, { "index": 1421, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1422, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/train/hyperbola/problem_1422.png", "answer": "2", "fact_expr": "P: Point;Coordinate(P) = (2, 3);PointOnCurve(P, C);C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;FocalLength(C) = 4", "coords": { "P": [ 2.0, 3.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 1423, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1423.png", "answer": "{(4*sqrt(5)/5)*b,(2*sqrt(5)/5)*a}", "fact_expr": "G: Ellipse;b: Number;a: Number;Q: Point;M: Point;P: Point;A: Point;B: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (0,b);Coordinate(Q) = (0,-b);Endpoint(MinorAxis(G))={P,Q};OneOf(Endpoint(MajorAxis(G)))=M;IsChordOf(LineSegmentOf(A,B),G);PointOnCurve(Center(G), LineSegmentOf(A,B));Negation(OverlappingLine(LineSegmentOf(A,B),axis));Slope(LineSegmentOf(P,A))*Slope(LineSegmentOf(P,B))=-1/4", "coords": {} }, { "index": 1424, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1424.png", "answer": "y^2 - x^2/3 = 1", "fact_expr": "G: Hyperbola;H: Parabola;O: Origin;Expression(H) = (y = x^2/8);Center(G) = O;OneOf(Focus(G)) = Focus(H);Eccentricity(G)=2", "coords": {} }, { "index": 1425, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1425.png", "answer": "sqrt(2)/2", "fact_expr": "l: Line;G: Ellipse;P: Point;Q: Point;M: Point;Expression(G) = (x^2 + y^2/2 = 1);PointOnCurve(Focus(G), l);Intersection(l, G) = {P, Q};Intersection(PerpendicularBisector(LineSegmentOf(P,Q)),xAxis) = M", "coords": {} }, { "index": 1426, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1426.png", "answer": "y^2 = 4*x", "fact_expr": "G: Parabola;p: Number;A: Point;F: Point;B: Point;O: Origin;Z: Line;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;PointOnCurve(F, Z);Intersection(Z, G) = {A, B};VectorOf(A, F) = 2*VectorOf(F, B);Area(TriangleOf(O, A, B)) = Abs(LineSegmentOf(A, B))*sqrt(2)/3", "coords": {} }, { "index": 1427, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 0.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1427.png", "answer": "{y=(1/2)*x - 1,y=-(1/2)*x+1}", "fact_expr": "l: Line;C: Hyperbola;P: Point;Expression(C) = (x^2 - 4*y^2 = 1);Coordinate(P) = (2, 0);PointOnCurve(P,l);NumIntersection(l,C)=1", "coords": { "P": [ 2.0, 0.0 ] } }, { "index": 1428, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1428.png", "answer": "64", "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (x^2/36 - y^2/m = 1);e:Number;Eccentricity(G)=e;e=5/3", "coords": {} }, { "index": 1429, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 0.929953396320343, "b": 2.629857301712036, "orientation": "horizontal" }, "optimization": { "final_loss": 4.594453912432073e-06, "converged": true, "iterations": 175 }, "output_path": "results/train/hyperbola/problem_1429.png", "answer": "1/3", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Eccentricity(C) = 3;G: Parabola;Expression(G) = (y = x^2/4)", "coords": {} }, { "index": 1430, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1430.png", "answer": "5/4", "fact_expr": "C: Parabola;G: Line;A: Point;B: Point;Expression(C) = (y^2 = x);PointOnCurve(Focus(C), G);Intersection(G, C) = {A, B};IsChordOf(LineSegmentOf(A, B), C);Length(LineSegmentOf(A, B)) = 2", "coords": {} }, { "index": 1431, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1431.png", "answer": "sqrt(a*b)", "fact_expr": "l: Line;G: Parabola;p: Number;P: Point;R: Point;Q: Point;S: Point;F: Point;M: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;PointOnCurve(F, l);Intersection(l,G)={P,Q};L1:Line;L2:Line;PointOnCurve(P,LineSegmentOf(P,R));PointOnCurve(Q,LineSegmentOf(Q,S));Directrix(G)={L1,L2};IsPerpendicular(L1,LineSegmentOf(P,R));IsPerpendicular(L2,LineSegmentOf(Q,S));FootPoint(L1,LineSegmentOf(P,R))=R;FootPoint(L2,LineSegmentOf(Q,S))=S;Abs(LineSegmentOf(P,F))=a;Abs(LineSegmentOf(Q, F)) = b;MidPoint(LineSegmentOf(R, S)) = M;a:Number;b:Number", "coords": {} }, { "index": 1432, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1432.png", "answer": "8", "fact_expr": "G: Ellipse;H: Line;F2: Point;A: Point;B: Point;F1: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(G) = {F1, F2};PointOnCurve(F1, H);Intersection(H, G) = {A,B};Abs(LineSegmentOf(F2, A)) + Abs(LineSegmentOf(F2, B)) = 12", "coords": {} }, { "index": 1433, "success": false, "conic_type": "parabola", "error": "validation: parabola_point_off_curve", "params": { "p": 3.9731175899505615, "direction": "right" }, "optimization": { "final_loss": 0.00018243117665406317, "converged": true, "iterations": 127 }, "output_path": "results/train/parabola/problem_1433.png", "answer": "-2", "fact_expr": "C: Parabola;p: Number;P: Point;A: Point;B: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(P) = (1, 4);PointOnCurve(P, C);Intersection(LineOf(P, A), C) = A;Intersection(LineOf(P, B), C) = B;Slope(LineOf(P, A)) + Slope(LineOf(P, B)) = 0", "coords": { "P": [ 1.0, 4.0 ] }, "validation_reasons": [ "parabola_point_off_curve" ] }, { "index": 1434, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1434.png", "answer": "9", "fact_expr": "G: Hyperbola;A: Point;B: Point;P: Point;Expression(G) = (x^2/4 - y^2/5 = 1);Coordinate(A) = (3, 0);Coordinate(B) = (9, 5);PointOnCurve(P, RightPart(G))", "coords": { "A": [ 3.0, 0.0 ], "B": [ 9.0, 5.0 ] } }, { "index": 1435, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1435.png", "answer": "13", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 16*x);P: Point;PointOnCurve(P, G);Distance(P, xAxis) = 12;F: Point;Focus(G) = F", "coords": {} }, { "index": 1436, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1436.png", "answer": "2", "fact_expr": "G: Ellipse;H: Circle;C:Circle;P: Point;Q: Point;R: Point;Expression(G) = (x^2/4 + y^2/3 = 1);Expression(H) = (y^2 + (x + 1)^2 = 1);Expression(C) = (y^2 + (x - 1)^2 = 1);PointOnCurve(P, G);PointOnCurve(Q, H);PointOnCurve(R, C)", "coords": {} }, { "index": 1437, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1437.png", "answer": "8", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/5 = 1);A: Point;B: Point;PointOnCurve(A, RightPart(G));PointOnCurve(B, RightPart(G));Distance(MidPoint(LineSegmentOf(A, B)), yAxis) = 4", "coords": {} }, { "index": 1438, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1438.png", "answer": "1", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2 = 1)", "coords": {} }, { "index": 1439, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.1503682136535645, "b": 2.5736796855926514, "orientation": "horizontal" }, "optimization": { "final_loss": 5.191216132516274e-06, "converged": true, "iterations": 118 }, "output_path": "results/train/hyperbola/problem_1439.png", "answer": "3", "fact_expr": "G: Hyperbola;b: Number;Expression(G) = (x^2/6 - y^2/b^2 = 1);Eccentricity(G) = sqrt(6)/2", "coords": {} }, { "index": 1440, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1440.png", "answer": "2*sqrt(5)/5", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;C: Point;F: Point;Expression(G) = (y^2 = 9*x);Focus(G) = F;Intersection(Directrix(G), xAxis) = C;PointOnCurve(F, H);Intersection(H, G) = {A, B};IsChordOf(LineSegmentOf(A, B), G);Distance(MidPoint(LineSegmentOf(A, B)), Directrix(G)) = 18", "coords": {} }, { "index": 1441, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1441.png", "answer": "9", "fact_expr": "PointOnCurve(Focus(Gamma), xAxis);Gamma: Ellipse;Expression(Gamma) = (x^2/16 + y^2/m = 1);m: Number;FocalLength(Gamma) = 2*sqrt(7)", "coords": {} }, { "index": 1442, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1443, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1443.png", "answer": "9", "fact_expr": "G: Hyperbola;H: Circle;A: Circle;P: Point;M: Point;N: Point;Expression(G) = (x^2/9 - y^2/16 = 1);Expression(H) = (y^2 + (x + 5)^2 = 4);Expression(A) = (y^2 + (x - 5)^2 = 1);PointOnCurve(P, G);PointOnCurve(M, H);PointOnCurve(N, A)", "coords": {} }, { "index": 1444, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1444.png", "answer": "sqrt(10) - 1", "fact_expr": "G: Parabola;A: Point;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (2, 3);PointOnCurve(P, G)", "coords": { "A": [ 2.0, 3.0 ] } }, { "index": 1445, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1445.png", "answer": "[4,+oo)", "fact_expr": "H: Line;Expression(H) = (x = t);t: Number;G: Parabola;Expression(G) = (y^2 = 4*x);A: Point;B: Point;Intersection(H, G) = {A, B};C: Point;PointOnCurve(C, G) = True;IsPerpendicular(LineSegmentOf(A, C), LineSegmentOf(B, C)) = True", "coords": {} }, { "index": 1446, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1446.png", "answer": "1", "fact_expr": "G: Parabola;F: Point;Focus(G) = F;Coordinate(F) = (0, -1/2);P: Point;Coordinate(P) = (1, t);t: Number;PointOnCurve(P, G)", "coords": { "F": [ 0.0, -0.5 ] } }, { "index": 1447, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1447.png", "answer": "3", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;PointOnCurve(P, C);IsPerpendicular(VectorOf(P, F1), VectorOf(P, F2));Area(TriangleOf(P, F1, F2)) = 9", "coords": {} }, { "index": 1448, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1448.png", "answer": "2/3", "fact_expr": "G: Ellipse;b: Number;a: Number;e: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);a <= sqrt(3)*b;Eccentricity(G) = e", "coords": {} }, { "index": 1449, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1449.png", "answer": "(3,4)+(4,25/4)", "fact_expr": "G: Ellipse;t: Number;P:Point;Expression(G)=(x^2/4 + y^2/t=1);PointOnCurve(P,G);Distance(P,OneOf(Focus(G)))>1", "coords": {} }, { "index": 1450, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1450.png", "answer": "2*sqrt(3)/3", "fact_expr": "G: Hyperbola;b: Number;a: Number;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (2, 0);RightFocus(G) = F;Distance(F,Asymptote(G))=1", "coords": { "F": [ 2.0, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 1451, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1451.png", "answer": "11/3", "fact_expr": "G: Ellipse;M: Point;A: Point;B: Point;Expression(G) = (x^2/16 + y^2/4 = 1);Coordinate(M) = (1, 0);PointOnCurve(A, G);PointOnCurve(B, G);IsPerpendicular(LineSegmentOf(M, A), LineSegmentOf(M, B))", "coords": { "M": [ 1.0, 0.0 ] } }, { "index": 1452, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1452.png", "answer": "4", "fact_expr": "C: Parabola;p: Number;p>0;Expression(C) = (y^2 = 2*(p*x));PointOnCurve(M,C);Focus(C)=F;M:Point;F:Point;Distance(M,F)-Distance(M,yAxis)=2", "coords": {} }, { "index": 1453, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1453.png", "answer": "9", "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;P: Point;F: Point;Expression(C) = (y^2 = 2*x);Coordinate(P) = (4, 1);PointOnCurve(P, l);Intersection(l, C) = {A, B};MidPoint(LineSegmentOf(A, B)) = P;Focus(C) = F", "coords": { "P": [ 4.0, 1.0 ] } }, { "index": 1454, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1454.png", "answer": "(1, 2)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P: Point;PointOnCurve(P, G);A: Point;Coordinate(A) = (2, 2);F: Point;Focus(G) = F;WhenMin(Abs(LineSegmentOf(P, A)) + Abs(LineSegmentOf(P, F)))", "coords": { "A": [ 2.0, 2.0 ] } }, { "index": 1455, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1455.png", "answer": "y=pm*sqrt(6)*x/2", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;l: Line;Expression(l)=(y=-sqrt(3)*a+sqrt(3)*x);Intersection(l, RightPart(C)) = {A, B};A: Point;B: Point;Intersection(l, yAxis) = M;M: Point;VectorOf(A, B) = -2*VectorOf(A, M)", "coords": {} }, { "index": 1456, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1456.png", "answer": "9", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/12 = 1);F: Point;LeftFocus(G) = F;A: Point;Coordinate(A) = (1, 4);P: Point;PointOnCurve(P, RightPart(G)) = True", "coords": { "A": [ 1.0, 4.0 ] } }, { "index": 1457, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1458, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1458.png", "answer": "sqrt(3)/3", "fact_expr": "F1: Point;F2: Point;C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;LeftFocus(C) = F1;RightFocus(C) = F2;l: Line;PointOnCurve(F1, l);Intersection(l, C) = {A, B};A: Point;B: Point;Intersection(l, yAxis) = P;P: Point;VectorOf(F1, P) = (3/2)*VectorOf(A, F1);AngleOf(P, F1, F2) = ApplyUnit(30, degree)", "coords": {} }, { "index": 1459, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 2.6457513110645907, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1459.png", "answer": "11", "fact_expr": "C: Hyperbola;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/9 - y^2/7 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) = 5", "coords": {} }, { "index": 1460, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1460.png", "answer": "x^2/4 - y^2/2 = 1", "fact_expr": "C: Hyperbola;b: Number;a: Number;O: Origin;P: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(OneOf(Asymptote(C))) = (x - sqrt(2)*y = 0);PointOnCurve(P, C);Min(Abs(LineSegmentOf(O, P))) = 2", "coords": {} }, { "index": 1461, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1462, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1463, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1463.png", "answer": "pi/2", "fact_expr": "l: Line;G: Parabola;P: Point;Q: Point;F: Point;Expression(G) = (y = -x^2/4);Coordinate(P) = (-4, -4);Focus(G) = F;TangentPoint(l, G) = P;Intersection(l, xAxis) = Q", "coords": { "P": [ -4.0, -4.0 ] } }, { "index": 1464, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1464.png", "answer": "{-1, 0}", "fact_expr": "E: Parabola;F: Line;k: Real;Expression(E) = (y^2 = 4*x);Expression(F) = (y = k*x - 1);NumIntersection(F, E) = 1", "coords": {} }, { "index": 1465, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.1503682136535645, "b": 2.5736796855926514, "orientation": "horizontal" }, "optimization": { "final_loss": 5.191216132516274e-06, "converged": true, "iterations": 118 }, "output_path": "results/train/hyperbola/problem_1465.png", "answer": "4*sqrt(2)", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = sqrt(6)/2", "coords": {} }, { "index": 1466, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1466.png", "answer": "4", "fact_expr": "G: Hyperbola;a: Number;Expression(G) = (y^2/9 - x^2/a = 1);Expression(Asymptote(G)) = (3*x+pm*2*y=0)", "coords": {} }, { "index": 1467, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1467.png", "answer": "4*sqrt(41)", "fact_expr": "G: Ellipse;Expression(G) = (y^2/25 + x^2/a^2 = 1);a: Number;a > 5;F1: Point;F2: Point;Focus(G) = {F1, F2};Abs(LineSegmentOf(F1, F2)) = 8;A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G) = True;PointOnCurve(F1, LineSegmentOf(A, B)) = True", "coords": {} }, { "index": 1468, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1468.png", "answer": "(2, +oo)", "fact_expr": "l: Line;Expression(l) = (x = m*y + 2);m: Number;C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(C) = F;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, C) = {A, B};Z: Line;O: Origin;PointOnCurve(O, Z);IsChordOf(LineSegmentOf(A, B), C);D: Point;MidPoint(LineSegmentOf(A, B)) = D;PointOnCurve(D, Z);E: Point;Negation(E=O);Intersection(Z, C) = E", "coords": {} }, { "index": 1469, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 2.449489742783178, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1469.png", "answer": "(\\sqrt{6}/3,\\sqrt{6})", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/2 - y^2/6 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;L: Line;PointOnCurve(F2, L) = True;Intersection(L, RightPart(G)) = {A, B};A: Point;B: Point;Quadrant(A) = 1;C: Circle;IsInTangent(C, TriangleOf(A, F1, F2)) = True;Radius(C) = r;r: Number", "coords": {} }, { "index": 1470, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.0625, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1470.png", "answer": "15/16", "fact_expr": "G: Parabola;M: Point;Expression(G) = (y = 4*x^2);PointOnCurve(M, G);Distance(M, Focus(G)) = 1", "coords": {} }, { "index": 1471, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1471.png", "answer": "pm*sqrt(3)", "fact_expr": "l: Line;G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;PointOnCurve(F, l) = True;Intersection(l, G) = {A, B};A: Point;B: Point;VectorOf(A, F) = 3*VectorOf(F, B)", "coords": {} }, { "index": 1472, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1472.png", "answer": "(-(a^2-b^2)/a, (a^2-b^2)/a)", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>b;b>0;A: Point;B: Point;PointOnCurve(A, G);PointOnCurve(B, G);x0: Number;P: Point;Coordinate(P) = (x0, 0);Intersection(PerpendicularBisector(LineSegmentOf(A, B)), xAxis) = P", "coords": {} }, { "index": 1473, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1473.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/5 + y^2/4 = 1);H: Circle;Expression(H) = (-6*x + x^2 + y^2 + 8 = 0);C: Point;Center(H) = C", "coords": {} }, { "index": 1474, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1475, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1475.png", "answer": "8*sqrt(5)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);O: Origin;A: Point;B: Point;PointOnCurve(A, G);PointOnCurve(B, G);IsPerpendicular(LineSegmentOf(O, A), LineSegmentOf(O, B));Inclination(LineOf(A, B)) = ApplyUnit(45, degree)", "coords": {} }, { "index": 1476, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1477, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "major_axis": "x", "x_coef": 3.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1477.png", "answer": "4*sqrt(3)", "fact_expr": "G: Ellipse;B: Point;C: Point;A: Point;Expression(G) = (x^2/3 + y^2 = 1);PointOnCurve(B, G);PointOnCurve(C, G);OneOf(Focus(G)) = A;F:Point;OneOf(Focus(G))=K;Negation(A = F);PointOnCurve(F, LineSegmentOf(B, C))", "coords": {} }, { "index": 1478, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_1478.png", "answer": "3/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;p: Number;A: Point;O: Origin;B: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);p>0;Expression(H) = (y^2 = 2*p*x);L1:Line;L2:Line;Asymptote(G)={L1,L2};Intersection(L1,Directrix(H))=A;Intersection(L2,Directrix(H))=B;Eccentricity(G)=2;Area(TriangleOf(A,O,B))=sqrt(3)", "coords": {} }, { "index": 1479, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1480, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1480.png", "answer": "1", "fact_expr": "l: Line;C: Hyperbola;F1: Point;B: Point;F2: Point;A: Point;L1:Line;L2:Line;Expression(C) = (x^2 - y^2/3 = 1);LeftFocus(C)=F1;RightFocus(C)=F2;PointOnCurve(F1,l);Asymptote(C)={L1,L2};Intersection(l,L1)=A;Intersection(l,L2)=B;DotProduct(VectorOf(F1,B),VectorOf(F2,B))=0;VectorOf(F1, A) = lambda*VectorOf(A, B);lambda:Number", "coords": {} }, { "index": 1481, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1481.png", "answer": "(-oo, -sqrt(2)/2]+[sqrt(2)/2, +oo)", "fact_expr": "l: Line;G: Ellipse;Expression(G) = (x^2/2 + y^2 = 1);A: Point;B: Point;Intersection(l, G) = {A, B};H: Line;Expression(H) = (x = -1/2);PointOnCurve(MidPoint(LineSegmentOf(A, B)), H)", "coords": {} }, { "index": 1482, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1482.png", "answer": "sqrt(5)/2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;H: Point;Coordinate(H) = (4, 2);PointOnCurve(H, OneOf(Asymptote(G))) = True", "coords": { "H": [ 4.0, 2.0 ] } }, { "index": 1483, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 0.7071067811865476, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1483.png", "answer": "sqrt(6)/2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - 2*y^2 = 1)", "coords": {} }, { "index": 1484, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1484.png", "answer": "1", "fact_expr": "F1: Point;F2: Point;Focus(M) = {F1,F2};Focus(N) = {F1,F2};M: Ellipse;Expression(M) = (y^2/2 + x^2/m^2 = 1);m: Number;N: Hyperbola;Expression(N) = (-y^2 + x^2/n^2 = 1);n: Number;P: Point;OneOf(Intersection(M,N)) = P;IsPerpendicular(LineSegmentOf(P,F1), LineSegmentOf(F1,F2)) = True", "coords": {} }, { "index": 1485, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1486, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1486.png", "answer": "-2*sqrt(2)", "fact_expr": "l: Line;C: Parabola;A: Point;F: Point;B: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F,l);Intersection(l,C)={A,B};Quadrant(A)=4;VectorOf(A, F) = 2*VectorOf(F, B)", "coords": {} }, { "index": 1487, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1487.png", "answer": "10 + 2*sqrt(10)", "fact_expr": "G: Ellipse;A: Point;B: Point;M: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Coordinate(A) = (4, 0);Coordinate(B) = (2, 2);PointOnCurve(M, G)", "coords": { "A": [ 4.0, 0.0 ], "B": [ 2.0, 2.0 ] } }, { "index": 1488, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1489, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1490, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 5.0, 0.0 ], "radius": 7.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_1490.png", "answer": "{(x^2/9-y^2/16=1)&(x>0)}", "fact_expr": "A: Circle;Expression(A) = (y^2 + (x + 5)^2 = 49);B:Circle;Expression(B)=((x-5)^2+y^2=49);H:Circle;IsOutTangent(H,A);IsOutTangent(H,B);P:Point;Center(H)=P", "coords": {} }, { "index": 1491, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1491.png", "answer": "8*sqrt(7)/7", "fact_expr": "G: Ellipse;L: Line;A: Point;B: Point;F2: Point;F1: Point;x1:Number;y1:Number;x2:Number;y2:Number;Expression(G) = (x^2/16 + y^2/9 = 1);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, L);Intersection(L, G) = {A, B};Area(InscribedCircle(TriangleOf(A,B,F2)))=pi", "coords": {} }, { "index": 1492, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1492.png", "answer": "{pm*1, pm*sqrt(5)}", "fact_expr": "G: Hyperbola;H: Line;k: Number;Expression(G) = (x^2 - y^2 = 1);Expression(H) = (y = k*x + 2);NumIntersection(H, G) = 1", "coords": {} }, { "index": 1493, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1493.png", "answer": "4", "fact_expr": "l: Line;E: Parabola;p: Number;B: Point;C: Point;F: Point;A: Point;p>0;Expression(E) = (y^2 = 2*(p*x));Focus(E) = F;PointOnCurve(F, l);Intersection(l, E) = {A, B};Intersection(l, Directrix(E)) = C;Between(B, A, C);Abs(LineSegmentOf(B, C)) = 3*Abs(LineSegmentOf(B, F));Abs(LineSegmentOf(A, B)) = 9", "coords": {} }, { "index": 1494, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1495, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1495.png", "answer": "3", "fact_expr": "m: Line;l:Line;G: Parabola;p: Number;A: Point;F: Point;C: Point;B: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;PointOnCurve(F, m);Intersection(m,G)={A,B};Intersection(m,l)=C;Abs(LineSegmentOf(A, F)) = 6;VectorOf(C, B) = 2*VectorOf(B, F);Directrix(G)=l", "coords": {} }, { "index": 1496, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1496.png", "answer": "3/2", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;LeftFocus(C) = F1;RightFocus(C) = F2;F1: Point;F2: Point;G: Line;Expression(G) = (y = sqrt(3)*b);Intersection(G, RightPart(C)) = P;P: Point;Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 1497, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1497.png", "answer": "sqrt(2)", "fact_expr": "C: Parabola;Expression(C) = (x^2 = 4*y);P: Point;PointOnCurve(P, C);Negation(P=Vertex(C));M: Point;Coordinate(M) = (0, -1);F: Point;Focus(C) = F;k: Real;Sin(AngleOf(P, F, M)) = k*Sin(AngleOf(P, M, F))", "coords": { "M": [ 0.0, -1.0 ] } }, { "index": 1498, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1498.png", "answer": "4*sqrt(10)", "fact_expr": "G: Parabola;H: Line;O: Origin;A: Point;B: Point;M: Point;Expression(G) = (y^2 = 4*x);Intersection(H, G) = {A, B};Slope(H) = 1;Circumcenter(TriangleOf(O, A, B)) = M;WhenMax(Abs(LineSegmentOf(A, B)))", "coords": {} }, { "index": 1499, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1499.png", "answer": "sqrt(5)/3", "fact_expr": "C: Ellipse;C1:Circle;b: Number;a: Number;c: Number;F: Point;A: Point;B: Point;a>0;b>0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Expression(C1) = (y^2 + (-c/2 + x)^2 = b^2/16);RightFocus(C) = F;PointOnCurve(A, C);TangentPoint(LineOf(A,F),C1)=B;VectorOf(A, F) = 4*VectorOf(B, F)", "coords": {} }, { "index": 1500, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1500.png", "answer": "[\\sqrt{3}/2, 1)", "fact_expr": "C: Ellipse;O: Circle;a: Number;b: Number;Expression(C) = (x^2/a^2 + y^2/b^2 = 1);a>b;b>0;Expression(O) = (x^2 + y^2 = b^2);P: Point;PointOnCurve(P, C);L1: Line;L2: Line;TangentOfPoint(P, O) = {L1, L2};A: Point;B: Point;TangentPoint(L1, O) = A;TangentPoint(L2, O) = B;AngleOf(A, P, B) = ApplyUnit(60, degree)", "coords": {} }, { "index": 1501, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1501.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;OneOf(Focus(G)) = F;Z: Line;PointOnCurve(F, Z);IsPerpendicular(Z, OneOf(Asymptote(G)));PointOnCurve(FootPoint(Z, OneOf(Asymptote(G))), PerpendicularBisector(LineSegmentOf(O, F)));O: Origin", "coords": {} }, { "index": 1502, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.4641016151377544, "major_axis": "y", "x_coef": 12.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1502.png", "answer": "y^2/2 - x^2/2 = 1", "fact_expr": "G: Hyperbola;C: Ellipse;H: Point;Expression(C) = (x^2/12 + y^2/16 = 1);Coordinate(H) = (1, sqrt(3));Focus(G)=Focus(C);PointOnCurve(H,G)", "coords": { "H": [ 1.0, 1.7320508075688772 ] } }, { "index": 1503, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1503.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/16 + y^2/9 = 1);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2;AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 1504, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1504.png", "answer": "y^2/8 - x^2/2 = 1", "fact_expr": "G: Hyperbola;C:Hyperbola;H: Point;Expression(G) = (x^2 - y^2/4 = 1);Coordinate(H) = (1, 2*sqrt(3));PointOnCurve(H, C);Asymptote(C)=Asymptote(G)", "coords": { "H": [ 1.0, 3.4641016151377544 ] } }, { "index": 1505, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1505.png", "answer": "6", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/4 - y^2/b^2 = 1);b: Number;b>0;Distance(Focus(C),Asymptote(C))=3", "coords": {} }, { "index": 1506, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1506.png", "answer": "5*sqrt(3)", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 - y^2 = 1);Coordinate(P) = (8, y0);Focus(G) = {F1, F2};PointOnCurve(P, G);y0:Number", "coords": {} }, { "index": 1507, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1507.png", "answer": "8", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 + y^2/3 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 1508, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1508.png", "answer": "9/2", "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/9 = 1);H:LineSegment;F: Point;IsChordOf(H,G);OneOf(Focus(G))=F;PointOnCurve(F,H);WhenMin(Length(H))", "coords": {} }, { "index": 1509, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1509.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/9 = 1)", "coords": {} }, { "index": 1510, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1510.png", "answer": "4*sqrt(2)", "fact_expr": "E: Hyperbola;b: Number;b>0;Expression(E) = (x^2/6 - y^2/b^2 = 1);Expression(Asymptote(E)) = (y = pm*(x*(sqrt(3)/3)))", "coords": {} }, { "index": 1511, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1512, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1512.png", "answer": "y^2 = x - 15/4", "fact_expr": "A: Point;Coordinate(A) = (5, 0);G: Parabola;Expression(G) = (y^2 = 4*x);P: Point;PointOnCurve(P, G);M: Point;PointOnCurve(M, LineSegmentOf(P, A));Abs(LineSegmentOf(P, M)) = 3*Abs(LineSegmentOf(M, A))", "coords": { "A": [ 5.0, 0.0 ] } }, { "index": 1513, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1513.png", "answer": "2*sqrt(3)/3", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;e: Number;Eccentricity(G)=e;e = 2", "coords": {} }, { "index": 1514, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1515, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "y", "x_coef": 16.0, "y_coef": 25.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1515.png", "answer": "10", "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/25 = 1)", "coords": {} }, { "index": 1516, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1517, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1517.png", "answer": "-16", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);A: Point;B: Point;PointOnCurve(A, G);PointOnCurve(B, G);Negation(A=B);O: Origin", "coords": {} }, { "index": 1518, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1519, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1519.png", "answer": "x^2 - y^2/4 = 1", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Ellipse;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2/9 + y^2/4 = 1);Focus(C) = Focus(G);Expression(Asymptote(C)) = (y = pm*(2*x))", "coords": {} }, { "index": 1520, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1520.png", "answer": "2*sqrt(2)", "fact_expr": "G: Parabola;H: Line;F: Point;A: Point;B: Point;k:Number;Expression(G) = (y^2 = 8*x);k>0;Expression(H) = (y = k*(x - 2));Intersection(H, G) = {A, B};Focus(G) = F;Abs(LineSegmentOf(F, A)) = 2*Abs(LineSegmentOf(F, B))", "coords": {} }, { "index": 1521, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1521.png", "answer": "8", "fact_expr": "G: Ellipse;A: Point;B: Point;F2: Point;F1: Point;Expression(G) = (x^2/25 + y^2/16 = 1);LeftFocus(G) = F1;PointOnCurve(F1, LineSegmentOf(A, B));IsChordOf(LineSegmentOf(A,B),G);Abs(LineSegmentOf(A, F2)) + Abs(LineSegmentOf(B, F2)) = 12;RightFocus(G) = F2", "coords": {} }, { "index": 1522, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1523, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1523.png", "answer": "3/2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);A: Point;B: Point;PointOnCurve(A, G);PointOnCurve(B, G);Distance(A, F) + Distance(B, F) = 5;F: Point;Focus(G) = F", "coords": {} }, { "index": 1524, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1524.png", "answer": "[1/2, 1)", "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;F: Point;P: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;Intersection(RightDirectrix(G), xAxis) = A;PointOnCurve(P, G);PointOnCurve(F, PerpendicularBisector(LineSegmentOf(A, P)))", "coords": {} }, { "index": 1525, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.826896789078654, "b": 2.826896789078654, "orientation": "horizontal" }, "optimization": { "final_loss": 7.081136889894256e-06, "converged": true, "iterations": 75 }, "output_path": "results/train/hyperbola/problem_1525.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Hyperbola;F1: Point;F2: Point;Focus(G) = {F1, F2};Coordinate(F1) = (-4, 0);Coordinate(F2) = (4, 0);Length(RealAxis(G)) = Length(ImageinaryAxis(G))", "coords": { "F1": [ -4.0, 0.0 ], "F2": [ 4.0, 0.0 ] } }, { "index": 1526, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1526.png", "answer": "[sqrt(2)/2, 1]", "fact_expr": "G: Parabola;N: Point;F: Point;M: Point;Expression(G) = (x^2 = 4*y);Focus(G) = F;Intersection(Directrix(G),SymmetryAxis(G))=M;PointOnCurve(N, G);lambda: Number;Abs(LineSegmentOf(N, F)) = lambda*Abs(LineSegmentOf(M, N))", "coords": {} }, { "index": 1527, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 0.8176298651812968, "b": 2.452642014841466, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4992967561765154e-06, "converged": true, "iterations": 283 }, "output_path": "results/train/hyperbola/problem_1527.png", "answer": "{(x^2-y^2/9=1),(y^2/9-x^2=1)}", "fact_expr": "G: Hyperbola;Expression(Asymptote(G)) = (y = pm*3*x);FocalLength(G)=2*sqrt(10)", "coords": {} }, { "index": 1528, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1528.png", "answer": "6", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;l1: Line;Directrix(C) = l1;G: Hyperbola;Expression(G) = (x^2 - y^2 = 1);PointOnCurve(OneOf(Vertex(G)), l1);P: Point;PointOnCurve(P, C);l2: Line;Expression(l2) = (3*x - 4*y + 27 = 0)", "coords": {} }, { "index": 1529, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 10.0, "b": 8.0, "major_axis": "x", "x_coef": 100.0, "y_coef": 64.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1529.png", "answer": "sqrt(15)", "fact_expr": "G: Ellipse;P: Point;A: Point;M: Point;x1: Number;y1: Number;Expression(G) = (x^2/100 + y^2/64 = 1);Coordinate(A) = (6, 0);Coordinate(P) = (x1, y1);PointOnCurve(P, G);Abs(VectorOf(A, M)) = 1;DotProduct(VectorOf(P, M), VectorOf(A, M)) = 0", "coords": { "A": [ 6.0, 0.0 ] } }, { "index": 1530, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1530.png", "answer": "26", "fact_expr": "l: Line;G: Hyperbola;A: Point;B: Point;F1: Point;F2: Point;Expression(G) = (x^2/16 - y^2/9 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, l);Intersection(l, RightPart(G)) = {A, B};Abs(LineSegmentOf(A, B)) = 5", "coords": {} }, { "index": 1531, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1532, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1532.png", "answer": "(2, +oo)", "fact_expr": "A: Point;B: Point;G: Parabola;M: Point;m:Number;Expression(G) = (y^2 = 4*x);Coordinate(M) = (m, 0);PointOnCurve(A,G);PointOnCurve(B,G);Negation(IsPerpendicular(LineOf(A,B),xAxis));Intersection(PerpendicularBisector(LineSegmentOf(A, B)), xAxis) = M", "coords": {} }, { "index": 1533, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1533.png", "answer": "2*sqrt(3)", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2 - y^2 = 1);Focus(G)={F1,F2};PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2))", "coords": {} }, { "index": 1534, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1534.png", "answer": "pi/4", "fact_expr": "G: Parabola;M: Point;Expression(G) = (y = x^2);Coordinate(M) = (1/2, 1/4);PointOnCurve(M,G)", "coords": { "M": [ 0.5, 0.25 ] } }, { "index": 1535, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1535.png", "answer": "-3/4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x);H: Line;PointOnCurve(Focus(G), H) ;Intersection(G, H) = {A, B};O: Origin;A: Point;B: Point", "coords": {} }, { "index": 1536, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1536.png", "answer": "2", "fact_expr": "C: Hyperbola;a: Number;b: Number;G: Circle;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y^2 + (x - 2)^2 = 4);Length(InterceptChord(OneOf(Asymptote(C)),G))=2", "coords": {} }, { "index": 1537, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1538, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1538.png", "answer": "sqrt(5)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;G: Parabola;Expression(G) = (y = x^2 + 1);IsTangent(Asymptote(C), G)", "coords": {} }, { "index": 1539, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 3.0, "major_axis": "x", "x_coef": 36.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1539.png", "answer": "x + 4*y - 5 = 0", "fact_expr": "G: Ellipse;H: LineSegment;A: Point;Expression(G) = (x^2/36 + y^2/9 = 1);Coordinate(A) = (1, 1);IsChordOf(H, G);MidPoint(H) = A", "coords": { "A": [ 1.0, 1.0 ] } }, { "index": 1540, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 2.0, "major_axis": "x", "x_coef": 8.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1540.png", "answer": "x^2/2 - y^2/2 = 1", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/8 + y^2/4 = 1);Eccentricity(G) = sqrt(2);Focus(G) = Focus(H)", "coords": {} }, { "index": 1541, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1541.png", "answer": "sqrt(5)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;A: Point;PointOnCurve(A, C);IsPerpendicular(LineSegmentOf(A, F2), LineSegmentOf(F1, F2));Intersection(LineOf(A, F1), C) = {A, B};B: Point;VectorOf(F1, A) = 4*VectorOf(F1, B)", "coords": {} }, { "index": 1542, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1543, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1543.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P: Point;PointOnCurve(P, G);M: Point;Projection(P, yAxis) = M;A: Point;a: Number;Coordinate(A) = (4, a);Abs(a)>4", "coords": {} }, { "index": 1544, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1544.png", "answer": "pm*6", "fact_expr": "C: Parabola;Expression(C) = (x^2 = 2*p*y);p: Number;p>0;F: Point;Focus(C) = F;M: Point;Coordinate(M) = (x0, 3);x0: Number;PointOnCurve(M, C) = True;Distance(M, F) = 6", "coords": {} }, { "index": 1545, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1546, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1546.png", "answer": "10", "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;P: Point;F: Point;Expression(G) = (x^2 = 4*y);Coordinate(P) = (1, 4);Focus(G) = F;PointOnCurve(P, l);Intersection(l, G) = {A,B};MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ 1.0, 4.0 ] } }, { "index": 1547, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1547.png", "answer": "2*sqrt(3)", "fact_expr": "G: Parabola;H: Line;F: Point;N: Point;M: Point;D: Point;l: Line;Expression(G) = (y^2 = 4*x);Focus(G) = F;Directrix(G) = l;PointOnCurve(F, H);Slope(H) = sqrt(3);Intersection(H, G) = M;IsPerpendicular(LineSegmentOf(M, N),l);FootPoint(LineSegmentOf(M, N),l)=N;Intersection(LineOf(N,F), yAxis) = D;Quadrant(M)=1", "coords": {} }, { "index": 1548, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1549, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1549.png", "answer": "1/2", "fact_expr": "G: Parabola;M: Point;Expression(G) = (y^2 = 2*x);PointOnCurve(M, G);Distance(M, Focus(G)) = 1", "coords": {} }, { "index": 1550, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1550.png", "answer": "1+sqrt(2)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;F2: Point;F1: Point;a>0;b>0;LeftFocus(C)= F1;RightFocus(C)= F2;l:Line;Slope(l)=1;PointOnCurve(F1,l);P:Point;Intersection(l,RightPart(C))=P;AngleOf(F1, F2, P)=ApplyUnit(90, degree)", "coords": {} }, { "index": 1551, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1551.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);M:Point;F1:Point;F2:Point;PointOnCurve(M,G);Focus(G)={F1,F2}", "coords": {} }, { "index": 1552, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1553, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1554, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1554.png", "answer": "4", "fact_expr": "C: Parabola;p: Number;M: Point;N: Point;F: Point;O: Origin;p>0;Expression(C) = (y^2 = 2*(p*x));PointOnCurve(M, C);Focus(C) = F;L:Line;PointOnCurve(M,L);IsPerpendicular(L,Directrix(C));FootPoint(L,Directrix(C))=N;AngleOf(M, F, O) = ApplyUnit(120, degree);Perimeter(TriangleOf(M, N, F)) = 12", "coords": {} }, { "index": 1555, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1555.png", "answer": "-3", "fact_expr": "G: Parabola;p: Number;H: Line;A: Point;F: Point;B: Point;P: Point;l: Line;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;Directrix(G) = l;PointOnCurve(F,H);Intersection(H,G)={A,B};Quadrant(B)=1;Intersection(H, l) = P;VectorOf(A, F) = VectorOf(F, B)/2;VectorOf(A, P) = lambda*VectorOf(A, F);lambda:Number", "coords": {} }, { "index": 1556, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1556.png", "answer": "sqrt(5)/5", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;F2: Point;A: Point;B: Point;C: Point;F1: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(F1, H);IsPerpendicular(H,xAxis);Intersection(H, G) = {A, B};Intersection(LineOf(A,F2),G)={A,C};Area(TriangleOf(A, B, C)) = 3*Area(TriangleOf(B, C, F2))", "coords": {} }, { "index": 1557, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1558, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1558.png", "answer": "2*sqrt(6)", "fact_expr": "G: Parabola;p: Number;H: Circle;p>0;l:Line;Expression(G) = (x^2 = 2*p*y);Expression(H) = (-6*x + x^2 + y^2 - 1 = 0);Directrix(G)=l;Length(InterceptChord(l,H))=4", "coords": {} }, { "index": 1559, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 4.47213595499958, "major_axis": "x", "x_coef": 36.0, "y_coef": 20.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1559.png", "answer": "[-12,84]", "fact_expr": "C: Ellipse;M: Circle;A: Point;B: Point;P: Point;Expression(C) = (x^2/36 + y^2/20 = 1);Expression(M) = (y^2 + x^2 - 8*x = 0);PointOnCurve(P, C);IsDiameter(LineSegmentOf(A, B), M)", "coords": {} }, { "index": 1560, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 16.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1560.png", "answer": "3+2*sqrt(2)", "fact_expr": "C: Parabola;G: Line;A: Point;B: Point;F: Point;Expression(C) = (y^2 = 64*x);Focus(C)=F;PointOnCurve(F,G);Intersection(G,C)={A,B};Abs(LineSegmentOf(F, A)) > Abs(LineSegmentOf(F, B));Slope(G)=1", "coords": {} }, { "index": 1561, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1561.png", "answer": "2*x-y-15=0", "fact_expr": "P: Point;Coordinate(P) = (8, 1);G: Hyperbola;Expression(G) = (x^2 - 4*y^2 = 4);H: LineSegment;IsChordOf(H, G);MidPoint(H) = P", "coords": { "P": [ 8.0, 1.0 ] } }, { "index": 1562, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1562.png", "answer": "64", "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;O: Origin;F:Point;Expression(C) = (y^2 = 16*x);Inclination(l)=pi/6;Focus(C)=F;PointOnCurve(F,l);Intersection(l,C) = {A, B}", "coords": {} }, { "index": 1563, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1563.png", "answer": "y^2=sqrt(2)*x", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(C) = F;l: Line;Directrix(C) = l;A: Point;PointOnCurve(A, C);G: Circle;Center(G) = F;Radius(G)=LineSegmentOf(F,A);B: Point;D: Point;Intersection(G, l) = {B, D};AngleOf(F, B, D) = ApplyUnit(30, degree);Area(TriangleOf(A, B, D)) = sqrt(3)", "coords": {} }, { "index": 1564, "success": false, "error": "Circle from diameter: cannot parse diameter pattern" }, { "index": 1565, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1565.png", "answer": "4", "fact_expr": "C: Parabola;l1:Line;l2:Line;A: Point;B: Point;D: Point;E: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F,l1);PointOnCurve(F,l2);IsPerpendicular(l1,l2);Intersection(l1, C) = {A, B};Intersection(l2, C) = {D, E}", "coords": {} }, { "index": 1566, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 0.7071067811865476, "b": 0.7071067811865476, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1566.png", "answer": "x^2/2 + y^2 = 1", "fact_expr": "G: Hyperbola;H: Ellipse;O: Origin;Expression(G) = (2*x^2 - 2*y^2 = 1);Center(H) = O;Focus(H) = Focus(G);InterReciprocal(Eccentricity(G), Eccentricity(H))", "coords": {} }, { "index": 1567, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1567.png", "answer": "y=pm*(3/2)*x", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;C1: Circle;Expression(C1) = (x^2 + y^2 - 4*x - 6*y = 0);PointOnCurve(Center(C1), OneOf(Asymptote(C)))", "coords": {} }, { "index": 1568, "success": false, "error": "Circle from diameter: missing point coordinates" }, { "index": 1569, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1569.png", "answer": "9", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(90, degree)", "coords": {} }, { "index": 1570, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1570.png", "answer": "3/2", "fact_expr": "G: Parabola;M: Point;N: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G)=F;PointOnCurve(M,G);PointOnCurve(N,G);Negation(M=N);Abs(LineSegmentOf(M, F)) + Abs(LineSegmentOf(N, F)) = 5", "coords": {} }, { "index": 1571, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1571.png", "answer": "x=-2", "fact_expr": "G: Parabola;p: Number;H: Ellipse;p>0;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (x^2/9 + y^2/5 = 1);PointOnCurve(OneOf(Focus(H)), Directrix(G))", "coords": {} }, { "index": 1572, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1572.png", "answer": "x+2*y-3=0", "fact_expr": "G: Ellipse;H: LineSegment;Expression(G) = (x^2/4 + y^2/2 = 1);Coordinate(MidPoint(H)) = (1, 1);IsChordOf(H,G)", "coords": {} }, { "index": 1573, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1573.png", "answer": "((2*x-1)^2)/2+y^2=1", "fact_expr": "G: Ellipse;Expression(G) = (x^2/2 + y^2 = 1);P:Point;PointOnCurve(P,G)", "coords": {} }, { "index": 1574, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1574.png", "answer": "7", "fact_expr": "C: Ellipse;Expression(C) = (y^2 + x^2/a^2 = 1);a: Number;a>0;Eccentricity(C) = sqrt(3)/2;F1: Point;F2: Point;Focus(C) = {F1, F2};l: Line;PointOnCurve(F1, l) = True;Intersection(l, C) = {A, B};A: Point;B: Point", "coords": {} }, { "index": 1575, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1575.png", "answer": "sqrt(3)/3", "fact_expr": "F: Point;G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;RightFocus(G) = F;A: Point;OneOf(Endpoint(MinorAxis(G))) = A;Intersection(LineOf(A, F), G) = {A,B};B: Point;VectorOf(A, F) = 2*VectorOf(F, B)", "coords": {} }, { "index": 1576, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1577, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1578, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1579, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1579.png", "answer": "(6*sqrt(5)/5,2)", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 - y^2/5 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P,G);Quadrant(P)=1;Area(TriangleOf(P, F1, F2)) = 6", "coords": {} }, { "index": 1580, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1580.png", "answer": "{(-oo,-2*sqrt(3)/3),(2*sqrt(3)/3,+oo)}", "fact_expr": "G: Hyperbola;H: Line;k:Number;Expression(G) = (x^2 - y^2 = 4);Expression(H) = (y=k*(x-1));NumIntersection(G,H)=0", "coords": {} }, { "index": 1581, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1581.png", "answer": "x^2/3+y^2/2=1", "fact_expr": "l: Line;C: Ellipse;b: Number;a: Number;F1: Point;B: Point;F2: Point;A: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;Eccentricity(C)=sqrt(3)/3;PointOnCurve(F2, l);Intersection(l,C)={A,B};Perimeter(TriangleOf(A,F1,B)) = 4*sqrt(3)", "coords": {} }, { "index": 1582, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1582.png", "answer": "(0,1/4)", "fact_expr": "G: Parabola;Expression(G) = (y = x^2)", "coords": {} }, { "index": 1583, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1583.png", "answer": "2", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;G: Circle;Expression(G) = (x^2 + y^2 = a^2);L1: Line;L2: Line;A: Point;B: Point;TangentOfPoint(OneOf(Focus(C)), G) = {L1, L2};TangentPoint(L1, G) = A;TangentPoint(L2, G) = B;O: Origin;AngleOf(A, O, B) = ApplyUnit(120, degree)", "coords": {} }, { "index": 1584, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1585, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1585.png", "answer": "sqrt(5)/2", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/4 - y^2 = 1)", "coords": {} }, { "index": 1586, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1586.png", "answer": "x^2 - y^2/3 = 1", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;A: Point;O: Origin;Z: Line;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(RightVertex(C), Z);IsPerpendicular(Z, xAxis);Intersection(Z, OneOf(Asymptote(C))) = A;Center(G) = RightFocus(C);Radius(G) = 2;PointOnCurve(A, G);PointOnCurve(O, G)", "coords": {} }, { "index": 1587, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1587.png", "answer": "pm*sqrt(3)", "fact_expr": "l: Line;G: Parabola;A: Point;F: Point;B: Point;C: Point;Expression(G) = (y^2 = 4*x);Focus(G)=F;PointOnCurve(F, l);Intersection(l,G)={A,B};Intersection(l,Directrix(G))=C;VectorOf(A, F) = (3/2)*VectorOf(B, C)", "coords": {} }, { "index": 1588, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1588.png", "answer": "2*sqrt(15)", "fact_expr": "G: Parabola;H: Line;k: Number;A: Point;B: Point;Expression(G) = (y^2 = 8*x);Expression(H) = (y = k*x - 2);Intersection(H, G) = {A, B};XCoordinate(MidPoint(LineSegmentOf(A,B))) = 2", "coords": {} }, { "index": 1589, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1589.png", "answer": "2", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(C) = F;A: Point;B: Point;PointOnCurve(F,LineSegmentOf(A,B)) = True;IsChordOf(LineSegmentOf(A,B),C) = True;VectorOf(F,A) + VectorOf(F,B) = 0;G: Circle;IsDiameter(LineSegmentOf(A,B),G) = True;PointOnCurve(LeftVertex(C),G) = True", "coords": {} }, { "index": 1590, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1590.png", "answer": "sqrt(15)", "fact_expr": "G: Hyperbola;a: Number;a>0;H:Parabola;Expression(G) = (-y^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 16*x);PointOnCurve(LeftFocus(G),Directrix(H))", "coords": {} }, { "index": 1591, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1591.png", "answer": "8", "fact_expr": "G: Ellipse;P: Point;F1: Point;M: Point;O: Origin;Expression(G) = (x^2/25 + y^2/16 = 1);LeftFocus(G) = F1;PointOnCurve(P, G);Negation(P=Vertex(G));MidPoint(LineSegmentOf(P, F1)) = M", "coords": {} }, { "index": 1592, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1593, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1593.png", "answer": "6", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 8*x);F: Point;M: Point;N: Point;Focus(C) = F;PointOnCurve(M, C);Intersection(OverlappingLine(LineSegmentOf(F, M)), yAxis) = N;MidPoint(LineSegmentOf(F, N)) = M", "coords": {} }, { "index": 1594, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.4641016151377544, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1594.png", "answer": "4", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/12 - y^2/4 = 1)", "coords": {} }, { "index": 1595, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 2.0, "major_axis": "x", "x_coef": 8.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1595.png", "answer": "4", "fact_expr": "G: Parabola;O: Origin;Vertex(G) = O;H: Ellipse;Expression(H) = (x^2 + 2*y^2 = 8);Focus(G) = OneOf(Focus(H))", "coords": {} }, { "index": 1596, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1596.png", "answer": "4*sqrt(3)/3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;L: Line;PointOnCurve(F, L);Inclination(L) = ApplyUnit(60, degree);Intersection(L, G) = {A, B};A: Point;B: Point;C1: Circle;IsDiameter(LineSegmentOf(A, F), C1) = True;C2: Circle;IsDiameter(LineSegmentOf(B, F), C2) = True;TangentPoint(C1, yAxis) = M;TangentPoint(C2, yAxis) = N;M: Point;N: Point", "coords": {} }, { "index": 1597, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1597.png", "answer": "[\\sqrt{2}-1, 1)", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;e: Number;Eccentricity(G) = e;P: Point;PointOnCurve(P, G);LineSegmentOf(P, F1)/LineSegmentOf(P, F2) = e", "coords": {} }, { "index": 1598, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1598.png", "answer": "10", "fact_expr": "l: Line;C: Parabola;p: Number;M: Point;A: Point;B: Point;F: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(M) = (-2, 2);Focus(C)=F;PointOnCurve(F, l);Intersection(l, C) = {A, B};PointOnCurve(M,Directrix(C));DotProduct(VectorOf(M,A),VectorOf(M,B))=0", "coords": { "M": [ -2.0, 2.0 ] } }, { "index": 1599, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1599.png", "answer": "2/3", "fact_expr": "G: Ellipse;a: Number;Expression(G) = (y^2/20 + x^2/a^2 = 1);PointOnCurve(Focus(G), xAxis);FocalLength(G) = 8", "coords": {} }, { "index": 1600, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1600.png", "answer": "4*x+y-3=0", "fact_expr": "I: Point;Coordinate(I) = (1, -1);G: Parabola;Expression(G) = (y^2 = 8*x);L: LineSegment;MidPoint(L) = I;IsChordOf(L, G)", "coords": { "I": [ 1.0, -1.0 ] } }, { "index": 1601, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1601.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1);F1: Point;F2: Point;Focus(G) = {F1,F2};P: Point;PointOnCurve(P,G) = True;AngleOf(F1, P, F2) = ApplyUnit(90, degree)", "coords": {} }, { "index": 1602, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1603, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1604, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1604.png", "answer": "y^2/3 - x^2 = 1", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;Expression(C) = (-x^2/b^2 + y^2/a^2 = 1);Coordinate(P) = (1, sqrt(3));FocalLength(C) = 4;PointOnCurve(P, Asymptote(C))", "coords": { "P": [ 1.0, 1.7320508075688772 ] } }, { "index": 1605, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1605.png", "answer": "0", "fact_expr": "F: Point;Focus(G) = F;G: Parabola;Expression(G) = (y = x^2/4);l: Line;PointOnCurve(F, l);Inclination(l) = ApplyUnit(150, degree);A: Point;B: Point;Intersection(l, G) = {A, B};l1: Line;l2: Line;TangentOnPoint(A, G) = l1;TangentOnPoint(B, G) = l2;C: Point;Intersection(l1, l2) = C", "coords": {} }, { "index": 1606, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1606.png", "answer": "1/2", "fact_expr": "F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;l: Line;PointOnCurve(F1,l) = True;Intersection(l,LeftPart(G)) = A;A: Point;B: Point;LineSegmentOf(A, F1) = 2*a;AngleOf(F1,A,F2) = 2*pi/3;Intersection(l,RightPart(G)) = B", "coords": {} }, { "index": 1607, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1608, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1608.png", "answer": "(2,2)", "fact_expr": "G: Parabola;M: Point;P: Point;F: Point;Expression(G) = (y^2 = 2*x);Coordinate(M) = (3, 2);Focus(G) = F;PointOnCurve(P, G);WhenMin(Abs(LineSegmentOf(P, F)) + Abs(LineSegmentOf(P, M)))", "coords": { "M": [ 3.0, 2.0 ] } }, { "index": 1609, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1610, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1610.png", "answer": "3", "fact_expr": "G: Hyperbola;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 - y^2/12 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(P,G);AngleOf(F1, P, F2) = ApplyUnit(90, degree)", "coords": {} }, { "index": 1611, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 2.23606797749979 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_1611.png", "answer": "sqrt(15)", "fact_expr": "C: Hyperbola;O: Origin;Center(C) = O;SymmetryAxis(C) = axis;O1: Circle;Expression(O1) = (x^2 + y^2 = 5);P: Point;Intersection(C, O1) = P;Coordinate(P) = (2, -1);IsParallel(TangentOnPoint(P, O1), OneOf(Asymptote(C)))", "coords": {} }, { "index": 1612, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1612.png", "answer": "1", "fact_expr": "G: Parabola;Q: Point;F: Point;O:Origin;Expression(G) = (y^2 = 5*x);PointOnCurve(Q, G);Focus(G) = F;Distance(Q, F) = 25/4", "coords": {} }, { "index": 1613, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.4641016151377544, "b": 2.8284271247461903, "major_axis": "x", "x_coef": 12.0, "y_coef": 8.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1613.png", "answer": "4*x - 3*y - 11 = 0", "fact_expr": "G: Ellipse;H: LineSegment;P: Point;Expression(G) = (x^2/12 + y^2/8 = 1);Coordinate(P) = (2, -1);IsChordOf(H, G);MidPoint(H) = P", "coords": { "P": [ 2.0, -1.0 ] } }, { "index": 1614, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.463192105293274, "b": 2.070901870727539, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4661259228887502e-06, "converged": true, "iterations": 106 }, "output_path": "results/train/hyperbola/problem_1614.png", "answer": "y=pm*sqrt(2)*x", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 4*x);Eccentricity(G) = sqrt(3);OneOf(Directrix(G))=Directrix(H)", "coords": {} }, { "index": 1615, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1615.png", "answer": "sqrt(2)/2", "fact_expr": "C: Ellipse;a: Number;b: Number;t:Number;P: Point;O: Origin;F1: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (t, sqrt(3)*t);PointOnCurve(P, C);LeftFocus(C) = F1;RightFocus(C) = F2;Abs(LineSegmentOf(O,P))=(sqrt(14)/7)*Abs(LineSegmentOf(F1,F2))", "coords": {} }, { "index": 1616, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1616.png", "answer": "8", "fact_expr": "G: Ellipse;H: Line;F2: Point;A: Point;B: Point;F1: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(G) = {F1,F2};PointOnCurve(F1, H);Intersection(H, G) = {A,B};Abs(LineSegmentOf(F2, A)) + Abs(LineSegmentOf(F2, B)) = 12", "coords": {} }, { "index": 1617, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2135674953460693, "b": 2.4270472526550293, "orientation": "horizontal" }, "optimization": { "final_loss": 4.453341261978494e-06, "converged": true, "iterations": 125 }, "output_path": "results/train/hyperbola/problem_1617.png", "answer": "x^2/5 - y^2/10 = 1", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (-10*x + x^2 + y^2 = 0);OneOf(Focus(G))=Center(H);Eccentricity(G)=sqrt(5)", "coords": {} }, { "index": 1618, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.8284271247461903, "major_axis": "x", "x_coef": 9.0, "y_coef": 8.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1618.png", "answer": "[3,15]", "fact_expr": "G: Ellipse;C: Circle;A: Point;B: Point;P: Point;Expression(G) = (x^2/9 + y^2/8 = 1);Expression(C) = (y^2 + (x - 1)^2 = 1);PointOnCurve(P, G);IsDiameter(LineSegmentOf(A,B),C)", "coords": {} }, { "index": 1619, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1620, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 4.898979485566356, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1620.png", "answer": "24", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2 - y^2/24 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);3*Abs(LineSegmentOf(P, F1)) = 4*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 1621, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1622, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1622.png", "answer": "sqrt(3)-1", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);AngleOf(P, F1, F2) = pi/6;AngleOf(P, F2, F1) = pi/3;Eccentricity(G) = e;e: Number", "coords": {} }, { "index": 1623, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2135674953460693, "b": 2.4270472526550293, "orientation": "horizontal" }, "optimization": { "final_loss": 4.453341261978494e-06, "converged": true, "iterations": 125 }, "output_path": "results/train/hyperbola/problem_1623.png", "answer": "5*x^2-(5/4)*y^2=1", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 4*x);OneOf(Focus(G)) = Focus(H);Eccentricity(G) = sqrt(5)", "coords": {} }, { "index": 1624, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1624.png", "answer": "2*sqrt(2)", "fact_expr": "E: Curve;A: Point;P: Point;Expression(E) = (y^2 = 4*x);Coordinate(A) = (-1, 2);PointOnCurve(P, E);l: Line;Expression(l) = (x = -1)", "coords": { "A": [ -1.0, 2.0 ] } }, { "index": 1625, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.125, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1625.png", "answer": "(1,2)", "fact_expr": "G: Parabola;H: Line;P: Point;Expression(G) = (y = 2*x^2);Expression(H) = (y = 4*x - 5);PointOnCurve(P, G);WhenMin(Distance(P,H))", "coords": {} }, { "index": 1626, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.4641016151377544, "b": 1.7320508075688772, "major_axis": "y", "x_coef": 3.0, "y_coef": 12.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1626.png", "answer": "2*x-y+4=0", "fact_expr": "C: Ellipse;Expression(C) = (x^2/3 + y^2/12 = 1);G: LineSegment;IsChordOf(G, C) = True;P: Point;Coordinate(P) = (-1, 2);PointOnCurve(P, G) = True;MidPoint(G) = P", "coords": { "P": [ -1.0, 2.0 ] } }, { "index": 1627, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1627.png", "answer": "14", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);H: Line;PointOnCurve(Focus(G), H);A: Point;B: Point;x1: Number;x2: Number;y1: Number;y2: Number;Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);Intersection(H, G) = {A, B};x1 + x2 = 12", "coords": {} }, { "index": 1628, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1628.png", "answer": "1/4", "fact_expr": "C: Ellipse;a: Number;b: Number;a>b;b>0;F1: Point;F2: Point;B1: Point;B2: Point;A: Point;D: Point;Expression(C) = (x^2/a^2+y^2/b^2=1);LeftFocus(C)=F1;RightFocus(C)=F2;UpperVertex(C)=B1;LowerVertex(C)=B2;RightVertex(C)=A;Intersection(LineOf(A,B1), LineOf(B2,F1)) = D;2*Abs(LineSegmentOf(A,B1)) = 3*Abs(LineSegmentOf(B1,D))", "coords": {} }, { "index": 1629, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1629.png", "answer": "x + pm*sqrt(2)*y = 0", "fact_expr": "C: Hyperbola;a: Number;b: Number;a>0;b>0;Expression(C) = (-x^2/b^2 + y^2/a^2 = 1);Distance(Focus(C), OneOf(Asymptote(C))) = sqrt(2)*a", "coords": {} }, { "index": 1630, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1630.png", "answer": "9", "fact_expr": "l: Line;E: Parabola;C: Point;F: Point;A: Point;B: Point;Expression(E) = (x^2 = 8*y);Focus(E) = F;PointOnCurve(F, l);Intersection(l, E) = {A, B};Intersection(l,xAxis)=C;MidPoint(LineSegmentOf(C,F)) = A", "coords": {} }, { "index": 1631, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1631.png", "answer": "{0, 1/2}", "fact_expr": "G: Line;k: Real;H: Curve;Expression(G) = (y = k*x + 1);Expression(H) = (y^2 = 2*x);NumIntersection(G, H) = 1", "coords": {} }, { "index": 1632, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1632.png", "answer": "2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);A: Point;B: Point;PointOnCurve(A, G);PointOnCurve(B, G);Abs(LineSegmentOf(A, B)) = 6;M: Point;MidPoint(LineSegmentOf(A, B)) = M", "coords": {} }, { "index": 1633, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1633.png", "answer": "(x^2+3*y^2=1)&Negation(y=0)", "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2/3 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C);I: Point;Incenter(TriangleOf(P, F1, F2)) = I", "coords": {} }, { "index": 1634, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1635, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1636, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1636.png", "answer": "10/3", "fact_expr": "G: Hyperbola;H: Line;A: Point;F: Point;B: Point;l1: Line;l2: Line;Expression(G) = (x^2/9 - y^2/16 = 1);RightVertex(G) = A;RightFocus(G) = F;PointOnCurve(F, H);IsParallel(H, l1);Intersection(H, l2) = B;OneOf(Asymptote(G))=L1;OneOf(Asymptote(G))=L2;Negation(L1=L2)", "coords": {} }, { "index": 1637, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1637.png", "answer": "2+sqrt(5)", "fact_expr": "C1: Hyperbola;C2: Parabola;M: Point;F1: Point;F2: Point;x0:Number;y0:Number;p:Number;p>0;a:Number;b:Number;a>0;b>0;Expression(C1)=(-y^2/b^2 + x^2/a^2 = 1);Expression(C2)=(y^2=2*p*x);Coordinate(M) = (x0, y0);LeftFocus(C1) =F1;RightFocus(C1)=F2;Quadrant(M)=1;PointOnCurve(M,Asymptote(C1));IsPerpendicular(LineSegmentOf(M, F1), LineSegmentOf(M, F2));Focus(C2)=F2;PointOnCurve(M,C2)", "coords": {} }, { "index": 1638, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1638.png", "answer": "2", "fact_expr": "G: Parabola;M: Point;F: Point;O: Origin;Expression(G) = (y^2 = 6*x);PointOnCurve(M, G);Focus(G) = F;AngleOf(M, F, O) = ApplyUnit(60, degree)", "coords": {} }, { "index": 1639, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1639.png", "answer": "4*x - y - 15 = 0", "fact_expr": "Q: Point;Coordinate(Q) = (4, 1);G: Parabola;Expression(G) = (y^2 = 8*x);A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G);PointOnCurve(Q, LineSegmentOf(A, B)) ;MidPoint(LineSegmentOf(A,B)) = Q", "coords": { "Q": [ 4.0, 1.0 ] } }, { "index": 1640, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1640.png", "answer": "0", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 - y^2/16 = 1);LeftFocus(G) =F1;RightFocus(G)=F2;Abs(LineSegmentOf(P, F1)) + Abs(LineSegmentOf(P, F2)) = 10;PointOnCurve(P,G)", "coords": {} }, { "index": 1641, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1642, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1642.png", "answer": "{sqrt(10)/10, 3*sqrt(10)/10}", "fact_expr": "G: Parabola;b: Number;H: Ellipse;a: Number;F1: Point;M: Point;F2: Point;Expression(G) = (y^2 = 4*(b*x));a > b;b > 0;Expression(H) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(H) = F1;RightFocus(H) = F2;Focus(G) = M;Abs(LineSegmentOf(F1, M)) = 2*Abs(LineSegmentOf(F2, M))", "coords": {} }, { "index": 1643, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1643.png", "answer": "y=pm*x\nsqrt(2)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2/3 = 1)", "coords": {} }, { "index": 1644, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1644.png", "answer": "sqrt(2)", "fact_expr": "G: Parabola;C: Circle;r: Number;P: Point;Expression(G) = (y = x^2/4);r>0;Expression(C) = ((x - 1)^2 + (y - 2)^2 = r^2);Intersection(C,G)=P;IsTangent(TangentOnPoint(P,G),C)", "coords": {} }, { "index": 1645, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1645.png", "answer": "sqrt(3)/2", "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/4 = 1)", "coords": {} }, { "index": 1646, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1646.png", "answer": "x^2/5+y^2/4=1", "fact_expr": "P: Point;Coordinate(P) = (1, 1/2);H: Circle;Expression(H) = (x^2 + y^2 = 1);L1: Line;L2: Line;A: Point;B: Point;TangentOfPoint(P, H) = {L1, L2};TangentPoint(L1, H) = A;TangentPoint(L2, H) = B;G: Ellipse;Expression(G)=(y^2/b^2+x^2/a^2=1);b: Number;a: Number;a>b;b>0;PointOnCurve(Focus(G), LineOf(A, B));PointOnCurve(UpperVertex(G), LineOf(A, B))", "coords": { "P": [ 1.0, 0.5 ] } }, { "index": 1647, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1648, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4506710767745972, "b": 2.0511727333068848, "orientation": "horizontal" }, "optimization": { "final_loss": 4.657696081267204e-06, "converged": true, "iterations": 116 }, "output_path": "results/train/hyperbola/problem_1648.png", "answer": "sqrt(3)", "fact_expr": "C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(OneOf(Asymptote(C))) = (y = sqrt(2)*x)", "coords": {} }, { "index": 1649, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1649.png", "answer": "2", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(C) = F;A: Point;RightVertex(C) = A;B: Point;PointOnCurve(B, C);IsPerpendicular(LineSegmentOf(B, F), xAxis);Slope(LineSegmentOf(A, B)) = 3", "coords": {} }, { "index": 1650, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1650.png", "answer": "x^2/4-y^2=1", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;FocalLength(G) = 2*sqrt(5);2*Length(ImageinaryAxis(G)) = Length(RealAxis(G))", "coords": {} }, { "index": 1651, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1651.png", "answer": "(x-2)^2+y^2=10", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);Center(H) = Focus(G);H: Circle;L: Line;Expression(L) = (3*x - y + 4 = 0);IsTangent(L,H)", "coords": {} }, { "index": 1652, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.552875593585339, "b": 2.3303634978370953, "orientation": "horizontal" }, "optimization": { "final_loss": 1.3057248743840209e-06, "converged": true, "iterations": 98 }, "output_path": "results/train/hyperbola/problem_1652.png", "answer": "{sqrt(13)/2,sqrt(13)/3}", "fact_expr": "G: Hyperbola;Expression(Asymptote(G)) = (y = pm*(3/2)*x)", "coords": {} }, { "index": 1653, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1653.png", "answer": "sqrt(3)-1", "fact_expr": "O: Origin;C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;G: Line;PointOnCurve(F1, G);Slope(G) = sqrt(3)/3;P: Point;Intersection(G, C) = P;Abs(LineSegmentOf(O, P)) = Abs(LineSegmentOf(O, F2))", "coords": {} }, { "index": 1654, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1654.png", "answer": "16", "fact_expr": "G: Parabola;C: Ellipse;A: Point;B: Point;O:Origin;Expression(C) = (x^2/4 + y^2/3 = 1);Vertex(G)=O;LeftDirectrix(C)=Directrix(G);Intersection(RightDirectrix(C),G)={A,B}", "coords": {} }, { "index": 1655, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1656, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1656.png", "answer": "y^2 = 4*x", "fact_expr": "G: Parabola;p: Number;H: Circle;A: Point;B: Point;P: Point;a: Number;p>0;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (y^2 + (x - 1)^2 = 1);Coordinate(P) = (1, a);a > 1;PointOnCurve(P, G);l1: Line;l2: Line;TangentOfPoint(P, H) = {l1, l2};Intersection(l1, G) = A;Intersection(l2, G) = B;Slope(LineOf(A, B)) = -1", "coords": {} }, { "index": 1657, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1657.png", "answer": "x = -1/2\n2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x);F: Point;Focus(G) = F;M: Point;Coordinate(M) = (x0, y0);x0: Number;y0: Number;PointOnCurve(M, G);Abs(LineSegmentOf(M, F)) = 5/2", "coords": {} }, { "index": 1658, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1658.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x + 2*y - 1 = 0);IsPerpendicular(OneOf(Asymptote(G)), H)", "coords": {} }, { "index": 1659, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1659.png", "answer": "sqrt(2)/2", "fact_expr": "G: Ellipse;a: Number;b: Number;O: Origin;B: Point;F: Point;a > b;b > 0;Expression(G) = (y^2/b + x^2/a = 1);RightFocus(G) = F;UpperVertex(G) = B;Abs(LineSegmentOf(O, B)) = Abs(LineSegmentOf(O, F))", "coords": {} }, { "index": 1660, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1661, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.125, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1661.png", "answer": "y=4*x^2-12*x+9", "fact_expr": "G: Parabola;Q: Point;P: Point;M:Point;Expression(G) = (y = 2*x^2);Coordinate(Q) = (3, 0);PointOnCurve(P, G);MidPoint(LineSegmentOf(P,Q))=M", "coords": { "Q": [ 3.0, 0.0 ] } }, { "index": 1662, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.125, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1662.png", "answer": "(1, 2)", "fact_expr": "G: Parabola;A: Point;P: Point;Expression(G) = (y = 2*x^2);Coordinate(A) = (1, 3);PointOnCurve(P, G);WhenMin(Distance(P, A) + Distance(P, Focus(G)))", "coords": { "A": [ 1.0, 3.0 ] } }, { "index": 1663, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1663.png", "answer": "sqrt(21)/3", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (2*x - sqrt(3)*y = 0);OneOf(Asymptote(G))=H", "coords": {} }, { "index": 1664, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1665, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1665.png", "answer": "y = pm*2*x", "fact_expr": "C: Hyperbola;b: Number;a: Number;l: Line;G:Line;M: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;OneOf(Asymptote(C)) = l;PointOnCurve(F2, G);IsParallel(l, G);Intersection(G, C) = M;Abs(LineSegmentOf(M, F1)) = 2*Abs(LineSegmentOf(M, F2))", "coords": {} }, { "index": 1666, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1666.png", "answer": "(0,2)", "fact_expr": "G: Parabola;Expression(G) = (y = x^2/8)", "coords": {} }, { "index": 1667, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.8284271247461903, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1667.png", "answer": "sqrt(6)/2; y=pm*(sqrt(2)/2)*x", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/8 - y^2/4 = 1)", "coords": {} }, { "index": 1668, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1668.png", "answer": "2", "fact_expr": "M: Point;N: Point;Coordinate(N)=(2,0);G:Curve;Expression(G)=(y^2 = 8*x);PointOnCurve(M,G)", "coords": { "N": [ 2.0, 0.0 ] } }, { "index": 1669, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1669.png", "answer": "(1,\\sqrt{2})", "fact_expr": "H: Line;Expression(H) = (y = k*x - 1);k: Number;G: Hyperbola;Expression(G) = (x^2 - y^2 = 1);P1: Point;P2: Point;Intersection(H,RightPart(G)) = {P1,P2};Negation(P1=P2)", "coords": {} }, { "index": 1670, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1670.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "l: Line;C: Parabola;A: Point;F: Point;B: Point;Expression(C) = (x^2 = 4*y);Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};Negation(A = B)", "coords": {} }, { "index": 1671, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1672, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1673, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1673.png", "answer": "4", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2:Point;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(P, G);OneOf(Focus(G)) = F1;OneOf(Focus(G))=F2;Distance(P, F1) = 6;Negation(F1=F2)", "coords": {} }, { "index": 1674, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1674.png", "answer": "(1+sqrt(5))/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;c: Number;I: Circle;F: Point;E: Point;O: Origin;P: Point;l1:Line;b > a;a > 0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 4*(c*x));Expression(I) = (x^2 + y^2 = a^2);Coordinate(F) = (-c, 0);LeftFocus(G) = F;TangentOfPoint(F, I) =l1;TangentPoint(l1,I)=E;Intersection(OverlappingLine(LineSegmentOf(F,E)), H) =P;VectorOf(O, E) = (VectorOf(O, F) + VectorOf(O, P))/2;c>0", "coords": {} }, { "index": 1675, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0707900524139404, "b": 1.5409371596115455, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 3.1717604031648706e-06, "converged": true, "iterations": 78 }, "output_path": "results/train/ellipse/problem_1675.png", "answer": "9", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2/5 + y^2/m = 1);PointOnCurve(Focus(G), yAxis);Eccentricity(G)=2/3", "coords": {} }, { "index": 1676, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7195594310760498, "b": 1.7224669456481934, "orientation": "horizontal" }, "optimization": { "final_loss": 2.391175712546101e-06, "converged": true, "iterations": 83 }, "output_path": "results/train/hyperbola/problem_1676.png", "answer": "x^2/8 - y^2/8 = 1", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;P: Point;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (0, 4);LeftFocus(G) = F;Eccentricity(G) = sqrt(2);PointOnCurve(F, H);PointOnCurve(P, H);IsParallel(H, OneOf(Asymptote(G)))", "coords": { "P": [ 0.0, 4.0 ] } }, { "index": 1677, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1678, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1678.png", "answer": "sqrt(5)/2", "fact_expr": "C2:Hyperbola;C1:Parabola;A: Point;B: Point;C: Point;D: Point;p:Number;p>0;a:Number;b:Number;a>0;b>0;L1:Line;L2:Line;O:Origin;Expression(C1)=(y^2=2*p*x);Expression(C2)=(x^2/a^2-y^2/b^2=1);Intersection(C1,C2)={A,B};Asymptote(C2)={L1,L2};Intersection(C1,L1)=C;Intersection(C1,L2)=D;Negation(O=C);Negation(O=D);PointOnCurve(Focus(C2),LineSegmentOf(A,B));PointOnCurve(Focus(C1),LineSegmentOf(C,D))", "coords": {} }, { "index": 1679, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1679.png", "answer": "x^2=sqrt(2)*y", "fact_expr": "C1: Parabola;Expression(C1) = (y^2 = 4*x);C2: Parabola;Expression(C2) = (x^2 = 2*p*y);p: Number;p>0;O: Origin;A: Point;Negation(A=O);Intersection(C1, C2) = A;Distance(A, Focus(C1)) = 3", "coords": {} }, { "index": 1680, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1681, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1682, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1682.png", "answer": "(2*sqrt(3)+sqrt(6))/6", "fact_expr": "F1: Point;F2: Point;G: Hyperbola;Expression(G) = (x^2/3 - y^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;H: Line;Expression(H) = (x - sqrt(2)*y + 4 = 0);PointOnCurve(P, H)", "coords": {} }, { "index": 1683, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1683.png", "answer": "x^2 = -8*y", "fact_expr": "G: Parabola;M: Point;O: Origin;m: Number;Coordinate(M) = (m, -2);Vertex(G) = O;PointOnCurve(Focus(G), yAxis);PointOnCurve(M, G);Distance(M, Focus(G)) = 4", "coords": {} }, { "index": 1684, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1684.png", "answer": "12+8*sqrt(2)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 16*x);F: Point;Focus(G) = F;PointOnCurve(F, L) = True;Intersection(L, G) = {A, B};L: Line;A: Point;B: Point", "coords": {} }, { "index": 1685, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1685.png", "answer": "{21/8, 51/8}", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;A: Point;PointOnCurve(A, G);H: Line;Expression(H) = (3*x + 4*y + 12 = 0);Min(Distance(A, H)) = 1", "coords": {} }, { "index": 1686, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1687, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1687.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;F: Point;Focus(G) = F;H: Line;Expression(H) = (y = sqrt(3)*x);Distance(F, H) = sqrt(3)", "coords": {} }, { "index": 1688, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1688.png", "answer": "3/5", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G)=(y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(60, degree);R:Number;r:Number;Radius(CircumCircle(TriangleOf(P,F1,F2)))=R;Radius(InscribedCircle(TriangleOf(P,F1,F2)))=r;R = 3*r;a: Number;b: Number;a>0;b>0", "coords": {} }, { "index": 1689, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1689.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;a: Number;H: Point;a>0;Expression(G) = (-y^2/4 + x^2/a^2 = 1);Coordinate(H) = (1, a);PointOnCurve(H, OneOf(Asymptote(G)))", "coords": {} }, { "index": 1690, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_focus_mismatch", "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1690.png", "answer": "x^2/6+y^2/2=1", "fact_expr": "F: Point;RightFocus(C) = F;Coordinate(F) = (2, 0);C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;l: Line;m: Number;Expression(l) = (y = m - x/3);A: Point;B: Point;Intersection(l, C) = {A, B};P: Point;MidPoint(LineSegmentOf(A, B)) = P;O: Origin;k: Number;Slope(LineOf(O, P)) = k;k = 1", "coords": { "F": [ 2.0, 0.0 ] }, "validation_reasons": [ "ellipse_focus_mismatch" ] }, { "index": 1691, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1692, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1693, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1693.png", "answer": "7", "fact_expr": "G: Parabola;H: Ellipse;m: Number;Expression(G) = (y^2 = 8*x);Expression(H) = (y^2/3 + x^2/m = 1);PointOnCurve(OneOf(Focus(H)), Directrix(G))", "coords": {} }, { "index": 1694, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1695, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1696, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1696.png", "answer": "(-\\infty,-1)", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;M: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);F1: Point;LeftFocus(G) = F1;F2: Point;RightFocus(G) = F2;PointOnCurve(F2, H);OneOf(Intersection(H,G)) = M;Inclination(H) = ApplyUnit(120, degree);IsPerpendicular(LineSegmentOf(M,F1),xAxis)", "coords": {} }, { "index": 1697, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1697.png", "answer": "x=-4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 16*x)", "coords": {} }, { "index": 1698, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1698.png", "answer": "4", "fact_expr": "H: Parabola;Expression(H) = (y^2 = 2*(p*x));p: Real;G: Hyperbola;Expression(G) = (x^2/3 - y^2 = 1);Focus(H) = RightFocus(G)", "coords": {} }, { "index": 1699, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1699.png", "answer": "{0, -1, 1/2}", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);l: Line;M: Point;Coordinate(M) = (-2, 1);PointOnCurve(M, l);k: Number;Slope(l) = k;NumIntersection(l, G) = 1", "coords": { "M": [ -2.0, 1.0 ] } }, { "index": 1700, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1701, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1701.png", "answer": "18", "fact_expr": "F: Point;G: Parabola;Expression(G) = (y^2 = 12*x);Focus(G) = F;A: Point;B: Point;C: Point;PointOnCurve(A, G) = True;PointOnCurve(B, G) = True;PointOnCurve(C, G) = True;VectorOf(F, A) + VectorOf(F, B) + VectorOf(F, C) = 0", "coords": {} }, { "index": 1702, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1703, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1703.png", "answer": "2", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);G: Circle;P: Point;Q: Point;F: Point;l:Line;Focus(C) = F;Negation(P=Q);PointOnCurve(P,l);TangentPoint(l,C)=Q;IsDiameter(LineSegmentOf(P,Q),G);PointOnCurve(F,G)", "coords": {} }, { "index": 1704, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1704.png", "answer": "2*sqrt(5)/5", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;a > b;b > 0;Expression(G) = (x^2/b^2 + y^2/a^2 = 1);Expression(H) = (x - 2*y + 2 = 0);PointOnCurve(OneOf(Focus(G)),H);PointOnCurve(OneOf(Vertex(G)),H)", "coords": {} }, { "index": 1705, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1706, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1706.png", "answer": "(-4,4*\\sqrt{3})", "fact_expr": "G: Parabola;M: Point;F: Point;Expression(G) = (y^2 = -12*x);Focus(G) = F;Abs(LineSegmentOf(M, F)) = 7;PointOnCurve(M,G);Quadrant(M)=2", "coords": {} }, { "index": 1707, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1707.png", "answer": "1/3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, G) = {A, B};C: Point;Intersection(OverlappingLine(LineSegmentOf(F, B)), Directrix(G)) = C;Abs(LineSegmentOf(B, C)) = 2*Abs(LineSegmentOf(B, F))", "coords": {} }, { "index": 1708, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1708.png", "answer": "1", "fact_expr": "l: Line;Expression(l) = (y = x - 1);G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;A: Point;B: Point;p>0;Intersection(l, G) = {A, B};XCoordinate(MidPoint(LineSegmentOf(A, B))) = 2;IsChordOf(LineSegmentOf(A, B), G)", "coords": {} }, { "index": 1709, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1709.png", "answer": "4", "fact_expr": "G: Parabola;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(A, G);PointOnCurve(B,G);VectorOf(A, F) = 3*VectorOf(F, B);IsChordOf(LineSegmentOf(A, B), G)", "coords": {} }, { "index": 1710, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1710.png", "answer": "x^2/4+y^2=1", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(OneOf(Focus(C))) = (sqrt(3), 0);Eccentricity(C)=sqrt(3)/2;a: Number;b: Number", "coords": {} }, { "index": 1711, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1712, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1712.png", "answer": "18", "fact_expr": "G: Parabola;p: Number;A: Point;B:Point;F: Point;p>0;y1:Number;y2:Number;Expression(G) = (y^2 = 2*(p*x));Coordinate(A) = (2, y1);Coordinate(B) = (1/2, y2);Focus(G) = F;PointOnCurve(A,G);PointOnCurve(B,G);Quadrant(A)=1;Quadrant(B)=4;Abs(LineSegmentOf(A, F)) = 20", "coords": {} }, { "index": 1713, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1713.png", "answer": "y=pm*(sqrt(2)/2)*x", "fact_expr": "H: Ellipse;Expression(H) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;G: Hyperbola;Expression(G) = (-y^2/n^2 + x^2/m^2 = 1);m: Number;n: Number;m>0;n>0;F1: Point;F2: Point;Focus(H) = {F1, F2};Focus(G) = {F1, F2};P: Point;OneOf(Intersection(H, G)) = P;AngleOf(F1, P, F2) = (2/3)*pi;WhenMin(Eccentricity(H)*Eccentricity(G))", "coords": {} }, { "index": 1714, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1715, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1716, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1717, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1717.png", "answer": "1/2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/4 = 1);F: Point;RightFocus(G) = F;H: Line;PointOnCurve(F, H);IsParallel(H, OneOf(Asymptote(G))) ;Intersection(H, G) = P;P: Point;M: Point;PointOnCurve(M, LineOf(P, F)) ;DotProduct(VectorOf(O, M), VectorOf(P, F)) = 0;O: Origin", "coords": {} }, { "index": 1718, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1718.png", "answer": "y=pm*x/2", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;RightFocus(C) = F2;F2: Point;L: Line;PointOnCurve(F2, L) = True;IsPerpendicular(L, OneOf(Asymptote(C))) = True;FootPoint(L, OneOf(Asymptote(C))) = P;P: Point;Intersection(L, C) = A;A: Point;VectorOf(F2, P) = 3*VectorOf(F2, A)", "coords": {} }, { "index": 1719, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1719.png", "answer": "2", "fact_expr": "C: Parabola;p: Number;G: Line;M: Point;A: Point;B: Point;l: Line;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(M) = (1, 0);Directrix(C) = l;PointOnCurve(M, G);Slope(G) = sqrt(3);Intersection(G, l) = A;OneOf(Intersection(G, C)) = B;VectorOf(A, M) = VectorOf(M, B)", "coords": { "M": [ 1.0, 0.0 ] } }, { "index": 1720, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1720.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F: Point;A: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftVertex(G)=A;RightFocus(G)=F;PointOnCurve(P,RightPart(G));AngleOf(P,F,A)=2*AngleOf(P,A,F)", "coords": {} }, { "index": 1721, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1721.png", "answer": "[\\sqrt{2} - 1, 1)", "fact_expr": "G: Ellipse;b: Number;a: Number;c: Number;F1: Point;F2: Point;P: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);c*VectorOf(P, F2) = a*VectorOf(P, F1)", "coords": {} }, { "index": 1722, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1723, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.0625, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1723.png", "answer": "1/16", "fact_expr": "G: Parabola;Expression(G) = (y = 4*x^2);P: Point;PointOnCurve(P, G)", "coords": {} }, { "index": 1724, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1724.png", "answer": "(\\sqrt{2}/2,1)", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);D:Circle;b: Number;a: Number;F: Point;a > b;b > 0;c:Number;Coordinate(F) = (-c, 0);LeftFocus(G) = F;Center(D)=F;Radius(D)=c;NumIntersection(H,LeftDirectrix(G))=2", "coords": {} }, { "index": 1725, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1725.png", "answer": "sqrt(6)/2", "fact_expr": "G: Hyperbola;a: Number;b: Number;P: Point;F1: Point;F2: Point;I: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, RightPart(G));LeftFocus(G) = F1;RightFocus(G) = F2;Incenter(TriangleOf(P, F1, F2)) = I;Area(TriangleOf(I, P, F1)) = (sqrt(6)/3)*Area(TriangleOf(I, F1, F2)) + Area(TriangleOf(I, P, F2))", "coords": {} }, { "index": 1726, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1727, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1727.png", "answer": "(-2, -1)", "fact_expr": "E: Ellipse;Expression(E) = (x^2/4 + y^2/3 = 1);H: Point;Coordinate(H) = (2, 0);OneOf(Vertex(E)) = H;P: Point;PointOnCurve(P, xAxis) ;Coordinate(P) = (t, 0);t: Real;M: Point;PointOnCurve(M, E);IsPerpendicular(LineSegmentOf(M, P), LineSegmentOf(M, H))", "coords": { "H": [ 2.0, 0.0 ] } }, { "index": 1728, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 5.0, "b": 12.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1728.png", "answer": "10", "fact_expr": "G: Hyperbola;P: Point;Expression(G) = (x^2/25 - y^2/144 = 1);PointOnCurve(P,LeftPart(G));Distance(P, LeftFocus(G)) = 16", "coords": {} }, { "index": 1729, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1729.png", "answer": "3*sqrt(2)/4", "fact_expr": "G: Parabola;p: Number;A: Point;F: Point;M: Point;B: Point;l: Line;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(A) = (0, 2);Focus(G) = F;Intersection(LineSegmentOf(F, A), G) = B;Directrix(G) = l;l1: Line;PointOnCurve(B, l1);IsPerpendicular(l1, l);FootPoint(l1, l) = M;IsPerpendicular(LineSegmentOf(A, M), LineSegmentOf(M, F));S: Number;Area(TriangleOf(A, F, M)) = S", "coords": { "A": [ 0.0, 2.0 ] } }, { "index": 1730, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1730.png", "answer": "y^2=2*(x-1)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;PointOnCurve(F,l) = True;l: Line;Intersection(l,G) = {A,B};A: Point;B: Point;IsChordOf(LineSegmentOf(A,B),G) = True", "coords": {} }, { "index": 1731, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1732, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1732.png", "answer": "sqrt(3)+1", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(C) = F;G: Line;Expression(G) = (y = sqrt(3)*x);A: Point;B: Point;Intersection(G, C) = {A, B};O: Origin;N: Point;M: Point;MidPoint(LineSegmentOf(A,F)) = M;MidPoint(LineSegmentOf(B,F)) = N;DotProduct(VectorOf(O,M),VectorOf(O,N)) = 0", "coords": {} }, { "index": 1733, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1733.png", "answer": "(-\\infty,-4)+(-4,0)", "fact_expr": "G: Ellipse;k: Real;Expression(G)=(x^2/4 - y^2/k = 1)", "coords": {} }, { "index": 1734, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2631799883090584, "b": 2.5275303393962245, "orientation": "horizontal" }, "optimization": { "final_loss": 4.481942266905428e-06, "converged": true, "iterations": 95 }, "output_path": "results/train/hyperbola/problem_1734.png", "answer": "4", "fact_expr": "G: Hyperbola;lambda: Number;Expression(G) = (x^2 - y^2/lambda = 1);lambda>0;Expression(OneOf(Asymptote(G))) = (y = 2*x)", "coords": {} }, { "index": 1735, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1736, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1737, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1737.png", "answer": "sqrt(5)/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;P: Point;A: Point;B: Point;a>0;b>0;Expression(G) = (-x^2/b^2 + y^2/a^2 = 1);PointOnCurve(P,G);PointOnCurve(P,H);IsParallel(xAxis,H);DotProduct(VectorOf(P, A), VectorOf(P, B)) = -a^2/4;L1:Line;L2:Line;Asymptote(G)={L1,L2};Intersection(H,L1)=A;Intersection(H,L2)=B", "coords": {} }, { "index": 1738, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1738.png", "answer": "3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;A: Point;x0: Number;Coordinate(A) = (x0, -3);PointOnCurve(A, G);Distance(A, Focus(G)) = 2*Distance(A, yAxis)", "coords": {} }, { "index": 1739, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1739.png", "answer": "sqrt(65)/5", "fact_expr": "G: Hyperbola;b: Number;a: Number;F1: Point;A: Point;F2: Point;B: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(A, RightPart(G));Intersection(LineSegmentOf(A,F1),LeftPart(G))=B;IsPerpendicular(LineSegmentOf(A, F2), LineSegmentOf(B, F2));Abs(LineSegmentOf(B, F1)) = Abs(LineSegmentOf(A, F2))/3", "coords": {} }, { "index": 1740, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1740.png", "answer": "sqrt(15)/3", "fact_expr": "C: Hyperbola;b: Number;a: Number;M: Point;F2: Point;N: Point;F1: Point;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;L:Line;PointOnCurve(F1, L);IsPerpendicular(L,xAxis);Intersection(L, C) = {M, N};Cos(AngleOf(M, F2, N)) = 7/8", "coords": {} }, { "index": 1741, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1741.png", "answer": "x^2/16+y^2/4=1", "fact_expr": "G: Ellipse;b: Number;a: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = sqrt(3)/2;Length(MinorAxis(G))=4", "coords": {} }, { "index": 1742, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1742.png", "answer": "4*sqrt(3)/3", "fact_expr": "G: Line;H: Curve;Expression(G) = (y = x);Expression(H) = (2*x^2 + y^2 = 2)", "coords": {} }, { "index": 1743, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1744, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1744.png", "answer": "2*sqrt(3)", "fact_expr": "Gamma: Ellipse;C: Circle;O: Origin;M: Point;N: Point;l:Ray;Expression(Gamma)=(x^2/2 + y^2 = 1);Expression(C) = ((x - 1)^2 + (y - 1)^2 = 2);PointOnCurve(O, l);Intersection(l,C)=M;Intersection(l,Gamma)=N;Negation(M=O)", "coords": {} }, { "index": 1745, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1745.png", "answer": "[10,+\\infty)", "fact_expr": "C: Parabola;p: Real;G: Line;A: Point;M: Point;B: Point;p>0;Expression(C) = (y^2 = 2*p*x);Expression(G) = (3*x + 4*y + 25 = 0);Focus(C) = A;Intersection(Directrix(C), xAxis) = B;PointOnCurve(M, G);AngleOf(A, M, B) = ApplyUnit(90, degree)", "coords": {} }, { "index": 1746, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1746.png", "answer": "(1, \\sqrt{5}]", "fact_expr": "H: Line;Expression(H) = (y = 2*x);G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;NumIntersection(H, G) = 0", "coords": {} }, { "index": 1747, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1747.png", "answer": "9", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 2*y);H: Line;PointOnCurve(Focus(G), H) = True;A: Point;B: Point;Intersection(H, G) = {A, B};YCoordinate(MidPoint(LineSegmentOf(A,B))) = 4", "coords": {} }, { "index": 1748, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1748.png", "answer": "sqrt(5)", "fact_expr": "C: Hyperbola;b: Number;a: Number;H: Circle;P: Point;F2: Point;F1: Point;G: Point;e: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P,RightPart(C));Expression(H) = (x^2 + y^2 = b^2);TangentPoint(LineSegmentOf(P, F2), H) = G;MidPoint(LineSegmentOf(P, F2)) = G;Eccentricity(C) = e", "coords": {} }, { "index": 1749, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1749.png", "answer": "3", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 8*x);F: Point;Focus(C) = F;l: Line;Directrix(C) = l;P: Point;PointOnCurve(P, l);Q: Point;OneOf(Intersection(LineOf(P, F), C)) = Q;O: Origin;VectorOf(F, P) = 4*VectorOf(F, Q)", "coords": {} }, { "index": 1750, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1750.png", "answer": "3", "fact_expr": "l: Line;E: Parabola;p: Number;B: Point;F: Point;A: Point;p>0;Expression(E) = (y^2 = 2*p*x);Focus(E) = F;PointOnCurve(F, l);Slope(l) = sqrt(3);Intersection(l, E) = {A, B};Abs(Abs(LineSegmentOf(A,F))-Abs(LineSegmentOf(B,F)))=4", "coords": {} }, { "index": 1751, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1752, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1752.png", "answer": "sqrt(2)/2", "fact_expr": "E: Ellipse;b: Number;a: Number;G: Line;A: Point;F1: Point;B: Point;F2: Point;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(E) = F1;RightFocus(E) = F2;PointOnCurve(F1, G);Intersection(G, E) = {A, B};Abs(LineSegmentOf(A, F1)) = 3*Abs(LineSegmentOf(B, F1));Cos(AngleOf(A, F2, B)) = 3/5", "coords": {} }, { "index": 1753, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_1753.png", "answer": "1", "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (-y^2/3 + x^2/m = 1);Eccentricity(G) = 2", "coords": {} }, { "index": 1754, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1754.png", "answer": "7/2", "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (y^2 = 2*x);Coordinate(A) = (3, 2);Focus(G) = F;PointOnCurve(P, G)", "coords": { "A": [ 3.0, 2.0 ] } }, { "index": 1755, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.0625, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1755.png", "answer": "(0, 1/16)", "fact_expr": "G: Parabola;Expression(G) = (y = 4*x^2)", "coords": {} }, { "index": 1756, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1757, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1757.png", "answer": "2", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;l: Line;Expression(l) = (y=4*x+m);m: Number;Intersection(l, C) = {A, B};PointOnCurve(Focus(C), l) = True;A: Point;B: Point;Abs(LineSegmentOf(A, B)) = 17/4", "coords": {} }, { "index": 1758, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1758.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;H: Circle;Expression(H) = (y^2 + (x - 2)^2 = 3);IsTangent(Asymptote(G), H) = True", "coords": {} }, { "index": 1759, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1759.png", "answer": "(0,pm*3)", "fact_expr": "H: ConicSection;Expression(H) = (x^2 + y^2/10 = 1)", "coords": {} }, { "index": 1760, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.449489742783178, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 6.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1760.png", "answer": "sqrt(3) + sqrt(6)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/6 + y^2/3 = 1);F: Point;LeftFocus(G) = F;PointOnCurve(A, G) = True;Negation(A = Vertex(G));A: Point;O: Origin;MidPoint(LineSegmentOf(A, F)) = B;B: Point", "coords": {} }, { "index": 1761, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1761.png", "answer": "x=-2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x)", "coords": {} }, { "index": 1762, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1762.png", "answer": "(\\sqrt{5}/5, 3/5)", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Circle;c: Number;e: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2 + y^2 = (b/2 + c)^2);Eccentricity(G) = e;HalfFocalLength(G)=c;NumIntersection(G,H)=4", "coords": {} }, { "index": 1763, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1763.png", "answer": "sqrt(10)/2", "fact_expr": "l: Line;E: Hyperbola;b: Number;a: Number;F2: Point;Q: Point;P: Point;F1: Point;a>0;b>0;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(E) = F1;RightFocus(E) = F2;PointOnCurve(F1, l);Intersection(l, LeftPart(E)) = {P, Q};Abs(LineSegmentOf(P, F1)) = 3*Abs(LineSegmentOf(F1, Q));IsPerpendicular(LineSegmentOf(F2, Q), LineSegmentOf(P, Q))", "coords": {} }, { "index": 1764, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1765, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1765.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;H: Line;Expression(H) = (x - y + 3 = 0);IsParallel(OneOf(Asymptote(G)),H) = True", "coords": {} }, { "index": 1766, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1767, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1767.png", "answer": "9", "fact_expr": "G: Hyperbola;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/16 - y^2/9 = 1);AngleOf(F1, P, F2) = pi/2;Focus(G) = {F1, F2};PointOnCurve(P, G)", "coords": {} }, { "index": 1768, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1768.png", "answer": "sqrt(5)/3", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;H: Line;Expression(H) = (y = x);M: Point;N: Point;Intersection(G, H) = {M, N};P: Point;PointOnCurve(P, G);Slope(LineOf(M, P))*Slope(LineOf(N, P)) = -4/9", "coords": {} }, { "index": 1769, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1769.png", "answer": "sqrt(3)+1", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;c: Number;FocalLength(C) = 2*c;G: Line;Expression(G) = (y = -sqrt(3)*(-c + x));P: Point;OneOf(Intersection(G, C)) = P;AngleOf(P, F2, F1) = 2*AngleOf(P, F1, F2)", "coords": {} }, { "index": 1770, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1771, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1771.png", "answer": "y = pm*sqrt(2)/2*x", "fact_expr": "C: Hyperbola;a: Number;b: Number;a > 0;b > 0;Expression(C) = (x^2/a^2 - y^2/b^2 = 1);A: Point;B: Point;PointOnCurve(A, C);PointOnCurve(B, C);VectorOf(A, O) = 1/2*VectorOf(A, B);O: Origin;M: Point;PointOnCurve(M, C);Negation(M = A);Negation(M = B);Slope(LineOf(M, A)) * Slope(LineOf(M, B)) = 1/2", "coords": {} }, { "index": 1772, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1773, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1773.png", "answer": "9", "fact_expr": "l: Line;G: Hyperbola;Expression(G) = (-y^2/7 + x^2/m = 1);m: Number;A: Point;B: Point;F2: Point;F1: Point;LeftFocus(G)=F1;PointOnCurve(F1,l);Intersection(l,LeftPart(G))={A,B};Abs(LineSegmentOf(A, B)) = 4;Perimeter(TriangleOf(A, B, F2)) = 20;RightFocus(G)=F2", "coords": {} }, { "index": 1774, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1774.png", "answer": "4", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/5 = 1);P: Point;PointOnCurve(P, G);l1: Line;l2: Line;Z1: Line;Z2: Line;Asymptote(G) = {Z1, Z2};PointOnCurve(P, l1);PointOnCurve(P, l2);IsParallel(Z1, l1);IsParallel(Z2, l2);M: Point;N: Point;Intersection(l1, xAxis) = M;Intersection(l2, xAxis) = N;O: Origin", "coords": {} }, { "index": 1775, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1775.png", "answer": "x^2+3*y^2/2=1", "fact_expr": "E: Ellipse;G: Line;A: Point;F1: Point;B: Point;F2: Point;b:Number;00;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;PointOnCurve(F, H);Slope(H) = 2*sqrt(2);Intersection(H, G) = {A, B};Quadrant(A)=1", "coords": {} }, { "index": 1777, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1778, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1778.png", "answer": "24/7", "fact_expr": "l: Line;G: Ellipse;a: Number;C: Point;D: Point;F: Point;a>0;Expression(G) = (y^2/3 + x^2/a^2 = 1);Coordinate(F) = (-1, 0);OneOf(Focus(G)) = F;PointOnCurve(F, l);Slope(l) = 1;Intersection(l, G) = {C, D}", "coords": { "F": [ -1.0, 0.0 ] } }, { "index": 1779, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1780, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1780.png", "answer": "6*sqrt(2)", "fact_expr": "G: Parabola;H: Line;P: Point;d1:Number;d2:Number;Expression(G) = (y^2 = -8*x);Expression(H) = (x + y - 10 = 0);PointOnCurve(P, G);Distance(P,Directrix(G)) = d1;Distance(P,H)=d2", "coords": {} }, { "index": 1781, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1781.png", "answer": "(0,4)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/m = 1);PointOnCurve(Focus(G), xAxis);m: Real", "coords": {} }, { "index": 1782, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1782.png", "answer": "sqrt(3)+1", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;A: Point;F: Point;B: Point;O:Origin;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;PointOnCurve(O,l);Inclination(l)=pi/3;Intersection(l,LeftPart(C))=A;Intersection(l,RightPart(C))=B;DotProduct(VectorOf(A, F), VectorOf(B, F)) = 0", "coords": {} }, { "index": 1783, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1783.png", "answer": "y+pm*sqrt(3)*x=0", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;LeftFocus(G) = F;A: Point;RightVertex(G) = A;H: Line;Expression(H) = (x = a);Intersection(H, OneOf(Asymptote(G))) = B;B: Point;AngleOf(B, F, A) = ApplyUnit(30, degree)", "coords": {} }, { "index": 1784, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1784.png", "answer": "5/3", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;Distance(RightFocus(G), Asymptote(G)) = (1/2)*Distance(RightFocus(G), LeftVertex(G));e: Number;Eccentricity(G) = e", "coords": {} }, { "index": 1785, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1785.png", "answer": "8/13", "fact_expr": "l: Line;G: Parabola;Expression(G) = (y^2 = 4*x);Expression(l)=(5*x+12*y+3=0)", "coords": {} }, { "index": 1786, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1786.png", "answer": "sqrt(17)/2", "fact_expr": "G: Parabola;H: Point;P: Point;Expression(G) = (y^2 = 2*x);Coordinate(H) = (0, 2);PointOnCurve(P, G)", "coords": { "H": [ 0.0, 2.0 ] } }, { "index": 1787, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1788, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1788.png", "answer": "sqrt(2)/2", "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;A: Point;F2: Point;O: Origin;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(A,G);IsPerpendicular(LineSegmentOf(A, F2), LineSegmentOf(F1, F2));Distance(O, LineOf(A,F1)) = Abs(LineSegmentOf(O, F1))/3", "coords": {} }, { "index": 1789, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1789.png", "answer": "7/6", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;a <= sqrt(3)*b;e: Number;Eccentricity(G) = e", "coords": {} }, { "index": 1790, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1791, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1791.png", "answer": "9", "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (x^2/16 - y^2/m = 1);FocalLength(G) = 10", "coords": {} }, { "index": 1792, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.125, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1792.png", "answer": "1/4", "fact_expr": "G: Parabola;F: Point;l: Line;Expression(G) = (x^2 = y/2);Focus(G) = F;Directrix(G) = l", "coords": {} }, { "index": 1793, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1793.png", "answer": "(1,2]", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);E: Hyperbola;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Negation(Distance(Focus(G), Asymptote(E)) > sqrt(3))", "coords": {} }, { "index": 1794, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1794.png", "answer": "4/3", "fact_expr": "C: Ellipse;P: Point;A: Point;F: Point;Expression(C) = (x^2/2 + y^2 = 1);RightFocus(C) = F;UpperVertex(C) = A;PointOnCurve(P, C);WhenMax(Perimeter(TriangleOf(P, A, F)))", "coords": {} }, { "index": 1795, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1796, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1797, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1797.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;OneOf(Focus(G)) = F;L: Line;PointOnCurve(F, L);IsPerpendicular(L, OneOf(Asymptote(G)));O: Origin;PointOnCurve(FootPoint(L, OneOf(Asymptote(G))), PerpendicularBisector(LineSegmentOf(O, F)))", "coords": {} }, { "index": 1798, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1798.png", "answer": "1/8", "fact_expr": "G: Ellipse;P: Point;F2: Point;F1: Point;Expression(G) = (x^2/25 + y^2/9 = 1);PointOnCurve(P, G);LeftFocus(G)=F1;RightFocus(G)=F2;Distance(P,F1)=4*Distance(P,F2)", "coords": {} }, { "index": 1799, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1799.png", "answer": "2*sqrt(3)/3", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Distance(RightVertex(C), OneOf(Asymptote(C)))=a/2", "coords": {} }, { "index": 1800, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1800.png", "answer": "x-y-3=0", "fact_expr": "G: Hyperbola;H: LineSegment;P: Point;Expression(G) = (x^2 - 4*y^2 = 4);Coordinate(P) = (4, 1);IsChordOf(H,G);MidPoint(H)=P", "coords": { "P": [ 4.0, 1.0 ] } }, { "index": 1801, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1802, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1802.png", "answer": "3", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F2: Point;F1: Point;O:Origin;b > a;a > 0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);Distance(P,O) = b;Sin(AngleOf(P, F2, F1)) = (5/3)*Sin(AngleOf(P, F1, F2))", "coords": {} }, { "index": 1803, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1803.png", "answer": "2", "fact_expr": "G: Ellipse;l1: Line;l2: Line;M: Point;N: Point;P: Point;Expression(G) = (x^2 + y^2/4 = 1);Expression(l1) = (y = 2*x);Expression(l2) = (y = -2*x);PointOnCurve(P, G);L1:Line;L2:Line;PointOnCurve(P, L1);PointOnCurve(P, L2);IsParallel(l1, L1);IsParallel(l2, L2);Intersection(L1, l2) = M;Intersection(L2, l1) = N", "coords": {} }, { "index": 1804, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -1.0, 0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_1804.png", "answer": "3", "fact_expr": "C: Ellipse;a: Number;M: Circle;A: Point;B: Point;P: Point;a>1;Expression(C) = (y^2/(a^2 - 1) + x^2/a^2 = 1);Expression(M) = (y^2 + (x - 1)^2 = 1);PointOnCurve(P, C);IsDiameter(LineSegmentOf(A,B),M);Max(DotProduct(VectorOf(P, A), VectorOf(P, B))) = 15", "coords": {} }, { "index": 1805, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1805.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, LeftPart(G));LeftFocus(G) = F1;RightFocus(G) = F2;Cos(AngleOf(P, F1, F2)) = Sin(AngleOf(P, F2, F1));Sin(AngleOf(P, F2, F1)) = sqrt(5)/5", "coords": {} }, { "index": 1806, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1807, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1808, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1808.png", "answer": "2", "fact_expr": "l: Line;C: Parabola;p: Number;A: Point;B: Point;D: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C)=F;PointOnCurve(F,l);Slope(l)=1/2;Intersection(l,C)={A,B};IsChordOf(LineSegmentOf(A,B),C);MidPoint(LineSegmentOf(A,B))=D;F:Point;Distance(D, Directrix(C)) = 10", "coords": {} }, { "index": 1809, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1809.png", "answer": "y=pm*(sqrt(3)*x/3)", "fact_expr": "P: Point;x0: Number;y0: Number;Coordinate(P) = (x0, y0);E: Hyperbola;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;PointOnCurve(P, E);M: Point;N: Point;LeftVertex(E) = M;RightVertex(E) = N;Slope(LineOf(P, M))*Slope(LineOf(P, N)) = 1/3", "coords": {} }, { "index": 1810, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1810.png", "answer": "x+2*y-3=0", "fact_expr": "G: Ellipse;H: LineSegment;P: Point;Expression(G) = (x^2/4 + y^2/2 = 1);Coordinate(P) = (1, 1);PointOnCurve(P,H);MidPoint(H)=P;IsChordOf(H,G)", "coords": { "P": [ 1.0, 1.0 ] } }, { "index": 1811, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1812, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1812.png", "answer": "5", "fact_expr": "C: Ellipse;m: Real;m>0;Expression(C) = (y^2/3 + x^2/m = 1);PointOnCurve(Focus(C) ,xAxis);FocalLength(C) = 2*sqrt(2)", "coords": {} }, { "index": 1813, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1813.png", "answer": "{pi/3,2*pi/3}", "fact_expr": "l: Line;G: Parabola;p: Number;P: Point;M: Point;F: Point;N: Point;p>0;Expression(G) = (y^2 = 2*p*x);Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {M, N};Intersection(l,Directrix(G)) = P;VectorOf(P, M) = 2*VectorOf(P, F)", "coords": {} }, { "index": 1814, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1814.png", "answer": "sqrt(2) + 1", "fact_expr": "C: Hyperbola;b: Number;a: Number;B: Point;F: Point;A: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightVertex(C) = A;RightFocus(C) = F;Intersection(LeftDirectrix(C), xAxis) = B ;MidPoint(LineSegmentOf(B, F)) = A", "coords": {} }, { "index": 1815, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1815.png", "answer": "(1,3/2)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);F: Point;RightFocus(G) = F;A: Point;PointOnCurve(A, G) = True;Quadrant(A) = 1;Intersection(OverlappingLine(LineSegmentOf(A, F)), G) = B;B: Point;Intersection(OverlappingLine(LineSegmentOf(A, O)), G) = C;C: Point;O: Origin;Area(TriangleOf(A, B, C)) = 3", "coords": {} }, { "index": 1816, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1816.png", "answer": "6", "fact_expr": "C: Ellipse;Expression(C) = (y^2/4 + x^2/a^2 = 1);a: Number;a>2;FocalLength(C) = 2*sqrt(5)", "coords": {} }, { "index": 1817, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.6457513110645907, "b": 2.23606797749979, "major_axis": "x", "x_coef": 7.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1817.png", "answer": "(-\\sqrt{2},0)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/7 + y^2/5 = 1)", "coords": {} }, { "index": 1818, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1818.png", "answer": "2*sqrt(3)", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;A: Point;B: Point;F2: Point;F1: Point;e:Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(C) = e;e=sqrt(3);LeftFocus(C)=F1;RightFocus(C)=F2;PointOnCurve(F1, G);Intersection(G,LeftPart(C))=A;Intersection(G,RightPart(C))=B;Abs(LineSegmentOf(A, B)) = 4;Abs(LineSegmentOf(A, F2)) = Abs(LineSegmentOf(B, F2))", "coords": {} }, { "index": 1819, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.4641016151377544, "b": 2.0, "major_axis": "x", "x_coef": 12.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1819.png", "answer": "{-1,-3}", "fact_expr": "l: Line;C: Ellipse;P: Point;A: Point;B: Point;x0:Number;m:Real;Expression(C) = (x^2/12 + y^2/4 = 1);Coordinate(P) = (x0, 2);Expression(l) = (y = x + m);Intersection(l, C) = {A, B};Negation(A=B);Abs(LineSegmentOf(A, B)) = 3*sqrt(2);Abs(VectorOf(P, A)) = Abs(VectorOf(P, B))", "coords": {} }, { "index": 1820, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1820.png", "answer": "2*sqrt(2)/3", "fact_expr": "G: Parabola;H: Line;B: Point;N: Point;A: Point;M: Point;k:Number;Expression(G) = (y^2 = 4*x);k>0;Expression(H) = (y = k*(x + 1));Intersection(H, G) = {A, B};Projection(A,Directrix(G))=M;Projection(B,Directrix(G))=N;Abs(LineSegmentOf(B, N)) = 2*Abs(LineSegmentOf(A, M))", "coords": {} }, { "index": 1821, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1821.png", "answer": "ApplyUnit(45, degree)", "fact_expr": "G: Parabola;P: Point;Q: Point;F: Point;l: Line;Expression(G) = (x^2 = 8*y);Focus(G) = F;Directrix(G) = l;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, Q), l);FootPoint(LineSegmentOf(P, Q), l) = Q;Abs(LineSegmentOf(P, F)) = 4", "coords": {} }, { "index": 1822, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1823, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1824, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1824.png", "answer": "2*sqrt(10) - 2", "fact_expr": "G: Parabola;A: Point;P: Point;M: Point;Expression(G) = (y^2 = 8*x);Coordinate(A) = (4, 6);PointOnCurve(P, G);Projection(P, yAxis) = M", "coords": { "A": [ 4.0, 6.0 ] } }, { "index": 1825, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1826, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1826.png", "answer": "(1, 2)", "fact_expr": "A: Point;B: Point;G: Parabola;H: Line;Expression(G) = (x^2 = 2*y);Expression(H) = (y = x + 1);Intersection(H, G) = {A, B}", "coords": {} }, { "index": 1827, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1828, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1828.png", "answer": "6", "fact_expr": "G: Parabola;P: Point;Expression(G) = (y = -x^2/4);Coordinate(P) = (m, n);PointOnCurve(P, G);m: Number;n: Number", "coords": {} }, { "index": 1829, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1830, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1830.png", "answer": "x^2 - y^2/4 = 1", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(H) = (sqrt(5), 0);Expression(OneOf(Asymptote(G))) = (2*x + y = 0);OneOf(Focus(G))=H", "coords": { "H": [ 2.23606797749979, 0.0 ] } }, { "index": 1831, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.872983346207417, "major_axis": "x", "x_coef": 16.0, "y_coef": 15.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1831.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = -4*x);P: Point;PointOnCurve(P, G);H: Ellipse;Expression(H) = (x^2/16 + y^2/15 = 1)", "coords": {} }, { "index": 1832, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1832.png", "answer": "-p^2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;H: Line;PointOnCurve(Focus(G), H);NumIntersection(H, G) = 2;A: Point;B: Point;Intersection(H, G) = {A, B};y1: Number;y2: Number;YCoordinate(A) = y1;YCoordinate(B) = y2", "coords": {} }, { "index": 1833, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1833.png", "answer": "x=3/2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 1)", "coords": {} }, { "index": 1834, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1834.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, G);Focus(G) = {F1, F2};IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2));Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 1835, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1836, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1836.png", "answer": "4", "fact_expr": "G: Hyperbola;b: Number;P: Point;b>0;Expression(G) = (x^2/4 - y^2/b^2 = 1);Coordinate(P) = (1, 2);PointOnCurve(P,OneOf(Asymptote(G)))", "coords": { "P": [ 1.0, 2.0 ] } }, { "index": 1837, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1837.png", "answer": "(1+sqrt(5))/2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(G) = F;A: Point;LeftVertex(G) = A;Endpoint(ImageinaryAxis(G)) = {B1, B2};B1: Point;B2: Point;PointOnCurve(F, H) ;PointOnCurve(A, H) ;PointOnCurve(B1, H) ;PointOnCurve(B2, H) ;H: Circle", "coords": {} }, { "index": 1838, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1839, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1839.png", "answer": "(1, 5/3]", "fact_expr": "G: Hyperbola;a: Number;b: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, RightPart(G));Abs(LineSegmentOf(P, F1)) = 4*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 1840, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1841, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1841.png", "answer": "(\\sqrt{21}+\\sqrt{3})/3", "fact_expr": "l: Line;W: Hyperbola;b: Number;a: Number;c:Number;F: Point;O: Origin;A: Point;C: Point;B: Point;a>0;b>0;Expression(W) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (-c, 0);LeftFocus(W) = F;Expression(l)=(y = (sqrt(3)/3)*(c + x));Intersection(l, LeftPart(W)) = A;Intersection(l, RightPart(W))=B;Intersection(l, yAxis)=C;DotProduct(VectorOf(O, A), VectorOf(C, F)) = c^2/3", "coords": {} }, { "index": 1842, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_1842.png", "answer": "(\\sqrt{2} - 1)/2", "fact_expr": "P: Point;m: Number;H: Circle;G: Parabola;Expression(G) = (x^2 = 4*(-m + y));Expression(H) = (x^2 + y^2 = 1);Quadrant(P) = 1;Intersection(H, G) = P;IsPerpendicular(TangentOnPoint(P, G), TangentOnPoint(P, H))", "coords": {} }, { "index": 1843, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1843.png", "answer": "7/2", "fact_expr": "G: Ellipse;M: Point;F1: Point;O: Origin;N: Point;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(M, G);LeftFocus(G)=F1;Distance(M, F1) = 3;MidPoint(LineSegmentOf(M, F1)) = N", "coords": {} }, { "index": 1844, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1845, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1846, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 1.7357404232025146, "b": 1.2960242636843804, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 3.0205966884629577e-05, "converged": true, "iterations": 91 }, "output_path": "results/train/ellipse/problem_1846.png", "answer": "2", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Eccentricity(C) = sqrt(2)/2;P: Point;Coordinate(P) = (sqrt(2)/2, sqrt(3)/2);PointOnCurve(P,C) = True;l: Line;Expression(l) = (y = k*x + m);m: Number;k: Number;Intersection(l, C) = {A, B};A: Point;B: Point;Negation(A=B);DotProduct(VectorOf(O, A), VectorOf(O, B)) = 0;O: Origin", "coords": { "P": [ 0.7071067811865476, 0.8660254037844386 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 1847, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1847.png", "answer": "9", "fact_expr": "G: Hyperbola;A: Point;P: Point;F: Point;Expression(G) = (x^2/4 - y^2/12 = 1);Coordinate(A) = (1, 4);LeftFocus(G) = F;PointOnCurve(P, RightPart(G))", "coords": { "A": [ 1.0, 4.0 ] } }, { "index": 1848, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1848.png", "answer": "sqrt(10)/3", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G))) = (y = -x/3)", "coords": {} }, { "index": 1849, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.039365291595459, "b": 2.936960895380947, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 2.390299779739223e-15, "converged": true, "iterations": 2 }, "output_path": "results/train/ellipse/problem_1849.png", "answer": "{x^2/25+y^2/9=1,y^2/25+x^2/9=1}", "fact_expr": "G: Ellipse;FocalLength(G)=8;Eccentricity(G)=0.8", "coords": {} }, { "index": 1850, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1850.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P: Point;PointOnCurve(P, G);d1: Number;Distance(P, Directrix(G)) = d1;d2: Number;A: Point;Coordinate(A) = (1, 4);Distance(P, A) = d2", "coords": { "A": [ 1.0, 4.0 ] } }, { "index": 1851, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1851.png", "answer": "4", "fact_expr": "A: Point;Coordinate(A) = (1, sqrt(3));C: Ellipse;Expression(C) = (x^2/9 + y^2/5 = 1);F: Point;LeftFocus(C) = F;P: Point;PointOnCurve(P, C)", "coords": { "A": [ 1.0, 1.7320508075688772 ] } }, { "index": 1852, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1852.png", "answer": "sqrt(10)/2", "fact_expr": "C: Hyperbola;b: Number;a: Number;F1: Point;F2: Point;B: Point;A: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);OneOf(Endpoint(ImageinaryAxis(C))) = B;LeftFocus(C) = F1;RightFocus(C) = F2;Intersection(LineSegmentOf(B, F2), C) = A;VectorOf(B, A) = 2*VectorOf(A, F2)", "coords": {} }, { "index": 1853, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1853.png", "answer": "3", "fact_expr": "G: Parabola;Q: Point;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(Q) = (2, -1);PointOnCurve(P, G)", "coords": { "Q": [ 2.0, -1.0 ] } }, { "index": 1854, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1854.png", "answer": "sqrt(13)/2", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;P: Point;F: Point;Q: Point;O:Origin;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;PointOnCurve(O, l);Intersection(l, C) = {P, Q};Abs(LineSegmentOf(P, F)) = 3*Abs(LineSegmentOf(Q, F));AngleOf(P, F, Q) = ApplyUnit(60, degree)", "coords": {} }, { "index": 1855, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2135674953460693, "b": 2.4270472526550293, "orientation": "horizontal" }, "optimization": { "final_loss": 4.453341261978494e-06, "converged": true, "iterations": 125 }, "output_path": "results/train/hyperbola/problem_1855.png", "answer": "x^2/5 - y^2/20 = 1", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (-10*x + x^2 + y^2 = 0);OneOf(Focus(G)) = Center(H);Eccentricity(G) = sqrt(5)", "coords": {} }, { "index": 1856, "success": false, "error": "Circle from diameter: missing point coordinates" }, { "index": 1857, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1857.png", "answer": "3/4", "fact_expr": "B: Point;A: Point;Length(LineSegmentOf(A, B)) = 2;Endpoint(LineSegmentOf(A,B))={A,B};G: Parabola;Expression(G) = (y^2 = x);PointOnCurve(A,G);PointOnCurve(B,G);M: Point;MidPoint(LineSegmentOf(A, B)) = M", "coords": {} }, { "index": 1858, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1858.png", "answer": "1/2", "fact_expr": "C: Ellipse;F: Point;Expression(C) = (x^2/2 + y^2 = 1);RightFocus(C) = F;l: Line;Expression(l) = (y = k*(x - 2));k: Number;Negation(k=0);A: Point;B: Point;Intersection(l, C) = {A, B};P: Point;MidPoint(LineSegmentOf(A, B)) = P;O: Origin;k1: Number;k2: Number;k3: Number;Slope(LineSegmentOf(O, P)) = k1;Slope(LineSegmentOf(A, F)) = k2;Slope(LineSegmentOf(B, F)) = k3;k2 = k1*k3", "coords": {} }, { "index": 1859, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1859.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;OneOf(Focus(G)) = F;l: Line;PointOnCurve(F, l) = True;IsPerpendicular(l, Asymptote(G)) = True;FootPoint(l, Asymptote(G)) = M;M: Point;Intersection(l, yAxis) = E;E: Point;VectorOf(F, M) = 3*VectorOf(M, E)", "coords": {} }, { "index": 1860, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.449489742783178, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1860.png", "answer": "6", "fact_expr": "G: Hyperbola;H: Parabola;p: Number;Expression(G) = (x^2/6 - y^2/3 = 1);Expression(H) = (y^2 = 2*(p*x));Focus(H) = RightFocus(G)", "coords": {} }, { "index": 1861, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1861.png", "answer": "16*sqrt(3)/3", "fact_expr": "C: Ellipse;F1: Point;P: Point;F2: Point;Expression(C) = (x^2/25 + y^2/16 = 1);Focus(C) = {F1, F2};PointOnCurve(P, C);AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 1862, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1862.png", "answer": "y^2=4*x", "fact_expr": "C: Parabola;G: Line;A: Point;B: Point;P: Point;O:Origin;Expression(G) = (x - y = 0);Coordinate(P) = (2, 2);Vertex(C)=O;PointOnCurve(Focus(C), xAxis);Intersection(G, C) = {A, B};MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ 2.0, 2.0 ] } }, { "index": 1863, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1864, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2631799883090584, "b": 2.5275303393962245, "orientation": "horizontal" }, "optimization": { "final_loss": 4.481942266905428e-06, "converged": true, "iterations": 95 }, "output_path": "results/train/hyperbola/problem_1864.png", "answer": "5*pi", "fact_expr": "G: Hyperbola;m: Number;H: Circle;O: Origin;F: Point;l: Line;m>0;Expression(G) = (-m*y^2 + x^2 = m);RightFocus(G)=F;IsPerpendicular(OneOf(Asymptote(G)),H);Center(H)=O;Radius(H)=Abs(LineSegmentOf(O,F));Expression(l)=(y=2*x)", "coords": {} }, { "index": 1865, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1866, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1866.png", "answer": "x = 4", "fact_expr": "G: Parabola;C: Circle;O: Origin;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Expression(C) = (y^2 + (x - 3)^2 = 4);l1: Line;l2: Line;PointOnCurve(O, l1);PointOnCurve(O, l2);IsPerpendicular(l1, l2);Negation(O = A);Negation(O = B);Intersection(l1, G) = A;Intersection(l2, G) = B;WhenMin(Length(InterceptChord(LineOf(A, B), C)))", "coords": {} }, { "index": 1867, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1867.png", "answer": "2", "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2/3 = 1)", "coords": {} }, { "index": 1868, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1868.png", "answer": "3", "fact_expr": "l: Line;G: Parabola;p: Number;A: Point;B: Point;F:Point;p>0;Expression(G) = (y^2 = 2*(p*x));Slope(l)=sqrt(3);PointOnCurve(F,l);Focus(G)=F;Intersection(l,G)={A,B};Abs(LineSegmentOf(A, B)) = 8", "coords": {} }, { "index": 1869, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1869.png", "answer": "y=pm*x\nx=-2", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/2 - y^2/2 = 1);G: Parabola;Expression(G) = (y^2 = 2*p*x);p:Number;RightFocus(C) = Focus(G)", "coords": {} }, { "index": 1870, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1870.png", "answer": "x-2*y-5=0", "fact_expr": "C: Parabola;B: Point;A: Point;M: Point;Expression(C) = (x^2 = 4*y);Coordinate(M) = (1, -2);PointOnCurve(A, C);PointOnCurve(B, C);VectorOf(A, M) = VectorOf(M, B)", "coords": { "M": [ 1.0, -2.0 ] } }, { "index": 1871, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1871.png", "answer": "4/5", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1);PointOnCurve(P,G);P:Point;L1:Line;L2:Line;Asymptote(G)={L1,L2};Distance(P,L1)*Distance(P,L2)=c;c:Number", "coords": {} }, { "index": 1872, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1872.png", "answer": "((3+sqrt(5))/2, sqrt(5)+1)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P: Point;PointOnCurve(P, G);Q: Point;Coordinate(Q) = (3, 4);WhenMin(Distance(P, Q)+Distance(P, Directrix(G)))", "coords": { "Q": [ 3.0, 4.0 ] } }, { "index": 1873, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1874, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_focus_mismatch;ellipse_point_off_curve", "params": { "a": 1.5952030420303345, "b": 1.581954950409272, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 5.265152936402376e-06, "converged": true, "iterations": 100 }, "output_path": "results/train/ellipse/problem_1874.png", "answer": "x^2/2 + y^2 = 1", "fact_expr": "C: Ellipse;a: Number;b: Number;F1: Point;F2: Point;P: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-1, 0);Coordinate(F2) = (1, 0);Coordinate(P) = (4/3, 1/3);Focus(C) = {F1, F2};PointOnCurve(P, C)", "coords": { "F1": [ -1.0, 0.0 ], "F2": [ 1.0, 0.0 ], "P": [ 1.3333333333333333, 0.3333333333333333 ] }, "validation_reasons": [ "ellipse_focus_mismatch", "ellipse_point_off_curve" ] }, { "index": 1875, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1875.png", "answer": "y=pm*x", "fact_expr": "l: Line;Inclination(l) = ApplyUnit(30, degree);G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, l);A: Point;B: Point;Intersection(l, G) = {A, B};PointOnCurve(F2, PerpendicularBisector(LineSegmentOf(A, B)))", "coords": {} }, { "index": 1876, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1876.png", "answer": "sqrt(5)/3", "fact_expr": "C: Ellipse;b: Number;a: Number;M: Point;F2: Point;O: Origin;A: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F2;PointOnCurve(M, yAxis);OneOf(Intersection(LineOf(M,F2),C))=A;Abs(LineSegmentOf(O,A))=Abs(LineSegmentOf(O,F2));Abs(LineSegmentOf(O,F2))=2*Abs(LineSegmentOf(O,M))", "coords": {} }, { "index": 1877, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1877.png", "answer": "3", "fact_expr": "C: Parabola;Expression(C) = (x^2 = 4*y);F: Point;Focus(C) = F;M: Point;PointOnCurve(M, C);N: Point;Intersection(OverlappingLine(LineSegmentOf(F, M)), xAxis) = N;MidPoint(LineSegmentOf(F, N)) = M", "coords": {} }, { "index": 1878, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1878.png", "answer": "sqrt(2)", "fact_expr": "l: Line;G: Ellipse;b: Number;B: Point;F2: Point;A: Point;F1: Point;0 < b;b < 2;Expression(G) = (x^2/4 + y^2/b^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, l);Intersection(l, G) = {A,B};Max(Abs(LineSegmentOf(A, F2)) + Abs(LineSegmentOf(B, F2))) = 6", "coords": {} }, { "index": 1879, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1879.png", "answer": "1/2", "fact_expr": "M: Point;Coordinate(M) = (1, -1);C: Parabola;Expression(C) = (x^2 = 4*y);k: Number;l: Line;Slope(l) = k;PointOnCurve(Focus(C), l);A: Point;B: Point;Intersection(l, C) = {A, B};N: Point;MidPoint(LineSegmentOf(A, B)) = N;2*Abs(LineSegmentOf(M, N)) = Abs(LineSegmentOf(A, B))", "coords": { "M": [ 1.0, -1.0 ] } }, { "index": 1880, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1881, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1881.png", "answer": "3", "fact_expr": "C: Parabola;G: Line;A: Point;F: Point;B: Point;lambda: Number;Expression(C) = (y^2 = 8*x);Focus(C) = F;PointOnCurve(F, G);Inclination(G) = ApplyUnit(60, degree);Intersection(G, C) = {A, B};LineSegmentOf(A, F) = LineSegmentOf(F, B)*lambda;Abs(LineSegmentOf(F, A)) > Abs(LineSegmentOf(F, B))", "coords": {} }, { "index": 1882, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1882.png", "answer": "2", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(C) = F;A: Point;LeftVertex(C) = A;B: Point;PointOnCurve(B, C);IsPerpendicular(LineSegmentOf(B, F), xAxis);Slope(LineOf(A, B)) = 1", "coords": {} }, { "index": 1883, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1883.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "C: Ellipse;a: Number;A: Point;F: Point;a>0;Expression(C) = (y^2/3 + x^2/a^2 = 1);Coordinate(A) = (3, 0);RightFocus(C) = F;PointOnCurve(MidPoint(LineSegmentOf(A,F)), C)", "coords": { "A": [ 3.0, 0.0 ] } }, { "index": 1884, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1884.png", "answer": "3*sqrt(2)/4", "fact_expr": "F: Point;A: Point;G: Parabola;p: Number;B: Point;p>0;Expression(G) = (y^2 = 2*p*x);Coordinate(A) = (0, 2);Focus(G) = F;MidPoint(LineSegmentOf(F, A))=B;PointOnCurve(B, G)", "coords": { "A": [ 0.0, 2.0 ] } }, { "index": 1885, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1886, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1886.png", "answer": "5", "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/9 = 1);P: Point;PointOnCurve(P, G);F1: Point;F2: Point;OneOf(Focus(G)) = F1;OneOf(Focus(G)) = F2;Negation(F1=F2);Distance(P, F1) = 3", "coords": {} }, { "index": 1887, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1888, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1889, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1889.png", "answer": "5/3", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;H: Line;Expression(H) = (3*x + 4*y - 1 = 0);IsPerpendicular(OneOf(Asymptote(G)), H)", "coords": {} }, { "index": 1890, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1891, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1892, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1893, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1893.png", "answer": "sqrt(3)", "fact_expr": "C: Hyperbola;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/3 - y^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 1894, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1894.png", "answer": "2", "fact_expr": "G: Hyperbola;a: Number;b: Number;H: Parabola;A: Point;B: Point;e: Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 4*x);l1:Line;l2:Line;Intersection(Directrix(H),l1) = A;Intersection(Directrix(H),l2) = B;Asymptote(G)={l1,l2};Abs(LineSegmentOf(A, B)) = 2*sqrt(3);Eccentricity(G) = e", "coords": {} }, { "index": 1895, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1896, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 2.0, "major_axis": "x", "x_coef": 8.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1896.png", "answer": "[4*sqrt(2)-sqrt(17),4*sqrt(2)+sqrt(17)]", "fact_expr": "C: Ellipse;A: Point;P: Point;F: Point;Expression(C) = (x^2/8 + y^2/4 = 1);Coordinate(A) = (2, 1);PointOnCurve(P, C);RightFocus(C) = F", "coords": { "A": [ 2.0, 1.0 ] } }, { "index": 1897, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1897.png", "answer": "{sqrt(6)-sqrt(3), sqrt(9-6*sqrt(2))}", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;A: Point;PointOnCurve(A, C) = True;DotProduct(VectorOf(A, F1), VectorOf(A, F2)) = 0;Intersection(LineOf(A, F2), C) = B;B: Point;Abs(VectorOf(A, B)) = Abs(VectorOf(A, F))", "coords": {} }, { "index": 1898, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1899, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1899.png", "answer": "pm*3*y+4*x=0", "fact_expr": "G: Hyperbola;m: Number;H: Circle;Expression(G) = (x^2/9 - y^2/m = 1);Expression(H) = (-4*x + x^2 + y^2 - 5 = 0);PointOnCurve(OneOf(Focus(G)),H)", "coords": {} }, { "index": 1900, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1900.png", "answer": "sqrt(2)*2/3", "fact_expr": "l: Line;Gamma: Parabola;A: Point;B: Point;C: Point;k: Number;p: Number;k > 0;Expression(l) = (y = k*(p/2 + x));Expression(Gamma) = (y^2 = 2*p*x);Intersection(Gamma, l) = {A, B};Intersection(Directrix(Gamma), xAxis) = C;VectorOf(A, B) + VectorOf(A, C) = 0", "coords": {} }, { "index": 1901, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1901.png", "answer": "sqrt(2) + 1", "fact_expr": "C2: Hyperbola;C1: Parabola;F: Point;A: Point;p: Number;p >0;Expression(C1) = (x^2 =2*p*y);Expression(C2) = (-x^2/b^2 + y^2/a^2 = 1);a: Number;b: Number;a > 0;b > 0;Focus(C1) = Focus(C2);Intersection(C1, C2) = A;Quadrant(A) = 1;LowerFocus(C2) = F;NumIntersection(LineOf(A, F), C1) = 1", "coords": {} }, { "index": 1902, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2631799883090584, "b": 2.5275303393962245, "orientation": "horizontal" }, "optimization": { "final_loss": 4.481942266905428e-06, "converged": true, "iterations": 95 }, "output_path": "results/train/hyperbola/problem_1902.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;b>0;Expression(G) = (x^2 - y^2/b^2 = 1);Expression(OneOf(Asymptote(G)))=(y=2*x)", "coords": {} }, { "index": 1903, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1903.png", "answer": "sqrt(5)/3", "fact_expr": "C: Ellipse;b: Number;a: Number;A: Point;B: Point;O: Origin;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;IsChordOf(LineSegmentOf(A,B),C);PointOnCurve(F,LineSegmentOf(A,B));Intersection(PerpendicularBisector(LineSegmentOf(A, B)), xAxis) = P;VectorOf(A, F) = 2*VectorOf(F, B);MidPoint(LineSegmentOf(O, F)) = P;F:Point;P:Point", "coords": {} }, { "index": 1904, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1905, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1905.png", "answer": "sqrt(3)", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F1: Point;O: Origin;F2: Point;l: Line;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F2,l);IsPerpendicular(l, OneOf(Asymptote(C)));FootPoint(l, OneOf(Asymptote(C)))=P;Abs(LineSegmentOf(P, F1)) = sqrt(6)*Abs(LineSegmentOf(O, P))", "coords": {} }, { "index": 1906, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1906.png", "answer": "2*x-y-15=0", "fact_expr": "P: Point;Coordinate(P) = (8, 1);H: Line;PointOnCurve(P, H);G: Hyperbola;Expression(G) = (x^2 - 4*y^2 = 4);A: Point;B: Point;Intersection(H, G) = {A, B};MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ 8.0, 1.0 ] } }, { "index": 1907, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.681220143106833, "b": 2.2412689930799665, "orientation": "horizontal" }, "optimization": { "final_loss": 4.985250663519028e-06, "converged": true, "iterations": 95 }, "output_path": "results/train/hyperbola/problem_1907.png", "answer": "16/3", "fact_expr": "G: Hyperbola;C: Circle;H: Point;Coordinate(H) = (4, (4*sqrt(7))/3);PointOnCurve(H, G);Expression(Asymptote(G)) = (y = pm*(4/3)*x);PointOnCurve(OneOf(Vertex(G)),C);PointOnCurve(OneOf(Focus(G)),C);PointOnCurve(Center(C),G)", "coords": { "H": [ 4.0, 3.5276684147527875 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 1908, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1908.png", "answer": "sqrt(3)/2+1", "fact_expr": "G: Parabola;H: Line;A: Point;F: Point;B: Point;Expression(G) = (y^2 = x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B}", "coords": {} }, { "index": 1909, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1909.png", "answer": "4", "fact_expr": "C: Parabola;p: Number;O: Origin;F: Point;A: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;OneOf(Intersection(PerpendicularBisector(LineSegmentOf(O,F)), C)) = A;Abs(LineSegmentOf(O, A)) = 3", "coords": {} }, { "index": 1910, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_1910.png", "answer": "y=pm*sqrt(3)*x", "fact_expr": "G: Hyperbola;a: Number;a>0;Expression(G) = (-y^2/12 + x^2/a^2 = 1);Eccentricity(G) = 2", "coords": {} }, { "index": 1911, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1911.png", "answer": "4*sqrt(2)", "fact_expr": "C: Hyperbola;b: Number;A: Point;P: Point;B: Point;b>0;Expression(C) = (x^2/4 - y^2/b^2 = 1);LeftVertex(C) = A;RightVertex(C) = B;PointOnCurve(P, C);Slope(LineOf(P, A)) * Slope(LineOf(P, B)) = 1", "coords": {} }, { "index": 1912, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1912.png", "answer": "(\\sqrt{10}-\\sqrt{2})/2", "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;F2: Point;P: Point;O: Origin;a>0;b>0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);Abs(LineSegmentOf(F1, F2)) = sqrt(2)*Abs(LineSegmentOf(P, F1));AngleOf(F1,P,F2)=ApplyUnit(120,degree)", "coords": {} }, { "index": 1913, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1913.png", "answer": "-4", "fact_expr": "G: Hyperbola;a: Number;Expression(G) = (y^2 + x^2/a = 1);Length(ImageinaryAxis(G)) = 2*Length(RealAxis(G))", "coords": {} }, { "index": 1914, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1914.png", "answer": "8/3", "fact_expr": "C: Parabola;F: Point;P: Point;Q: Point;l: Line;Expression(C)=(x^2=4*y);Focus(C)=F;Directrix(C) = l;PointOnCurve(P, l);OneOf(Intersection(LineOf(P,F),C))=Q;VectorOf(P, Q) = 4*VectorOf(F, Q)", "coords": {} }, { "index": 1915, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1915.png", "answer": "12", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);H: Line;PointOnCurve(Center(G), H);A: Point;B: Point;Intersection(H, G) = {A, B};F1: Point;OneOf(Focus(G)) = F1", "coords": {} }, { "index": 1916, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1917, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1918, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1918.png", "answer": "6", "fact_expr": "l: Line;G: Parabola;M: Point;O: Origin;A: Point;B: Point;F: Point;t:Number;Expression(G) = (y^2 = 4*x);Coordinate(M) = (2, t);Focus(G)=F;PointOnCurve(F, l);Intersection(l, G) = {A, B};VectorOf(O, M)=(1/2)*(VectorOf(O, A) + VectorOf(O, B))", "coords": {} }, { "index": 1919, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1919.png", "answer": "4/3", "fact_expr": "G: Hyperbola;b: Number;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 - y^2/b^2 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(P,G);Min(DotProduct(VectorOf(P, F1), VectorOf(P, F2))) = (-1)*7", "coords": {} }, { "index": 1920, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1921, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1922, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.58257569495584, "major_axis": "x", "x_coef": 25.0, "y_coef": 21.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1922.png", "answer": "20", "fact_expr": "G: Ellipse;H: Line;A: Point;B: Point;F: Point;Expression(G) = (x^2/25 + y^2/21 = 1);Expression(H) = (x = -2);Intersection(H, G) = {A, B};RightFocus(G) = F", "coords": {} }, { "index": 1923, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1923.png", "answer": "sqrt(13)/3", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(C) = F;O: Origin;G: Line;PointOnCurve(F, G);A: Point;B: Point;Intersection(G, RightPart(C)) = {A, B};P: Point;Intersection(OverlappingLine(LineSegmentOf(A, O)), C) = P;Abs(LineSegmentOf(A, F)) = 2*Abs(LineSegmentOf(B, F));AngleOf(P, F, B) = ApplyUnit(60, degree)", "coords": {} }, { "index": 1924, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1924.png", "answer": "(2, 4)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 8*x);F: Point;Focus(C) = F;P: Point;PointOnCurve(P, C);A: Point;Coordinate(A) = (-2, 0);WhenMax(Abs(LineSegmentOf(P, A)))", "coords": { "A": [ -2.0, 0.0 ] } }, { "index": 1925, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 8.0, "b": 4.0, "major_axis": "x", "x_coef": 64.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1925.png", "answer": "32", "fact_expr": "G: Ellipse;Expression(G) = (x^2/64 + y^2/16 = 1);F2: Point;RightFocus(G) = F2;PointOnCurve(F2, LineOf(A, B)) = True;IsPerpendicular(LineOf(A, B), xAxis) = True;Intersection(LineOf(A, B), G) = {A, B};A: Point;B: Point;F1: Point;LeftFocus(G) = F1", "coords": {} }, { "index": 1926, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1926.png", "answer": "2", "fact_expr": "C: Hyperbola;a: Number;b: Number;M: Point;P: Point;F1: Point;F2: Point;m:Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(M) = (m, 0);PointOnCurve(P, RightPart(C));Negation(PointOnCurve(P, xAxis));LeftFocus(C) = F1;RightFocus(C) = F2;TangentPoint(InscribedCircle(TriangleOf(P,F1,F2)),xAxis)=M;(sqrt(3)/3)*b<=m;m<=2*b", "coords": {} }, { "index": 1927, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1927.png", "answer": "y=pm*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/4 = 1)", "coords": {} }, { "index": 1928, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.6457513110645907, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 7.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1928.png", "answer": "2", "fact_expr": "G: Ellipse;H: Circle;I: Line;n: Number;m: Number;P:Point;Expression(G) = (x^2/7 + y^2/3 = 1);Expression(H) = (x^2 + y^2 = 3);Expression(I) = (m*x + n*y - 3 = 0);Coordinate(P) = (m, n);NumIntersection(I,H)=0;L:Line;PointOnCurve(P,L)", "coords": {} }, { "index": 1929, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1929.png", "answer": "sqrt(2)/2", "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;F2: Point;P: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, RightDirectrix(G));c: Number;HalfFocalLength(G) = c;YCoordinate(P) = sqrt(3)*c;LineSegmentOf(F1, F2) = LineSegmentOf(F2, P)", "coords": {} }, { "index": 1930, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1930.png", "answer": "sqrt(57)/4", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);H: Circle;PointOnCurve(RightVertex(G), H) ;PointOnCurve(RightFocus(G), H);PointOnCurve(Center(H), G)", "coords": {} }, { "index": 1931, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1931.png", "answer": "2", "fact_expr": "l: Line;G: Hyperbola;b: Number;a: Number;A: Point;B: Point;b > a;a > 0;c: Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (a, 0);Coordinate(B) = (0, b);FocalLength(G) = 2*c;PointOnCurve(A, l);PointOnCurve(B, l);O: Origin;Distance(O, l) = c*(sqrt(3)/4)", "coords": {} }, { "index": 1932, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1932.png", "answer": "3/2", "fact_expr": "G: Parabola;H: Line;A: Point;F: Point;B: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, F)) = 3", "coords": {} }, { "index": 1933, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1934, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1935, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1936, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1936.png", "answer": "3/5", "fact_expr": "G: Line;Expression(G) = (5*x + 4*y - 1 = 0);C: Ellipse;Expression(C) = (x^2/b^2 + y^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Intersection(G, C) = {M, N};M: Point;N: Point;MidPoint(LineSegmentOf(M, N)) = P;P: Point;O: Origin;Slope(LineOf(O, P)) = 5/4", "coords": {} }, { "index": 1937, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 2.8706156235768385, "b": 0.9134866531697662, "major_axis": "x", "x_coef": 9.0, "y_coef": 1.0 }, "optimization": { "final_loss": 8.509210183089842e-07, "converged": true, "iterations": 93 }, "output_path": "results/train/ellipse/problem_1937.png", "answer": "{x^2/9 + y^2 = 1, x^2/9 + y^2/81 = 1}", "fact_expr": "G: Ellipse;A: Point;Coordinate(A) = (3, 0);Length(MajorAxis(G)) = 3 * Length(MinorAxis(G));OverlappingLine(MajorAxis(G), axis);OverlappingLine(MinorAxis(G), axis);PointOnCurve(A, G)", "coords": { "A": [ 3.0, 0.0 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 1938, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1939, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1939.png", "answer": "6", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/9 = 1)", "coords": {} }, { "index": 1940, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1941, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1941.png", "answer": "5", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 + y^2/4 = 1);PointOnCurve(P, G);Focus(G) = {F1, F2}", "coords": {} }, { "index": 1942, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1942.png", "answer": "4", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2/3 = 1);PointOnCurve(P, G);Focus(G)={F1,F2};Abs(LineSegmentOf(P,F1))*Abs(LineSegmentOf(P,F2))=k;k:Number", "coords": {} }, { "index": 1943, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1943.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 + (x - 2)^2 = 2);IsTangent(H, Asymptote(G))", "coords": {} }, { "index": 1944, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1944.png", "answer": "1/16", "fact_expr": "A: Point;B: Point;C: Point;D: Point;G: Parabola;Expression(G) = (y^2 = 8*x);F: Point;Focus(G) = F;PointOnCurve(F,LineSegmentOf(A,B));IsChordOf(LineSegmentOf(A,B), G) ;PointOnCurve(F, LineSegmentOf(C,D));IsChordOf(LineSegmentOf(C,D),G) ;IsPerpendicular(LineSegmentOf(A,B), LineSegmentOf(C,D))", "coords": {} }, { "index": 1945, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1945.png", "answer": "(4*sqrt(5)/5)-1", "fact_expr": "G: Parabola;H: Line;P: Point;Q: Point;M: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (2*x - y + 2 = 0);PointOnCurve(P,G);L:Line;PointOnCurve(P,L);IsPerpendicular(L,yAxis);FootPoint(L,yAxis)=Q;PointOnCurve(M, H)", "coords": {} }, { "index": 1946, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1946.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "H: Ellipse;O: Origin;Center(H) = O;G: Parabola;Expression(G) = (y^2 = 8*x);OneOf(Focus(H)) = Focus(G);Coordinate(OneOf(Vertex(H))) = (0, 2)", "coords": {} }, { "index": 1947, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1947.png", "answer": "pm*4", "fact_expr": "G: Parabola;P: Point;O: Origin;F:Point;Focus(G)=F;Coordinate(P) = (m,-2);Vertex(G) = O;PointOnCurve(Focus(G),yAxis);PointOnCurve(P,G);Distance(P,F)=4;m:Number", "coords": {} }, { "index": 1948, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1948.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Hyperbola;F1: Point;M: Point;F2: Point;Expression(G) = (x^2/4 - y^2/9 = 1);Focus(G) = {F1, F2};PointOnCurve(M, G);AngleOf(F1, M, F2) = ApplyUnit(120, degree)", "coords": {} }, { "index": 1949, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1949.png", "answer": "[1/3,1)", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);e: Number;Abs(LineSegmentOf(P,F1))=(3/2)*e*Abs(LineSegmentOf(P,F2));Eccentricity(G) = e", "coords": {} }, { "index": 1950, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1950.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "P: Point;F: Point;C: Parabola;p: Number;Q: Point;M: Point;l: Line;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(P) = (0, 2);Focus(C) = F;Intersection(LineSegmentOf(P, F), C) = M;PointOnCurve(M, l);IsPerpendicular(l, Directrix(C));FootPoint(l, Directrix(C)) = Q;AngleOf(P, Q, F) = ApplyUnit(90, degree)", "coords": { "P": [ 0.0, 2.0 ] } }, { "index": 1951, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1951.png", "answer": "[1/2, sqrt(2)/2]", "fact_expr": "l: Line;Expression(l) = (-2*k + k*x - y + 1 = 0);C1: Ellipse;Expression(C1) = (y^2/b^2 + x^2/a^2 = 1);Intersection(l, C1) = {A, B};A: Point;B: Point;C2: Circle;Expression(C2) = ((x - 2)^2 + (y - 1)^2 = 1);Intersection(l, C2) = {C, D};C: Point;D: Point;k: Number;In(k, [-3/2, -1]);VectorOf(A, C) = VectorOf(D, B);b:Number;a>b;b>0", "coords": {} }, { "index": 1952, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1952.png", "answer": "y=pm*sqrt(3)*x", "fact_expr": "C: Hyperbola;a: Number;b: Number;G: Circle;F1: Point;F2: Point;P: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;IsDiameter(LineSegmentOf(F1,F2),G);Intersection(G,Asymptote(C))=P;Abs(LineSegmentOf(P, F1)) = 2*b;Quadrant(P)=1", "coords": {} }, { "index": 1953, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 5.0, 0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_1953.png", "answer": "(x^2/9-y^2/16=1)&(x<=-3)", "fact_expr": "G1: Circle;G2: Circle;Expression(G2) = (y^2 + (x + 5)^2 = 1);G3: Circle;Expression(G3) = (y^2 + (x - 5)^2 = 49);IsOutTangent(G1, G2) = True;IsOutTangent(G1, G3) = True", "coords": {} }, { "index": 1954, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1954.png", "answer": "x^2/8 + y^2/4 = 1", "fact_expr": "C: Ellipse;b: Number;a: Number;G: Circle;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2 + y^2 = 4);PointOnCurve(Focus(C), G);PointOnCurve(Endpoint(MinorAxis(C)), G)", "coords": {} }, { "index": 1955, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1955.png", "answer": "4*sqrt(3)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p>0;F: Point;Focus(C) = F;l: Line;Directrix(C) = l;Expression(l) = (x=-1);M: Point;PointOnCurve(M, C) = True;A: Point;Projection(M,l) = A;Slope(LineOf(A,F)) = -sqrt(3);p: Number", "coords": {} }, { "index": 1956, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1956.png", "answer": "3*sqrt(5)/5", "fact_expr": "G: Hyperbola;b: Number;P: Point;O: Origin;F: Point;H: Line;b>0;Expression(G) = (x^2/5 - y^2/b^2 = 1);RightFocus(G) = F;PointOnCurve(F, H);IsPerpendicular(Asymptote(G), H);FootPoint(Asymptote(G), H) = P;Area(TriangleOf(P, O, F)) = sqrt(5)", "coords": {} }, { "index": 1957, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1957.png", "answer": "9", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/4 - y^2 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;G: Line;PointOnCurve(F1, G) = True;Intersection(G, LeftPart(C)) = {A, B};A: Point;B: Point", "coords": {} }, { "index": 1958, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1958.png", "answer": "1\n2", "fact_expr": "C1: Hyperbola;C2: Hyperbola;F: Point;a: Number;b: Number;a > 0;b > 0;Expression(C1) = (x^2/a^2 - y^2/b^2 = 1);Expression(C2) = (x^2/4 - y^2/16 = 1);Asymptote(C1) = Asymptote(C2);Coordinate(F) = (sqrt(5), 0);RightFocus(C1) = F", "coords": { "F": [ 2.23606797749979, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 1959, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1959.png", "answer": "2", "fact_expr": "G: Parabola;p: Number;M: Point;O: Origin;F: Point;p>0;Expression(G) = (y^2 = 2*p*x);Coordinate(M) = (4, y1);Focus(G)=F;Distance(M,F)=5;y1:Number;PointOnCurve(M,G)", "coords": {} }, { "index": 1960, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1960.png", "answer": "{(4,-4),(16,8)}", "fact_expr": "G: Parabola;H: Line;P: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (y = x);PointOnCurve(P, G);Distance(P, H) = 4*sqrt(2)", "coords": {} }, { "index": 1961, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1962, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1962.png", "answer": "sqrt(2)", "fact_expr": "G: Parabola;H: Point;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(H) = (0, -1);PointOnCurve(P, G)", "coords": { "H": [ 0.0, -1.0 ] } }, { "index": 1963, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1963.png", "answer": "sqrt(5)/3", "fact_expr": "E: Ellipse;b: Number;a: Number;G: Line;A: Point;F1: Point;F2: Point;B: Point;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(E) = F1;RightFocus(E) = F2;PointOnCurve(F2, G);Intersection(G, E) = {A, B};DotProduct(VectorOf(A, F1), VectorOf(A, F2)) = 0;VectorOf(A, F2) = 2*VectorOf(F2, B)", "coords": {} }, { "index": 1964, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1965, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1965.png", "answer": "sqrt(3)/2", "fact_expr": "G: Hyperbola;H: Parabola;I: Ellipse;O: Origin;Expression(G) = (x^2/16 - y^2 = 1);Expression(H) = (x^2 = 8*y);Center(I) = O;SymmetryAxis(I) = axis;PointOnCurve(Focus(H), I);PointOnCurve(Vertex(G), I)", "coords": {} }, { "index": 1966, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1966.png", "answer": "3", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l: Line;Directrix(C) = l;P: Point;PointOnCurve(P, l);Q: Point;OneOf(Intersection(LineOf(P, F), C)) = Q;VectorOf(Q, P) = 3*VectorOf(Q, F)", "coords": {} }, { "index": 1967, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1967.png", "answer": "25/4", "fact_expr": "l: Line;G: Parabola;Expression(G) = (y^2 = 8*x);A: Point;B: Point;Intersection(l, G) = {A, B};F: Point;Focus(G) = F;PointOnCurve(F, l);Coordinate(A) = (8, 8)", "coords": { "A": [ 8.0, 8.0 ] } }, { "index": 1968, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1968.png", "answer": "2*sqrt(2)/3", "fact_expr": "l: Line;G: Parabola;A: Point;F: Point;B: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B};Abs(LineSegmentOf(A, F)) = 3;O:Origin", "coords": {} }, { "index": 1969, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1970, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1970.png", "answer": "4*sqrt(3)/3", "fact_expr": "G: Hyperbola;b2: Number;a2: Number;H: Ellipse;a1: Number;b1: Number;F1: Point;P: Point;F2: Point;a2>0;b2>0;Expression(G) = (-y^2/b2^2 + x^2/a2^2 = 1);a1 > b1;b1 > 0;Expression(H) = (y^2/b1^2 + x^2/a1^2 = 1);OneOf(Intersection(H, G)) = P;Focus(G) = {F1, F2};Focus(H) = {F1, F2};e1: Number;e2: Number;Eccentricity(H) = e1;Eccentricity(G) = e2;AngleOf(F1, P, F2) = pi/3", "coords": {} }, { "index": 1971, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1972, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1973, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1973.png", "answer": "-1/2", "fact_expr": "m: Line;G: Ellipse;O: Origin;P: Point;P1: Point;P2: Point;M: Point;k1: Number;k2: Number;Negation(k1 = 0);Expression(G) = (x^2/2 + y^2 = 1);Coordinate(M) = (-2, 0);PointOnCurve(M, m);Intersection(m, G) = {P1, P2};MidPoint(LineSegmentOf(P1, P2)) = P;Slope(m) = k1;Slope(LineOf(O, P)) = k2", "coords": { "M": [ -2.0, 0.0 ] } }, { "index": 1974, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1974.png", "answer": "sqrt(21)", "fact_expr": "G: Ellipse;H: Line;P0: Point;Expression(G) = (x^2/9 + y^2/4 = 1);Expression(H) = (2*x - sqrt(3)*y + 3*sqrt(3) = 0);PointOnCurve(P0, G)", "coords": {} }, { "index": 1975, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1976, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1976.png", "answer": "1/2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;l: Line;PointOnCurve(F, l);Negation(IsPerpendicular(l,axis));A: Point;B: Point;Intersection(l, G) = {A, B};M: Point;Intersection(PerpendicularBisector(LineSegmentOf(A, B)), xAxis) = M", "coords": {} }, { "index": 1977, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1978, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1978.png", "answer": "(x-1/2)^2+y^2=1", "fact_expr": "G: Parabola;H: Circle;Expression(G) = (y^2 = 2*x);Focus(G)=Center(H);IsTangent(Directrix(G),H)", "coords": {} }, { "index": 1979, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1979.png", "answer": "26", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, H) = True;H: Line;Intersection(H, LeftPart(G)) = {A, B};A: Point;B: Point;Length(LineSegmentOf(A, B)) = 5;a = 4", "coords": {} }, { "index": 1980, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1980.png", "answer": "2*sqrt(5)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2/2 = 1)", "coords": {} }, { "index": 1981, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1982, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 9.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1982.png", "answer": "2*pi/3", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 + y^2/2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 4", "coords": {} }, { "index": 1983, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1983.png", "answer": "4", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G) = True;Area(TriangleOf(F1, P, F2)) = 2", "coords": {} }, { "index": 1984, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1985, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1985.png", "answer": "-1/2", "fact_expr": "m: Line;G: Ellipse;Expression(G) = (x^2/2 + y^2 = 1);P1: Point;P2: Point;Intersection(m, G) = {P1, P2};P: Point;MidPoint(LineSegmentOf(P1, P2)) = P;k1: Number;Negation(k1=0);Slope(m) = k1;O: Origin;k2: Number;Slope(LineOf(O, P)) = k2", "coords": {} }, { "index": 1986, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1986.png", "answer": "(x-5)^2+y^2=4^2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);H: Circle;Center(H) = RightFocus(G);IsTangent(Asymptote(G), H) = True", "coords": {} }, { "index": 1987, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1988, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1988.png", "answer": "3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);P: Point;PointOnCurve(P, G);Q: Point;H: Circle;PointOnCurve(Q, H);C: Point;Coordinate(C) = (2, 0);Center(H) = C;Length(Radius(H)) = 1", "coords": { "C": [ 2.0, 0.0 ] } }, { "index": 1989, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1989.png", "answer": "2", "fact_expr": "C: Hyperbola;b: Number;a: Number;M: Point;l1: Line;l2: Line;F1: Point;F2: Point;G: Circle;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;IsDiameter(LineSegmentOf(F1, F2), G);OneOf(Asymptote(C)) = l1;OneOf(Asymptote(C)) = l2;Negation(l1 = l2);Intersection(G, l1) = M;Abs(LineSegmentOf(M, F1)) > Abs(LineSegmentOf(M, F2));PointOnCurve(MidPoint(LineSegmentOf(M, F1)), l2)", "coords": {} }, { "index": 1990, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1990.png", "answer": "(8/3,+oo)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F2, l) ;l: Line;Intersection(l, RightPart(C)) = {A, B};A: Point;B: Point;VectorOf(A, F2) = 2*VectorOf(F2, B);Perimeter(TriangleOf(A, B, F1)) = FocalLength(C) * 3;AngleOf(A, F1, B) > AngleOf(A, B, F1)", "coords": {} }, { "index": 1991, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 1992, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 4.47213595499958, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1992.png", "answer": "17", "fact_expr": "G: Hyperbola;F2: Point;P: Point;F1: Point;Expression(G) = (x^2/16 - y^2/20 = 1);Focus(G)={F1,F2};PointOnCurve(P,G);Distance(P,F1) = 9", "coords": {} }, { "index": 1993, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1993.png", "answer": "6", "fact_expr": "G: Parabola;p: Number;H: Line;p>0;Expression(G) = (x^2=2*p*y);Expression(H) = (2*x-y+3=0);PointOnCurve(Focus(G), H)", "coords": {} }, { "index": 1994, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_1994.png", "answer": "21/10", "fact_expr": "l: Line;G: Parabola;p: Number;H: Line;B: Point;A: Point;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (x = -1/2);alpha:Number;Coordinate(F) = (-Sin(alpha)*Cos(alpha), 0);Tan(alpha) = -2;Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B};Abs(LineSegmentOf(A, B)) = 4", "coords": {} }, { "index": 1995, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1995.png", "answer": "6*sqrt(7)/7", "fact_expr": "G: Hyperbola;L1: Line;L2: Line;Expression(G) = (x^2/3 - y^2/4 = 1);Directrix(G) = {L1, L2}", "coords": {} }, { "index": 1996, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_1996.png", "answer": "5", "fact_expr": "G: Parabola;Expression(G) = (y^2 = -4*x);F: Point;Focus(G) = F;A: Point;Coordinate(A) = (3, -3);P: Point;PointOnCurve(P, G)", "coords": { "A": [ 3.0, -3.0 ] } }, { "index": 1997, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_1997.png", "answer": "(1,0)", "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;e:Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Focus(G)={F1,F2};Eccentricity(G)=e;e=sqrt(2)/2;UpperVertex(G)=P;Area(TriangleOf(P, F1, F2)) = 1;RightFocus(G)=F2", "coords": {} }, { "index": 1998, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_1998.png", "answer": "8", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/5 + x^2/m = 1);m: Number;m>0;Expression(OneOf(Asymptote(G))) = (sqrt(5)*x + 2*y = 0);F: Point;LeftFocus(G) = F;P: Point;PointOnCurve(P, RightPart(G));H: Circle;Expression(H) = (x^2 + (y - 4)^2 = 1);Q: Point;PointOnCurve(Q, H)", "coords": {} }, { "index": 1999, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2000, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2000.png", "answer": "5*sqrt(3)/3", "fact_expr": "G: Ellipse;P: Point;F2: Point;F1: Point;Expression(G) = (x^2/9 + y^2/5 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 2001, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.3300262689590454, "b": 2.30381441116333, "orientation": "horizontal" }, "optimization": { "final_loss": 6.658357051492203e-06, "converged": true, "iterations": 122 }, "output_path": "results/train/hyperbola/problem_2001.png", "answer": "x^2/4 - y^2/12 = 1", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 16*x);Expression(OneOf(Asymptote(G)))=(y=sqrt(3)*x);OneOf(Focus(G))=Focus(H)", "coords": {} }, { "index": 2002, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2002.png", "answer": "4", "fact_expr": "G: Hyperbola;H: Parabola;p: Number;Expression(G) = (x^2 - y^2/3 = 1);Expression(H) = (y^2 = 2*(p*x));Focus(H) = RightFocus(G)", "coords": {} }, { "index": 2003, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2003.png", "answer": "sqrt(5)/2", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(C) = F;L: Line;PointOnCurve(F,L) = True;IsPerpendicular(L,OneOf(Asymptote(C))) = True;FootPoint(L,OneOf(Asymptote(C))) = A;O: Origin;Abs(LineSegmentOf(O, A)) = 2*Abs(LineSegmentOf(A, F));A: Point", "coords": {} }, { "index": 2004, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2004.png", "answer": "sqrt(10)", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(l) = (x + 3*y + 2022 = 0);IsPerpendicular(OneOf(Asymptote(C)), l)", "coords": {} }, { "index": 2005, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2005.png", "answer": "12", "fact_expr": "l: Line;C: Ellipse;A: Point;B: Point;F2: Point;F1: Point;Expression(C) = (x^2/9 + y^2/4 = 1);Focus(C) = {F1, F2};PointOnCurve(F1, l);Intersection(l, C) = {A, B}", "coords": {} }, { "index": 2006, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_2006.png", "answer": "((x-2)^2-y^2/3=1)&(x>2)", "fact_expr": "P: Circle;O1: Circle;Expression(O1) = (x^2 + y^2 = 1);O2: Circle;Expression(O2) = ((x - 4)^2 + y^2 = 9);IsInTangent(P, O1);IsInTangent(P, O2)", "coords": {} }, { "index": 2007, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2007.png", "answer": "b", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1)", "coords": {} }, { "index": 2008, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2008.png", "answer": "3/5", "fact_expr": "C: Parabola;p: Number;P: Point;Q: Point;F: Point;O: Origin;p>0;Expression(C) = (y^2 = 2*(p*x));PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P, Q), xAxis);FootPoint(LineSegmentOf(P, Q), xAxis) = Q;Focus(C) = F ;AngleOf(P, O, Q) = ApplyUnit(45, degree)", "coords": {} }, { "index": 2009, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2010, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 5.196152422706632, "major_axis": "x", "x_coef": 36.0, "y_coef": 27.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2010.png", "answer": "2", "fact_expr": "C: Ellipse;P: Point;F1: Point;O: Origin;M: Point;F2: Point;Expression(C) = (x^2/36 + y^2/27 = 1);Focus(C) = {F1, F2};PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) = 8;MidPoint(LineSegmentOf(P,F1))=M", "coords": {} }, { "index": 2011, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2011.png", "answer": "3*sqrt(5)/5", "fact_expr": "G: Hyperbola;a: Number;b: Number;H: Circle;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (-6*x + x^2 + y^2 + 5 = 0);RightFocus(G) = F;Center(H)=F;IsTangent(Asymptote(G),H)", "coords": {} }, { "index": 2012, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2012.png", "answer": "(1, 5/3]", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, RightPart(G));Abs(LineSegmentOf(P, F1)) = 4*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 2013, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 12.0, "b": 6.0, "major_axis": "x", "x_coef": 144.0, "y_coef": 36.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2013.png", "answer": "14", "fact_expr": "G: Ellipse;Expression(G) = (x^2/144 + y^2/36 = 1);P: Point;PointOnCurve(P, G);F1: Point;F2: Point;Distance(P, F1) = 10;OneOf(Focus(G))=F1;OneOf(Focus(G))=F2;Negation(F1=F2)", "coords": {} }, { "index": 2014, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2014.png", "answer": "1/4", "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F: Point;A:Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F;RightVertex(G) = A;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P,F),xAxis);Abs(LineSegmentOf(P,F))=(3/4)*Abs(LineSegmentOf(A,F))", "coords": {} }, { "index": 2015, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2015.png", "answer": "2", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2 + 2*y^2 = 2);LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(P,G)", "coords": {} }, { "index": 2016, "success": false, "error": "Circle from diameter: missing point coordinates" }, { "index": 2017, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.4641016151377544, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 12.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2017.png", "answer": "x+2*y-4=0", "fact_expr": "G: Ellipse;Expression(G) = (x^2/12 + y^2/3 = 1);H: LineSegment;IsChordOf(H, G);I: Point;Coordinate(I) = (2, 1);MidPoint(H) = I", "coords": { "I": [ 2.0, 1.0 ] } }, { "index": 2018, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2019, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 5.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2019.png", "answer": "4", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/25 - y^2/9 = 1);M: Point;PointOnCurve(M, RightPart(G));F1: Point;LeftFocus(G) = F1;Abs(LineSegmentOf(M, F1)) = 18;N: Point;MidPoint(LineSegmentOf(M, F1)) = N;O: Origin", "coords": {} }, { "index": 2020, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2020.png", "answer": "4*c*x-2*b*y+a=0", "fact_expr": "H: Line;G: Parabola;P: Point;Q: Point;F: Point;a: Number;b: Number;c: Number;Negation(c=0);M: Point;N: Point;Expression(H) = (a*x + b*y + c = 0);Expression(G) = (y^2 = 2*x);Intersection(H, G)={P,Q};Focus(G) = F;Intersection(LineOf(P, F), G) = M;Intersection(LineOf(Q, F), G) = N", "coords": {} }, { "index": 2021, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2021.png", "answer": "sqrt(5)/3", "fact_expr": "E: Ellipse;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(E) = F1;RightFocus(E) = F2;G: Line;PointOnCurve(F2, G);A: Point;B: Point;Intersection(G, E) = {A, B};IsPerpendicular(LineSegmentOf(A, F1), LineSegmentOf(A, B));VectorOf(A, B) = 3*VectorOf(F2, B)", "coords": {} }, { "index": 2022, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2022.png", "answer": "(1, -1/4)", "fact_expr": "G: Parabola;Expression(G) = (x^2 = -4*y);P: Point;PointOnCurve(P, G);F: Point;Focus(G) = F;A: Point;Coordinate(A) = (1, -2);WhenMin(Abs(LineSegmentOf(P, A))+Abs(LineSegmentOf(P, F)))", "coords": { "A": [ 1.0, -2.0 ] } }, { "index": 2023, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2023.png", "answer": "8", "fact_expr": "C: Parabola;A: Point;M: Point;F: Point;Expression(C) = (y^2 = 16*x);Coordinate(A) = (4, 1);Focus(C) = F;PointOnCurve(M, C)", "coords": { "A": [ 4.0, 1.0 ] } }, { "index": 2024, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2024.png", "answer": "(41/16, 3/4)", "fact_expr": "l: Line;P: Point;Coordinate(P) = (2, 0);Slope(l) = 4/3;PointOnCurve(P, l) = True;G: Parabola;Expression(G) = (y^2 = 2*x);Intersection(l, G) = {A, B};B: Point;A: Point;MidPoint(LineSegmentOf(A, B)) = M;M: Point", "coords": { "P": [ 2.0, 0.0 ] } }, { "index": 2025, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2025.png", "answer": "y = pm*x*sqrt(3)/3", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y^2 + (x - 2)^2 = 1);IsTangent(G, Asymptote(C))", "coords": {} }, { "index": 2026, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2026.png", "answer": "32", "fact_expr": "C: Parabola;G: Line;A: Point;B: Point;F: Point;Expression(C) = (y^2 = 16*x);Focus(C) = F;PointOnCurve(F, G);Inclination(G) = ApplyUnit(45, degree);Intersection(G, C) = {A, B}", "coords": {} }, { "index": 2027, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2028, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2029, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2029.png", "answer": "2", "fact_expr": "G: Parabola;L1: Line;L2: Line;P: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(P, G);Expression(L1)=(4*x-3*y+6=0);Expression(L2)=(x=-1)", "coords": {} }, { "index": 2030, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.125, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2030.png", "answer": "(0,1/8)", "fact_expr": "G: Parabola;Expression(G) = (y = 2*x^2)", "coords": {} }, { "index": 2031, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2031.png", "answer": "{1,2}", "fact_expr": "G: Parabola;p: Number;B: Point;A: Point;M: Point;l1: Line;l2:Line;p>0;Expression(G) = (x^2=2*p*y);Coordinate(M) = (2,-2*p);TangentOfPoint(M,G)={l1,l2};TangentPoint(l1,G)=A;TangentPoint(l2,G)=B;YCoordinate(MidPoint(LineSegmentOf(A,B)))=6", "coords": {} }, { "index": 2032, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2032.png", "answer": "[2,+oo)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F:Point;RightFocus(G) = F;L: Line;PointOnCurve(F,L) = True;Inclination(L) = ApplyUnit(60,degree);NumIntersection(L,RightPart(G)) = 1", "coords": {} }, { "index": 2033, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2033.png", "answer": "[2/5, 5/8]", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Length(MinorAxis(G)) = 8;A: Point;UpperVertex(G) = A;B: Point;LeftVertex(G) = B;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;Area(TriangleOf(F1, A, B)) = 4;P: Point;PointOnCurve(P, G)", "coords": {} }, { "index": 2034, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2034.png", "answer": "[\\sqrt{2}/2,1)", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;O: Origin;M: Point;N: Point;A: Point;F1: Point;B: Point;l:Line;LeftFocus(G) = F1;Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A, F1)) = M;MidPoint(LineSegmentOf(B, F1)) = N;IsPerpendicular(LineSegmentOf(O, M), LineSegmentOf(O, N));PointOnCurve(O,l)", "coords": {} }, { "index": 2035, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2035.png", "answer": "2", "fact_expr": "l: Line;C: Parabola;G: Line;B: Point;D: Point;P: Point;A: Point;O:Origin;Expression(C) = (y^2 = 8*x);Coordinate(P) = (m, 0);m:Number;m>0;PointOnCurve(P, l);Intersection(l, C) = {A, B};PointOnCurve(A,G);PointOnCurve(O,G);Vertex(C)=O;Intersection(G,Directrix(C))=D;IsParallel(LineSegmentOf(B,D),SymmetryAxis(C))", "coords": {} }, { "index": 2036, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.656854249492381, "b": 4.0, "major_axis": "y", "x_coef": 16.0, "y_coef": 32.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2036.png", "answer": "8", "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/32 = 1)", "coords": {} }, { "index": 2037, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_asymptote_mismatch", "params": { "a": 1.1532304286956787, "b": 2.5773532390594482, "orientation": "vertical" }, "optimization": { "final_loss": 8.412490387854632e-06, "converged": true, "iterations": 114 }, "output_path": "results/train/hyperbola/problem_2037.png", "answer": "3*sqrt(5)/5", "fact_expr": "C: Hyperbola;Expression(C) = (-x^2/b^2 + y^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Expression(Asymptote(C)) = (y = pm*(sqrt(5)*x)/2)", "coords": {}, "validation_reasons": [ "hyperbola_asymptote_mismatch" ] }, { "index": 2038, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 5.477225575051661, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2038.png", "answer": "139", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/30 = 1);P: Point;x1: Number;y1: Number;Coordinate(P) = (x1, y1);PointOnCurve(P, G);Distance(P, OneOf(Focus(G))) = 9", "coords": {} }, { "index": 2039, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2040, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2040.png", "answer": "sqrt(2)", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2/2 = 1);Focus(G)={F1,F2};PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) - Abs(LineSegmentOf(P, F2)) = 2", "coords": {} }, { "index": 2041, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 5.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2041.png", "answer": "-1", "fact_expr": "G: Hyperbola;H: Circle;P: Point;F1: Point;M: Point;N: Point;O: Origin;Expression(G) = (x^2/16 - y^2/25 = 1);Expression(H) = (x^2 + y^2 = 16);LeftFocus(G) = F1;PointOnCurve(P, RightPart(G));TangentPoint(LineSegmentOf(P, F1), H) = N;MidPoint(LineSegmentOf(P, F1)) = M", "coords": {} }, { "index": 2042, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2043, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2043.png", "answer": "[1/2,1)", "fact_expr": "G: Ellipse;a: Number;b: Number;A: Point;P: Point;F: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;Intersection(RightDirectrix(G), xAxis) = A;PointOnCurve(P,G);PointOnCurve(F,PerpendicularBisector(LineSegmentOf(A,P)))", "coords": {} }, { "index": 2044, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2044.png", "answer": "1", "fact_expr": "C: Parabola;m: Number;P: Point;m>0;Expression(C) = (x^2 = m*y);PointOnCurve(P, C);Distance(P, Focus(C)) = 1;Distance(P, xAxis) = 3/4", "coords": {} }, { "index": 2045, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2045.png", "answer": "2*sqrt(15)", "fact_expr": "H: Line;Expression(H) = (y = k*x - 2);k: Number;G: Parabola;Expression(G) = (y^2 = 8*x);Intersection(H, G) = {A, B};A: Point;B: Point;XCoordinate(MidPoint(LineSegmentOf(A, B))) = 2;IsChordOf(LineSegmentOf(A, B), G) = True", "coords": {} }, { "index": 2046, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2046.png", "answer": "2*sqrt(2)", "fact_expr": "l: Line;Expression(l) = (y = x - 1);C: Parabola;Expression(C) = (y^2 = 4*x);Intersection(l, C) = {A, B};A: Point;B: Point;M: Point;MidPoint(LineSegmentOf(A, B)) = M;Intersection(PerpendicularBisector(LineSegmentOf(A, B)), xAxis) = N;N: Point;Negation(A=B)", "coords": {} }, { "index": 2047, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2047.png", "answer": "9/32", "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/9 = 1);H: LineSegment;IsChordOf(H, G);M: Point;Coordinate(M) = (-1, 2);MidPoint(H) = M", "coords": { "M": [ -1.0, 2.0 ] } }, { "index": 2048, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/train/hyperbola/problem_2048.png", "answer": "x^2 - y^2 / 3 = 1", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (2, -3);PointOnCurve(P, G);Focus(G)={F1, F2} ;Distance(F1, F2) = 4", "coords": { "P": [ 2.0, -3.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 2049, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2049.png", "answer": "y=pm*(1/2)*x", "fact_expr": "C: Hyperbola;a: Number;b: Number;F: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;Distance(F, OneOf(Asymptote(C))) = a/2", "coords": {} }, { "index": 2050, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.8284271247461903, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2050.png", "answer": "(2*sqrt(5))/5", "fact_expr": "H: Parabola;Expression(H) = (y^2 = 8*x);G: Hyperbola;Expression(G) = (x^2/8 - y^2/2 = 1)", "coords": {} }, { "index": 2051, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2051.png", "answer": "[-1, 3]", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;Z: Line;PointOnCurve(F, Z);Slope(Z) = 1;A: Point;B: Point;Intersection(Z, C) = {A, B};E: Circle;IsDiameter(LineSegmentOf(A, B), E);P: Point;Q: Point;PointOnCurve(P, E);PointOnCurve(Q, E);U: Circle;IsDiameter(LineSegmentOf(P, Q), U);D: Point;t: Real;Coordinate(D) = (-2, t);PointOnCurve(D, U)", "coords": {} }, { "index": 2052, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2053, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2053.png", "answer": "pm*2*sqrt(6)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;l: Line;PointOnCurve(F, l);Intersection(l, G) = {A, B};A: Point;B: Point;Abs(LineSegmentOf(A, F)) = (3/2)*Abs(LineSegmentOf(B, F));k: Number;Slope(l) = k", "coords": {} }, { "index": 2054, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2054.png", "answer": "sqrt(2)", "fact_expr": "F: Point;P: Point;G: Hyperbola;a: Number;b: Number;H: Circle;M: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2 + y^2 = a^2);RightFocus(G) = F;TangentOfPoint(F, H) = LineOf(F, M);TangentPoint(LineOf(F,M),H)= M ;Intersection(LineOf(F, M), yAxis) = P;MidPoint(LineSegmentOf(F, P)) = M", "coords": {} }, { "index": 2055, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 4.898979485566356, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2055.png", "answer": "24", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2 - y^2/24 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, LeftPart(G));Abs(LineSegmentOf(P, F1)) = (3/5)*Abs(LineSegmentOf(F1, F2))", "coords": {} }, { "index": 2056, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2057, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2058, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2058.png", "answer": "8*sqrt(3)/3", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);A: Point;B: Point;F: Point;P: Point;Q: Point;Focus(C) = F;l: Line;PointOnCurve(F, l);Intersection(l, C) = {A, B};L1: Line;L2: Line;Z: Line;Directrix(C) = Z;PointOnCurve(A,L1);PointOnCurve(B,L2);IsPerpendicular(Z,L1);IsPerpendicular(Z,L2);FootPoint(Z,L1) = P;FootPoint(Z,L2) = Q;Abs(LineSegmentOf(A, F)) = 3*Abs(LineSegmentOf(B, F))", "coords": {} }, { "index": 2059, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2059.png", "answer": "x^2/16 - y^2/9 = 1", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Slope(OneOf(Asymptote(C))) = 3/4;FocalLength(C) = 10", "coords": {} }, { "index": 2060, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2061, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2061.png", "answer": "sqrt(2)+1", "fact_expr": "G: Hyperbola;H: Parabola;P: Point;F: Point;A: Point;Expression(H) = (y = x^2/4);Intersection(SymmetryAxis(H),Directrix(H))=A;Focus(H) = F;PointOnCurve(P, H);Abs(LineSegmentOf(P, F)) = m*Abs(LineSegmentOf(P, A));WhenMin(m);m:Number;Focus(G)={A,F};PointOnCurve(P, G)", "coords": {} }, { "index": 2062, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2062.png", "answer": "2", "fact_expr": "G: Parabola;p: Number;H: Circle;p>0;Expression(G) = (y^2 = 2*p*x);Expression(H) = (y^2 + (x - 3)^2 = 16);IsTangent(Directrix(G),H)", "coords": {} }, { "index": 2063, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2063.png", "answer": "pi/3", "fact_expr": "G: Parabola;p: Number;M: Point;N: Point;F: Point;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;Intersection(Directrix(G), xAxis) = M;PointOnCurve(N, G);Abs(LineSegmentOf(M, N)) = 2*Abs(LineSegmentOf(N, F))", "coords": {} }, { "index": 2064, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "y", "x_coef": 9.0, "y_coef": 25.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2064.png", "answer": "20", "fact_expr": "G: Ellipse;H: Line;A: Point;B: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 + y^2/25 = 1);Focus(G) = {F1,F2};PointOnCurve(F1, H);Intersection(H, G) = {A, B}", "coords": {} }, { "index": 2065, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2065.png", "answer": "12", "fact_expr": "G: Parabola;F: Point;A: Point;B: Point;C: Point;Expression(G) = (x^2 = 8*y);Focus(G) = F;PointOnCurve(A, G);PointOnCurve(B, G);PointOnCurve(C, G);VectorOf(F, A) + VectorOf(F, B) + VectorOf(F, C) = 0", "coords": {} }, { "index": 2066, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2066.png", "answer": "5", "fact_expr": "C: Parabola;M: Point;P: Point;F: Point;Expression(C) = (x^2 = 8*y);Coordinate(M) = (-4, 3);Focus(C) = F;PointOnCurve(P, C)", "coords": { "M": [ -4.0, 3.0 ] } }, { "index": 2067, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2067.png", "answer": "4*sqrt(2)", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/4 - y^2/b^2 = 1);b: Number;P: Point;B: Point;A: Point;b>0;LeftVertex(C) = A;RightVertex(C) = B;PointOnCurve(P, C);AngleOf(P, B, A) = pi/2 + AngleOf(P, A, B)", "coords": {} }, { "index": 2068, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2068.png", "answer": "9", "fact_expr": "G: Hyperbola;Z1: Circle;Z2: Circle;P: Point;M: Point;N: Point;Expression(G) = (x^2/9 - y^2/16 = 1);Expression(Z1) = (y^2 + (x + 5)^2 = 4);Expression(Z2) = (y^2 + (x - 5)^2 = 1);PointOnCurve(P, RightPart(G));PointOnCurve(M, Z1);PointOnCurve(N, Z2)", "coords": {} }, { "index": 2069, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2069.png", "answer": "5", "fact_expr": "C: Parabola;p: Number;G: Line;A: Point;F: Point;B: Point;O: Origin;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;PointOnCurve(F, G);Intersection(G, C) = {A, B};Abs(LineSegmentOf(A, F)) = 4*Abs(LineSegmentOf(B, F))", "coords": {} }, { "index": 2070, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "y", "x_coef": 2.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2070.png", "answer": "x^2/8+y^2/4=1", "fact_expr": "M: Ellipse;b: Number;a: Number;a > b;b > 0;Expression(M) = (y^2/b^2 + x^2/a^2 = 1);Length(MajorAxis(M))=4*sqrt(2);G:Ellipse;Expression(G)=(y^2/4 + x^2/2= 1);Eccentricity(M)=Eccentricity(G)", "coords": {} }, { "index": 2071, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2071.png", "answer": "2", "fact_expr": "G: Parabola;H: Line;k: Number;A: Point;B: Point;Expression(G) = (y^2 = 8*x);Expression(H) = (y = k*x - 2);Intersection(H, G) = {A, B};XCoordinate(MidPoint(LineSegmentOf(A, B))) = 2", "coords": {} }, { "index": 2072, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2073, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2073.png", "answer": "2", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;A: Point;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2 + (y - 2*sqrt(3))^2 = 4);Intersection(OneOf(Asymptote(C)),G) = {A, B};Abs(LineSegmentOf(A, B)) = 2", "coords": {} }, { "index": 2074, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2075, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2075.png", "answer": "1", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x)", "coords": {} }, { "index": 2076, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2076.png", "answer": "8", "fact_expr": "C: Hyperbola;a: Number;O: Origin;A: Point;B: Point;F: Point;Expression(C) = (-y^2/4 + x^2/a^2 = 1);RightFocus(C) = F;L: Line;PointOnCurve(F, L);IsPerpendicular(L, xAxis);L1: Line;L2: Line;Asymptote(C) = {L1, L2};Intersection(L, L1) = A;Intersection(L, L2) = B", "coords": {} }, { "index": 2077, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2077.png", "answer": "1/4", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2 + y^2/m = 1);PointOnCurve(Focus(G), xAxis);Eccentricity(G) = (sqrt(3)/2)", "coords": {} }, { "index": 2078, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 4.242640687119285, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2078.png", "answer": "9", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/16 - y^2/18 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C);Q: Point;VectorOf(F1, Q) = VectorOf(Q, P);O: Origin;Abs(LineSegmentOf(P, F1)) = 10", "coords": {} }, { "index": 2079, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2079.png", "answer": "10", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1)", "coords": {} }, { "index": 2080, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2080.png", "answer": "pm*2*sqrt(5)/5", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);PointOnCurve(Focus(G), l) = True;l: Line;H: Circle;Expression(H) = (y^2 + (x - 4)^2 = 4);IsTangent(l, H) = True", "coords": {} }, { "index": 2081, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2081.png", "answer": "sqrt(26)-1", "fact_expr": "G: Parabola;H: Circle;I: Line;P: Point;Q: Point;Expression(G) = (y^2 = 4*x);Expression(H) = ((x - 2)^2 + (y - 5)^2 = 1);Expression(I) = (x = -1);PointOnCurve(P, H);PointOnCurve(Q, G)", "coords": {} }, { "index": 2082, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2083, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0010501146316528, "direction": "right" }, "optimization": { "final_loss": 3.455598402979376e-07, "converged": true, "iterations": 81 }, "output_path": "results/train/parabola/problem_2083.png", "answer": "-4", "fact_expr": "C: Parabola;p: Number;H: Line;A: Point;P: Point;B: Point;Q: Point;p>0;Expression(C) = (y^2 = 2*p*x);Coordinate(A) = (1, 2);Coordinate(B) = (2,-2);PointOnCurve(A, C);PointOnCurve(B, H);Intersection(H, C) = {P, Q};k1:Number;k2:Number;Slope(LineOf(A,P))=k1;Slope(LineOf(A,Q))=k2", "coords": { "A": [ 1.0, 2.0 ], "B": [ 2.0, -2.0 ] } }, { "index": 2084, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2084.png", "answer": "sqrt(2)/2", "fact_expr": "E: Ellipse;A: Point;B: Point;M: Point;a:Number;b:Number;a>b;b>0;Coordinate(M) = (2, 1);Expression(E) = (y^2/b^2 + x^2/a^2 = 1);IsChordOf(LineSegmentOf(A,B),E);Inclination(LineSegmentOf(A,B))=ApplyUnit(135,degree);MidPoint(LineSegmentOf(A, B)) = M", "coords": { "M": [ 2.0, 1.0 ] } }, { "index": 2085, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 5.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2085.png", "answer": "-1", "fact_expr": "G: Hyperbola;H: Circle;P: Point;F1: Point;M: Point;N: Point;O: Origin;Expression(G) = (x^2/16 - y^2/25 = 1);Expression(H) = (x^2 + y^2 = 16);LeftFocus(G) = F1;PointOnCurve(P, RightPart(G));TangentPoint(LineSegmentOf(P,F1),H)=N;MidPoint(LineSegmentOf(P,F1))=M", "coords": {} }, { "index": 2086, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2087, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2087.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = -8*x)", "coords": {} }, { "index": 2088, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2089, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2089.png", "answer": "sqrt(3)", "fact_expr": "l: Line;C: Parabola;A: Point;F: Point;B: Point;Expression(C) = (y^2 = 4*x);Focus(C)=F;PointOnCurve(F,l);Intersection(l,C)={A,B};Quadrant(A)=1;Quadrant(B)=4;Abs(LineSegmentOf(A,F))=3*Abs(LineSegmentOf(F,B))", "coords": {} }, { "index": 2090, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2090.png", "answer": "[\\sqrt{2}, +\\infty)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(C) = F;G: Line;PointOnCurve(F, G);Inclination(G) = ApplyUnit(45, degree);NumIntersection(G, RightPart(C)) = 1", "coords": {} }, { "index": 2091, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2092, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2092.png", "answer": "4/3", "fact_expr": "G: Parabola;p: Number;A: Point;B: Point;F: Point;p>0;Expression(G) = (y^2 = 2*p*x);Focus(G) = F;PointOnCurve(A, G);PointOnCurve(B, G);Quadrant(A) = 1;Quadrant(B) = 4;VectorOf(A, F) = 4*VectorOf(F, B)", "coords": {} }, { "index": 2093, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.5, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2093.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Hyperbola;a: Number;H: Parabola;a>0;Expression(G) = (-y^2 + x^2/a^2 = 1);Expression(H) = (y^2 = -6*x);OneOf(Directrix(G)) = Directrix(H)", "coords": {} }, { "index": 2094, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2094.png", "answer": "(1, \\sqrt{5})", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;H: Line;PointOnCurve(RightVertex(G), H);Slope(H) = 2;NumIntersection(H, RightPart(G)) = 2", "coords": {} }, { "index": 2095, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2095.png", "answer": "2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;H: Line;PointOnCurve(F, H);A: Point;B: Point;Intersection(H, G) = {A, B};LineSegmentOf(A, F)/LineSegmentOf(B, F) = 4;O: Origin", "coords": {} }, { "index": 2096, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2096.png", "answer": "sqrt(2)/2", "fact_expr": "C: Parabola;p: Number;G: Line;F: Point;B: Point;A: Point;p>0;Expression(C) = (y^2 = 2*p*x);PointOnCurve(A, Directrix(C));Focus(C) = F;PointOnCurve(A,C);IsPerpendicular(G,Directrix(C));FootPoint(G,Directrix(C))=B;Inclination(LineOf(F,B))=ApplyUnit(150, degree);Area(TriangleOf(F, A, B)) = sqrt(3)/18", "coords": {} }, { "index": 2097, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2098, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2098.png", "answer": "3*sqrt(2)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;F: Point;Focus(G) = F;A: Point;B: Point;PointOnCurve(A, G);PointOnCurve(B, G);Abs(VectorOf(A, B)) = 6;DotProduct(VectorOf(A, F), VectorOf(B, F)) = 0;M: Point;MidPoint(LineSegmentOf(A, B)) = M;N: Point;Projection(M, Directrix(G)) = N", "coords": {} }, { "index": 2099, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2099.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 - y^2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);XCoordinate(P) = 2*sqrt(2)", "coords": {} }, { "index": 2100, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2100.png", "answer": "y=2*x-1", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*x);l: Line;PointOnCurve(Focus(C), l);A: Point;B: Point;Intersection(l, C) = {A, B};G: Circle;IsDiameter(LineSegmentOf(A, B), G);M: Point;Coordinate(M) = (-1/2, 1/2);TangentPoint(G, Directrix(C)) = M", "coords": { "M": [ -0.5, 0.5 ] } }, { "index": 2101, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2101.png", "answer": "1/3", "fact_expr": "G: Ellipse;Expression(G) = (x^2/b^2 + y^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;F: Point;UpperFocus(G) = F;B1: Point;B2: Point;LeftVertex(G) = B1;RightVertex(G) = B2;A: Point;LowerVertex(G) = A;P: Point;Intersection(LineOf(A, B2), LineOf(B1, F)) = P;VectorOf(A, P) = 2*VectorOf(A, B2)", "coords": {} }, { "index": 2102, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.449489742783178, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 6.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2102.png", "answer": "4", "fact_expr": "G: Parabola;p: Number;H: Ellipse;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (x^2/6 + y^2/2 = 1);Focus(G) = RightFocus(H)", "coords": {} }, { "index": 2103, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2103.png", "answer": "sqrt(2)/2", "fact_expr": "C: Parabola;Q: Point;P: Point;F: Point;Expression(C) = (x^2 = 4*y);Coordinate(Q) = (0, -1);Focus(C)=F;PointOnCurve(P, C)", "coords": { "Q": [ 0.0, -1.0 ] } }, { "index": 2104, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2104.png", "answer": "1", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;e: Number;Eccentricity(G) = e;e = sqrt(3)/2;A: Point;B: Point;LeftVertex(G) = A;RightVertex(G) = B;P: Point;PointOnCurve(P, G);Negation(P=A);Negation(P=B);alpha: Number;beta: Number;Inclination(LineOf(P, A)) = alpha;Inclination(LineOf(P, B)) = beta", "coords": {} }, { "index": 2105, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2105.png", "answer": "2*x+y+1=0", "fact_expr": "G: Parabola;Expression(G) = (y = x^2);H: Point;Coordinate(H) = (-1, 1)", "coords": { "H": [ -1.0, 1.0 ] } }, { "index": 2106, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2106.png", "answer": "12", "fact_expr": "G: Ellipse;H: Line;A: Point;B: Point;F1: Point;Expression(G) = (x^2/25 + y^2/9 = 1);PointOnCurve(Center(G), H);Intersection(H, G) = {A, B};RightFocus(G) = F1", "coords": {} }, { "index": 2107, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.1622776601683795, "b": 2.23606797749979, "major_axis": "x", "x_coef": 10.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2107.png", "answer": "5*x^2/32-5*y^2/18=1", "fact_expr": "G: Hyperbola;Expression(Asymptote(G)) = (3*x + pm*4*y = 0);H: Ellipse;Expression(H) = (x^2/10 + y^2/5 = 1);Focus(G) = Endpoint(MajorAxis(H))", "coords": {} }, { "index": 2108, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2108.png", "answer": "sqrt(5)/2", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (2, 1);PointOnCurve(P, Asymptote(C))", "coords": { "P": [ 2.0, 1.0 ] } }, { "index": 2109, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2109.png", "answer": "pm*4", "fact_expr": "G: Parabola;Vertex(G) = O;O: Origin;PointOnCurve(Focus(G), yAxis);P: Point;Coordinate(P) = (m, -2);m: Real;PointOnCurve(P, G);Distance(P, Focus(G)) = 4", "coords": {} }, { "index": 2110, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2110.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Parabola;p: Number;A: Point;B: Point;F: Point;M: Point;l:Line;M1:Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;PointOnCurve(A,G);PointOnCurve(B, G);AngleOf(A, F, B) = ApplyUnit(120, degree);IsChordOf(LineSegmentOf(A,B),G);MidPoint(LineSegmentOf(A,B))=M;Directrix(G)=l;L:Line;PointOnCurve(M,L);IsPerpendicular(l,L);FootPoint(l,L)=M1", "coords": {} }, { "index": 2111, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2111.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1);P: Point;PointOnCurve(P, G);M: Point;Coordinate(M) = (3, 0)", "coords": { "M": [ 3.0, 0.0 ] } }, { "index": 2112, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2112.png", "answer": "sqrt(3)", "fact_expr": "C: Ellipse;F1: Point;P: Point;F2: Point;Expression(C) = (x^2/4 + y^2/3 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);AngleOf(F1, P, F2) = pi/3", "coords": {} }, { "index": 2113, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2113.png", "answer": "sqrt(2)/2", "fact_expr": "M: Ellipse;b: Number;a: Number;N: Circle;Expression(M) = (y^2/b^2 + x^2/a^2 = 1);Expression(N) = (x^2 + y^2 = b^2);PointOnCurve(Focus(M), N)", "coords": {} }, { "index": 2114, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 8.0, "b": 4.0, "major_axis": "y", "x_coef": 16.0, "y_coef": 64.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2114.png", "answer": "y^2/36 - x^2/12 = 1", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (4*x^2 + y^2 = 64);Focus(G) = Focus(H);InterReciprocal(Eccentricity(G), Eccentricity(H))", "coords": {} }, { "index": 2115, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2115.png", "answer": "-1", "fact_expr": "G: Parabola;C: Circle;a: Real;l: Line;Expression(G) = (y^2 = 4*x);Expression(C) = (-2*sqrt(3)*y + 2*a*x + x^2 + y^2 = 0);IsTangent(l,C);Directrix(G)=l", "coords": {} }, { "index": 2116, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2116.png", "answer": "2*sqrt(7)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/9 + x^2/a^2 = 1);a: Number;H: Parabola;Expression(H) = (y^2 = 16*x);Focus(H)=OneOf(Focus(G))", "coords": {} }, { "index": 2117, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2117.png", "answer": "16*sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1) ;LeftFocus(G) = F1;RightFocus(G) = F2;F1: Point;F2: Point;P: Point;PointOnCurve(P, G) = True;Abs(LineSegmentOf(P, F1))*Abs(LineSegmentOf(P, F2)) = 64;Area(TriangleOf(P, F1, F2)) = S;S: Number", "coords": {} }, { "index": 2118, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2118.png", "answer": "y = pm*sqrt(3)*x", "fact_expr": "G: Hyperbola;a: Number;H: Parabola;a>0;Expression(G) = (-y^2/12 + x^2/a^2 = 1);Expression(H) = (y^2 = 16*x);Focus(H) = OneOf(Focus(G))", "coords": {} }, { "index": 2119, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2119.png", "answer": "x+4*y-5=0", "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/4 = 1);A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G);M: Point;Coordinate(M) = (1, 1);MidPoint(LineSegmentOf(A, B)) = M", "coords": { "M": [ 1.0, 1.0 ] } }, { "index": 2120, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2120.png", "answer": "2*sqrt(2)", "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;Q: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;Slope(l)>0;PointOnCurve(F,l);Intersection(l,C)={A,B};Intersection(l,Directrix(C))=Q;VectorOf(A, B) = VectorOf(B, Q)", "coords": {} }, { "index": 2121, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2122, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2122.png", "answer": "sqrt(10)/4", "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;O: Origin;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;Abs(VectorOf(P, F1)) = 3*Abs(VectorOf(P, F2))", "coords": {} }, { "index": 2123, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2123.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Ellipse;a: Number;a>0;Expression(G) = (y^2/3 + x^2/a^2 = 1);Expression(OneOf(Directrix(G))) = (x = 4)", "coords": {} }, { "index": 2124, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2124.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Circle;L: Line;M: Point;N: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(RightFocus(G),L);IsPerpendicular(L,xAxis);Intersection(L,G) = {M, N};IsDiameter(LineSegmentOf(M,N),H);PointOnCurve(LeftFocus(G),H)", "coords": {} }, { "index": 2125, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2125.png", "answer": "x + 2*y - 4 = 0", "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/4 + y^2/3 = 1);Coordinate(P) = (1, 3/2);PointOnCurve(P, G)", "coords": { "P": [ 1.0, 1.5 ] } }, { "index": 2126, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2127, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2127.png", "answer": "sqrt(2)/2", "fact_expr": "M: Point;N: Point;G: Line;a: Number;H: Curve;C:Curve;Expression(G) = (y = a);Expression(H) = (y^2 = x);Expression(C) = (y = e^x);Intersection(G, H) = M;Intersection(G, C) = N;WhenMin(LineSegmentOf(M, N))", "coords": {} }, { "index": 2128, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_2128.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;a: Number;a>0;Expression(G) = (-y^2/9 + x^2/a^2 = 1);Eccentricity(G) = 2", "coords": {} }, { "index": 2129, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2130, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2130.png", "answer": "4", "fact_expr": "G: Ellipse;Expression(G) = (x^2 + 2*y^2 = 4)", "coords": {} }, { "index": 2131, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2132, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2133, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2133.png", "answer": "32/2", "fact_expr": "l: Line;C: Parabola;A: Point;M: Point;B: Point;F: Point;Expression(C) = (y^2 = 8*x);Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};Intersection(l, Directrix(C)) = M;MidPoint(LineSegmentOf(A, M)) = F", "coords": {} }, { "index": 2134, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2134.png", "answer": "pm*sqrt(2)", "fact_expr": "F1: Point;LeftFocus(C) = F1;C: Ellipse;Expression(C) = (x^2/4 + y^2 = 1);l: Line;PointOnCurve(F1, l);A: Point;B: Point;Intersection(l, C) = {A, B};VectorOf(B, F1) = 3*VectorOf(F1, A)", "coords": {} }, { "index": 2135, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2135.png", "answer": "[13/12, +oo)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;A: Point;B: Point;C: Point;D: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(RightFocus(G), H);IsPerpendicular(H,xAxis);Intersection(H, G) = {A, B};Intersection(H,Asymptote(G))={C,D};Abs(LineSegmentOf(A, B)) >= (5/13)*Abs(LineSegmentOf(C, D))", "coords": {} }, { "index": 2136, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2136.png", "answer": "sqrt(13)/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Ellipse;P: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2/4 + y^2/3 = 1);Intersection(H,OneOf(Asymptote(G)))=P;XCoordinate(P) = 1", "coords": {} }, { "index": 2137, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2137.png", "answer": "sqrt(30)/3", "fact_expr": "G: Ellipse;H: Point;Expression(G) = (x^2 + 2*y^2 = 4);Coordinate(H) = (1, 1);L: LineSegment;IsChordOf(L, G);MidPoint(L) = H", "coords": { "H": [ 1.0, 1.0 ] } }, { "index": 2138, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2138.png", "answer": "4*sqrt(3)", "fact_expr": "G: Parabola;M: Point;F: Point;l: Line;K: Point;Z: Line;Expression(G) = (y^2 = 4*x);PointOnCurve(M, G);Focus(G) = F;LineSegmentOf(M, F) = 4;PointOnCurve(M, Z);Directrix(G) = l;IsPerpendicular(Z, l);FootPoint(Z, l) = K", "coords": {} }, { "index": 2139, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2140, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2140.png", "answer": "sqrt(19)/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;A: Point;O: Origin;C: Point;B: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = 2*b);Intersection(H,LeftPart(G))=B;Intersection(H,RightPart(G))=C;RightVertex(G)=A;AngleOf(A, O, C) = AngleOf(B, O, C)", "coords": {} }, { "index": 2141, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2141.png", "answer": "{5, 3}", "fact_expr": "E: Ellipse;m: Number;Expression(E) = (y^2/4 + x^2/m = 1);FocalLength(E) = 2", "coords": {} }, { "index": 2142, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.0625, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2142.png", "answer": "17/8", "fact_expr": "G: Parabola;Expression(G) = (y = 4*x^2);l: Line;PointOnCurve(Focus(G), l);A: Point;B: Point;x1: Number;x2: Number;y1: Number;y2: Number;Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);Intersection(l, G) = {A, B};y1 + y2 = 2", "coords": {} }, { "index": 2143, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2143.png", "answer": "x^2/10 - y^2/5 = 1", "fact_expr": "G: Hyperbola;C: Hyperbola;H: Point;Expression(G) = (x^2/2 - y^2 = 1);Coordinate(H) = (4, sqrt(3));PointOnCurve(H, C);Asymptote(C)=Asymptote(G)", "coords": { "H": [ 4.0, 1.7320508075688772 ] } }, { "index": 2144, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2144.png", "answer": "1", "fact_expr": "C: Parabola;Q: Point;P: Point;Expression(C) = (y^2 = 8*x);Coordinate(Q) = (1, 0);PointOnCurve(P, C)", "coords": { "Q": [ 1.0, 0.0 ] } }, { "index": 2145, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2631799883090584, "b": 2.5275303393962245, "orientation": "horizontal" }, "optimization": { "final_loss": 4.481942266905428e-06, "converged": true, "iterations": 95 }, "output_path": "results/train/hyperbola/problem_2145.png", "answer": "3", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/(a + 3)^2 + x^2/a^2 = 1);a: Number;a>0;Expression(Asymptote(G)) = (y = pm*2*x)", "coords": {} }, { "index": 2146, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2146.png", "answer": "10*sqrt(2)", "fact_expr": "G: Hyperbola;M: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 - y^2/5 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(M, G);Perimeter(TriangleOf(M, F1, F2)) = 20", "coords": {} }, { "index": 2147, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2147.png", "answer": "sqrt(2)-1", "fact_expr": "G: Ellipse;Expression(G) = (x^2/2 + y^2 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;M: Point;Negation(M = Endpoint(MajorAxis(G)));PointOnCurve(M, G);I: Point;Incenter(TriangleOf(M, F1, F2)) = I", "coords": {} }, { "index": 2148, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2148.png", "answer": "(2,2)", "fact_expr": "C: Parabola;p: Number;A: Point;p>0;Expression(C) = (y^2 = 2*p*x);Expression(Directrix(C)) = (x = -1/2);PointOnCurve(A,C);Quadrant(A)=1;Distance(A, Focus(C)) = 5/2", "coords": {} }, { "index": 2149, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2149.png", "answer": "-3/4", "fact_expr": "G: Parabola;H: Line;O: Origin;A:Point;B: Point;Expression(G) = (y^2 = 2*x);Intersection(G, H) = {A,B};PointOnCurve(Focus(G), H)", "coords": {} }, { "index": 2150, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2150.png", "answer": "5", "fact_expr": "C: Parabola;G: Line;M: Point;A: Point;B: Point;Expression(C) = (y^2 = 4*x);Coordinate(M) = (-1, 1);PointOnCurve(Focus(C), G);Intersection(G, C) = {A, B};AngleOf(A, M, B) = ApplyUnit(90, degree);IsChordOf(LineSegmentOf(A,B),C)", "coords": { "M": [ -1.0, 1.0 ] } }, { "index": 2151, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2152, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -2.0, 0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_2152.png", "answer": "y = pm*(sqrt(3)/3)*x", "fact_expr": "C: Hyperbola;a: Number;G: Circle;a>0;Expression(C) = (-y^2 + x^2/a^2 = 1);Expression(G) = (y^2 + (x - 2)^2 = 1);IsTangent(G, Asymptote(C))", "coords": {} }, { "index": 2153, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2154, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2154.png", "answer": "-3/4", "fact_expr": "G: Parabola;H: Line;O: Origin;A: Point;B: Point;Expression(G) = (y^2 = 2*x);PointOnCurve(Focus(G),H);Intersection(G, H) = {A, B}", "coords": {} }, { "index": 2155, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2155.png", "answer": "sqrt(2)", "fact_expr": "H: Parabola;Expression(H) = (y^2 = 2*(p*x));p: Number;p>0;G: Hyperbola;Expression(G) = (x^2 - y^2 = 1);Directrix(H) = OneOf(Directrix(G))", "coords": {} }, { "index": 2156, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2156.png", "answer": "2", "fact_expr": "G: Parabola;M: Point;N: Point;x0: Number;y0: Number;Coordinate(N) = (2*sqrt(3), 0);Coordinate(M) = (x0, y0);Expression(G) = (y = x^2/8);PointOnCurve(M, G)", "coords": { "N": [ 3.4641016151377544, 0.0 ] } }, { "index": 2157, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2158, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 2.043849229812622, "b": 1.7850669475408654, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 3.888911938768068e-06, "converged": true, "iterations": 97 }, "output_path": "results/train/ellipse/problem_2158.png", "answer": "2*sqrt(3)", "fact_expr": "G: Ellipse;m: Number;H: Point;Expression(G) = (x^2/9 + y^2/m = 1);Coordinate(H) = (sqrt(3), sqrt(2));PointOnCurve(H, G)", "coords": { "H": [ 1.7320508075688772, 1.4142135623730951 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 2159, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2159.png", "answer": "1/2", "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2/b^2 = 1);b: Number;0 < b;b < 2;F: Point;LeftFocus(C) = F;M: Point;PointOnCurve(M, C) = True;N: Point;Coordinate(N) = (0, sqrt(3));Max(Abs(LineSegmentOf(M, F)) + Abs(LineSegmentOf(M, N))) = 6", "coords": { "N": [ 0.0, 1.7320508075688772 ] } }, { "index": 2160, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2160.png", "answer": "(1+sqrt(5))/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;c: Number;D: Circle;F: Point;E: Point;O: Origin;P: Point;l1:Line;b > a;a > 0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 4*c*x);Expression(D) = (x^2 + y^2 = a^2);Coordinate(F) = (-c, 0);LeftFocus(G)=F;TangentOfPoint(F,D)=l1;TangentPoint(l1,D)=E;Intersection(OverlappingLine(LineSegmentOf(F,E)),H)=P;c>0;VectorOf(O,E)=(VectorOf(O,F)+VectorOf(O,P))/2", "coords": {} }, { "index": 2161, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2631799883090584, "b": 2.5275303393962245, "orientation": "horizontal" }, "optimization": { "final_loss": 4.481942266905428e-06, "converged": true, "iterations": 95 }, "output_path": "results/train/hyperbola/problem_2161.png", "answer": "(pm*5,0)\nx^2/5-y^2/20=1", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/49 + y^2/24 = 1);Expression(Asymptote(G)) = (y = pm*2*x);Focus(G) = Focus(H)", "coords": {} }, { "index": 2162, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2163, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2163.png", "answer": "sqrt(3)", "fact_expr": "G: Parabola;H: Line;Expression(G) = (x^2 = 8*y);Expression(H) = (x - sqrt(3)*y = 0)", "coords": {} }, { "index": 2164, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2164.png", "answer": "4", "fact_expr": "G: Ellipse;m: Real;Expression(G) = (y^2/3 + x^2/m = 1);PointOnCurve(Focus(G), xAxis);Eccentricity(G) = (1/2)", "coords": {} }, { "index": 2165, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2165.png", "answer": "sqrt(5)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P: Point;PointOnCurve(P, G) = True;H: Line;Expression(H) = (x = -1);Q: Point;Coordinate(Q) = (2, 2)", "coords": { "Q": [ 2.0, 2.0 ] } }, { "index": 2166, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2167, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2167.png", "answer": "13", "fact_expr": "G: Hyperbola;m: Number;H: Parabola;Expression(G) = (y^2/3 - x^2/m = 1);Expression(H) = (y = x^2/16);Focus(H) = UpperFocus(G)", "coords": {} }, { "index": 2168, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2168.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "l: Line ;H:LineSegment;G: Hyperbola;A: Point;Coordinate(A) = (2, 1);Expression(G) = (x^2 - y^2/3 = 1);IsChordOf(H, G);MidPoint(H) = A;OverlappingLine(H,l)", "coords": { "A": [ 2.0, 1.0 ] } }, { "index": 2169, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2169.png", "answer": "{x^2/8+y^2/6=1,3*y^2/25+4*y^2/25=1}", "fact_expr": "G: Ellipse;H: Point;C:Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);Coordinate(H) = (2, -sqrt(3));Eccentricity(G)=Eccentricity(C);PointOnCurve(H,C)", "coords": { "H": [ 2.0, -1.7320508075688772 ] } }, { "index": 2170, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2170.png", "answer": "10", "fact_expr": "l: Line;G: Hyperbola;B: Point;F2: Point;A: Point;F1: Point;Expression(G) = (x^2/4 - y^2/2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, l);Intersection(l, LeftPart(G)) = {A, B}", "coords": {} }, { "index": 2171, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2171.png", "answer": "5/4", "fact_expr": "G: Parabola;A: Point;B: Point;F: Point;Expression(G) = (y^2 = x);Focus(G) = F;PointOnCurve(A, G);PointOnCurve(B, G);Abs(LineSegmentOf(A, F)) + Abs(LineSegmentOf(B, F)) = 3", "coords": {} }, { "index": 2172, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2172.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Parabola;H: Line;A: Point;F: Point;B: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, F)) = 4", "coords": {} }, { "index": 2173, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.8284271247461903, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2173.png", "answer": "(4/7,1/7)", "fact_expr": "G: Hyperbola;H: Line;A: Point;B: Point;Expression(G) = (x^2/8 - y^2/4 = 1);Expression(H) = (y = 2*x - 1);Intersection(H,G)={A,B}", "coords": {} }, { "index": 2174, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2174.png", "answer": "10*sqrt(3)", "fact_expr": "l: Line;G: Ellipse;b: Number;a: Number;L: Line;A: Point;O: Origin;B: Point;F: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F;PointOnCurve(F, l);Slope(l) = 1;Intersection(l, G) = {A, B};Abs(LineSegmentOf(A, B)) = 8;IsTangent(L,G);PointOnCurve(A,L);Slope(L)=k1;k1:Number;Slope(LineOf(O,A))=k2;k2:Number;k1*k2=-1/4", "coords": {} }, { "index": 2175, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2175.png", "answer": "sqrt(6)/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;F2: Point;M: Point;O: Origin;N: Point;L1:Line;L2:Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F2;PointOnCurve(M,L1);PointOnCurve(N,L1);IsParallel(LineOf(F2,M),L2);IsPerpendicular(LineSegmentOf(O, F2), LineSegmentOf(F2, N));Abs(LineSegmentOf(F2, M)) = (sqrt(3)/2)*Abs(LineSegmentOf(F2, N));OneOf(Asymptote(G)) = L1;OneOf(Asymptote(G)) = L2;Negation(L1=L2)", "coords": {} }, { "index": 2176, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2177, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2177.png", "answer": "(1/2,1)", "fact_expr": "C: Ellipse;G: Line;B: Point;A: Point;M: Point;F: Point;N:Point;Expression(C) = (x^2/2 + y^2 = 1);LeftFocus(C)=F;PointOnCurve(F, G);Intersection(G, C) = {A, B};Intersection(PerpendicularBisector(LineSegmentOf(A,B)),xAxis)=M;Intersection(PerpendicularBisector(LineSegmentOf(A,B)),yAxis)=N", "coords": {} }, { "index": 2178, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_2178.png", "answer": "2*sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/b^2 = 1);b: Number;b>0;Eccentricity(G) = 2", "coords": {} }, { "index": 2179, "success": false, "conic_type": "parabola", "error": "validation: parabola_focus_mismatch", "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2179.png", "answer": "5/2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Coordinate(F) = (2, 0);Focus(G) = F;A: Point;Coordinate(A) = (3, 2);L: Line;PointOnCurve(A,L);IsPerpendicular(L, Directrix(G));E: Point;Intersection(L, G) = E", "coords": { "F": [ 2.0, 0.0 ], "A": [ 3.0, 2.0 ] }, "validation_reasons": [ "parabola_focus_mismatch" ] }, { "index": 2180, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2180.png", "answer": "24", "fact_expr": "A: Point;B: Point;G: Hyperbola;Expression(G) = (x^2/4 - y^2/12 = 1);LeftFocus(G) = A;RightFocus(G) = B;C: Point;PointOnCurve(C, G) = True;Cos(AngleOf(A, C, B)) = 3/5", "coords": {} }, { "index": 2181, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2181.png", "answer": "4*sqrt(7)/7", "fact_expr": "G: Hyperbola;a: Number;H: Parabola;O: Origin;a>0;Expression(G) = (-y^2/9 + x^2/a^2 = 1);Expression(H) = (y^2 = 16*x);Center(G) = O;RightFocus(G) = Focus(H)", "coords": {} }, { "index": 2182, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2183, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.463192105293274, "b": 2.070901870727539, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4661259228887502e-06, "converged": true, "iterations": 106 }, "output_path": "results/train/hyperbola/problem_2183.png", "answer": "(8/3)*sqrt(3)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;Length(RealAxis(C)) = 4;Eccentricity(C) = sqrt(3);P: Point;PointOnCurve(P, C);AngleOf(F1, P, F2) = ApplyUnit(120, degree)", "coords": {} }, { "index": 2184, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2184.png", "answer": "sqrt(3)", "fact_expr": "Gamma: Ellipse;Expression(Gamma) = (y^2 + x^2/a^2 = 1);a: Number;a>1;LeftVertex(Gamma) = A;A: Point;PointOnCurve(A, l);l: Line;Intersection(l, Gamma) = {A,B};B: Point;Intersection(l, yAxis) = C;C: Point;Abs(LineSegmentOf(O, A)) = Abs(LineSegmentOf(O, C));Abs(LineSegmentOf(A, B)) = Abs(LineSegmentOf(B, C));O: Origin", "coords": {} }, { "index": 2185, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2186, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2186.png", "answer": "sqrt(10)/3", "fact_expr": "l: Line;C: Hyperbola;a: Number;b: Number;a>0;b>0;Expression(C) = (-x^2/b^2 + y^2/a^2 = 1);Expression(l)=(3*x-y+10=0);IsParallel(OneOf(Asymptote(C)),l)", "coords": {} }, { "index": 2187, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_2187.png", "answer": "2*sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;H: Point;PointOnCurve(H, G);L1: Line;L2: Line;Asymptote(G) = {L1, L2};Distance(H, L1)*Distance(H, L2) = 3/4;Eccentricity(G) = 2", "coords": {} }, { "index": 2188, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2188.png", "answer": "sqrt(5)/2", "fact_expr": "C: Hyperbola;Expression(C) = (-x^2/b^2 + y^2/a^2 = 1);b: Number;a: Number;a>0;b>0;P: Point;PointOnCurve(P, xAxis);G: Line;Expression(G) = (y = m + 2*x);m: Number;Negation(m=0);PointOnCurve(P, G);NumIntersection(G, C) = 1", "coords": {} }, { "index": 2189, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2189.png", "answer": "1/2", "fact_expr": "G: Parabola;H: Line;A: Point;F: Point;B: Point;Expression(G) = (y^2 = -2*x);Focus(G) = F;Slope(H) = sqrt(3);PointOnCurve(F, H);Intersection(H, G) = {A, B}", "coords": {} }, { "index": 2190, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "y", "x_coef": 3.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2190.png", "answer": "(-3/2, 1)", "fact_expr": "C: Ellipse;P: Point;Q: Point;F: Point;Expression(C) = (x^2/3 + y^2/4 = 1);Coordinate(Q) = (1, 1);LowerFocus(C)=F;PointOnCurve(P, C);WhenMax(Abs(LineSegmentOf(P,Q))+Abs(LineSegmentOf(P,F)))", "coords": { "Q": [ 1.0, 1.0 ] } }, { "index": 2191, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.5, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2191.png", "answer": "2/5", "fact_expr": "G: Parabola;H: Line;m: Number;Expression(G) = (x^2 = -10*y);Expression(H) = (2*(m*x) + m*y + 1 = 0);PointOnCurve(Focus(G), H)", "coords": {} }, { "index": 2192, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2192.png", "answer": "sqrt(3)/3", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;L: Line;PointOnCurve(F2, L) = True;IsPerpendicular(L, xAxis) = True;Intersection(L, C) = {A, B};A: Point;B: Point;Intersection(LineSegmentOf(F1, B), yAxis) = D;D: Point;IsPerpendicular(LineSegmentOf(A, D), LineSegmentOf(F1, B)) = True", "coords": {} }, { "index": 2193, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2193.png", "answer": "sqrt(3) + 1", "fact_expr": "G: Hyperbola;a: Number;b: Number;H: Circle;F1: Point;F2: Point;P: Point;a>0;b>0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;IsDiameter(LineSegmentOf(F1, F2), H);OneOf(Intersection(H, RightPart(G))) = P;AngleOf(P, F2, F1) = 2*AngleOf(P, F1, F2)", "coords": {} }, { "index": 2194, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2194.png", "answer": "4*sqrt(2)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);A: Point;PointOnCurve(A, G);F: Point;Focus(G) = F;Distance(A, F) = 5;O: Origin;Distance(A, O) = Abs(LineSegmentOf(O, A))", "coords": {} }, { "index": 2195, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2195.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;B: Point;Intersection(LeftDirectrix(G), xAxis) = B;A: Point;Coordinate(A) = (0, b);P: Point;VectorOf(A, P) = 2*VectorOf(A, B);PointOnCurve(P, G)", "coords": {} }, { "index": 2196, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2196.png", "answer": "sqrt(3)/2", "fact_expr": "C: Parabola;G: Line;F: Point;A: Point;B: Point;k:Number;Expression(C) = (y^2 = 12*x);k>0;Expression(G) = (y = k*(x + 3));Intersection(G, C) = {A, B};Focus(C) = F;Abs(LineSegmentOf(F, A)) = 3*Abs(LineSegmentOf(F, B))", "coords": {} }, { "index": 2197, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2197.png", "answer": "(1, 3]", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F2: Point;F1: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, LeftPart(G));Abs(LineSegmentOf(P, F2))^2/Abs(LineSegmentOf(P, F1))^2 = 8*a", "coords": {} }, { "index": 2198, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2198.png", "answer": "1", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(90, degree)", "coords": {} }, { "index": 2199, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2199.png", "answer": "y^2=20*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);H: Parabola;RightFocus(G) = Focus(H)", "coords": {} }, { "index": 2200, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2200.png", "answer": "4", "fact_expr": "G: Hyperbola;C:Hyperbola;b: Number;a: Number;a>b;b>0;e1:Number;e2:Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(C)=(x^2/a^2-y^2/b^2=-1);Eccentricity(G)=e1;Eccentricity(C)=e2", "coords": {} }, { "index": 2201, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2201.png", "answer": "y=pm*(sqrt(3)/3)*x", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Eccentricity(G) = (2*sqrt(3))/3", "coords": {} }, { "index": 2202, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2202.png", "answer": "6*x+3*y+4=0", "fact_expr": "G: Parabola;H: Circle;A: Point;B: Point;C: Point;Expression(G) = (x^2 = 2*y);Expression(H) = (x^2 + (y - 2)^2 = 1);Coordinate(A) = (2, 2);PointOnCurve(A,LineOf(A,B));PointOnCurve(A,LineOf(A,C));IsTangent(LineOf(A,B),H);IsTangent(LineOf(A,C),H);Intersection(LineOf(A,C),G)={B,C}", "coords": { "A": [ 2.0, 2.0 ] } }, { "index": 2203, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2204, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2205, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.449489742783178, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2205.png", "answer": "6", "fact_expr": "G: Hyperbola;C: Parabola;p: Number;Expression(G) = (x^2/6 - y^2/3 = 1);Expression(C) = (y^2 = 2*p*x);Focus(C) = RightFocus(G)", "coords": {} }, { "index": 2206, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2206.png", "answer": "13", "fact_expr": "G: Parabola;P: Point;F: Point;Expression(G) = (y^2 = 16*x);PointOnCurve(P, G);Distance(P, xAxis) = 12;Focus(G)=F;Distance(P,F)=Abs(LineSegmentOf(P,F))", "coords": {} }, { "index": 2207, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2208, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -4.0, 0.0 ], "radius": 1.4142135623730951 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_2208.png", "answer": "{(x^2/2 - y^2/14 = 1) & (x >= sqrt(2))}", "fact_expr": "M: Circle;A:Circle;B:Circle;Expression(A)=((x+4)^2+y^2=2);Expression(B)=((x-4)^2+y^2=2);IsOutTangent(M,A);IsInTangent(M,B);M1:Point;Center(M)=M1", "coords": {} }, { "index": 2209, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4149149626832767, "b": 2.450642358201144, "orientation": "horizontal" }, "optimization": { "final_loss": 1.0319236343469396e-05, "converged": true, "iterations": 94 }, "output_path": "results/train/hyperbola/problem_2209.png", "answer": "x^2/4 - y^2/12 = 1", "fact_expr": "G: Hyperbola;Coordinate(Focus(G)) = (pm*4, 0);Expression(OneOf(Asymptote(G)))=(y=sqrt(3)*x)", "coords": {} }, { "index": 2210, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2210.png", "answer": "sqrt(10)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/4 = 1);Z: Line;Expression(Z) = (x + 2*y - sqrt(2) = 0);K: Point;PointOnCurve(K, G)", "coords": {} }, { "index": 2211, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2211.png", "answer": "3/2", "fact_expr": "C: Parabola;F: Point;M: Point;N: Point;Expression(C) = (y^2 = 2*x);Focus(C) = F;PointOnCurve(M, C);Intersection(OverlappingLine(LineSegmentOf(F, M)), yAxis) = N;MidPoint(LineSegmentOf(F, N)) = M", "coords": {} }, { "index": 2212, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2212.png", "answer": "1/2", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1)", "coords": {} }, { "index": 2213, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2214, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2214.png", "answer": "sqrt(5)/3", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;OneOf(Focus(G)) = F1;P: Point;PointOnCurve(P, G) = True;H: Circle;IsDiameter(MinorAxis(G),H) ;TangentPoint(H,LineSegmentOf(P,F1)) = MidPoint(LineSegmentOf(P,F1))", "coords": {} }, { "index": 2215, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2215.png", "answer": "5", "fact_expr": "P: Point;PointOnCurve(P, G);G: Ellipse;Expression(G) = (x^2/9 + y^2/5 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};AngleOf(F1, P, F2) = ApplyUnit(90, degree)", "coords": {} }, { "index": 2216, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2216.png", "answer": "5/3", "fact_expr": "G: Ellipse;I: Point;P: Point;F1: Point;F2: Point;D: Point;Expression(G) = (x^2/25 + y^2/16 = 1);Focus(G) = {F1, F2};PointOnCurve(P,G);Negation(P=Endpoint(MajorAxis(G)));Incenter(TriangleOf(P, F1, F2)) = I;Intersection(LineOf(P,I), xAxis) = D", "coords": {} }, { "index": 2217, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.1622776601683795, "b": 2.449489742783178, "major_axis": "x", "x_coef": 10.0, "y_coef": 6.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2217.png", "answer": "2*sqrt(5)-2", "fact_expr": "G: Parabola;p: Number;H: Ellipse;A: Point;M: Point;N: Point;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (x^2/10 + y^2/6 = 1);Coordinate(A) = (4, 4);Focus(G) = RightFocus(H);PointOnCurve(M, G);Projection(M,yAxis)=N", "coords": { "A": [ 4.0, 4.0 ] } }, { "index": 2218, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2219, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2220, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2220.png", "answer": "2", "fact_expr": "A: Point;B: Point;Vertex(MinorAxis(E)) = {A, B};E: Ellipse;Expression(E) = (y^2 + x^2/m = 1);m: Number;m>1;P: Point;PointOnCurve(P, E);Negation(P = A);Negation(P = B);Slope(LineOf(A, P))*Slope(LineOf(B, P)) = -m/4", "coords": {} }, { "index": 2221, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2222, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.552875593585339, "b": 2.3303634978370953, "orientation": "horizontal" }, "optimization": { "final_loss": 1.3057248743840209e-06, "converged": true, "iterations": 98 }, "output_path": "results/train/hyperbola/problem_2222.png", "answer": "sqrt(13)/3", "fact_expr": "G: Hyperbola;Expression(Asymptote(G)) = (y = pm*(3/2)*x)", "coords": {} }, { "index": 2223, "success": false, "error": "\n= 1 $ $ (a > b > 0) $, the length of the major axis is $ \\frac{3\n ^\nParseException: Expected end of text, found '$' (at char 4), (line:1, col:5)" }, { "index": 2224, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2225, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2225.png", "answer": "x^2/16 - y^2/9 = 1", "fact_expr": "C: Hyperbola;b: Number;a: Number;F2: Point;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F2) = (5, 0);e:Number;Eccentricity(C) = e;e = 5/4;RightFocus(C) = F2", "coords": { "F2": [ 5.0, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 2226, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "y", "x_coef": 2.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2226.png", "answer": "2*x+y-3=0", "fact_expr": "G: Ellipse;H: LineSegment;M: Point;Expression(G) = (x^2/2 + y^2/4 = 1);Coordinate(M) = (1, 1);IsChordOf(H, G);MidPoint(H) = M", "coords": { "M": [ 1.0, 1.0 ] } }, { "index": 2227, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2227.png", "answer": "4+sqrt(61)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);F1: Point;F2:Point;LeftFocus(G) = F1;RightFocus(G) = F2;H: Circle;Expression(H) = (x^2 + (y - 6)^2 = 4);A: Point;PointOnCurve(A, H);M: Point;PointOnCurve(M, RightPart(G))", "coords": {} }, { "index": 2228, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2228.png", "answer": "-1/4", "fact_expr": "l: Line;C: Ellipse;b: Number;a: Number;M: Point;P: Point;Q: Point;F: Point;O: Origin;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;PointOnCurve(O, l);Intersection(l, C) = {P, Q};PointOnCurve(M, C);Negation(M = P);Negation(M = Q);Slope(LineOf(M, P))=k1;Slope(LineOf(M, Q))=k2;k1*k2 = -5/9;Abs(LineSegmentOf(P, F)) = 2*Abs(LineSegmentOf(Q, F));k1:Number;k2:Number", "coords": {} }, { "index": 2229, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2229.png", "answer": "\\sqrt{21}/2", "fact_expr": "C: Parabola;p: Number;G: Circle;H: Point;F: Point;A: Point;O: Origin;B: Point;l: Line;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(H) = (1, sqrt(2));Directrix(C) = l;Focus(C) = F;Center(G) = H;IsTangent(G, l);PointOnCurve(A, C);PointOnCurve(B, l);VectorOf(A, O) = 3*VectorOf(O, B);PointOnCurve(F, G)", "coords": { "H": [ 1.0, 1.4142135623730951 ] } }, { "index": 2230, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2230.png", "answer": "y^2=8*x", "fact_expr": "G: Parabola;p: Number;M: Point;p>0;Expression(G) = (y^2 = 2*(p*x));PointOnCurve(M, G);XCoordinate(M)=3;Distance(M,Focus(G))=5", "coords": {} }, { "index": 2231, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2231.png", "answer": "x^2/16+y^2/12=1", "fact_expr": "C: Ellipse;Center(C) = O;O: Origin;PointOnCurve(Focus(C), xAxis) = True;Eccentricity(C) = 1/2;G: Parabola;Expression(G) = (x^2 = 8*(sqrt(3)*y));OneOf(Vertex(C)) = Focus(G)", "coords": {} }, { "index": 2232, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2232.png", "answer": "2", "fact_expr": "G: Hyperbola;a: Number;Expression(G)=(x^2/a-y^2/9=1);Inclination(OneOf(Asymptote(G)))=ApplyUnit(60,degree)", "coords": {} }, { "index": 2233, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 1.4142164151010692, "b": 1.4142164151010692, "orientation": "horizontal" }, "optimization": { "final_loss": 2.2160979279094996e-06, "converged": true, "iterations": 95 }, "output_path": "results/train/hyperbola/problem_2233.png", "answer": "2", "fact_expr": "C: Hyperbola;O: Origin;PointOnCurve(O, C);F1: Point;F2: Point;Coordinate(F1) = (-1, 0);Coordinate(F2) = (3, 0);Focus(C) = {F1, F2}", "coords": { "F1": [ -1.0, 0.0 ], "F2": [ 3.0, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 2234, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0010501146316528, "direction": "right" }, "optimization": { "final_loss": 3.455598402979376e-07, "converged": true, "iterations": 81 }, "output_path": "results/train/parabola/problem_2234.png", "answer": "1/4", "fact_expr": "C: Parabola;p: Number;A: Point;B: Point;P: Point;F: Point;p>0;Expression(C) = (y^2 = 2*p*x);Coordinate(A) = (1, 2);Coordinate(B) = (2, 0);Focus(C) = F;PointOnCurve(A,C);PointOnCurve(P,C)", "coords": { "A": [ 1.0, 2.0 ], "B": [ 2.0, 0.0 ] } }, { "index": 2235, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2235.png", "answer": "2*sqrt(3)/3", "fact_expr": "C: Hyperbola;a: Number;G: Parabola;a>0;Expression(C) = (-y^2 + x^2/a^2 = 1);Expression(G) = (y^2 = -8*x);Focus(G) = LeftFocus(C)", "coords": {} }, { "index": 2236, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2237, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2237.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "l: Line;G: Ellipse;A: Point;B: Point;C: Point;Expression(G) = (x^2/2 + y^2 = 1);Expression(l)=(x - y = 0);Intersection(l, G) = {A, B};PointOnCurve(C, G)", "coords": {} }, { "index": 2238, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 10.0, "b": 8.0, "major_axis": "x", "x_coef": 100.0, "y_coef": 64.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2238.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Ellipse;Expression(G) = (x^2/100 + y^2/64 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 2239, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2239.png", "answer": "x+2*y-2=0", "fact_expr": "l: Line;C: Ellipse;M: Point;A: Point;B: Point;F1: Point;Expression(C) = (x^2/9 + y^2/5 = 1);Coordinate(M) = (0, 1);PointOnCurve(M, l);LeftFocus(C)=F1;Intersection(l, C) = {A, B};WhenMax(Perimeter(TriangleOf(A,B,F1)))", "coords": { "M": [ 0.0, 1.0 ] } }, { "index": 2240, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/train/hyperbola/problem_2240.png", "answer": "x^2+(y-3)^2=10", "fact_expr": "C: Hyperbola;m: Number;P: Point;A: Point;B: Point;Expression(C) = (-y^2/2 + x^2/m = 1);Coordinate(P) = (3, 4);LeftVertex(C) = A;RightVertex(C) = B;PointOnCurve(P, C)", "coords": { "P": [ 3.0, 4.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 2241, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2241.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Hyperbola;m: Number;L: Line;A: Point;B: Point;F2: Point;F1: Point;Expression(G) = (-y^2/7 + x^2/m = 1);LeftFocus(G) = F1;PointOnCurve(F1,L);Intersection(L,LeftPart(G))={A,B};Abs(LineSegmentOf(A, B)) = 4;RightFocus(G)=F2;Perimeter(TriangleOf(A, B, F2)) = 20", "coords": {} }, { "index": 2242, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2242.png", "answer": "1/2", "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;I: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2;Incenter(TriangleOf(P, F1, F2)) = I;Area(TriangleOf(P,F1,F2))=3*Area(TriangleOf(I,F1,F2))", "coords": {} }, { "index": 2243, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_focus_mismatch", "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2243.png", "answer": "3", "fact_expr": "G: Ellipse;k: Number;Expression(G) = (x^2 + y^2/k = 1);Coordinate(F)=(0,sqrt(2));F:Point;OneOf(Focus(G))=F", "coords": { "F": [ 0.0, 1.4142135623730951 ] }, "validation_reasons": [ "ellipse_focus_mismatch" ] }, { "index": 2244, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2244.png", "answer": "1/7", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/b^2 = 1);b: Number;2 > b;b > 0;H: Line;Expression(H) = (x + y = 1);Intersection(H, G) = {P, Q};P: Point;Q: Point;M: Point;MidPoint(LineSegmentOf(P, Q)) = M;O: Origin;IsPerpendicular(LineSegmentOf(O, P), LineSegmentOf(O, Q)) = True", "coords": {} }, { "index": 2245, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2245.png", "answer": "8*sqrt(7)/7", "fact_expr": "G: Ellipse;A: Point;B: Point;F2: Point;F1: Point;x1:Number;x2:Number;y1:Number;y2:Number;Expression(G) = (x^2/16 + y^2/9 = 1);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);LeftFocus(G) = F1;RightFocus(G) = F2;I:Line;PointOnCurve(F1, I);Intersection(I, G) = {A, B};Area(InscribedCircle(TriangleOf(A,B,F2)))=pi", "coords": {} }, { "index": 2246, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2247, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2247.png", "answer": "(-4,0)+(0,4)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/m^2 = 1);PointOnCurve(Focus(G), xAxis);m:Number", "coords": {} }, { "index": 2248, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2248.png", "answer": "4*sqrt(2)", "fact_expr": "G: Hyperbola;Q: Point;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/2 - y^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, RightPart(G));PointOnCurve(Q, RightPart(G));PointOnCurve(F2, LineOf(P, Q))", "coords": {} }, { "index": 2249, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_focus_mismatch", "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2249.png", "answer": "(3*x^2)/16+(3*y^2)/4=1", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;RightFocus(C) = F;Coordinate(F) = (2, 0);G: Line;PointOnCurve(F, G);A: Point;B: Point;Intersection(G, C) = {A, B};Coordinate(MidPoint(LineSegmentOf(A, B))) = (1, 1/2)", "coords": { "F": [ 2.0, 0.0 ] }, "validation_reasons": [ "ellipse_focus_mismatch" ] }, { "index": 2250, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2251, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2251.png", "answer": "2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G) = True;P: Point;Coordinate(P) = (1, 1);MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ 1.0, 1.0 ] } }, { "index": 2252, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2253, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2253.png", "answer": "2\nx = -1", "fact_expr": "G: Parabola;p: Number;Expression(G) = (y^2 = 2*(p*x));Coordinate(Focus(G)) = (1, 0)", "coords": {} }, { "index": 2254, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2255, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2255.png", "answer": "4/3", "fact_expr": "G: Ellipse;Expression(G) = (x^2/2 + y^2 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;H: Line;PointOnCurve(F2, H);Inclination(H) = pi/4;A: Point;B: Point;Intersection(H, G) = {A, B}", "coords": {} }, { "index": 2256, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2256.png", "answer": "4*sqrt(2)", "fact_expr": "G: Parabola;A: Point;Expression(G) = (y^2 = -4*x);PointOnCurve(A, G);Distance(A, Focus(G)) = 5;O:Origin", "coords": {} }, { "index": 2257, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2258, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2258.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "H: Line;Expression(H) = (y = k*x - 1);k: Real;G: Hyperbola;Expression(G) = (x^2 - y^2 = 4);NumIntersection(H, G) = 0", "coords": {} }, { "index": 2259, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2260, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2260.png", "answer": "2,6", "fact_expr": "G: Parabola;p: Number;H: Circle;p>0;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (4*x + x^2 + y^2 + 3 = 0);IsTangent(H, Directrix(G))", "coords": {} }, { "index": 2261, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2262, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2262.png", "answer": "(2,0)", "fact_expr": "C:Circle;G: Parabola;Expression(G) = (y^2 = 8*x);PointOnCurve(Center(C),G);L: Line;Expression(L) = (x + 2 = 0);IsTangent(C,L);P:Point;PointOnCurve(P,C)", "coords": {} }, { "index": 2263, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2263.png", "answer": "(1+sqrt(13))/2", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;c: Number;Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);LeftFocus(C) = F1;RightFocus(C) = F2;G: Line;PointOnCurve(F1, G);Slope(G) = sqrt(3);P: Point;Q: Point;Intersection(G, LeftPart(C)) = P;Intersection(G, RightPart(C)) = Q;LineSegmentOf(Q, P) = LineSegmentOf(Q, F2)", "coords": {} }, { "index": 2264, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.2352789464429823, "b": 2.2352789464429823, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4312591569844774e-06, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_2264.png", "answer": "x^2/9 - y^2 = 1", "fact_expr": "C: Hyperbola;F1: Point;F2: Point;Coordinate(F1) = (-sqrt(10), 0);Coordinate(F2) = (sqrt(10), 0);Focus(C) = {F1, F2};M: Point;PointOnCurve(M, C);Abs(LineSegmentOf(M, F1)) - Abs(LineSegmentOf(M, F2)) = 6", "coords": { "F1": [ -3.1622776601683795, 0.0 ], "F2": [ 3.1622776601683795, 0.0 ] } }, { "index": 2265, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2265.png", "answer": "8", "fact_expr": "G: Parabola;a: Real;Expression(G) = (y^2 = a*x);Coordinate(Focus(G)) = (2, 0)", "coords": {} }, { "index": 2266, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2266.png", "answer": "4*sqrt(2)", "fact_expr": "P: Point;PointOnCurve(P, G);G: Parabola;Expression(G) = (y^2 = 4*x);Focus(G) = F;F: Point;H: Line;Expression(H) = (x = -2);Z: Line;PointOnCurve(P, Z) ;IsPerpendicular(H, Z) ;FootPoint(H, Z) = M ;M: Point;Q: Point;PointOnCurve(Q, C) ;C: Circle;Expression(C) = ((x + 3)^2 + (y - 4)^2 = 1)", "coords": {} }, { "index": 2267, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2268, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2269, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2269.png", "answer": "(0, 1/2)", "fact_expr": "l: Line;Expression(l) = (x=m*y+2);m: Number;C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(C) = F;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, C) = {A, B};G: Line;O: Origin;PointOnCurve(O, G);IsChordOf(LineSegmentOf(A, B), C);D: Point;MidPoint(LineSegmentOf(A, B)) = D;PointOnCurve(D, G);E: Point;Intersection(G, C) = E;Negation(E=O)", "coords": {} }, { "index": 2270, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 1.0, 2.5 ], "radius": 1.8027756377319946 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_2270.png", "answer": "y^2 - x^2/15 = 1", "fact_expr": "G: Hyperbola;C: Circle;Expression(C) = (-5*y - 2*x + x^2 + y^2 + 4 = 0);P:Point;P1:Point;Intersection(axis,C)={P,P1};OneOf(Focus(G))=P;OneOf(Vertex(G))=P1", "coords": {} }, { "index": 2271, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 3.0, "b": 2.449489742783178, "major_axis": "y", "x_coef": 6.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2271.png", "answer": "x^2/3 + y^2/6 = 1", "fact_expr": "G: Ellipse;A: Point;H:Ellipse;Expression(H) = (x^2/6 + y^2/9 = 1);Coordinate(A) = (-1, -2);PointOnCurve(A, G);Focus(G)=Focus(H)", "coords": { "A": [ -1.0, -2.0 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 2272, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2273, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2273.png", "answer": "(1, pm*3/2)", "fact_expr": "C: Ellipse;P: Point;F: Point;Expression(C) = (x^2/4 + y^2/3 = 1);RightFocus(C) = F;PointOnCurve(P, C);Abs(LineSegmentOf(P, F)) = 3/2", "coords": {} }, { "index": 2274, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2274.png", "answer": "2*sqrt(2)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/2 + y^2 = 1);F: Point;LeftFocus(G) = F;M: Point;N: Point;PointOnCurve(F, LineOf(M, N));Intersection(LineOf(M, N), G) = {M, N};P: Point;Q: Point;O: Origin;PointOnCurve(O, LineOf(P, Q));IsParallel(LineOf(P, Q), LineSegmentOf(M, N));Intersection(LineSegmentOf(P, Q), G) = {P, Q}", "coords": {} }, { "index": 2275, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2275.png", "answer": "{2*sqrt(3)/3, sqrt(3)}", "fact_expr": "C: Hyperbola;b: Number;a: Number;O: Origin;B: Point;A: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightVertex(C) = A;L1: Line;L2: Line;L: Line;PointOnCurve(A, L);IsPerpendicular(L, L1);Intersection(L, L2) = B;Abs(LineSegmentOf(O, B)) = sqrt(3)*Abs(LineSegmentOf(O, A));OneOf(Asymptote(C))=L1;OneOf(Asymptote(C))=L2;Negation(L1=L2)", "coords": {} }, { "index": 2276, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2276.png", "answer": "2", "fact_expr": "M: Point;N: Point;G: Hyperbola;b: Number;a: Number;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, LineOf(M, N));Intersection(LineOf(M,N),RightPart(G))={M,N};Cos(AngleOf(F1, M, N)) = Cos(AngleOf(F1, F2, M));Abs(LineSegmentOf(F1, M))/Abs(LineSegmentOf(F1, N)) = 1/2", "coords": {} }, { "index": 2277, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2277.png", "answer": "1", "fact_expr": "G: Hyperbola;H: Circle;F: Point;P: Point;T: Point;M: Point;O: Origin;l: Line;Expression(G) = (x^2/9 - y^2/16 = 1);Expression(H) = (x^2 + y^2 = 9);LeftFocus(G) = F;TangentOfPoint(F, H) = l;TangentPoint(l, H) = T;Intersection(OverlappingLine(LineSegmentOf(F,T)),RightPart(G))=P;MidPoint(LineSegmentOf(F, P)) = M", "coords": {} }, { "index": 2278, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2278.png", "answer": "2", "fact_expr": "G: Parabola;P: Point;F: Point;Expression(G) = (x^2 = 4*y);Focus(G) = F;PointOnCurve(P, G);Distance(P, yAxis) = 2", "coords": {} }, { "index": 2279, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2280, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2280.png", "answer": "sqrt(2)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);H: Line;Expression(H) = (y = x + 3);A: Point;PointOnCurve(A, G)", "coords": {} }, { "index": 2281, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2281.png", "answer": "2*sqrt(3)/3", "fact_expr": "M: Hyperbola;Expression(M) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;G: Circle;Expression(G) = (x^2 + (-2*b + y)^2 = a^2);IsTangent(Asymptote(M), G)", "coords": {} }, { "index": 2282, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2282.png", "answer": "(\\sqrt{2}/2,1)", "fact_expr": "C: Ellipse;b: Number;a: Number;G: Circle;H: Line;A: Point;F: Point;B: Point;M: Point;N: Point;O: Origin;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;PointOnCurve(O,H);Intersection(H,C)={A,B};MidPoint(LineSegmentOf(A,F))=M;MidPoint(LineSegmentOf(B,F))=N;IsDiameter(LineSegmentOf(M,N),G);PointOnCurve(O,G)", "coords": {} }, { "index": 2283, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2284, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2284.png", "answer": "2*pi/3", "fact_expr": "G: Parabola;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 8*x);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);Focus(G) = F;x1:Number;y1:Number;x2:Number;y2:Number;PointOnCurve(A,G);PointOnCurve(B, G);x1 + x2 + 4 = (2*sqrt(3)/3)*Abs(LineSegmentOf(A, B))", "coords": {} }, { "index": 2285, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2285.png", "answer": "2", "fact_expr": "C: Ellipse;Expression(C) = (2*x^2 + y^2 = 2)", "coords": {} }, { "index": 2286, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2286.png", "answer": "2*sqrt(3)/3", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - 3*y^2 = 3)", "coords": {} }, { "index": 2287, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2287.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "F1: Point;F2: Point;G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;LeftFocus(G) = F1;RightFocus(G) = F2;H: Line;Expression(H) = (x = a^2/c);c: Number;P: Point;PointOnCurve(P, H);PointOnCurve(F2, PerpendicularBisector(LineSegmentOf(P, F1)))", "coords": {} }, { "index": 2288, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2288.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;P: Point;PointOnCurve(P, G);Coordinate(P) = (2, m);m: Number;F: Point;Focus(G) = F;Distance(P, F) = 4", "coords": {} }, { "index": 2289, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "y", "x_coef": 4.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2289.png", "answer": "9*x + 4*y - 13 = 0", "fact_expr": "l: Line;G: Ellipse;A: Point;B: Point;M: Point;Expression(G) = (x^2/4 + y^2/9 = 1);Coordinate(M) = (1, 1);PointOnCurve(M, l);Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A, B)) = M", "coords": { "M": [ 1.0, 1.0 ] } }, { "index": 2290, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2290.png", "answer": "sqrt(3)", "fact_expr": "C1:Parabola;C2:Hyperbola;a:Number;b:Number;p:Number;a>0;b>0;p>0;O:Origin;A:Point;Expression(C1)=(y^2=2*p*x);Expression(C2)=(x^2/a^2-y^2/b^2=1);Intersection(C1,OneOf(Asymptote(C2)))=A;Negation(A=O);Distance(A,Directrix(C1))=(3/2)*p", "coords": {} }, { "index": 2291, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2292, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2292.png", "answer": "sqrt(2)*2", "fact_expr": "G: Parabola;P: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(P, G);Distance(P,Directrix(G)) = 3", "coords": {} }, { "index": 2293, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2293.png", "answer": "3*sqrt(5)/5", "fact_expr": "G: Parabola;A: Point;B: Point;P0: Point;Expression(G) = (x^2 = y);Coordinate(A) = (0, -4);Coordinate(B)=(3, 2);PointOnCurve(P0, G)", "coords": { "A": [ 0.0, -4.0 ], "B": [ 3.0, 2.0 ] } }, { "index": 2294, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2294.png", "answer": "6", "fact_expr": "G: Parabola;M: Point;N: Point;P: Point;Expression(G) = (y = x^2/12);Coordinate(M) = (0, 3);Coordinate(N) = (4, 3);PointOnCurve(P, G)", "coords": { "M": [ 0.0, 3.0 ], "N": [ 4.0, 3.0 ] } }, { "index": 2295, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2295.png", "answer": "sqrt(5)/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;F2: Point;A: Point;B: Point;F1: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, H);Slope(H) = 1/3;Intersection(H, l1)=A;Abs(LineSegmentOf(F2, A)) = Abs(LineSegmentOf(F2, B));l1:Line;l2:Line;Intersection(H, l2)=B;Asymptote(G)={l1,l2}", "coords": {} }, { "index": 2296, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2296.png", "answer": "(1/2,0)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x)", "coords": {} }, { "index": 2297, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2297.png", "answer": "x^2/16+y^2/12=1", "fact_expr": "G: Parabola;H: Ellipse;m: Number;n: Number;Expression(G) = (y^2 = 8*x);m>0;n>0;Expression(H) = (y^2/n^2 + x^2/m^2 = 1);RightFocus(H) = Focus(G);Eccentricity(H)=(1/2)", "coords": {} }, { "index": 2298, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2298.png", "answer": "sqrt(3)/2", "fact_expr": "G: Ellipse;Expression(G) = (y^2 + x^2/a^2 = 1);a: Number;FocalLength(G) = 2*sqrt(3)", "coords": {} }, { "index": 2299, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 2.8284271247461903 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_2299.png", "answer": "x^2/8 + y^2/2 = 1", "fact_expr": "G: Circle;P: Point;H: Point;L: Line;M: Point;Expression(G) = (x^2 + y^2 = 8);PointOnCurve(P, G);FootPoint(L, xAxis) = H;IsPerpendicular(L, xAxis);PointOnCurve(P, L);MidPoint(LineSegmentOf(P, H)) = M", "coords": {} }, { "index": 2300, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7195594310760498, "b": 1.7224669456481934, "orientation": "horizontal" }, "optimization": { "final_loss": 2.391175712546101e-06, "converged": true, "iterations": 83 }, "output_path": "results/train/hyperbola/problem_2300.png", "answer": "{0}", "fact_expr": "Gamma: Hyperbola;H: Line;A: Point;B: Point;C: Point;m: Number;a: Number;b: Number;a > 0;b > 0;Expression(Gamma) = (-y^2/b^2 + x^2/a^2 = 1);RightVertex(Gamma) = A;IsParallel(H, xAxis);Intersection(H, Gamma) = {B, C};DotProduct(VectorOf(A, B), VectorOf(A, C)) = m;Eccentricity(Gamma) = sqrt(2)", "coords": {} }, { "index": 2301, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2301.png", "answer": "4", "fact_expr": "G: Ellipse;F1: Point;P: Point;O: Origin;M: Point;F2: Point;Expression(G) = (x^2/25 + y^2/16 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);MidPoint(LineSegmentOf(F1, P)) = M;Abs(LineSegmentOf(O, M)) = 3", "coords": {} }, { "index": 2302, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2302.png", "answer": "8", "fact_expr": "H: Line;Expression(H) = (y = k*(x - 1));G: Parabola;Expression(G) = (y^2 = 4*x);A: Point;B: Point;x1: Number;x2: Number;y1: Number;y2: Number;Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);Intersection(H, G) = {A, B};x1 + x2 = 6;k: Number", "coords": {} }, { "index": 2303, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2304, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2304.png", "answer": "1\n2", "fact_expr": "C1: Hyperbola;Expression(C1) = (x^2/a^2 - y^2/b^2 = 1);b: Number;a: Number;C2: Hyperbola;Expression(C2) = (x^2/4 - y^2/16 = 1);Asymptote(C1) = Asymptote(C2);F: Point;Coordinate(F) = (sqrt(5), 0);RightFocus(C1) = F", "coords": { "F": [ 2.23606797749979, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 2305, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2305.png", "answer": "10", "fact_expr": "G: Parabola;A: Point;B: Point;Expression(G) = (y^2 = 8*x);IsChordOf(LineSegmentOf(A, B), G);PointOnCurve(Focus(G), LineSegmentOf(A, B));XCoordinate(MidPoint(LineSegmentOf(A, B))) = 3", "coords": {} }, { "index": 2306, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2306.png", "answer": "20/3", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/9 + y^2/5 = 1);PointOnCurve(P, G);Focus(G)={F1,F2};AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 2307, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2308, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2308.png", "answer": "81*pi/16", "fact_expr": "G: Ellipse;H: Circle;L: Line;A: Point;B: Point;F: Point;Expression(G) = (x^2/16 + y^2/9 = 1);RightFocus(G) = F;PointOnCurve(F, L);IsPerpendicular(L,xAxis);Intersection(L, G) = {A, B};IsDiameter(LineSegmentOf(A,B),H)", "coords": {} }, { "index": 2309, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2309.png", "answer": "2*sqrt(3)/3", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;A: Point;F: Point;B: Point;l1: Line;l2: Line;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2 + y^2 = a^2);OneOf(Focus(C)) = F;TangentOfPoint(F, G) = {l1, l2};TangentPoint(l1,G)=A;TangentPoint(l2,G)=B;AngleOf(A, F, B) = ApplyUnit(120, degree)", "coords": {} }, { "index": 2310, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2310.png", "answer": "2*sqrt(3)/3", "fact_expr": "E: Parabola;p: Number;p>0;Expression(E) = (y^2 = 2*(p*x));F: Point;Focus(E) = F;M: Point;PointOnCurve(M, E);O: Origin", "coords": {} }, { "index": 2311, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2311.png", "answer": "-4", "fact_expr": "G: Parabola;p: Number;H: Circle;p<0;Expression(G) = (y^2 = p*x);Expression(H) = (y^2 + (x + 1)^2 = 1);Center(H)=Focus(G)", "coords": {} }, { "index": 2312, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2312.png", "answer": "pi/6", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;M: Point;Intersection(Directrix(G), xAxis) = M;N: Point;PointOnCurve(N, G);Abs(LineSegmentOf(N, F)) = (sqrt(3)/2)*Abs(LineSegmentOf(M, N))", "coords": {} }, { "index": 2313, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2314, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 3.0, "major_axis": "x", "x_coef": 36.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2314.png", "answer": "x + 2*y - 8 = 0", "fact_expr": "l: Line;G: Ellipse;A: Point;B: Point;P: Point;Expression(G) = (x^2/36 + y^2/9 = 1);Coordinate(P) = (4, 2);PointOnCurve(P, l);Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ 4.0, 2.0 ] } }, { "index": 2315, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2315.png", "answer": "3", "fact_expr": "G: Parabola;C: Circle;A: Point;B: Point;P: Point;Expression(C) = (y^2 + (x-2)^2 = 1);Expression(G) = (y^2 = 4*x);PointOnCurve(A, C);PointOnCurve(B, C);PointOnCurve(P, G);Abs(LineSegmentOf(A, B)) = 2", "coords": {} }, { "index": 2316, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2316.png", "answer": "sqrt(3)/2", "fact_expr": "l: Line;O: Origin;PointOnCurve(O, l);Negation(l = axis);C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;P: Point;Q: Point;Intersection(l, C) = {P, Q};B: Point;PointOnCurve(B, C);Negation(B = P);Negation(B = Q);Slope(LineOf(B, P)) * Slope(LineOf(B, Q)) = -1/4", "coords": {} }, { "index": 2317, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2318, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2319, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2319.png", "answer": "y=pm*sqrt(3)*x", "fact_expr": "G: Hyperbola;a: Number;H: Parabola;a>0;Expression(G) = (-y^2/3 + x^2/a^2 = 1);Expression(H) = (y^2 = 8*x);Focus(H) = RightFocus(G)", "coords": {} }, { "index": 2320, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2320.png", "answer": "8", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;x1:Number;y1:Number;x2:Number;y2:Number;Expression(G) = (y^2 = 4*x);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);Focus(G)=F;PointOnCurve(F,H);Intersection(H,G) = {B, A};x1+x2=6;F:Point", "coords": {} }, { "index": 2321, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2321.png", "answer": "sqrt(5)/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;m: Number;P: Point;A: Point;B: Point;a>0;b>0;Negation(m=0);l1:Line;l2:Line;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (m + x - 3*y = 0);Coordinate(P) = (m, 0);Asymptote(G)={l1,l2};Intersection(H,l1)=A;Intersection(H,l2)=B;Abs(LineSegmentOf(P, A)) = Abs(LineSegmentOf(P, B))", "coords": {} }, { "index": 2322, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2323, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2323.png", "answer": "5*x+9*y-14=0", "fact_expr": "G: Ellipse;H: LineSegment;I: Point;Expression(G) = (x^2/9 + y^2/5 = 1);Coordinate(I) = (1, 1);IsChordOf(H,G);MidPoint(H)=I", "coords": { "I": [ 1.0, 1.0 ] } }, { "index": 2324, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2325, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2325.png", "answer": "(x+2)^2+y^2=16", "fact_expr": "G: Parabola;Expression(G) = (y^2 = -8*x);H:Circle;Center(H)=Focus(G);IsTangent(Directrix(G),H)", "coords": {} }, { "index": 2326, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2327, "success": false, "error": "Circle from diameter: missing point coordinates" }, { "index": 2328, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2328.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;Coordinate(P) = (x0, y0);x0: Number;y0: Number;0 < x0^2/4 + y0^2/3;x0^2/4 + y0^2/3 < 1", "coords": {} }, { "index": 2329, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2329.png", "answer": "4*x^2-4*y^2=1", "fact_expr": "P: Point;PointOnCurve(P, G);G: Hyperbola;Expression(G) = (x^2 - y^2 = 1);O: Origin;M: Point;MidPoint(LineSegmentOf(P, O)) = M", "coords": {} }, { "index": 2330, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2330.png", "answer": "(1,\\sqrt{2}]", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;y0: Number;x0: Number;H: Line;P: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = ((x - x0)^2 + (y - y0)^2 = 2);Expression(H) = (2*a - a*y + b*x = 0);Coordinate(P) = (x0, y0);PointOnCurve(P, H);NumIntersection(G, RightPart(C))=0", "coords": {} }, { "index": 2331, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2332, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2332.png", "answer": "sqrt(5)/3", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C) = True;G: Circle;Expression(G) = (x^2 + y^2 = b^2);TangentPoint(LineSegmentOf(P, F2), G) = Q;Q: Point;MidPoint(LineSegmentOf(P, F2)) = Q;e: Number;Eccentricity(C) = e", "coords": {} }, { "index": 2333, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2334, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2334.png", "answer": "5*sqrt(2)/3", "fact_expr": "F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;G: Ellipse;Expression(G) = (x^2/2 + y^2 = 1);A: Point;LowerVertex(G) = A;B: Point;Intersection(OverlappingLine(LineSegmentOf(A,F2)),G)=B", "coords": {} }, { "index": 2335, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2336, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2336.png", "answer": "sqrt(7)/3", "fact_expr": "P: Point;G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;PointOnCurve(P, G) = True;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;AngleOf(F1, P, F2) = ApplyUnit(120, degree);Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 2337, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2337.png", "answer": "2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;H: Line;PointOnCurve(F, H);A: Point;B: Point;Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, F))/Abs(LineSegmentOf(B, F)) = 4;O: Origin", "coords": {} }, { "index": 2338, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2338.png", "answer": "9", "fact_expr": "G: Ellipse;H: Circle;C:Circle;P: Point;Q: Point;R: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Expression(H) = (y^2 + (x + 4)^2 = 1/4);Expression(C) = (y^2 + (x - 4)^2 = 1/4);PointOnCurve(P, G);PointOnCurve(Q, H);PointOnCurve(R, C)", "coords": {} }, { "index": 2339, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2339.png", "answer": "ApplyUnit(120,degree)", "fact_expr": "F1: Point;F2: Point;G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1))*Abs(LineSegmentOf(P, F2)) = 64/3", "coords": {} }, { "index": 2340, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2340.png", "answer": "sqrt(2)+1", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;M: Point;Coordinate(M) = (1, 2);PointOnCurve(M, RightPart(C)) = True;PointOnCurve(F2, G) = True;IsDiameter(LineSegmentOf(M, F1), G) = True;G: Circle", "coords": { "M": [ 1.0, 2.0 ] } }, { "index": 2341, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.0625, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2341.png", "answer": "(0, 1/16)", "fact_expr": "G: Parabola;Expression(G) = (y = 4*x^2)", "coords": {} }, { "index": 2342, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2342.png", "answer": "5/2", "fact_expr": "F: Point;RightFocus(C) = F;C: Ellipse;Expression(C) = (x^2/4 + y^2/3 = 1);A: Point;Coordinate(A) = (-1, 1);M: Point;PointOnCurve(M, C)", "coords": { "A": [ -1.0, 1.0 ] } }, { "index": 2343, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2343.png", "answer": "-1/4", "fact_expr": "l: Line;C: Parabola;O: Origin;A: Point;B: Point;Expression(C) = (y^2 = x);Intersection(C, l) = {A, B}", "coords": {} }, { "index": 2344, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2345, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2345.png", "answer": "5*pi", "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2/3 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;A: Point;B: Point;LeftVertex(C) = A;RightVertex(C) = B;H: Line;PointOnCurve(F1, H);P: Point;Intersection(H, RightPart(C)) = P;IsPerpendicular(LineSegmentOf(P, F2), LineSegmentOf(A, F2))", "coords": {} }, { "index": 2346, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2347, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2347.png", "answer": "6", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(C) = F;G: Line;PointOnCurve(F, G);A: Point;B: Point;Intersection(G, C) = {A, B};l: Line;Directrix(C) = l;L1: Line;L2: Line;PointOnCurve(A, L1);PointOnCurve(B, L2);IsPerpendicular(L1, l);IsPerpendicular(L2, l);A1: Point;B1: Point;FootPoint(L1, l) = A1;FootPoint(L2, l) = B1;Area(TriangleOf(A, A1, F)) = 9;Area(TriangleOf(B, B1, F)) = 1", "coords": {} }, { "index": 2348, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.826896789078654, "b": 2.826896789078654, "orientation": "horizontal" }, "optimization": { "final_loss": 7.081136889894256e-06, "converged": true, "iterations": 75 }, "output_path": "results/train/hyperbola/problem_2348.png", "answer": "x^2/4 - y^2/12 = 1", "fact_expr": "G: Hyperbola;F1: Point;F2: Point;P: Point;Coordinate(F1) = (-4, 0);Coordinate(F2) = (4, 0);PointOnCurve(P,G);Abs(Abs(LineSegmentOf(P,F1))-Abs(LineSegmentOf(P,F2)))=4;LeftFocus(G) = F1;RightFocus(G) = F2", "coords": { "F1": [ -4.0, 0.0 ], "F2": [ 4.0, 0.0 ] } }, { "index": 2349, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2350, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "y", "x_coef": 4.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2350.png", "answer": "2*sqrt(5)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/9 = 1)", "coords": {} }, { "index": 2351, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2351.png", "answer": "4/5", "fact_expr": "C: Ellipse;Expression(C) = (x^2/b^2 + y^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;PointOnCurve(P, C);Perimeter(TriangleOf(P, F1, F2)) = 18;Min(Abs(LineSegmentOf(P, F1))) = 1", "coords": {} }, { "index": 2352, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2352.png", "answer": "sqrt(10)/2", "fact_expr": "H: Line;Expression(H) = (y = 1 - x);G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;A: Point;B: Point;Intersection(H, G) = {A, B};O: Origin;IsPerpendicular(LineSegmentOf(O, A), LineSegmentOf(O, B));e: Number;Eccentricity(G) = e;In(e, [1/2, sqrt(3)/2])", "coords": {} }, { "index": 2353, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2353.png", "answer": "(4,\\pm 4)", "fact_expr": "G: Parabola;P: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(P, G);Distance(P, Focus(G))=5", "coords": {} }, { "index": 2354, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2354.png", "answer": "2", "fact_expr": "G: Hyperbola;a: Number;b: Number;H: Circle;M: Point;N: Point;Z: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(LeftFocus(G),Z) ;IsPerpendicular(Z, xAxis);Intersection(Z, G) = {M, N};IsDiameter(LineSegmentOf(M, N), H);PointOnCurve(RightVertex(G), H)", "coords": {} }, { "index": 2355, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2356, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.8284271247461903, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2356.png", "answer": "8*sqrt(3)", "fact_expr": "F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;G: Hyperbola;Expression(G) = (x^2/4 - y^2/8 = 1);P: Point;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 2357, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_2357.png", "answer": "y=pm*sqrt(3)*x", "fact_expr": "G: Hyperbola;a: Number;a>0;Expression(G) = (-y^2/3 + x^2/a^2 = 1);Eccentricity(G) = 2*a", "coords": {} }, { "index": 2358, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.125, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2358.png", "answer": "15/8", "fact_expr": "G: Parabola;Expression(G) = (x^2 = y/2);M: Point;PointOnCurve(M, G);Distance(M, Focus(G)) = 2", "coords": {} }, { "index": 2359, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2360, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2360.png", "answer": "(0,4-2*sqrt(3)]+[4+2*sqrt(3),+oo)", "fact_expr": "C: Ellipse;m: Number;A: Point;M: Point;B: Point;m>0;Expression(C) = (x^2/4 + y^2/m^2 = 1);Endpoint(MajorAxis(C)) = {A, B};PointOnCurve(M,C);AngleOf(A,M,B)=ApplyUnit(150,degree)", "coords": {} }, { "index": 2361, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2361.png", "answer": "y^2=4*x", "fact_expr": "C: Parabola;G: Line;A: Point;B: Point;P: Point;O:Origin;Expression(G) = (x - y = 0);Vertex(C)=O;Coordinate(P) = (2, 2);PointOnCurve(Focus(C), xAxis);Intersection(G, C) = {A, B};MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ 2.0, 2.0 ] } }, { "index": 2362, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2362.png", "answer": "3*x+4*y-5=0", "fact_expr": "G: Hyperbola;L: LineSegment;M: Point;Expression(G) = (x^2/4 - y^2 = 1);Coordinate(M) = (3, -1);IsChordOf(L,G);MidPoint(L)=M;PointOnCurve(M,L)", "coords": { "M": [ 3.0, -1.0 ] } }, { "index": 2363, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 3.0, "major_axis": "x", "x_coef": 36.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2363.png", "answer": "(x^2/4 + y^2 = 1)&Negation(x = pm*2)", "fact_expr": "G: Ellipse;P: Point;F2: Point;Q: Point;F1: Point;Expression(G) = (x^2/36 + y^2/9 = 1);PointOnCurve(Q, G);Negation(Q = Endpoint(MajorAxis(G)));Focus(G) = {F1, F2};VectorOf(P, F1) + VectorOf(P, F2) + VectorOf(P, Q) = 0", "coords": {} }, { "index": 2364, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2364.png", "answer": "4*sqrt(22)/11", "fact_expr": "P: Point;G: Ellipse;Expression(G) = (x^2/16 + y^2/4 = 1);PointOnCurve(P, G) ;H: Circle;Expression(H) = (y^2 + (x - 1)^2 = 1);l1: Line;l2: Line;TangentOfPoint(P, H) = {l1, l2};TangentPoint(l1, H) = A;TangentPoint(l2, H) = B;A: Point;B: Point", "coords": {} }, { "index": 2365, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2366, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2366.png", "answer": "x \\pm \\left(\\sqrt{3} \\cdot y\\right) = 0", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2 = 1)", "coords": {} }, { "index": 2367, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2367.png", "answer": "4/5", "fact_expr": "G: Ellipse;e: Number;Expression(G) = (x^2/25 + y^2/9 = 1);Eccentricity(G) = e", "coords": {} }, { "index": 2368, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2368.png", "answer": "8*x-y-31=0", "fact_expr": "l: Line;M: Point;Coordinate(M) = (4, 1);PointOnCurve(M, l);G: Hyperbola;Expression(G) = (x^2 - y^2/2 = 1);A: Point;B: Point;Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A, B)) = M", "coords": { "M": [ 4.0, 1.0 ] } }, { "index": 2369, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2370, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2370.png", "answer": "1", "fact_expr": "G: Ellipse;M: Point;N: Point;m: Number;0 < m;m < 2;x1: Number;y1: Number;Expression(G) = (x^2/4 + y^2/2 = 1);Coordinate(M) = (x1, y1);Coordinate(N) = (m, 0);PointOnCurve(M, G);Distance(M, N) = Abs(LineSegmentOf(M, N));Min(Abs(LineSegmentOf(M, N))) = 1", "coords": {} }, { "index": 2371, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2371.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (y = -x^2/4);M: Point;PointOnCurve(M,G) = True;P1: Point;Coordinate(P1) = (0, -1);P2: Point;Coordinate(P2) = (1, -3)", "coords": { "P1": [ 0.0, -1.0 ], "P2": [ 1.0, -3.0 ] } }, { "index": 2372, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2372.png", "answer": "9", "fact_expr": "G: Hyperbola;a: Number;P: Point;F1: Point;F2: Point;a>0;Expression(G) = (-y^2/9 + x^2/a^2 = 1);PointOnCurve(P, G);Expression(OneOf(Asymptote(G))) = (3*x - 2*y = 0);LeftFocus(G) = F1;RightFocus(G) = F2;Abs(LineSegmentOf(P, F1)) = 5", "coords": {} }, { "index": 2373, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2373.png", "answer": "x^2+(y-1)^2=10", "fact_expr": "G: Parabola;C:Circle;H: Line;A: Point;B: Point;Expression(G) = (x^2 = 4*y);Expression(H) = (4*x - 3*y - 2 = 0);Center(C)=Focus(G);Intersection(H, C) = {A, B};Abs(LineSegmentOf(A, B)) = 6", "coords": {} }, { "index": 2374, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_2374.png", "answer": "y=pm*sqrt(3)*x/3", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Eccentricity(G) = 2*sqrt(3)/3", "coords": {} }, { "index": 2375, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2375.png", "answer": "13/5", "fact_expr": "C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(OneOf(Asymptote(C))) = (y = (12/5)*x)", "coords": {} }, { "index": 2376, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_2376.png", "answer": "x^2+y^2+3*x-4*y+6=0", "fact_expr": "P: Circle;B: Point;A: Point;Expression(P) = (x^2 + y^2 = 1);Coordinate(B) = (-3, 4);PointOnCurve(A, P);Q:Point;MidPoint(Distance(A,B))=Q", "coords": {} }, { "index": 2377, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2378, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2378.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;F: Point;A: Point;P: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (-2, 0);Coordinate(A) = (0, sqrt(5));LeftFocus(G) = F;PointOnCurve(P, RightPart(G));Min(Perimeter(TriangleOf(A, P, F))) = 8", "coords": { "F": [ -2.0, 0.0 ], "A": [ 0.0, 2.23606797749979 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 2379, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2379.png", "answer": "sqrt(3)/2", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>b;b>0;e: Number;Eccentricity(G) = e;WhenMin(a^2 + 16/(b*(a - b)))", "coords": {} }, { "index": 2380, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2381, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2381.png", "answer": "4", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1)", "coords": {} }, { "index": 2382, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2382.png", "answer": "sqrt(3)/2", "fact_expr": "G: Hyperbola;P: Point;Expression(G) = (x^2/4 - y^2/12 = 1);Intersection(LeftDirectrix(G), xAxis) = P", "coords": {} }, { "index": 2383, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2384, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2384.png", "answer": "8/3", "fact_expr": "PointOnCurve(Focus(G), yAxis);G: Ellipse;Expression(G) = (x^2/2 + y^2/m = 1);m: Number;e: Number;Eccentricity(G) = e;e = 1/2", "coords": {} }, { "index": 2385, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2385.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Parabola;H: Line;P: Point;Expression(G) = (y = x^2);Expression(H) = (x - y + 2 = 0);PointOnCurve(P, G);WhenMin(Distance(P,H))", "coords": {} }, { "index": 2386, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2386.png", "answer": "3", "fact_expr": "G: Parabola;M: Point;F: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(M, G);Focus(G) = F;Distance(M, F) = Abs(LineSegmentOf(M, F));Abs(LineSegmentOf(M, F)) = 4;x_: Number;XCoordinate(M) = x_", "coords": {} }, { "index": 2387, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.23606797749979, "b": 2.0, "major_axis": "x", "x_coef": 5.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2387.png", "answer": "5*sqrt(5)/3", "fact_expr": "G: Ellipse;H: Line;A: Point;B: Point;O: Origin;Expression(G) = (x^2/5 + y^2/4 = 1);PointOnCurve(RightFocus(G), H);Intersection(H,G) = {A,B};Slope(H)=2;IsChordOf(LineSegmentOf(A,B),G)", "coords": {} }, { "index": 2388, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2388.png", "answer": "x^2+y^2=1", "fact_expr": "G: Ellipse;Expression(G) = (x^2/2 + y^2 = 1);F1: Point;F2: Point;B2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;UpperVertex(G) = B2;H: Circle;PointOnCurve(F1,H) = True;PointOnCurve(F2,H) = True;PointOnCurve(B2,H) = True", "coords": {} }, { "index": 2389, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "y", "x_coef": 9.0, "y_coef": 25.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2389.png", "answer": "{(3,0),(-3,0)}", "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/9 + y^2/25 = 1);PointOnCurve(P, G);F1:Point;F2:Point;Focus(G)={F1,F2};Distance(P,F1)*Distance(P,F2) = m;WhenMax(m);m:Number", "coords": {} }, { "index": 2390, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2390.png", "answer": "4/3", "fact_expr": "G: Parabola;P: Point;A: Point;F: Point;O: Origin;l: Line;Expression(G) = (x^2 = 4*y);Focus(G) = F;Directrix(G) = l;PointOnCurve(P, G);PointOnCurve(P, LineSegmentOf(P,A));IsPerpendicular(LineSegmentOf(P, A),l);FootPoint(LineSegmentOf(P, A),l)=A;AngleOf(A, F, O) = ApplyUnit(30, degree)", "coords": {} }, { "index": 2391, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2392, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2392.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;F: Point;Focus(G) = F;l: Line;Directrix(G) = l;L: Line;PointOnCurve(Focus(G), L);Intersection(L, G) = {A, B};A: Point;B: Point;L1: Line;L2: Line;PointOnCurve(A, L1);PointOnCurve(B, L2);IsPerpendicular(L1, l) ;IsPerpendicular(L2, l);FootPoint(L1, l) = C;FootPoint(L2, l) = D;C: Point;D: Point;Abs(LineSegmentOf(A, F)) = 3*Abs(LineSegmentOf(B, F));Area(TriangleOf(C, D, F)) = sqrt(3)", "coords": {} }, { "index": 2393, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 3.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2393.png", "answer": "6*sqrt(13)/13", "fact_expr": "P: Point;Coordinate(P) = (1, 1);H: Line;PointOnCurve(P, H);G: Ellipse;Expression(G) = (x^2/3 + y^2/2 = 1);A: Point;B: Point;Intersection(H, G) = {A, B};lambda: Number;VectorOf(A, P) = lambda*VectorOf(P, B);Q: Point;VectorOf(A, Q) = -lambda*VectorOf(Q, B);O: Origin", "coords": { "P": [ 1.0, 1.0 ] } }, { "index": 2394, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2394.png", "answer": "4/5", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1)", "coords": {} }, { "index": 2395, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.3300262689590454, "b": 2.30381441116333, "orientation": "horizontal" }, "optimization": { "final_loss": 6.658357051492203e-06, "converged": true, "iterations": 122 }, "output_path": "results/train/hyperbola/problem_2395.png", "answer": "x^2/4 - y^2/12 = 1", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x);H: Parabola;Expression(H) = (y^2 = 16*x);OneOf(Focus(G)) = Focus(H)", "coords": {} }, { "index": 2396, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2396.png", "answer": "[3/5, \\sqrt{2}/2)", "fact_expr": "G: Ellipse;b: Number;a: Number;c: Number;H: Circle;F2: Point;P: Point;T: Point;F1: Point;l: Line;a > b;b > c;c > 0;a^2=b^2+c^2;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;Center(H) = F2;Radius(H) = b - c;PointOnCurve(P, G);TangentOfPoint(P, H) = l;TangentPoint(l, H) = T;Min(Abs(LineSegmentOf(P, T))) = (sqrt(3)/2)*(a - c)", "coords": {} }, { "index": 2397, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.3166247903554, "b": 3.1622776601683795, "major_axis": "y", "x_coef": 10.0, "y_coef": 11.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2397.png", "answer": "(pm*1,0)\n2*sqrt(11)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/10 + y^2/11 = 1)", "coords": {} }, { "index": 2398, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2398.png", "answer": "4", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/4 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 2399, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2399.png", "answer": "3", "fact_expr": "G: Ellipse;O: Origin;M: Point;P: Point;F: Point;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(P, G);LeftFocus(G) = F;Distance(P, F) = 4;VectorOf(O, M) = (VectorOf(O, F) + VectorOf(O, P))/2", "coords": {} }, { "index": 2400, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2400.png", "answer": "x^2=12*y", "fact_expr": "G: Parabola;P: Point;O: Origin;Coordinate(P) = (m, 1);Vertex(G) = O;PointOnCurve(Focus(G), yAxis);PointOnCurve(P, G);Distance(P, Focus(G)) = 4;m:Number", "coords": {} }, { "index": 2401, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2401.png", "answer": "2", "fact_expr": "C: Parabola;p: Number;G: Line;M: Point;A: Point;B: Point;l: Line;p>0;Expression(C) = (y^2 = 2*p*x);Coordinate(M) = (1, 0);PointOnCurve(M,G);Slope(G)=sqrt(3);Intersection(G,l)=A;OneOf(Intersection(G,C))=B;VectorOf(A, M) = VectorOf(M, B);Directrix(C)=l", "coords": { "M": [ 1.0, 0.0 ] } }, { "index": 2402, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2402.png", "answer": "0", "fact_expr": "P: Point;PointOnCurve(P, C);C: Ellipse;Expression(C) = (x^2/4 + y^2/3 = 1);A: Point;LeftVertex(C) = A;F: Point;RightFocus(C) = F", "coords": {} }, { "index": 2403, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2403.png", "answer": "2", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);A: Point;B: Point;PointOnCurve(A, C);PointOnCurve(B, C);Abs(LineSegmentOf(A, B)) = 6;M: Point;MidPoint(LineSegmentOf(A, B)) = M", "coords": {} }, { "index": 2404, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2404.png", "answer": "pm*4*sqrt(2)", "fact_expr": "G: Parabola;p: Number;A: Point;Expression(G) = (y^2 = 2*p*x);Coordinate(A) = (4, m);PointOnCurve(A, G);Distance(A,Directrix(G)) = 6;m:Number", "coords": {} }, { "index": 2405, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2405.png", "answer": "20", "fact_expr": "G: Ellipse;A: Point;B: Point;F2: Point;F1: Point;Expression(G)=(x^2/a^2+y^2/9=1);a:Number;Focus(G)={F1,F2};Abs(LineSegmentOf(F1, F2)) = 8;IsChordOf(LineSegmentOf(A,B),G);PointOnCurve(F1, LineSegmentOf(A, B))", "coords": {} }, { "index": 2406, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2407, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2408, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2409, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2409.png", "answer": "2", "fact_expr": "C: Parabola;p: Number;F: Point;P: Point;O: Origin;Q: Point;A: Point;G:Circle;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;PointOnCurve(P, C);Center(G)=F;Radius(G)=p;Intersection(LineSegmentOf(P,F),G)=Q;TangentOfPoint(P,G)=L;L:Line;TangentPoint(L,G)=A;Abs(LineSegmentOf(P, A)) = sqrt(3)*p;Area(TriangleOf(O, P, Q)) = sqrt(3)/2", "coords": {} }, { "index": 2410, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2410.png", "answer": "4", "fact_expr": "G: Parabola;H: Circle;L: Line;Q: Point;F: Point;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;Intersection(Directrix(G), xAxis) = Q;PointOnCurve(F,L);Intersection(L, G) = {A, B};IsDiameter(LineSegmentOf(Q, F),H);PointOnCurve(B,H)", "coords": {} }, { "index": 2411, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2412, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2412.png", "answer": "4", "fact_expr": "G: Parabola;H: Circle;F: Point;Q: Point;P: Point;l: Line;Expression(G) = (y^2 = 4*x);Focus(G) = F;Directrix(G) = l;PointOnCurve(P,G);IsPerpendicular(LineSegmentOf(P,Q),l);Intersection(LineSegmentOf(P,Q),l)=Q;IsDiameter(LineSegmentOf(F,Q),H);Length(InterceptChord(l,H))=2*sqrt(3)", "coords": {} }, { "index": 2413, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2413.png", "answer": "-1", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);H: Line;F: Point;Focus(G) = F;PointOnCurve(F, H);k: Number;Slope(H) = k;A: Point;B: Point;Intersection(H, G) = {A, B};A1: Point;B1: Point;Z: Line;Expression(Z) = (x = -1);L1: Line;L2: Line;PointOnCurve(A, L1);PointOnCurve(B, L2);IsPerpendicular(L1, Z);IsPerpendicular(L2, Z);FootPoint(L1, Z) = A1;FootPoint(L2, Z) = B1", "coords": {} }, { "index": 2414, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2415, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2415.png", "answer": "5", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;M: Point;PointOnCurve(M, C);N: Point;Intersection(OverlappingLine(LineSegmentOf(F, M)), yAxis) = N;LineSegmentOf(F, M) = (1/2)*LineSegmentOf(M, N)", "coords": {} }, { "index": 2416, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2416.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/4 = 1)", "coords": {} }, { "index": 2417, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2418, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2418.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;F: Point;M: Point;N: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;PointOnCurve(F,H);IsPerpendicular(H,OneOf(Asymptote(G)));FootPoint(H,OneOf(Asymptote(G)))=M;Quadrant(M)=1;Intersection(LineSegmentOf(M,F),G) = N;VectorOf(M,N)=VectorOf(N,F)/2", "coords": {} }, { "index": 2419, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2419.png", "answer": "y=pm*x", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;FocalLength(G) = 2*c;c: Number;A: Point;RightVertex(G) = A;H: Parabola;Expression(H) = (x^2 = 2*p*y);p: Number;p>0;F: Point;Focus(H) = F;Length(InterceptChord(Directrix(H), G)) = 2*c;Abs(LineSegmentOf(P, A)) = c;P: Point", "coords": {} }, { "index": 2420, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2420.png", "answer": "((x-2*m)^2+y^2=m^2)&Negation(m=0)", "fact_expr": "G: Hyperbola;H: Circle;Expression(G) = (x^2/3 - y^2 = 1);PointOnCurve(Center(H),xAxis);IsTangent(Asymptote(G),H)", "coords": {} }, { "index": 2421, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2422, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2422.png", "answer": "Negation(k=pm*(2*sqrt(3)/3))", "fact_expr": "G: Hyperbola;H: Line;k: Real;Expression(G) = (x^2/3 - y^2/4 = 1);Expression(H) = (y = k*x - 2*k);NumIntersection(H, G) = 2", "coords": {} }, { "index": 2423, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2423.png", "answer": "(2*sqrt(5))/5", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;A: Point;RightVertex(C) = A;P: Point;PointOnCurve(P, C);O: Origin;AngleOf(P, O, A) = ApplyUnit(60, degree);IsPerpendicular(LineSegmentOf(O, P), LineSegmentOf(A, P))", "coords": {} }, { "index": 2424, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2424.png", "answer": "3", "fact_expr": "P: Point;G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);PointOnCurve(P, G);l: Line;PointOnCurve(P, l);H: Circle;Expression(H) = (x^2 + y^2 = 4);A: Point;B: Point;Intersection(l, H) = {A, B}", "coords": {} }, { "index": 2425, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 2.8706156235768385, "b": 0.9134866531697662, "major_axis": "x", "x_coef": 9.0, "y_coef": 1.0 }, "optimization": { "final_loss": 8.509210183089842e-07, "converged": true, "iterations": 93 }, "output_path": "results/train/ellipse/problem_2425.png", "answer": "{x^2/9+y^2=1,y^2/81+x^2/9=1}", "fact_expr": "G: Ellipse;P: Point;Coordinate(P) = (3, 0);Length(MajorAxis(G))=3*Length(MinorAxis(G));PointOnCurve(P,G)", "coords": { "P": [ 3.0, 0.0 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 2426, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2426.png", "answer": "sqrt(5)/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Distance(RightFocus(G),OneOf(Asymptote(G)))=a/2", "coords": {} }, { "index": 2427, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 2.6457513110645907, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2427.png", "answer": "3", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/7 = 1);P: Point;PointOnCurve(P, G);Distance(P, RightFocus(G)) = 1;O: Origin", "coords": {} }, { "index": 2428, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2428.png", "answer": "2*sqrt(3)/3", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F2: Point;F1: Point;a>0;b>0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;L:Line;PointOnCurve(F1, L);IsPerpendicular(L, OneOf(Asymptote(G)));FootPoint(L, OneOf(Asymptote(G)))=P;Abs(LineSegmentOf(P, F2)) = a*(sqrt(39)/3)", "coords": {} }, { "index": 2429, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2429.png", "answer": "1", "fact_expr": "C: Parabola;G: Line;M: Point;A: Point;B: Point;Expression(C) = (y^2 = 2*x);Coordinate(M) = (-1/2, 1);PointOnCurve(Focus(C),G);Slope(G)=k;k:Number;Intersection(G,C)={A,B};AngleOf(A,M,B)=ApplyUnit(90,degree)", "coords": { "M": [ -0.5, 1.0 ] } }, { "index": 2430, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2431, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.9560877084732056, "b": 1.622148733645621, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 2.7588219936233933e-05, "converged": true, "iterations": 62 }, "output_path": "results/train/ellipse/problem_2431.png", "answer": "[1/2, sqrt(3)/2]", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;H: Point;Coordinate(H) = (1, sqrt(2));PointOnCurve(H, G) = True;Range(Length(MinorAxis(G))) = [3, sqrt(11)]", "coords": { "H": [ 1.0, 1.4142135623730951 ] } }, { "index": 2432, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2432.png", "answer": "(\\sqrt{2}, 3]", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;b > a;a > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, LeftPart(G));Max(2*Abs(LineSegmentOf(P, F1))/Abs(LineSegmentOf(P, F2))^2) = 1/(4*a)", "coords": {} }, { "index": 2433, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2433.png", "answer": "sqrt(5)", "fact_expr": "C: Hyperbola;a: Number;b: Number;P: Point;F2: Point;O: Origin;F1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Focus(C) = {F1, F2};PointOnCurve(P, C);Abs(LineSegmentOf(O, P)) = Abs(LineSegmentOf(O, F1));IsParallel(LineSegmentOf(P, F2), OneOf(Asymptote(C)))", "coords": {} }, { "index": 2434, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2434.png", "answer": "(\\sqrt{2},2)", "fact_expr": "C: Hyperbola;F: Point;R: Point;M: Point;l1:Line;l2:Line;l:Line;lambda:Number;a:Number;b:Number;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a>0;b>0;Asymptote(C) = {l1,l2};RightFocus(C) = F;PointOnCurve(F, l);IsParallel(l,l2) = True;Intersection(l, C) = R;Intersection(l,l1) = M;VectorOf(F, R) = lambda*VectorOf(F, M);In(lambda,(1/2,3/4))", "coords": {} }, { "index": 2435, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2435.png", "answer": "[16,26]", "fact_expr": "G: Ellipse;H: Circle;A: Point;B: Point;F1: Point;F2: Point;P: Point;Expression(G) = (x^2/9 + y^2/4 = 1);Focus(G)={F1,F2};PointOnCurve(P,G);Center(H)=P;Radius(H)=1;IsDiameter(LineSegmentOf(A,B),H)", "coords": {} }, { "index": 2436, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2436.png", "answer": "sqrt(2)/2", "fact_expr": "G: Ellipse;Expression(G) = (x^2/2 + y^2 = 1);P: Point;Coordinate(P) = (2, 0);l: Line;PointOnCurve(P, l);k: Number;Slope(l) = k;A: Point;B: Point;Intersection(l, G) = {A, B};Negation(A=B);O: Origin;S: Number;Area(TriangleOf(A, O, B)) = S", "coords": { "P": [ 2.0, 0.0 ] } }, { "index": 2437, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2437.png", "answer": "3", "fact_expr": "G: Parabola;M: Point;P: Point;F: Point;Expression(G) = (x^2 = 4*y);Coordinate(M) = (1, 2);Focus(G) = F;PointOnCurve(P, G)", "coords": { "M": [ 1.0, 2.0 ] } }, { "index": 2438, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2438.png", "answer": "16", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);A: Point;B: Point;PointOnCurve(A, C);PointOnCurve(B, C);O: Origin;Negation(O=A);Negation(O=B);IsPerpendicular(LineSegmentOf(O, A), LineSegmentOf(O, B))", "coords": {} }, { "index": 2439, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2439.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;l: Line;Directrix(G) = l;P: Point;PointOnCurve(P, G);A: Point;IsPerpendicular(LineSegmentOf(P, A), l);FootPoint(LineSegmentOf(P, A), l) = A;Inclination(LineOf(A, F)) = ApplyUnit(120, degree)", "coords": {} }, { "index": 2440, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2441, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 4.898979485566356, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2441.png", "answer": "pi/2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/24 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};H: Ellipse;Expression(H) = (x^2/49 + y^2/24 = 1);P: Point;OneOf(Intersection(G, H)) = P", "coords": {} }, { "index": 2442, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2442.png", "answer": "6", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);P: Point;PointOnCurve(P, G) = True;Distance(P, yAxis) = 4", "coords": {} }, { "index": 2443, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2444, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2445, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2445.png", "answer": "2*sqrt(3)/3", "fact_expr": "C: Hyperbola;b: Number;a: Number;H: Circle;F2: Point;O: Origin;F1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;Center(H)=F2;IsTangent(Asymptote(C),H);PointOnCurve(MidPoint(LineSegmentOf(O,F2)),H)", "coords": {} }, { "index": 2446, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2446.png", "answer": "3", "fact_expr": "G: Hyperbola;m: Number;H: Parabola;Expression(G) = (-y^2 + x^2/m = 1);Expression(H) = (y^2 = 8*x);RightFocus(G) =Focus(H)", "coords": {} }, { "index": 2447, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2447.png", "answer": "5", "fact_expr": "E: Parabola;Expression(E) = (y^2 = 4*x);P: Point;PointOnCurve(P, E);F: Point;Focus(E) = F;B: Point;Coordinate(B) = (4, 2)", "coords": { "B": [ 4.0, 2.0 ] } }, { "index": 2448, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2448.png", "answer": "2", "fact_expr": "l: Line;C: Parabola;p: Number;A: Point;F: Point;M: Point;B: Point;N: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C)=F;PointOnCurve(F,l);Intersection(l,C) = {B, A};Projection(A,Directrix(C))=M;Projection(B,Directrix(C))=N;InterReciprocal(Area(TriangleOf(A,F,M)),Area(TriangleOf(B,F,N)))", "coords": {} }, { "index": 2449, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2449.png", "answer": "sqrt(6)/3", "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;F1: Point;F2: Point;B: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;OneOf(Vertex(MajorAxis(G)))=A;OneOf(Vertex(MinorAxis(G)))=B;Eccentricity(G)=e;e:Number;DotProduct(VectorOf(A, F1), VectorOf(A, F2)) + DotProduct(VectorOf(B, F1), VectorOf(B, F2)) = 0", "coords": {} }, { "index": 2450, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "y", "x_coef": 9.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2450.png", "answer": "sqrt(7)/4", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/16 = 1)", "coords": {} }, { "index": 2451, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2451.png", "answer": "(0, 4)", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 16*y)", "coords": {} }, { "index": 2452, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2452.png", "answer": "3/5", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2 - 4*y^2 = 4);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P,RightPart(G));Abs(LineSegmentOf(P, F1)) + Abs(LineSegmentOf(P, F2)) = 6", "coords": {} }, { "index": 2453, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2454, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2455, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2455.png", "answer": "{y^2=8*x, x^2=-12*y}", "fact_expr": "G: Parabola;O: Origin;Vertex(G) = O;SymmetryAxis(G) = axis;C: Curve;Expression(C) = (3*x - 2*y - 6 = 0);PointOnCurve(Focus(G), C)", "coords": {} }, { "index": 2456, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2456.png", "answer": "sqrt(10)/10", "fact_expr": "C: Ellipse;b: Number;a: Number;G: Circle;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Expression(G) = ((x + 1)^2 + (y - 1)^2 = 5);F: Point;B: Point;RightFocus(C) = F;UpperVertex(C) = B;PointOnCurve(F, G);PointOnCurve(B, G)", "coords": {} }, { "index": 2457, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2457.png", "answer": "4", "fact_expr": "G: Parabola;A: Point;M: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (3, 2);PointOnCurve(M, G);Focus(G) = F", "coords": { "A": [ 3.0, 2.0 ] } }, { "index": 2458, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2458.png", "answer": "1", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;x1:Number;y1:Number;x2:Number;y2:Number;Expression(G) = (y^2 = 4*x);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);Focus(G)=F;PointOnCurve(F,H);Intersection(H,G) = {B, A};F:Point", "coords": {} }, { "index": 2459, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2459.png", "answer": "14", "fact_expr": "G: Parabola;A: Point;B: Point;x1: Number;y1: Number;x2: Number;y2: Number;Expression(G) = (y^2 = 8*x);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);x1 + x2 = 10;PointOnCurve(Focus(G), LineSegmentOf(A, B));IsChordOf(LineSegmentOf(A, B), G)", "coords": {} }, { "index": 2460, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2460.png", "answer": "{2/3,-2/3}", "fact_expr": "G: Hyperbola;H: Curve;a: Number;Expression(G) = (x^2 - y^2 = 1);Expression(H) = (y = a + x^3/3);IsTangent(OneOf(Asymptote(G)), H)", "coords": {} }, { "index": 2461, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 10.0, "b": 8.0, "major_axis": "x", "x_coef": 100.0, "y_coef": 64.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2461.png", "answer": "12", "fact_expr": "C: Ellipse;Expression(C) = (x^2/100 + y^2/64 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) = 8", "coords": {} }, { "index": 2462, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 25.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2462.png", "answer": "5", "fact_expr": "P: Point;PointOnCurve(P, G);G: Ellipse;Expression(G) = (x^2/25 + y^2/5 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0", "coords": {} }, { "index": 2463, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2463.png", "answer": "14", "fact_expr": "G: Ellipse;A: Point;P: Point;F: Point;Expression(G) = (x^2/9 + y^2/5 = 1);Coordinate(A) = (0, 2*sqrt(3));PointOnCurve(P, G);RightFocus(G)=F", "coords": { "A": [ 0.0, 3.4641016151377544 ] } }, { "index": 2464, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2464.png", "answer": "{4,16}", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);P: Point;PointOnCurve(P, G) = True;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;Abs(LineSegmentOf(P, F1)) = 10", "coords": {} }, { "index": 2465, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0010501146316528, "direction": "right" }, "optimization": { "final_loss": 3.455598402979376e-07, "converged": true, "iterations": 81 }, "output_path": "results/train/parabola/problem_2465.png", "answer": "(1,0)", "fact_expr": "H: Point;Coordinate(H) = (1, 2);G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;PointOnCurve(H, G)", "coords": { "H": [ 1.0, 2.0 ] } }, { "index": 2466, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2467, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2468, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_2468.png", "answer": "12", "fact_expr": "G: Hyperbola;m: Number;n: Number;m>0;n>0;H:Parabola;Expression(G) = (-y^2/n + x^2/m = 1);Eccentricity(G) = 2;Expression(H) =(y^2 = 16*x);OneOf(Focus(G))=Focus(H)", "coords": {} }, { "index": 2469, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2469.png", "answer": "5/4", "fact_expr": "G: Hyperbola;e: Number;Expression(G) = (9*x^2 - 16*y^2 = 144);Eccentricity(G) = e", "coords": {} }, { "index": 2470, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2470.png", "answer": "3/4", "fact_expr": "C: Parabola;M: Circle;P: Point;A: Point;L: Line;Expression(C) = (y = x^2);Expression(M) = (x^2 + (y-2)^2 = 1);PointOnCurve(P, C);TangentOfPoint(P, M) = L;TangentPoint(L,M) = A", "coords": {} }, { "index": 2471, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2472, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2472.png", "answer": "(1,(2+sqrt(7))/3)", "fact_expr": "M: Hyperbola;b: Number;a: Number;F1: Point;F2: Point;P: Point;a>0;b>0;Expression(M) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(M)=F1;RightFocus(M)=F2;Abs(LineSegmentOf(F1, F2)) = 2*c;PointOnCurve(P,RightPart(M));a/Sin(AngleOf(P, F1, F2)) = (3*c)/Sin(AngleOf(P, F2, F1));c:Number", "coords": {} }, { "index": 2473, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.3300262689590454, "b": 2.30381441116333, "orientation": "horizontal" }, "optimization": { "final_loss": 6.658357051492203e-06, "converged": true, "iterations": 122 }, "output_path": "results/train/hyperbola/problem_2473.png", "answer": "(0, 1/2]", "fact_expr": "D: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(D) = (-y^2/b^2 + x^2/a^2 = 1);Expression(Asymptote(D)) = (y = pm*sqrt(3)*x);LeftFocus(D) = F1;RightFocus(D) = F2;PointOnCurve(P, RightPart(D))", "coords": {} }, { "index": 2474, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2475, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2476, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2476.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;H: Parabola;Expression(H) = (y = x^2 + 1);NumIntersection(OneOf(Asymptote(G)), H) = 1", "coords": {} }, { "index": 2477, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2478, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2478.png", "answer": "13", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 - y^2/16 = 1);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2;Abs(LineSegmentOf(P, F1)) = 7", "coords": {} }, { "index": 2479, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2480, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2480.png", "answer": "(4*sqrt(2),4*sqrt(6))", "fact_expr": "l: Line;G: Parabola;O: Point;A: Point;P: Point;Q: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (2, 0);Vertex(G)=O;Inclination(l)=pi/4;IsIntersect(l,LineSegmentOf(O,A));Negation(PointOnCurve(O,l));Negation(PointOnCurve(A,l));Intersection(l,G)={P,Q}", "coords": { "A": [ 2.0, 0.0 ] } }, { "index": 2481, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.8284271247461903, "major_axis": "x", "x_coef": 9.0, "y_coef": 8.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2481.png", "answer": "8 + 2*sqrt(2)", "fact_expr": "G: Ellipse;P: Point;M: Point;F: Point;Expression(G) = (x^2/9 + y^2/8 = 1);Coordinate(P) = (1, -2);RightFocus(G) = F;PointOnCurve(M, G)", "coords": { "P": [ 1.0, -2.0 ] } }, { "index": 2482, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2483, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_2483.png", "answer": "x^2 + 4*y^2 = 4", "fact_expr": "G: Circle;N: Point;M: Point;Expression(G) = (x^2 + y^2 = 4);PointOnCurve(M, G);L: Line;PointOnCurve(M, L);IsPerpendicular(L, xAxis);FootPoint(L, xAxis) = N", "coords": {} }, { "index": 2484, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2485, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2485.png", "answer": "1", "fact_expr": "G: Hyperbola;L:Line;Expression(G) = (x^2/4 - y^2 = 1);IsPerpendicular(L,xAxis);PointOnCurve(RightFocus(G),L)", "coords": {} }, { "index": 2486, "success": false, "conic_type": "parabola", "error": "validation: parabola_point_off_curve", "params": { "p": 3.9731175899505615, "direction": "up" }, "optimization": { "final_loss": 0.00018243117665406317, "converged": true, "iterations": 127 }, "output_path": "results/train/parabola/problem_2486.png", "answer": "5", "fact_expr": "C: Parabola;p: Number;P: Point;F: Point;p>0;Expression(C) = (x^2 = 2*(p*y));Coordinate(P) = (4, 1);PointOnCurve(P, C);Focus(C) = F", "coords": { "P": [ 4.0, 1.0 ] }, "validation_reasons": [ "parabola_point_off_curve" ] }, { "index": 2487, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2488, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2489, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2489.png", "answer": "x^2/15+y^2/10=1", "fact_expr": "G: Ellipse;H: Point;C:Ellipse;Expression(G) = (x^2/9 + y^2/4 = 1);Coordinate(H) = (3, 2);PointOnCurve(H, C);Focus(G)=Focus(C)", "coords": { "H": [ 3.0, 2.0 ] } }, { "index": 2490, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2491, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2491.png", "answer": "pm*sqrt(3)", "fact_expr": "l: Line;C: Parabola;p: Number;M: Point;N: Point;F: Point;p>0;Expression(C) = (y^2 = 2*p*x);Focus(C) = F;PointOnCurve(F,l);Intersection(l, C) = {M, N};VectorOf(M, N) = 4*VectorOf(M, F)", "coords": {} }, { "index": 2492, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2492.png", "answer": "x^2 = -4*y", "fact_expr": "G: Parabola;M: Point;O: Origin;F: Point;Coordinate(M) = (a,-4);Vertex(G) = O;PointOnCurve(Focus(G), yAxis);PointOnCurve(M, G);Focus(G) = F;Distance(M, F) = 5;a:Number;a>0", "coords": {} }, { "index": 2493, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2494, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2494.png", "answer": "4", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 4);L:LineSegment;IsChordOf(L,G);PointOnCurve(Focus(G),L);IsParallel(L,ImageinaryAxis(G))", "coords": {} }, { "index": 2495, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2495.png", "answer": "3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 6*x)", "coords": {} }, { "index": 2496, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2135674953460693, "b": 2.4270472526550293, "orientation": "horizontal" }, "optimization": { "final_loss": 4.453341261978494e-06, "converged": true, "iterations": 125 }, "output_path": "results/train/hyperbola/problem_2496.png", "answer": "5*x^2 - (5/4)*y^2 = 1", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 4*x);OneOf(Focus(G)) = Focus(H);Eccentricity(G) = sqrt(5)", "coords": {} }, { "index": 2497, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2497.png", "answer": "0", "fact_expr": "Gamma: Hyperbola;Expression(Gamma) = (x^2/3-y^2/2=1);F1: Point;F2: Point;LeftFocus(Gamma) = F1;RightFocus(Gamma) = F2;P: Point;PointOnCurve(P, Gamma);Abs(LineSegmentOf(P, F1))*Abs(LineSegmentOf(P, F2)) = 4", "coords": {} }, { "index": 2498, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2498.png", "answer": "5/4", "fact_expr": "M: Point;N: Point;G: Parabola;F: Point;Expression(G) = (y = x^2);Focus(G) = F;PointOnCurve(M, G);PointOnCurve(N, G);Abs(LineSegmentOf(M, F)) + Abs(LineSegmentOf(N, F)) = 3", "coords": {} }, { "index": 2499, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2499.png", "answer": "3/2", "fact_expr": "C: Parabola;A: Point;F: Point;O: Origin;Expression(C) = (y^2 = 4*x);Focus(C)=F;PointOnCurve(A, C);Abs(LineSegmentOf(A, O)) = Abs(LineSegmentOf(A, F))", "coords": {} }, { "index": 2500, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/train/hyperbola/problem_2500.png", "answer": "2", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (2, 3);PointOnCurve(P, C);FocalLength(C) = 4", "coords": { "P": [ 2.0, 3.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 2501, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2501.png", "answer": "sqrt(5)", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C)=F1;RightFocus(C)=F2;PointOnCurve(P,C);Abs(VectorOf(P, F1) + VectorOf(P, F2)) = sqrt(Abs(VectorOf(P, F1))^2 + Abs(VectorOf(P, F2))^2);Abs(VectorOf(P, F1)) = 2*Abs(VectorOf(P, F2))", "coords": {} }, { "index": 2502, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2502.png", "answer": "sqrt(5)/5", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, G) = True;G: Line;Intersection(G, C) = {A, B};A: Point;B: Point;VectorOf(A, F1) = (3/2)*VectorOf(F1, B);AngleOf(A, F2, B) = ApplyUnit(90, degree)", "coords": {} }, { "index": 2503, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2503.png", "answer": "x^2/4 - y^2/5 = 1", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;F: Point;Focus(C) = F;Coordinate(F) = (3, 0);Distance(F, Asymptote(C)) = sqrt(5)", "coords": { "F": [ 3.0, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 2504, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2504.png", "answer": "2*p", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;F: Point;Focus(G) = F;Intersection(Directrix(G), xAxis) = C;C: Point;PointOnCurve(F, LineSegmentOf(A, B)) = True;IsChordOf(LineSegmentOf(A, B), G) = True;A: Point;B: Point;AngleOf(C, B, F) = ApplyUnit(90, degree)", "coords": {} }, { "index": 2505, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2505.png", "answer": "y=pm*(sqrt(5)/2)*x", "fact_expr": "C: Hyperbola;m: Number;m>0;Expression(C) = (x^2/4 - y^2/m = 1);Coordinate(OneOf(Focus(C))) = (3, 0)", "coords": {} }, { "index": 2506, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2507, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2507.png", "answer": "sqrt(13)/2", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 6*x);F: Point;Focus(C) = F;l: Line;PointOnCurve(F,l) = True;Intersection(C,l) = A;Intersection(Directrix(C),l) = B;B: Point;A: Point;VectorOf(F, B) = 3*(VectorOf(F, A));O: Origin", "coords": {} }, { "index": 2508, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2508.png", "answer": "[1, 4/3]", "fact_expr": "Gamma: Ellipse;H: Line;A: Point;F: Point;B: Point;Expression(Gamma) = (x^2/4 + y^2/3 = 1);RightFocus(Gamma)=F;O: Origin;PointOnCurve(O, H);Intersection(H, Gamma) = {A, B}", "coords": {} }, { "index": 2509, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2509.png", "answer": "12*sqrt(3)", "fact_expr": "G: Hyperbola;F1: Point;M: Point;F2: Point;Expression(G) = (x^2/16 - y^2/12 = 1);LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(M,G);AngleOf(F1, M, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 2510, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2510.png", "answer": "2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);C: Circle;P: Point;Q: Point;F: Point;Expression(C) = ((x - 2)^2 + (y - 1)^2 = 1);PointOnCurve(P, G);Focus(G) = F;PointOnCurve(Q, C)", "coords": {} }, { "index": 2511, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2511.png", "answer": "2", "fact_expr": "G: Hyperbola;F2: Point;P: Point;A: Point;F1: Point;Expression(G) = (x^2/4 - y^2/9 = 1);PointOnCurve(P, RightPart(G));LeftFocus(G) = F1;RightFocus(G) = F2;DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;Intersection(LineOf(P,F2), yAxis) = A", "coords": {} }, { "index": 2512, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.6457513110645907, "major_axis": "x", "x_coef": 16.0, "y_coef": 7.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2512.png", "answer": "y^2=12*x", "fact_expr": "O: Origin;Vertex(C) = O;C: Parabola;G: Ellipse;Expression(G) = (x^2/16 + y^2/7 = 1);Focus(C) = RightFocus(G)", "coords": {} }, { "index": 2513, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2514, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2514.png", "answer": "2", "fact_expr": "G: Parabola;Q: Point;P: Point;x1: Number;y1: Number;Expression(G) = (y = x^2/4);Coordinate(Q) = (2*sqrt(2), 0);Coordinate(P) = (x1, y1);PointOnCurve(P, G)", "coords": { "Q": [ 2.8284271247461903, 0.0 ] } }, { "index": 2515, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2516, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2516.png", "answer": "5/4", "fact_expr": "G: Parabola;b: Number;A: Point;F: Point;B: Point;Expression(G) = (x^2 = 4*y);Expression(H) = (y = b + 2*x);TangentPoint(H, G) = A;Focus(G)=F;Intersection(LineOf(A,F),G)={A,B};H:Line", "coords": {} }, { "index": 2517, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2518, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2518.png", "answer": "ApplyUnit(90,degree)", "fact_expr": "G: Parabola;H: Line;C:Line;M: Point;F: Point;N: Point;A: Point;B:Point;Expression(G) = (y^2 = x);Expression(C) = (x = 1/4);Focus(G)=F;PointOnCurve(F,H);Intersection(H,G) = {A,B};Projection(A,C)=M;Projection(B,C)=N", "coords": {} }, { "index": 2519, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 2.6457513110645907, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2519.png", "answer": "6", "fact_expr": "l: Line;C: Hyperbola;A: Point;P: Point;O: Origin;M: Point;N: Point;F: Point;Expression(C) = (x^2/9 - y^2/7 = 1);Coordinate(A)=(3, 0);Coordinate(F)=(4,0);PointOnCurve(F, l);Intersection(l, C) = {M, N};PointOnCurve(P,l);Abs(VectorOf(A,P)+VectorOf(O,P))=4", "coords": { "A": [ 3.0, 0.0 ], "F": [ 4.0, 0.0 ] } }, { "index": 2520, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2520.png", "answer": "5*sqrt(2)", "fact_expr": "l: Line;C: Parabola;p: Number;N: Point;M: Point;F: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;PointOnCurve(F, l);Slope(l) = sqrt(5);Intersection(l, C) = {M, N};YCoordinate(MidPoint(LineSegmentOf(M, N))) = sqrt(10)", "coords": {} }, { "index": 2521, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2521.png", "answer": "y^2 = 12*x", "fact_expr": "G: Hyperbola;H: Parabola;a: Number;Expression(G) = (x^2/7 - y^2/2 = 1);a>0;Expression(H) = (y^2 = a*x);Focus(H) = OneOf(Focus(G))", "coords": {} }, { "index": 2522, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2522.png", "answer": "y=pm*sqrt(3)*x", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;P: Point;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 8*x);OneOf(Focus(G)) = F;Focus(H) = F;OneOf(Intersection(H,G))=P;Abs(LineSegmentOf(P, F)) = 5", "coords": {} }, { "index": 2523, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2523.png", "answer": "x - y - 3 = 0", "fact_expr": "l: Line;G: Hyperbola;M: Point;Z: LineSegment;Coordinate(M) = (4, 1);Expression(G) = (x^2 - 4*y^2 = 4);IsChordOf(Z, G);MidPoint(Z) = M;OverlappingLine(Z, l)", "coords": { "M": [ 4.0, 1.0 ] } }, { "index": 2524, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2524.png", "answer": "ApplyUnit(30, degree)", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);F: Point;Focus(G) = F;Intersection(Directrix(G),yAxis) = M;M: Point;PointOnCurve(N,G) = True;N: Point;Abs(LineSegmentOf(N, F)) = (sqrt(3)/2)*Abs(LineSegmentOf(M, N))", "coords": {} }, { "index": 2525, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2526, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2526.png", "answer": "2*sqrt(2)", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;Expression(G) = (y^2 = 2*x);Expression(H) = (y = x);Intersection(H, G) = {A, B}", "coords": {} }, { "index": 2527, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2528, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2528.png", "answer": "6", "fact_expr": "G: Parabola;A: Point;B: Point;Expression(G) = (y^2 = 4*x);IsChordOf(LineSegmentOf(A,B),G);XCoordinate(MidPoint(LineSegmentOf(A,B)))=2", "coords": {} }, { "index": 2529, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2529.png", "answer": "{y^2=8*x,y^2=-16*x}", "fact_expr": "G: Parabola;m: Number;H: Line;Expression(G) = (y^2 = m*x);Expression(H) = (x = 1);Distance(Directrix(G),H) = 3", "coords": {} }, { "index": 2530, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2530.png", "answer": "y = pm*sqrt(6)*x", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);2*Abs(LineSegmentOf(P, F1)) = 3*Abs(LineSegmentOf(P, F2));AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 2531, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2531.png", "answer": "(4, 4), (4, -4)", "fact_expr": "G: Parabola;P: Point;F: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(P, G);Abs(LineSegmentOf(P, F)) = 5;Focus(G) = F", "coords": {} }, { "index": 2532, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2532.png", "answer": "48", "fact_expr": "G: Hyperbola;P: Point;F2: Point;F1: Point;Expression(G) = (x^2/9 - y^2/16 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(P,RightPart(G));Abs(LineSegmentOf(P,F2))=Abs(LineSegmentOf(F1,F2))", "coords": {} }, { "index": 2533, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.75, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2533.png", "answer": "9/4", "fact_expr": "C: Parabola;G: Line;A: Point;B: Point;O: Origin;F: Point;Expression(C) = (y^2 = 3*x);Focus(C) = F;PointOnCurve(F, G);Inclination(G) = ApplyUnit(30, degree);Intersection(G, C) = {A, B}", "coords": {} }, { "index": 2534, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2534.png", "answer": "x^2/36+y^2/9=1", "fact_expr": "G: Ellipse;a: Number;b: Number;H: Line;P: Point;Q: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x + 2*y + 8 = 0);Eccentricity(G) = sqrt(3)/2;Intersection(G, H) = {P, Q};Abs(LineSegmentOf(P, Q)) = sqrt(10)", "coords": {} }, { "index": 2535, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2535.png", "answer": "13", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 16*x);P: Point;PointOnCurve(P, G);Distance(P, xAxis) = 12;F: Point;Focus(G) = F;Distance(P, F) = Abs(LineSegmentOf(P, F))", "coords": {} }, { "index": 2536, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2536.png", "answer": "9/2", "fact_expr": "G: Parabola;A: Point;M: Point;F: Point;Expression(G) = (y^2 = 2*x);Coordinate(A) = (4, 2);Focus(G) = F;PointOnCurve(M, G);WhenMin(Abs(LineSegmentOf(M,A))+Abs(LineSegmentOf(M,F)))", "coords": { "A": [ 4.0, 2.0 ] } }, { "index": 2537, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2537.png", "answer": "sqrt(2)+1", "fact_expr": "G: Hyperbola;b: Number;a: Number;F1: Point;P: Point;F2: Point;H:Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(F1,H);IsPerpendicular(H,xAxis);Intersection(H,G)=P;AngleOf(F1, P, F2) = ApplyUnit(45, degree)", "coords": {} }, { "index": 2538, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2539, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2539.png", "answer": "{x+y+1=0,x-y+1=0}", "fact_expr": "C: Parabola;A: Point;P: Point;F: Point;Expression(C) = (y^2 = 4*x);Coordinate(A) = (-1, 0);Focus(C) = F;PointOnCurve(P, C);WhenMin(Abs(LineSegmentOf(P,F))/Abs(LineSegmentOf(P,A)))", "coords": { "A": [ -1.0, 0.0 ] } }, { "index": 2540, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2541, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2542, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2542.png", "answer": "6", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);D: Point;PointOnCurve(D, G) = True;Distance(D, LeftDirectrix(G)) = 5", "coords": {} }, { "index": 2543, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2544, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2545, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2545.png", "answer": "sqrt(3)+1", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;Coordinate(F) = (-c, 0);LeftFocus(G) = F;c: Number;c>0;PointOnCurve(F, LineOf(F, E));E: Point;Inclination(LineOf(F, E)) = pi/6;P: Point;Intersection(LineOf(F, E), RightPart(G)) = P;O: Origin;VectorOf(O, E) = (1/2)*(VectorOf(O, F) + VectorOf(O, P));DotProduct(VectorOf(O, E), VectorOf(E, F)) = 0", "coords": {} }, { "index": 2546, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2546.png", "answer": "sqrt(10) - 1", "fact_expr": "G: Parabola;P: Point;A: Point;M: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (2, 3);PointOnCurve(P, G);Projection(P, yAxis) = M", "coords": { "A": [ 2.0, 3.0 ] } }, { "index": 2547, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 9.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2547.png", "answer": "ApplyUnit(120,degree)", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 + y^2/2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 4", "coords": {} }, { "index": 2548, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2549, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.449489742783178, "major_axis": "x", "x_coef": 9.0, "y_coef": 6.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2549.png", "answer": "y=pm*(sqrt(2)/2)*x", "fact_expr": "G: Hyperbola;a: Number;b:Number;H: Ellipse;a>0;b>0;Expression(G) = (-y^2 + x^2/a^2 = 1);Expression(H) = (x^2/9 + y^2/6 = 1);Focus(G) = Focus(H)", "coords": {} }, { "index": 2550, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.242640687119285, "b": 3.0, "major_axis": "x", "x_coef": 18.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2550.png", "answer": "12", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/18 + y^2/9 = 1);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2;Area(TriangleOf(F1, P, F2)) = 3*sqrt(3)", "coords": {} }, { "index": 2551, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2552, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2553, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2553.png", "answer": "4", "fact_expr": "G: Hyperbola;H: Parabola;p: Number;Expression(G) = (x^2 - y^2/3 = 1);p>0;Expression(H) = (y^2 = 2*(p*x));Focus(H) = OneOf(Focus(G))", "coords": {} }, { "index": 2554, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2554.png", "answer": "2*sqrt(3)/3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;F: Point;Focus(G) = F;l: Line;Directrix(G) = l;H: Line;PointOnCurve(F,H);A: Point;B: Point;Intersection(H, G) = {A, B};L1: Line;L2: Line;PointOnCurve(A,L1);PointOnCurve(B,L2);IsPerpendicular(l,L1);IsPerpendicular(l,L2);C: Point;D: Point;FootPoint(l,L1) = C;FootPoint(l,L2) = D;Abs(LineSegmentOf(A, F)) = 2*Abs(LineSegmentOf(B, F));Area(TriangleOf(C, D, F)) = sqrt(2)", "coords": {} }, { "index": 2555, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2555.png", "answer": "2*sqrt(3)/3", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;A: Point;B: Point;F: Point;a>0;b>0;L1:Line;L2:Line;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;PointOnCurve(F,l);IsPerpendicular(l,L1);FootPoint(l,L1)=A;Intersection(l,L2)=B;Abs(LineSegmentOf(A, B)) = sqrt(3)*a;OneOf(Asymptote(C)) = L1;OneOf(Asymptote(C)) = L2;Negation(L1=L2)", "coords": {} }, { "index": 2556, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2556.png", "answer": "[4*sqrt(2), +oo)", "fact_expr": "C: Parabola;p: Real;G: Line;F: Point;M: Point;A: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Expression(G) = (x + y + 4 = 0);Focus(C) = F;Intersection(Directrix(C), xAxis) = A;PointOnCurve(M, G);AngleOf(F, M, A) = ApplyUnit(90, degree)", "coords": {} }, { "index": 2557, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -5.0, 0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_2557.png", "answer": "10", "fact_expr": "C1: Curve;C2: Curve;C3: Curve;P: Point;Q: Point;R: Point;Expression(C1) = (x^2/16 - y^2/9 = 1);Expression(C2) = (y^2 + (x - 5)^2 = 1);Expression(C3) = (y^2 + (x + 5)^2 = 1);PointOnCurve(P, C1);PointOnCurve(Q, C2);PointOnCurve(R, C3)", "coords": {} }, { "index": 2558, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2559, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2559.png", "answer": "pm*sqrt(3)", "fact_expr": "G: Parabola;p: Number;I: Line;B: Point;F: Point;A: Point;C: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;PointOnCurve(F, I);Intersection(I, G) = {A, B};Intersection(I,Directrix(G)) = C;LineSegmentOf(C, B) = 2 * LineSegmentOf(B, F)", "coords": {} }, { "index": 2560, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2560.png", "answer": "sqrt(2)/2", "fact_expr": "l: Line;G: Ellipse;b: Number;a: Number;A: Point;F: Point;E: Point;B: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F;PointOnCurve(F, l);Slope(l) = sqrt(2)/2;Intersection(l, G) = {A, B};Quadrant(B) = 1;Intersection(l, yAxis) = E;VectorOf(A, F) = VectorOf(E, B)", "coords": {} }, { "index": 2561, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2561.png", "answer": "8/5", "fact_expr": "C: Parabola;p: Number;A: Point;P: Point;H: Point;0 < p;p < 4;Expression(C) = (y^2 = 2*(p*x));Coordinate(A) = (4, 0);l: Line;Directrix(C) = (l);PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P, H), l);FootPoint(LineSegmentOf(P, H), l) = H;Abs(LineSegmentOf(P, H)) = Abs(LineSegmentOf(P, A));AngleOf(A, P, H) = ApplyUnit(120, degree)", "coords": { "A": [ 4.0, 0.0 ] } }, { "index": 2562, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2562.png", "answer": "(\\sqrt{2}/4,\\sqrt{2}/2]", "fact_expr": "l: Line;Negation(Slope(l) = 0);C: Ellipse;Expression(C) = (x^2/2 + y^2 = 1);F: Point;LeftFocus(C) = F;PointOnCurve(F, l) = True;Intersection(l, C) = {A, B};A: Point;B: Point;PointOnCurve(M, yAxis) = True;M: Point;Abs(LineSegmentOf(M, A)) = Abs(LineSegmentOf(M, B))", "coords": {} }, { "index": 2563, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2563.png", "answer": "x^2/3-y^2=1", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;F: Point;Coordinate(F) = (2, 0);OneOf(Focus(C)) = F;Distance(F, Asymptote(C)) = 1", "coords": { "F": [ 2.0, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 2564, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 2.0, "major_axis": "x", "x_coef": 8.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2564.png", "answer": "[0, 2*sqrt(2)+2]", "fact_expr": "G: Ellipse;Expression(G) = (x^2/8 + y^2/4 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G)", "coords": {} }, { "index": 2565, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2565.png", "answer": "7", "fact_expr": "G: Parabola;B: Point;A: Point;C: Point;F: Point;Expression(G) = (y = x^2/2);Coordinate(C) = (0, 4);Focus(G)=F;PointOnCurve(A,G);PointOnCurve(B,G);Abs(LineSegmentOf(A, F)) + Abs(LineSegmentOf(B, F)) = n;Intersection(PerpendicularBisector(LineSegmentOf(A,B)), yAxis) = C;n:Number", "coords": { "C": [ 0.0, 4.0 ] } }, { "index": 2566, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2567, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4149149626832767, "b": 2.450642358201144, "orientation": "horizontal" }, "optimization": { "final_loss": 1.0319236343469396e-05, "converged": true, "iterations": 94 }, "output_path": "results/train/hyperbola/problem_2567.png", "answer": "x^2 - y^2/3 = 1", "fact_expr": "G: Hyperbola;H: Parabola;Expression(H) = (y^2 = 8*x);Expression(Asymptote(G)) = (y = pm*sqrt(3)*x);Center(G) = Vertex(H);Focus(H) = RightFocus(G)", "coords": {} }, { "index": 2568, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2568.png", "answer": "x^2/4+y^2=1", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;LeftFocus(G) = F1;RightFocus(G) = F2;F1: Point;F2: Point;PointOnCurve(P, G) = True;P: Point;IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2)) = True;Abs(LineSegmentOf(P, F1))*Abs(LineSegmentOf(P, F2)) = 2;a = 2*b", "coords": {} }, { "index": 2569, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2569.png", "answer": "4*sqrt(7)/7", "fact_expr": "G: Hyperbola;a: Number;H: Parabola;O:Origin;a>0;Expression(G) = (-y^2/9 + x^2/a^2 = 1);Expression(H) = (y^2 = 16*x);RightFocus(G) = Focus(H);Center(G)=O", "coords": {} }, { "index": 2570, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2570.png", "answer": "1", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/2 + y^2 = 1);PointOnCurve(P, G);Focus(G)={F1,F2};AngleOf(F1, P, F2) = ApplyUnit(90, degree)", "coords": {} }, { "index": 2571, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.6457513110645907, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2571.png", "answer": "y^2=12*x", "fact_expr": "C: Parabola;Vertex(C) = O;O: Origin;G: Hyperbola;Expression(G) = (x^2/7 - y^2/2 = 1);Focus(C) = RightFocus(G)", "coords": {} }, { "index": 2572, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_ecc_mismatch", "params": { "a": 1.9257268905639648, "b": 1.7993911322594118, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 2.5576271215184017e-15, "converged": true, "iterations": 2 }, "output_path": "results/train/ellipse/problem_2572.png", "answer": "{3,16/3}", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (y^2/4 + x^2/m = 1);Eccentricity(G) = 1/2", "coords": {}, "validation_reasons": [ "ellipse_ecc_mismatch" ] }, { "index": 2573, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2573.png", "answer": "4", "fact_expr": "G: Ellipse;m: Number;m>0;P: Point;F1: Point;F2: Point;Expression(G) = (y^2/3 + x^2/m = 1);Focus(G)={F1,F2};PointOnCurve(F1,xAxis);PointOnCurve(F2,xAxis);Quadrant(P)=1;PointOnCurve(P, G);(Sin(AngleOf(P, F1, F2)) + Sin(AngleOf(P, F2, F1)))/(Sin(AngleOf(F1, P, F2)))= 2", "coords": {} }, { "index": 2574, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2574.png", "answer": "2*x-y-2=0", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l: Line;Slope(l) = 2;Intersection(l, C) = {A, B};A: Point;B: Point;Abs(LineSegmentOf(A, F)) + Abs(LineSegmentOf(B, F)) = 5", "coords": {} }, { "index": 2575, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2575.png", "answer": "2*sqrt(2)", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;O: Origin;F: Point;A: Point;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;IsDiameter(LineSegmentOf(O, F), G);l1: Line;l2: Line;Asymptote(C) = {l1, l2};Intersection(G, l1) = A;Intersection(G, l2) = B;Negation(A = O);Negation(B = O);Area(TriangleOf(O, A, B)) = (a*b)/8", "coords": {} }, { "index": 2576, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2576.png", "answer": "2*sqrt(5)", "fact_expr": "G: Hyperbola;Expression(G) = (2*x^2 - 3*y^2 = 6)", "coords": {} }, { "index": 2577, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2577.png", "answer": "3", "fact_expr": "l: Line;C: Hyperbola;G: Circle;Q: Point;P: Point;M: Point;F: Point;Expression(C) = (x^2 - y^2/3 = 1);RightFocus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {P, Q};IsDiameter(LineSegmentOf(P,Q),G);PointOnCurve(M, G)", "coords": {} }, { "index": 2578, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.125, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2578.png", "answer": "15/8", "fact_expr": "Length(LineSegmentOf(A, B)) = 4;A: Point;B: Point;Endpoint(LineSegmentOf(A, B)) = {A, B};G: Parabola;Expression(G) = (y = 2*x^2);PointOnCurve(A, G);PointOnCurve(B, G);M: Point;MidPoint(LineSegmentOf(A, B)) = M", "coords": {} }, { "index": 2579, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2579.png", "answer": "3", "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/4 + y^2 = 1);PointOnCurve(P, C);LeftFocus(C) = F1;RightFocus(C) = F2;Abs(LineSegmentOf(P, F1)) = 3*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 2580, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2580.png", "answer": "sqrt(5)", "fact_expr": "C: Parabola;B: Point;A: Point;M: Point;Expression(C) = (y^2 = x);Coordinate(M) = (1, -1);PointOnCurve(M,C);PointOnCurve(A, C);PointOnCurve(B, C);DotProduct(VectorOf(M, A), VectorOf(M, B)) = 0", "coords": { "M": [ 1.0, -1.0 ] } }, { "index": 2581, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 0.9257914695554953, "b": 0.6016238374279513, "major_axis": "y", "x_coef": 0.19999999999999993, "y_coef": 0.25 }, "optimization": { "final_loss": 5.471293296806409e-05, "converged": true, "iterations": 67 }, "output_path": "results/train/ellipse/problem_2581.png", "answer": "5*x^2+4*y^2=1", "fact_expr": "G: Ellipse;P1: Point;P2: Point;Coordinate(P1) = (1/3, 1/3);Coordinate(P2) = (0, -1/2);PointOnCurve(P1,G);PointOnCurve(P2,G)", "coords": { "P1": [ 0.3333333333333333, 0.3333333333333333 ], "P2": [ 0.0, -0.5 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 2582, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 5.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2582.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/25 - y^2/9 = 1);P:Point;F1:Point;F2:Point;PointOnCurve(P,G);OneOf(Focus(G))=F1;OneOf(Focus(G))=F2;Negation(F1=F2);Distance(P,F1)=12", "coords": {} }, { "index": 2583, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2583.png", "answer": "3*sqrt(2)/4", "fact_expr": "G: Hyperbola;m: Number;n: Number;H: Parabola;A: Point;F: Point;B: Point;n>0;Expression(G) = (-y^2/n^2 + x^2/m^2 = 1);m>0;Expression(H) = (x^2 = 2*(m*y));Focus(H) = F;Intersection(Directrix(H),G)={A,B};AngleOf(A, F, B) = ApplyUnit(120, degree)", "coords": {} }, { "index": 2584, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2584.png", "answer": "sqrt(3)/2", "fact_expr": "C: Parabola;p: Number;B: Point;A: Point;F: Point;M: Point;N: Point;l: Line;p>0;Expression(C) = (y^2 = 2*p*x);Focus(C) = F;Directrix(C) = l;PointOnCurve(A,C);PointOnCurve(B, C);IsPerpendicular(LineSegmentOf(A,F),LineSegmentOf(A,B));AngleOf(A,B,F)=ApplyUnit(30,degree);MidPoint(LineSegmentOf(A,B))=M;Projection(M,l)=N", "coords": {} }, { "index": 2585, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2585.png", "answer": "1/2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x);A: Point;PointOnCurve(A, G);Distance(A, Focus(G)) = 1", "coords": {} }, { "index": 2586, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2586.png", "answer": "2*sqrt(2)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);P: Point;A: Point;Coordinate(A) = (3, 0);PointOnCurve(P, C)", "coords": { "A": [ 3.0, 0.0 ] } }, { "index": 2587, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.4641016151377544, "b": 2.0, "major_axis": "x", "x_coef": 12.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2587.png", "answer": "-1/3", "fact_expr": "P: Point;A: Point;G: Ellipse;B: Point;Expression(G) = (x^2/12 + y^2/4 = 1);PointOnCurve(P,G);LeftVertex(G)=A;RightVertex(G)=B;Negation(P=A);Negation(P=B);k1:Number;k2:Number;Slope(LineOf(P, A)) = k1;Slope(LineOf(P, B)) = k2", "coords": {} }, { "index": 2588, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2588.png", "answer": "-8", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2 - y^2 = 16);PointOnCurve(P, LeftPart(G));LeftFocus(G) = F1;RightFocus(G) = F2", "coords": {} }, { "index": 2589, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2589.png", "answer": "8*sqrt(3)/3", "fact_expr": "G: Hyperbola;H: Line;F: Point;P: Point;Q: Point;Expression(G) = (x^2 - y^2 = 4);RightFocus(G) = F;PointOnCurve(F, H);Inclination(H) = ApplyUnit(105, degree);Intersection(H,G) = {P,Q}", "coords": {} }, { "index": 2590, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2590.png", "answer": "x^2/3 - y^2/6 = 1", "fact_expr": "G: Hyperbola;Center(G) = O;O: Origin;Eccentricity(G) = sqrt(3);H: Parabola;Expression(H) = (y^2 = 4*x);OneOf(Directrix(G)) = Directrix(H)", "coords": {} }, { "index": 2591, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2591.png", "answer": "sqrt(3)", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Focus(C) = {F1, F2};PointOnCurve(P, C);LineSegmentOf(P, F1) = 2*LineSegmentOf(P, F2);AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 2592, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2593, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2593.png", "answer": "y^2 = 2*sqrt(2)*x", "fact_expr": "C: Parabola;A: Point;F: Point;M: Point;l: Line;B: Point;H: Line;p: Number;Coordinate(A) = (0, 2);Expression(C) = (y^2 = 2*p*x);p > 0;Focus(C) = F;Directrix(C) = l;Intersection(LineSegmentOf(F,A), C) = B;PointOnCurve(B, H);IsPerpendicular(H,l);FootPoint(H,l) = M;IsPerpendicular(LineSegmentOf(A, M), LineSegmentOf(M, F))", "coords": { "A": [ 0.0, 2.0 ] } }, { "index": 2594, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2594.png", "answer": "(0, \\sqrt{3}/3)", "fact_expr": "C1:Ellipse;C2:Circle;P: Point;L1: Line;L2: Line;Expression(C1)=(y^2/b^2 + x^2/a^2 = 1);Expression(C2)=(x^2 + y^2 = (3*b^2)/4);Negation(PointOnCurve(P,C1));TangentOfPoint(P,C2)={L1,L2};IsPerpendicular(L1,L2);a:Number;b:Number;a>b;b>0", "coords": {} }, { "index": 2595, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 2.8284271247461903 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_2595.png", "answer": "(x-2)^2+y^2=2", "fact_expr": "G: Curve;A: Point;B: Point;P:Point;Expression(G) = (x^2 + y^2 = 8);Coordinate(A) = (4, 0);PointOnCurve(B, G);MidPoint(LineSegmentOf(A,B))=P", "coords": {} }, { "index": 2596, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2596.png", "answer": "-1/12", "fact_expr": "G: Hyperbola;H: Parabola;a: Number;Expression(G) = (-x^2/6 + y^2/3 = 1);Expression(H) = (y = a*x^2);LowerFocus(G) = Focus(H)", "coords": {} }, { "index": 2597, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2597.png", "answer": "48", "fact_expr": "G: Hyperbola;Expression(G) = (y^2/16 - x^2/m = 1);m: Number;e: Number;Eccentricity(G) = e;e = 2", "coords": {} }, { "index": 2598, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2599, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2599.png", "answer": "y=pm*(sqrt(10)/2)*x", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;F: Point;RightFocus(C) = F;L1: Line;L2: Line;OneOf(Asymptote(C))=L1;OneOf(Asymptote(C))=L2;Negation(L1=L2);L: Line;M: Point;PointOnCurve(F, L);IsPerpendicular(L, L1);FootPoint(L, L1) = M;N: Point;Intersection(L, L2) = N;7*VectorOf(F, M) = 3*VectorOf(F, N)", "coords": {} }, { "index": 2600, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2600.png", "answer": "(2, 2)", "fact_expr": "P: Point;C: Parabola;Expression(C) = (y^2 = 2*x);PointOnCurve(P, C) = True;F: Point;Focus(C) = F;A: Point;Coordinate(A) = (7/2, 2);WhenMin(LineSegmentOf(P, A) + LineSegmentOf(P, F))", "coords": { "A": [ 3.5, 2.0 ] } }, { "index": 2601, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2601.png", "answer": "sqrt(2)/2", "fact_expr": "C: Ellipse;b: Number;a: Number;G: Line;A: Point;F1: Point;F2: Point;B: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F2, G);Intersection(G, C) = {A, B};DotProduct(VectorOf(A, F1), VectorOf(A, F2)) = 0;VectorOf(A, F2) = 3*VectorOf(F2, B)", "coords": {} }, { "index": 2602, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2602.png", "answer": "4/9", "fact_expr": "G: Hyperbola;H: Line;A: Point;P: Point;B: Point;k1: Number;k2: Number;Expression(G) = (x^2/9 - y^2/4 = 1);Expression(H) = (y = x/2);Intersection(H, G) = {A, B};PointOnCurve(P, G);Negation(A = P);Negation(B = P);Slope(LineOf(P,A))=k1;Slope(LineOf(P,B))=k2", "coords": {} }, { "index": 2603, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.463192105293274, "b": 2.070901870727539, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4661259228887502e-06, "converged": true, "iterations": 106 }, "output_path": "results/train/hyperbola/problem_2603.png", "answer": "y = pm*x/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Ellipse;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a > b;b > 0;Expression(H) = (y^2/b^2 + x^2/a^2 = 1);Eccentricity(H) = sqrt(3)/2", "coords": {} }, { "index": 2604, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2604.png", "answer": "9/41", "fact_expr": "C: Ellipse;P: Point;A: Point;B: Point;alpha: Number;beta: Number;Expression(C) = (x^2/25 + y^2/16 = 1);PointOnCurve(P, C);UpperVertex(C) = A;LowerVertex(C) = B;Inclination(LineOf(P, A)) = alpha;Inclination(LineOf(P, B)) = beta", "coords": {} }, { "index": 2605, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 1.9248086180933468, "b": 0.924587494146203, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 1.8526520682049785e-05, "converged": true, "iterations": 80 }, "output_path": "results/train/ellipse/problem_2605.png", "answer": "{x^2/4+y^2=1,y^2/16+x^2/4=1}", "fact_expr": "G: Ellipse;H: Point;Coordinate(H) = (2, 0);Length(MajorAxis(G)) = 2*Length(MinorAxis(G));PointOnCurve(H,G)", "coords": { "H": [ 2.0, 0.0 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 2606, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2606.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;H: Line;Expression(H) = (y = x*(b/a));IsPerpendicular(LineSegmentOf(F2, A), H);A: Point;FootPoint(LineSegmentOf(F2, A), H) = A;B: Point;Intersection(LineSegmentOf(F2, A), G) = B;VectorOf(F2, B) = VectorOf(B, A)", "coords": {} }, { "index": 2607, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 7.0, "b": 4.898979485566356, "major_axis": "x", "x_coef": 49.0, "y_coef": 24.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2607.png", "answer": "24", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/49 + y^2/24 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 6", "coords": {} }, { "index": 2608, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2608.png", "answer": "{pm*1, pm*2*sqrt(3)/3}", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 4);l: Line;Expression(l) = (y = k*(x - 1));k: Number;NumIntersection(l, G) = 1", "coords": {} }, { "index": 2609, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2609.png", "answer": "5/3", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1)", "coords": {} }, { "index": 2610, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2610.png", "answer": "Sin(alpha)/(Sin(gamma)+Sin(beta))", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Focus(G) = {F1,F2};F1: Point;F2: Point;P: Point;PointOnCurve(P,G) = True;Negation(P = Endpoint(MajorAxis(G)));alpha: Number;AngleOf(F1,P,F2) = alpha;beta: Number;AngleOf(P,F1,F2) = beta;gamma: Number;AngleOf(F1,F2,P) = gamma;e: Number;Eccentricity(G) = e", "coords": {} }, { "index": 2611, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2611.png", "answer": "{x-y-1=0, x+y+1=0}", "fact_expr": "G: Parabola;H: Line;P: Point;F: Point;Expression(G) = (x^2 = 4*y);Focus(G)=F;PointOnCurve(F,H);IsPerpendicular(H,yAxis);Intersection(H, G) = P;TangentOnPoint(P, G)=l;l:Line", "coords": {} }, { "index": 2612, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2612.png", "answer": "3", "fact_expr": "l: Line;G: Parabola;p: Number;H: Line;B: Point;A: Point;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (x = -3);Focus(G) = F;PointOnCurve(F, l);Intersection(l,G)={A,B};Abs(LineSegmentOf(A, B)) = 7;Distance(MidPoint(LineSegmentOf(A,B)), H) = 5", "coords": {} }, { "index": 2613, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2613.png", "answer": "2*sqrt(5)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/4 = 1)", "coords": {} }, { "index": 2614, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2614.png", "answer": "pi/6", "fact_expr": "G: Parabola;N: Point;F: Point;M: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;Intersection(Directrix(G), xAxis) = M;PointOnCurve(N, G);Abs(LineSegmentOf(N, F)) = (sqrt(3)/2)*Abs(LineSegmentOf(M, N))", "coords": {} }, { "index": 2615, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2615.png", "answer": "-5", "fact_expr": "G: Hyperbola;b1: Number;a1: Number;H: Ellipse;A: Point;P: Point;B: Point;M: Point;k1:Number;k2:Number;k3:Number;k4:Number;a1>0;b1>0;Expression(G) = (-y^2/b1^2 + x^2/a1^2 = 1);a> b;b> 0;Expression(H) = (y^2/b^2 + x^2/a^2 = 1);Vertex(G)={A,B};Vertex(H)={A,B};PointOnCurve(P,G);PointOnCurve(M,H);Negation(P=A);Negation(P=B);Negation(M=A);Negation(M=B);lambda:Real;VectorOf(A, P) + VectorOf(B, P) = lambda*(VectorOf(A, M) + VectorOf(B, M));Slope(LineOf(A,P))=k1;Slope(LineOf(B,P))=k2;Slope(LineOf(A,M))=k3;Slope(LineOf(B,M))=k4;k1+k2=5;a:Number;b:Number", "coords": {} }, { "index": 2616, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.4142135623730951, "major_axis": "y", "x_coef": 2.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2616.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Ellipse;Expression(G) = (x^2/2 + y^2/3 = 1);L: Line;PointOnCurve(LowerFocus(G), L);H: Circle;Expression(H) = (y - 3*x + x^2 + y^2 + 3/2 = 0);IsTangent(L, H)", "coords": {} }, { "index": 2617, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2617.png", "answer": "1", "fact_expr": "G: Hyperbola;b: Number;b>0;Expression(G) = (x^2/4 - y^2/b^2 = 1);Expression(Asymptote(G)) = (y = pm*(x/2))", "coords": {} }, { "index": 2618, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2619, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2619.png", "answer": "5/2", "fact_expr": "M: Point;Coordinate(M) = (-3, 2);G: Parabola;Expression(G) = (y^2 = 2*x);F: Point;Focus(G) = F;Q: Point;PointOnCurve(Q, G)", "coords": { "M": [ -3.0, 2.0 ] } }, { "index": 2620, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2620.png", "answer": "sqrt(2)+1", "fact_expr": "H: Parabola;Expression(H) = (x^2 = 4*y);A: Point;Intersection(SymmetryAxis(H), Directrix(H)) = A;B: Point;Focus(H) = B;P: Point;PointOnCurve(P, H);IsTangent(LineSegmentOf(P, A), H);G: Hyperbola;PointOnCurve(P, G);Focus(G) = {A, B}", "coords": {} }, { "index": 2621, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_2621.png", "answer": "2*sqrt(3)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = 2;PointOnCurve(RightFocus(G),H);IsPerpendicular(H,xAxis);Length(InterceptChord(H,G))=m;m:Number", "coords": {} }, { "index": 2622, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2623, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2624, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2624.png", "answer": "8", "fact_expr": "G: Ellipse;C: Point;F1: Point;F2: Point;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(C, G);LeftFocus(G) = F1;RightFocus(G) = F2;Abs(LineSegmentOf(C, F1)) = 2", "coords": {} }, { "index": 2625, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2625.png", "answer": "x^2/15 + y^2/10 = 1", "fact_expr": "G: Ellipse;H: Point;Coordinate(H) = (3,-2);C:Curve;Expression(C)=(x^2/9+y^2/4=1);Focus(C)=Focus(G);PointOnCurve(H,G)", "coords": { "H": [ 3.0, -2.0 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 2626, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2626.png", "answer": "34/3", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);P: Point;PointOnCurve(P, G);Distance(P, LeftDirectrix(G)) = 16/5", "coords": {} }, { "index": 2627, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 2.8284271247461903 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_2627.png", "answer": "x^2/8+y^2/2=1", "fact_expr": "G: Circle;Q: Point;P: Point;M:Point;Expression(G) = (x^2 + y^2 = 8);PointOnCurve(P,G);L:Line;PointOnCurve(P,L);IsPerpendicular(L,xAxis);FootPoint(L,xAxis)=Q;MidPoint(LineSegmentOf(P,Q))=M", "coords": {} }, { "index": 2628, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2628.png", "answer": "(-5, 0)", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Circle;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (-6*x + x^2 + y^2 + 8 = 0);Center(H) = OneOf(Focus(G));Length(MinorAxis(G)) = 8", "coords": {} }, { "index": 2629, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2629.png", "answer": "y=pm*sqrt(3)*x", "fact_expr": "H: Parabola;Expression(H) = (y^2 = 2*x);G: Hyperbola;Expression(G) = (-y^2/3 + x^2/m = 1);m: Number;Directrix(H) = OneOf(Directrix(G))", "coords": {} }, { "index": 2630, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2630.png", "answer": "x^2/4 + y^2/2 = 1", "fact_expr": "G: Hyperbola;H: Parabola;C:Ellipse;b: Number;a: Number;Expression(G) = (x^2 - y^2 = 1);Expression(H) = (y^2 = 8*x);Expression(C) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(Focus(H),C);Focus(G)=Focus(C)", "coords": {} }, { "index": 2631, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2631.png", "answer": "sqrt(2)/2", "fact_expr": "C: Ellipse;a: Number;G: Point;Expression(C) = (y^2/4 + x^2/a^2 = 1);Coordinate(G) = (2, 0);OneOf(Focus(C)) = G", "coords": { "G": [ 2.0, 0.0 ] } }, { "index": 2632, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2632.png", "answer": "4", "fact_expr": "G: Parabola;P: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(P) = (m, 2*sqrt(3));PointOnCurve(P, G);Focus(G) = F;m:Number", "coords": {} }, { "index": 2633, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2633.png", "answer": "x = 1-\\sqrt{2}", "fact_expr": "G: Parabola;H: Ellipse;b: Number;P: Point;F1: Point;F2: Point;O:Origin;0 < b;b < 1;Expression(H) = (x^2 + y^2/b^2 = 1);LeftFocus(H) = F1;RightFocus(H)=F2;Vertex(G) = O;Focus(G)=F2;Intersection(G, H) = P;AngleOf(P, F1, F2) = ApplyUnit(45, degree)", "coords": {} }, { "index": 2634, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 8.0, "b": 6.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2634.png", "answer": "33", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/64 - y^2/36 = 1);P: Point;PointOnCurve(P, G);F1: Point;F2: Point;Focus(G) = {F1, F2};Abs(LineSegmentOf(P, F1)) = 17", "coords": {} }, { "index": 2635, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2636, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2636.png", "answer": "sqrt(2)", "fact_expr": "H: Parabola;Expression(H) = (y^2 = 8*x);G: Hyperbola;Expression(G) = (x^2/2 - y^2/2 = 1)", "coords": {} }, { "index": 2637, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2637.png", "answer": "2*sqrt(6)", "fact_expr": "l: Line;G: Parabola;p: Number;A: Point;F: Point;B: Point;p>0;k:Number;k>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;Slope(l)=k;PointOnCurve(F, l);Intersection(l, G) = {A, B};Abs(LineSegmentOf(A, F)) = (3/2)*Abs(LineSegmentOf(B, F))", "coords": {} }, { "index": 2638, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2638.png", "answer": "[2, 2*sqrt(2))\n0", "fact_expr": "C: Ellipse;H: Line;x0: Number;y0: Number;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/2 + y^2 = 1);Expression(H) = (y*y0 + (x*x0)/2 = 1);Coordinate(P) = (x0, y0);Focus(C) = {F1, F2};0 < x0^2/2 + y0^2", "coords": {} }, { "index": 2639, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2639.png", "answer": "6", "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2/3 = 1);F: Point;RightFocus(C) = F;P: Point;PointOnCurve(P, C);A: Point;Coordinate(A) = (5, 8)", "coords": { "A": [ 5.0, 8.0 ] } }, { "index": 2640, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2641, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2641.png", "answer": "2*sqrt(2)-2", "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (4, 0);Focus(G)=F;PointOnCurve(P,G);Abs(LineSegmentOf(P,A))=2*Abs(LineSegmentOf(P,F))", "coords": { "A": [ 4.0, 0.0 ] } }, { "index": 2642, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2643, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2643.png", "answer": "4*sqrt(3)", "fact_expr": "G: Parabola;M: Point;Expression(G) = (y^2 = 8*x);Coordinate(M) = (x0, y0);PointOnCurve(M, G);Distance(M, Focus(G)) = 6;O:Origin;x0:Number;y0:Number", "coords": {} }, { "index": 2644, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2644.png", "answer": "sqrt(2)", "fact_expr": "O: Origin;F: Point;G: Hyperbola;b: Number;a: Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);OneOf(Focus(G))=F;L:Line;PointOnCurve(F,L);IsPerpendicular(L,OneOf(Asymptote(G)));PointOnCurve(FootPoint(L,OneOf(Asymptote(G))),PerpendicularBisector(LineSegmentOf(O,F)))", "coords": {} }, { "index": 2645, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2645.png", "answer": "(3+2*sqrt(2),2+2*sqrt(2))", "fact_expr": "G: Parabola;M: Point;P: Point;Q: Point;Expression(G) = (y^2 = 4*x);Coordinate(M) = (7, 6);PointOnCurve(P, G);Negation(P=Vertex(G));L:Line;PointOnCurve(P,L);IsPerpendicular(L,yAxis);FootPoint(L,yAxis)=Q;WhenMin(Length(LineSegmentOf(P,M))+Length(LineSegmentOf(P,Q)))", "coords": { "M": [ 7.0, 6.0 ] } }, { "index": 2646, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2647, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2648, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2649, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2649.png", "answer": "6", "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2/3 = 1);F1: Point;LeftFocus(C) = F1;Q: Point;Vertex(C) = Q;Coordinate(Q) = (0, 2*sqrt(3));P: Point;PointOnCurve(P, RightPart(C))", "coords": { "Q": [ 0.0, 3.4641016151377544 ] } }, { "index": 2650, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2650.png", "answer": "2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;A: Point;PointOnCurve(A, G) = True;F: Point;Focus(G) = F;Distance(A, F) = 2;IsDiameter(LineSegmentOf(A, F), V) = True;V: Circle;Intersection(V, yAxis) = H;H: Point;Coordinate(H) = (0, 1)", "coords": { "H": [ 0.0, 1.0 ] } }, { "index": 2651, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.463192105293274, "b": 2.070901870727539, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4661259228887502e-06, "converged": true, "iterations": 106 }, "output_path": "results/train/hyperbola/problem_2651.png", "answer": "sqrt(5)/2", "fact_expr": "H: Ellipse;Expression(H) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Eccentricity(H) = sqrt(3)/2;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1)", "coords": {} }, { "index": 2652, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2652.png", "answer": "sqrt(30)/3", "fact_expr": "G: Ellipse;Expression(G) = (x^2 + 2*y^2 = 4);IsChordOf(LineSegmentOf(A, B), G);Coordinate(MidPoint(LineSegmentOf(A, B))) = (-1, -1);A:Point;B:Point", "coords": {} }, { "index": 2653, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2653.png", "answer": "x^2 + y^2/5 = 1", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Circle;B: Point;A: Point;P:Point;L1:Line;L2:Line;Expression(G) = (x^2/b^2 + y^2/a^2 = 1);Expression(H) = (x^2 + y^2 = 1);Coordinate(P) = (1, 1/2);PointOnCurve(Focus(G), yAxis);TangentOfPoint(P,H)={L1,L2};TangentPoint(L1,H)=A;TangentPoint(L2,H)=B;PointOnCurve(UpperFocus(G),LineOf(A,B));PointOnCurve(RightVertex(G),LineOf(A,B))", "coords": { "P": [ 1.0, 0.5 ] } }, { "index": 2654, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2655, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.463192105293274, "b": 2.070901870727539, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4661259228887502e-06, "converged": true, "iterations": 106 }, "output_path": "results/train/hyperbola/problem_2655.png", "answer": "sqrt(5)/2, sqrt(5)", "fact_expr": "H: Ellipse;Expression(H) = (x^2/4 + y^2/b^2 = 1);Eccentricity(H) = sqrt(3)/2;b: Number;b>0;G: Hyperbola;Expression(G) = (x^2/4 - y^2/b^2 = 1)", "coords": {} }, { "index": 2656, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2656.png", "answer": "2*sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/2 - y^2 = 1)", "coords": {} }, { "index": 2657, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.2352789464429823, "b": 2.2352789464429823, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4312591569844774e-06, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_2657.png", "answer": "x**2/9-y**2=1", "fact_expr": "E: Hyperbola;F1: Point;F2: Point;Coordinate(F1) = (-sqrt(10), 0);Coordinate(F2) = (sqrt(10), 0);P: Point;PointOnCurve(P, E);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;Abs(LineSegmentOf(P, F1)) * Abs(LineSegmentOf(P, F2)) = 2;Focus(G)={F1,F2}", "coords": { "F1": [ -3.1622776601683795, 0.0 ], "F2": [ 3.1622776601683795, 0.0 ] } }, { "index": 2658, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2658.png", "answer": "pi/2", "fact_expr": "l: Line;G: Parabola;P: Point;Q: Point;F: Point;Expression(G) = (y = -x^2/4);Coordinate(P) = (-4, -4);Focus(G) = F;TangentPoint(l,G)=P;Intersection(l,xAxis)=Q", "coords": { "P": [ -4.0, -4.0 ] } }, { "index": 2659, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2659.png", "answer": "x^2/20 + y^2/15 = 1", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/4 = 1);E: Ellipse;Focus(G) = Focus(E);Length(MajorAxis(E)) = 4*sqrt(5)", "coords": {} }, { "index": 2660, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 2.7608087062835693, "b": 2.1130559371771285, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 1.5615703236659225e-06, "converged": true, "iterations": 104 }, "output_path": "results/train/ellipse/problem_2660.png", "answer": "1", "fact_expr": "m: Real;b: Real;P: Point;Coordinate(P) = (sqrt(10)/3, 8/3);H: Ellipse;Expression(H) = (x^2/10 + y^2/m = 1);PointOnCurve(P, H);G: Hyperbola;Expression(G) = (x^2 - y^2/b = 1);Focus(H) = Focus(G)", "coords": { "P": [ 1.0540925533894598, 2.6666666666666665 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 2661, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2661.png", "answer": "(pm*sqrt(10), 0)\n(pm*1, 0)\n2\n6\ny=pm*3*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/9 = 1)", "coords": {} }, { "index": 2662, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2662.png", "answer": "4", "fact_expr": "G: Hyperbola;H: Parabola;p: Real;Expression(G) = (x^2 - y^2/3 = 1);Expression(H) = (y^2 = 2*(p*x));PointOnCurve(LeftFocus(G), Directrix(H))", "coords": {} }, { "index": 2663, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2663.png", "answer": "16", "fact_expr": "C: Parabola;p: Number;G: Circle;H: Line;A: Point;B: Point;p>0;Expression(C) = (x^2 = 2*p*y);Expression(G) = (x^2 + (y - 1)^2 = 2);Center(G)=Focus(C);PointOnCurve(Focus(C),H);Inclination(H)=ApplyUnit(60,degree);Intersection(H,C)={A,B}", "coords": {} }, { "index": 2664, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2664.png", "answer": "x-y+2=0", "fact_expr": "IsChordOf(LineSegmentOf(A, B), G) = True;G: Parabola;Expression(G) = (x^2 = y);Expression(PerpendicularBisector(LineSegmentOf(A, B))) = (y = -x + 3);A: Point;B: Point", "coords": {} }, { "index": 2665, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2665.png", "answer": "1", "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2/b^2 = 1);b: Number;b>0;P: Point;PointOnCurve(P, C);O: Origin;Min(Abs(LineSegmentOf(O, P))) = 1", "coords": {} }, { "index": 2666, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2666.png", "answer": "sqrt(3) + 1", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;c: Number;P: Point;F2: Point;F1: Point;e: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y = (sqrt(3)/3)*(c + x));LeftFocus(C) = F1;RightFocus(C) = F2;FocalLength(C) = 2*c;AngleOf(P, F2, F1) = 2*AngleOf(P, F1, F2);OneOf(Intersection(G, C)) = P;Eccentricity(C) = e", "coords": {} }, { "index": 2667, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2667.png", "answer": "y^2/12 - x^2/24 = 1", "fact_expr": "G: Hyperbola;H: Point;C:Hyperbola;Expression(G) = (x^2/2 - y^2 = 1);Coordinate(H) = (0, 6);Focus(C)=H;Asymptote(C)=Asymptote(G)", "coords": { "H": [ 0.0, 6.0 ] } }, { "index": 2668, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2668.png", "answer": "y=4", "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;Expression(G) = (x^2 = 4*y);IsParallel(xAxis, l);Intersection(l, G) = {A, B};Abs(LineSegmentOf(A, B)) = 8", "coords": {} }, { "index": 2669, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2669.png", "answer": "4*sqrt(41)", "fact_expr": "G: Ellipse;A: Point;B: Point;F1: Point;F2: Point;a:Number;a>5;Expression(G)=(x^2/a^2+y^2/25=1);Focus(G)={F1,F2};PointOnCurve(A,G);PointOnCurve(B,G);PointOnCurve(F1,LineSegmentOf(A,B));Abs(LineSegmentOf(F1, F2)) = 8;IsChordOf(LineSegmentOf(A, B), G)", "coords": {} }, { "index": 2670, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2670.png", "answer": "2/3", "fact_expr": "C: Ellipse;A: Point;P: Point;F1: Point;F2: Point;B: Point;Expression(C) = (x^2/9 + y^2/5 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);Incenter(TriangleOf(P, F1, F2)) = A;Intersection(OverlappingLine(LineSegmentOf(P,A)),LineSegmentOf(F1,F2)) = B", "coords": {} }, { "index": 2671, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2671.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Ellipse;Expression(G) = (x^2 + 2*y^2 = 2);H: Line;PointOnCurve(Focus(G), H);Inclination(H) = ApplyUnit(45, degree);A: Point;B: Point;Intersection(H, G) = {A, B};O: Origin;Center(G) = O", "coords": {} }, { "index": 2672, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2672.png", "answer": "{y=sqrt(2)*x + pm*sqrt(2), y=(sqrt(2)/4)*x+pm*(sqrt(3)/2)}", "fact_expr": "l: Line;C: Ellipse;Expression(C) = (x^2/2 + y^2 = 1);P: Point;Q: Point;Intersection(l, C) = {P, Q};Slope(LineOf(P, Q))>0;O: Origin;IsPerpendicular(LineSegmentOf(O, P), LineSegmentOf(O, Q));Area(TriangleOf(O, P, Q)) = (2*sqrt(3))/5", "coords": {} }, { "index": 2673, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2673.png", "answer": "1", "fact_expr": "G: Hyperbola;a: Number;a>0;Expression(G) = (-y^2/4 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G))) = (y = -2*x)", "coords": {} }, { "index": 2674, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 1.0, "b": 0.28867513459481287, "major_axis": "x", "x_coef": 1.0, "y_coef": 0.08333333333333333 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2674.png", "answer": "19/20", "fact_expr": "G: Line;H: Curve;P: Point;M: Point;N: Point;Expression(H) = (x^2 + 12*y^2 = 1);Coordinate(P) = (sqrt(10)/2, 0);PointOnCurve(P, G);Intersection(G, H) = {M, N}", "coords": { "P": [ 1.5811388300841898, 0.0 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 2675, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2675.png", "answer": "(2, 2)", "fact_expr": "M: Point;Coordinate(M) = (3, 2);F: Point;Focus(G) = F;G: Parabola;Expression(G) = (y^2 = 2*x);P: Point;PointOnCurve(P, G);WhenMin(Perimeter(TriangleOf(P, M, F)))", "coords": { "M": [ 3.0, 2.0 ] } }, { "index": 2676, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2676.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;a: Number;H: Parabola;a>0;Expression(G) = (-y^2/2 + x^2/a = 1);Expression(H) = (y^2 = 8*x);Focus(H)=RightFocus(G)", "coords": {} }, { "index": 2677, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2677.png", "answer": "(3, \\sqrt{10}]", "fact_expr": "C: Hyperbola;A: Point;F1: Point;F2: Point;D: Point;Expression(C) = (x^2 - y^2/3 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(A, RightPart(C));Center(InscribedCircle(TriangleOf(A,F1,F2)))=D;r:Number;Radius(InscribedCircle(TriangleOf(A,F1,F2)))=r;In(r,(0,1])", "coords": {} }, { "index": 2678, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2678.png", "answer": "sqrt(15)/2", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);Area(TriangleOf(P, F1, F2)) = 1;Tan(AngleOf(P, F1, F2)) = 1/2;Tan(AngleOf(P, F2, F1)) = -2", "coords": {} }, { "index": 2679, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2680, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2681, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2681.png", "answer": "4*sqrt(3)", "fact_expr": "E: Hyperbola;Center(E) = O;O: Origin;Eccentricity(E) = 2;OneOf(Vertex(E)) = Focus(G);G: Parabola;Expression(G) = (y^2 = 8*x)", "coords": {} }, { "index": 2682, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2682.png", "answer": "sqrt(2)", "fact_expr": "C: Parabola;G: Line;A: Point;B: Point;F: Point;M: Point;Expression(C) = (y^2 = 4*x);Intersection(Directrix(C), xAxis) = M;PointOnCurve(M, G);Intersection(G, C) = {A, B};Abs(LineSegmentOf(M, A)) = Abs(LineSegmentOf(A, B));Focus(C) = F", "coords": {} }, { "index": 2683, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2683.png", "answer": "sqrt(2)/2", "fact_expr": "G: Parabola;b: Number;H: Ellipse;a: Number;F1: Point;F: Point;F2: Point;Expression(G) = (y^2 = 2*(b*x));a > b;b > 0;Expression(H) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(H) = F1;RightFocus(H) = F2;Focus(G) = F;VectorOf(F1, F) = 3*VectorOf(F, F2)", "coords": {} }, { "index": 2684, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2684.png", "answer": "sqrt(2)", "fact_expr": "F1: Point;F2: Point;C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, L) = True;J: Line;OneOf(Asymptote(C)) = J;Expression(J) = (y = b*x/a);IsPerpendicular(L, J) = True;L: Line;FootPoint(L, J) = P;P: Point;Intersection(L, LeftPart(C)) = Q;Q: Point;IsParallel(LineSegmentOf(O, Q), LineSegmentOf(P, F2)) = True;O: Origin", "coords": {} }, { "index": 2685, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4506710767745972, "b": 2.0511727333068848, "orientation": "horizontal" }, "optimization": { "final_loss": 4.657696081267204e-06, "converged": true, "iterations": 116 }, "output_path": "results/train/hyperbola/problem_2685.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;b: Number;a: Real;H: Parabola;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = -12*x);LeftFocus(G) = Focus(H);Expression(Asymptote(G)) = (y = pm*(sqrt(2)*x))", "coords": {} }, { "index": 2686, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2687, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.4641016151377544, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 12.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2687.png", "answer": "pm*(sqrt(3)/4)", "fact_expr": "G: Ellipse;F1: Point;P: Point;M: Point;Expression(G) = (x^2/12 + y^2/3 = 1);OneOf(Focus(G)) = F1;PointOnCurve(P, G);MidPoint(LineSegmentOf(P,F1))=M;PointOnCurve(M, yAxis)", "coords": {} }, { "index": 2688, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2688.png", "answer": "-3/13", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;F: Point;Focus(G) = F;H: Line;A: Point;B: Point;O: Origin;PointOnCurve(F, H);Slope(H) = 1/3;Intersection(H, G) = {A,B}", "coords": {} }, { "index": 2689, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2689.png", "answer": "2*sqrt(5)/5", "fact_expr": "alpha: Number;Inclination(H) = alpha;H: Line;G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1);Slope(H) = Eccentricity(G)", "coords": {} }, { "index": 2690, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2690.png", "answer": "sqrt(5)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C)=F;O: Origin;P: Point;PointOnCurve(P,C);IsPerpendicular(LineSegmentOf(P,F),LineSegmentOf(O,F))", "coords": {} }, { "index": 2691, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2691.png", "answer": "x^2/9 - y^2/7 = 1", "fact_expr": "H: Ellipse;Expression(H) = (x^2/25 + y^2/9 = 1);Focus(H) = Focus(G);G: Hyperbola;e: Number;Eccentricity(G) = e;e = 4/3", "coords": {} }, { "index": 2692, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2692.png", "answer": "2*sqrt(5)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/4 = 1)", "coords": {} }, { "index": 2693, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2694, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2694.png", "answer": "4", "fact_expr": "G: Parabola;p: Number;H: Circle;p>0;Expression(G) = (y^2 = 2*p*x);Expression(H) = (2*x + x^2 + y^2 = 0);IsTangent(Directrix(G),H)", "coords": {} }, { "index": 2695, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2695.png", "answer": "2*sqrt(2)", "fact_expr": "a: Number;b: Number;e1: Number;e2: Number;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = e1;Z: Hyperbola;Expression(Z) = (x^2/b^2 - y^2/a^2 = 1);Eccentricity(Z) = e2", "coords": {} }, { "index": 2696, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2696.png", "answer": "(pm*2,2)", "fact_expr": "G: Parabola;M: Point;F: Point;Expression(G) = (x^2 = 2*y);PointOnCurve(M, G);Focus(G) = F;Distance(M, F) = Abs(LineSegmentOf(M, F)) ;Abs(LineSegmentOf(M, F)) = 5/2", "coords": {} }, { "index": 2697, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2697.png", "answer": "sqrt(3)", "fact_expr": "C1: Hyperbola;C2: Parabola;P: Point;F2: Point;F1: Point;O: Origin;Expression(C1)=(-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C1)=F1;RightFocus(C1)=F2;Vertex(C2) = O;Directrix(C2)=LeftDirectrix(C1);Intersection(C1,C2)=P;IsPerpendicular(LineSegmentOf(P, F2), LineSegmentOf(F1, F2));a: Number;b: Number;a > 0;b > 0", "coords": {} }, { "index": 2698, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2175992727279663, "b": 2.434081792831421, "orientation": "horizontal" }, "optimization": { "final_loss": 3.452532837400213e-06, "converged": true, "iterations": 124 }, "output_path": "results/train/hyperbola/problem_2698.png", "answer": "1", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/4 + x^2/a^2 = 1);a: Number;a>0;Expression(OneOf(Asymptote(G))) = (y = 2*x)", "coords": {} }, { "index": 2699, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2699.png", "answer": "4", "fact_expr": "C: Hyperbola;b: Number;a: Number;A1: Point;A2: Point;P: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftVertex(C)=A1;RightVertex(C)=A2;Coordinate(A1) = (-3, 0);Coordinate(A2) = (3, 0);PointOnCurve(P,Asymptote(C));Abs(LineSegmentOf(P,A1))=2*Abs(LineSegmentOf(P,A2))", "coords": { "A1": [ -3.0, 0.0 ], "A2": [ 3.0, 0.0 ] } }, { "index": 2700, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2700.png", "answer": "(3,2*sqrt(6)), (3,-2*sqrt(6))", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);F: Point;Focus(G) = F;P: Point;PointOnCurve(P, G) = True;Abs(LineSegmentOf(P, F)) = 5", "coords": {} }, { "index": 2701, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2702, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/train/hyperbola/problem_2702.png", "answer": "2", "fact_expr": "C: Hyperbola;a: Number;G: Point;a>0;Expression(C) = (-y^2/12 + x^2/a^2 = 1);Coordinate(G) = (3, sqrt(15));PointOnCurve(G, C)", "coords": { "G": [ 3.0, 3.872983346207417 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 2703, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2704, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2705, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2706, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2707, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2707.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;a: Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(Asymptote(G)) = (y = pm*(2*x))", "coords": {} }, { "index": 2708, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 0.9550795555114746, "b": 2.8647773265838623, "orientation": "horizontal" }, "optimization": { "final_loss": 1.6550993677810766e-05, "converged": true, "iterations": 144 }, "output_path": "results/train/hyperbola/problem_2708.png", "answer": "y = pm*3*x", "fact_expr": "C: Hyperbola;a: Number;b: Number;a > 0;b > 0;Expression(C) = (x^2/a^2 - y^2/b^2 = 1);Eccentricity(C) = sqrt(10)", "coords": {} }, { "index": 2709, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2709.png", "answer": "sqrt(10)", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G))) = (3*x + y = 0)", "coords": {} }, { "index": 2710, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2710.png", "answer": "sqrt(2)", "fact_expr": "C: Ellipse;b: Number;a: Number;G: Line;A: Point;F: Point;B: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Eccentricity(C) = sqrt(3)/2;RightFocus(C)=F;PointOnCurve(F, G);Slope(G)=k;k:Number;k>0;Intersection(G, C) = {A,B};VectorOf(A, F) = 3*VectorOf(F, B)", "coords": {} }, { "index": 2711, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 3.872983346207417, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2711.png", "answer": "5", "fact_expr": "G: Hyperbola;H: Circle;Z: Circle;P: Point;M: Point;N: Point;Expression(G) = (x^2 - y^2/15 = 1);Expression(H) = (y^2 + (x + 4)^2 = 4);Expression(Z) = (y^2 + (x - 4)^2 = 1);PointOnCurve(P, RightPart(G));PointOnCurve(M, H);PointOnCurve(N, Z)", "coords": {} }, { "index": 2712, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2712.png", "answer": "sqrt(2)", "fact_expr": "C: Parabola;p: Number;F: Point;P: Point;M: Point;p>0;Expression(C) = (x^2 = 2*(p*y));Intersection(SymmetryAxis(C),Directrix(C))=M;Focus(C) = F;PointOnCurve(P, C);Sin(AngleOf(P, F, M)) = lambda*Sin(AngleOf(P, M, F));lambda:Number", "coords": {} }, { "index": 2713, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2713.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;L1: Line;L2: Line;Asymptote(G) = {L1, L2};IsPerpendicular(L1, L2)", "coords": {} }, { "index": 2714, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "y", "x_coef": 3.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2714.png", "answer": "1/2", "fact_expr": "C: Ellipse;Expression(C) = (x^2/3 + y^2/4 = 1)", "coords": {} }, { "index": 2715, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2715.png", "answer": "y=pm*(sqrt(3)/3)*x", "fact_expr": "C: Hyperbola;m: Number;G: Circle;m>0;Expression(C) = (-x^2/9 + y^2/m^2 = 1);Expression(G) = (y^2 + (x - 2)^2 = 1);IsTangent(G, Asymptote(C))", "coords": {} }, { "index": 2716, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2717, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2717.png", "answer": "6", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);M: Point;PointOnCurve(M, G);F1: Point;LeftFocus(G) = F1;Distance(M, F1) = 4;F2: Point;RightFocus(G) = F2", "coords": {} }, { "index": 2718, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2719, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2719.png", "answer": "(-\\infty,0] \\cup (16/9,+\\infty)", "fact_expr": "C: Hyperbola;M: Point;F1: Point;F2: Point;x1:Number;y1:Number;Negation(x1=pm*5);k1:Number;k2:Number;Expression(C) = (x^2/9 - y^2/16 = 1);Coordinate(M) = (x1, y1);LeftFocus(C) =F1;RightFocus(G)=F2;PointOnCurve(M,C);Slope(LineOf(M,F1))=k1;Slope(LineOf(M,F2))=k2", "coords": {} }, { "index": 2720, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.4641016151377544, "major_axis": "x", "x_coef": 16.0, "y_coef": 12.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2720.png", "answer": "(x+1)^2/4+y^2/3=1", "fact_expr": "C: Ellipse;F1: Point;M: Point;P: Point;Expression(C) = (x^2/16 + y^2/12 = 1);LeftFocus(C) = F1;PointOnCurve(M, C);MidPoint(LineSegmentOf(F1,M)) = P", "coords": {} }, { "index": 2721, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2721.png", "answer": "4/3", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;A: Point;B: Point;LeftVertex(G) = A;RightVertex(G) = B;P: Point;PointOnCurve(P, G);Negation(P=A);Negation(P=B);O: Origin;Slope(LineOf(P, A))*Slope(LineOf(P, B)) = 7/9", "coords": {} }, { "index": 2722, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2722.png", "answer": "3*sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/3 = 1);P: Point;PointOnCurve(P, G);F1: Point;F2: Point;Focus(G) = {F1, F2};AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 2723, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2723.png", "answer": "2*sqrt(3)-3", "fact_expr": "G: Hyperbola;H: Circle;M: Point;T: Point;F: Point;P: Point;O: Origin;Expression(G) = (x^2/9 - y^2/12 = 1);Expression(H) = (x^2 + y^2 = 9);IsTangent(LineOf(M,T),H);LeftFocus(G)=F;PointOnCurve(F,LineOf(M,T));TangentPoint(LineOf(M,T),H)=T;Intersection(LineOf(M,N),RightPart(G))=M;MidPoint(LineSegmentOf(M, F)) = P", "coords": {} }, { "index": 2724, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2725, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2726, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2727, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2727.png", "answer": "-1/2", "fact_expr": "H: Line;Expression(H) = (y = k*x + 1);k: Number;G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);A: Point;B: Point;Intersection(H, G) = {A, B};XCoordinate(MidPoint(LineSegmentOf(A, B))) = 1", "coords": {} }, { "index": 2728, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2728.png", "answer": "4*sqrt(2)", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 2*(p*y));p: Number;p>0;PointOnCurve(A,G) = True;A: Point;YCoordinate(A) = 4;F: Point;Focus(G) = F;Distance(A,F) = 5;O: Origin", "coords": {} }, { "index": 2729, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2729.png", "answer": "2", "fact_expr": "S: Hyperbola;Expression(S) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;C: Circle;Expression(C) = (x^2 + (y - 4)^2 = 4);IsTangent(Asymptote(S),C)", "coords": {} }, { "index": 2730, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2730.png", "answer": "sqrt(10)/5", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;A: Point;B: Point;F2: Point;F1: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A,B))=Abs(LineSegmentOf(B,F2));Abs(LineSegmentOf(B,F2))=4*Abs(LineSegmentOf(B,F1))", "coords": {} }, { "index": 2731, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.125, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2731.png", "answer": "y = -1/8", "fact_expr": "G: Parabola;Expression(G) = (y = 2*x^2)", "coords": {} }, { "index": 2732, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2732.png", "answer": "4", "fact_expr": "A: Point;Coordinate(A) = (3, -1);F: Point;Focus(G) = F;G: Parabola;Expression(G) = (y^2 = 4*x);M: Point;PointOnCurve(M, G)", "coords": { "A": [ 3.0, -1.0 ] } }, { "index": 2733, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2734, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2734.png", "answer": "-50/17", "fact_expr": "H: Line;Expression(H) = (x = t);t: Number;G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);P: Point;Q: Point;Intersection(H, G) = {P, Q};F: Point;LeftFocus(G) = F;WhenMin(DotProduct(VectorOf(F, P), VectorOf(F, Q)))", "coords": {} }, { "index": 2735, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2735.png", "answer": "-1/2", "fact_expr": "M: Point;Coordinate(M) = (0, 1);l: Line;PointOnCurve(M, l);C: Ellipse;Expression(C) = (x^2/9 + y^2/5 = 1);A: Point;B: Point;Intersection(l, C) = {A, B};F1: Point;LeftFocus(C) = F1;WhenMax(Perimeter(TriangleOf(A, B, F1)))", "coords": { "M": [ 0.0, 1.0 ] } }, { "index": 2736, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2736.png", "answer": "3", "fact_expr": "l: Line;C: Hyperbola;k:Number;B: Point;A: Point;M: Point;Expression(C) = (x^2/2 - y^2/3 = 1);Coordinate(M) = (2, 1);Slope(l)=k;Intersection(l,C) = {A, B};MidPoint(LineSegmentOf(A,B))=M", "coords": { "M": [ 2.0, 1.0 ] } }, { "index": 2737, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2738, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2738.png", "answer": "2+sqrt(3)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;G: Line;PointOnCurve(RightFocus(C), G);IsParallel(G, Asymptote(C));P: Point;Intersection(G, C) = P;XCoordinate(P) = 2*a", "coords": {} }, { "index": 2739, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2739.png", "answer": "x^2/9+y^2/4=1", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;H: Circle;IsDiameter(LineSegmentOf(F1, F2), H);P: Point;Coordinate(P) = ((3*sqrt(5))/5, (4*sqrt(5))/5);Intersection(H, G) = P", "coords": { "P": [ 1.3416407864998738, 1.788854381999832 ] } }, { "index": 2740, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2740.png", "answer": "[2,4)", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2/3 = 1);Coordinate(P) = (x0, y0);Focus(G)={F1,F2};x0:Number;y0:Number;0 b;b > 0;Expression(D) = (x^2/b^2 + y^2/a^2 = 1);M: Point;PointOnCurve(M, D);C: Circle;Expression(C) = (x^2 + y^2 = b^2/2);l1, l2: Line;TangentOfPoint(M, C) = {l1, l2};P: Point;Q: Point;TangentPoint(l1, C) = P;TangentPoint(l2, C) = Q;E: Point;F: Point;Intersection(LineOf(P, Q), xAxis) = E;Intersection(LineOf(P, Q), yAxis) = F;O: Origin", "coords": {} }, { "index": 2744, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2744.png", "answer": "y^2=4*x-4", "fact_expr": "G: Parabola;F: Point;P: Point;Expression(G) = (y^2 = 8*x);Focus(G)=F;PointOnCurve(P, G)", "coords": {} }, { "index": 2745, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2745.png", "answer": "1", "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/4 + y^2/2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) = 3", "coords": {} }, { "index": 2746, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2747, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2747.png", "answer": "sqrt(2)", "fact_expr": "G: Parabola;H: Point;P: Point;Expression(G) = (y^2 = x);Coordinate(H) = (9/4, 0);Coordinate(P) = (x1, y1);x1:Number;y1:Number;PointOnCurve(P,G)", "coords": { "H": [ 2.25, 0.0 ] } }, { "index": 2748, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2748.png", "answer": "4", "fact_expr": "M: Point;Coordinate(M) = (-2, 8);G: Parabola;Expression(G) = (y^2 = 2*p*x);PointOnCurve(M, Directrix(G));p: Real", "coords": { "M": [ -2.0, 8.0 ] } }, { "index": 2749, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2749.png", "answer": "2/3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;I: Point;Coordinate(I) = (2, 0);H: Line;PointOnCurve(I, H) = True;Intersection(H, G) = {A, B};A: Point;B: Point", "coords": { "I": [ 2.0, 0.0 ] } }, { "index": 2750, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2750.png", "answer": "4*sqrt(41)", "fact_expr": "G: Ellipse;A: Point;B: Point;F2: Point;F1: Point;a: Number;a > 5;Expression(G) = (y^2/25 + x^2/a^2 = 1);Abs(LineSegmentOf(F1, F2)) = 8;Focus(G) = {F1, F2};IsChordOf(LineSegmentOf(A, B), G);PointOnCurve(F1, LineSegmentOf(A, B))", "coords": {} }, { "index": 2751, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2751.png", "answer": "y=pm*(sqrt(3)/3)*x", "fact_expr": "G: Hyperbola;a: Number;b: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Distance(OneOf(Focus(G)),OneOf(Asymptote(G)))=(1/4)*FocalLength(G)", "coords": {} }, { "index": 2752, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2753, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2754, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2754.png", "answer": "2", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;Eccentricity(G) = sqrt(2)/2;A: Point;B: Point;C: Point;PointOnCurve(Vertex(TriangleOf(A, B, C)), G);D: Point;E: Point;F: Point;MidPoint(LineSegmentOf(A, B)) = D;MidPoint(LineSegmentOf(B, C)) = E;MidPoint(LineSegmentOf(A, C)) = F;k1: Number;k2: Number;k3: Number;Slope(OverlappingLine(LineSegmentOf(A, B))) = k1;Slope(OverlappingLine(LineSegmentOf(B, C))) = k2;Slope(OverlappingLine(LineSegmentOf(A, C))) = k3;Negation(k1*k2*k3=0);O: Origin;Slope(LineOf(O, D)) + Slope(LineOf(O, E)) + Slope(LineOf(O, F)) = -1", "coords": {} }, { "index": 2755, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2755.png", "answer": "pm*sqrt(5)/4", "fact_expr": "G: Ellipse;H: Line;k: Number;Expression(G) = (x^2/16 + y^2/4 = 1);Expression(H) = (k*x - y + 3 = 0);IsTangent(H,G)", "coords": {} }, { "index": 2756, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2756.png", "answer": "x^2+(y-1)^2=2", "fact_expr": "C: Parabola;P: Point;Q: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;Quadrant(P) = 1;PointOnCurve(P, C);Projection(P, Directrix(C)) = Q;Abs(LineSegmentOf(P, Q)) = 2", "coords": {} }, { "index": 2757, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2757.png", "answer": "4", "fact_expr": "E: Parabola;A: Point;A1: Point;F: Point;S: Point;T: Point;l: Line;Expression(E) = (y^2 = 4*x);Focus(E) = F;Directrix(E) = l;Intersection(l, xAxis) = T;PointOnCurve(A, E);IsPerpendicular(LineSegmentOf(A, A1), l);FootPoint(LineSegmentOf(A, A1), l) = A1;Intersection(LineSegmentOf(A1, F), yAxis) = S;IsParallel(LineSegmentOf(S, T), LineSegmentOf(A, F))", "coords": {} }, { "index": 2758, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2759, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2759.png", "answer": "{x^2/4+y^2=1, y^2/4+x^2=1}", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Length(MajorAxis(C)) = 4;Eccentricity(C) = sqrt(3)/2", "coords": {} }, { "index": 2760, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 3.6924539658341695, "b": 2.223882945759832, "major_axis": "x", "x_coef": 14.999999999999998, "y_coef": 5.000000000000001 }, "optimization": { "final_loss": 1.845880297380376e-05, "converged": true, "iterations": 93 }, "output_path": "results/train/ellipse/problem_2760.png", "answer": "x^2/15 + y^2/5 = 1", "fact_expr": "C: Ellipse;PointOnCurve(Focus(C), axis) = True;A: Point;Coordinate(A) = (-sqrt(3), -2);B: Point;Coordinate(B) = (-2*sqrt(3), 1);PointOnCurve(A, C) = True;PointOnCurve(B, C) = True", "coords": { "A": [ -1.7320508075688772, -2.0 ], "B": [ -3.4641016151377544, 1.0 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 2761, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2761.png", "answer": "y^2 = 8*x", "fact_expr": "T: Parabola;p: Number;C: Circle;p>0;Expression(T) = (y^2 = 2*(p*x));Expression(C) = (-4*y + x^2 + y^2 - 4 = 0);Length(InterceptChord(Directrix(T), C)) = 4", "coords": {} }, { "index": 2762, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2762.png", "answer": "3/2", "fact_expr": "G: Parabola;O: Origin;P: Point;F: Point;Expression(G) = (x^2 = 4*y);Focus(G)=F;PointOnCurve(P,G);IsTangent(CircumCircle(TriangleOf(O,P,F)),Directrix(G))", "coords": {} }, { "index": 2763, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2764, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.6457513110645907, "major_axis": "x", "x_coef": 9.0, "y_coef": 7.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2764.png", "answer": "(-sqrt(2)/2, sqrt(2)/2)", "fact_expr": "G: Ellipse;H: Line;M: Point;A: Point;B: Point;Expression(G) = (x^2/9 + y^2/7 = 1);Slope(H)= -1/3;Intersection(H,G)={A,B};Negation(A=B);PointOnCurve(M,yAxis);Abs(LineSegmentOf(M,A))=Abs(LineSegmentOf(M,B))", "coords": {} }, { "index": 2765, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2765.png", "answer": "(3/2)*p", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;A: Point;B: Point;Length(LineSegmentOf(A, B)) = 4*p;Endpoint(LineSegmentOf(A, B)) = {A, B};PointOnCurve(A, C);PointOnCurve(B, C);M: Point;MidPoint(LineSegmentOf(A, B)) = M", "coords": {} }, { "index": 2766, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2766.png", "answer": "{4,1/4}", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2 + y^2/m = 1);Eccentricity(G) = sqrt(3)/2", "coords": {} }, { "index": 2767, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2767.png", "answer": "x^2/4+y^2/3=1", "fact_expr": "G: Ellipse;Expression(G) = (y^2/3 + x^2/a^2 = 1);a: Number;a>sqrt(3);F: Point;RightFocus(G) = F;A: Point;RightVertex(G) = A;e: Number;1/Abs(LineSegmentOf(O, F)) + 1/Abs(LineSegmentOf(O, A)) = (3*e)/Abs(LineSegmentOf(F, A));O: Origin;Eccentricity(G) = e", "coords": {} }, { "index": 2768, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2768.png", "answer": "(1, 3]", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;e: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, RightPart(C));Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2));Eccentricity(C) = e", "coords": {} }, { "index": 2769, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2769.png", "answer": "sqrt(5)", "fact_expr": "C: Hyperbola;b: Number;a: Number;F: Point;M: Point;P: Point;L:Line;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C)=F;PointOnCurve(F,L);IsPerpendicular(Asymptote(C),L);FootPoint(Asymptote(C),L)=M;Intersection(OverlappingLine(LineSegmentOf(F,M)),yAxis)=P;Abs(LineSegmentOf(F, M)) = 4*Abs(LineSegmentOf(P, M))", "coords": {} }, { "index": 2770, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.4641016151377544, "major_axis": "x", "x_coef": 16.0, "y_coef": 12.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2770.png", "answer": "1", "fact_expr": "G: Ellipse;M: Point;F1: Point;O: Origin;N: Point;Expression(G) = (x^2/16 + y^2/12 = 1);PointOnCurve(M,G);LeftFocus(G)=F1;Distance(M,F1)=6;MidPoint(LineSegmentOf(M,F1))=N", "coords": {} }, { "index": 2771, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2772, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.449489742783178, "b": 2.449489742783178, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2772.png", "answer": "1", "fact_expr": "G: Hyperbola;P: Point;A1: Point;A2:Point;Expression(G) = (x^2 - y^2 = 6);LeftVertex(G)=A1;RightVertex(G)=A2;PointOnCurve(P,RightPart(G));Quadrant(P)=1;k1:Number;k2:Number;Slope(LineOf(P,A1))=k1;Slope(LineOf(P,A2))=k2", "coords": {} }, { "index": 2773, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2774, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2774.png", "answer": "2", "fact_expr": "C: Ellipse;C1: Circle;A: Point;P: Point;B: Point;Expression(C) = (x^2/4 + y^2/3 = 1);Expression(C1)=((x-1)^2+y^2=1);Coordinate(A) = (-3, -sqrt(5));PointOnCurve(P, C);PointOnCurve(B,C1)", "coords": { "A": [ -3.0, -2.23606797749979 ] } }, { "index": 2775, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2775.png", "answer": "8", "fact_expr": "C: Parabola;Expression(C) = (y = x^2/8);F: Point;Focus(C) = F;P: Point;PointOnCurve(P, C);Abs(LineSegmentOf(P, F)) = 10;O: Origin", "coords": {} }, { "index": 2776, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2776.png", "answer": "x^2/5+y^2=1", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x - 2*y + 2 = 0);PointOnCurve(OneOf(Vertex(G)),H);PointOnCurve(OneOf(Focus(G)),H)", "coords": {} }, { "index": 2777, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2777.png", "answer": "y^2 = 8*x", "fact_expr": "G: Parabola;p: Number;p>0;Expression(G) = (y^2 = 2*(p*x));P: Point;PointOnCurve(P, G);XCoordinate(P) = 1;Distance(P, Vertex(G)) = Distance(P, Directrix(G))", "coords": {} }, { "index": 2778, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2778.png", "answer": "x = pm*1/2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1)", "coords": {} }, { "index": 2779, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2779.png", "answer": "[(sqrt(5)+1)/2,sqrt(3))", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;O: Origin;B: Point;A: Point;F: Point;e: Number;b > a;a > 0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;PointOnCurve(F, l);Intersection(l, RightPart(C)) = {A, B};DotProduct(VectorOf(O, A), VectorOf(O, B)) = 0;Eccentricity(C) = e", "coords": {} }, { "index": 2780, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2780.png", "answer": "4", "fact_expr": "G: Ellipse;F1: Point;F2: Point;M: Point;Expression(G) = (x^2/9 + y^2/4 = 1);Focus(G) ={F1,F2};PointOnCurve(M, G);Abs(LineSegmentOf(M, F1)) - Abs(LineSegmentOf(M, F2)) = 2", "coords": {} }, { "index": 2781, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2781.png", "answer": "y = pm*sqrt(2)*x", "fact_expr": "G: Hyperbola;b: Number;a: Number;Q: Point;M: Point;N: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Endpoint(RealAxis(G)) = {M, N};PointOnCurve(Q, G);Negation(Q = M);Negation(Q = N);Tan(AngleOf(Q, M, N))*Tan(AngleOf(Q, N, M)) = -2", "coords": {} }, { "index": 2782, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2782.png", "answer": "x^2/3 - y^2/3 = 1", "fact_expr": "G: Hyperbola;P: Point;C:Hyperbola;Expression(G) = (x^2/4 - y^2/2 = 1);Coordinate(P) = (2, 1);Focus(G)=Focus(C);PointOnCurve(P,C)", "coords": { "P": [ 2.0, 1.0 ] } }, { "index": 2783, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2783.png", "answer": "y=pm*2*sqrt(2)*x", "fact_expr": "LeftFocus(C) = F1;RightFocus(C) = F2;F1: Point;F2: Point;C: Hyperbola;Expression(C) = (x^2/a^2 - y^2/b^2 = 1);a: Number;b: Number;a>0;b>0;P: Point;Quadrant(P) = 1;PointOnCurve(P, Asymptote(C)) = True;Abs(LineSegmentOf(P, F1)) = lambda*Abs(LineSegmentOf(P, F2));lambda: Number;Min(lambda) = sqrt(2)", "coords": {} }, { "index": 2784, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2785, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2785.png", "answer": "5", "fact_expr": "C: Parabola;p: Number;G: Line;F: Point;P: Point;A: Point;B: Point;Expression(C) = (y^2 = 2*p*x);Coordinate(F) = (1, 0);Focus(C) = F;PointOnCurve(F,G);Intersection(G, C) = {A, B};Intersection(TangentOnPoint(A,C),Directrix(C))=P;Abs(LineSegmentOf(P, F)) = sqrt(5)", "coords": { "F": [ 1.0, 0.0 ] } }, { "index": 2786, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2786.png", "answer": "8", "fact_expr": "G: Hyperbola;H: Line;M: Point;F2: Point;N: Point;F: Point;Expression(G) = (x^2/4 - y^2/3 = 1);LeftFocus(G)=F;PointOnCurve(F, H);Intersection(H,LeftPart(G)) = {M, N};RightFocus(G) = F2", "coords": {} }, { "index": 2787, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_focus_mismatch", "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2787.png", "answer": "8+sqrt(2)", "fact_expr": "F: Point;Coordinate(F) = (2, 0);RightFocus(G) = F;G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;L: LineSegment;PointOnCurve(F,L) = True;IsPerpendicular(L,xAxis) = True;IsChordOf(L,G) = True;Length(L) = 6;A: Point;Coordinate(A) = (-2, sqrt(2));M: Point;PointOnCurve(M, G) = True", "coords": { "F": [ 2.0, 0.0 ], "A": [ -2.0, 1.4142135623730951 ] }, "validation_reasons": [ "ellipse_focus_mismatch" ] }, { "index": 2788, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2789, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2790, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2790.png", "answer": "sqrt(7)/2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/3 = 1)", "coords": {} }, { "index": 2791, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2791.png", "answer": "4", "fact_expr": "G: Parabola;H: Line;C: Point;B: Point;F: Point;A: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H,G)={A,B};PointOnCurve(C,Directrix(G));VectorOf(C,B)=2*VectorOf(B,F)", "coords": {} }, { "index": 2792, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2793, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2793.png", "answer": "9", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);P: Point;PointOnCurve(P, RightPart(G));H: Circle;Expression(H) = (y^2 + (x + 5)^2 = 4);K:Circle;Expression(K) = (y^2 + (x - 5)^2 = 1);M: Point;N: Point;PointOnCurve(M, H);PointOnCurve(N, K)", "coords": {} }, { "index": 2794, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2794.png", "answer": "x=-1", "fact_expr": "H: Line;Expression(H) = (x - y - 2 = 0);G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;A: Point;B: Point;Intersection(H, G) = {A, B};M: Point;Coordinate(M) = (4, 2);MidPoint(LineSegmentOf(A, B)) = M", "coords": { "M": [ 4.0, 2.0 ] } }, { "index": 2795, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2795.png", "answer": "20/3", "fact_expr": "G: Ellipse;P: Point;F1: Point;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(P, G);LeftFocus(G) = F1;Distance(P, F1) = 6", "coords": {} }, { "index": 2796, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2796.png", "answer": "pm*4", "fact_expr": "C: Parabola;p: Number;P: Point;F: Point;p>0;y0: Number;Expression(C) = (y^2 = 2*(p*x));Coordinate(P) = (2, y0);PointOnCurve(P, C);Focus(C) = F;Abs(LineSegmentOf(P, F)) = 4", "coords": {} }, { "index": 2797, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2797.png", "answer": "x^2/20+y^2/16=1", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Circle;P:Point;B: Point;A: Point;L1:Line;L2:Line;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2 + y^2 = 4);PointOnCurve(Focus(G), xAxis);Coordinate(P)=(2,1);TangentOfPoint(P,H)={L1,L2};TangentPoint(L1,H)=A;TangentPoint(L2,H)=B;PointOnCurve(RightFocus(G),LineOf(A,B));PointOnCurve(UpperVertex(G),LineOf(A,B))", "coords": { "P": [ 2.0, 1.0 ] } }, { "index": 2798, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.8284271247461903, "major_axis": "x", "x_coef": 16.0, "y_coef": 8.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2798.png", "answer": "(0,pm*2*sqrt(2))", "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/8 = 1);P: Point;PointOnCurve(P, G) = True;F1: Point;F2: Point;Focus(G) = {F1, F2};IsPerpendicular(LineSegmentOf(P,F1), LineSegmentOf(P, F2)) = True", "coords": {} }, { "index": 2799, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.0, "b": 0.5, "major_axis": "x", "x_coef": 1.0, "y_coef": 0.25 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2799.png", "answer": "sqrt(3)/2", "fact_expr": "G: Ellipse;Expression(G) = (x^2 + 4*y^2 = 1)", "coords": {} }, { "index": 2800, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2800.png", "answer": "12/5", "fact_expr": "l: Line;G: Ellipse;P: Point;A: Point;B: Point;Q: Point;O: Origin;Expression(G) = (x^2/4 + y^2/3 = 1);Coordinate(P) = (1, 1);PointOnCurve(P, l);Intersection(l, G) = {A, B};VectorOf(A,P)=lambda*VectorOf(P,B);lambda:Number;VectorOf(A,Q)=-lambda*VectorOf(Q,B)", "coords": { "P": [ 1.0, 1.0 ] } }, { "index": 2801, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 3.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2801.png", "answer": "3", "fact_expr": "l: Line;C: Ellipse;P: Point;Q: Point;O: Origin;Expression(C) = (x^2/3 + y^2/2 = 1);Coordinate(P) = (x1,y1);Coordinate(Q) = (x2,y2);Intersection(l, C) = {P,Q};Negation(P=Q);Area(TriangleOf(O, P, Q)) = sqrt(6)/2;x1:Number;y1:Number;x2:Number;y2:Number", "coords": {} }, { "index": 2802, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2802.png", "answer": "4", "fact_expr": "C: Parabola;p: Number;O: Origin;F: Point;M: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C)=F;PointOnCurve(M, C);IsTangent(CircumCircle(TriangleOf(O,F,M)),Directrix(C));Area(CircumCircle(TriangleOf(O,F,M))) = 9*pi", "coords": {} }, { "index": 2803, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2803.png", "answer": "(-sqrt(3), -3/2) + (3/2, sqrt(3))", "fact_expr": "l: Line;C: Hyperbola;b: Number;b>0;Expression(C) = (x^2/4 - y^2/b^2 = 1);Expression(l) = (y = sqrt(3)*(x -1 ));NumIntersection(l, RightPart(C)) = 2;k: Number;Slope(Asymptote(C)) = k", "coords": {} }, { "index": 2804, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2804.png", "answer": "sqrt(5)/2\ny=pm*(1/2)*x", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/4 - y^2 = 1)", "coords": {} }, { "index": 2805, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2805.png", "answer": "5", "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/25 + y^2/16 = 1);Coordinate(P) = (x1, y1);PointOnCurve(P, G);x1:Number;y1:Number", "coords": {} }, { "index": 2806, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2806.png", "answer": "2*sqrt(5)/5", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;l: Line;PointOnCurve(F, l);Slope(l) = sqrt(2);A: Point;B: Point;Intersection(l, G) = {A, B};S: Circle;IsDiameter(LineSegmentOf(A, B), S) = True;M: Point;N: Point;Intersection(S, yAxis) = {M, N};H: Point;MidPoint(LineSegmentOf(A,B)) = H", "coords": {} }, { "index": 2807, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2807.png", "answer": "(0,1/4)", "fact_expr": "G: Parabola;Expression(G) = (x^2 = y)", "coords": {} }, { "index": 2808, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2808.png", "answer": "sqrt(3)", "fact_expr": "l: Line;G: Ellipse;b: Number;B: Point;F2: Point;A: Point;F1: Point;0 < b;b < 2;Expression(G) = (x^2/4 + y^2/b^2 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(F1, l);Intersection(l, G) = {A, B};Max(Abs(LineSegmentOf(A, F2)) + Abs(LineSegmentOf(B, F2))) = 5", "coords": {} }, { "index": 2809, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.23606797749979, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2809.png", "answer": "6", "fact_expr": "G: Hyperbola;H: Parabola;p: Number;Expression(G) = (x^2/5 - y^2/4 = 1);p>0;Expression(H) = (y^2 = 2*(p*x));Focus(H) = RightFocus(G)", "coords": {} }, { "index": 2810, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2810.png", "answer": "-9", "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (x^2/16 + y^2/m = 1);Eccentricity(G) = 5/4", "coords": {} }, { "index": 2811, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 7.0, "b": 4.898979485566356, "major_axis": "x", "x_coef": 49.0, "y_coef": 24.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2811.png", "answer": "10", "fact_expr": "C: Ellipse;Expression(C) = (x^2/49 + y^2/24 = 1)", "coords": {} }, { "index": 2812, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2813, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2814, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2814.png", "answer": "4", "fact_expr": "l: Line;G: Hyperbola;B: Point;A: Point;Expression(G) = (-x^2/8 + y^2/16 = 1);PointOnCurve(OneOf(Focus(G)),l);IsPerpendicular(l,RealAxis(G));Intersection(l, G) = {A, B}", "coords": {} }, { "index": 2815, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2815.png", "answer": "y^2 = 16*x", "fact_expr": "G: Ellipse;H: Line;M: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Expression(H) = (x = -4);Distance(M, RightFocus(G)) = Distance(M, H)", "coords": {} }, { "index": 2816, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2816.png", "answer": "[3, 4]", "fact_expr": "C: Ellipse;F1: Point;A: Point;F2: Point;B: Point;lambda: Number;Expression(C) = (x^2/4 + y^2/3 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(A, C);PointOnCurve(B, C);VectorOf(F1, A) = lambda*VectorOf(F2, B);In(lambda, [1/3, 3])", "coords": {} }, { "index": 2817, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2817.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;m: Number;m>0;Expression(G) = (x^2/2 - y^2/m = 1);Distance(OneOf(Vertex(G)), Asymptote(G)) = 1", "coords": {} }, { "index": 2818, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2819, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2819.png", "answer": "5", "fact_expr": "G: Parabola;p: Number;H: Circle;m: Number;A: Point;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (8*y - 4*x + x^2 + y^2 + 19 = 0);Coordinate(A) = (3, m);Focus(G) = F;PointOnCurve(A, G);PointOnCurve(Center(H), G)", "coords": {} }, { "index": 2820, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2821, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2822, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2823, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2824, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2825, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2825.png", "answer": "sqrt(10)", "fact_expr": "G: Parabola;Q: Point;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(Q) = (0, 3);PointOnCurve(P, G)", "coords": { "Q": [ 0.0, 3.0 ] } }, { "index": 2826, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2827, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2827.png", "answer": "pm*sqrt(2)/2", "fact_expr": "E: Parabola;p: Number;p > 0;Expression(E) = (y^2 = 2*p*x);F: Point;Focus(E) = F;L: Line;PointOnCurve(F, L);A: Point;B: Point;Intersection(L, E) = {A, B};H: Circle;Expression(H) = ((x-p/2)^2 + y^2 = p^2);C: Point;D: Point;Abs(LineSegmentOf(A, B)) = 3*Abs(LineSegmentOf(C, D))", "coords": {} }, { "index": 2828, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2828.png", "answer": "sqrt(3)/2", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);H: Line;Expression(H) = (y = sqrt(3)*x)", "coords": {} }, { "index": 2829, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2829.png", "answer": "sqrt(3)/2", "fact_expr": "G: Ellipse;b: Number;a: Number;B1: Point;B2:Point;P: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LowerVertex(G)=B1;UpperVertex(G)=B2;PointOnCurve(P, G);Slope(LineOf(P,B1))=1/4;Slope(LineOf(P,B2))=-1", "coords": {} }, { "index": 2830, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2831, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2831.png", "answer": "2", "fact_expr": "H: Ellipse;Expression(H) = (x^2/5 + y^2/p^2 = 1);G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;PointOnCurve(LeftFocus(H), Directrix(G)) = True", "coords": {} }, { "index": 2832, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 10.0, "b": 6.0, "major_axis": "x", "x_coef": 100.0, "y_coef": 36.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2832.png", "answer": "34/5", "fact_expr": "G: Ellipse;Expression(G) = (x^2/100 + y^2/36 = 1);P: Point;PointOnCurve(P, G);Distance(P, RightDirectrix(G)) = 17/2", "coords": {} }, { "index": 2833, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2833.png", "answer": "{(6,6*sqrt(2)),(6,-6*sqrt(2))}", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 12*x);P:Point;PointOnCurve(P,G);Distance(P,Focus(G))=9", "coords": {} }, { "index": 2834, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2834.png", "answer": "3*sqrt(2)/4", "fact_expr": "G: Parabola;p: Number;F: Point;A: Point;B: Point;p>0;Expression(G) = (y^2 = 2*p*x);Coordinate(A) = (0, 2);Focus(G) = F;MidPoint(LineSegmentOf(F,A))=B;PointOnCurve(B, G)", "coords": { "A": [ 0.0, 2.0 ] } }, { "index": 2835, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2835.png", "answer": "5/3", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (1, 4/3);PointOnCurve(P, Asymptote(G))", "coords": { "P": [ 1.0, 1.3333333333333333 ] } }, { "index": 2836, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2837, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/train/hyperbola/problem_2837.png", "answer": "8", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Inclination(OneOf(Asymptote(G))) = ApplyUnit(45, degree);H: Point;Coordinate(H) = (3, 1);PointOnCurve(H, G)", "coords": { "H": [ 3.0, 1.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 2838, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2838.png", "answer": "{0,pm*1}", "fact_expr": "l: Line;G: Parabola;H: Point;Expression(G) = (y^2 = 4*x);Coordinate(H) = (-1, 0);PointOnCurve(H,l);NumIntersection(l, G) = 1", "coords": { "H": [ -1.0, 0.0 ] } }, { "index": 2839, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2839.png", "answer": "(pm*5,0)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1)", "coords": {} }, { "index": 2840, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2841, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2842, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.8284271247461903, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2842.png", "answer": "3", "fact_expr": "H: Ellipse;Expression(H) = (x^2/12 + y^2/m = 1);m: Number;G: Hyperbola;Expression(G) = (x^2 - 8*y^2 = 8);Focus(H) =Focus(G)", "coords": {} }, { "index": 2843, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2843.png", "answer": "2", "fact_expr": "l: Line;G: Parabola;p: Number;M: Point;A: Point;B: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(M) = (2, 1);Slope(l)= 2;Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A,B))=M", "coords": { "M": [ 2.0, 1.0 ] } }, { "index": 2844, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 2.8284271247461903, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2844.png", "answer": "1, 9", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/8 = 1);P: Point;PointOnCurve(P, G);F1: Point;F2: Point;OneOf(Focus(G)) = F1;OneOf(Focus(G)) = F2;Negation(F1=F2);Distance(P, F1) = 10;N: Point;MidPoint(LineSegmentOf(P, F1)) = N;O: Origin", "coords": {} }, { "index": 2845, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2845.png", "answer": "6", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(m*x));m: Number;m>0;F: Point;Focus(G) = F;L: Line;PointOnCurve(F, L) = True;Intersection(L, G) = {A, B};A: Point;B: Point;H: Circle;IsDiameter(LineSegmentOf(A,B),H) = True;Expression(H) = (x^2+y^2-2*x-2*t*y+t^2-15=0);t: Number", "coords": {} }, { "index": 2846, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2846.png", "answer": "2*sqrt(3)/3", "fact_expr": "C: Parabola;O: Origin;P: Point;F: Point;Expression(C) = (y^2 = 4*x);PointOnCurve(P, C);Focus(C) = F", "coords": {} }, { "index": 2847, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2847.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;I: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2;Incenter(TriangleOf(P, F1, F2)) = I;2*(Area(TriangleOf(P, F1, I)) - Area(TriangleOf(P, F2, I))) = Area(TriangleOf(F1, F2, I))", "coords": {} }, { "index": 2848, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2848.png", "answer": "16/3", "fact_expr": "l: Line;G: Parabola;P: Point;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Coordinate(P) = (1, 0);Inclination(l) = ApplyUnit(60, degree);Intersection(l, G) = {A, B};IsChordOf(LineSegmentOf(A, B), G);PointOnCurve(P, l)", "coords": { "P": [ 1.0, 0.0 ] } }, { "index": 2849, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2849.png", "answer": "1", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 6*x);H: Circle;P: Point;Coordinate(P) = (-2, 1);Center(H) = P;Length(InterceptChord(Directrix(G), H)) = sqrt(3)", "coords": { "P": [ -2.0, 1.0 ] } }, { "index": 2850, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2850.png", "answer": "4*sqrt(2)", "fact_expr": "G: Parabola;p: Number;O: Origin;M: Point;F: Point;y1:Number;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(M) = (4, y1);PointOnCurve(M, G);Focus(G) = F;Distance(M, F) = 5", "coords": {} }, { "index": 2851, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2852, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2852.png", "answer": "sqrt(2)/2", "fact_expr": "G: Parabola;p: Number;B: Point;A: Point;F: Point;M: Point;N: Point;l: Line;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;Directrix(G) = l;PointOnCurve(A, G);PointOnCurve(B, G);AngleOf(A, F, B) = pi/2;MidPoint(LineSegmentOf(A, B)) = M;Projection(M, l) = N", "coords": {} }, { "index": 2853, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2853.png", "answer": "4", "fact_expr": "C: Parabola;Expression(C)=(x^2 = 2*(p*y));F:Point;Focus(C)=F;M: Point;N: Point;L:Line;PointOnCurve(F,L);Intersection(L, C) = {M, N};Abs(LineSegmentOf(M, N)) = 8;p:Number;p>0", "coords": {} }, { "index": 2854, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2854.png", "answer": "y = pm*x", "fact_expr": "C: Hyperbola;a: Number;b: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Length(RealAxis(C)) = Length(ImageinaryAxis(C))", "coords": {} }, { "index": 2855, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2856, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.8284271247461903, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2856.png", "answer": "9*sqrt(6)+6", "fact_expr": "C: Hyperbola;A: Point;P: Point;F: Point;Expression(C) = (x^2 - y^2/8 = 1);Coordinate(A) = (0, 6*sqrt(6));RightFocus(C) = F;PointOnCurve(P, LeftPart(C));PointOnCurve(A, yAxis)", "coords": { "A": [ 0.0, 14.696938456699067 ] } }, { "index": 2857, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2631799883090584, "b": 2.5275303393962245, "orientation": "horizontal" }, "optimization": { "final_loss": 4.481942266905428e-06, "converged": true, "iterations": 95 }, "output_path": "results/train/hyperbola/problem_2857.png", "answer": "y^2/4-x^2=1", "fact_expr": "G: Hyperbola;Expression(Asymptote(G)) = (y = pm*2*x);Coordinate(OneOf(Focus(G))) = (0, sqrt(5))", "coords": {} }, { "index": 2858, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -1.0, 0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_2858.png", "answer": "x^2/9+y^2/8=1", "fact_expr": "M: Circle;M1:Point;C1:Circle;C2:Circle;Expression(C1)=((x+1)^2+y^2=1);Expression(C2)=((x-1)^2+y^2=25);IsOutTangent(M,C1);IsInTangent(M,C2);Center(M)=M1", "coords": {} }, { "index": 2859, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2859.png", "answer": "(pm*2, 0)", "fact_expr": "G: Ellipse;M: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2/3 = 1);PointOnCurve(M, G);LeftFocus(G) = F1;RightFocus(G) = F2;Abs(LineSegmentOf(M, F1)) = 3*Abs(LineSegmentOf(M, F2))", "coords": {} }, { "index": 2860, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 3.3559478327666556, "b": 2.4609801193339877, "major_axis": "y", "x_coef": 5.999999999999999, "y_coef": 10.000000000000002 }, "optimization": { "final_loss": 2.7979567391100454e-05, "converged": true, "iterations": 81 }, "output_path": "results/train/ellipse/problem_2860.png", "answer": "y^2/10 + x^2/6 = 1", "fact_expr": "G: Ellipse;H: Point;I: Point;O: Origin;Coordinate(H) = (-3/2, 5/2);Coordinate(I) = (sqrt(3), sqrt(5));Center(G) = O;SymmetryAxis(G)=axis;PointOnCurve(H,G);PointOnCurve(I,G)", "coords": { "H": [ -1.5, 2.5 ], "I": [ 1.7320508075688772, 2.23606797749979 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 2861, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2861.png", "answer": "1/2", "fact_expr": "C: Ellipse;b: Number;a: Number;F1: Point;D: Point;A: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) =F1;RightFocus(C)=F2;UpperVertex(C)=A;PointOnCurve(F1,LineSegmentOf(F1,D));IsPerpendicular(LineSegmentOf(F1,D),LineSegmentOf(A,F2));FootPoint(LineSegmentOf(F1,D),LineSegmentOf(A,F2))=D;Abs(LineSegmentOf(F2, D)) = Abs(LineSegmentOf(F1, F2))/2", "coords": {} }, { "index": 2862, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2862.png", "answer": "{1,4/5}", "fact_expr": "C: Ellipse;Expression(C) = (x^2/9 + y^2/b^2 = 1);b: Number;3 > b;b > 0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;O: Origin;PointOnCurve(P, C) = True;P: Point;Intersection(OverlappingLine(LineSegmentOf(P,F2)),C) = A;A: Point;Abs(LineSegmentOf(O, F1)) = Abs(LineSegmentOf(O, A));Area(TriangleOf(O,F1,A)) = 2", "coords": {} }, { "index": 2863, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2864, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2865, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2865.png", "answer": "[1, 3/2)", "fact_expr": "l: Line;C: Hyperbola;A: Point;F: Point;Expression(C) = (x^2 - y^2/3 = 1);RightFocus(C) = F;PointOnCurve(F, l);Intersection(l, RightPart(C)) = A;NumIntersection(l, LeftPart(C)) = 1", "coords": {} }, { "index": 2866, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2866.png", "answer": "sqrt(13)/2", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;G: Circle;F1: Point;T: Point;P: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2 + y^2 = a^2);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, l);TangentPoint(l, G) = T;Intersection(l, RightPart(C)) = P;VectorOf(F1, P) = 3*VectorOf(F1, T)", "coords": {} }, { "index": 2867, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2867.png", "answer": "{4*sqrt(3)/3,4*sqrt(3)}", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/16 + y^2/4 = 1);PointOnCurve(P, G);LeftFocus(G)=F1;RightFocus(G)=F2;Radius(CircumCircle(TriangleOf(F1,P,F2)))=4", "coords": {} }, { "index": 2868, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2869, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2869.png", "answer": "4/3", "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C);AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 2870, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2870.png", "answer": "4", "fact_expr": "G: Ellipse;Expression(G) = (x^2 + y^2/m = 1);m: Number;PointOnCurve(Focus(G), yAxis);Length(MajorAxis(G)) = 2*Length(MinorAxis(G))", "coords": {} }, { "index": 2871, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2871.png", "answer": "2", "fact_expr": "C: Parabola;p: Number;M: Point;N: Point;H:LineSegment;p>0;Expression(C) = (y^2 = 2*p*x);Coordinate(P) = (1, 1);IsChordOf(H,C);Intersection(H, C) = {M, N};Abs(LineSegmentOf(M, N)) = sqrt(15);MidPoint(H)=P;P:Point", "coords": { "P": [ 1.0, 1.0 ] } }, { "index": 2872, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2872.png", "answer": "y^2 = 32*x/5", "fact_expr": "C2: Parabola;p: Number;C1: Circle;A: Point;B: Point;p > 0;Expression(C2) = (y^2 = 2*(p*x));Expression(C1) = (x^2 + (y - 2)^2 = 4);Intersection(C1, C2) = {A, B};Abs(LineSegmentOf(A, B)) = 8*sqrt(5)/5", "coords": {} }, { "index": 2873, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2873.png", "answer": "{17,1}", "fact_expr": "G: Hyperbola;H: Point;I: Point;P: Point;Expression(G) = (x^2/16 - y^2/9 = 1);Coordinate(H) = (5, 0);Coordinate(I) = (-5, 0);PointOnCurve(P, G);Distance(P, H) = 9", "coords": { "H": [ 5.0, 0.0 ], "I": [ -5.0, 0.0 ] } }, { "index": 2874, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2874.png", "answer": "6", "fact_expr": "H: Circle;Expression(H) = (x^2 + y^2 - 9 = 0);G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;IsTangent(H, Directrix(G))", "coords": {} }, { "index": 2875, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.581065485469099, "b": 1.581065485469099, "orientation": "horizontal" }, "optimization": { "final_loss": 1.6812139285133018e-06, "converged": true, "iterations": 95 }, "output_path": "results/train/hyperbola/problem_2875.png", "answer": "x^2/4-y^2=1", "fact_expr": "G: Hyperbola;F1: Point;Coordinate(F1) = (-sqrt(5), 0);F2: Point;Coordinate(F2) = (sqrt(5), 0);Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2));Abs(LineSegmentOf(P, F1))*Abs(LineSegmentOf(P, F2)) = 2", "coords": { "F1": [ -2.23606797749979, 0.0 ], "F2": [ 2.23606797749979, 0.0 ] } }, { "index": 2876, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2877, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 1.0, -0.0 ], "radius": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_2877.png", "answer": "x^2/4+y^2/3=1", "fact_expr": "G: Ellipse;PointOnCurve(Focus(G), xAxis);Eccentricity(G) = 1/2;Length(MajorAxis(G)) = Radius(H);H: Circle;Expression(H) = (-2*x + x^2 + y^2 - 15 = 0)", "coords": {} }, { "index": 2878, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2879, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2880, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2880.png", "answer": "2*sqrt(3)/3", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - 16*y^2/p^2 = 1);p: Number;p>0;H: Parabola;Expression(H) = (y^2 = 2*p*x);PointOnCurve(LeftFocus(G), Directrix(H))", "coords": {} }, { "index": 2881, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2881.png", "answer": "2", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(C) = F;IsDiameter(LineSegmentOf(O, F), G) = True;G: Circle;O: Origin;L1: Line;L2: Line;Asymptote(C) = {L1, L2};Intersection(G, L1) = A;Intersection(G, L2) = B;Negation(A = O);Negation(B = O);A: Point;B: Point;Area(TriangleOf(O, A, B)) = a*b/4", "coords": {} }, { "index": 2882, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 7.0, "b": 4.898979485566356, "major_axis": "x", "x_coef": 49.0, "y_coef": 24.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2882.png", "answer": "12/5", "fact_expr": "C: Ellipse;Expression(C) = (x^2/49 + y^2/24 = 1);LeftFocus(C) = F1;F1: Point;RightFocus(C) = F2;F2: Point;P: Point;PointOnCurve(P,C) = True;Quadrant(P) = 1;IsPerpendicular(LineSegmentOf(P,F1),LineSegmentOf(P,F2)) = True;Intersection(OverlappingLine(LineSegmentOf(P,F2)),C) = Q;Q: Point", "coords": {} }, { "index": 2883, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2884, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2884.png", "answer": "(\\sqrt{5}, +\\infty)", "fact_expr": "l: Line;G: Hyperbola;b: Number;a: Number;e: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Slope(l)=2;PointOnCurve(RightFocus(G),l);IsIntersect(l,LeftPart(G));IsIntersect(l,RightPart(G));Eccentricity(G) = e", "coords": {} }, { "index": 2885, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2886, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.449489742783178, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2886.png", "answer": "(3,0)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - 2*y^2 = 6)", "coords": {} }, { "index": 2887, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2887.png", "answer": "3/5", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;H: Line;PointOnCurve(F1, H);IsIntersect(H, G);M: Point;N: Point;Min(Length(InterceptChord(H, G))) = Abs(LineSegmentOf(M, N));Abs(LineSegmentOf(M, N)) = 32/5;Perimeter(TriangleOf(M, F2, N)) = 20;e: Number;Eccentricity(G) = e", "coords": {} }, { "index": 2888, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2888.png", "answer": "6", "fact_expr": "C: Parabola;E: Ellipse;A: Point;B: Point;O:Origin;Expression(C) = (y^2 = 8*x);Center(E)=O;Eccentricity(E)=1/2;RightFocus(E) = Focus(C);Intersection(Directrix(C),E)={A,B}", "coords": {} }, { "index": 2889, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2889.png", "answer": "5/4", "fact_expr": "G: Hyperbola;Expression(G)=(x^2/16 - y^2/9 = 1)", "coords": {} }, { "index": 2890, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2890.png", "answer": "8", "fact_expr": "C: Parabola;Expression(C) = (x^2 = 8*y);P: Point;PointOnCurve(P, Directrix(C));A: Point;B: Point;TangentOfPoint(P, C) = {LineOf(P, A), LineOf(P, B)};TangentPoint(LineOf(P, A), C) = A;TangentPoint(LineOf(P, B), C) = B", "coords": {} }, { "index": 2891, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2891.png", "answer": "3/4", "fact_expr": "G: Hyperbola;P: Point;M: Point;N: Point;Expression(G) = (x^2/4 - y^2/3 = 1);Negation(P=M);Negation(P=N);Endpoint(RealAxis(G)) = {M, N};PointOnCurve(P, G)", "coords": {} }, { "index": 2892, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2893, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2894, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2894.png", "answer": "sqrt(10)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l: Line;Directrix(C) = l;A: Point;Coordinate(A) = (0, -3);P: Point;PointOnCurve(P, C);M: Point;IsPerpendicular(LineSegmentOf(P, M), l);FootPoint(LineSegmentOf(P, M), l) = M", "coords": { "A": [ 0.0, -3.0 ] } }, { "index": 2895, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2895.png", "answer": "[1,2]", "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2 + y^2/2 = 1);Coordinate(P) = (m, n);PointOnCurve(P, G);m:Number;n:Number", "coords": {} }, { "index": 2896, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2897, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_focus_mismatch", "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2897.png", "answer": "1", "fact_expr": "C: Ellipse;a: Number;N: Point;M: Point;F: Point;a>0;Expression(C) = (y^2 + x^2/a^2 = 1);Coordinate(F) = (-1, 0);Coordinate(N) = (3, 2);OneOf(Focus(C))=F;PointOnCurve(M,C);WhenMax(Abs(LineSegmentOf(M, F)) + Abs(LineSegmentOf(M, N)))", "coords": { "F": [ -1.0, 0.0 ], "N": [ 3.0, 2.0 ] }, "validation_reasons": [ "ellipse_focus_mismatch" ] }, { "index": 2898, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2899, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2899.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "l: Line;G: Hyperbola;b: Number;M: Point;P: Point;A: Point;B: Point;F1:Point;F2: Point;l1:Line;l2:Line;x1:Number;y1:Number;Expression(G) = (x^2/4 - y^2/b^2 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(F2,l);IsPerpendicular(l,xAxis);Asymptote(G)={l1,l2};Intersection(l,l1)=A;Intersection(l,l2)=B;Coordinate(M) = (4, 1);Coordinate(P) = (x1, y1);Abs(LineSegmentOf(A, B)) = 3*sqrt(5);PointOnCurve(P, G)", "coords": { "M": [ 4.0, 1.0 ] } }, { "index": 2900, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2900.png", "answer": "sqrt(5)", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F2: Point;F1: Point;N: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, LeftPart(C));LeftFocus(C) = F1;RightFocus(C) = F2;DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;MidPoint(LineSegmentOf(P, F2)) = N;Quadrant(N) = 1;PointOnCurve(N, OneOf(Asymptote(C)))", "coords": {} }, { "index": 2901, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 2.6457513110645907, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2901.png", "answer": "pm*sqrt(15)/7", "fact_expr": "G: Hyperbola;H: Circle;F1: Point;P: Point;M: Point;Expression(G) = (x^2/9 - y^2/7 = 1);Expression(H) = (x^2 + y^2 = 16);LeftFocus(G) = F1;PointOnCurve(P,RightPart(G));Intersection(LineSegmentOf(P,F1),H) = M;VectorOf(F1, M) = VectorOf(M, P)", "coords": {} }, { "index": 2902, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2902.png", "answer": "sqrt(5)", "fact_expr": "C: Hyperbola;b: Number;a: Number;E: Circle;c: Number;P: Point;F: Point;M: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(E) = (y^2 + (-c/3 + x)^2 = b^2/9);Coordinate(F) = (c, 0);RightFocus(C) = F;PointOnCurve(P, LeftPart(C));TangentPoint(LineOf(F, P), E) = M;VectorOf(P, M) = 2*VectorOf(M, F)", "coords": {} }, { "index": 2903, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2903.png", "answer": "10", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);H: Line;PointOnCurve(Focus(G), H);A: Point;B: Point;x1: Number;x2: Number;y1: Number;y2: Number;Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);Intersection(H, G) = {A,B};x1 + x2 =6", "coords": {} }, { "index": 2904, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2904.png", "answer": "x^2/20+y^2/4=1", "fact_expr": "G: Ellipse;H: Point;C:Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);Coordinate(H) = (sqrt(5), sqrt(3));Focus(G)=Focus(C);PointOnCurve(H,C)", "coords": { "H": [ 2.23606797749979, 1.7320508075688772 ] } }, { "index": 2905, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2905.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;F: Point;Focus(G) = F;H: Line;PointOnCurve(F,H) = True;Intersection(H,G) = {M,N};M: Point;N: Point;Quadrant(M) = 1;Slope(LineOf(M,N)) = sqrt(3);XCoordinate(M) = 6", "coords": {} }, { "index": 2906, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2906.png", "answer": "3", "fact_expr": "C: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Focus(C)={F1,F2};PointOnCurve(P, C);IsPerpendicular(VectorOf(P, F2),VectorOf(P, F1));Area(TriangleOf(P, F1, F2)) = 9", "coords": {} }, { "index": 2907, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2907.png", "answer": "4", "fact_expr": "G: Parabola;a: Number;H: Line;A: Point;B: Point;x1: Number;x2: Number;y1: Number;y2: Number;Expression(G) = (y^2 = a*x);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);PointOnCurve(Focus(G), H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 10;x1 + x2 = 8", "coords": {} }, { "index": 2908, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2909, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2909.png", "answer": "-1", "fact_expr": "G: Parabola;H: Line;a: Real;Expression(G) = (y^2 = 4*x);Expression(H) = (a*x - y + 1 = 0);PointOnCurve(Focus(G), H)", "coords": {} }, { "index": 2910, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2910.png", "answer": "sqrt(3)-1", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G) = True;P: Point;AngleOf(P, F1, F2) = ApplyUnit(30, degree);AngleOf(P, F2, F1) = ApplyUnit(60, degree);e: Number;Eccentricity(G) = e", "coords": {} }, { "index": 2911, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "major_axis": "x", "x_coef": 3.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2911.png", "answer": "2*sqrt(2)", "fact_expr": "G: Ellipse;F1: Point;F2: Point;Expression(G) = (x^2/3 + y^2 = 1);Focus(G) = {F1, F2}", "coords": {} }, { "index": 2912, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2913, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_focus_mismatch", "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2913.png", "answer": "x^2/8+y^2/4=1", "fact_expr": "G: Ellipse;Expression(G) = (y^2/n^2 + x^2/m^2 = 1);m: Number;n: Number;m>0;n>0;RightFocus(G) = Z;Eccentricity(G) = sqrt(2)/2;Z: Point;Coordinate(Z) = (2,0)", "coords": { "Z": [ 2.0, 0.0 ] }, "validation_reasons": [ "ellipse_focus_mismatch" ] }, { "index": 2914, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2915, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2915.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;A: Point;PointOnCurve(A, G);Distance(A, yAxis) = 3", "coords": {} }, { "index": 2916, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2916.png", "answer": "(0, 0)", "fact_expr": "G: Ellipse;H: Circle;B: Point;P: Point;A: Point;M: Point;Q: Point;Expression(G) = (x^2/4 + y^2/2 = 1);LeftVertex(G)=A;RightVertex(G)=B;IsPerpendicular(LineSegmentOf(M, B),LineSegmentOf(A, B));Intersection(LineSegmentOf(A,M),G)=P;PointOnCurve(Q,xAxis);Negation(Q = B);Negation(Q=A);IsDiameter(LineSegmentOf(M,P),H);PointOnCurve(Intersection(LineSegmentOf(B,P),LineSegmentOf(M,Q)),H)", "coords": {} }, { "index": 2917, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2917.png", "answer": "2", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;F1: Point;F2: Point;P: Point;e: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;Eccentricity(C) = e;IsDiameter(LineSegmentOf(F1, F2), G);L1: Line;L2: Line;Intersection(G, L1) = P;Quadrant(P) = 1;Q: Point;MidPoint(LineSegmentOf(P, F1)) = Q;PointOnCurve(Q, L2);OneOf(Asymptote(G)) = L1;OneOf(Asymptote(G)) = L2;Negation(L1=L2)", "coords": {} }, { "index": 2918, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2918.png", "answer": "1", "fact_expr": "G: Parabola;A: Point;B: Point;Expression(G) = (y^2 = 8*x);IsChordOf(LineSegmentOf(A,B),G);IsPerpendicular(LineSegmentOf(A,B),xAxis);Abs(LineSegmentOf(A,B))=4*sqrt(6)", "coords": {} }, { "index": 2919, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2920, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2921, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2922, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2922.png", "answer": "x^2/16 + y^2/11 = 1", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/4 = 1);H: Point;Coordinate(H) = (4, 0);Z: Ellipse;Focus(G) = Focus(Z);PointOnCurve(H, Z)", "coords": { "H": [ 4.0, 0.0 ] } }, { "index": 2923, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2923.png", "answer": "-3", "fact_expr": "G: Parabola;A: Point;B: Point;O: Origin;Expression(G) = (y^2 = 4*x);IsChordOf(LineSegmentOf(A,B),G);PointOnCurve(F,LineSegmentOf(A,B));Focus(G)=F;F: Point", "coords": {} }, { "index": 2924, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2924.png", "answer": "1/2", "fact_expr": "C: Parabola;Expression(C) = (y = x^2);M: Point;PointOnCurve(M,C) = True;Intersection(L1 , xAxis) = A;Intersection(L1 , yAxis) = B;A: Point;B: Point;L1: Line;TangentOnPoint(M,C) = L1;Negation(M = Vertex(C))", "coords": {} }, { "index": 2925, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2925.png", "answer": "a^2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;M: Point;PointOnCurve(M, G);H: Line;PointOnCurve(M, H);IsParallel(H, RealAxis(G));P: Point;Q: Point;Intersection(H, Asymptote(G)) = {P, Q}", "coords": {} }, { "index": 2926, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2926.png", "answer": "(\\sqrt{2}-1,1)", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;c: Number;Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);a/Sin(AngleOf(P, F1, F2)) = c/Sin(AngleOf(P, F2, F1))", "coords": {} }, { "index": 2927, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2927.png", "answer": "1+2*sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;M: Point;N: Point;PointOnCurve(M, Asymptote(G));PointOnCurve(N, LeftPart(G))", "coords": {} }, { "index": 2928, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2928.png", "answer": "-3", "fact_expr": "l: Line;G: Parabola;H: Point;P: Point;Q: Point;x1: Number;x2: Number;y1: Number;y2: Number;Expression(G) = (y^2 = 4*x);Coordinate(H) = (1, 0);Coordinate(P) = (x1, y1);Coordinate(Q) = (x2, y2);PointOnCurve(H, l);Intersection(l, G) = {P, Q}", "coords": { "H": [ 1.0, 0.0 ] } }, { "index": 2929, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2929.png", "answer": "1/2", "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;B: Point;F: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F;RightVertex(G) = A;PointOnCurve(B, G);IsPerpendicular(LineSegmentOf(B,F),xAxis);Intersection(LineOf(A,B),yAxis)=P;VectorOf(A,P)=2*VectorOf(P,B);P:Point", "coords": {} }, { "index": 2930, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2930.png", "answer": "{(-oo, -3), (0, +oo)}", "fact_expr": "l: Line;C: Parabola;G: Circle;M: Point;N: Point;t: Real;k: Number;Expression(C) = (x^2 = 4*y);Expression(G) = (x^2 + (y + 1)^2 = 1);Expression(l) = (y = k*x + t);Intersection(l, C) = {M, N};IsTangent(l, G);Negation(M = N)", "coords": {} }, { "index": 2931, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2931.png", "answer": "13", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);F1: Point;F2: Point;Focus(G) = {F1,F2};P: Point;PointOnCurve(P,G) = True;Abs(LineSegmentOf(P, F1)) = 7", "coords": {} }, { "index": 2932, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2933, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2933.png", "answer": "20", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);F: Point;Focus(G) = F;H: Line;PointOnCurve(F, H);A: Point;B: Point;x1: Number;x2: Number;y1: Number;y2: Number;Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);Intersection(H, G) = {A, B}", "coords": {} }, { "index": 2934, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2934.png", "answer": "2", "fact_expr": "G: Parabola;p: Number;H: Line;O: Origin;A: Point;B: Point;p>0;Expression(G) = (y^2 = 2*p*x);Expression(H) = (y = x - 4);Intersection(H, G) = {A, B};IsPerpendicular(LineSegmentOf(O,A),LineSegmentOf(O,B))", "coords": {} }, { "index": 2935, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2935.png", "answer": "3/4", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);F2: Point;RightFocus(G) = F2;H: Line;PointOnCurve(F2, H);M: Point;N: Point;Intersection(H, G) = {M, N};m: Number;n: Number;Abs(LineSegmentOf(F2, M)) = m;Abs(LineSegmentOf(F2, N)) = n", "coords": {} }, { "index": 2936, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2936.png", "answer": "sqrt(5)-1", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);P: Point;PointOnCurve(P, G);M: Point;Projection(P, xAxis) = M;A: Point;Coordinate(A) = (2, 0)", "coords": { "A": [ 2.0, 0.0 ] } }, { "index": 2937, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 0.7841060161590576, "b": 0.3689577579498291, "orientation": "horizontal" }, "optimization": { "final_loss": 5.885034397579147e-07, "converged": true, "iterations": 122 }, "output_path": "results/train/hyperbola/problem_2937.png", "answer": "3*sqrt(5)/5", "fact_expr": "G: Hyperbola;b: Number;a: Number;F1: Point;F2: Point;P: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-sqrt(3)/2, 0);Coordinate(F2) = (sqrt(3)/2, 0);Focus(G) = {F1,F2};Quadrant(P) = 1;PointOnCurve(P, G);Tan(AngleOf(P,F1,F2))=1/2;Tan(AngleOf(P, F2, F1)) = -2", "coords": { "F1": [ -0.8660254037844386, 0.0 ], "F2": [ 0.8660254037844386, 0.0 ] } }, { "index": 2938, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2938.png", "answer": "x^2/9 - y^2/16 = 1", "fact_expr": "G: Hyperbola;a: Number;b: Number;H: Circle;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;IsDiameter(LineSegmentOf(F1, F2), H);Coordinate(OneOf(Intersection(H, Asymptote(G)))) = (3, 4)", "coords": {} }, { "index": 2939, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2939.png", "answer": "5/3", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;A:Point;F1: Point;P: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, RightPart(G));LeftFocus(G) = F1;RightFocus(G) = F2;Center(H)=O;O:Origin;Radius(H)=a;TangentPoint(LineOf(P,F1),H)=A;PointOnCurve(F2,PerpendicularBisector(LineSegmentOf(P,F1)))", "coords": {} }, { "index": 2940, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2941, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2941.png", "answer": "6", "fact_expr": "G: Hyperbola;a: Number;H: Line;a>0;Expression(G) = (-y^2/4 + x^2/a^2 = 1);Expression(H) = (y = (3/2)*x);IsPerpendicular(OneOf(Asymptote(G)), H)", "coords": {} }, { "index": 2942, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2943, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.8284271247461903, "major_axis": "x", "x_coef": 9.0, "y_coef": 8.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2943.png", "answer": "4*x^2-4*y^2/3=1", "fact_expr": "G: Hyperbola;H: Ellipse;P: Point;Expression(H) = (x^2/9 + y^2/8 = 1);Coordinate(P) = (2, 3*sqrt(5)/2);PointOnCurve(P, G);Focus(G)=Focus(H)", "coords": { "P": [ 2.0, 3.3541019662496847 ] } }, { "index": 2944, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2944.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;a: Number;B: Point;F: Point;O: Origin;A: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;PointOnCurve(B,LeftPart(G));Intersection(LineSegmentOf(B,F),OneOf(Asymptote(G)))=A;DotProduct((-VectorOf(O, B) + VectorOf(O, F)),VectorOf(O, A)) = 0;2*VectorOf(O, A) = VectorOf(O, B) + VectorOf(O, F)", "coords": {} }, { "index": 2945, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2945.png", "answer": "sqrt(3)-1", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;Coordinate(F2) = (1, 0);PointOnCurve(F2, G) = True;Slope(G) = 1;G: Line;Intersection(G, C) = {A, B};A: Point;B: Point;Area(TriangleOf(F1, A, B)) = sqrt(2)*b^2", "coords": { "F2": [ 1.0, 0.0 ] } }, { "index": 2946, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2947, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2947.png", "answer": "[6-sqrt(10),6+sqrt(10)]", "fact_expr": "G: Ellipse;F1: Point;A: Point;P: Point;Coordinate(F1) = (2, 0);Coordinate(A) = (1, 1);Expression(G) = (x^2/9 + y^2/5 = 1);PointOnCurve(P, G)", "coords": { "F1": [ 2.0, 0.0 ], "A": [ 1.0, 1.0 ] } }, { "index": 2948, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 6.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2948.png", "answer": "2", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 24*y);x0: Number;y0: Number;H: Point;Coordinate(H) = (x0, y0);PointOnCurve(H, G);Distance(H, Focus(G)) = 4*Distance(H, xAxis)", "coords": {} }, { "index": 2949, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2949.png", "answer": "(5,2*sqrt(2)), (5,-2*sqrt(2))", "fact_expr": "G: Parabola;H: Line;A: Point;F: Point;B: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};LineSegmentOf(A, F) = 2*LineSegmentOf(B, F)", "coords": {} }, { "index": 2950, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2950.png", "answer": "x+pm*y=0;pm*3", "fact_expr": "l: Line;C: Hyperbola;P: Point;A: Point;Q: Point;Expression(C) = (x^2 - y^2 = 1);RightVertex(C)=A;PointOnCurve(A, l);Intersection(l,J2)=Q;VectorOf(P, A) = 2*VectorOf(A, Q);J1:Line;J2:Line;Asymptote(C)={J1,J2}", "coords": {} }, { "index": 2951, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2951.png", "answer": "{sqrt(3),-sqrt(3)}", "fact_expr": "G: Parabola;p: Number;M: Point;F: Point;p>0;Expression(G) = (y^2 = 2*p*x);PointOnCurve(M, G);Focus(G) = F;Distance(M, F) = 2*p", "coords": {} }, { "index": 2952, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2952.png", "answer": "{18, 2}", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/16 - y^2/9 = 1);F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) = 10", "coords": {} }, { "index": 2953, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2954, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2954.png", "answer": "2/3", "fact_expr": "P: Point;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;PointOnCurve(P, G);Negation(Vertex(G) = P);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;c: Number;HalfFocalLength(G) = c;TangentPoint(InscribedCircle(TriangleOf(P, F1, F2)), LineSegmentOf(F1, F2)) = M;M: Point", "coords": {} }, { "index": 2955, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2955.png", "answer": "sqrt(3)+2", "fact_expr": "C: Parabola;M: Point;N: Point;P: Point;F: Point;Q: Point;D: Point;l:Line;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F, LineOf(M,N));Intersection(LineOf(M,N),C) = {M,N};PointOnCurve(P,l);Directrix(C)=l;IsPerpendicular(LineSegmentOf(P,F),LineSegmentOf(M,N));Intersection(LineSegmentOf(P,M),yAxis)=Q;PointOnCurve(Q,LineSegmentOf(Q,D));IsPerpendicular(LineSegmentOf(Q,D),LineSegmentOf(M,F));Abs(LineSegmentOf(M, D)) = 2*Abs(LineSegmentOf(F, N));FootPoint(LineSegmentOf(Q,D),LineSegmentOf(M,F))=D", "coords": {} }, { "index": 2956, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "y", "x_coef": 3.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2956.png", "answer": "8", "fact_expr": "G: Ellipse;Expression(G) = (x^2/3 + y^2/4 = 1);F: Point;UpperFocus(G) = F;Z1: Line;Z2: Line;Expression(Z1) = (x + y + 1 = 0);Expression(Z2) = (x + y - 1 = 0);A: Point;B: Point;C: Point;D: Point;Intersection(Z1, G) = {A, B};Intersection(Z2, G) = {C, D}", "coords": {} }, { "index": 2957, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2957.png", "answer": "y=-1", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 2*p*y);p: Number;p > 0;H: Circle;Expression(H) = (2*x + x^2 + y^2 - 1 = 0);PointOnCurve(Focus(G), H) = True", "coords": {} }, { "index": 2958, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2958.png", "answer": "4", "fact_expr": "G: Parabola;F: Point;E: Point;M: Point;PointOnCurve(M, G);Expression(G) = (y = -x^2/4);Coordinate(F) = (0, -1);Coordinate(E) = (1, -3)", "coords": { "F": [ 0.0, -1.0 ], "E": [ 1.0, -3.0 ] } }, { "index": 2959, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2959.png", "answer": "2", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;F1: Point;P: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2 + y^2 = a^2);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, Asymptote(C));DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;IsTangent(LineOf(P,F1),G)", "coords": {} }, { "index": 2960, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2960.png", "answer": "-4", "fact_expr": "G: Parabola;p: Number;H: Ellipse;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (x^2/9 + y^2/5 = 1);Focus(G) = LeftFocus(H)", "coords": {} }, { "index": 2961, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2961.png", "answer": "2", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*x);F: Point;Focus(C) = F;l: Line;Directrix(C) = l;A: Point;In(A, l);Intersection(LineSegmentOf(A,F), C) = B;B: Point;VectorOf(F, A) = 3*VectorOf(F, B)", "coords": {} }, { "index": 2962, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2962.png", "answer": "y=pm*16/5", "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/9 + y^2/16 = 1)", "coords": {} }, { "index": 2963, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2964, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2964.png", "answer": "sqrt(2)/2", "fact_expr": "C: Ellipse;b: Number;a: Number;G: Circle;A: Point;F: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2 + (y - 3)^2 = 1);LeftFocus(C) = F;LowerVertex(C) = A;L:Line;IsParallel(LineSegmentOf(A,F),L);Intercept(L,yAxis)=(3-sqrt(2));IsTangent(L,G)", "coords": {} }, { "index": 2965, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2966, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2967, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2967.png", "answer": "2", "fact_expr": "E: Hyperbola;b: Number;a: Number;G: Circle;c: Number;a>0;b>0;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y^2 + (-c + x)^2 = 9*a^2);HalfFocalLength(E) = c;NumIntersection(E, G)=3", "coords": {} }, { "index": 2968, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2968.png", "answer": "sqrt(14)/2", "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;O: Origin;d:Number;Expression(C)=(x^2/4+y^2/3=1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P,C);Distance(O,TangentOnPoint(P,C))=d;Abs(LineSegmentOf(P,F1))*Abs(LineSegmentOf(P,F2))=24/7", "coords": {} }, { "index": 2969, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "y", "x_coef": 3.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2969.png", "answer": "4", "fact_expr": "G: Ellipse;Expression(G) = (x^2/3 + y^2/4 = 1)", "coords": {} }, { "index": 2970, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2970.png", "answer": "2*sqrt(5)/5", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1)", "coords": {} }, { "index": 2971, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.0625, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2971.png", "answer": "(1/2, 1)", "fact_expr": "G: Parabola;Expression(G) = (x^2 = y/4);H: Line;Expression(H) = (y = 4*x - 5);P: Point;PointOnCurve(P, G);WhenMin(Distance(P, H))", "coords": {} }, { "index": 2972, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2972.png", "answer": "8", "fact_expr": "H: Circle;Expression(H) = (y^2 + (x + 1)^2 = 4);G: Parabola;Expression(G) = (y^2 = m*x);m: Number;Negation(m = 0);Intersection(H, Directrix(G)) = {A, B};A: Point;B: Point;Abs(LineSegmentOf(A, B)) = 2*sqrt(3)", "coords": {} }, { "index": 2973, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 5.656854249492381, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2973.png", "answer": "2", "fact_expr": "A: Point;x0: Number;y0: Number;Coordinate(A) = (x0, y0);PointOnCurve(A, RightPart(G));G: Hyperbola;Expression(G) = (x^2/4 - y^2/32 = 1);Distance(A, LeftFocus(G)) = 4*x0", "coords": {} }, { "index": 2974, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 2.449489742783178, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2974.png", "answer": "16", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/6 = 1);A: Point;F1: Point;B: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;l:Line;PointOnCurve(F2, l);Intersection(l, RightPart(G)) = {A, B}", "coords": {} }, { "index": 2975, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.4641016151377544, "major_axis": "x", "x_coef": 16.0, "y_coef": 12.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2975.png", "answer": "6", "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/12 = 1);H:LineSegment;PointOnCurve(F,H);Focus(G)=F;IsChordOf(H,G);WhenMin(Length(H));F:Point", "coords": {} }, { "index": 2976, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2976.png", "answer": "x^2/3-y^2=1", "fact_expr": "C: Hyperbola;b: Number;a: Number;A: Point;F: Point;B:Point;O:Origin;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;L1:Line;L2:Line;Asymptote(C)={L1,L2};PointOnCurve(A,L1);PointOnCurve(B,L2);IsPerpendicular(LineSegmentOf(A,F),xAxis);IsPerpendicular(LineSegmentOf(A,B),LineSegmentOf(O,B));IsParallel(LineSegmentOf(B,F),LineSegmentOf(O,A));FocalLength(C)=4", "coords": {} }, { "index": 2977, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2977.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;l: Line;Directrix(G) = l;A: Point;PointOnCurve(A, G);B: Point;Projection(A, l) = B;Distance(B, LineOf(A, F)) = 2*sqrt(3)", "coords": {} }, { "index": 2978, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2978.png", "answer": "2/3", "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;B: Point;F: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F;RightVertex(G)=A;UpperVertex(G)=B;Distance(F,LineOf(A,B))=(5*sqrt(14)/14)*b", "coords": {} }, { "index": 2979, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2979.png", "answer": "y^2 = 4*sqrt(3)*x", "fact_expr": "G: Parabola;H: Ellipse;F: Point;Expression(H) = (x^2/4 + y^2 = 1);RightFocus(H) = F;Focus(G) = F", "coords": {} }, { "index": 2980, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2980.png", "answer": "pm*1, pm*sqrt(5)", "fact_expr": "G: Hyperbola;H: Line;k: Number;Expression(G) = (x^2 - y^2 = 1);Expression(H) = (y = k*x - 2);NumIntersection(H, G) = 1", "coords": {} }, { "index": 2981, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2981.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "C: Ellipse;M: Point;F1: Point;Expression(C) = (x^2/4 + y^2/3 = 1);LeftFocus(C)=F1;PointOnCurve(M,C)", "coords": {} }, { "index": 2982, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_ecc_mismatch", "params": { "a": 1.9257268905639648, "b": 1.7993911322594118, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 2.5576271215184017e-15, "converged": true, "iterations": 2 }, "output_path": "results/train/ellipse/problem_2982.png", "answer": "3", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/m = 1);m: Number;m<4;Eccentricity(G) = 1/2", "coords": {}, "validation_reasons": [ "ellipse_ecc_mismatch" ] }, { "index": 2983, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2983.png", "answer": "3", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;G: Line;PointOnCurve(F, G);Slope(G) = sqrt(3);A: Point;B: Point;Intersection(G, C) = {A, B};Abs(LineSegmentOf(F, A)) > Abs(LineSegmentOf(F, B))", "coords": {} }, { "index": 2984, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2985, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2986, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2987, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2988, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2989, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2989.png", "answer": "2-sqrt(2)", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;Quadrant(P) = 1;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2))", "coords": {} }, { "index": 2990, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2991, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2991.png", "answer": "7", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/5 = 1);F: Point;LeftFocus(G) = F;A: Point;Coordinate(A) = (1, sqrt(5));P: Point;PointOnCurve(P, RightPart(G))", "coords": { "A": [ 1.0, 2.23606797749979 ] } }, { "index": 2992, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_2992.png", "answer": "2", "fact_expr": "C: Parabola;p: Number;P: Point;p>0;y0:Number;Expression(C) = (y^2 = 2*(p*x));Coordinate(P) = (1, y0);PointOnCurve(P, C);Distance(P, Focus(C)) = 2;y0>0", "coords": {} }, { "index": 2993, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_2993.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;a: Number;a>0;Expression(G) = (-y^2/9 + x^2/a^2 = 1);Eccentricity(G) = 2", "coords": {} }, { "index": 2994, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2994.png", "answer": "2", "fact_expr": "C: Parabola;G: Curve;k: Number;P: Point;F: Point;Expression(C) = (y^2 = 4*x);k>0;Expression(G) = (y = k/x);Focus(C) = F;Intersection(G, C) = P;IsPerpendicular(LineSegmentOf(P,F),xAxis)", "coords": {} }, { "index": 2995, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 2996, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2996.png", "answer": "6", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1)", "coords": {} }, { "index": 2997, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_2997.png", "answer": "a", "fact_expr": "G: Parabola;A: Point;P: Point;a: Number;0 < a;a < 1;Expression(G) = (y^2 = 2*x);Coordinate(A) = (a, 0);PointOnCurve(P, G)", "coords": {} }, { "index": 2998, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_2998.png", "answer": "sqrt(3)+1", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;O: Origin;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;FocalLength(G) = 4;PointOnCurve(P, RightPart(G));IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2));DotProduct(VectorOf(F1, P), VectorOf(F1, O)) = 6", "coords": {} }, { "index": 2999, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_2999.png", "answer": "pi/3", "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F2: Point;F1: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Focus(G)={F1,F2};PointOnCurve(P,G);Abs(LineSegmentOf(P, F2)) = (sqrt(3)/2)*Abs(LineSegmentOf(P, F1))", "coords": {} }, { "index": 3000, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3000.png", "answer": "sqrt(34)/5", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;c:Number;c>0;Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);Focus(C)={F1,F2};M: Point;Coordinate(M) = (3*c, 0);l1: Line;PointOnCurve(M, l1);IsPerpendicular(l1,Asymptote(C));P: Point;FootPoint(l1,Asymptote(C))=P;O: Origin;Abs(LineSegmentOf(O, P)) = 5*b", "coords": {} }, { "index": 3001, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3001.png", "answer": "3/2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(G) = F;A: Point;RightVertex(G) = A;B: Point;OneOf(Endpoint(ImageinaryAxis(G))) = B;Distance(F, LineOf(A, B)) = b/3", "coords": {} }, { "index": 3002, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3002.png", "answer": "6*sqrt(2)", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 8*y);F: Point;Focus(G) = F;H: Line;PointOnCurve(F, H);A: Point;B: Point;Intersection(H, G) = {A, B};O: Origin;Abs(LineSegmentOf(A, F))/Abs(LineSegmentOf(B, F)) = 1/2", "coords": {} }, { "index": 3003, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3003.png", "answer": "2", "fact_expr": "H: Line;Expression(H) = (y = k*x - 2);k: Number;G: Parabola;Expression(G) = (y^2 = 8*x);A: Point;B: Point;Negation(A=B);Intersection(H, G) = {A, B};XCoordinate(MidPoint(LineSegmentOf(A, B))) = 2", "coords": {} }, { "index": 3004, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3004.png", "answer": "8", "fact_expr": "C: Hyperbola;A: Point;P: Point;F: Point;Expression(C) = (x^2/16 - y^2/9 = 1);Coordinate(A) = (2, 4);LeftFocus(C) = F;PointOnCurve(P, RightPart(C));WhenMin(Perimeter(TriangleOf(A, P, F)))", "coords": { "A": [ 2.0, 4.0 ] } }, { "index": 3005, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3005.png", "answer": "pm*y+2*x=0;sqrt(5)", "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2/4 = 1);Eccentricity(C) = e ;e: Number", "coords": {} }, { "index": 3006, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3006.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;F1: Point;A: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = x*(b/a));LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(A, LeftPart(G));IsParallel(LineOf(A,F1),H);Perimeter(TriangleOf(A,F1,F2))=9*a", "coords": {} }, { "index": 3007, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3008, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3008.png", "answer": "[\\sqrt{2}/2,1)", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, l1) = True;PointOnCurve(F2, l2) = True;IsPerpendicular(l1, l2) = True;l1: Line;l2: Line;PointOnCurve(Intersection(l1, l2), G) = True", "coords": {} }, { "index": 3009, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3009.png", "answer": "3*sqrt(5)/5", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;C: Circle;Expression(C) = (y^2 + (x - 3)^2 = 4);l1:Line;l2:Line;Asymptote(G)={l1,l2};IsTangent(l1,C);IsTangent(l2,C)", "coords": {} }, { "index": 3010, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3010.png", "answer": "6;2", "fact_expr": "G: Hyperbola;a: Number;H: Parabola;F: Point;P: Point;F2: Point;a>0;Expression(G) = (-y^2/5 + x^2/a^2 = 1);Expression(H) = (y^2 = 12*x);OneOf(Focus(G)) = F;OneOf(Focus(G)) = F2;Negation(F = F2);Focus(H) = F;PointOnCurve(P,G);Distance(P,F) = 2", "coords": {} }, { "index": 3011, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3011.png", "answer": "2*sqrt(2)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/2 + y^2 = 1)", "coords": {} }, { "index": 3012, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3012.png", "answer": "0", "fact_expr": "G: Parabola;H: Line;C:Line;P: Point;A: Point;F: Point;B: Point;Expression(G) = (y^2 = 4*x);Expression(C) = (x = -1);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};Intersection(H,C)=P;VectorOf(P, A) = lambda*VectorOf(A, F);VectorOf(P, B) = mu*VectorOf(B, F);lambda:Real;mu:Real", "coords": {} }, { "index": 3013, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3014, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3014.png", "answer": "(0, \\sqrt{2}/2)", "fact_expr": "G: Ellipse;m: Number;n: Number;H: Curve;e: Number;m > n;n > 0;Expression(G) = (y^2/n + x^2/m = 1);Expression(H) = (x^2 + y^2 = m - n);NumIntersection(G, H)=0;Eccentricity(G) = e", "coords": {} }, { "index": 3015, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3016, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3016.png", "answer": "(3-sqrt(3))/3", "fact_expr": "C: Ellipse;b: Number;a: Number;A: Point;P: Point;F: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;LeftVertex(C) = A;PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P,F),xAxis);Slope(LineOf(A,P)) = sqrt(3)/3", "coords": {} }, { "index": 3017, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3017.png", "answer": "2*sqrt(5)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, B)) = 6;G: Circle;IsDiameter(LineSegmentOf(A, B), G)", "coords": {} }, { "index": 3018, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3018.png", "answer": "2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;H: Line;PointOnCurve(F, H);Intersection(H, G) = {A, B};A: Point;B: Point;Abs(LineSegmentOf(F, A))*Abs(LineSegmentOf(F, B)) = 6;Abs(LineSegmentOf(A, B)) = 6", "coords": {} }, { "index": 3019, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3019.png", "answer": "y^2=8*x", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;O: Origin;M: Point;PointOnCurve(M, G);Abs(LineSegmentOf(M, F)) = 4*Abs(LineSegmentOf(O, F));Area(TriangleOf(M, F, O)) = 4*sqrt(3)", "coords": {} }, { "index": 3020, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3021, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3021.png", "answer": "sqrt(6)/3", "fact_expr": "l: Line;C: Ellipse;b: Number;a: Number;A: Point;B: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Slope(l) = -1/3;Intersection(l, C) = {A, B};Coordinate(MidPoint(LineSegmentOf(A,B))) = (1, 1)", "coords": {} }, { "index": 3022, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3023, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3023.png", "answer": "5/2", "fact_expr": "C: Parabola;F: Point;Expression(C) = (y^2 = 2*x);Focus(C) = F;P: Point;PointOnCurve(P, C);a: Number;Abs(LineSegmentOf(P, F)) = a;a > 1/2;l: Line;Directrix(C) = l;M: Point;PointOnCurve(P,LineSegmentOf(P,M));IsPerpendicular(LineSegmentOf(P,M),l);FootPoint(LineSegmentOf(P,M),l) = M;Area(InscribedCircle(TriangleOf(M,P,F))) = (3-sqrt(5))*a*pi/4", "coords": {} }, { "index": 3024, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 1.4142135623730951 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_3024.png", "answer": "0", "fact_expr": "G: Circle;H: Line;O: Origin;A: Point;B: Point;Expression(G) = (x^2 + y^2 = 2);Expression(H) = (x + 2*y + sqrt(5) = 0);Intersection(H, G) = {A, B}", "coords": {} }, { "index": 3025, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3025.png", "answer": "4", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;HalfFocalLength(G) = c;c: Number;H: Line;Expression(H) = (a*y + b*x = a*b);O: Origin;Distance(O, H) = c/4 + 1", "coords": {} }, { "index": 3026, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 3.0, "major_axis": "x", "x_coef": 36.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3026.png", "answer": "-1/2", "fact_expr": "G: Ellipse;H: LineSegment;I: Point;Expression(G) = (x^2/36 + y^2/9 = 1);Coordinate(I) = (4, 2);IsChordOf(H, G);MidPoint(H) = I", "coords": { "I": [ 4.0, 2.0 ] } }, { "index": 3027, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3027.png", "answer": "(1, 2)", "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (3, 2);Focus(G) = F;PointOnCurve(P, G);WhenMin(Abs(LineSegmentOf(P, A))+Abs(LineSegmentOf(P, F)))", "coords": { "A": [ 3.0, 2.0 ] } }, { "index": 3028, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3028.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Point;F: Point;A: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;PointOnCurve(F,LineSegmentOf(F,H));IsPerpendicular(LineSegmentOf(F,H),Asymptote(G));FootPoint(LineSegmentOf(F,H),Asymptote(G))=H;Intersection(LineSegmentOf(F,H),RightPart(C))=A;MidPoint(LineSegmentOf(F,H))=A", "coords": {} }, { "index": 3029, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3030, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3030.png", "answer": "5/3", "fact_expr": "G: Hyperbola;b: Number;a: Number;I: Line;A: Point;F1: Point;F2: Point;b > a;a > 0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(RightFocus(G), I);IsParallel(I, OneOf(Asymptote(G)));Intersection(I, G) = A;Radius(InscribedCircle(TriangleOf(A, F1, F2))) = b/4", "coords": {} }, { "index": 3031, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3031.png", "answer": "3*sqrt(3)/2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 9*x);F: Point;Focus(G) = F;Intersection(Directrix(G), xAxis) = M;M: Point;PointOnCurve(N, G) = True;N: Point;sqrt(6)*Abs(LineSegmentOf(N, F)) = 2*Abs(LineSegmentOf(M, N))", "coords": {} }, { "index": 3032, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 9.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3032.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 + y^2/2 = 1);Focus(G)={F1,F2};PointOnCurve(P,G);Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 3033, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3033.png", "answer": "5", "fact_expr": "G: Ellipse;A: Point;x1:Number;y1:Number;Expression(G) = (x^2/9 + y^2/4 = 1);Coordinate(A) = (x1, y1);PointOnCurve(A, G)", "coords": {} }, { "index": 3034, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3034.png", "answer": "2*sqrt(2)", "fact_expr": "G: Parabola;H: Line;P: Point;O: Origin;Q: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(Focus(G), H);Inclination(H)=pi/4;Intersection(H, G) = {P, Q}", "coords": {} }, { "index": 3035, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "y", "x_coef": 4.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3035.png", "answer": "2*sqrt(10)/5", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 + y^2/9 = 1);PointOnCurve(P, G);Focus(G) = {F1,F2};Cos(AngleOf(F1, P, F2)) = 1/3", "coords": {} }, { "index": 3036, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3036.png", "answer": "5*sqrt(3)/3", "fact_expr": "G: Ellipse;P: Point;F: Point;Expression(G) = (x^2/4 + y^2 = 1);PointOnCurve(P, G);RightFocus(G)=F;Distance(P, F) = 3/2", "coords": {} }, { "index": 3037, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3037.png", "answer": "sqrt(3)-1", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2));AngleOf(P, F1, F2) = ApplyUnit(30, degree)", "coords": {} }, { "index": 3038, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3038.png", "answer": "2", "fact_expr": "G: Ellipse;Expression(G)=(x^2/9+y^2/4=1);H:Line;Expression(H)=(m*x-n*y=4);m:Number;n:Number;C:Circle;Expression(C)=(x^2+y^2=4);NumIntersection(H,C)=0;P:Point;Coordinate(P)=(m,n);L:Line;PointOnCurve(P,L)", "coords": {} }, { "index": 3039, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3039.png", "answer": "12", "fact_expr": "C: Hyperbola;a: Number;G: Line;P: Point;F1: Point;F2: Point;a>0;Expression(C) = (-y^2/3 + x^2/a^2 = 1);Expression(G) = (x = 2);Inclination(OneOf(Asymptote(C))) = ApplyUnit(60, degree);LeftFocus(C) =F1;RightFocus(C)=F2;Intersection(G, C) = P", "coords": {} }, { "index": 3040, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3040.png", "answer": "3+2*sqrt(2)", "fact_expr": "C: Parabola;p: Number;G: Line;F: Point;A: Point;B: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;PointOnCurve(F, G);Slope(G) = 1;Intersection(G, C) = {A, B};Abs(LineSegmentOf(F, A)) > Abs(LineSegmentOf(F, B))", "coords": {} }, { "index": 3041, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3041.png", "answer": "2*sqrt(7)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 8*x);G: Circle;F:Point;Focus(C) = F;H: Line;M: Point;Center(G)=M;PointOnCurve(M, C);Expression(H) = (x = -1);Radius(G)=Abs(LineSegmentOf(M,F));Abs(LineSegmentOf(M,F))=4", "coords": {} }, { "index": 3042, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3043, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3043.png", "answer": "sqrt(2)", "fact_expr": "C: Parabola;p: Number;G: Line;A: Point;B: Point;p>0;Expression(C) = (x^2 = 2*p*y);Intersection(Directrix(C),axis)=A;PointOnCurve(A,G);TangentPoint(G,C)=B;Abs(LineSegmentOf(A, B))=2", "coords": {} }, { "index": 3044, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3044.png", "answer": "8", "fact_expr": "G: Parabola;A: Point;F: Point;P: Point;l: Line;Expression(G) = (y^2 = 8*x);Focus(G) = F;Directrix(G)=l;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P,A),l);FootPoint(LineSegmentOf(P,A),l)=A;Inclination(LineOf(A,F)) = (2*pi)/3", "coords": {} }, { "index": 3045, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3045.png", "answer": "16/3", "fact_expr": "G: Hyperbola;F: Point;P: Point;O: Origin;H:Circle;Expression(G) = (x^2/4 - y^2/5 = 1);LeftFocus(G)=F;PointOnCurve(P,RightPart(G));Radius(H)=Abs(LineSegmentOf(O,F));PointOnCurve(MidPoint(LineSegmentOf(F,P)),H)", "coords": {} }, { "index": 3046, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3047, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3047.png", "answer": "(-12,0)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 + y^2/k = 1);k: Real;e:Number;Eccentricity(G)=e;In(e, (1, 2))", "coords": {} }, { "index": 3048, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3048.png", "answer": "sqrt(15)/15", "fact_expr": "F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;C: Hyperbola;Expression(C) = (x^2/4 - y^2 = 1);a: Number;b: Number;a>0;b>0;P: Point;PointOnCurve(P, C) = True;AngleOf(F1, P, F2) = ApplyUnit(120, degree)", "coords": {} }, { "index": 3049, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.2352789464429823, "b": 2.2352789464429823, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4312591569844774e-06, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_3049.png", "answer": "x^2/9-y^2=1", "fact_expr": "G: Hyperbola;F1: Point;F2: Point;M: Point;Coordinate(F1) = (-sqrt(10), 0);Coordinate(F2) = (sqrt(10), 0);Focus(G) = {F1,F2};PointOnCurve(M, G);DotProduct(VectorOf(M, F1), VectorOf(M, F2)) = 0;Abs(VectorOf(M, F1))*Abs(VectorOf(M, F2)) = 2", "coords": { "F1": [ -3.1622776601683795, 0.0 ], "F2": [ 3.1622776601683795, 0.0 ] } }, { "index": 3050, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3050.png", "answer": "12", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/5 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;l: Line;PointOnCurve(F1, l);M: Point;N: Point;Intersection(l, G) = {M, N}", "coords": {} }, { "index": 3051, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3052, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3053, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_ecc_mismatch", "params": { "a": 1.9257268905639648, "b": 1.7993911322594118, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 2.5576271215184017e-15, "converged": true, "iterations": 2 }, "output_path": "results/train/ellipse/problem_3053.png", "answer": "3/2", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2/2 + y^2/m = 1);PointOnCurve(Focus(G), xAxis);Eccentricity(G) = 1/2", "coords": {}, "validation_reasons": [ "ellipse_ecc_mismatch" ] }, { "index": 3054, "success": false, "error": "Circle from diameter: missing point coordinates" }, { "index": 3055, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3056, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3056.png", "answer": "sqrt(15)", "fact_expr": "G: Parabola;H: Line;Expression(G) = (y^2 = 12*x);Expression(H) = (y = 2*x + 1)", "coords": {} }, { "index": 3057, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3057.png", "answer": "2", "fact_expr": "G: Parabola;p: Number;H: Line;I: Point;P:Point;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (x = -1);Coordinate(I) = (1, 0);PointOnCurve(P,G);Distance(P,I)=Distance(P,H)", "coords": { "I": [ 1.0, 0.0 ] } }, { "index": 3058, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3058.png", "answer": "y=pm*sqrt(2)*x", "fact_expr": "G: Hyperbola;a: Number;Expression(G) = (-y^2/2 + x^2/a = 1);Coordinate(OneOf(Focus(G))) = (-sqrt(3), 0)", "coords": {} }, { "index": 3059, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3059.png", "answer": "x^2/2 - y^2 = 1", "fact_expr": "G: Hyperbola;H: Ellipse;P: Point;Expression(H) = (x^2/4 + y^2 = 1);Coordinate(P) = (2, 1);Focus(H) = Focus(G);PointOnCurve(P, G)", "coords": { "P": [ 2.0, 1.0 ] } }, { "index": 3060, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3060.png", "answer": "2*sqrt(3)", "fact_expr": "C: Hyperbola;P: Point;F: Point;O: Origin;Expression(C) = (x^2/4 - y^2/2 = 1);RightFocus(C) = F;PointOnCurve(P, OneOf(Asymptote(C)));Abs(LineSegmentOf(O, P)) = 2*Abs(LineSegmentOf(O, F))", "coords": {} }, { "index": 3061, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3061.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "F1: Point;F2: Point;C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Focus(C) = {F1, F2};P: Point;PointOnCurve(P, C) = True;IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2)) = True;AngleOf(P, F1, F2) = ApplyUnit(30, degree)", "coords": {} }, { "index": 3062, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3062.png", "answer": "2", "fact_expr": "G: Parabola;p: Number;H: Line;p>0;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (y = x + 1);PointOnCurve(OneOf(Intersection(H,axis)),Directrix(G))", "coords": {} }, { "index": 3063, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.47213595499958, "b": 4.0, "major_axis": "x", "x_coef": 20.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3063.png", "answer": "20", "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/20 + y^2/16 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);m:Number;Abs(VectorOf(P,F1))*Abs(VectorOf(P,F2))=m", "coords": {} }, { "index": 3064, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3064.png", "answer": "8", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);PointOnCurve(Focus(G), H);Intersection(H, G) = {A,B};x1+x2=6;x1:Number;x2:Number;y1:Number;y2:Number", "coords": {} }, { "index": 3065, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3065.png", "answer": "5+sqrt(17)", "fact_expr": "C: Parabola;M: Point;P: Point;F: Point;Expression(C) = (x^2 = 8*y);Coordinate(M) = (-4, 3);Focus(C) = F;PointOnCurve(P, C)", "coords": { "M": [ -4.0, 3.0 ] } }, { "index": 3066, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3066.png", "answer": "2*sqrt(2)", "fact_expr": "l: Line;E: Parabola;A: Point;C: Point;B: Point;Expression(E) = (y^2 = 4*x);PointOnCurve(Focus(E), l);Intersection(l, E) = {A, B};Intersection(l,Directrix(E))=C;Quadrant(A)=1;MidPoint(LineSegmentOf(A, C)) = B", "coords": {} }, { "index": 3067, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3067.png", "answer": "16.5", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1);P: Point;PointOnCurve(P, G);A: Point;Coordinate(A) = (5, 0);Distance(P, A) = 8.5;B: Point;Coordinate(B) = (-5, 0)", "coords": { "A": [ 5.0, 0.0 ], "B": [ -5.0, 0.0 ] } }, { "index": 3068, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3068.png", "answer": "4", "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/16 + y^2/4 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2))", "coords": {} }, { "index": 3069, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3069.png", "answer": "sqrt(13)/3", "fact_expr": "C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-x^2/b^2 + y^2/a^2 = 1);Expression(OneOf(Asymptote(C))) = (3*x - 2*y = 0)", "coords": {} }, { "index": 3070, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3070.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, L) = True;Intersection(L, LeftPart(G)) = M;Intersection(L, RightPart(G)) = N;L: Line;M: Point;N: Point;IsDiameter(LineSegmentOf(M, N), H) = True;H: Circle;PointOnCurve(F2, H) = True;Abs(LineSegmentOf(M, F2)) = Abs(LineSegmentOf(N, F2))", "coords": {} }, { "index": 3071, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3071.png", "answer": "6", "fact_expr": "l: Line;C: Parabola;p: Number;p>0;Expression(C) = (y^2 = 2*(p*x));Expression(l)=(x + 2*y - 3 = 0);PointOnCurve(Focus(C),l)", "coords": {} }, { "index": 3072, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3073, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3073.png", "answer": "(0,-2)", "fact_expr": "G: Parabola;Expression(G) = (y = -x^2/8)", "coords": {} }, { "index": 3074, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3074.png", "answer": "sqrt(3)/2", "fact_expr": "E: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(E) = F1;RightFocus(E) = F2;PointOnCurve(P, E);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;Area(TriangleOf(P, F1, F2)) = (a*b)/2", "coords": {} }, { "index": 3075, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3075.png", "answer": "y=pm*sqrt(3)*x", "fact_expr": "C: Hyperbola;a: Number;F: Point;a>0;Expression(C) = (-y^2/3 + x^2/a^2 = 1);Coordinate(F) = (2, 0);OneOf(Focus(C)) = F", "coords": { "F": [ 2.0, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 3076, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 5.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3076.png", "answer": "x^2=20*y", "fact_expr": "G: Parabola;O: Origin;Vertex(G) = O;F: Point;Coordinate(F) = (0, 5);Focus(G) = F", "coords": { "F": [ 0.0, 5.0 ] } }, { "index": 3077, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3077.png", "answer": "24", "fact_expr": "l: Line;PointOnCurve(RightFocus(G), l);G: Hyperbola;Expression(G) = (x^2/16 - y^2/4 = 1);A: Point;B: Point;Intersection(l, RightPart(G)) = {A, B};Abs(LineSegmentOf(A, B)) = 4;F: Point;LeftFocus(G) = F", "coords": {} }, { "index": 3078, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3079, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3079.png", "answer": "-9/16", "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/9 = 1);A: Point;B: Point;LeftVertex(G) = A;RightVertex(G) = B;P: Point;PointOnCurve(P, G)", "coords": {} }, { "index": 3080, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3080.png", "answer": "4*sqrt(3)", "fact_expr": "C: Parabola;P: Point;F: Point;O: Origin;Expression(C) = (y^2 = 8*x);Focus(C) = F;PointOnCurve(P, C);Abs(LineSegmentOf(P, F)) = 6", "coords": {} }, { "index": 3081, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3081.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;A: Point;P: Point;B: Point;Q: Point;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F;LeftVertex(G)=A;RightVertex(G)=B;PointOnCurve(F,H);IsPerpendicular(H,xAxis);Intersection(H,G)={P,Q};IsPerpendicular(LineSegmentOf(A, P), LineSegmentOf(B, Q))", "coords": {} }, { "index": 3082, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "major_axis": "x", "x_coef": 3.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3082.png", "answer": "[2*sqrt(2),2*sqrt(3))\n0", "fact_expr": "C: Ellipse;G: Line;y0: Number;x0: Number;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/3 + y^2 = 1);Expression(G) = (y*y0 + (x*x0)/3 = 1);Coordinate(P) = (x0, y0);Focus(C) = {F1,F2};0 b;b > 0;M: Circle;Expression(M) = (y^2 + (x - 3)^2 = 1);OneOf(Focus(C)) = Center(M);Length(MajorAxis(C)) = 10", "coords": {} }, { "index": 3086, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3087, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3087.png", "answer": "x^2/2-y^2=1", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;F: Point;A: Point;B: Point;O: Origin;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (-sqrt(3), 0);LeftFocus(C) = F;Intersection(l,LeftPart(C)) = A;Intersection(l,RightPart(C)) = B;Abs(LineSegmentOf(F, A)) + Abs(LineSegmentOf(F, B)) = 4;DotProduct(VectorOf(F, A), VectorOf(F,B)) = 0;PointOnCurve(O,l)", "coords": { "F": [ -1.7320508075688772, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 3088, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3088.png", "answer": "(x-1)^2+y^2=1", "fact_expr": "H: Circle;Expression(H) = ((-a + x)^2 + (-b + y)^2 = r^2);b: Number;a: Number;r: Number;G: Parabola;Expression(G) = (y^2 = 4*x);Center(H) = Focus(G);L: Line;Expression(L) = (3*x + 4*y + 2 = 0);IsTangent(H, L)", "coords": {} }, { "index": 3089, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3089.png", "answer": "sqrt(34) - 1", "fact_expr": "G: Parabola;P: Point;Q: Point;M: Point;Expression(G) = (y^2 = 4*x);Coordinate(M) = (4, 5);PointOnCurve(P, G);Projection(P, yAxis) = Q", "coords": { "M": [ 4.0, 5.0 ] } }, { "index": 3090, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3091, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3092, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3092.png", "answer": "y^2 = pm*6*x", "fact_expr": "G: Hyperbola;H: Parabola;Expression(G) = (x^2/3 - y^2 = 1);O: Origin;Directrix(H) = OneOf(Directrix(G));Vertex(H) = O", "coords": {} }, { "index": 3093, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 10.0, "b": 4.0, "major_axis": "x", "x_coef": 100.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3093.png", "answer": "14", "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/100 + y^2/16 = 1);PointOnCurve(P, G);Distance(P, LeftFocus(G)) = 6", "coords": {} }, { "index": 3094, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 8.0, "b": 6.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3094.png", "answer": "50", "fact_expr": "l: Line;G: Hyperbola;A: Point;B: Point;F2: Point;F1: Point;Expression(G) = (x^2/64 - y^2/36 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, l);Intersection(l,LeftPart(G))={A, B};Abs(LineSegmentOf(A, B)) = 9", "coords": {} }, { "index": 3095, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 0.9220545291900635, "b": 2.765991687774658, "orientation": "horizontal" }, "optimization": { "final_loss": 4.6528161874448415e-06, "converged": true, "iterations": 186 }, "output_path": "results/train/hyperbola/problem_3095.png", "answer": "2/3", "fact_expr": "G: Hyperbola;m: Real;Expression(G) = (-y^2/6 + x^2/m = 1);Expression(OneOf(Asymptote(G))) = (y = 3*x)", "coords": {} }, { "index": 3096, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3096.png", "answer": "sqrt(2)", "fact_expr": "C: Parabola;p: Number;O: Origin;H:Circle;P: Point;F: Point;Q: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Expression(H) = (x^2 + y^2 = 12);Focus(C) = F;Intersection(C, H) = {P, Q};Abs(LineSegmentOf(P, Q)) = 4*sqrt(2)", "coords": {} }, { "index": 3097, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3097.png", "answer": "sqrt(3)/3", "fact_expr": "G: Parabola;p: Number;O: Origin;M: Point;P: Point;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;PointOnCurve(P, G);PointOnCurve(M, LineSegmentOf(P, F));Abs(LineSegmentOf(P, M)) = 3*Abs(LineSegmentOf(M, F))", "coords": {} }, { "index": 3098, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3098.png", "answer": "[\\sqrt{2}-1, 1)", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;e: Number;Eccentricity(G) = e;P: Point;PointOnCurve(P, G);LineSegmentOf(P, F1)/LineSegmentOf(P, F2) = e", "coords": {} }, { "index": 3099, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3100, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3100.png", "answer": "pm*6", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;M: Point;y0: Number;Coordinate(M) = (3, y0);PointOnCurve(M, G);Distance(M, F) = 6", "coords": {} }, { "index": 3101, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3102, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3102.png", "answer": "1", "fact_expr": "l: Line;C: Parabola;M: Point;N: Point;F: Point;a:Number;b:Number;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F, l);Slope(l) = 1;Intersection(l, C) = {M, N};b = (Abs(LineSegmentOf(M, F)) + Abs(LineSegmentOf(N, F)))/2;XCoordinate(Intersection(PerpendicularBisector(LineSegmentOf(M,N))), xAxis) = a", "coords": {} }, { "index": 3103, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 3.4641016151377544, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3103.png", "answer": "y=pm*(sqrt(3)/2)*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/12 - y^2/9 = -1)", "coords": {} }, { "index": 3104, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3105, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 5.656854249492381, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3105.png", "answer": "2", "fact_expr": "G: Hyperbola;A: Point;x0:Number;y0:Number;Expression(G) = (x^2/4 - y^2/32 = 1);Coordinate(A) = (x0, y0);PointOnCurve(A, RightPart(G));Distance(A,RightFocus(G)) = 2*x0", "coords": {} }, { "index": 3106, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3106.png", "answer": "x \\pm \\sqrt{3} y - 1 = 0", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l: Line;PointOnCurve(F, l) = True;Intersection(l, C) = {A, B};A: Point;B: Point;Projection(MidPoint(LineSegmentOf(A, F)), yAxis) = P;P: Point;Projection(MidPoint(LineSegmentOf(B, F)), yAxis) = Q;Q: Point;Abs(LineSegmentOf(P, Q)) = 4", "coords": {} }, { "index": 3107, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 5.656854249492381, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3107.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/32 = 1);A: Point;x0: Number;y0: Number;Coordinate(A) = (x0, y0);PointOnCurve(A, RightPart(G));Distance(A, RightFocus(G)) = 2*x0", "coords": {} }, { "index": 3108, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3108.png", "answer": "-1/2", "fact_expr": "M: Point;Coordinate(M) = (-2, 0);m: Line;PointOnCurve(M, m) = True;G: Ellipse;Expression(G) = (x^2/2 + y^2 = 1);Intersection(m, G) = {P1, P2};P1: Point;P2: Point;MidPoint(LineSegmentOf(P1,P2)) = P;P: Point;Slope(m) = k1;k1: Number;O: Origin;Slope(LineOf(O,P)) = k2;k2: Number", "coords": { "M": [ -2.0, 0.0 ] } }, { "index": 3109, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3109.png", "answer": "1", "fact_expr": "l: Line;C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;p>0;F: Point;Coordinate(F) = (1, 0);Focus(C) = F;PointOnCurve(F, l) = True;Intersection(l, C) = {A, B};A: Point;B: Point", "coords": { "F": [ 1.0, 0.0 ] } }, { "index": 3110, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3110.png", "answer": "sqrt(2) + 1 - sqrt(3)", "fact_expr": "E: Hyperbola;a: Number;b: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(E) = F1;RightFocus(E) = F2;PointOnCurve(P, RightPart(E));Abs(LineSegmentOf(P, F1)) = Abs(LineSegmentOf(F1, F2));AngleOf(P, F1, F2) = ApplyUnit(30, degree);Area(TriangleOf(P, F1, F2)) = 2", "coords": {} }, { "index": 3111, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3111.png", "answer": "2", "fact_expr": "H: Circle;Expression(H) = (y^2 + x^2 - 2*x = 0);C: Point;Center(H) = C;G: Parabola;Expression(G) = (y^2 = 4*x);l: Line;Directrix(G) = l", "coords": {} }, { "index": 3112, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.732385540716972, "b": 2.4497476064802375, "orientation": "horizontal" }, "optimization": { "final_loss": 2.601017381791051e-06, "converged": true, "iterations": 91 }, "output_path": "results/train/hyperbola/problem_3112.png", "answer": "x^2/3 - y^2/6 = 1", "fact_expr": "G: Hyperbola;F1: Point;F2: Point;Coordinate(F1) = (-3, 0);Coordinate(F2) = (3, 0);Focus(G) = {F1, F2};Expression(OneOf(Asymptote(G))) = (y = sqrt(2)*x)", "coords": { "F1": [ -3.0, 0.0 ], "F2": [ 3.0, 0.0 ] } }, { "index": 3113, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3113.png", "answer": "(1, 2]", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, RightPart(G));Abs(LineSegmentOf(P, F1)) = 3*Abs(LineSegmentOf(P, F2));e: Number;Eccentricity(G) = e", "coords": {} }, { "index": 3114, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/train/hyperbola/problem_3114.png", "answer": "x^2/4-y^2=1", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;Expression(Asymptote(G)) = (y = pm*(x/2));H: Point;Coordinate(H) = (10/3, 4/3);PointOnCurve(H, G)", "coords": { "H": [ 3.3333333333333335, 1.3333333333333333 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 3115, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3115.png", "answer": "(-oo, sqrt(2)]", "fact_expr": "G: Hyperbola;H: Line;P: Point;m: Real;Expression(G) = (x^2 - y^2 = 1);Expression(H) = (y = x + 2);PointOnCurve(P, RightPart(G));Distance(P, H) > m", "coords": {} }, { "index": 3116, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_asymptote_mismatch", "params": { "a": 1.2175992727279663, "b": 2.434081792831421, "orientation": "vertical" }, "optimization": { "final_loss": 3.452532837400213e-06, "converged": true, "iterations": 124 }, "output_path": "results/train/hyperbola/problem_3116.png", "answer": "sqrt(5)/2", "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (y^2/4 - x^2/m = 1);Expression(OneOf(Asymptote(G))) = (y = 2*x)", "coords": {}, "validation_reasons": [ "hyperbola_asymptote_mismatch" ] }, { "index": 3117, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3117.png", "answer": "(x-2)^2+(y-2*sqrt(2))^2=9", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);C: Circle;Expression(C) = ((-a + x)^2 + (-b + y)^2 = r^2);r: Number;b: Number;a: Number;A: Point;b>0;PointOnCurve(A,C);Coordinate(A) = (3, 0);PointOnCurve(Center(C),G);IsTangent(C,Directrix(G))", "coords": { "A": [ 3.0, 0.0 ] } }, { "index": 3118, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3118.png", "answer": "A straight line does not exist", "fact_expr": "l: Line;G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1);A: Point;B: Point;Intersection(l, G) = {A, B};P: Point;Coordinate(P) = (6, 3);MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ 6.0, 3.0 ] } }, { "index": 3119, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3119.png", "answer": "sqrt(5)/3", "fact_expr": "P: Point;PointOnCurve(P,G) = True;G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Focus(G) = {F1,F2};F1: Point;F2: Point;DotProduct(VectorOf(P,F1),VectorOf(P,F2))=0;Tan(AngleOf(P,F1,F2)) = 1/2", "coords": {} }, { "index": 3120, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3120.png", "answer": "1", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);DotProduct(VectorOf(F1, P), VectorOf(P, F2)) = 0", "coords": {} }, { "index": 3121, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3121.png", "answer": "-2, -1/2", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(C) = F;O: Origin;Center(G) = F;Radius(G) = 2*sqrt(3)*a;G: Circle;Intersection(G, OneOf(Asymptote(C))) = {P, Q};P: Point;Q: Point;DotProduct(VectorOf(F, P), VectorOf(F, Q)) = -6*a^2;VectorOf(O, P) = lambda*VectorOf(O, Q);lambda: Number", "coords": {} }, { "index": 3122, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3122.png", "answer": "3/8", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C) = True;Negation(P = LeftVertex(C));Negation(P = RightVertex(C));M: Point;Incenter(TriangleOf(P, F1, F2)) = M;5*VectorOf(M, F1) + 3*VectorOf(M, F2) + 3*VectorOf(M, P) = 0", "coords": {} }, { "index": 3123, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3123.png", "answer": "-6", "fact_expr": "G: Hyperbola;p: Number;H: Parabola;Expression(G) = (x^2/3 + y^2/p = 1);Expression(H) = (y^2 = 2*(p*x));PointOnCurve(OneOf(Focus(G)), Directrix(H))", "coords": {} }, { "index": 3124, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3125, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3126, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3127, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3127.png", "answer": "(1/4, -1)", "fact_expr": "M: Point;Coordinate(M) = (0, 2);G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;A: Point;B: Point;PointOnCurve(F, LineOf(A, B));Intersection(LineOf(A, B), G) = {A, B};AngleOf(A, M, F) = pi/2", "coords": { "M": [ 0.0, 2.0 ] } }, { "index": 3128, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3128.png", "answer": "4.5", "fact_expr": "G: Parabola;H: Line;A: Point;F: Point;B: Point;Expression(G) = (y^2 = 2*x);Focus(G) = F;PointOnCurve(F,H);Intersection(H,G)={A,B}", "coords": {} }, { "index": 3129, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3129.png", "answer": "-4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = a*x);a: Number;Coordinate(Focus(G)) = (-1, 0)", "coords": {} }, { "index": 3130, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3130.png", "answer": "sqrt(5)/2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1)", "coords": {} }, { "index": 3131, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.47213595499958, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 20.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3131.png", "answer": "x^2/16 - y^2/2 = 1", "fact_expr": "G: Hyperbola;H: Ellipse;C:Curve;Expression(H) = (x^2/20 + y^2/2 = 1);Expression(C)= (x^2/8 - y^2 = 1);Asymptote(G)=Asymptote(C);Focus(G) = Focus(H)", "coords": {} }, { "index": 3132, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3133, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3133.png", "answer": "(4,\\pm 4)", "fact_expr": "G: Parabola;P:Point;Expression(G) = (y^2 = 4*x);PointOnCurve(P,G);Distance(P,Focus(G))=5", "coords": {} }, { "index": 3134, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3134.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;F2: Point;a>0;b>0;RightFocus(G) = F2;H: Line;PointOnCurve(F2,H) = True;IsPerpendicular(H,xAxis) = True;P: Point;Intersection(H,Asymptote(G)) = P;Quadrant(P) = 1;F1: Point;LeftFocus(G) = F1;Inclination(LineOf(F1,P)) = pi/4;e: Number;Eccentricity(G) = e", "coords": {} }, { "index": 3135, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3135.png", "answer": "{ApplyUnit(60, degree), ApplyUnit(120, degree)}", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);M: Point;Coordinate(M) = (1, 0);A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G);PointOnCurve(M, LineSegmentOf(A, B));Abs(LineSegmentOf(A, B)) = (8*sqrt(10))/3", "coords": { "M": [ 1.0, 0.0 ] } }, { "index": 3136, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3137, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3137.png", "answer": "8", "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;G: Line;PointOnCurve(F1, G);A: Point;B: Point;Intersection(G, C) = {A, B}", "coords": {} }, { "index": 3138, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3139, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3139.png", "answer": "5*sqrt(2)", "fact_expr": "C: Ellipse;Expression(C) = (x^2/2 + y^2 = 1);F: Point;LeftFocus(C) = F;P: Point;PointOnCurve(P, C);Q: Point;Coordinate(Q) = (4, 3)", "coords": { "Q": [ 4.0, 3.0 ] } }, { "index": 3140, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3140.png", "answer": "(0, 2)", "fact_expr": "C: Parabola;M: Point;N: Point;F: Point;l: Line;Expression(C) = (y^2 = 4*x);Focus(C) = F;Directrix(C) = l;Quadrant(M) = 1;PointOnCurve(M, C);PointOnCurve(N, l);IsPerpendicular(LineSegmentOf(M, F), LineSegmentOf(N, F));Abs(LineSegmentOf(M, F)) = 5;Intersection(LineOf(M,N),yAxis)=P;P:Point", "coords": {} }, { "index": 3141, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3141.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;P: Point;PointOnCurve(P, G);F1: Point;F2: Point;Focus(G) = {F1, F2};IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2));LineSegmentOf(P, F1) = 2*LineSegmentOf(P, F2)", "coords": {} }, { "index": 3142, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.463192105293274, "b": 2.070901870727539, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4661259228887502e-06, "converged": true, "iterations": 106 }, "output_path": "results/train/hyperbola/problem_3142.png", "answer": "(0,2*sqrt(2))", "fact_expr": "C: Hyperbola;b: Number;a: Number;A: Point;P: Point;B: Point;O: Origin;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(C) = sqrt(3);LeftVertex(C)=A;RightVertex(C)=B;PointOnCurve(P, C);Quadrant(P)=1;Slope(LineOf(P,A))=k1;Slope(LineOf(P,B))=k2;Slope(LineOf(P,O))=k3;k1:Number;k2:Number;k3:Number;m:Number;m=k1*k2*k3", "coords": {} }, { "index": 3143, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3143.png", "answer": "4*sqrt(3)/3", "fact_expr": "l: Line;G: Parabola;O: Origin;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G)=F;PointOnCurve(F, l);Intersection(l, G) = {A, B};Abs(LineSegmentOf(A, F)) = 4", "coords": {} }, { "index": 3144, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3144.png", "answer": "4", "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(P, G);Distance(P, RightDirectrix(G)) = 10", "coords": {} }, { "index": 3145, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.125, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3145.png", "answer": "{(5/4, +oo), 0}", "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;F: Point;x1: Number;x2: Number;y1: Number;y2: Number;Expression(G) = (y = 2*x^2);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);PointOnCurve(A, G);PointOnCurve(B, G);PerpendicularBisector(LineSegmentOf(A, B)) = l;Focus(G) = F;PointOnCurve(F, l)", "coords": {} }, { "index": 3146, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3147, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3147.png", "answer": "{5/2, pm*2}", "fact_expr": "l: Line;G: Hyperbola;P: Point;Expression(G) = (x^2 - y^2/4 = 1);Coordinate(P) = (1, 1);PointOnCurve(P, l);Slope(l) = k;NumIntersection(l, G) = 1;k: Number", "coords": { "P": [ 1.0, 1.0 ] } }, { "index": 3148, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3149, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3149.png", "answer": "(\\sqrt{3} - 1)/2", "fact_expr": "G: Ellipse;b: Number;a: Number;c: Number;F1: Point;F2: Point;P: Point;e: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 2*c;AngleOf(F1, P, F2) = ApplyUnit(30, degree);Eccentricity(G) = e", "coords": {} }, { "index": 3150, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3150.png", "answer": "sqrt(3)/2", "fact_expr": "L: Line;Slope(L) = -1/2;C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;A: Point;B: Point;Intersection(L, C) = {A, B};P: Point;Coordinate(P) = (2, 1);MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ 2.0, 1.0 ] } }, { "index": 3151, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3152, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3152.png", "answer": "x=3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = -12*x)", "coords": {} }, { "index": 3153, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3153.png", "answer": "4", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/3 - y^2 = 1)", "coords": {} }, { "index": 3154, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3154.png", "answer": "2*sqrt(3)", "fact_expr": "G: Ellipse;b: Number;H: Line;B: Point;A: Point;C: Point;F1: Point;F2: Point;0 < b;b < 3;Expression(G) = (x^2/9 + y^2/b^2 = 1);Expression(H) = (x + 3*y - 7 = 0);Coordinate(C) = (1, 2);Intersection(H, G) = {A, B};Focus(G)={F1,F2};MidPoint(LineSegmentOf(A,B)) = C", "coords": { "C": [ 1.0, 2.0 ] } }, { "index": 3155, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 1.0, -0.0 ], "radius": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_3155.png", "answer": "1/16", "fact_expr": "G: Parabola;p: Number;H: Circle;p>0;Expression(G) = (y = 2*(p*x^2));Expression(H) = (-2*x + x^2 + y^2 - 3 = 0);IsTangent(H, Directrix(G))", "coords": {} }, { "index": 3156, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3156.png", "answer": "b*sqrt(a^2-b^2)", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;RightFocus(G) = F;A: Point;B: Point;O: Origin;Center(G) = O;PointOnCurve(O, LineSegmentOf(A, B));IsChordOf(LineSegmentOf(A, B), G)", "coords": {} }, { "index": 3157, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3157.png", "answer": "3", "fact_expr": "G: Parabola;P: Point;A: Point;F: Point;Expression(G) = (x^2 = 4*y);Coordinate(A) = (0, -1);Focus(G) = F;PointOnCurve(P, G)", "coords": { "A": [ 0.0, -1.0 ] } }, { "index": 3158, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -1.0, 0.0 ], "radius": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_3158.png", "answer": "x-y-3=0", "fact_expr": "G: Circle;Expression(G) = (y^2 + (x - 1)^2 = 25);P: Point;Coordinate(P) = (2, -1);A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G);MidPoint(LineSegmentOf(A, B)) = P", "coords": {} }, { "index": 3159, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3160, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2631799883090584, "b": 2.5275303393962245, "orientation": "horizontal" }, "optimization": { "final_loss": 4.481942266905428e-06, "converged": true, "iterations": 95 }, "output_path": "results/train/hyperbola/problem_3160.png", "answer": "{sqrt(5),sqrt(5)/2}", "fact_expr": "G: Hyperbola;Expression(OneOf(Asymptote(G))) = (y = 2*x)", "coords": {} }, { "index": 3161, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3161.png", "answer": "sqrt(5)/3", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/m = 1);m: Number;K: Point;PointOnCurve(K, G);F1: Point;F2: Point;Focus(G) = {F1, F2};Distance(K, F1)+Distance(K, F2) = m-3", "coords": {} }, { "index": 3162, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3162.png", "answer": "x \\pm \\sqrt{3} y = 0", "fact_expr": "H: Parabola;Expression(H) = (y^2 = 8*x);G: Hyperbola;Expression(G) = (-y^2 + x^2/a^2 = 1);a: Number;Focus(H) = RightFocus(G)", "coords": {} }, { "index": 3163, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3163.png", "answer": "pm*2", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(C) = F;l: Line;PointOnCurve(F,l) = True;Slope(l) = k;k: Number;Intersection(l, C) = {A, B};A: Point;B: Point;Abs(LineSegmentOf(A, B)) = (5/2)*p", "coords": {} }, { "index": 3164, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 2.0, "major_axis": "x", "x_coef": 8.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3164.png", "answer": "[0, \\sqrt{2}]", "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;O: Origin;Expression(C) = (x^2/8 + y^2/4 = 1);PointOnCurve(P, C);LeftFocus(C) = F1;RightFocus(C) = F2", "coords": {} }, { "index": 3165, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3165.png", "answer": "sqrt(2)/2", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;LeftFocus(C) = F;G: Line;PointOnCurve(F, G);B: Point;UpperVertex(C) = B;PointOnCurve(B, G);A: Point;Intersection(G, C) = {A,B};Abs(LineSegmentOf(B, F)) = 3*Abs(LineSegmentOf(A, F))", "coords": {} }, { "index": 3166, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3167, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3167.png", "answer": "(2/5)*sqrt(5)", "fact_expr": "G: Hyperbola;H: Parabola;F: Point;Expression(G) = (x^2/4 - y^2 = 1);Expression(H) = (y = x^2/4);Focus(H) = F", "coords": {} }, { "index": 3168, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3168.png", "answer": "8", "fact_expr": "Gamma: Parabola;Expression(Gamma) = (y^2 = 6*x);l: Line;PointOnCurve(Focus(Gamma), l);Inclination(l) = ApplyUnit(120, degree)", "coords": {} }, { "index": 3169, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3169.png", "answer": "3", "fact_expr": "G: Ellipse;A: Point;Expression(G) = (x^2/4 + y^2/3 = 1);PointOnCurve(A, G);Distance(A, LeftFocus(G)) = 5/2", "coords": {} }, { "index": 3170, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3170.png", "answer": "(1, 2]", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, RightPart(C));Abs(LineSegmentOf(P, F1))*Abs(LineSegmentOf(P, F2)) = 3*a^2", "coords": {} }, { "index": 3171, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3172, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "y", "x_coef": 5.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3172.png", "answer": "10", "fact_expr": "G: Ellipse;A: Point;M: Point;F: Point;Expression(G) = (x^2/5 + y^2/9 = 1);Coordinate(A) = (2*sqrt(3), 0);UpperFocus(G) = F;PointOnCurve(M, G)", "coords": { "A": [ 3.4641016151377544, 0.0 ] } }, { "index": 3173, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3173.png", "answer": "sqrt(3)/3", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);PointOnCurve(P, G);Focus(G) = {F1, F2};AngleOf(F1,P,F2)=pi/3", "coords": {} }, { "index": 3174, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 3.0, "major_axis": "x", "x_coef": 36.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3174.png", "answer": "x+2*y-8=0", "fact_expr": "l: Line;G: Ellipse;Expression(G) = (x^2/36 + y^2/9 = 1);Coordinate(MidPoint(InterceptChord(l,G))) = (4, 2)", "coords": {} }, { "index": 3175, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 5.916079783099616, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3175.png", "answer": "6", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/35 = 1)", "coords": {} }, { "index": 3176, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3176.png", "answer": "pm*x-3*y=0", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2 = 1)", "coords": {} }, { "index": 3177, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3177.png", "answer": "sqrt(10)/2", "fact_expr": "C: Hyperbola;b: Number;a: Number;A: Point;B: Point;F2: Point;F1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;IsChordOf(LineSegmentOf(A,B),RightPart(C));PointOnCurve(F2,LineSegmentOf(A,B));VectorOf(A,F2)=(3/4)*VectorOf(A,B);Abs(LineSegmentOf(A,F1))=Abs(LineSegmentOf(A,F2))+(1/2)*Abs(LineSegmentOf(A,B))", "coords": {} }, { "index": 3178, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3179, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3180, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3180.png", "answer": "2", "fact_expr": "Q: Point;Coordinate(Q) = (2*sqrt(2), 0);P: Point;x0: Number;y0: Number;Coordinate(P) = (x0, y0);G: Parabola;Expression(G) = (y = x^2/4);PointOnCurve(P, G)", "coords": { "Q": [ 2.8284271247461903, 0.0 ] } }, { "index": 3181, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3181.png", "answer": "sqrt(13)", "fact_expr": "C: Hyperbola;b: Number;a: Number;O: Origin;P: Point;F2: Point;F1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) =F1;RightFocus(C)=F2;PointOnCurve(P,RightPart(C));DotProduct((VectorOf(O, F2) + VectorOf(O, P)),VectorOf(P, F2) )= 0;2*Abs(VectorOf(P, F1)) = 3*Abs(VectorOf(P, F2))", "coords": {} }, { "index": 3182, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3182.png", "answer": "13", "fact_expr": "E: Ellipse;A: Point;P: Point;F1: Point;F2: Point;Expression(E) = (x^2/25 + y^2/9 = 1);Coordinate(A) = (2, sqrt(5));LeftFocus(E) = F1;RightFocus(E) = F2;PointOnCurve(P, E)", "coords": { "A": [ 2.0, 2.23606797749979 ] } }, { "index": 3183, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3183.png", "answer": "2/3", "fact_expr": "G: Ellipse;b: Number;a: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Eccentricity(G)=e;a <=sqrt(3)*b;e:Number", "coords": {} }, { "index": 3184, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3184.png", "answer": "3", "fact_expr": "C: Parabola;P: Point;F: Point;Q: Point;l:Line;Expression(C) = (y^2 = 8*x);Focus(C) = F;Directrix(C) = l;PointOnCurve(P, l);OneOf(Intersection(LineOf(P,F),C))=Q;VectorOf(P, F) = 4*VectorOf(Q, F)", "coords": {} }, { "index": 3185, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3186, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.4641016151377544, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 12.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3186.png", "answer": "7*sqrt(3)/2", "fact_expr": "F1: Point;F2: Point;Focus(G) = {F1, F2};G: Ellipse;Expression(G) = (x^2/12 + y^2/3 = 1);P: Point;PointOnCurve(P, G);PointOnCurve(MidPoint(LineSegmentOf(P, F1)), yAxis)", "coords": {} }, { "index": 3187, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3187.png", "answer": "sqrt(13)/2", "fact_expr": "E: Hyperbola;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(E) = F1;RightFocus(E) = F2;A: Point;PointOnCurve(A, LeftPart(E));AngleOf(F1, A, F2) = ApplyUnit(120, degree);Abs(LineSegmentOf(A, F2)) = 3*Abs(LineSegmentOf(A, F1))", "coords": {} }, { "index": 3188, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3188.png", "answer": "x^2+(y-1)^2=2", "fact_expr": "G: Parabola;H: Circle;L: Line;Center(H)=Focus(G);Expression(G) = (x^2 = 4*y);Expression(L) = (y = x + 3);IsTangent(L,H)", "coords": {} }, { "index": 3189, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3189.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(RightFocus(G), H);Inclination(H)=ApplyUnit(60,degree);NumIntersection(H,G)=1", "coords": {} }, { "index": 3190, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3191, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3192, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3192.png", "answer": "15", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);M: Point;Coordinate(M) = (6, 4)", "coords": { "M": [ 6.0, 4.0 ] } }, { "index": 3193, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3194, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3195, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3195.png", "answer": "6", "fact_expr": "G: Ellipse;P: Point;F: Point;Expression(G) = (x^2/9 + y^2/5 = 1);PointOnCurve(P, G);LeftFocus(G) = F;Distance(P, F) = 2", "coords": {} }, { "index": 3196, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3196.png", "answer": "4*sqrt(5)/5-1", "fact_expr": "G: Ellipse;H: Circle;P: Point;Q: Point;Expression(G) = (x^2/9 + y^2/4 = 1);Expression(H) = (y^2 + (x - 1)^2 = 1);PointOnCurve(P, H);PointOnCurve(Q, G)", "coords": {} }, { "index": 3197, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3197.png", "answer": "y=pm*((sqrt(15)/5)*x)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;A: Point;PointOnCurve(A, G);O: Origin;Slope(LineOf(O, A)) = sqrt(3)/3;PointOnCurve(Vertex(G), PerpendicularBisector(LineSegmentOf(O, A)))", "coords": {} }, { "index": 3198, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.23606797749979, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3198.png", "answer": "sqrt(3)", "fact_expr": "C: Hyperbola;F: Point;Expression(C) = (x^2/5 - y^2/3 = 1);RightFocus(C) = F", "coords": {} }, { "index": 3199, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3199.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1)", "coords": {} }, { "index": 3200, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3201, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -0.0, 2.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_3201.png", "answer": "(x^2 + (y-1)^2 = 1)&(3/2 < y <= 2)", "fact_expr": "l: Line;G: Circle;A: Point;B: Point;O: Origin;M: Point;Expression(G) = (-4*y + x^2 + y^2 + 3 = 0);PointOnCurve(O, l);Intersection(l, G) = {A, B};IsChordOf(LineSegmentOf(A, B), G);MidPoint(LineSegmentOf(A,B)) = M", "coords": {} }, { "index": 3202, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3202.png", "answer": "y=pm*sqrt(2)*x", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;FocalLength(C) = sqrt(3)*Length(RealAxis(C))", "coords": {} }, { "index": 3203, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3203.png", "answer": "5", "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (x^2/4 - y^2/m = 1);FocalLength(G) = 6", "coords": {} }, { "index": 3204, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3204.png", "answer": "4/9", "fact_expr": "M: Parabola;Expression(M) = (x^2 = 2*p*y);p: Number;p>0;F: Point;Focus(M) = F;PointOnCurve(F, l) = True;Slope(l) = 5/12;l: Line;Intersection(l, M) = {A, B};A: Point;B: Point;Quadrant(A) = 2", "coords": {} }, { "index": 3205, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3205.png", "answer": "8", "fact_expr": "C: Parabola;G: Line;A: Point;B: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F, G);Slope(G) = 1;Intersection(G, C) = {A, B}", "coords": {} }, { "index": 3206, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3206.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Parabola;A: Point;B: Point;M: Point;Expression(G) = (y = x^2);IsChordOf(LineSegmentOf(A, B), G);Length(LineSegmentOf(A, B)) = 2;MidPoint(LineSegmentOf(A, B)) = M", "coords": {} }, { "index": 3207, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 4.47213595499958, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3207.png", "answer": "14", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/20 = 1);M: Point;PointOnCurve(M, G);F1: Point;F2: Point;OneOf(Focus(G)) = F1;OneOf(Focus(G)) = F2;Negation(F1=F2);Distance(M, F1) = 6", "coords": {} }, { "index": 3208, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 3.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3208.png", "answer": "pm*sqrt(6)/3", "fact_expr": "H: Line;Expression(H) = (y = k*x + 2);k: Number;G: Ellipse;Expression(G) = (x^2/3 + y^2/2 = 1);NumIntersection(H, G) = 1;Slope(H) = k", "coords": {} }, { "index": 3209, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3210, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3210.png", "answer": "y=pm*(2*sqrt(2)/3)*x", "fact_expr": "H: Ellipse;Expression(H) = (y^2/b^2 + x^2/a^2 = 1);Eccentricity(H) = 1/3;b: Number;a: Number;a > b;b > 0;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1)", "coords": {} }, { "index": 3211, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3211.png", "answer": "1/2", "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;B: Point;F: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(A)=(-a,0);Coordinate(B) = (0, b);LeftFocus(G)=F;In(A,Vertex(G));Distance(F, LineSegmentOf(A, B)) = b/sqrt(7);In(B,Vertex(G))", "coords": {} }, { "index": 3212, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3213, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3213.png", "answer": "2", "fact_expr": "G: Ellipse;Expression(G) = (x^2 + y^2/16 = 1)", "coords": {} }, { "index": 3214, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3215, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3216, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3217, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3217.png", "answer": "(1,1)", "fact_expr": "G: Parabola;Expression(G) = (y = x^2);H: Line;Expression(H) = (2*x - y - 4 = 0);P: Point;PointOnCurve(P,G) = True;WhenMin(Distance(P,H)) = True", "coords": {} }, { "index": 3218, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3218.png", "answer": "sqrt(2)", "fact_expr": "C: Ellipse;b: Number;a: Number;G: Line;A: Point;F: Point;B: Point;a > b;b > 0;k:Number;k>0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Eccentricity(C) = sqrt(3)/2;RightFocus(C)=F;PointOnCurve(F, G);Slope(G) = k;Intersection(G, C) = {A, B};VectorOf(A, F) = 3*VectorOf(F, B)", "coords": {} }, { "index": 3219, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3219.png", "answer": "{3, 5}", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2/4 + y^2/m = 1);FocalLength(G) = 2", "coords": {} }, { "index": 3220, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3220.png", "answer": "y=pm*(sqrt(2)/2)*x", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Ellipse;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1/3);a1 > b1;b1 > 0;Expression(H) = (y^2/b1^2 + x^2/a1^2 = 1);Focus(G) = Focus(H);a1:Number;b1:Number", "coords": {} }, { "index": 3221, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3221.png", "answer": "3/2", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 6*x);l: Line;PointOnCurve(P, l) = True;Coordinate(P) = (2, 2);P: Point;Intersection(l,C) = {M,N};N: Point;M: Point;MidPoint(LineSegmentOf(M,N)) = P", "coords": { "P": [ 2.0, 2.0 ] } }, { "index": 3222, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3223, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3223.png", "answer": "5", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);A: Point;PointOnCurve(A, G);YCoordinate(A) = 4", "coords": {} }, { "index": 3224, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3225, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3225.png", "answer": "(\\sqrt{5}+1)/2", "fact_expr": "G: Hyperbola;a: Number;b: Number;H: Circle;c: Number;C:Parabola;F: Point;P: Point;L:Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2 + y^2 = c^2);Coordinate(F) = (-c, 0);LeftFocus(G) = F;PointOnCurve(F,L);IsParallel(L,Asymptote(G));Intersection(L,H)=P;PointOnCurve(P,C);Expression(C)=(y^2 = 4*(c*x));c>0", "coords": {} }, { "index": 3226, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3226.png", "answer": "-4", "fact_expr": "G: Ellipse;Expression(G) = (x^2 + y^2/4 = 1);A: Point;B: Point;M: Point;IsChordOf(LineSegmentOf(A,B),G) = True;MidPoint(LineSegmentOf(A,B)) = M;O: Origin", "coords": {} }, { "index": 3227, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3227.png", "answer": "{2*x - y + 4 = 0, 2*x - y - 6 = 0}", "fact_expr": "G: Curve;Expression(G) = (y = x^2);P: Point;Coordinate(P) = (1, 1);l: Line;IsParallel(TangentOnPoint(P, G), l);Distance(TangentOnPoint(P, G), l) = sqrt(5)", "coords": { "P": [ 1.0, 1.0 ] } }, { "index": 3228, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3228.png", "answer": "9", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;G: Line;PointOnCurve(F, G);A: Point;B: Point;Intersection(G, C) = {A, B};N: Point;MidPoint(LineSegmentOf(A, B)) = N;P: Point;Coordinate(P) = (12, 4);WhenMin(Abs(LineSegmentOf(N, A))+Abs(LineSegmentOf(N, P)))", "coords": { "P": [ 12.0, 4.0 ] } }, { "index": 3229, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3229.png", "answer": "(\\sqrt{5}-1)/2", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Z: Ellipse;Expression(Z) = (y^2/a^2+x^2/b^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;RightFocus(G) = F;P: Point;PointOnCurve(P, G);H: Circle;Center(H) = P;IsTangent(H, yAxis);TangentPoint(H, xAxis) = F", "coords": {} }, { "index": 3230, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.0, "major_axis": "y", "x_coef": 4.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3230.png", "answer": "{(0, -2*sqrt(3)), (0, 2*sqrt(3))}", "fact_expr": "G: Ellipse;Expression(G) = (4*x^2 + y^2 = 16)", "coords": {} }, { "index": 3231, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3231.png", "answer": "3/2", "fact_expr": "A: Point;B: Point;PointOnCurve(A, G) = True;PointOnCurve(B, G) = True;G: Parabola;Expression(G) = (y = x^2/4);Abs(LineSegmentOf(A, B)) = 5;MidPoint(LineSegmentOf(A,B)) = M;M1: Point;Projection(M,xAxis) = M1;M: Point", "coords": {} }, { "index": 3232, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3233, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3233.png", "answer": "2*sqrt(3)/3", "fact_expr": "A: Point;B: Point;LeftVertex(G) = A;RightVertex(G) = B;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;P: Point;PointOnCurve(P, G);Negation(P=A);Negation(P=B);m: Number;n: Number;Slope(LineOf(A, P)) = m;Slope(LineOf(B, P)) = n;WhenMin(sqrt(1/(m*n)) + (3*b)/a)", "coords": {} }, { "index": 3234, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 2.3612699570633513, "b": 1.5726654040333643, "orientation": "horizontal" }, "optimization": { "final_loss": 6.189494861157194e-07, "converged": true, "iterations": 81 }, "output_path": "results/train/hyperbola/problem_3234.png", "answer": "y^2/4 - x^2/9 = 1", "fact_expr": "C: Hyperbola;G: Point;Coordinate(G) = (3, 2*sqrt(2));PointOnCurve(G, C);Expression(Asymptote(C)) = (y = pm*(2/3)*x)", "coords": { "G": [ 3.0, 2.8284271247461903 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 3235, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3235.png", "answer": "(sqrt(6) - sqrt(2))/2", "fact_expr": "G: Ellipse;b: Number;a: Number;M: Point;P: Point;Q: Point;F: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(M, G);Z: Circle;Center(Z) = M;Focus(G) = F;TangentPoint(Z, xAxis) = F;Intersection(Z, yAxis) = {P, Q};Negation(P = Q);Abs(VectorOf(M, P) + VectorOf(M, Q)) = Abs(VectorOf(P, Q))", "coords": {} }, { "index": 3236, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3236.png", "answer": "sqrt(3)/2", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>b;b>0;WhenMin(a^2 + 16/(b*(a - b)));e: Number;Eccentricity(G) = e", "coords": {} }, { "index": 3237, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3237.png", "answer": "3/2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a:Number;b:Number;a>0;b>0;Distance(OneOf(Focus(G)),OneOf(Asymptote(G)))=(sqrt(5)/3)*c;c:Number;HalfFocalLength(G)=c", "coords": {} }, { "index": 3238, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3239, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3239.png", "answer": "81/4", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1);F1: Point;LeftFocus(G) = F1;l: Line;PointOnCurve(F2, l) = True;F2: Point;RightFocus(G) = F2;Intersection(l, RightPart(G)) = {A, B};A: Point;B: Point", "coords": {} }, { "index": 3240, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3240.png", "answer": "5", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;Z: Line;PointOnCurve(F1, Z);A: Point;B: Point;Intersection(Z, G) = {A, B};Area(InscribedCircle(TriangleOf(A, B, F2))) = 4*pi;x1: Number;x2: Number;y1: Number;y2: Number;Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2)", "coords": {} }, { "index": 3241, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3242, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3242.png", "answer": "1/9", "fact_expr": "A: Point;M: Point;G: Hyperbola;a: Real;H: Parabola;p: Number;Expression(G) = (-y^2 + x^2/a = 1);p>0;Expression(H) = (y^2 = 2*(p*x));Coordinate(M) = (1, m);PointOnCurve(M, H);Distance(M, Focus(H)) = 5;LeftVertex(G) = A;m:Number;m>0;IsParallel(OneOf(Asymptote(G)), LineOf(A, M))", "coords": {} }, { "index": 3243, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3244, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3245, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3245.png", "answer": "y=pm*sqrt(3)*x", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Ellipse;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2/25 + y^2/9 = 1);Eccentricity(G) = 2;Focus(G) = Focus(H)", "coords": {} }, { "index": 3246, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3247, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3248, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3249, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3249.png", "answer": "sqrt(10)/2", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;A: Point;F1: Point;B: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, G);Intersection(G, LeftPart(C)) = {A, B};VectorOf(A, F1) = 3*VectorOf(F1, B);AngleOf(A, B, F2) = ApplyUnit(90, degree)", "coords": {} }, { "index": 3250, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3250.png", "answer": "(0,2)", "fact_expr": "G: Parabola;Expression(G) = (y = x^2/8)", "coords": {} }, { "index": 3251, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3251.png", "answer": "[7/9, 1)", "fact_expr": "C: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) = 8*Abs(LineSegmentOf(P, F2));LeftFocus(C) = F1;RightFocus(C) = F2", "coords": {} }, { "index": 3252, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.6277244091033936, "b": 1.818325400352478, "orientation": "horizontal" }, "optimization": { "final_loss": 6.648646149187698e-07, "converged": true, "iterations": 99 }, "output_path": "results/train/hyperbola/problem_3252.png", "answer": "x^2/4 - y^2/5 = 1", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Ellipse;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2/12 + y^2/3 = 1);Eccentricity(C) = 3/2;Focus(C) = Focus(G)", "coords": {} }, { "index": 3253, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 8.0, "b": 4.0, "major_axis": "x", "x_coef": 64.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3253.png", "answer": "28", "fact_expr": "l: Line;G: Ellipse;A: Point;F2: Point;B: Point;F1: Point;Expression(G) = (x^2/64 + y^2/16 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, l);Intersection(l, G) = {A, B}", "coords": {} }, { "index": 3254, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3254.png", "answer": "-13/4", "fact_expr": "P: Point;G: Hyperbola;Expression(G) = (x^2 - y^2/12 = 1);PointOnCurve(P, G) = True;F1: Point;F2: Point;Focus(G) = {F1, F2};Abs(LineSegmentOf(P, F1)) = (3/2)*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 3255, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3255.png", "answer": "sqrt(2)/2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 1)", "coords": {} }, { "index": 3256, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3256.png", "answer": "64", "fact_expr": "C: Parabola;p: Number;A: Point;B: Point;T: Point;l1: Line;l2: Line;F: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(T) = (-1, 2);Focus(C)=F;TangentOfPoint(T, C) = {l1, l2};TangentPoint(l1,C)=A;TangentPoint(l2,C)=B;PointOnCurve(F, LineOf(A,B))", "coords": { "T": [ -1.0, 2.0 ] } }, { "index": 3257, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3257.png", "answer": "32/3", "fact_expr": "G: Parabola;p: Number;H: Circle;L: Line;F: Point;A: Point;B: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (y^2 + x^2 - 2*x - 8 = 0);Center(H)=F;Focus(G)=F;PointOnCurve(F, L);Intersection(L, Directrix(G)) = A;OneOf(Intersection(L,G))=B;VectorOf(F, A) = -3*VectorOf(F, B)", "coords": {} }, { "index": 3258, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3258.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "F: Point;Focus(C) = F;C: Parabola;Expression(C) = (y^2 = 4*x);P: Point;Coordinate(P) = (-1, 0);PointOnCurve(P,l) = True;l: Line;Intersection(l, C) = {A, B};A: Point;B: Point;MidPoint(LineSegmentOf(A,B)) = Q;Q: Point;Abs(LineSegmentOf(F, Q)) = 2", "coords": { "P": [ -1.0, 0.0 ] } }, { "index": 3259, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3260, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3261, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.3300262689590454, "b": 2.30381441116333, "orientation": "horizontal" }, "optimization": { "final_loss": 6.658357051492203e-06, "converged": true, "iterations": 122 }, "output_path": "results/train/hyperbola/problem_3261.png", "answer": "x^2/4 - y^2/12 = 1", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;J: Line;OneOf(Asymptote(G)) = J;Expression(J) = (y = sqrt(3)*x);Distance(RightVertex(G), J) = sqrt(3)", "coords": {} }, { "index": 3262, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3262.png", "answer": "2*sqrt(3)", "fact_expr": "C: Hyperbola;b: Number;a: Number;M: Point;F1: Point;O: Origin;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(M, C);(1/3)*Abs(LineSegmentOf(M,F1))=Abs(LineSegmentOf(M,O));Abs(LineSegmentOf(M,O))=Abs(LineSegmentOf(M,F2))", "coords": {} }, { "index": 3263, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3263.png", "answer": "(-4, 9/5)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);P: Point;PointOnCurve(P, G);H: Line;Expression(H) = (4*x - 5*y + 40 = 0);WhenMin(Distance(P,H))", "coords": {} }, { "index": 3264, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3265, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3265.png", "answer": "sqrt(21)/7", "fact_expr": "C: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P, F2), xAxis);Q: Point;OneOf(Intersection(LineOf(P, F1), C)) = Q;Abs(LineSegmentOf(P, F1)) = 4*Abs(LineSegmentOf(F1, Q))", "coords": {} }, { "index": 3266, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3266.png", "answer": "4", "fact_expr": "G: Parabola;p: Real;M: Point;Expression(G) = (y^2 = 2*(p*x));Coordinate(M) = (-2, 8);PointOnCurve(M, Directrix(G))", "coords": { "M": [ -2.0, 8.0 ] } }, { "index": 3267, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3268, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3268.png", "answer": "8", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 12*y);F: Point;Focus(G) = F;P: Point;Coordinate(P) = (2, 1);l: Line;PointOnCurve(P, l);A: Point;B: Point;Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ 2.0, 1.0 ] } }, { "index": 3269, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3269.png", "answer": "2*pi/3", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/3 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G) = True;Area(TriangleOf(P,F1,F2))=sqrt(3)", "coords": {} }, { "index": 3270, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3270.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Slope(OneOf(Asymptote(G))) = sqrt(2);Eccentricity(G)=e;e:Number", "coords": {} }, { "index": 3271, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3271.png", "answer": "18", "fact_expr": "C: Ellipse;A: Point;F1: Point;F2: Point;Expression(C) = (x^2/25 + y^2/9 = 1);LeftFocus(C)=F1;RightFocus(C)=F2;PointOnCurve(A,C)", "coords": {} }, { "index": 3272, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3272.png", "answer": "sqrt(2)", "fact_expr": "C: Ellipse;A: Point;F: Point;B: Point;l: Line;Expression(C) = (x^2/2 + y^2 = 1);RightFocus(C) = F;RightDirectrix(C) = l;In(A, l);Intersection(LineSegmentOf(A, F), C) = B;VectorOf(F, A) = 3*VectorOf(F, B)", "coords": {} }, { "index": 3273, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3273.png", "answer": "2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;H: Line;PointOnCurve(F, H);Slope(H) = 1;A: Point;B: Point;Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, F))*Abs(LineSegmentOf(B, F)) = 8", "coords": {} }, { "index": 3274, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.463192105293274, "b": 2.070901870727539, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4661259228887502e-06, "converged": true, "iterations": 106 }, "output_path": "results/train/hyperbola/problem_3274.png", "answer": "x^2/3 - y^2/6 = 1", "fact_expr": "C: Hyperbola;a: Number;b: Number;a > 0;b > 0;Expression(C) = (x^2/a^2 - y^2/b^2 = 1);Eccentricity(C) = sqrt(3);E: Parabola;Expression(E) = (y^2 = 4*x);OneOf(Directrix(C)) = Directrix(E)", "coords": {} }, { "index": 3275, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3275.png", "answer": "3*sqrt(3)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);M: Point;PointOnCurve(M, G);F1: Point;F2: Point;Focus(G) = {F1, F2};AngleOf(F1, M, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 3276, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3277, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3277.png", "answer": "3", "fact_expr": "G: Hyperbola;A: Point;B: Point;F1: Point;Expression(G) = (x^2 - y^2/3 = 1);LeftFocus(G) = F1;PointOnCurve(F1,LineOf(A,B));Intersection(LineOf(A,B),G)={A,B};Inclination(LineOf(A,B))=pi/6", "coords": {} }, { "index": 3278, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3278.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (2, 0);Distance(P, OneOf(Asymptote(G))) = sqrt(2)", "coords": { "P": [ 2.0, 0.0 ] } }, { "index": 3279, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3279.png", "answer": "1/2", "fact_expr": "G: Parabola;Expression(G) = (x^2 = y)", "coords": {} }, { "index": 3280, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3280.png", "answer": "3*sqrt(21)/14", "fact_expr": "G: Ellipse;A: Point;B: Point;F: Point;C: Point;D: Point;Expression(G) = (x^2/4 + y^2/3 = 1);LeftFocus(G) = F;UpperVertex(G)=A;LowerVertex(G)=C;LeftVertex(G)=B;Intersection(LineOf(A,B), LineOf(F, C)) = D", "coords": {} }, { "index": 3281, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 10.0, "b": 8.0, "major_axis": "x", "x_coef": 100.0, "y_coef": 64.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3281.png", "answer": "(pm*6,0)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/100 + y^2/64 = 1)", "coords": {} }, { "index": 3282, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3282.png", "answer": "9", "fact_expr": "G: Parabola;M: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(M, G);Distance(M, Focus(G)) = 10", "coords": {} }, { "index": 3283, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3283.png", "answer": "sqrt(5)/2", "fact_expr": "G: Parabola;H: Circle;C:Line;P: Point;F: Point;Expression(G) = (y^2 = 4*x);Expression(C) = (x + y = 2);Focus(G)=F;PointOnCurve(P,G);Quadrant(P)=1;IsDiameter(LineSegmentOf(P,F),H);PointOnCurve(Center(H),C)", "coords": {} }, { "index": 3284, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3284.png", "answer": "2*sqrt(5)-1", "fact_expr": "G: Parabola;H: Line;A: Point;P: Point;Expression(G) = (y^2 = 12*x);Expression(H) = (x = -2);Coordinate(A) = (1, 4);PointOnCurve(P, G);Distance(P, H) = d;d:Number", "coords": { "A": [ 1.0, 4.0 ] } }, { "index": 3285, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3285.png", "answer": "16/3", "fact_expr": "l: Line;G: Parabola;Expression(G) = (y^2 = 8*x);F: Point;Focus(G) = F;PointOnCurve(F, l) = True;Intersection(l,G) = {A,B};B: Point;A: Point;Intersection(l,Directrix(G)) = C;C: Point;MidPoint(LineSegmentOf(A, C)) = F", "coords": {} }, { "index": 3286, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3286.png", "answer": "{5/4, 3}", "fact_expr": "C: Parabola;p: Number;P: Point;F: Point;p>0;m:Number;Expression(C) = (y^2 = 2*(p*x));Coordinate(P) = (2, m);Focus(C) = F;PointOnCurve(P, C);O:Origin;d:Number;Distance(P,O)=d;d/Abs(LineSegmentOf(P, F)) = 8/7", "coords": {} }, { "index": 3287, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.449489742783178, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 6.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3287.png", "answer": "sqrt(2)/2", "fact_expr": "G: Ellipse;Expression(G) = (x^2/6 + y^2/3 = 1)", "coords": {} }, { "index": 3288, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3288.png", "answer": "(5*sqrt(3))/2", "fact_expr": "C: Ellipse;Expression(C) = (x^2/9 + y^2/5 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C);AngleOf(P, F1, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 3289, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_3289.png", "answer": "(pm*4,0)\ny=pm*sqrt(3)*x", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 16*x);Eccentricity(G) = 2;OneOf(Focus(G)) = Focus(H)", "coords": {} }, { "index": 3290, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3290.png", "answer": "sqrt(5)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);K: Point;PointOnCurve(K, G);H: Point;Coordinate(H) = (0, 2)", "coords": { "H": [ 0.0, 2.0 ] } }, { "index": 3291, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3291.png", "answer": "4", "fact_expr": "G: Ellipse;F1: Point;F2:Point;P: Point;O: Origin;M: Point;Expression(G) = (y^2/16 + y^2/16 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);MidPoint(LineSegmentOf(F1, P)) = M;Abs(LineSegmentOf(O, M)) = 3", "coords": {} }, { "index": 3292, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3292.png", "answer": "(1/4, -1)", "fact_expr": "G: Parabola;P: Point;Q: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;Coordinate(Q) = (2, -1);PointOnCurve(P, G);WhenMin(Abs(LineSegmentOf(P, Q)) + Abs(LineSegmentOf(P, F)))", "coords": { "Q": [ 2.0, -1.0 ] } }, { "index": 3293, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3293.png", "answer": "4/3", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/2 = 1);P:Point;PointOnCurve(P,G);L1:Line;L2:Line;Asymptote(G)={L1,L2};Distance(P,L1)=d1;Distance(P,L2)=d2;d1:Number;d2:Number", "coords": {} }, { "index": 3294, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3294.png", "answer": "x^2=2*y-2", "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;Expression(G) = (x^2 = 4*y);Focus(G)=F;PointOnCurve(F,l);Intersection(l,G) = {A,B};IsChordOf(LineSegmentOf(A,B),G);MidPoint(LineSegmentOf(A,B))=M;M:Point;F:Point", "coords": {} }, { "index": 3295, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3296, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3296.png", "answer": "2", "fact_expr": "G: Hyperbola;a: Number;H: Circle;a>0;Expression(G) = (-y^2/3 + x^2/a^2 = 1);Expression(H) = (y^2 + (x - 2)^2 = 4);Length(InterceptChord(OneOf(Asymptote(G)),H))=2", "coords": {} }, { "index": 3297, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3298, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3298.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;c: Number;FocalLength(G) = 2*c;F: Point;RightFocus(G) = F;O: Origin;P: Point;PointOnCurve(P, G);Abs(LineSegmentOf(P, O)) = c;Area(TriangleOf(P, O, F)) = (1/2)*(a*b)", "coords": {} }, { "index": 3299, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3299.png", "answer": "sqrt(3)", "fact_expr": "G: Ellipse;C2: Circle;P: Point;M: Point;A: Point;Expression(G) = (x^2/25 + y^2/16 = 1);Expression(C2) = ((x-3)^2+y^2=1);Coordinate(A) = (3, 0);PointOnCurve(P, G);PointOnCurve(M, C2);IsPerpendicular(LineSegmentOf(P, M), LineSegmentOf(A, M))", "coords": { "A": [ 3.0, 0.0 ] } }, { "index": 3300, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3301, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3301.png", "answer": "2*sqrt(2)", "fact_expr": "H: Ellipse;Expression(H) = (y^2/4 + x^2/a^2 = 1);F1: Point;F2: Point;Focus(H) = {F1, F2};a: Number;a>0;P: Point;PointOnCurve(P, H);G: Parabola;Expression(G) = (y^2 = x);PointOnCurve(P, G);F3: Point;Focus(G) = F3;Abs(LineSegmentOf(P, F3)) = 9/4", "coords": {} }, { "index": 3302, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3302.png", "answer": "x=3", "fact_expr": "l: Line;G: Parabola;Expression(G) = (y^2 = 4*x);IsPerpendicular(l,xAxis);Length(InterceptChord(l,G))=4*sqrt(3)", "coords": {} }, { "index": 3303, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3303.png", "answer": "(x-2)^2+(y-4)^2=16", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;P: Point;Coordinate(P) = (2, m);m: Number;m>0;PointOnCurve(P, C);F: Point;Focus(C)=F;Distance(P,F) = 4;G: Circle;Center(G) = P;IsTangent(G, Directrix(C))", "coords": {} }, { "index": 3304, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3304.png", "answer": "2", "fact_expr": "G: Ellipse;a: Number;b: Number;O: Origin;D: Point;A: Point;B: Point;C: Point;E: Point;F: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (1, 0);Eccentricity(G) = sqrt(2)/2;RightFocus(G) = F;PointOnCurve(Vertex(TriangleOf(A, B, C)), G);MidPoint(LineSegmentOf(A, B)) = D;MidPoint(LineSegmentOf(B, C)) = E;MidPoint(LineSegmentOf(A, C)) = F;k1: Number;k2: Number;k3: Number;Slope(OverlappingLine(LineSegmentOf(A, B))) = k1;Slope(OverlappingLine(LineSegmentOf(B, C))) = k2;Slope(OverlappingLine(LineSegmentOf(A, C))) = k3;Negation(k1*k2*k3 = 0);Slope(LineOf(O, D)) + Slope(LineOf(O, E)) + Slope(LineOf(O, F)) = -1", "coords": { "F": [ 1.0, 0.0 ] } }, { "index": 3305, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3305.png", "answer": "3", "fact_expr": "C: Parabola;p: Number;F: Point;l: Line;A: Point;B: Point;p > 0;Expression(C) = (y^2 = 2*p*x);Focus(C) = F;PointOnCurve(F,l);Intersection(l,C) = {A,B};Min(Abs(LineSegmentOf(A,B))) = 6", "coords": {} }, { "index": 3306, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3306.png", "answer": "5/3", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2 + y^2 = a^2);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, RightPart(C));IsTangent(LineOf(P, F1), G);Abs(LineSegmentOf(P, F2)) = Abs(LineSegmentOf(F1, F2))", "coords": {} }, { "index": 3307, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3308, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3308.png", "answer": "x^2/8 - y^2/6 = 1", "fact_expr": "H: Point;Coordinate(H) = (3, 2);G: Hyperbola;Expression(G) = (-x^2/3 + y^2/4 = 1);Z: Hyperbola;Asymptote(Z) = Asymptote(G);PointOnCurve(H, Z)", "coords": { "H": [ 3.0, 2.0 ] } }, { "index": 3309, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0010501146316528, "direction": "right" }, "optimization": { "final_loss": 3.455598402979376e-07, "converged": true, "iterations": 81 }, "output_path": "results/train/parabola/problem_3309.png", "answer": "y^2=4*x", "fact_expr": "G: Parabola;p: Number;H: Point;p>0;Expression(G) = (y^2 = 2*p*x);Coordinate(H) = (1, 2);PointOnCurve(H,G)", "coords": { "H": [ 1.0, 2.0 ] } }, { "index": 3310, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 3.0, "major_axis": "x", "x_coef": 36.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3310.png", "answer": "-1/2", "fact_expr": "G: Ellipse;Expression(G) = (x^2/36 + y^2/9 = 1);P: Point;Coordinate(P) = (4, 2);l: Line;PointOnCurve(P, l);Intersection(l, G) = {A, B};A: Point;B: Point;MidPoint(LineSegmentOf(A, B)) = P;OverlappingLine(LineSegmentOf(A, B)) = l", "coords": { "P": [ 4.0, 2.0 ] } }, { "index": 3311, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 2.8284271247461903, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3311.png", "answer": "8", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - 2*y^2 = 16)", "coords": {} }, { "index": 3312, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_3312.png", "answer": "sqrt(6)/3", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/2 + x^2/a^2 = 1);a: Number;a>0;Eccentricity(G) = 2", "coords": {} }, { "index": 3313, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 2.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3313.png", "answer": "20", "fact_expr": "l: Line;G: Ellipse;A: Point;B: Point;F2: Point;F1: Point;Expression(G) = (x^2/25 + y^2/4 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, l);Intersection(l, G) = {A, B}", "coords": {} }, { "index": 3314, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3314.png", "answer": "sqrt(3)/2", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);l: Line;H: Point;Coordinate(H) = (1, 0);PointOnCurve(H, l) = True;Intersection(l, G) = {A, B};A: Point;B: Point;O: Origin", "coords": { "H": [ 1.0, 0.0 ] } }, { "index": 3315, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3315.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;e: Number;Expression(G) = (x^2/4 - y^2/4 = 1);Eccentricity(G) = e", "coords": {} }, { "index": 3316, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3317, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3317.png", "answer": "x^2/12 + y^2/4 = 1", "fact_expr": "G: Ellipse;a: Number;b: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, G);Focus(G) = {F1, F2};Eccentricity(G) = sqrt(6)/3;DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;Area(TriangleOf(P, F1, F2)) = 4", "coords": {} }, { "index": 3318, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3318.png", "answer": "1/3", "fact_expr": "H: Ellipse;Expression(H) = (x^2/6 + y^2/2 = 1);G: Hyperbola;Expression(G) = (x^2/3 - y^2 = 1);F1: Point;F2: Point;Focus(H) = {F1, F2};Focus(G) = {F1, F2};P: Point;OneOf(Intersection(H, G)) = P", "coords": {} }, { "index": 3319, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3319.png", "answer": "pm*4", "fact_expr": "G: Parabola;P: Point;F: Point;Expression(G) = (x^2 = 4*y);PointOnCurve(P, G);Focus(G) = F;Distance(P, F) = 5", "coords": {} }, { "index": 3320, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3320.png", "answer": "3*sqrt(2)/4", "fact_expr": "G: Hyperbola;Expression(G) = (y^2/9 - x^2/m^2 = 1);m: Number;m>0;Distance(OneOf(Vertex(G)), OneOf(Asymptote(G))) = 1", "coords": {} }, { "index": 3321, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3321.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/4 = 1)", "coords": {} }, { "index": 3322, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3323, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3324, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 5.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3324.png", "answer": "y=pm*(4/5)*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/25 - y^2/16 = 1)", "coords": {} }, { "index": 3325, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3325.png", "answer": "b^2", "fact_expr": "G: Hyperbola;a: Number;b: Number;F1: Point;F2: Point;P: Point;M: Point;c: Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, G);Negation(P = Vertex(G));LeftFocus(G) = F1;RightFocus(G) = F2;HalfFocalLength(G) = c;TangentPoint(InscribedCircle(TriangleOf(P, F1, F2)), LineSegmentOf(F1, F2)) = M", "coords": {} }, { "index": 3326, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3327, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3327.png", "answer": "9/2", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*x);F: Point;Focus(C) = F;B: Point;A: Point;PointOnCurve(A, C);PointOnCurve(B, C);Negation(A = B);PointOnCurve(F, LineOf(A, B))", "coords": {} }, { "index": 3328, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3328.png", "answer": "4*a", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;A: Point;F2: Point;B: Point;F1: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Focus(G)={F1,F2};PointOnCurve(F1,H);Intersection(H,LeftPart(G)) = {A, B};Abs(LineSegmentOf(A, F2)) + Abs(LineSegmentOf(B, F2)) = 2*Abs(LineSegmentOf(A, B));LeftFocus(G)=F1", "coords": {} }, { "index": 3329, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3330, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3330.png", "answer": "2*sqrt(2)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, C) = {A, B};M: Point;Intersection(Directrix(C), xAxis) = M;Abs(LineSegmentOf(A, B)) = 8", "coords": {} }, { "index": 3331, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3331.png", "answer": "(-4, 2)", "fact_expr": "C: Parabola;M: Point;P: Point;F: Point;Expression(C) = (x^2 = 8*y);Coordinate(M) = (-4, 3);Focus(C) = F;PointOnCurve(P, C);WhenMin(Perimeter(TriangleOf(P, M, F)))", "coords": { "M": [ -4.0, 3.0 ] } }, { "index": 3332, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3332.png", "answer": "2", "fact_expr": "l: Line;C: Parabola;p: Number;A: Point;F: Point;B: Point;M: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;PointOnCurve(F,l);Intersection(l, C) = {A,B};Intersection(l, yAxis) = M;Abs(LineSegmentOf(A, F)) = Abs(LineSegmentOf(B, M));Abs(LineSegmentOf(A, B)) = 2 + sqrt(5)", "coords": {} }, { "index": 3333, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3333.png", "answer": "(0,-1)", "fact_expr": "G: Parabola;Expression(G) = (y = -x^2/4)", "coords": {} }, { "index": 3334, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3334.png", "answer": "x^2/16+y^2/4=1", "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;B: Point;F1: Point;F2: Point;e:Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Focus(G) = {F1, F2};PointOnCurve(F2, LineSegmentOf(A, B));IsChordOf(LineSegmentOf(A, B), G);Perimeter(TriangleOf(A, F1, B)) = 16;Eccentricity(G) = e;e=sqrt(3)/2", "coords": {} }, { "index": 3335, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3335.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;k: Number;x1:Number;x2:Number;y2:Number;Expression(G) = (x^2 = 4*y);Coordinate(A) = (x1,y1);Coordinate(B)=(x2,y2);PointOnCurve(Focus(G), l);Slope(l)=k;Intersection(l, G)={A,B};Abs(LineSegmentOf(A, B)) = 8;k>0;x10;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;Inclination(H)=ApplyUnit(60, degree) ;Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, F)) > Abs(LineSegmentOf(B, F));Abs(LineSegmentOf(A, F)) = 2;PointOnCurve(F,H)", "coords": {} }, { "index": 3341, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3341.png", "answer": "a/sqrt(a^2-b^2)", "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;I: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2;Incenter(TriangleOf(P, F1, F2)) = I;Area(TriangleOf(P,I,F1))+Area(TriangleOf(P,I,F2))=lambda*Area(TriangleOf(F1,I,F2));lambda: Number", "coords": {} }, { "index": 3342, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3342.png", "answer": "y=x+1", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/2 = 1);A: Point;B: Point;PointOnCurve(A, G) = True;PointOnCurve(B, G) = True;M: Point;Coordinate(M) = (1, 2);MidPoint(LineSegmentOf(A, B)) = M", "coords": { "M": [ 1.0, 2.0 ] } }, { "index": 3343, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3343.png", "answer": "2\nx=-1", "fact_expr": "G: Parabola;p: Number;Expression(G) = (y^2 = 2*(p*x));Coordinate(Focus(G)) = (1, 0)", "coords": {} }, { "index": 3344, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3344.png", "answer": "16/3", "fact_expr": "C: Parabola;A: Point;F: Point;B: Point;Expression(C) = (y^2 = 4*x);Focus(C)=F;PointOnCurve(A,C);PointOnCurve(B,C);VectorOf(A,F)=3*VectorOf(F,B)", "coords": {} }, { "index": 3345, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3345.png", "answer": "7/5", "fact_expr": "l: Line;Expression(l) = (4*x - 3*y + 8 = 0);G: Parabola;Expression(G) = (y^2 = 4*x);P: Point;PointOnCurve(P, G)", "coords": {} }, { "index": 3346, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3346.png", "answer": "[3+2*\\sqrt{3},+\\infty)", "fact_expr": "G: Hyperbola;O: Point;P: Point;F: Point;Expression(G) = (x^2/3 - y^2 = 1);PointOnCurve(P, RightPart(G));Center(G)=O;LeftFocus(G)=F", "coords": {} }, { "index": 3347, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3348, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3349, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3349.png", "answer": "[\\sqrt{2}-1,1)", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G) = True;Abs(LineSegmentOf(P, F1)) = e*Abs(LineSegmentOf(P, F2));e: Number;Eccentricity(G) = e", "coords": {} }, { "index": 3350, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3350.png", "answer": "{2, 6}", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(C) = F;A: Point;Coordinate(A) = (0, sqrt(3));B: Point;PointOnCurve(B, C);IsPerpendicular(LineSegmentOf(A, B), LineSegmentOf(A, F));Abs(LineSegmentOf(B, F)) = 4", "coords": { "A": [ 0.0, 1.7320508075688772 ] } }, { "index": 3351, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3352, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_focus_mismatch", "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3352.png", "answer": "2*sqrt(10)/5", "fact_expr": "C: Ellipse;a: Number;b: Number;A1: Circle;G: Line;A: Point;P: Point;Q: Point;O: Origin;F: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y = x*(b/a));Coordinate(F) = (sqrt(3), 0);OneOf(Focus(C)) = F;RightVertex(C) = A;Center(A1) = A;Intersection(A1, G) = {P, Q};DotProduct(VectorOf(A, Q), VectorOf(A, P)) = 0;VectorOf(O, P) = 3*VectorOf(O, Q)", "coords": { "F": [ 1.7320508075688772, 0.0 ] }, "validation_reasons": [ "ellipse_focus_mismatch" ] }, { "index": 3353, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.4641016151377544, "major_axis": "x", "x_coef": 16.0, "y_coef": 12.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3353.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Ellipse;A: Point;B: Point;M: Point;Expression(G) = (x^2/16 + y^2/12 = 1);Coordinate(A) = (-2,sqrt(3));Coordinate(B) = (2, 0);PointOnCurve(M, G);WhenMax(Abs(LineSegmentOf(M,A))+Abs(LineSegmentOf(M,B)))", "coords": { "A": [ -2.0, 1.7320508075688772 ], "B": [ 2.0, 0.0 ] } }, { "index": 3354, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3354.png", "answer": "(-oo, -2-sqrt(2)) + (2, +oo)", "fact_expr": "l: Line;G: Hyperbola;H: Line;k: Number;I: Point;A: Point;B: Point;Expression(G) = (x^2 - y^2 = 1);Expression(H) = (y = k*x + 1);Coordinate(I) = (-2, 0);Intersection(H, LeftPart(G)) = {A, B};PointOnCurve(I, l);PointOnCurve(MidPoint(LineSegmentOf(A, B)), l);b: Number;Intercept(l, yAxis) = b", "coords": { "I": [ -2.0, 0.0 ] } }, { "index": 3355, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3355.png", "answer": "3 + 2*sqrt(2)", "fact_expr": "G: Ellipse;b: Number;a: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);-3*c^2 + a^2 + b^2 = 0;c: Number;HalfFocalLength(G) = c", "coords": {} }, { "index": 3356, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3356.png", "answer": "8", "fact_expr": "G: Ellipse;H: Line;A: Point;B: Point;F2: Point;F1: Point;Expression(G) = (x^2/4 + y^2/2 = 1);PointOnCurve(F1, H);Intersection(H, G) = {A, B};OneOf(Focus(G))=F1;OneOf(Focus(G))=F2;Negation(F1=F2)", "coords": {} }, { "index": 3357, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3357.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;F: Point;P: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 8*x);OneOf(Focus(G)) = F;Focus(H) = F;OneOf(Intersection(G,H))=P;Abs(LineSegmentOf(F, P)) = 5", "coords": {} }, { "index": 3358, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3358.png", "answer": "13", "fact_expr": "P: Point;PointOnCurve(P, RightPart(G)) = True;G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1);C1: Circle;Expression(C1) = (y^2 + (x + 5)^2 = 9);PointOnCurve(M,C1) = True;M: Point;C2: Circle;Expression(C2) = (y^2 + (x - 5)^2 = 4);PointOnCurve(N,C2) = True;N: Point", "coords": {} }, { "index": 3359, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3359.png", "answer": "(1, \\sqrt{2}]", "fact_expr": "G: Ellipse;b: Number;a: Number;a>b;b>0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);HalfFocalLength(G) = c;c: Number", "coords": {} }, { "index": 3360, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3361, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3361.png", "answer": "[1/2, 1)", "fact_expr": "E: Ellipse;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;F: Point;c: Number;Coordinate(F) = (c, 0);RightFocus(E) = F;G: Line;Expression(G) = (x = a^2/c);A: Point;Intersection(G, xAxis) = A;P: Point;PointOnCurve(P, E);PointOnCurve(F, PerpendicularBisector(LineSegmentOf(A, P)))", "coords": {} }, { "index": 3362, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3362.png", "answer": "8/5", "fact_expr": "H: Line;Slope(H) = 1;G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);PointOnCurve(RightFocus(G), H);A: Point;B: Point;Intersection(H, G) = {A, B};IsChordOf(LineSegmentOf(A, B), G)", "coords": {} }, { "index": 3363, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0010501146316528, "direction": "right" }, "optimization": { "final_loss": 3.455598402979376e-07, "converged": true, "iterations": 81 }, "output_path": "results/train/parabola/problem_3363.png", "answer": "x=-1", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;P: Point;Coordinate(P) = (1, 2);PointOnCurve(P, C)", "coords": { "P": [ 1.0, 2.0 ] } }, { "index": 3364, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3364.png", "answer": "-1/3", "fact_expr": "C: Parabola;G: Line;F: Point;A: Point;B: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F, G);Slope(G) = sqrt(3);Intersection(G, C) = {A, B};Abs(LineSegmentOf(F, A)) > Abs(LineSegmentOf(F, B))", "coords": {} }, { "index": 3365, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3365.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Distance(Focus(G),OneOf(Asymptote(G))) = Length(RealAxis(G))", "coords": {} }, { "index": 3366, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3367, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3367.png", "answer": "x-y-3=0", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1);M: Point;Coordinate(M) = (4, 1);PointOnCurve(M, H);H: LineSegment;IsChordOf(H, G);MidPoint(H) = M", "coords": { "M": [ 4.0, 1.0 ] } }, { "index": 3368, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3369, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3370, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3370.png", "answer": "8*x-y-15=0", "fact_expr": "G: Parabola;H: LineSegment;I: Point;Expression(G) = (y^2 = 16*x);Coordinate(I) = (2, 1);PointOnCurve(I,H);MidPoint(H)=I;IsChordOf(H,G)", "coords": { "I": [ 2.0, 1.0 ] } }, { "index": 3371, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3371.png", "answer": "y=pm*(sqrt(3)/3)*x", "fact_expr": "G: Hyperbola;a: Number;b: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);FocalLength(G) = 2 * Length(ImageinaryAxis(G))", "coords": {} }, { "index": 3372, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3372.png", "answer": "16", "fact_expr": "G: Ellipse;Expression(G) = (x^2/10 + y^2/m = 1);m: Number;PointOnCurve(Focus(G), yAxis);Length(MajorAxis(G)) = 8", "coords": {} }, { "index": 3373, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3373.png", "answer": "8", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);l: Line;PointOnCurve(Focus(G), l);P: Point;Q: Point;x1: Number;x2: Number;y1: Number;y2: Number;Coordinate(P) = (x1, y1);Coordinate(Q) = (x2, y2);Intersection(l, G) = {P, Q};x1 + x2 = 6", "coords": {} }, { "index": 3374, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3374.png", "answer": "3", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2 + x^2/m = 1);m: Number;m>0;H: Parabola;Expression(H) = (y^2 = 8*x);RightFocus(G) = Focus(H)", "coords": {} }, { "index": 3375, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 3.3166247903554, "major_axis": "x", "x_coef": 36.0, "y_coef": 11.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3375.png", "answer": "2", "fact_expr": "C: Ellipse;F1: Point;F2:Point;M: Point;N: Point;O: Origin;Expression(C)=(x^2/36 + y^2/11 = 1);LeftFocus(C) =F1;RightFocus(C)=F2;PointOnCurve(M, C);MidPoint(LineSegmentOf(M, F1)) = N;Abs(LineSegmentOf(M, N)) = 4", "coords": {} }, { "index": 3376, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3376.png", "answer": "9/8", "fact_expr": "P: Point;G: Hyperbola;Expression(G) = (x^2/9 - y^2/3 = 1);PointOnCurve(P, RightPart(G)) = True;l1: Line;l2: Line;M: Point;N: Point;l3: Line;l4: Line;Asymptote(G) = {l3,l4};PointOnCurve(P,l1) =True;PointOnCurve(P,l2) = True;IsPerpendicular(l1,l3) = True;IsPerpendicular(l2,l4) = True;FootPoint(l1, l3) = M;FootPoint(l2, l4) = N", "coords": {} }, { "index": 3377, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3378, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3379, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4149149626832767, "b": 2.450642358201144, "orientation": "horizontal" }, "optimization": { "final_loss": 1.0319236343469396e-05, "converged": true, "iterations": 94 }, "output_path": "results/train/hyperbola/problem_3379.png", "answer": "x^2 - y^2/3 = 1", "fact_expr": "G: Hyperbola;Expression(Asymptote(G)) = (y = pm*sqrt(3)*x)", "coords": {} }, { "index": 3380, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3380.png", "answer": "5/2", "fact_expr": "C: Parabola;P: Point;F: Point;Expression(C) = (y^2 = 2*x);Focus(C) = F;PointOnCurve(P,C);XCoordinate(P)=2", "coords": {} }, { "index": 3381, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3381.png", "answer": "-4", "fact_expr": "l: Line;G: Parabola;Expression(G) = (x^2 = 4*y);A: Point;B:Point;PointOnCurve(Focus(G),l);Coordinate(A) = (x1,y1);Coordinate(B)=(x2,y2);Intersection(l, G) = {A,B};x1:Number;x2:Number;y1:Number;y2:Number", "coords": {} }, { "index": 3382, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3383, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3383.png", "answer": "sqrt(3)", "fact_expr": "C: Hyperbola;Expression(C)=(x^2/a^2-y^2/b^2=1);G: Circle;a:Number;b:Number;a>0;b>0;F1: Point;F2:Point;M: Point;LeftFocus(C)=F1;RightFocus(C)=F2;Coordinate(M) = (sqrt(3), 2);PointOnCurve(M, RightPart(C));PointOnCurve(F2,G);IsDiameter(LineSegmentOf(M,F1),G)", "coords": { "M": [ 1.7320508075688772, 2.0 ] } }, { "index": 3384, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3384.png", "answer": "2*sqrt(3)", "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2 = 1);F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2))", "coords": {} }, { "index": 3385, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3386, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 1.4149149626832767, "b": 2.450642358201144, "orientation": "horizontal" }, "optimization": { "final_loss": 1.0319236343469396e-05, "converged": true, "iterations": 94 }, "output_path": "results/train/hyperbola/problem_3386.png", "answer": "x^2 - y^2 / 3 = 1", "fact_expr": "G: Hyperbola;F: Point;Coordinate(F) = (2, 0);OneOf(Focus(G))=F;Expression(Asymptote(G))=(y=pm*sqrt(3)*x)", "coords": { "F": [ 2.0, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 3387, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3387.png", "answer": "x^2/4+y^2/3=1", "fact_expr": "C: Ellipse;b: Number;a: Number;F1: Point;P: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;UpperVertex(C)=P;AngleOf(F1, P, F2) = pi/3;Area(TriangleOf(F1,P,F2))=sqrt(3)", "coords": {} }, { "index": 3388, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3388.png", "answer": "3*sqrt(3)/5", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);P: Point;PointOnCurve(P, G) = True;F1: Point;F2: Point;Focus(G) = {F1,F2};AngleOf(P, F1, F2) = ApplyUnit(120, degree)", "coords": {} }, { "index": 3389, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3390, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.6457513110645907, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 7.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3390.png", "answer": "2*sqrt(7)/7", "fact_expr": "G: Ellipse;Expression(G) = (x^2/7 + y^2/3 = 1)", "coords": {} }, { "index": 3391, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3391.png", "answer": "sqrt(2)", "fact_expr": "F: Point;P: Point;G: Hyperbola;b: Number;a: Number;H: Circle;M: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2 + y^2 = a^2);RightFocus(G)=F;TangentOfPoint(F,H)=LineOf(F,M);TangentPoint(LineOf(F,M),H)=M;Intersection(LineOf(F,M), yAxis)=P;MidPoint(LineSegmentOf(F, P)) = M", "coords": {} }, { "index": 3392, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3392.png", "answer": "(0, pm*sqrt(13))", "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/4 + y^2/9 = 1)", "coords": {} }, { "index": 3393, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3394, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3395, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3395.png", "answer": "arccos15/17", "fact_expr": "G: Hyperbola;H: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/16 - y^2/9 = 1);Expression(H) = (x^2/169 + y^2/144 = 1);Focus(G) = {F1, F2};Focus(H) = {F1, F2};OneOf(Intersection(G,H))=P", "coords": {} }, { "index": 3396, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3396.png", "answer": "2*sqrt(5)", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C) = True;Cos(AngleOf(F1, P, F2)) = 11/14;Area(TriangleOf(F1, P, F2)) = sqrt(3)", "coords": {} }, { "index": 3397, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3397.png", "answer": "{(11*sqrt(5)/15-2)*a, (11/6-sqrt(5))*b}", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Eccentricity(G) = sqrt(5)/5;F: Point;RightFocus(G) = F;A: Point;PointOnCurve(A, G);l: Line;Expression(l) = (b*x + a*y - a^2 - b^2 = 0);d: Number;Distance(A, l) = d", "coords": {} }, { "index": 3398, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3398.png", "answer": "{sqrt(5), sqrt(5)/2}", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(C) = F;L: Line;PointOnCurve(F, L);IsPerpendicular(L, OneOf(Asymptote(C)));A: Point;FootPoint(L, OneOf(Asymptote(C))) = A;O: Origin;Area(TriangleOf(O, A, F)) = (a^2+b^2)/5", "coords": {} }, { "index": 3399, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3399.png", "answer": "pm*2*sqrt(2)", "fact_expr": "G: Parabola;p: Number;H: Point;I: Point;t: Number;Expression(G) = (y^2 = 2*(p*x));Coordinate(H) = (2, t);Coordinate(I) = (p/2, 0);PointOnCurve(H, G);Distance(H, I) = 3", "coords": {} }, { "index": 3400, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3400.png", "answer": "2", "fact_expr": "E: Hyperbola;b: Number;a: Number;C: Circle;a>0;b>0;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);Expression(C) = (x^2 + (y - 4)^2 = 4);IsTangent(C, Asymptote(E))", "coords": {} }, { "index": 3401, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3401.png", "answer": "sqrt(2)-1", "fact_expr": "C: Ellipse;a: Number;b: Number;G: Circle;P: Point;F1: Point;a > b;b > 0;k:Number;k>=1;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2 + y^2 = a^2 - b^2);PointOnCurve(P, C);LeftFocus(C) = F1;PointOnCurve(MidPoint(LineSegmentOf(P,F1)), G);Slope(LineOf(P,F1)) = k", "coords": {} }, { "index": 3402, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3402.png", "answer": "ApplyUnit(60, degree)", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (16*x^2 - 9*y^2 = 144);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1))*Abs(LineSegmentOf(P, F2)) = 64", "coords": {} }, { "index": 3403, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3403.png", "answer": "{-1, 2}", "fact_expr": "C: Parabola;G: Line;P: Point;A: Point;B: Point;k: Number;Expression(C) = (y = x^2/4);Coordinate(P) = (2, -2);PointOnCurve(Focus(C), G);Slope(G) = k;Intersection(G, C) = {A, B};DotProduct(VectorOf(P, A), VectorOf(P, B)) = 25", "coords": { "P": [ 2.0, -2.0 ] } }, { "index": 3404, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.3166247903554, "b": 2.449489742783178, "major_axis": "x", "x_coef": 11.0, "y_coef": 6.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3404.png", "answer": "2*sqrt(11)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/11 + y^2/6 = 1);H: Point;PointOnCurve(H, G);F1: Point;F2: Point;Focus(G) = {F1, F2}", "coords": {} }, { "index": 3405, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3405.png", "answer": "pm*3*sqrt(5)/5", "fact_expr": "l: Line;G: Ellipse;H: Circle;B: Point;A: Point;Expression(G) = (x^2 + y^2/2 = 1);Expression(H) = (x^2 + y^2 = 1);Expression(l) = (m + x - y = 0);Intersection(l, G) = {A, B};PointOnCurve(MidPoint(LineSegmentOf(A,B)), H);m:Number", "coords": {} }, { "index": 3406, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3406.png", "answer": "1/2", "fact_expr": "G: Parabola;H: Ellipse;m: Number;Expression(G) = (y^2 = 4*x);Expression(H) = (m*x^2 + y^2 = 1);OneOf(Focus(H)) = Focus(G)", "coords": {} }, { "index": 3407, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.23606797749979, "b": 2.0, "major_axis": "x", "x_coef": 5.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3407.png", "answer": "1/2", "fact_expr": "G: Ellipse;F: Point;M: Point;A: Point;l: Line;Expression(G) = (x^2/5 + y^2/4 = 1);RightFocus(G) = F;RightDirectrix(G) = l;UpperVertex(G)=A;PointOnCurve(A,LineSegmentOf(A,M));IsPerpendicular(LineSegmentOf(A,M),l);FootPoint(LineSegmentOf(A,M),l)=M", "coords": {} }, { "index": 3408, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3408.png", "answer": "x^2=16*y", "fact_expr": "C1: Hyperbola;Expression(C1) = ((x^2/a^2) - (y^2/b^2) = 1);a: Number;b: Number;a>0;Eccentricity(C1)=2;C2: Parabola;Expression(C2)=(x^2=2*p*y);p:Number;p>0;Distance(Focus(C2),Asymptote(C1))=2;b>0", "coords": {} }, { "index": 3409, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3409.png", "answer": "4", "fact_expr": "M: Point;Coordinate(M) = (4, 0);P: Point;PointOnCurve(P, G);G: Curve;Expression(G) = (y^2 = 8*x);Q: Point;G1:Curve;Expression(G1) = (y^2 + (x - 2)^2 = 1);PointOnCurve(Q, G1)", "coords": { "M": [ 4.0, 0.0 ] } }, { "index": 3410, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3410.png", "answer": "2+sqrt(5)", "fact_expr": "G: Parabola;P: Point;x1:Number;y1:Number;Expression(G) = (y^2 = 8*x);Coordinate(P) = (x1, y1);PointOnCurve(P, G)", "coords": {} }, { "index": 3411, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3412, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3412.png", "answer": "{3, 7}", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2 - y^2/2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 5", "coords": {} }, { "index": 3413, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3413.png", "answer": "4", "fact_expr": "C: Parabola;A: Point;F: Point;P: Point;Expression(C) = (y = x^2/4);Coordinate(A) = (4, 1);Focus(C)=F;PointOnCurve(P, C)", "coords": { "A": [ 4.0, 1.0 ] } }, { "index": 3414, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_3414.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;G: Parabola;Expression(G) = (y^2 = 8*x);OneOf(Focus(C)) = Focus(G);Eccentricity(C) = 2", "coords": {} }, { "index": 3415, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3415.png", "answer": "9/2", "fact_expr": "G: Parabola;A: Point;P: Point;M: Point;Expression(G) = (y^2 = 2*x);Coordinate(A) = (7/2, -4);PointOnCurve(P, G);Projection(P, yAxis) = M", "coords": { "A": [ 3.5, -4.0 ] } }, { "index": 3416, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3417, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3417.png", "answer": "1/2", "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;I: Point;a>0;b>0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2;Incenter(TriangleOf(P,F1,F2))=I;Area(TriangleOf(I,P,F1))+Area(TriangleOf(I,P,F2))=2*Area(TriangleOf(I,F1,F2))", "coords": {} }, { "index": 3418, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3418.png", "answer": "-8*sqrt(3)/9", "fact_expr": "G: Parabola;p: Number;A: Point;B: Point;O: Origin;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;IsChordOf(LineSegmentOf(A,B),G);PointOnCurve(F, LineSegmentOf(A, B));DotProduct(VectorOf(O, A), VectorOf(O, B)) =-3;Inclination(LineSegmentOf(A,B))=ApplyUnit(60,degree)", "coords": {} }, { "index": 3419, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3419.png", "answer": "{2*sqrt(3)/3, 2}", "fact_expr": "C: Hyperbola;a: Number;b: Number;F1: Point;F2: Point;P: Point;A: Point;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Focus(C) = {F1, F2};PointOnCurve(P, C);l: Line;l1: Line;l2: Line;Asymptote(C) = {l1, l2};PointOnCurve(P, l);IsPerpendicular(l, l1);IsPerpendicular(l, l2);FootPoint(l, l1) = A;FootPoint(l, l2) = B;3*Abs(LineSegmentOf(F1, F2))^2 = (64*Abs(LineSegmentOf(P, A)))*Abs(LineSegmentOf(P, B))", "coords": {} }, { "index": 3420, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -3.0, 0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_3420.png", "answer": "y=x^2/2", "fact_expr": "G: Parabola;a: Number;C: Circle;l: Line;a>0;Expression(G) = (y = a*x^2);Expression(C) = (y^2 + (x - 3)^2 = 1);Directrix(G) = l;Length(InterceptChord(l,C))=sqrt(3)", "coords": {} }, { "index": 3421, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3421.png", "answer": "2*sqrt(13)", "fact_expr": "G: Parabola;A: Point;F: Point;P: Point;O: Origin;Expression(G) = (y^2 = -8*x);Focus(G) = F;PointOnCurve(P,Directrix(G));PointOnCurve(A, G);Abs(LineSegmentOf(A, F)) = 4", "coords": {} }, { "index": 3422, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3422.png", "answer": "5/2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);IsChordOf(LineSegmentOf(A, B), G) = True;A: Point;B: Point;Length(LineSegmentOf(A, B)) = 9", "coords": {} }, { "index": 3423, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3423.png", "answer": "x^2/20+y^2/16=1", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;PointOnCurve(Focus(G), xAxis);I: Point;Coordinate(I) = (2, 1);L1: Line;L2: Line;TangentOfPoint(I, H) = {L1, L2};H: Circle;Expression(H) = (x^2 + y^2 = 4);TangentPoint(L1, H) = A;TangentPoint(L2, H) = B;A: Point;B: Point;PointOnCurve(RightFocus(G), LineOf(A, B));PointOnCurve(UpperVertex(G), LineOf(A, B))", "coords": { "I": [ 2.0, 1.0 ] } }, { "index": 3424, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3424.png", "answer": "[3/2, +\\infty)", "fact_expr": "C: Hyperbola;b: Number;a: Number;M: Point;F1: Point;F2: Point;O: Origin;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) =F1;RightFocus(C) = F2;PointOnCurve(M, C);Abs(LineSegmentOf(M, F1)) + Abs(LineSegmentOf(M, F2)) = 3*Abs(LineSegmentOf(M, O))", "coords": {} }, { "index": 3425, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3426, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3427, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3427.png", "answer": "5", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/4 = 1);P: Point;PointOnCurve(P, G);F1: Point;F2: Point;Focus(G) = {F1, F2}", "coords": {} }, { "index": 3428, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3428.png", "answer": "2*sqrt(5)/5", "fact_expr": "C: Hyperbola;P: Point;A: Point;Expression(C) = (x^2/4 - y^2 = 1);Coordinate(A) = (3, 0);PointOnCurve(P, C)", "coords": { "A": [ 3.0, 0.0 ] } }, { "index": 3429, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3430, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3431, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3431.png", "answer": "(-2,1/2)", "fact_expr": "G: Parabola;M: Point;P: Point;F: Point;Expression(G) = (y = x^2/8);Coordinate(M) = (-2, 4);Focus(G)=F;PointOnCurve(P,G);WhenMin(Abs(LineSegmentOf(P,M))+Abs(LineSegmentOf(P,F)))", "coords": { "M": [ -2.0, 4.0 ] } }, { "index": 3432, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "y", "x_coef": 9.0, "y_coef": 25.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3432.png", "answer": "y^2 - x^2/15 = 1", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/9 + y^2/25 = 1);Focus(H) = Focus(G);Eccentricity(G) + Eccentricity(H) = 24/5", "coords": {} }, { "index": 3433, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3433.png", "answer": "y^2=4*sqrt(2)*x", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;F: Point;Focus(G) = F;M: Point;PointOnCurve(M, G);N: Point;Intersection(Directrix(G),axis)=N;Abs(LineSegmentOf(M,F))=(7/2)*p;Area(TriangleOf(M,N,F))=4*sqrt(6)", "coords": {} }, { "index": 3434, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3434.png", "answer": "8", "fact_expr": "E: Point;PointOnCurve(E, yAxis);F: Point;Focus(G) = F;G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;M: Point;N: Point;Intersection(LineOf(E, F), G) = {M, N};MidPoint(LineSegmentOf(E, F)) = M;Abs(LineSegmentOf(N, F)) = 12", "coords": {} }, { "index": 3435, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3435.png", "answer": "pi/2", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 - y^2/16 = 1);Focus(G)={F1,F2};PointOnCurve(P, G);Abs(LineSegmentOf(P, F1))*Abs(LineSegmentOf(P, F2)) = 32", "coords": {} }, { "index": 3436, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3436.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;PointOnCurve(LeftFocus(G), L) = True;IsPerpendicular(L, xAxis) = True;L: Line;Intersection(L, G) = {M, N};M: Point;N: Point;Q: Circle;IsDiameter(LineSegmentOf(M, N), Q) = True;PointOnCurve(RightVertex(G), Q) = True", "coords": {} }, { "index": 3437, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3437.png", "answer": "3-(3*sqrt(3)/2)", "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/4 + y^2 = 1);Focus(C) = {F1, F2};PointOnCurve(P,C);Quadrant(P)=1;XCoordinate(P)=1", "coords": {} }, { "index": 3438, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3438.png", "answer": "y^2 = 6*x", "fact_expr": "C: Parabola;p: Number;M: Point;H: Point;F: Point;l: Line;l1: Line;p>0;Expression(C) = (y^2 = 2*p*x);Focus(C) = F;Directrix(C) = l;PointOnCurve(M, C);PointOnCurve(M, l1);IsPerpendicular(l,l1);Intersection(l,l1)=H;Abs(LineSegmentOf(M, H)) = 2;AngleOf(H, F, M) = ApplyUnit(30, degree)", "coords": {} }, { "index": 3439, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3440, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.4641016151377544, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 12.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3440.png", "answer": "7", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/12 + y^2/3 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);PointOnCurve(MidPoint(LineSegmentOf(P,F1)), yAxis);LineSegmentOf(P, F1) = t*LineSegmentOf(P, F2);t:Number", "coords": {} }, { "index": 3441, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3442, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3442.png", "answer": "sqrt(15)/2", "fact_expr": "G: Parabola;A: Point;M: Point;Coordinate(A) = (4, 0);Expression(G) = (y^2 = x);PointOnCurve(M, G);Distance(A,M)=Abs(LineSegmentOf(M,A))", "coords": { "A": [ 4.0, 0.0 ] } }, { "index": 3443, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3443.png", "answer": "24", "fact_expr": "G: Parabola;Expression(G) = (y^2 = -12*x);l: Line;PointOnCurve(Focus(G), l);A: Point;B: Point;Intersection(l, G) = {A, B};XCoordinate(MidPoint(LineSegmentOf(A, B))) = -9", "coords": {} }, { "index": 3444, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_ecc_mismatch", "params": { "a": 1.9257268905639648, "b": 1.7993911322594118, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 2.5576271215184017e-15, "converged": true, "iterations": 2 }, "output_path": "results/train/ellipse/problem_3444.png", "answer": "3", "fact_expr": "C1: Ellipse;Expression(C1) = (x^2/4 + y^2/b^2 = 1);b: Number;0 b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;RightVertex(C)=A;PointOnCurve(P,C);IsPerpendicular(LineSegmentOf(P,F),LineSegmentOf(P,A));Eccentricity(C) = e", "coords": {} }, { "index": 3453, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3454, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3454.png", "answer": "8", "fact_expr": "C: Hyperbola;G: Circle;Expression(C) = (x^2/4 - y^2/9 = 1);OneOf(Focus(C))=Center(G);Radius(G)=5", "coords": {} }, { "index": 3455, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3456, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3456.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;a: Number;b: Number;P: Point;F2: Point;F1: Point;PointOnCurve(P,G);Focus(G)={F1,F2};Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);IsPerpendicular(LineSegmentOf(P, F2), LineSegmentOf(F1, F2));Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 3457, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3457.png", "answer": "[b^2,a^2]", "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F2: Point;F1: Point;a > b;b > 0;x1:Number;y1:Number;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (x1, y1);PointOnCurve(P, G);Focus(G)={F1,F2}", "coords": {} }, { "index": 3458, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3458.png", "answer": "3", "fact_expr": "C: Hyperbola;b: Number;a: Number;A: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightVertex(C)=A;Distance(A, OneOf(Asymptote(C))) = a*(2*sqrt(2))/3", "coords": {} }, { "index": 3459, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3459.png", "answer": "sqrt(6)/3", "fact_expr": "C: Ellipse;a: Number;G: Line;P: Point;A: Point;B: Point;a>0;Expression(C) = (y^2 + x^2/a^2 = 1);Expression(G) = (a*x + y - 4 = 0);PointOnCurve(P, G);IsTangent(LineOf(P, A), C);IsTangent(LineOf(P, B), C);IsPerpendicular(LineSegmentOf(P, A),LineSegmentOf(P, B))", "coords": {} }, { "index": 3460, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3460.png", "answer": "16", "fact_expr": "E: Ellipse;Expression(E) = (x^2/25 + y^2/16 = 1);F1: Point;F2: Point;LeftFocus(E) = F1;RightFocus(E) = F2;P: Point;PointOnCurve(P, E)", "coords": {} }, { "index": 3461, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3461.png", "answer": "17/4", "fact_expr": "H: Line;I: Point;Coordinate(I) = (0, 1);PointOnCurve(I, H);G: Parabola;Expression(G) = (x^2 = 4*y);Intersection(H, G) = {A, B};A: Point;Coordinate(A) = (x1, y1);x1: Number;y1: Number;B: Point;Coordinate(B) = (x2, y2);x2: Number;y2: Number;y1 + y2 = 9/4", "coords": { "I": [ 0.0, 1.0 ] } }, { "index": 3462, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3462.png", "answer": "y=pm*(sqrt(3)+1)*x", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;Q: Point;F1: Point;F2: Point;P: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F2,l);IsPerpendicular(l,OneOf(Asymptote(C)));Intersection(l,LeftPart(C))=P;Intersection(l,RightPart(C))=Q;PointOnCurve(P, PerpendicularBisector(LineSegmentOf(Q, F1)))", "coords": {} }, { "index": 3463, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3463.png", "answer": "6", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;H: Line;M: Point;A: Point;B: Point;F: Point;p>0;Coordinate(M) = (-p/2, 0);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 24;Tan(AngleOf(A, M, B)) = 2*sqrt(2)", "coords": {} }, { "index": 3464, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3464.png", "answer": "(3*sqrt(5)+9)/2", "fact_expr": "P: Point;PointOnCurve(P, C);C: Ellipse;Expression(C) = (x^2/4 + y^2 = 1);l: Line;Expression(l) = (x - y + 3 = 0);A: Point;B: Point;Intersection(l, xAxis) = A;Intersection(l, yAxis) = B", "coords": {} }, { "index": 3465, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.449489742783178, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 6.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3465.png", "answer": "sqrt(2)", "fact_expr": "C1: Curve;C2: Curve;Expression(C1)=(x^2/6+y^2/2=1);Expression(C2)=(x^2/3-y^2=1);P: Point;F1: Point;F2: Point;Focus(C1)={F1,F2};OneOf(Intersection(C1,C2))=P", "coords": {} }, { "index": 3466, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3466.png", "answer": "3*sqrt(2)+2*sqrt(10)", "fact_expr": "G: Ellipse;a: Number;P: Point;Q: Point;F: Point;a>1;Expression(G) = (y^2 + x^2/a^2 = 1);Coordinate(P) = (0, 3);RightFocus(G)=F;PointOnCurve(Q,G);Negation(Max(Distance(P,Q))>2*sqrt(5));WhenMax(Eccentricity(G))", "coords": { "P": [ 0.0, 3.0 ] } }, { "index": 3467, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3467.png", "answer": "y=pm*4*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/16 = 1)", "coords": {} }, { "index": 3468, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3468.png", "answer": "5/6", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 2*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 25/12;Abs(LineSegmentOf(A, F)) < Abs(LineSegmentOf(B, F))", "coords": {} }, { "index": 3469, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3469.png", "answer": "x^2/4+y^2=1", "fact_expr": "C: Ellipse;b: Number;a: Number;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Length(MajorAxis(C))=4;Eccentricity(C)=sqrt(3)/2", "coords": {} }, { "index": 3470, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3471, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3471.png", "answer": "2", "fact_expr": "G: Parabola;l1: Line;l2: Line;Expression(G) = (y^2 = 4*x);P: Point;Expression(l1) = (4*x - 3*y + 6 = 0);Expression(l2) = (x = -1);PointOnCurve(P, G)", "coords": {} }, { "index": 3472, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3473, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3473.png", "answer": "sqrt(a^2+9)-1", "fact_expr": "P: Point;PointOnCurve(P, G) = True;G: Parabola;Expression(G) = (y^2 = 4*x);Projection(P, yAxis) = M;M: Point;A: Point;Coordinate(A) = (4, a);a: Number;Abs(a)>4", "coords": {} }, { "index": 3474, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3474.png", "answer": "9", "fact_expr": "P: Point;PointOnCurve(P, RightPart(G)) = True;G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);M: Point;PointOnCurve(M,H1) = True;H1: Circle;Expression(H1) = (y^2 + (x + 5)^2 = 4);N: Point;PointOnCurve(N,H2) = True;H2: Circle;Expression(H2) = (y^2 + (x - 5)^2 = 1)", "coords": {} }, { "index": 3475, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3476, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.3300262689590454, "b": 2.30381441116333, "orientation": "horizontal" }, "optimization": { "final_loss": 6.658357051492203e-06, "converged": true, "iterations": 122 }, "output_path": "results/train/hyperbola/problem_3476.png", "answer": "(0, 1/2]", "fact_expr": "Expression(Asymptote(D)) = (y = pm*sqrt(3)*x);D: Hyperbola;Expression(D) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(D) = F1;RightFocus(D) = F2;P: Point;PointOnCurve(P, RightPart(D)) = True", "coords": {} }, { "index": 3477, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3478, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3478.png", "answer": "sqrt(5)/2", "fact_expr": "C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(Asymptote(C)) = (y = pm*(x/2))", "coords": {} }, { "index": 3479, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3479.png", "answer": "(0,1/2)", "fact_expr": "G: Ellipse;b: Number;a: Number;M: Point;F: Point;A: Point;B: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (a, 0);Coordinate(B) = (0, b);LeftFocus(G) = F;VectorOf(B, M) = 2*VectorOf(M, A)", "coords": {} }, { "index": 3480, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3480.png", "answer": "8", "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/9 + y^2/16 = 1)", "coords": {} }, { "index": 3481, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3481.png", "answer": "x^2/16+y^2/7=1", "fact_expr": "G: Parabola;H: Ellipse;Expression(G) = (y^2 = 12*x);Length(MajorAxis(H)) = 8;Focus(G)=OneOf(Focus(H))", "coords": {} }, { "index": 3482, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3482.png", "answer": "1/3", "fact_expr": "H: Ellipse;Expression(H) = (x^2/6 + y^2/2 = 1);G: Hyperbola;Expression(G) = (x^2/3 - y^2/b^2 = 1);b: Number;F1: Point;F2: Point;Focus(H) = {F1, F2};Focus(G) = {F1, F2};P: Point;OneOf(Intersection(H, G)) = P", "coords": {} }, { "index": 3483, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.4641016151377544, "major_axis": "x", "x_coef": 16.0, "y_coef": 12.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3483.png", "answer": "8", "fact_expr": "G: Curve;A: Point;B: Point;P: Point;x1: Number;y1: Number;Expression(G) = (x^2/16 + y^2/12 = 1);Coordinate(A) = (-2, 0);Coordinate(B) = (2, 0);Coordinate(P) = (x1, y1);PointOnCurve(P, G)", "coords": { "A": [ -2.0, 0.0 ], "B": [ 2.0, 0.0 ] } }, { "index": 3484, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.3300262689590454, "b": 2.30381441116333, "orientation": "horizontal" }, "optimization": { "final_loss": 6.658357051492203e-06, "converged": true, "iterations": 122 }, "output_path": "results/train/hyperbola/problem_3484.png", "answer": "x^2/4 - y^2/12 = 1", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x);H: Parabola;Expression(H) = (y^2 = 16*x);OneOf(Focus(G)) = Focus(H)", "coords": {} }, { "index": 3485, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3486, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_focus_mismatch;ellipse_point_off_curve", "params": { "a": 2.5848636627197266, "b": 2.4273612433297647, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 1.9079626818174794e-06, "converged": true, "iterations": 111 }, "output_path": "results/train/ellipse/problem_3486.png", "answer": "x^2/18+y^2/9=1", "fact_expr": "C: Ellipse;b: Number;a: Number;G: Point;F: Point;a>0;b>0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);RightFocus(C)=F;Coordinate(F) = (3, 0);PointOnCurve(G, C);Coordinate(G)=(-3,3*sqrt(2)/2)", "coords": { "F": [ 3.0, 0.0 ], "G": [ -3.0, 2.121320343559643 ] }, "validation_reasons": [ "ellipse_focus_mismatch", "ellipse_point_off_curve" ] }, { "index": 3487, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3488, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3488.png", "answer": "-3*sqrt(3)", "fact_expr": "G: Ellipse;b: Number;a: Number;C: Point;F: Point;A: Point;B: Point;D: Point;a>0;b>0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (-a, 0);Coordinate(B) = (0, b);Coordinate(C) = (0, -b);LeftFocus(G) = F;Vertex(G)={A,B,C};Intersection(LineOf(C,F), LineOf(A, B)) = D;Eccentricity(G) = e ;e:Number;e=1/2", "coords": {} }, { "index": 3489, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3490, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3490.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);H: Line;PointOnCurve(Focus(G), H);A: Point;B: Point;Intersection(H, G) = {A, B};Negation(A=B)", "coords": {} }, { "index": 3491, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3491.png", "answer": "28", "fact_expr": "G: Hyperbola;A: Point;B: Point;F2: Point;F1: Point;Expression(G) = (x^2/16 - y^2/9 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;Abs(LineSegmentOf(A, B)) = 6;IsChordOf(LineSegmentOf(A, B),LeftPart(G)) = True;PointOnCurve(F1,LineSegmentOf(A,B))", "coords": {} }, { "index": 3492, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3492.png", "answer": "16.5", "fact_expr": "G: Hyperbola;H: Point;I: Point;P: Point;Expression(G) = (x^2/16 - y^2/9 = 1);Coordinate(H) = (5, 0);Coordinate(I) = (-5, 0);PointOnCurve(P, G);Distance(P, H) = 8.5", "coords": { "H": [ 5.0, 0.0 ], "I": [ -5.0, 0.0 ] } }, { "index": 3493, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3493.png", "answer": "2*sqrt(3)", "fact_expr": "G: Ellipse;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2", "coords": {} }, { "index": 3494, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3494.png", "answer": "(2, 4)", "fact_expr": "H: Line;Expression(H) = (y = k*x + 2);k: Number;G: Ellipse;Expression(G) = (x^2/16 + y^2/b^2 = 1);b: Real;b>0;PointOnCurve(Focus(G), xAxis);NumIntersection(H, G) = 2", "coords": {} }, { "index": 3495, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3495.png", "answer": "y=x+1/4", "fact_expr": "G: Parabola;H: Line;Expression(G) = (y^2 = x);Slope(H)=1;NumIntersection(H,G)=1", "coords": {} }, { "index": 3496, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3496.png", "answer": "(pm*sqrt(5), 0)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1)", "coords": {} }, { "index": 3497, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3497.png", "answer": "-83", "fact_expr": "H: Line;Expression(H) = (y = -x - 4);G: Parabola;Expression(G) = (y^2 = -12*x);A: Point;B: Point;Intersection(H, G) = {A, B};F: Point;Focus(G) = F", "coords": {} }, { "index": 3498, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3498.png", "answer": "sqrt(2)/2", "fact_expr": "G: Parabola;Expression(G) = (y^2=2*p*x);p: Number;p>0;F: Point;Focus(G) = F;L1: Ray;L2: Ray;PointOnCurve(F, L1);PointOnCurve(F, L2);IsPerpendicular(L1, L2);M: Point;N: Point;Intersection(L1, G) = M;Intersection(L2, G) = N;IsChordOf(LineSegmentOf(M, N), G);P: Point;MidPoint(LineSegmentOf(M, N)) = P;Q: Point;PointOnCurve(P, LineSegmentOf(P, Q));IsPerpendicular(LineSegmentOf(P, Q), Directrix(G));FootPoint(LineSegmentOf(P, Q), Directrix(G)) = Q", "coords": {} }, { "index": 3499, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3499.png", "answer": "2", "fact_expr": "G: Ellipse;Expression(G) = (y^2 + x^2/a = 1);FocalLength(G) = 2;a: Number;a>1", "coords": {} }, { "index": 3500, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3501, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3501.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;F: Point;B: Point;A: Point;e: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G)=F;Endpoint(ImageinaryAxis(G))=B;RightVertex(G)=A;Distance(A, LineOf(F,B)) = b/sqrt(7);Eccentricity(G) = e;L:Line;PointOnCurve(F,L);PointOnCurve(B,L)", "coords": {} }, { "index": 3502, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3502.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;P: Point;F2: Point;F1:Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Slope(H)=sqrt(3)/3;LeftFocus(G)=F1;PointOnCurve(F1,H);Intersection(H,RightPart(G))=P;IsPerpendicular(LineSegmentOf(P,F2),xAxis);RightFocus(G)=F2", "coords": {} }, { "index": 3503, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3504, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3504.png", "answer": "(16,4)", "fact_expr": "G: Parabola;C: Circle;C1:Point;H: Point;A: Point;B: Point;O:Origin;Expression(G) = (y^2 = x);Coordinate(H) = (0, 2);Center(C)=C1;PointOnCurve(H, C);PointOnCurve(C1,G);Negation(C1=O);Intersection(C, yAxis) = {A, B};Abs(LineSegmentOf(A, B)) = 4", "coords": { "H": [ 0.0, 2.0 ] } }, { "index": 3505, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3505.png", "answer": "2*sqrt(2)", "fact_expr": "C: Parabola;p: Number;M: Point;N: Point;Expression(C) = (y^2 = 2*(p*x));Coordinate(N) = (3, 0);Expression(Directrix(C)) = (x = -1);PointOnCurve(M, C)", "coords": { "N": [ 3.0, 0.0 ] } }, { "index": 3506, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3506.png", "answer": "1", "fact_expr": "G: Parabola;a: Number;Expression(G) = (x^2 = a*y);Expression(Directrix(G)) = (y = -1/4)", "coords": {} }, { "index": 3507, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3508, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 1.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3508.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "M: Point;PointOnCurve(M, G);G: Ellipse;Expression(G) = (x^2/9 + y^2 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2}", "coords": {} }, { "index": 3509, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3509.png", "answer": "x = -1/4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = x)", "coords": {} }, { "index": 3510, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3511, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3512, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3512.png", "answer": "sqrt(2)/2", "fact_expr": "E: Ellipse;m: Number;A: Point;P: Point;B: Point;m>1;Expression(E) = (x^2 + y^2/m = 1);Vertex(MinorAxis(E)) = {A, B};PointOnCurve(P, E);Negation(P = A);Negation(P = B);Slope(LineOf(A, P))*Slope(LineOf(B, P)) = -4/m", "coords": {} }, { "index": 3513, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3513.png", "answer": "pm*2*sqrt(2)/3", "fact_expr": "C: Parabola;G: Line;F: Point;A: Point;B: Point;k: Number;Expression(C) = (y^2 = 8*x);Expression(G) = (y = k*(x + 2));Intersection(G, C) = {A, B};Focus(C) = F;Abs(LineSegmentOf(F, A)) = 2*Abs(LineSegmentOf(F, B))", "coords": {} }, { "index": 3514, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3514.png", "answer": "sqrt(2)", "fact_expr": "l: Line;E: Hyperbola;a: Number;b: Number;P: Point;Q: Point;C: Point;F2: Point;O: Origin;M: Point;F1: Point;a>0;b>0;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(E) = F2;LeftFocus(E) = F1;PointOnCurve(F1, l);Intersection(l, LeftPart(E)) = P;Intersection(l, RightPart(E)) = Q;Intersection(yAxis, l) = C;MidPoint(LineSegmentOf(P, Q)) = M;IsPerpendicular(LineSegmentOf(C, F2), LineSegmentOf(O, M))", "coords": {} }, { "index": 3515, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3516, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3516.png", "answer": "4", "fact_expr": "G: Parabola;M: Point;F: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(M,G);Focus(G)=F;Distance(M,F)=Abs(LineSegmentOf(M,F));Abs(LineSegmentOf(M,F))=5", "coords": {} }, { "index": 3517, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3517.png", "answer": "(0, 1)", "fact_expr": "G: Parabola;p: Number;M: Point;F: Point;p>0;Expression(G) = (x^2 = 2*(p*y));y0: Number;Coordinate(M) = (4, y0);PointOnCurve(M, G);Focus(G) = F;Distance(M, F) = Abs(LineSegmentOf(M, F)) ;Abs(LineSegmentOf(M, F)) = (5/4)*y0", "coords": {} }, { "index": 3518, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3519, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3520, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3520.png", "answer": "ApplyUnit(45,degree)", "fact_expr": "l: Line;C: Parabola;B: Point;A: Point;Expression(C) = (y^2 = 6*x);Intersection(l,C) = {A, B};YCoordinate(MidPoint(LineSegmentOf(A,B))) = 3", "coords": {} }, { "index": 3521, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3521.png", "answer": "sqrt(3)", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y^2 + (x - 2)^2 = 4);Length(InterceptChord(OneOf(Asymptote(C)),G))=2", "coords": {} }, { "index": 3522, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3522.png", "answer": "3", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;P: Point;PointOnCurve(P, C);LineSegmentOf(P, F) = 3;G: Line;Expression(G) = (x = -1)", "coords": {} }, { "index": 3523, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.449489742783178, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 6.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3523.png", "answer": "1/3", "fact_expr": "C1: Curve;C2: Curve;F1: Point;P: Point;F2: Point;Expression(C1)= (x^2/6 + y^2/2 = 1);Expression(C2)=( x^2/3 - y^2 = 1);Focus(C1)={F1,F2};OneOf(Intersection(C1,C2))=P", "coords": {} }, { "index": 3524, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3524.png", "answer": "sqrt(5)", "fact_expr": "H: Ellipse;Expression(H) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(H) = F1;RightFocus(H) = F2;G: Parabola;Expression(G) = (y^2 = 2*(b*x));F: Point;Focus(G) = F;VectorOf(F1, F) = (5/3)*VectorOf(F, F2)", "coords": {} }, { "index": 3525, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3526, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3526.png", "answer": "8", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (y = x - 1);Intersection(H, G) = {A, B};IsChordOf(LineSegmentOf(A,B),G)", "coords": {} }, { "index": 3527, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3527.png", "answer": "{2, 6}", "fact_expr": "C: Parabola;p: Number;A: Point;B: Point;F: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(A) = (0, sqrt(3));Focus(C) = F;PointOnCurve(B, C);IsPerpendicular(LineSegmentOf(A, B), LineSegmentOf(A, F));Abs(LineSegmentOf(B, F)) = 4", "coords": { "A": [ 0.0, 1.7320508075688772 ] } }, { "index": 3528, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3528.png", "answer": "6", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 6*x);F: Point;Focus(G) = F;PointOnCurve(F, l) = True;l: Line;Intersection(l, G) = {A, B};B: Point;A: Point;O: Origin;VectorOf(O, M) = VectorOf(O, B)/4;VectorOf(O, A) = VectorOf(O, N)/4;L1: Line;L2: Line;PointOnCurve(M, L1) = True;PointOnCurve(N, L2) = True;M: Point;N: Point;IsPerpendicular(L1, Directrix(G)) = True;IsPerpendicular(L2, Directrix(G)) = True;FootPoint(L1, Directrix(G)) = C;FootPoint(L2, Directrix(G)) = D;C: Point;D: Point", "coords": {} }, { "index": 3529, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3530, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.125, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3530.png", "answer": "pm*4*sqrt(3)", "fact_expr": "C: Parabola;G: Line;k: Number;A: Point;B: Point;N: Point;M: Point;Expression(C) = (y = 2*x^2);Expression(G) = (y = k*x + 2);Intersection(C, G) = {A, B};MidPoint(LineSegmentOf(A, B)) = M;l: Line;PointOnCurve(M, l);IsPerpendicular(l, xAxis);FootPoint(l, xAxis) = N;DotProduct(VectorOf(N, A), VectorOf(N, B)) = 0", "coords": {} }, { "index": 3531, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3531.png", "answer": "42", "fact_expr": "G: Parabola;p: Number;A: Point;B: Point;F: Point;p>0;y1:Number;y2:Number;Expression(G) = (y^2 = 2*p*x);Coordinate(A) = (2, y1);Coordinate(B) = (1/2, y2);Focus(G) = F;Quadrant(A)=1;Quadrant(B)=4;Abs(LineSegmentOf(A, F)) = 10;PointOnCurve(A,G);PointOnCurve(B,G)", "coords": {} }, { "index": 3532, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3532.png", "answer": "x^2/2-y^2=1", "fact_expr": "G: Hyperbola;H: Ellipse;Q: Point;Expression(H) = (x^2/4 + y^2 = 1);Coordinate(Q) = (2, 1);Focus(G)=Focus(H);PointOnCurve(Q,G)", "coords": { "Q": [ 2.0, 1.0 ] } }, { "index": 3533, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3534, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3535, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3535.png", "answer": "sqrt(5)", "fact_expr": "P: Point;PointOnCurve(P, G);F1: Point;F2: Point;Focus(G) = {F1, F2};G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2));Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 3536, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3537, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3537.png", "answer": "sqrt(3)/2", "fact_expr": "G: Parabola;H: Ellipse;k: Number;Expression(G) = (y^2 = 12*x);k>0;Expression(H) = (k*y^2 + x^2 = 3*k);OneOf(Focus(H))=Focus(G)", "coords": {} }, { "index": 3538, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3538.png", "answer": "4", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G)", "coords": {} }, { "index": 3539, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3539.png", "answer": "16/3", "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F,l);Slope(l)=sqrt(3);Intersection(l, C) = {A, B}", "coords": {} }, { "index": 3540, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3540.png", "answer": "2", "fact_expr": "G:Hyperbola;p: Number;H: Parabola;Expression(G) = (-x^2 + 2*y^2 = 2*p^2);p>0;Expression(H) = (y^2 = 2*p*x);Distance(Focus(H),OneOf(Focus(G))) = sqrt(13)", "coords": {} }, { "index": 3541, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3541.png", "answer": "sqrt(3)/2", "fact_expr": "G: Hyperbola;a: Number;H: Ellipse;e: Number;Expression(G) = (-y^2/2 + x^2/a = 1);Expression(H) = (x^2/4 + y^2/a^2 = 1);Focus(H) = Focus(G);Eccentricity(H) = e", "coords": {} }, { "index": 3542, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3542.png", "answer": "5/3", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;l: Line;PointOnCurve(F2, l) ;Intersection(l, RightPart(C)) = {A, B};A: Point;B: Point;Abs(LineSegmentOf(A, F1)) = Abs(LineSegmentOf(F1, F2));Abs(LineSegmentOf(B, F2)) = 2*Abs(LineSegmentOf(A, F2))", "coords": {} }, { "index": 3543, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3543.png", "answer": "y = pm*(x- 1)", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Abs(LineSegmentOf(A, B)) = 8;NumIntersection(H, G) = 2;IsChordOf(LineSegmentOf(A, B), G);PointOnCurve(Focus(G), H)", "coords": {} }, { "index": 3544, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3544.png", "answer": "(2*sqrt(3))/3", "fact_expr": "G: Hyperbola;a: Number;b: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Distance(OneOf(Focus(G)),OneOf(Asymptote(G)))=1/4*FocalLength(G)", "coords": {} }, { "index": 3545, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3545.png", "answer": "2", "fact_expr": "C: Parabola;H: Line;A: Point;F: Point;B: Point;K: Point;Expression(C)=(y^2=4*x);Focus(C)=F;Intersection(Directrix(C), xAxis) = K;PointOnCurve(F,H);Intersection(H, C) = {A, B};Abs(LineSegmentOf(A, F)) - Abs(LineSegmentOf(B, F)) = 3/2", "coords": {} }, { "index": 3546, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3546.png", "answer": "2*sqrt(3)/3", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Expression(OneOf(Asymptote(C))) = (x - sqrt(3)*y = 0)", "coords": {} }, { "index": 3547, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3547.png", "answer": "5/4", "fact_expr": "G: Parabola;A: Point;B: Point;F: Point;M: Point;Expression(G) = (y^2 = x);Focus(G) = F;PointOnCurve(F, LineOf(A, B));Intersection(LineOf(A, B), G) = {A, B};Abs(LineSegmentOf(A, B)) = 3;MidPoint(LineSegmentOf(A, B)) = M", "coords": {} }, { "index": 3548, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3548.png", "answer": "3", "fact_expr": "G: Parabola;P: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(P, G);O:Origin;Distance(P,O)=2*sqrt(3)", "coords": {} }, { "index": 3549, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3549.png", "answer": "(2,+oo)", "fact_expr": "M: Point;Coordinate(M) = (x0, y0);x0: Number;y0: Number;C: Parabola;Expression(C) = (x^2 = 8*y);PointOnCurve(M, C) = True;Focus(C) = F;F:Point;G: Circle;Center(G) = F;Radius(G) = Abs(LineSegmentOf(F, M));IsIntersect(G , Directrix(C)) = True", "coords": {} }, { "index": 3550, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_3550.png", "answer": "(\\sqrt{2}-1)/2", "fact_expr": "P: Point;m: Number;H: Circle;G: Parabola;Expression(G) = (x^2 = 4*(y-m));Expression(H) = (x^2 + y^2 = 1);Quadrant(P) = 1;Intersection(H, G)=P;IsPerpendicular(TangentOnPoint(P, G), TangentOnPoint(P, H))", "coords": {} }, { "index": 3551, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3551.png", "answer": "1", "fact_expr": "G: Hyperbola;M: Point;P: Point;Q: Point;L:Line;Expression(G) = (x^2 - y^2/4 = 1);PointOnCurve(M,G);PointOnCurve(M,L);IsParallel(L,RealAxis(G));Intersection(L,Asymptote(G))={P,Q}", "coords": {} }, { "index": 3552, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3552.png", "answer": "y^2 = 8*x", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p<0;F: Point;Focus(G) = F;PointOnCurve(F, LineSegmentOf(A, B)) = True;PointOnCurve(F, LineSegmentOf(C, D)) = True;IsChordOf(LineSegmentOf(A, B), G) = True;IsChordOf(LineSegmentOf(C, D), G) = True;IsPerpendicular(LineSegmentOf(A, B), LineSegmentOf(C, D)) = True;A: Point;B: Point;C: Point;D: Point;Min(Area(TriangleOf(A, C, F)) + Area(TriangleOf(B, D, F))) = 32", "coords": {} }, { "index": 3553, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3553.png", "answer": "8", "fact_expr": "C: Parabola;G: Line;F: Point;P: Point;Q: Point;Expression(C) = (x^2 = 12*y);Expression(G) = (y=-3);Focus(C) = F;PointOnCurve(P, C);Intersection(LineOf(F,P),G) = Q;VectorOf(P, Q) = 2*VectorOf(F, Q)", "coords": {} }, { "index": 3554, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3555, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3555.png", "answer": "sqrt(3) - 1", "fact_expr": "C1: Ellipse;Expression(C1) = (x^2/a^2+y^2/b^2 = 1);a: Number;b: Number;a>b;b>0;C2: Circle;Expression(C2) = (y^2+x^2 = a^2-b^2);P: Point;OneOf(Intersection(C1,C2)) = P;2*AngleOf(P,F1,F2) = AngleOf(P,F2,F1);F1: Point;F2: Point;LeftFocus(C1) = F1;RightFocus(C1) = F2", "coords": {} }, { "index": 3556, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3557, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3558, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3558.png", "answer": "(0,4]", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);D: Point;Coordinate(D) = (4, 0);F: Point;Focus(C) = F;PointOnCurve(F, l);Slope(l) = k;k: Number;l: Line;Intersection(l, C) = {A, B};A: Point;B: Point;DotProduct(VectorOf(A, D), VectorOf(B, D)) <= 1", "coords": { "D": [ 4.0, 0.0 ] } }, { "index": 3559, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3559.png", "answer": "{1, 17}", "fact_expr": "G: Hyperbola;F1: Point;F2: Point;P: Point;Expression(G) = (x^2/16 - y^2/9 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);Distance(P, F1) = 9", "coords": {} }, { "index": 3560, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3561, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3562, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3562.png", "answer": "3/2", "fact_expr": "G: Parabola;M: Point;O: Origin;Expression(G) = (y^2 = 2*x);PointOnCurve(M, G);Distance(M, O) = sqrt(3)", "coords": {} }, { "index": 3563, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3563.png", "answer": "{y^2 = 8*x, y^2 = -16*x}", "fact_expr": "G: Parabola;m: Number;H: Line;Expression(G) = (y^2 = m*x);Expression(H) = (x = 1);Distance(Directrix(G), H) = 3", "coords": {} }, { "index": 3564, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3564.png", "answer": "sqrt(17)/2", "fact_expr": "G: Parabola;Q: Point;P: Point;Expression(G) = (y^2 = 2*x);Coordinate(Q) = (0, 2);PointOnCurve(P,G)", "coords": { "Q": [ 0.0, 2.0 ] } }, { "index": 3565, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3565.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1);P: Point;PointOnCurve(P, LeftPart(G));F: Point;RightFocus(G) = F;M: Point;MidPoint(LineSegmentOf(F, P)) = M;O: Origin;Distance(M, O) = 7", "coords": {} }, { "index": 3566, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3567, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3567.png", "answer": "1/2", "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/4 + y^2/3 = 1);Focus(C) = {F1, F2};Quadrant(P) = 1;PointOnCurve(P, C);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 9/4", "coords": {} }, { "index": 3568, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3568.png", "answer": "x=-1", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;H: Line;Slope(H) = 1;PointOnCurve(Focus(G), H);A: Point;B: Point;Intersection(H, G) = {A, B};YCoordinate(MidPoint(LineSegmentOf(A, B))) = 2", "coords": {} }, { "index": 3569, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.0, "b": 0.5, "major_axis": "y", "x_coef": 0.25, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3569.png", "answer": "1 - sqrt(3)/2", "fact_expr": "G: Ellipse;P: Point;F: Point;Expression(G) = (4*x^2 + y^2 = 1);PointOnCurve(P, G);OneOf(Focus(G))=F", "coords": {} }, { "index": 3570, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3570.png", "answer": "sqrt(6)-sqrt(3)", "fact_expr": "C: Ellipse;b: Number;a: Number;P: Point;F1: Point;O: Origin;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C)=F1;RightFocus(C)=F2;PointOnCurve(P,C);DotProduct(VectorOf(P,F1),(VectorOf(O,F1)+VectorOf(O,P)))=0;Abs(LineSegmentOf(P,F1))=sqrt(2)*Abs(LineSegmentOf(P,F2))", "coords": {} }, { "index": 3571, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3571.png", "answer": "[0, 1]", "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2/m^2 = 1);m: Number;0 < m;m < 2;F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;x0: Number;y0: Number;Coordinate(P) = (x0, y0);PointOnCurve(P, C);Area(TriangleOf(P, F1, F2)) = sqrt(3)", "coords": {} }, { "index": 3572, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3572.png", "answer": "2", "fact_expr": "G: Parabola;p: Number;H: Line;M: Point;N: Point;A: Point;B: Point;p>0;Expression(G) = (x^2 = 2*(p*y));Coordinate(M) = (0, 1);PointOnCurve(M, H);Intersection(H, G) = {A, B};Negation(A=B);Intersection(TangentOnPoint(A,G),TangentOnPoint(B,G))=N;DotProduct(VectorOf(N, A), VectorOf(N, B)) = 0", "coords": { "M": [ 0.0, 1.0 ] } }, { "index": 3573, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3574, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3574.png", "answer": "5/3", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;H: Parabola;Expression(H) = (y^2 = 20*x);OneOf(Focus(G)) = Focus(H);Distance(OneOf(Focus(G)), Asymptote(G)) = 4", "coords": {} }, { "index": 3575, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3575.png", "answer": "pi/2", "fact_expr": "E: Point;Coordinate(E) = (2, 2);C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;H: Point;Coordinate(H) = (2, 0);l: Line;PointOnCurve(H, l) = True;Intersection(l, C) = {A, B};A: Point;B: Point;Negation(A = E);Negation(B = E);G: Line;Expression(G) = (x = -2);Intersection(LineOf(E, A), G) = M;M: Point;Intersection(LineOf(E, B), G) = N;N: Point;O: Origin", "coords": { "E": [ 2.0, 2.0 ], "H": [ 2.0, 0.0 ] } }, { "index": 3576, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.463192105293274, "b": 2.070901870727539, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4661259228887502e-06, "converged": true, "iterations": 106 }, "output_path": "results/train/hyperbola/problem_3576.png", "answer": "x^2/(1/3)-y^2/(2/3)=1", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 4*x);Eccentricity(G) = sqrt(3);OneOf(Focus(G)) = Focus(H)", "coords": {} }, { "index": 3577, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3577.png", "answer": "y^2 = 8*x", "fact_expr": "l: Line;G: Parabola;a: Number;O: Origin;A: Point;F: Point;a>0;Expression(G) = (y^2 = a*x);Slope(l) = 2;Focus(G) = F;PointOnCurve(F, l);Intersection(l,yAxis) = A;Area(TriangleOf(O, A, F)) = 4", "coords": {} }, { "index": 3578, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3578.png", "answer": "y^2=20*x", "fact_expr": "H: Parabola;Expression(H) = (y^2 = 4*(m*x));m: Number;m>0;G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1);Distance(Focus(H), OneOf(Asymptote(G))) = 3", "coords": {} }, { "index": 3579, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3579.png", "answer": "-3", "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (x^2 + y^2/m = 1);Eccentricity(G) = 2", "coords": {} }, { "index": 3580, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3580.png", "answer": "sqrt(13)", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;F1: Point;F2: Point;A: Point;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;IsDiameter(LineSegmentOf(F1,F2),G);Intersection(G,LeftPart(C))=A;Intersection(LineSegmentOf(A,F2),RightPart(C))=B;Cos(AngleOf(F1, B, F2)) = -3/5", "coords": {} }, { "index": 3581, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3581.png", "answer": "{(sqrt(3),0),(-sqrt(3),0)}\nsqrt(6)/2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - 2*y^2 = 2)", "coords": {} }, { "index": 3582, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3582.png", "answer": "4*sqrt(5)/5", "fact_expr": "G: Ellipse;H1: Line;H2: Line;P: Point;d1: Number;d2: Number;Expression(G) = (x^2 + y^2/4 = 1);Expression(H1) = (2*x + y = 0);Expression(H2) = (2*x - y = 0);PointOnCurve(P, G);Distance(P, H1) = d1;Distance(P, H2) = d2", "coords": {} }, { "index": 3583, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3583.png", "answer": "sqrt(5)/2", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(C) = F;L: Line;PointOnCurve(F,L) = True;Intersection(L,C) = {A,B};A: Point;B: Point;Quadrant(A) = 1;VectorOf(A, F) = 5*VectorOf(F, B)", "coords": {} }, { "index": 3584, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3584.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;b: Number;H: Parabola;Expression(G) = (x^2/8 - y^2/b^2 = 1);Expression(H) = (y^2 = 8*x);OneOf(Directrix(G))=Directrix(H)", "coords": {} }, { "index": 3585, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3586, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3586.png", "answer": "42", "fact_expr": "G: Hyperbola;P: Point;Expression(G) = (-x^2/36 + y^2/64 = 1);PointOnCurve(P, G);Distance(P, OneOf(Focus(G))) = 3;F1:Point;F2:Point;Focus(G)={F1,F2}", "coords": {} }, { "index": 3587, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3588, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3588.png", "answer": "2*sqrt(2)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/2 - y^2 = 1)", "coords": {} }, { "index": 3589, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3589.png", "answer": "[1/3, 3]", "fact_expr": "G: Ellipse;M: Point;D: Point;C: Point;lambda: Real;Expression(G) = (x^2/4 + y^2 = 1);Coordinate(M) = (0, 2);PointOnCurve(C, G);PointOnCurve(D, G);VectorOf(M, D) = lambda*VectorOf(M, C)", "coords": { "M": [ 0.0, 2.0 ] } }, { "index": 3590, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3590.png", "answer": "3*sqrt(2)/4", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;P: Point;F1: Point;e: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = x*(b/(3*a)));LeftFocus(G) = F1;Intersection(H,LeftPart(G))=P;IsPerpendicular(LineSegmentOf(P,F1),xAxis);Eccentricity(G) = e", "coords": {} }, { "index": 3591, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3592, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3593, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3593.png", "answer": "25/8", "fact_expr": "l: Line;G: Parabola;A: Point;F: Point;B: Point;Expression(G) = (y^2 = 2*x);Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B};Abs(LineSegmentOf(A, F)) = 4*Abs(LineSegmentOf(B, F));IsChordOf(LineSegmentOf(A, B), G)", "coords": {} }, { "index": 3594, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3594.png", "answer": "17/2", "fact_expr": "C: Parabola;G: Line;P: Point;B: Point;A: Point;F: Point;Expression(C) = (y^2 = 12*x);Coordinate(P) = (1, 0);Focus(C) = F;PointOnCurve(P,G);Intersection(G,C) = {A, B};2*VectorOf(B, P) = VectorOf(P, A)", "coords": { "P": [ 1.0, 0.0 ] } }, { "index": 3595, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3595.png", "answer": "x^2/5 - y^2/20 = 1", "fact_expr": "l: Line;G: Hyperbola;a: Number;b: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(l) = (y = 2*x - 10);IsParallel(l, OneOf(Asymptote(G)));PointOnCurve(OneOf(Focus(G)), l)", "coords": {} }, { "index": 3596, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3596.png", "answer": "y=pm*(sqrt(3)/3)*x", "fact_expr": "G: Hyperbola;a: Number;H: Point;Expression(G) = (y^2 + x^2/a = 1);Coordinate(H) = (0,-2);OneOf(Focus(G))=H", "coords": { "H": [ 0.0, -2.0 ] } }, { "index": 3597, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.681220143106833, "b": 2.2412689930799665, "orientation": "horizontal" }, "optimization": { "final_loss": 4.985250663519028e-06, "converged": true, "iterations": 95 }, "output_path": "results/train/hyperbola/problem_3597.png", "answer": "x^2/9 - y^2/16 = 1", "fact_expr": "G: Hyperbola;Coordinate(Focus(G)) = (pm*5, 0);Expression(Asymptote(G)) = (y = pm*(4/3)*x)", "coords": {} }, { "index": 3598, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3598.png", "answer": "sqrt(5)*2/5", "fact_expr": "G: Hyperbola;H: Point;Expression(G) = (x^2/4 - y^2 = 1);Coordinate(H) = (2, 0)", "coords": { "H": [ 2.0, 0.0 ] } }, { "index": 3599, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3599.png", "answer": "4", "fact_expr": "H: Parabola;Expression(H) = (y^2 = 2*(p*x));p: Real;p>0;G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1);Focus(H) = RightFocus(G)", "coords": {} }, { "index": 3600, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3600.png", "answer": "(2, 2*sqrt(2))", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;P: Point;x1: Number;y1: Number;Coordinate(P) = (x1, y1);Quadrant(P) = 1;PointOnCurve(P, C);Abs(LineSegmentOf(P, F)) = 3", "coords": {} }, { "index": 3601, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 7.0, "b": 4.898979485566356, "major_axis": "x", "x_coef": 49.0, "y_coef": 24.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3601.png", "answer": "x^2/9 - y^2/16 = 1", "fact_expr": "G: Hyperbola;H: Ellipse;e:Number;Expression(H) = (x^2/49 + y^2/24 = 1);Focus(G)=Focus(H);Eccentricity(G)=e;e=5/3", "coords": {} }, { "index": 3602, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3602.png", "answer": "y=pm*sqrt(2)*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/2 - y^2/4 = -1)", "coords": {} }, { "index": 3603, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3603.png", "answer": "2*sqrt(3)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/m^2 = 1);m: Number;0 < m;m < 4;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;l: Line;PointOnCurve(F1, l);A: Point;B: Point;Intersection(l, G) = {A, B};Max(Abs(LineSegmentOf(A, F2)) + Abs(LineSegmentOf(B, F2))) = 10", "coords": {} }, { "index": 3604, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "major_axis": "x", "x_coef": 3.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3604.png", "answer": "sqrt(2)", "fact_expr": "G: Ellipse;H: Line;P:Point;Expression(G) = (x^2/3 + y^2 = 1);Expression(H) = (x + y = 4);PointOnCurve(P,G)", "coords": {} }, { "index": 3605, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3606, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3607, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3607.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;A: Point;B: Point;O: Origin;F: Point;l: Line;a>0;b>0;L1:Line;L2:Line;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 4*x);Focus(H) = F;Directrix(H) = l;Intersection(l,L1)=A;Intersection(l,L2)=B;Abs(LineSegmentOf(A, B)) = 4*Abs(LineSegmentOf(O, F));Asymptote(G)={L1,L2}", "coords": {} }, { "index": 3608, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3608.png", "answer": "{2,1/2}", "fact_expr": "F: Point;Focus(C) = F;C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;p>0;l: Line;Intersection(l, C) = {M, N};M: Point;N: Point;AngleOf(M, F, N) = ApplyUnit(60, degree);MidPoint(LineSegmentOf(M, N)) = P;P: Point;Distance(P, Directrix(C)) = d;d: Number;d = Abs(LineSegmentOf(M, N))*sqrt(3)/2", "coords": {} }, { "index": 3609, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3610, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_focus_mismatch;ellipse_point_off_curve", "params": { "a": 1.8644779920578003, "b": 1.6090798277777116, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 1.9372904104920558e-05, "converged": true, "iterations": 60 }, "output_path": "results/train/ellipse/problem_3610.png", "answer": "x^2/4 + y^2/2 = 1", "fact_expr": "E: Ellipse;b: Number;a: Number;G: Point;F: Point;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(G) = (sqrt(2), 1);Coordinate(F) = (sqrt(2), 0);RightFocus(E) = F;PointOnCurve(G, E)", "coords": { "G": [ 1.4142135623730951, 1.0 ], "F": [ 1.4142135623730951, 0.0 ] }, "validation_reasons": [ "ellipse_focus_mismatch", "ellipse_point_off_curve" ] }, { "index": 3611, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 1.2135674953460693, "b": 2.4270472526550293, "orientation": "horizontal" }, "optimization": { "final_loss": 4.453341261978494e-06, "converged": true, "iterations": 125 }, "output_path": "results/train/hyperbola/problem_3611.png", "answer": "5*sqrt(5)/8", "fact_expr": "l: Line;E: Hyperbola;b: Number;a: Number;F: Point;A: Point;C: Point;B: Point;D: Point;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (-5, 0);Eccentricity(E) = sqrt(5)/2;LeftFocus(E) = F;PointOnCurve(F,l);Intersection(l, E) = {A, B};L1:Line;L2:Line;Asymptote(E)={L1,L2};Intersection(l,L1)=C;Intersection(l,L2)=D;VectorOf(F, A) = VectorOf(A, C)", "coords": { "F": [ -5.0, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 3612, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3613, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3614, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3614.png", "answer": "y=pm*(1/2)*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1)", "coords": {} }, { "index": 3615, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_3615.png", "answer": "y = pm*sqrt(3)*x", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = 2", "coords": {} }, { "index": 3616, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3616.png", "answer": "[\\sqrt{3}, +\\infty)", "fact_expr": "G: Hyperbola;a: Number;b: Number;c: Number;F1: Point;F2: Point;P: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = -2*c^2/3", "coords": {} }, { "index": 3617, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3617.png", "answer": "y=pm*2*x", "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2/4 = 1)", "coords": {} }, { "index": 3618, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3618.png", "answer": "2", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;c: Number;F: Point;a>0;b>0;L:Line;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y^2 + (-a + x)^2 = c^2/16);RightFocus(C) = F;TangentOfPoint(F,G)=L;IsPerpendicular(L,OneOf(Asymptote(C)))", "coords": {} }, { "index": 3619, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3619.png", "answer": "2*sqrt(2)", "fact_expr": "C: Parabola;p: Number;G: Line;Q: Point;P: Point;F: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;PointOnCurve(F,G);Intersection(G, C) = {P, Q};Quadrant(Q)=1;2*VectorOf(P, F) = VectorOf(F, Q)", "coords": {} }, { "index": 3620, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.9970735311508179, "direction": "right" }, "optimization": { "final_loss": 2.85904548036342e-06, "converged": true, "iterations": 88 }, "output_path": "results/train/parabola/problem_3620.png", "answer": "4", "fact_expr": "G: Parabola;p: Real;H: Point;Expression(G) = (y^2 = 2*(p*x));Coordinate(H) = (2, 4);PointOnCurve(H, G)", "coords": { "H": [ 2.0, 4.0 ] } }, { "index": 3621, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3621.png", "answer": "7", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 12*y);A: Point;P: Point;B: Point;Coordinate(A) = (0, 5);PointOnCurve(P,G);l1:Line;Expression(l1)=(y=-3);l2:Line;IsPerpendicular(l2,l1);PointOnCurve(P,l2);FootPoint(l1,l2)=B;Abs(LineSegmentOf(P, B)) = Abs(LineSegmentOf(P, A))", "coords": { "A": [ 0.0, 5.0 ] } }, { "index": 3622, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3622.png", "answer": "4", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;HalfFocalLength(G)=c;c:Number;O:Origin;l:Line;Expression(l) = (a*y + b*x = a*b);Distance(O,l)=(c/4+1)", "coords": {} }, { "index": 3623, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3623.png", "answer": "2*sqrt(7)", "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (x^2/4 - y^2/m = 1);Expression(Asymptote(G)) = (y = pm*(x*(sqrt(3)/2)))", "coords": {} }, { "index": 3624, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3624.png", "answer": "4", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1)", "coords": {} }, { "index": 3625, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3625.png", "answer": "-3", "fact_expr": "l: Line;C: Parabola;P: Point;Q: Point;A: Point;B: Point;lambda:Real;mu:Real;Expression(C) = (x^2 = y);Expression(l) = (y = x + 1);Intersection(l, C) = {A, B};Coordinate(P) = (0, 1);Coordinate(Q) = (-1, 0);VectorOf(P, Q) = lambda*VectorOf(Q, A);lambda*VectorOf(Q, A) = mu*VectorOf(Q, B)", "coords": { "P": [ 0.0, 1.0 ], "Q": [ -1.0, 0.0 ] } }, { "index": 3626, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 5.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3626.png", "answer": "(5,0)", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/25 - y^2/9 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 9*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 3627, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3628, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3628.png", "answer": "6*sqrt(2)", "fact_expr": "G: Parabola;H: Line;P: Point;d1: Number;d2: Number;Expression(G) = (y^2 = -8*x);Expression(H) = (x + y - 10 = 0);PointOnCurve(P, G);Distance(P, Directrix(G)) = d1;Distance(P, H) = d2", "coords": {} }, { "index": 3629, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3629.png", "answer": "5/7", "fact_expr": "C: Ellipse;b: Number;a: Number;G: Line;A: Point;F: Point;B: Point;O: Origin;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;PointOnCurve(O, G);Intersection(C, G) = {A, B};LineSegmentOf(A, B) = 15;LineSegmentOf(B, F) = 12;Sin(AngleOf(A, B, F)) = 3/5", "coords": {} }, { "index": 3630, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3630.png", "answer": "-4", "fact_expr": "G: Hyperbola;H: Parabola;p: Real;Expression(G) = (x^2/3 - y^2 = 1);Expression(H) = (y^2 = 2*(p*x));Focus(H) = LeftFocus(G)", "coords": {} }, { "index": 3631, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3631.png", "answer": "x^2/3-y^2=1", "fact_expr": "C1: Hyperbola;Expression(C1) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;FocalLength(C1) = 4;C2: Circle;Expression(C2) = ((x - 2)^2 + y^2 = 1);IsTangent(Asymptote(C1), C2)", "coords": {} }, { "index": 3632, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3633, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3633.png", "answer": "3*sqrt(5)/5", "fact_expr": "G: Hyperbola;b: Number;a: Number;A: Point;P: Point;B: Point;a>0;b>0;Expression(G) = (-x^2/b^2 + y^2/a^2 = 1);Intersection(LeftDirectrix(G), xAxis) = B;Coordinate(A) = (0, b);VectorOf(A, P) = 3*VectorOf(A, B);PointOnCurve(P, G)", "coords": {} }, { "index": 3634, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3634.png", "answer": "(\\sqrt{2}/2, 1)", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);DotProduct(VectorOf(P, F1), VectorOf(P, F2))<0", "coords": {} }, { "index": 3635, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3635.png", "answer": "sqrt(5)/2", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;O: Origin;F: Point;M: Point;N: Point;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;IsDiameter(LineSegmentOf(O,F),G);Intersection(OneOf(Asymptote(C)),G)=M;Negation(M=O);L:Line;PointOnCurve(F,L);IsPerpendicular(L,xAxis);Intersection(L,OneOf(Asymptote(C)))=N;Area(TriangleOf(O,M,F))=4*Area(TriangleOf(M,N,F))", "coords": {} }, { "index": 3636, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3636.png", "answer": "y=pm*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/2 - y^2/m = 1);m: Number;H: Circle;Expression(H) = (-4*y - 3*x + x^2 + y^2 + 2 = 0);F: Point;RightFocus(G) = F;PointOnCurve(F, H)", "coords": {} }, { "index": 3637, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3637.png", "answer": "3*x-2*y-3=0", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);l: Line;PointOnCurve(Focus(G), l);H: Line;Expression(H) = (3*x - 2*y = 0);IsParallel(H, l)", "coords": {} }, { "index": 3638, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3638.png", "answer": "x^2/(4*a^2)+y^2/(4*b^2)=1", "fact_expr": "G: Ellipse;b: Number;a: Number;O: Origin;Q: Point;P: Point;F1: Point;F2:Point;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P,G);Focus(G)={F1,F2};VectorOf(O, Q) = VectorOf(P, F1) + VectorOf(P, F2)", "coords": {} }, { "index": 3639, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.3484079837799072, "b": 1.8638464212417603, "orientation": "horizontal" }, "optimization": { "final_loss": 4.620347226591548e-06, "converged": true, "iterations": 88 }, "output_path": "results/train/hyperbola/problem_3639.png", "answer": "2*sqrt(2)", "fact_expr": "L: Hyperbola;b: Number;a: Number;F1: Point;F2: Point;P: Point;Expression(L) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-3, 0);Coordinate(F2) = (3, 0);LeftFocus(L) = F1;RightFocus(L) = F2;PointOnCurve(P, L);4*Abs(LineSegmentOf(P, F1)) = 5*Abs(LineSegmentOf(P, F2));DotProduct(VectorOf(P, F2), VectorOf(F1, F2)) = 0", "coords": { "F1": [ -3.0, 0.0 ], "F2": [ 3.0, 0.0 ] } }, { "index": 3640, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3640.png", "answer": "8", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;M: Point;N: Point;F:Point;Expression(G) = (y^2 = 8*x);PointOnCurve(F,H);Focus(G)=F;Intersection(H,G)={A,B};MidPoint(LineSegmentOf(A,B))=M;L:Line;PointOnCurve(M,L);IsPerpendicular(L,yAxis);Intersection(L,yAxis)=N;Abs(LineSegmentOf(M, N)) = 2", "coords": {} }, { "index": 3641, "success": false, "error": "Circle from diameter: missing point coordinates" }, { "index": 3642, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4506710767745972, "b": 2.0511727333068848, "orientation": "horizontal" }, "optimization": { "final_loss": 4.657696081267204e-06, "converged": true, "iterations": 116 }, "output_path": "results/train/hyperbola/problem_3642.png", "answer": "3", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/4 - y^2/m = 1);m: Number;m>0;Expression(Asymptote(C)) = (y = pm*(sqrt(2)*x));F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, RightPart(C)) = True;Abs(LineSegmentOf(P, F1)) = m - 1", "coords": {} }, { "index": 3643, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3643.png", "answer": "2*sqrt(2)", "fact_expr": "C: Hyperbola;b: Number;a: Number;G:Line;k: Number;A: Point;F2: Point;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y = k*x);RightFocus(C) = F2;Intersection(G, C) = {A, B};AngleOf(A, F2, B) = (2*pi)/3;Area(TriangleOf(A,F2,B))=2*sqrt(3)", "coords": {} }, { "index": 3644, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3644.png", "answer": "(x+1)^2+(y-1/2)^2=1/2", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 2*y);H: Circle;PointOnCurve(Center(H),G) = True;I: Line;Expression(I) = (2*x + 2*y + 3 = 0);IsTangent(I,H) = True;WhenMin(Area(H))", "coords": {} }, { "index": 3645, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3646, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3646.png", "answer": "2", "fact_expr": "G: Parabola;A: Point;B: Point;O: Origin;Expression(G) = (y^2 = 4*x);PointOnCurve(Focus(G), LineSegmentOf(A, B));IsChordOf(LineSegmentOf(A, B), G);LineSegmentOf(A, B) = 4", "coords": {} }, { "index": 3647, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3647.png", "answer": "4", "fact_expr": "C:Hyperbola;a:Number;b:Number;a>0;b>0;F:Point;Expression(C)=(x^2/a^2-y^2/b^2=1);OneOf(Focus(C))=F;Coordinate(F)=(5,0);Slope(OneOf(Asymptote(C)))=3/4", "coords": { "F": [ 5.0, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 3648, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3648.png", "answer": "(y-x^2)*(1+4*x^2)=1", "fact_expr": "A: Point;B: Point;l: Line;G: Parabola;M: Point;Expression(G) = (y = x^2);Intersection(l, G) = {A, B};Abs(LineSegmentOf(A, B)) = 2;MidPoint(LineSegmentOf(A, B)) = M", "coords": {} }, { "index": 3649, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3650, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3650.png", "answer": "sqrt(17)/2", "fact_expr": "G: Parabola;H: Point;P: Point;Expression(G) = (y^2 = 2*x);Coordinate(H) = (0, 2);PointOnCurve(P, G)", "coords": { "H": [ 0.0, 2.0 ] } }, { "index": 3651, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3651.png", "answer": "y^2=12*x", "fact_expr": "H: Ellipse;Expression(H) = (x^2/25 + y^2/16 = 1);Vertex(G) = Center(H);G: Parabola;RightFocus(H) = Focus(G)", "coords": {} }, { "index": 3652, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3653, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3654, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3654.png", "answer": "3", "fact_expr": "l: Line;G: Ellipse;F1: Point;P: Point;Q: Point;F2: Point;Expression(G) = (x^2/4 + y^2/3 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, l);Intersection(l, G) = {P, Q}", "coords": {} }, { "index": 3655, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3655.png", "answer": "{0, 1}", "fact_expr": "H: Line;Expression(H) = (y = k*x + 2);k: Number;G: Parabola;Expression(G) = (y^2 = 8*x);NumIntersection(H, G) = 1", "coords": {} }, { "index": 3656, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3656.png", "answer": "4", "fact_expr": "G: Parabola;H: Line;P: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (x + 2 = 0);PointOnCurve(P, G);Distance(P, H) = 5;Focus(G)=F;F:Point", "coords": {} }, { "index": 3657, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3658, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3658.png", "answer": "(a-p)/2", "fact_expr": "A: Point;B: Point;Length(LineSegmentOf(A, B)) = a;a: Number;Endpoint(LineSegmentOf(A, B)) = {A, B};G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;a>2*p;PointOnCurve(A,G) = True;PointOnCurve(B,G) = True;MidPoint(LineSegmentOf(A,B)) = M;M: Point", "coords": {} }, { "index": 3659, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3659.png", "answer": "(0,12)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/k = 1);k: Number;e: Number;In(e, (1, 2));Eccentricity(G) = e", "coords": {} }, { "index": 3660, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3660.png", "answer": "x^2/5+y^2=1\n2*sqrt(5)/5", "fact_expr": "H: Line;Expression(H) = (x - 2*y + 2 = 0);G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;PointOnCurve(OneOf(Vertex(G)), H);PointOnCurve(OneOf(Focus(G)), H)", "coords": {} }, { "index": 3661, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 1.9161535892818333, "b": 0.8358714748346172, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 1.4021570127665244e-06, "converged": true, "iterations": 98 }, "output_path": "results/train/ellipse/problem_3661.png", "answer": "x^2/4+y^2=1", "fact_expr": "G: Ellipse;H: Point;I: Point;Coordinate(H) = (2, 0);Coordinate(I) = (0, 1);PointOnCurve(H,G);PointOnCurve(I,G)", "coords": { "H": [ 2.0, 0.0 ], "I": [ 0.0, 1.0 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 3662, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3662.png", "answer": "sqrt(2)/2", "fact_expr": "G: Ellipse;a: Number;b: Number;F1: Point;F2: Point;P: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P,RightDirectrix(G));YCoordinate(P)=sqrt(3)*c;c:Number;HalfFocalLength(G)=c;LineSegmentOf(F1, F2) = LineSegmentOf(F2, P)", "coords": {} }, { "index": 3663, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3663.png", "answer": "x^2/4 - y^2/2 = 1", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;O: Origin;F: Point;a > b;b > 0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;PointOnCurve(P,L);Expression(L)=(x+sqrt(2)*y=0);Abs(LineSegmentOf(O,F))=Abs(LineSegmentOf(P,F));Area(TriangleOf(P, O, F)) = 2*sqrt(2);L:Line;OneOf(Asymptote(C))=L", "coords": {} }, { "index": 3664, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_3664.png", "answer": "[2*sqrt(3)-1,2*sqrt(3)+1]", "fact_expr": "C1: Circle;C2: Circle;B: Point;A: Point;M: Point;Expression(C1) = (x^2 + y^2 = 9);Expression(C2) = (x^2 + y^2 = 4);Coordinate(M) = (1, 0);PointOnCurve(A, C2);PointOnCurve(B, C1);AngleOf(A, M, B) = ApplyUnit(90, degree)", "coords": {} }, { "index": 3665, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.1503682136535645, "b": 2.5736796855926514, "orientation": "horizontal" }, "optimization": { "final_loss": 5.191216132516274e-06, "converged": true, "iterations": 118 }, "output_path": "results/train/hyperbola/problem_3665.png", "answer": "4*sqrt(2)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Eccentricity(G) = sqrt(6)/2", "coords": {} }, { "index": 3666, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3666.png", "answer": "(1, \\sqrt{2}+1)", "fact_expr": "G: Hyperbola;b: Number;a: Number;F1: Point;F2: Point;P: Point;a>0;b>0;c:Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);LeftFocus(G) = F1;RightFocus(G)=F2;PointOnCurve(P,G);Sin(AngleOf(P,F1,F2))/Sin(AngleOf(P,F2,F1))=a/c", "coords": {} }, { "index": 3667, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3667.png", "answer": "y = 1", "fact_expr": "G: Parabola;Expression(G) = (y = x^2/4);l:Line;PointOnCurve(Focus(G),l);IsPerpendicular(l,SymmetryAxis(G))", "coords": {} }, { "index": 3668, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3669, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3669.png", "answer": "(p, +\\infty)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;A: Point;B: Point;Negation(A=B);PointOnCurve(A, G);PointOnCurve(B, G);P: Point;x0: Number;Coordinate(P) = (x0, 0);Intersection(PerpendicularBisector(LineSegmentOf(A, B)), xAxis) = P", "coords": {} }, { "index": 3670, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3671, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3671.png", "answer": "sqrt(29)", "fact_expr": "G: Parabola;P: Point;A: Point;F: Point;Expression(G) = (y^2 = 2*x);Coordinate(A) = (5/2, 5);Focus(G) = F;PointOnCurve(P, G)", "coords": { "A": [ 2.5, 5.0 ] } }, { "index": 3672, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3672.png", "answer": "4*sqrt(3)", "fact_expr": "G: Parabola;A: Point;B: Point;O: Origin;F: Point;C: Point;Expression(G) = (y^2 = 8*x);Focus(G) = F;PointOnCurve(F, LineSegmentOf(A, B));IsChordOf(LineSegmentOf(A,B),G);Intersection(Directrix(G), xAxis) = C;AngleOf(O, F, A) = 2*pi/3", "coords": {} }, { "index": 3673, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3673.png", "answer": "11", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/9 - y^2 = 1);F1: Point;F2: Point;F1 = LeftFocus(C);F2 = RightFocus(C);A: Point;PointOnCurve(A, LeftPart(C));Abs(LineSegmentOf(A, F1)) = 5", "coords": {} }, { "index": 3674, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3674.png", "answer": "2", "fact_expr": "C: Ellipse;b: Number;a: Number;F1: Point;P: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Focus(C) = {F1, F2};PointOnCurve(P, C);AngleOf(F1, P, F2) = ApplyUnit(120, degree);Area(TriangleOf(F1, P, F2)) = 4*sqrt(3)", "coords": {} }, { "index": 3675, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 3.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3675.png", "answer": "3*sqrt(2)/2", "fact_expr": "G: Ellipse;A: Point;B: Point;C: Point;Expression(G) = (x^2/3 + y^2/2 = 1);LeftFocus(G) = A;UpperVertex(G)=B;PointOnCurve(C, G)", "coords": {} }, { "index": 3676, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3676.png", "answer": "(pm*3,0)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1)", "coords": {} }, { "index": 3677, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3677.png", "answer": "(3,2)", "fact_expr": "G: Parabola;H: Line;Expression(G) = (y^2 = 4*x);Expression(H) = (y = x - 1)", "coords": {} }, { "index": 3678, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3679, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3679.png", "answer": "pm*(4/3)", "fact_expr": "l: Line;C: Parabola;A: Point;M: Point;F: Point;N: Point;k:Number;Expression(C) = (y^2 = 8*x);Coordinate(A) = (-1, 0);Focus(C) = F;PointOnCurve(A, l);Slope(l) = k;Intersection(l, C) = {M, N};Area(TriangleOf(A,M,F))=2*Area(TriangleOf(A,N,F))", "coords": { "A": [ -1.0, 0.0 ] } }, { "index": 3680, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3680.png", "answer": "sqrt(2)", "fact_expr": "G: Parabola;A: Point;Expression(G) = (x^2 = y);PointOnCurve(A, G);Distance(A, Directrix(G)) = 5/4", "coords": {} }, { "index": 3681, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3681.png", "answer": "y=x+1", "fact_expr": "l: Line;G: Ellipse;B: Point;A: Point;Expression(G) = (x^2/4 + y^2 = 1);Intersection(l, G) = {A, B};Coordinate(MidPoint(LineSegmentOf(A,B)))=(-4/5,1/5)", "coords": {} }, { "index": 3682, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3683, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3683.png", "answer": "sqrt(5)/5", "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;P: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P,G);Focus(G)={F1,F2};WhenMax(AngleOf(F1,P,F2));Tan(AngleOf(P, F1, F2)) = 2", "coords": {} }, { "index": 3684, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3684.png", "answer": "{y=2*x,y=-2*x}", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;O: Origin;F: Point;A: Point;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;Center(G)=F;Radius(G)=LineSegmentOf(O,F);Intersection(G, xAxis) = {O, A};Intersection(G,OneOf(Asymptote(C)))={O,B};LineSegmentOf(A,B)=4*a", "coords": {} }, { "index": 3685, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3685.png", "answer": "y=(pm*1/2)*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1)", "coords": {} }, { "index": 3686, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3686.png", "answer": "3", "fact_expr": "G: Parabola;l: Line;P: Point;Expression(G) = (y^2 = 4*x);Expression(l) = (4*x - 3*y + 11 = 0);PointOnCurve(P, G);Distance(P, Directrix(G)) = d1;Distance(P,l)=d2;d1:Number;d2:Number", "coords": {} }, { "index": 3687, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3688, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3688.png", "answer": "x + pm*y = 0", "fact_expr": "l: Line;C: Hyperbola;P: Point;A: Point;Q: Point;Expression(C) = (x^2 - y^2 = 1);RightVertex(C) = A;PointOnCurve(A, l);l1: Line;l2: Line;Asymptote(C) = {l1, l2};Intersection(l, l1) = P;Intersection(l, l2) = Q;VectorOf(P, A) = 2*VectorOf(A, Q)", "coords": {} }, { "index": 3689, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3689.png", "answer": "y^2/5 - x^2/4 = 1", "fact_expr": "G: Hyperbola;H: Ellipse;A: Point;Expression(H) = (x^2/16 + y^2/9 = 1);Coordinate(A) = (4, -5);PointOnCurve(A, G);Focus(G)=Endpoint(MinorAxis(H))", "coords": { "A": [ 4.0, -5.0 ] } }, { "index": 3690, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3690.png", "answer": "sqrt(2) - 1", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;c: Number;HalfFocalLength(G) = c;H: Line;Expression(H) = (y = 2*x);XCoordinate(OneOf(Intersection(H, G))) = c", "coords": {} }, { "index": 3691, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3691.png", "answer": "2/3", "fact_expr": "G: Ellipse;b: Number;H: Circle;I: Line;F1: Point;F2: Point;3 > b;b > 0;Expression(G) = (x^2/9 + y^2/b^2 = 1);Expression(I) = (x + y + 2 = 0);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1,H);PointOnCurve(F2,H);IsTangent(I,H)", "coords": {} }, { "index": 3692, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.4970548748970032, "direction": "right" }, "optimization": { "final_loss": 0.00010081290383823216, "converged": true, "iterations": 80 }, "output_path": "results/train/parabola/problem_3692.png", "answer": "sqrt(2)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;A: Point;Coordinate(A) = (1, sqrt(2));PointOnCurve(A, G);B: Point;C: Point;PointOnCurve(Vertex(TriangleOf(A, B, C)), G);k1: Number;k2: Number;k3: Number;Slope(OverlappingLine(LineSegmentOf(A, B))) = k1;Slope(OverlappingLine(LineSegmentOf(B, C))) = k2;Slope(OverlappingLine(LineSegmentOf(C, A))) = k3", "coords": { "A": [ 1.0, 1.4142135623730951 ] } }, { "index": 3693, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3694, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3694.png", "answer": "5/4", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;Area(TriangleOf(P, F1, F2)) = 9;a + b = 7", "coords": {} }, { "index": 3695, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3695.png", "answer": "x^2/8 - y^2/8 = 1", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;M: Point;A: Point;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x - 2*y = 0);Coordinate(M) = (-a, 2*b);FocalLength(C) = 8;Intersection(G, C) = {A, B};Abs(LineSegmentOf(M, A)) = Abs(LineSegmentOf(M, B))", "coords": {} }, { "index": 3696, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3696.png", "answer": "[sqrt(3)/3, sqrt(2)/2]", "fact_expr": "F1: Point;F2: Point;Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);Focus(G) = {F1, F2};c: Number;G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;P: Point;PointOnCurve(P, G);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = c^2", "coords": {} }, { "index": 3697, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3697.png", "answer": "2", "fact_expr": "E: Hyperbola;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;Coordinate(F) = (-2, 0);LeftFocus(E) = F;l: Line;A: Point;Coordinate(A) = (2*a, 0);PointOnCurve(A, l);B: Point;C: Point;L1: Line;L2: Line;Asymptote(E) = {L1, L2};Intersection(l, L1) = B;Intersection(l, L2) = C ;VectorOf(A, B) = VectorOf(B, C);IsPerpendicular(LineSegmentOf(C, F), l)", "coords": { "F": [ -2.0, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 3698, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3699, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3699.png", "answer": "4/5", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 + y^2/5 = 1);PointOnCurve(P, G);Focus(G) = {F1, F2};Area(TriangleOf(P, F1, F2)) = 5/2", "coords": {} }, { "index": 3700, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3700.png", "answer": "sqrt(5)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Expression(OneOf(Asymptote(C))) = (2*x - y = 0)", "coords": {} }, { "index": 3701, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.9959803819656372, "direction": "right" }, "optimization": { "final_loss": 3.6780886603082763e-07, "converged": true, "iterations": 56 }, "output_path": "results/train/parabola/problem_3701.png", "answer": "5/4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = a*x);a: Number;A: Point;Coordinate(A) = (1/4, 1);PointOnCurve(A, G) = True", "coords": { "A": [ 0.25, 1.0 ] } }, { "index": 3702, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3702.png", "answer": "4+sqrt(5)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;P: Point;PointOnCurve(P, G);A: Point;Coordinate(A) = (3, 1)", "coords": { "A": [ 3.0, 1.0 ] } }, { "index": 3703, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 0.929953396320343, "b": 2.629857301712036, "orientation": "horizontal" }, "optimization": { "final_loss": 4.594453912432073e-06, "converged": true, "iterations": 175 }, "output_path": "results/train/hyperbola/problem_3703.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Eccentricity(G) = 3;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;H: Line;PointOnCurve(F2, H);IsParallel(H, OneOf(Asymptote(G)));A: Point;Intersection(H, G) = A;r: Number;Radius(InscribedCircle(TriangleOf(A, F1, F2))) = r", "coords": {} }, { "index": 3704, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3704.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0", "coords": {} }, { "index": 3705, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_3705.png", "answer": "3", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(G) = (-2*sqrt(3), 0);Eccentricity(C) = 2", "coords": { "G": [ -3.4641016151377544, 0.0 ] } }, { "index": 3706, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3707, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3707.png", "answer": "sqrt(5)/2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Slope(OneOf(Asymptote(G))) = 1/2", "coords": {} }, { "index": 3708, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3708.png", "answer": "3*sqrt(3)", "fact_expr": "P: Point;C: Ellipse;Expression(C) = (x^2/25 + y^2/9 = 1);PointOnCurve(P, C);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 3709, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 10.0, "b": 6.0, "major_axis": "x", "x_coef": 100.0, "y_coef": 36.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3709.png", "answer": "14", "fact_expr": "G: Ellipse;Expression(G) = (x^2/100 + y^2/36 = 1);P: Point;PointOnCurve(P, G) = True;F1: Point;F2: Point;OneOf(Focus(G)) = F1;OneOf(Focus(G)) = F2;Negation(F1 = F2);Distance(P, F1) = 6", "coords": {} }, { "index": 3710, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3710.png", "answer": "2", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x + sqrt(3)*y = 0);IsPerpendicular(OneOf(Asymptote(C)), G)", "coords": {} }, { "index": 3711, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3711.png", "answer": "4", "fact_expr": "G: Hyperbola;p: Number;H: Parabola;Expression(G) = (x^2/3 - 16*y^2/p^2 = 1);Expression(H) = (y^2 = 2*(p*x));PointOnCurve(LeftFocus(G), Directrix(H))", "coords": {} }, { "index": 3712, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3712.png", "answer": "2", "fact_expr": "G: Parabola;p: Number;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));PointOnCurve(F, G);XCoordinate(F)=3;Distance(F, Focus(G)) = 4", "coords": {} }, { "index": 3713, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3713.png", "answer": "3*sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/b^2 = 1);b: Number;b>0;Inclination(OneOf(Asymptote(G))) = pi/3", "coords": {} }, { "index": 3714, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3714.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "l: Line;G: Ellipse;Expression(G) = (x^2/4 + y^2/2 = 1);A: Point;B: Point;P: Point;Coordinate(P) = (1, 1);PointOnCurve(P,l);Intersection(l,G)={A,B};MidPoint(LineSegmentOf(A,B)) = P", "coords": { "P": [ 1.0, 1.0 ] } }, { "index": 3715, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3715.png", "answer": "16/5", "fact_expr": "l: Line;G: Ellipse;A: Point;B: Point;F1: Point;Expression(G) = (x^2 + 4*y^2 = 16);PointOnCurve(F1,l);Intersection(l,G)={A,B};LeftFocus(G) = F1;Slope(l) = 1;IsChordOf(LineSegmentOf(A, B),G)", "coords": {} }, { "index": 3716, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3716.png", "answer": "y^2=4*x", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(C) = F;l: Line;Directrix(C) = l;M: Point;PointOnCurve(M, C);H: Point;IsPerpendicular(LineSegmentOf(M, H), l);FootPoint(LineSegmentOf(M, H), l) = H;Abs(LineSegmentOf(M, H)) = 4;AngleOf(H, F, M) = ApplyUnit(60, degree)", "coords": {} }, { "index": 3717, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3717.png", "answer": "1/3", "fact_expr": "G: Hyperbola;n: Real;m: Real;H: Parabola;Expression(G) = (y^2/n + x^2/m = 1);Expression(H) = (y^2 = 4*x);Eccentricity(G)=2;RightFocus(G)=Focus(H)", "coords": {} }, { "index": 3718, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3719, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3720, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3720.png", "answer": "(-6, \\pm 6\\sqrt{2})", "fact_expr": "G: Parabola;Expression(G) = (y^2 = -12*x);P:Point;Distance(P,Focus(G))=9;PointOnCurve(P,G)", "coords": {} }, { "index": 3721, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3721.png", "answer": "2*x-y-15=0", "fact_expr": "A: Point;B: Point;G: Hyperbola;H: Line;Expression(G) = (x^2 - 4*y^2 = 4);Intersection(H, G) = {A,B};Coordinate(MidPoint(LineSegmentOf(A,B))) =(8,1)", "coords": {} }, { "index": 3722, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.0625, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3722.png", "answer": "65/16", "fact_expr": "l: Line;G: Parabola;Expression(G) = (y = 4*x^2);Directrix(G) = l;A: Point;Coordinate(A) = (1, 4)", "coords": { "A": [ 1.0, 4.0 ] } }, { "index": 3723, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3723.png", "answer": "(pm*4, 0)\nsqrt(3)*x+pm*y=0", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Ellipse;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2/25 + y^2/9 = 1);Eccentricity(G) = 2;Focus(G) = Focus(H)", "coords": {} }, { "index": 3724, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3724.png", "answer": "1", "fact_expr": "G: Hyperbola;a: Number;H: Ellipse;Expression(G) = (-y^2/2 + x^2/a = 1);Expression(H) = (x^2/4 + y^2/a^2 = 1);Focus(H) = Focus(G)", "coords": {} }, { "index": 3725, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_ecc_mismatch", "params": { "a": 1.9257268905639648, "b": 1.7993911322594118, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 2.5576271215184017e-15, "converged": true, "iterations": 2 }, "output_path": "results/train/ellipse/problem_3725.png", "answer": "3", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2/4 + y^2/m = 1);PointOnCurve(Focus(G), xAxis);Eccentricity(G)=1/2", "coords": {}, "validation_reasons": [ "ellipse_ecc_mismatch" ] }, { "index": 3726, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3726.png", "answer": "sqrt(21)/7", "fact_expr": "C: Ellipse;G: Line;A: Point;F1: Point;B: Point;F2: Point;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, G);Intersection(G, C) = {A, B};Abs(LineSegmentOf(A, F1)) = 4*Abs(LineSegmentOf(B, F1));AngleOf(A, F2, F1) = ApplyUnit(90, degree);b:Number;a:Number", "coords": {} }, { "index": 3727, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "y", "x_coef": 2.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3727.png", "answer": "x^2/2+y^2=1", "fact_expr": "G: Ellipse;Expression(G) = (2*x^2 + y^2 = 4);D: Point;P: Point;M: Point;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P,D),xAxis);FootPoint(LineSegmentOf(P,D),xAxis)=D;MidPoint(LineSegmentOf(P,D))=M;PointOnCurve(P,LineSegmentOf(P,D))", "coords": {} }, { "index": 3728, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3729, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3729.png", "answer": "4/9", "fact_expr": "l: Line;G: Hyperbola;M: Point;O: Origin;A: Point;B: Point;k1: Number;k2: Number;Expression(G) = (x^2/9 - y^2/4 = 1);Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A, B)) = M;Slope(l) = k1;Negation(k1 = 0);Slope(LineOf(O, M)) = k2", "coords": {} }, { "index": 3730, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3730.png", "answer": "{-14,2}", "fact_expr": "G: Parabola;a: Real;H: Circle;Expression(G) = (y^2 = 2*(a*x));Expression(H) = (-6*x + x^2 + y^2 - 7 = 0);IsTangent(Directrix(G),H)", "coords": {} }, { "index": 3731, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3731.png", "answer": "(\\sqrt{5}+1)/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;O: Origin;M: Point;N: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(LeftFocus(G), H);IsPerpendicular(H,xAxis);Intersection(H, G) = {M, N};Center(G) = O;DotProduct(VectorOf(O, M), VectorOf(O, N)) = 0", "coords": {} }, { "index": 3732, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3732.png", "answer": "2*sqrt(2)", "fact_expr": "Gamma: Parabola;Expression(Gamma) = (y^2 = 4*x);A: Point;B: Point;PointOnCurve(A, Gamma);PointOnCurve(B, Gamma);M: Point;PointOnCurve(M, Directrix(Gamma));PointOnCurve(MidPoint(LineSegmentOf(M, A)), Gamma);PointOnCurve(MidPoint(LineSegmentOf(M, B)), Gamma);N: Point;n: Number;Coordinate(N) = (0, n);Intersection(LineOf(A, B), yAxis) = N", "coords": {} }, { "index": 3733, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3734, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3734.png", "answer": "y = pm*2*x", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;OneOf(Focus(C)) = F1;A1: Point;A2: Point;Vertex(C) = {A1,A2};M: Point;PointOnCurve(M, C);G: Circle;IsDiameter(LineSegmentOf(A1,A2),G);TangentPoint(LineSegmentOf(M, F1), G) = N;MidPoint(LineSegmentOf(M, F1)) = N;N:Point", "coords": {} }, { "index": 3735, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3736, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3736.png", "answer": "x^2/8+y^2/2=1", "fact_expr": "C: Ellipse;b: Number;a: Number;G: Line;P: Point;A: Point;B: Point;a > b;b > 0;y0:Number;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y = -x);Coordinate(P) = (2*sqrt(10)/5, y0);PointOnCurve(P,C);Eccentricity(C)=e;Intersection(G, C) = {B, A};Negation(A=P);Negation(B=P);Abs(LineSegmentOf(P, A)) = Abs(LineSegmentOf(P, B));e:Number;e=sqrt(3)/2", "coords": {} }, { "index": 3737, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3737.png", "answer": "{y^2=4*x,y^2=36*x}", "fact_expr": "G: Parabola;p: Number;H: Line;p>0;M:Point;Expression(G) = (y^2 = 2*p*x);Expression(H) = (x = -p/2);PointOnCurve(M,G);Distance(M,H)=10;Distance(M,SymmetryAxis(G))=6", "coords": {} }, { "index": 3738, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3738.png", "answer": "5/3", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;M: Point;F2: Point;F1: Point;N: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, G) = True;Intersection(G, LeftPart(C)) = {M, N};Abs(LineSegmentOf(M, F2)) = Abs(LineSegmentOf(F1, F2));2*Abs(LineSegmentOf(M, F1)) = Abs(LineSegmentOf(N, F1))", "coords": {} }, { "index": 3739, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3739.png", "answer": "3/5", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 + y^2/4 = 1);Focus(G)={F1,F2};PointOnCurve(P,G);Abs(LineSegmentOf(P, F1)) = 5", "coords": {} }, { "index": 3740, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3740.png", "answer": "[\\sqrt{14}/2,2)", "fact_expr": "C: Circle;Expression(C) = (y^2 + (x - 3)^2 = 1);M: Point;T: Parabola;Expression(T) = (y^2 = 4*x);PointOnCurve(M, T);l1: Line;l2: Line;TangentPoint(l1, C) = A;TangentPoint(l2, C) = B;A: Point;B: Point;TangentOfPoint(M,C)={l1,l2}", "coords": {} }, { "index": 3741, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3741.png", "answer": "sqrt(2)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;G: Parabola;Expression(G) = (y^2 = 4*x);OneOf(Directrix(C)) = Directrix(G);WhenMin((a^4 + 4)/sqrt(a^2 + b^2))", "coords": {} }, { "index": 3742, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3742.png", "answer": "6", "fact_expr": "A: Point;Coordinate(A) = (2, -3);G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;Distance(A, Focus(G)) = 5", "coords": { "A": [ 2.0, -3.0 ] } }, { "index": 3743, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3743.png", "answer": "8", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);H: Line;F: Point;Focus(G) = F;PointOnCurve(F, H);P1: Point;P2: Point;x1: Number;x2: Number;y1: Number;y2: Number;Coordinate(P1) = (x1, y1);Coordinate(P2) = (x2, y2);Intersection(H, G) = {P1, P2};y1 + y2 = 6", "coords": {} }, { "index": 3744, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3744.png", "answer": "x^2/6 + y^2/2 = 1", "fact_expr": "G: Ellipse;a: Number;H: Point;Expression(G) = (y^2/2 + x^2/a^2 = 1);Coordinate(H) = (2, 0);OneOf(Focus(G))=H", "coords": { "H": [ 2.0, 0.0 ] } }, { "index": 3745, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3745.png", "answer": "sqrt(2)", "fact_expr": "l: Line;C: Parabola;p: Number;B: Point;F: Point;A: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;Expression(l)=(-p + 2*x + 2*y = 0);Intersection(l, C) = {A, B};Abs(LineSegmentOf(B, F)) = Abs(LineSegmentOf(A, F)) + 1", "coords": {} }, { "index": 3746, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3746.png", "answer": "sqrt(3)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;A: Point;B: Point;L: Line;PointOnCurve(F2,L) = True;IsPerpendicular(L,xAxis) = True;Intersection(L,C) = {A,B};D: Point;Intersection(LineSegmentOf(F1,B),yAxis) = D;IsPerpendicular(LineSegmentOf(A,D),LineSegmentOf(F1,B)) = True", "coords": {} }, { "index": 3747, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3747.png", "answer": "x=3", "fact_expr": "G: Parabola;A: Point;B: Point;Expression(G) = (y^2 = 8*x);PointOnCurve(A, G);PointOnCurve(B, G);Distance(MidPoint(LineSegmentOf(A,B)),Directrix(G))=5", "coords": {} }, { "index": 3748, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3748.png", "answer": "4", "fact_expr": "l: Line;G: Parabola;A: Point;O: Origin;B: Point;Expression(G) = (y^2 = 2*x);Intersection(l,G)={A,B};AngleOf(A, O, B) = ApplyUnit(90, degree)", "coords": {} }, { "index": 3749, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3750, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3750.png", "answer": "4", "fact_expr": "G: Parabola;M: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(M, G);Distance(M, Focus(G)) = 5", "coords": {} }, { "index": 3751, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3751.png", "answer": "8/3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;A: Point;B: Point;PointOnCurve(A, G);PointOnCurve(B, G);VectorOf(A, F) = 3*VectorOf(F, B);IsChordOf(LineSegmentOf(A, B), G)", "coords": {} }, { "index": 3752, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.449489742783178, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 6.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3752.png", "answer": "4", "fact_expr": "G: Parabola;p: Number;H: Ellipse;p>0;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (x^2/6 + y^2/2 = 1);Focus(G) = RightFocus(H)", "coords": {} }, { "index": 3753, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3753.png", "answer": "sqrt(2)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P:Point;PointOnCurve(P,G);A: Point;Coordinate(A) = (0, 1);l:Line;Expression(l) = (x = -1)", "coords": { "A": [ 0.0, 1.0 ] } }, { "index": 3754, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3755, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.0, "b": 0.3333333333333333, "major_axis": "y", "x_coef": 0.1111111111111111, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3755.png", "answer": "-9", "fact_expr": "l: Line;C: Ellipse;O: Origin;M:Point;A: Point;B: Point;Expression(C) = (9*x^2 + y^2 = 1);Negation(PointOnCurve(O,l));Negation(IsParallel(l,axis));Intersection(l, C) = {A, B};MidPoint(LineSegmentOf(A,B))=M", "coords": {} }, { "index": 3756, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3757, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3757.png", "answer": "8", "fact_expr": "G: Hyperbola;H: Parabola;a: Number;Expression(G) = (x^2 - y^2 = 2);Expression(H) = (y^2 = a*x);Focus(H) = RightFocus(G)", "coords": {} }, { "index": 3758, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.4641016151377544, "major_axis": "x", "x_coef": 16.0, "y_coef": 12.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3758.png", "answer": "y=pm*sqrt(3)*x", "fact_expr": "G: Hyperbola;O: Origin;Center(G) = O;H: Ellipse;Expression(H) = (x^2/16 + y^2/12 = 1);Focus(G) = Focus(H);InterReciprocal(Eccentricity(G), Eccentricity(H))", "coords": {} }, { "index": 3759, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3759.png", "answer": "16", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/12 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;H: Line;PointOnCurve(F1, H);M: Point;N: Point;Intersection(H, LeftPart(G)) = {M, N}", "coords": {} }, { "index": 3760, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3761, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3761.png", "answer": "9", "fact_expr": "P: Point;PointOnCurve(P, RightPart(G)) = True;G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);M: Point;C1: Circle;Expression(C1) = (y^2 + (x + 5)^2 = 4);PointOnCurve(M,C1) = True;N: Point;C2: Circle;Expression(C2) = (y^2 + (x - 5)^2 = 1);PointOnCurve(N,C2) = True", "coords": {} }, { "index": 3762, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.5, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3762.png", "answer": "(x+3/2)^2+y^2=9", "fact_expr": "G: Parabola;H: Circle;Expression(G) = (y^2 = -6*x);Focus(G) = Center(H);IsTangent(Directrix(G),H)", "coords": {} }, { "index": 3763, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.7195594310760498, "b": 1.7224669456481934, "orientation": "horizontal" }, "optimization": { "final_loss": 2.391175712546101e-06, "converged": true, "iterations": 83 }, "output_path": "results/train/hyperbola/problem_3763.png", "answer": "8", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Eccentricity(G) = sqrt(2);H: Point;Coordinate(H) = (3, 1);PointOnCurve(H, G)", "coords": { "H": [ 3.0, 1.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 3764, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3764.png", "answer": "2/3", "fact_expr": "C: Ellipse;l: Line;O: Origin;A: Point;B: Point;Expression(C) = (x^2/2 + y^2 = 1);Expression(l) = (y = x - 1);PointOnCurve(RightFocus(C),l);Intersection(l, C) = {A, B}", "coords": {} }, { "index": 3765, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3765.png", "answer": "4", "fact_expr": "M: Point;Coordinate(M) = (2, 1);l: Line;G: Hyperbola;Expression(G) = (x^2 - y^2/2 = 1);PointOnCurve(M, l);A: Point;B: Point;Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A, B)) = M", "coords": { "M": [ 2.0, 1.0 ] } }, { "index": 3766, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3766.png", "answer": "sqrt(6)/2", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/4 + x^2/a^2 = 1);a: Number;FocalLength(C) = 4*sqrt(3)", "coords": {} }, { "index": 3767, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3767.png", "answer": "5", "fact_expr": "C: Hyperbola;a: Number;b: Number;G: Line;P: Point;B: Point;A: Point;E: Point;Q: Point;F: Point;M: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftVertex(C) = A;RightVertex(C) = B ;LeftFocus(C) = F;PointOnCurve(F, G);IsPerpendicular(xAxis, G);Intersection(G, C) = {P, Q};Quadrant(P) = 2;Intersection(LineSegmentOf(P, B), yAxis) = E;Intersection(LineSegmentOf(A, E), LineSegmentOf(Q, F)) = M;VectorOf(F, M) = 2*VectorOf(M, Q)", "coords": {} }, { "index": 3768, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3768.png", "answer": "y=pm*x/2", "fact_expr": "C: Curve;Expression(C) = (2/m + 1/n = 1);m: Number;n: Number;m > 0;n > 0;WhenMin(m*n);G: Hyperbola;Expression(G) = (-y^2/n^2 + x^2/m^2 = 1)", "coords": {} }, { "index": 3769, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3769.png", "answer": "pm*sqrt(2)/2", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;H: Line;k: Number;a > b;b > 0;Expression(H) = (y = k*x);Eccentricity(G) = sqrt(2)/2;XCoordinate(OneOf(Intersection(H,G)))=b", "coords": {} }, { "index": 3770, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3770.png", "answer": "9", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 6*x);F: Point;Focus(G) = F;A: Point;B: Point;C: Point;PointOnCurve(A, G) = True;PointOnCurve(B, G) = True;PointOnCurve(C, G) = True;VectorOf(F, A) + VectorOf(F, B) = -VectorOf(F, C)", "coords": {} }, { "index": 3771, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3772, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3772.png", "answer": "{(x^2/8+y^2/2=1),(x^2/3-y^2/3=1)}", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/2 = 1);Focus(G) = Focus(H);Q: Point;Coordinate(Q) = (2, 1);PointOnCurve(Q, H) = True;H: ConicSection", "coords": { "Q": [ 2.0, 1.0 ] } }, { "index": 3773, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3773.png", "answer": "sqrt(5)/3", "fact_expr": "G: Ellipse;b: Number;a: Number;O: Origin;F1: Point;P: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);DotProduct((VectorOf(O, F1) - VectorOf(O, P)), (VectorOf(O, F1) + VectorOf(O, P))) = 0;Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2));LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G)", "coords": {} }, { "index": 3774, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3774.png", "answer": "9", "fact_expr": "C: Hyperbola;P: Point;F1: Point;F2: Point;Expression(C) = (x^2 - y^2/3 = 1);LeftFocus(C)=F1;RightFocus(C)=F2;PointOnCurve(P, C);Area(TriangleOf(P, F1, F2)) = 6", "coords": {} }, { "index": 3775, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3775.png", "answer": "9*sqrt(3)", "fact_expr": "G: Hyperbola;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/16 - y^2/9 = 1);PointOnCurve(P,G);Focus(G)={F1,F2};AngleOf(F1, P, F2) = pi/3", "coords": {} }, { "index": 3776, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3777, "success": false, "conic_type": "circle", "error": "validation: circle_point_off_curve", "params": { "center": [ 3.0, 0.0 ], "radius": 8.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_3777.png", "answer": "x^2/16+y^2/7=1", "fact_expr": "M: Circle;A: Point;B:Circle;Coordinate(A) = (-3, 0);PointOnCurve(A, M);IsInTangent(M,B);Expression(B)=((x-3)^2+y^2=64);M1:Point;Center(M)=M1", "coords": { "A": [ -3.0, 0.0 ] }, "validation_reasons": [ "circle_point_off_curve" ] }, { "index": 3778, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3778.png", "answer": "2*sqrt(2) - 2", "fact_expr": "G: Parabola;O1: Circle;M: Point;N: Point;m: Number;Expression(G) = (y^2 = -4*x);PointOnCurve(N, G);Expression(O1) = ((x - 1)^2 + (y - 2)^2 = 1);PointOnCurve(M, O1);Distance(N, yAxis) = m", "coords": {} }, { "index": 3779, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.3300262689590454, "b": 2.30381441116333, "orientation": "horizontal" }, "optimization": { "final_loss": 6.658357051492203e-06, "converged": true, "iterations": 122 }, "output_path": "results/train/hyperbola/problem_3779.png", "answer": "x^2/4 - y^2/12 = 1", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x);H: Parabola;Expression(H) = (y^2 = 16*x);OneOf(Focus(G)) = Focus(H)", "coords": {} }, { "index": 3780, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3780.png", "answer": "6*sqrt(2)/7", "fact_expr": "E: Ellipse;Expression(E) = (x^2/4 + y^2/3 = 1);B: Point;RightVertex(E) = B;PointOnCurve(RightFocus(E), L) = True;Slope(L) = 1;L: Line;M: Point;N: Point;Intersection(L, E) = {M, N}", "coords": {} }, { "index": 3781, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -3.0, 3.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_3781.png", "answer": "4", "fact_expr": "P: Point;G: Parabola;Expression(G) = (y^2 = 4);PointOnCurve(P, G);d1: Number;d2: Number;Distance(P, Directrix(G)) = d1;Distance(P, Q) = d2;H: Circle;Expression(H) = ((x + 3)^2 + (y - 3)^2 = 1);Q: Point;PointOnCurve(Q, H)", "coords": {} }, { "index": 3782, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3782.png", "answer": "4*sqrt(2)", "fact_expr": "G: Parabola;B: Point;A: Point;M: Point;Expression(G) = (x^2 = 4*y);Coordinate(M) = (2, 2);PointOnCurve(A, G);PointOnCurve(B, G);MidPoint(LineSegmentOf(A,B)) = M", "coords": { "M": [ 2.0, 2.0 ] } }, { "index": 3783, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3783.png", "answer": "5", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/9 + x^2/a^2 = 1);a: Number;a>0;P: Point;PointOnCurve(P, RightPart(G));F1: Point;LeftFocus(G)=F1;F2: Point;RightFocus(G)=F2;Expression(OneOf(Asymptote(G)))= (3*x - y = 0);Abs(LineSegmentOf(P, F2)) = 3", "coords": {} }, { "index": 3784, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3785, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3785.png", "answer": "[-2,1]", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G)", "coords": {} }, { "index": 3786, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3786.png", "answer": "2", "fact_expr": "G: Parabola;H: Line;F: Point;M: Point;N: Point;Expression(G) = (y^2 = x);Expression(H) = (x = -1/4);Focus(G) = F;PointOnCurve(M, G);Intersection(OverlappingLine(LineSegmentOf(M, F)), H) = N", "coords": {} }, { "index": 3787, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3787.png", "answer": "3/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F2: Point;F1: Point;a > b;b > 0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P,G);IsPerpendicular(LineSegmentOf(P,F2),xAxis);Abs(LineSegmentOf(F1,F2)) = 12;Abs(LineSegmentOf(P,F2)) = 5", "coords": {} }, { "index": 3788, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 2.23606797749979, "b": 2.0, "major_axis": "x", "x_coef": 5.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3788.png", "answer": "{(x^2/10+y^2/8=1),(41*y^2/4+41*x^2/5=1)}", "fact_expr": "C: Curve;Expression(C) = (x^2/5 + y^2/4 = 1);G: Ellipse;Eccentricity(C) = Eccentricity(G);H: Point;Coordinate(H) = (sqrt(5), 2);PointOnCurve(H, G) = True", "coords": { "H": [ 2.23606797749979, 2.0 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 3789, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3789.png", "answer": "(-\\infty,-1)+(1,+\\infty)", "fact_expr": "Gamma:Parabola;H: Line;A: Point;B: Point;t: Real;Expression(Gamma) = (x^2=2*y);Coordinate(A) = (0,-2);Coordinate(B) = (t, 0);PointOnCurve(A,H);PointOnCurve(B,H);NumIntersection(H,Gamma)=0", "coords": { "A": [ 0.0, -2.0 ] } }, { "index": 3790, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2631799883090584, "b": 2.5275303393962245, "orientation": "horizontal" }, "optimization": { "final_loss": 4.481942266905428e-06, "converged": true, "iterations": 95 }, "output_path": "results/train/hyperbola/problem_3790.png", "answer": "{sqrt(5),sqrt(5)/2}", "fact_expr": "C: Hyperbola;O: Origin;Expression(Asymptote(C)) = (y = pm*2*x);PointOnCurve(Focus(C),axis);Center(C)=O", "coords": {} }, { "index": 3791, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3791.png", "answer": "2", "fact_expr": "G: Parabola;A: Point;B: Point;D: Point;C: Point;L1:Line;L2:Line;H:LineSegment;Expression(G) = (y^2 = 4*x);Focus(G)=F;PointOnCurve(F,H);Intersection(H,G)={A,B};PointOnCurve(A,L1);PointOnCurve(B,L2);IsPerpendicular(L1,yAxis);IsPerpendicular(L2,yAxis);FootPoint(L1,yAxis)=C;FootPoint(L2,yAxis)=D;IsChordOf(H,G);F:Point", "coords": {} }, { "index": 3792, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3793, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3793.png", "answer": "2*sqrt(5)-6", "fact_expr": "C1: Ellipse;Expression(C1) = (x^2/4 + y^2/3 = 1);P: Point;PointOnCurve(P, C1);F: Point;RightFocus(C1) = F;C2: Circle;Expression(C2) = (-8*y + 6*x + x^2 + y^2 + 21 = 0);Q: Point;PointOnCurve(Q, C2)", "coords": {} }, { "index": 3794, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3794.png", "answer": "8", "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/4^2 + y^2/3^2 = 1);PointOnCurve(P, G);F1: Point;F2: Point;Focus(G) = {F1, F2}", "coords": {} }, { "index": 3795, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3795.png", "answer": "7, 23", "fact_expr": "G: Hyperbola;H: Point;I: Point;A: Point;Expression(G) = (x^2/16 - y^2/9 = 1);Coordinate(H) = (5, 0);Coordinate(I) = (-5, 0);PointOnCurve(A, G);Distance(A, H) = 15", "coords": { "H": [ 5.0, 0.0 ], "I": [ -5.0, 0.0 ] } }, { "index": 3796, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3797, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3798, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3798.png", "answer": "2", "fact_expr": "G: Hyperbola;P: Point;Expression(G) = (x^2/4 - y^2/5 = 1);PointOnCurve(P,RightPart(G));Distance(P,RightFocus(G)) = 1", "coords": {} }, { "index": 3799, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3799.png", "answer": "(0,1/2)", "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;O: Origin;P: Point;Q: Point;F: Point;Expression(C) = (y^2 = 8*x);Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};MidPoint(LineSegmentOf(A,B)) = P;Intersection(OverlappingLine(LineSegmentOf(O,P)),C)=Q", "coords": {} }, { "index": 3800, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3801, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3801.png", "answer": "5/2", "fact_expr": "l: Line;C: Parabola;F: Point;A: Point;B: Point;P:Point;Q:Point;Expression(C) = (x^2 = 2*y);Focus(C) = F;Intersection(l,C)={A,B};PointOnCurve(F, l);Intersection(TangentOnPoint(A, C),xAxis)=P;Intersection(TangentOnPoint(A, C),yAxis)=Q;Area(TriangleOf(P,O,Q))=1;O:Origin", "coords": {} }, { "index": 3802, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3802.png", "answer": "1/2", "fact_expr": "P: Point;G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;PointOnCurve(P, G);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;I: Point;Incenter(TriangleOf(P,F1,F2))=I;Area(TriangleOf(I, P, F1)) + Abs(TriangleOf(I, P, F2)) = 2*Area(TriangleOf(I, F1, F2))", "coords": {} }, { "index": 3803, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3803.png", "answer": "{y^2=2*x, y^2=10*x}", "fact_expr": "C: Circle;Expression(C) = (y^2 + (x - 3)^2 = 4);G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;PointOnCurve(Focus(G), C)", "coords": {} }, { "index": 3804, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3805, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3805.png", "answer": "(2,4]", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/12 = 1);P: Point;PointOnCurve(P, G);F1: Point;F2: Point;LeftFocus(G)=F1;RightFocus(G)=F2;O: Origin", "coords": {} }, { "index": 3806, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3807, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.3300262689590454, "b": 2.30381441116333, "orientation": "horizontal" }, "optimization": { "final_loss": 6.658357051492203e-06, "converged": true, "iterations": 122 }, "output_path": "results/train/hyperbola/problem_3807.png", "answer": "3", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/3 + x^2/a = 1);a: Number;Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x);H: Parabola;Expression(H) = (y^2 = 4*a*x);M: Point;Coordinate(M) = (2, y0);y0: Number;PointOnCurve(M, H);F: Point;Focus(H) = F", "coords": {} }, { "index": 3808, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3809, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3810, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.75, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3810.png", "answer": "{pi/3, 2*pi/3}", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 3*x);F: Point;Focus(C) = F;A: Point;PointOnCurve(A, C);Abs(LineSegmentOf(F, A)) = 3", "coords": {} }, { "index": 3811, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3811.png", "answer": "(0,1)", "fact_expr": "G: Parabola;p: Number;H: Line;p>0;Expression(G) = (x^2 = 2*(p*y));Expression(H) = (y = -1);Directrix(G) = H", "coords": {} }, { "index": 3812, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.0625, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3812.png", "answer": "15/16", "fact_expr": "G: Parabola;Expression(G) = (y = 4*x^2);M: Point;PointOnCurve(M, G);Distance(M, Focus(G)) = 1", "coords": {} }, { "index": 3813, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.449489742783178, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3813.png", "answer": "24", "fact_expr": "G: Hyperbola;M: Point;F2: Point;F1: Point;N: Point;Expression(G) = (x^2/4 - y^2/6 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(M,RightPart(G));DotProduct(VectorOf(M,F1),VectorOf(M,F2))=0;OneOf(Intersection(LineOf(M,F2),G))=N", "coords": {} }, { "index": 3814, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3814.png", "answer": "4", "fact_expr": "G: Ellipse;M: Point;F1: Point;O: Origin;N: Point;Expression(G) = (x^2/25 + y^2/9 = 1);OneOf(Focus(G))=F1;Distance(M,F1)=2;MidPoint(LineSegmentOf(M,F1))=N;PointOnCurve(M,G)", "coords": {} }, { "index": 3815, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3815.png", "answer": "5/4", "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/b^2 + y^2/a^2 = 1);b: Number;a: Number;Expression(OneOf(Asymptote(G))) = (y = (4/3)*x)", "coords": {} }, { "index": 3816, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -1.0, 0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_3816.png", "answer": "x^2/9+y^2/8=1", "fact_expr": "C1: Circle;Expression(C1) = ((x+1)^2 + y^2 = 1);C2: Circle;Expression(C2) = ((x-1)^2 + y^2 = 25);M: Circle;IsOutTangent(M, C1) ;IsInTangent(M, C2);M1: Point;Center(M) = M1", "coords": {} }, { "index": 3817, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3817.png", "answer": "(1/2, 0) \nx = -1/2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x)", "coords": {} }, { "index": 3818, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3818.png", "answer": "2*sqrt(10)", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2 - y^2/9 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0", "coords": {} }, { "index": 3819, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/train/hyperbola/problem_3819.png", "answer": "y=pm*3/4*x", "fact_expr": "E: Hyperbola;a: Number;b: Number;a>0;b>0;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(E) = 5/4", "coords": {} }, { "index": 3820, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3820.png", "answer": "-3/4", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);M: Point;N: Point;Endpoint(MajorAxis(G)) = {M, N};P: Point;Negation(P=M);Negation(P=N);PointOnCurve(P, G)", "coords": {} }, { "index": 3821, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3822, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3822.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/12 = 1)", "coords": {} }, { "index": 3823, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3823.png", "answer": "x^2/2+y^2=1", "fact_expr": "C: Ellipse;O: Origin;Center(C) = O;PointOnCurve(Focus(C), xAxis);Eccentricity(C) = sqrt(2)/2;G: Parabola;Expression(G) = (x^2 = 4*y);OneOf(Vertex(C)) = Focus(G)", "coords": {} }, { "index": 3824, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3824.png", "answer": "y=-1/4", "fact_expr": "G: Parabola;Expression(G) = (x^2 = y)", "coords": {} }, { "index": 3825, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3826, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_focus_mismatch", "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3826.png", "answer": "x^2/5+y^2/4=1", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;PointOnCurve(Focus(G), xAxis);H: Circle;Expression(H) = (x^2 + y^2 = 1);F: Point;Coordinate(F) = (1, 1/2);L1: Line;L2: Line;TangentOfPoint(F, H) = {L1, L2};A: Point;B: Point;TangentPoint(L1, H) = A;TangentPoint(L2, H) = B;PointOnCurve(RightFocus(G), LineOf(A, B));PointOnCurve(UpperVertex(G), LineOf(A, B))", "coords": { "F": [ 1.0, 0.5 ] }, "validation_reasons": [ "ellipse_focus_mismatch" ] }, { "index": 3827, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3827.png", "answer": "6", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;A: Point;PointOnCurve(A, C);Distance(A, Focus(C)) = 12;Distance(A, yAxis) = 9", "coords": {} }, { "index": 3828, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3828.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;B: Point;A: Point;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;RightVertex(G)=A;OneOf(Endpoint(ImageinaryAxis(G)))=B;Distance(F,LineOf(A,B)) = b/2", "coords": {} }, { "index": 3829, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3830, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3830.png", "answer": "2", "fact_expr": "E: Hyperbola;b: Number;a: Number;T: Ellipse;a>0;b>0;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);Expression(T) = (x^2/4 + y^2/3 = 1);LeftVertex(E) = LeftFocus(T);RightVertex(E) = RightFocus(T);LeftVertex(T) = LeftFocus(E);RightVertex(T) = RightFocus(E)", "coords": {} }, { "index": 3831, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3831.png", "answer": "8", "fact_expr": "G: Ellipse;H: Line;F2: Point;A: Point;B: Point;F1: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(G) = {F1, F2};PointOnCurve(F1, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(F2, A)) + Abs(LineSegmentOf(F2, B)) = 12", "coords": {} }, { "index": 3832, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 3.0, "major_axis": "x", "x_coef": 36.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3832.png", "answer": "x+2*y-8=0", "fact_expr": "M: Point;Coordinate(M) = (4, 2);l: Line;G: Ellipse;Expression(G) = (x^2 + 4*y^2 = 36);A: Point;B: Point;MidPoint(LineSegmentOf(A, B)) = M;InterceptChord(l, G) = LineSegmentOf(A, B)", "coords": { "M": [ 4.0, 2.0 ] } }, { "index": 3833, "success": false, "error": "Circle from diameter: missing point coordinates" }, { "index": 3834, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3834.png", "answer": "3/4", "fact_expr": "C: Hyperbola;P: Point;F1: Point;F2: Point;Expression(C) = (x^2 - y^2 = 2);LeftFocus(C) =F1;RightFocus(C)=F2;PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 3835, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3835.png", "answer": "{2, 18}", "fact_expr": "G: Hyperbola;P: Point;Expression(G) = (x^2/16 - y^2/4 = 1);PointOnCurve(P, G);Distance(P, F1) = 10;F1: Point;F2: Point;OneOf(Focus(G)) = F1;OneOf(Focus(G)) = F2;Negation(F1 = F2)", "coords": {} }, { "index": 3836, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3836.png", "answer": "sqrt(5)", "fact_expr": "C1:Parabola;C2:Hyperbola;F: Point;A:Point;a:Number;b:Number;a>0;b>0;Expression(C1)=(y^2=4*x);Expression(C2)=(x^2/a^2-y^2/b^2=1);Focus(C1)=F;OneOf(Intersection(C1,OneOf(Asymptote(C2))))=A;IsPerpendicular(LineSegmentOf(A,F),xAxis)", "coords": {} }, { "index": 3837, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3837.png", "answer": "y=pm*sqrt(2)*(x-sqrt(2))", "fact_expr": "l: Line;G: Ellipse;H: Circle;A: Point;B: Point;F1: Point;F2: Point;O: Origin;Expression(G) = (x^2/4 + y^2/2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2,l);Intersection(l, G) = {A,B};IsDiameter(LineSegmentOf(A, B),H);PointOnCurve(O, H)", "coords": {} }, { "index": 3838, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3838.png", "answer": "sqrt(5)/2", "fact_expr": "E: Hyperbola;b: Number;a: Number;P: Point;F: Point;Q: Point;a>0;b>0;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(E) = F;L1:Line;L2:Line;L:Line;OneOf(Asymptote(E))=L1;OneOf(Asymptote(E))=L2;Negation(L1=L2);PointOnCurve(F,L);IsPerpendicular(L1,L);FootPoint(L1,L)=P;Intersection(L2,L)=Q;5*VectorOf(P, F) = 3*VectorOf(F, Q)", "coords": {} }, { "index": 3839, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3839.png", "answer": "1", "fact_expr": "C: Parabola;p: Number;G: Circle;F: Point;A: Point;B: Point;D: Point;l: Line;p>0;Expression(C) = (y^2 = 2*p*x);Focus(C) = F;Directrix(C) = l;PointOnCurve(A, C);Center(G)=F;Radius(G)=LineSegmentOf(F,A);Intersection(G,l)={B,D};AngleOf(B, F, D) = ApplyUnit(120, degree);Area(TriangleOf(A,B,D))=2*sqrt(3)", "coords": {} }, { "index": 3840, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/train/hyperbola/problem_3840.png", "answer": "2*sqrt(13)", "fact_expr": "G: Hyperbola;a: Number;H: Point;Expression(G) = (-y^2/4 + x^2/a^2 = 1);Coordinate(H) = (-3*sqrt(2), 2);PointOnCurve(H, G)", "coords": { "H": [ -4.242640687119286, 2.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 3841, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3842, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3843, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3844, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3844.png", "answer": "(\\sqrt{5}-1)/2", "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;B: Point;F: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftVertex(G)=A;Endpoint(MinorAxis(G))=B;RightFocus(G)=F;IsPerpendicular(LineSegmentOf(A, B), LineSegmentOf(B, F))", "coords": {} }, { "index": 3845, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3846, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3846.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Hyperbola;C:Hyperbola;A: Point;Expression(G) = (x^2/16 - y^2/9 = 1);Coordinate(A) = (2*sqrt(3), -3);Asymptote(G) = Asymptote(C);PointOnCurve(A, C)", "coords": { "A": [ 3.4641016151377544, -3.0 ] } }, { "index": 3847, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3847.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F2: Point;F1: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, LeftPart(G));LeftFocus(G) = F1;RightFocus(G) = F2;PerpendicularBisector(LineSegmentOf(P, F2)) = OneOf(Asymptote(G))", "coords": {} }, { "index": 3848, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3848.png", "answer": "2*sqrt(2)-sqrt(6)", "fact_expr": "G: Ellipse;O: Origin;A: Point;B: Point;E: Point;P: Point;Expression(G) = (x^2/4 + y^2 = 1);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);x1:Number;x2:Number;y1:Number;y2:Number;Coordinate(E) = (sqrt(6), 0);PointOnCurve(A, G);PointOnCurve(B, G);H:Line;C:Line;PointOnCurve(A,H);PointOnCurve(B,C);Slope(H)=x1/(4*y1);Slope(C)=x2/(4*y2);Intersection(H,C) = P;Slope(LineOf(O,A))*Slope(LineOf(O,B))=-1/4", "coords": { "E": [ 2.449489742783178, 0.0 ] } }, { "index": 3849, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 1.8537487983703613, "b": 1.6118527266061986, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 1.4665974334810584e-06, "converged": true, "iterations": 90 }, "output_path": "results/train/ellipse/problem_3849.png", "answer": "2", "fact_expr": "C1: Parabola;Expression(C1) = (y = a*x^2);a: Number;a>0;F: Point;Focus(C1) = F;C2: Ellipse;Expression(C2) = (y^2/4 + x^2/b^2 = 1);b: Number;b>0;OneOf(Focus(C2)) = F;M: Point;PointOnCurve(M, C1);P: Point;Coordinate(P) = (3/2, 1);PointOnCurve(P, C2)", "coords": { "P": [ 1.5, 1.0 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 3850, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3851, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3851.png", "answer": "sqrt(3)/3", "fact_expr": "A: Point;Coordinate(A) = (0, b);b: Number;B: Point;G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);a: Number;a > b;b > 0;Intersection(LeftDirectrix(G), xAxis) = B;MidPoint(LineSegmentOf(A, B)) = C;C: Point;PointOnCurve(C, G) = True", "coords": {} }, { "index": 3852, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3852.png", "answer": "sqrt(10)/3", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;LeftFocus(G) = F;l: Line;PointOnCurve(F, l);IsPerpendicular(OneOf(Asymptote(G)), l);A: Point;B: Point;Intersection(l, LeftPart(G)) = {A, B};VectorOf(A, F) = 2*VectorOf(F, B);Negation(A=B)", "coords": {} }, { "index": 3853, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3854, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3855, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3856, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_3856.png", "answer": "1", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;p: Number;A: Point;O: Origin;B: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);p>0;Expression(H) = (y^2 = 2*(p*x));Eccentricity(G) = 2;L1: Line;L2: Line;Asymptote(G) = {L1, L2};Intersection(L1, Directrix(H)) = A;Intersection(L2, Directrix(H)) = B;Area(TriangleOf(A, O, B)) = sqrt(3)/4", "coords": {} }, { "index": 3857, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3857.png", "answer": "6", "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (x^2 = 4*y);Coordinate(A) = (-1, 5);Focus(G) = F;PointOnCurve(P, G)", "coords": { "A": [ -1.0, 5.0 ] } }, { "index": 3858, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3858.png", "answer": "2/3", "fact_expr": "G: Parabola;H: Ellipse;a: Number;Expression(G) = (y^2 = 8*x);a>sqrt(5);Expression(H) = (y^2/5 + x^2/a^2 = 1);PointOnCurve(OneOf(Focus(H)), Directrix(G))", "coords": {} }, { "index": 3859, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3859.png", "answer": "sqrt(10)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);P: Point;PointOnCurve(P, G);H: Line;Expression(H) = (x - y + 3*sqrt(5) = 0)", "coords": {} }, { "index": 3860, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.449489742783178, "b": 3.1622776601683795, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3860.png", "answer": "y^2=16*x", "fact_expr": "G: Hyperbola;H: Parabola;Expression(G) = (x^2/6 - y^2/10 = 1);Vertex(H) = Center(G);Focus(H) = RightFocus(G)", "coords": {} }, { "index": 3861, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3861.png", "answer": "sqrt(2)-1", "fact_expr": "C: Ellipse;b: Number;a: Number;F: Point;P: Point;l: Line;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;RightDirectrix(C) = l;PointOnCurve(P, l);PointOnCurve(MidPoint(LineSegmentOf(P,F)), C)", "coords": {} }, { "index": 3862, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3863, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3864, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3864.png", "answer": "sqrt(5)/3", "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F1: Point;O: Origin;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(P,G);DotProduct(VectorOf(P,F1),(VectorOf(O,F1)+VectorOf(O,P)))=0;Abs(LineSegmentOf(P,F1))=2*Abs(LineSegmentOf(P,F2))", "coords": {} }, { "index": 3865, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3866, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3866.png", "answer": "{4, 1}", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (y^2/2 + x^2/m = 1);Eccentricity(G) = sqrt(2)/2", "coords": {} }, { "index": 3867, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3867.png", "answer": "3/2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;b: Number;a: Number;P: Point;PointOnCurve(P,RightPart(G));Abs(LineSegmentOf(P,F1))=3*Abs(LineSegmentOf(P,F2));DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = a^2/2", "coords": {} }, { "index": 3868, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3868.png", "answer": "(\\sqrt{5}-1)/2", "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;B: Point;F: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);RightVertex(G) = A;UpperVertex(G) = B;LeftFocus(G) = F;AngleOf(A, B, F) = ApplyUnit(90, degree)", "coords": {} }, { "index": 3869, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3869.png", "answer": "1/2", "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;M: Point;k: Number;x0: Number;Expression(G) = (y^2 = 4*x);Coordinate(M) = (x0, 4);Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A, B)) = M;Slope(l) = k;Negation(A = B)", "coords": {} }, { "index": 3870, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3870.png", "answer": "3*sqrt(2)/4", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;H:Line;F1: Point;e: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y =(b/(3*a))*x);Intersection(H,LeftPart(G))=P;LeftFocus(G) = F1;IsPerpendicular(LineSegmentOf(P,F1),xAxis);Eccentricity(G) = e", "coords": {} }, { "index": 3871, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3871.png", "answer": "sqrt(2)/2", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;L:Line;A: Point;B: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = 1 - x);Expression(L) = (x - 2*y = 0);Intersection(H, G) = {A, B};PointOnCurve(MidPoint(LineSegmentOf(A,B)), L)", "coords": {} }, { "index": 3872, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3872.png", "answer": "(1,0)\n7", "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (6, 3);Focus(G)=F;PointOnCurve(P,G)", "coords": { "A": [ 6.0, 3.0 ] } }, { "index": 3873, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3873.png", "answer": "12", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);F: Point;Focus(G) = F;A: Point;B: Point;C: Point;PointOnCurve(A, G);PointOnCurve(B, G);PointOnCurve(C, G);VectorOf(F, A) + VectorOf(F, B) + VectorOf(F, C) = 0", "coords": {} }, { "index": 3874, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3874.png", "answer": "y=pm*(sqrt(3)/3)*x", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;N: Point;F1: Point;M: Point;F2: Point;a > b;b > 0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1,l);L1:Line;L2:Line;Asymptote(C)={L1,L2};IsPerpendicular(l,L1);Intersection(l,L1)=M;Intersection(l,L2)=N;Abs(LineSegmentOf(N, F1)) = 2*Abs(LineSegmentOf(M, F1))", "coords": {} }, { "index": 3875, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3875.png", "answer": "(1,1)", "fact_expr": "C: Parabola;M: Point;F: Point;Expression(C) = (y^2 = x);Focus(C) = F;Quadrant(M)=1;PointOnCurve(M,C);Abs(LineSegmentOf(M, F)) = 5/4", "coords": {} }, { "index": 3876, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3876.png", "answer": "y^2 = 8*x", "fact_expr": "C: Parabola;p: Number;G: Point;y0: Number;Expression(C) = (y^2 = 2*(p*x));Coordinate(G) = (1, y0);PointOnCurve(G, C);Distance(G, Focus(C)) = 3", "coords": {} }, { "index": 3877, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3877.png", "answer": "9", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);P: Point;PointOnCurve(P, G);F1: Point;F2: Point;Focus(G) = {F1, F2};IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2))", "coords": {} }, { "index": 3878, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3879, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3880, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3881, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3881.png", "answer": "12*sqrt(2)", "fact_expr": "G: Parabola;Expression(G) = (y = x^2/8);P: Point;M: Point;FootPoint(l, Directrix(G)) = M;F: Point;Abs(LineSegmentOf(P, M)) = 6;Focus(G) = F;l:Line;PointOnCurve(P,l);IsPerpendicular(l,Directrix(G));PointOnCurve(P,G)", "coords": {} }, { "index": 3882, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -4.0, 0.0 ], "radius": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_3882.png", "answer": "(x^2/4-y^2/12=1)&(x>=2)", "fact_expr": "M: Circle;C1: Circle;C2: Circle;M1: Point;Expression(C1)=((x+4)^2+y^2=4);Expression(C2)=((x-4)^2+y^2=4);IsOutTangent(M,C1);IsInTangent(M,C2);Center(M)=M1;Locus(M1)=C;C:Curve", "coords": {} }, { "index": 3883, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3884, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3884.png", "answer": "(0,15)", "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (y^2/5 - x^2/m = 1);Eccentricity(G)=e;e:Number;In(e,(1,2))", "coords": {} }, { "index": 3885, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3885.png", "answer": "2", "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/16 + y^2/4 = 1);PointOnCurve(P, G);F1:Point;F2:Point;OneOf(Focus(G))=F1;OneOf(Focus(G))=F2;Negation(F1=F2);Distance(P, F1) = 6", "coords": {} }, { "index": 3886, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3886.png", "answer": "4", "fact_expr": "G: Hyperbola;m: Number;H: Ellipse;Expression(G) = (-y^2/5 + x^2/m = 1);Expression(H) = (x^2/25 + y^2/16 = 1);Focus(H) = Focus(G)", "coords": {} }, { "index": 3887, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3887.png", "answer": "-4", "fact_expr": "G: Parabola;p: Number;H: Line;A: Point;B: Point;p>0;x1:Number;y1:Number;x2:Number;y2:Number;Expression(G) = (y^2 = 2*(p*x));Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);PointOnCurve(Focus(G), H);Intersection(H, G) = {A,B}", "coords": {} }, { "index": 3888, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3888.png", "answer": "6\nx=-3", "fact_expr": "C: Parabola;p: Real;G: Line;Expression(C) = (y^2 = 2*(p*x));Expression(G) = (x + y - 3 = 0);PointOnCurve(Focus(C), G)", "coords": {} }, { "index": 3889, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.23606797749979, "major_axis": "y", "x_coef": 5.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3889.png", "answer": "8", "fact_expr": "G: Ellipse;Expression(G) = (x^2/5 + y^2/16 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G)", "coords": {} }, { "index": 3890, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3890.png", "answer": "1", "fact_expr": "G: Parabola;Q: Point;P: Point;x1: Number;y1: Number;Expression(G) = (x^2 = 2*y);Coordinate(Q) = (sqrt(2), 0);Coordinate(P) = (x1, y1);PointOnCurve(P, G)", "coords": { "Q": [ 1.4142135623730951, 0.0 ] } }, { "index": 3891, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3891.png", "answer": "1", "fact_expr": "G: Parabola;p: Number;H: Circle;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (y^2 + (x - 2)^2 = 3);NumIntersection(G,H)=2", "coords": {} }, { "index": 3892, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3893, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3893.png", "answer": "sqrt(22)/2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;H: Circle;Expression(H) = (x^2 + y^2 = b^2);Intersection(H,G) = {A,C};A: Point;C: Point;Quadrant(A) = 2;Quadrant(C) = 4;RightFocus(G) = F;F: Point;Abs(LineSegmentOf(A, F)) = 2*Abs(LineSegmentOf(C, F))", "coords": {} }, { "index": 3894, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3895, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3895.png", "answer": "4*sqrt(5)/5", "fact_expr": "E: Parabola;p: Number;G: Circle;Q: Point;P: Point;F: Point;p>0;Expression(E) = (y^2 = 2*(p*x));Focus(E) = F;Center(G)=F;Radius(G)=3*p;Intersection(G, E) = {P, Q};I:Point;Coordinate(I)=(0,-1);C:Circle;IsDiameter(LineSegmentOf(P,F),C);PointOnCurve(I,C)", "coords": { "I": [ 0.0, -1.0 ] } }, { "index": 3896, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3896.png", "answer": "[\\pi/6, 5\\pi/6]", "fact_expr": "G: Ellipse;a: Number;b: Number;H: Line;A: Point;F: Point;B: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);theta: Number;Inclination(H) = theta;O: Origin;PointOnCurve(O, H);Intersection(H, G) = {A, B};LeftFocus(G) = F;IsPerpendicular(LineSegmentOf(A, F), LineSegmentOf(B, F));Eccentricity(G) = e;e: Number;In(e, [sqrt(2)/2, sqrt(6)/3])", "coords": {} }, { "index": 3897, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.9876673221588135, "b": 1.5893483050269506, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 3.171300880237459e-05, "converged": true, "iterations": 68 }, "output_path": "results/train/ellipse/problem_3897.png", "answer": "20", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F2: Point;F1: Point;LeftFocus(G) = F1;RightFocus(G) = F2;b = 4;Eccentricity(G) = 3/5;H: Line;PointOnCurve(F1,H) = True;Intersection(H,G) = {A,B};A: Point;B: Point", "coords": {} }, { "index": 3898, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3898.png", "answer": "2", "fact_expr": "G: Hyperbola;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 - y^2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(90, degree)", "coords": {} }, { "index": 3899, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3899.png", "answer": "1", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);AngleOf(F1, P, F2) = pi/2", "coords": {} }, { "index": 3900, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3900.png", "answer": "2", "fact_expr": "G: Parabola;P: Point;y0:Number;Coordinate(P) = (3/2, y0);Expression(G) = (y^2 = 2*x);PointOnCurve(P, G);Focus(G) = F;F:Point", "coords": {} }, { "index": 3901, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3901.png", "answer": "8", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;H: Ellipse;Expression(H) = (x^2/((3*p)) + y^2/p = 1);Focus(G) = OneOf(Focus(H))", "coords": {} }, { "index": 3902, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3902.png", "answer": "x^2/4+y^2/3=1", "fact_expr": "l: Line;G: Ellipse;b: Number;A: Point;F2: Point;B: Point;F1: Point;Expression(G) = (x^2/4 + y^2/b^2 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(F1, l);Intersection(l, G) = {A, B};Max(LineSegmentOf(A, F2) + LineSegmentOf(B, F2)) = 5", "coords": {} }, { "index": 3903, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2135674953460693, "b": 2.4270472526550293, "orientation": "horizontal" }, "optimization": { "final_loss": 4.453341261978494e-06, "converged": true, "iterations": 125 }, "output_path": "results/train/hyperbola/problem_3903.png", "answer": "16", "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (x^2/4 - y^2/m = 1);Eccentricity(G) = sqrt(5)", "coords": {} }, { "index": 3904, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3904.png", "answer": "8*sqrt(2)/3", "fact_expr": "l: Line;C: Ellipse;F1: Point;A: Point;B: Point;Expression(C) = (x^2/2 + y^2 = 1);LeftFocus(C) = F1;Expression(l) = (y = x - 1);Intersection(l, C) = {A, B}", "coords": {} }, { "index": 3905, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3905.png", "answer": "1", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G) = True;LineSegmentOf(P, F1)*LineSegmentOf(P, F2) = 0", "coords": {} }, { "index": 3906, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 1.0, "major_axis": "x", "x_coef": 8.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3906.png", "answer": "15*sqrt(7)/7", "fact_expr": "P: Point;Q: Point;H: Circle;Expression(H) = (x^2 + (y + 1)^2 = 7);G: Ellipse;Expression(G) = (x^2/8 + y^2 = 1);PointOnCurve(P, H) = True;PointOnCurve(Q, G) = True", "coords": {} }, { "index": 3907, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3907.png", "answer": "16", "fact_expr": "G: Ellipse;H: Line;A: Point;B: Point;F1: Point;F2: Point;Expression(G) = (x^2/16 + y^2/9 = 1);Focus(G)={F1,F2};PointOnCurve(F1, H);Intersection(H, G) = {A, B}", "coords": {} }, { "index": 3908, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3908.png", "answer": "[-3*sqrt(10)/2,3*sqrt(10)/2]&Negation(b=0)", "fact_expr": "G: Ellipse;b: Number;Expression(G) = (x^2/45 + y^2/b^2 = 1);PointOnCurve(Focus(G),xAxis);P:Point;PointOnCurve(P,G);F1:Point;F2:Point;IsPerpendicular(LineSegmentOf(P,F1),LineSegmentOf(P,F2));Focus(G)={F1,F2}", "coords": {} }, { "index": 3909, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3909.png", "answer": "9/4", "fact_expr": "G: Ellipse;Expression(G) = (y^2 + x^2/a^2 = 1);a: Number;a>1;A: Point;UpperVertex(G) = A;B: Point;LeftVertex(G) = B;P: Point;PointOnCurve(P, G);Max(Area(TriangleOf(P, A, B))) = 1 + sqrt(2);M: Point;Coordinate(M) = (-sqrt(3), 0);N: Point;Coordinate(N) = (sqrt(3), 0);Q: Point;PointOnCurve(Q, G)", "coords": { "M": [ -1.7320508075688772, 0.0 ], "N": [ 1.7320508075688772, 0.0 ] } }, { "index": 3910, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3911, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3911.png", "answer": "2*sqrt(3)/3", "fact_expr": "C: Hyperbola;b: Number;a: Number;A: Point;F: Point;B: Point;L1: Line;L2: Line;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;L:Line;OneOf(Asymptote(C))=L1;OneOf(Asymptote(C))=L2;Negation(L1=L2);PointOnCurve(F,L);IsPerpendicular(L,L1);FootPoint(L,L1)=A;Intersection(L,L2)=B;2*VectorOf(A, F) = VectorOf(F, B)", "coords": {} }, { "index": 3912, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3912.png", "answer": "11/4", "fact_expr": "G: Parabola;Expression(G) = (y = x^2);B: Point;A: Point;Length(LineSegmentOf(A,B)) = 6;PointOnCurve(Endpoint(LineSegmentOf(A,B)), G);MidPoint(LineSegmentOf(A,B)) = M;M: Point", "coords": {} }, { "index": 3913, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3913.png", "answer": "16", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l1: Line;l2: Line;PointOnCurve(F, l1);PointOnCurve(F, l2);IsPerpendicular(l1, l2);A: Point;B: Point;Intersection(l1, C) = {A, B};D: Point;E: Point;Intersection(l2, C) = {D, E}", "coords": {} }, { "index": 3914, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3914.png", "answer": "sqrt(2)", "fact_expr": "P: Point;M: Point;N: Point;G: Hyperbola;H: Parabola;a: Number;b: Number;I: Line;Z: Line;A: Point;F: Point;B: Point;O: Origin;a>0;b>0;k1: Number;k2: Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = x);Focus(H) = F;PointOnCurve(F, I);Intersection(I, H) = {A, B};PointOnCurve(O, Z);Intersection(Z, G) = {M, N};PointOnCurve(P, G);Slope(LineOf(P, M)) = k1;Slope(LineOf(P, N)) = k2;(Abs(k1) + 4*Abs(k2))*(Abs(LineSegmentOf(A, F))*Abs(LineSegmentOf(B, F))) >= Abs(LineSegmentOf(A, F)) + Abs(LineSegmentOf(B, F))", "coords": {} }, { "index": 3915, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3915.png", "answer": "{sqrt(2),sqrt(17)}", "fact_expr": "l: Line;G: Hyperbola;b: Number;a: Number;F1: Point;P: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(F1,l);Intersection(l,G)=P;Sin(AngleOf(F1, P, F2)) = 4/5", "coords": {} }, { "index": 3916, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.6259966743174588, "b": 2.298015331604694, "orientation": "horizontal" }, "optimization": { "final_loss": 1.93011531933813e-06, "converged": true, "iterations": 91 }, "output_path": "results/train/hyperbola/problem_3916.png", "answer": "16*sqrt(3)", "fact_expr": "G: Hyperbola;Expression(Asymptote(G)) = (y = pm*(sqrt(2)*x));K: Point;Coordinate(K) = (3, -2*sqrt(3));PointOnCurve(K, G);H: Line;F: Point;RightFocus(G) = F;PointOnCurve(F, H);Inclination(H) = ApplyUnit(60, degree);A: Point;B: Point;Intersection(H, G) = {A, B}", "coords": { "K": [ 3.0, -3.4641016151377544 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 3917, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3917.png", "answer": "3", "fact_expr": "G: Hyperbola;H: Parabola;M: Point;N: Point;F: Point;Expression(H) = (x^2 = 8*y);Coordinate(M) = (0, -2);Focus(H) = F;PointOnCurve(N, H);WhenMin(Abs(LineSegmentOf(N,F))/Abs(LineSegmentOf(N,M)));Focus(G)={M,F};PointOnCurve(N,G)", "coords": { "M": [ 0.0, -2.0 ] } }, { "index": 3918, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2631799883090584, "b": 2.5275303393962245, "orientation": "horizontal" }, "optimization": { "final_loss": 4.481942266905428e-06, "converged": true, "iterations": 95 }, "output_path": "results/train/hyperbola/problem_3918.png", "answer": "-1\n4", "fact_expr": "m: Number;n: Number;Negation(m*n=0);H: Curve;Expression(H) = (x^2/m+y^2/n=1);G: Hyperbola;H=G;PointOnCurve(Focus(G), yAxis);Expression(OneOf(Asymptote(G))) = (y = 2*x)", "coords": {} }, { "index": 3919, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3919.png", "answer": "40", "fact_expr": "C: Hyperbola;P: Point;Q: Point;A: Point;F: Point;Expression(C) = (x^2/16 - y^2/9 = 1);Coordinate(A) = (5, 0);LeftFocus(C) = F;PointOnCurve(A, LineSegmentOf(P,Q));PointOnCurve(P,C);PointOnCurve(Q,C);Length(LineSegmentOf(P,Q))=2*Length(ImageinaryAxis(C))", "coords": { "A": [ 5.0, 0.0 ] } }, { "index": 3920, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3920.png", "answer": "(0, \\sqrt{3}/2)", "fact_expr": "C: Ellipse;m: Number;l1: Line;l2: Line;A: Point;m>1;Expression(C) = (y^2 + x^2/m = 1);Coordinate(A) = (1, 2);PointOnCurve(A, l1);PointOnCurve(A, l2);IsPerpendicular(l1, l2);NumIntersection(l1,C)=0;Negation(IsIntersect(l2, C))", "coords": { "A": [ 1.0, 2.0 ] } }, { "index": 3921, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3922, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3922.png", "answer": "y = x", "fact_expr": "A: Point;C: Point;G: Parabola;H: Line;B: Point;l: Line;Z: Line;Expression(G) = (y^2 = 4*x);Coordinate(A) = (4, 4);PointOnCurve(Focus(G), H);Intersection(H, G) = {A, B};PointOnCurve(B, Z);IsPerpendicular(Z, l);Directrix(G) = l;FootPoint(Z, l) = C", "coords": { "A": [ 4.0, 4.0 ] } }, { "index": 3923, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3924, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3925, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3926, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3926.png", "answer": "3/5", "fact_expr": "C: Ellipse;b: Number;a: Number;I: Point;P: Point;F1: Point;F2: Point;e: Number;s:Number;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C)=F1;RightFocus(C)=F2;PointOnCurve(P, C);Incenter(TriangleOf(P, F1, F2)) = I;Coordinate(I)=(s,1);Area(TriangleOf(P, F1, F2)) = 2*b;Eccentricity(C) = e", "coords": {} }, { "index": 3927, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3927.png", "answer": "8*sqrt(3)/9", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;F: Point;Focus(G) = F;A: Point;Coordinate(A) = (0, 2);Intersection(LineSegmentOf(F, A), G) = B;B: Point;VectorOf(F, B) = 2*VectorOf(B, A)", "coords": { "A": [ 0.0, 2.0 ] } }, { "index": 3928, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.463192105293274, "b": 2.070901870727539, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4661259228887502e-06, "converged": true, "iterations": 106 }, "output_path": "results/train/hyperbola/problem_3928.png", "answer": "4", "fact_expr": "G: Hyperbola;m: Real;Expression(G) = (-y^2/8 + x^2/m = 1);Eccentricity(G) = sqrt(3)", "coords": {} }, { "index": 3929, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3930, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3930.png", "answer": "(2,2)", "fact_expr": "G: Parabola;H: Line;B: Point;A: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (y = x);Intersection(H, G) = {A, B}", "coords": {} }, { "index": 3931, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3932, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3932.png", "answer": "4 + 2*sqrt(2)", "fact_expr": "C: Hyperbola;B: Point;P: Point;F: Point;Expression(C) = (x^2/2 - y^2 = 1);RightFocus(C) = F;OneOf(Endpoint(ImageinaryAxis(C))) = B;PointOnCurve(P, LeftPart(C))", "coords": {} }, { "index": 3933, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3933.png", "answer": "-2*sqrt(2)", "fact_expr": "G: Hyperbola;H: Parabola;p: Number;A: Point;F: Point;Expression(G) = (x^2/3 - y^2 = 1);p>0;Expression(H) = (y^2 = 2*(p*x));Focus(H) = F ;RightFocus(G) = F;Quadrant(A) = 1;PointOnCurve(A, H);Abs(LineSegmentOf(A, F)) = 3", "coords": {} }, { "index": 3934, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3934.png", "answer": "1/2", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;LeftFocus(G) = F;A: Point;RightVertex(G) = A;B: Point;PointOnCurve(B, G);IsPerpendicular(LineSegmentOf(B, F), xAxis);P: Point;Intersection(LineOf(A, B), yAxis) = P;Abs(LineSegmentOf(A, P)) = 2*Abs(LineSegmentOf(P, B))", "coords": {} }, { "index": 3935, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3936, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3936.png", "answer": "pi/2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/12 = 1);P: Point;PointOnCurve(P, G);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;Area(TriangleOf(P, F1, F2)) = 12", "coords": {} }, { "index": 3937, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3937.png", "answer": "{42, 22}", "fact_expr": "G: Parabola;p: Number;M: Point;P: Point;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(M) = (20, 40);Focus(G) = F;PointOnCurve(P, G);Min(Abs(LineSegmentOf(P, F)) + Abs(LineSegmentOf(P, M))) = 41", "coords": { "M": [ 20.0, 40.0 ] } }, { "index": 3938, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3938.png", "answer": "sqrt(3)", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;G: Circle;F2: Point;A: Point;B: Point;F1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2 + y^2 = a^2);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1,l);IsTangent(l,G);L1:Line;L2:Line;Asymptote(C)={L1,L2};Intersection(L1,l)=A;Intersection(L2,l)=B;DotProduct((VectorOf(F2, A) + VectorOf(F2, B)),VectorOf(A, B)) = 0", "coords": {} }, { "index": 3939, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3940, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3940.png", "answer": "y=pm*x", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, RightPart(C)) = True;Cos(AngleOf(F1, P, F2)) = 3/4;Abs(LineSegmentOf(P, F2)) = Length(ImageinaryAxis(C))", "coords": {} }, { "index": 3941, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3941.png", "answer": "6", "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;F: Point;O: Origin;MidPoint(LineSegmentOf(A, B)) = Z;Z: Point;Coordinate(Z) = (2, m);Coordinate(F) = (1, 0);Vertex(C) = O;Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};IsChordOf(LineSegmentOf(A, B),C)", "coords": { "F": [ 1.0, 0.0 ] } }, { "index": 3942, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3942.png", "answer": "16", "fact_expr": "G: Parabola;a: Number;a>0;P:Point;Expression(G) = (y^2 = a*x);PointOnCurve(P,G);XCoordinate(P)=6;Distance(P, Focus(G)) = 10", "coords": {} }, { "index": 3943, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3943.png", "answer": "-8; 2*sqrt(2)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;P: Point;Coordinate(P) = (2, 0);H: Line;PointOnCurve(P,H);A: Point;B: Point;Intersection(H, G) = {A, B};x1: Number;x2: Number;y1: Number;y2: Number;Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2)", "coords": { "P": [ 2.0, 0.0 ] } }, { "index": 3944, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.125, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3944.png", "answer": "-1/4", "fact_expr": "G: Parabola;H:Line;m: Number;Expression(G) = (y = 2*x^2);Expression(H) = (m + x + 2*y = 0);PointOnCurve(Focus(G),H)", "coords": {} }, { "index": 3945, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3945.png", "answer": "(1, \\sqrt{5}]", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;Coordinate(F) = (3*sqrt(5), 0);RightFocus(C) = F;N: Point;Coordinate(N) = (0, 2);M: Point;PointOnCurve(M, LeftPart(C));Negation(Perimeter(TriangleOf(M, N, F)) < 20)", "coords": { "F": [ 6.708203932499369, 0.0 ], "N": [ 0.0, 2.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 3946, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3946.png", "answer": "3/2", "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2/3 = 1);F: Point;OneOf(Focus(C))=F;P: Point;PointOnCurve(P, C);O: Origin;Abs(LineSegmentOf(O, P)) = Abs(LineSegmentOf(O, F))", "coords": {} }, { "index": 3947, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3947.png", "answer": "[2,+oo)", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;F: Point;O: Origin;P: Point;Q: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = ((x + 2)^2 + (y - 4)^2 = 4);Coordinate(F) = (2, 0);PointOnCurve(P, G);RightFocus(C) = F;PointOnCurve(Q,Asymptote(C));VectorOf(O,P)+VectorOf(O,F)=2*VectorOf(O,Q)", "coords": { "F": [ 2.0, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 3948, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3948.png", "answer": "17/4", "fact_expr": "G: Parabola;A: Point;M: Point;F:Point;Coordinate(A) = (4, 1);Expression(G) = (y^2 = x);PointOnCurve(M, G);Focus(G)=F", "coords": { "A": [ 4.0, 1.0 ] } }, { "index": 3949, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3950, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3950.png", "answer": "14", "fact_expr": "C: Hyperbola;Expression(C) = (-x^2/64 + y^2/16 = 1);F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) = 6", "coords": {} }, { "index": 3951, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3952, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3953, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3954, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3955, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3955.png", "answer": "sqrt(2)", "fact_expr": "P: Point;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;PointOnCurve(P, RightPart(G)) = True;Negation(Vertex(G) = P);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;M: Point;Incenter(TriangleOf(P, F1, F2)) = M;Area(TriangleOf(M, P, F1)) - Area(TriangleOf(M, P, F2)) = sqrt(2)*Area(TriangleOf(M, F1, F2))/2", "coords": {} }, { "index": 3956, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3957, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3957.png", "answer": "sqrt(3)/3", "fact_expr": "l: Line;C: Ellipse;b: Number;a: Number;A: Point;F1: Point;B: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;FocalLength(C) = 2;PointOnCurve(F2, l);Intersection(l, C) = {A, B};Perimeter(TriangleOf(A, F1, B)) = 4*sqrt(3)", "coords": {} }, { "index": 3958, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3958.png", "answer": "5/3", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, RightPart(C));Abs(LineSegmentOf(P, F2)) = Abs(LineSegmentOf(F1, F2));Distance(F2,LineOf(P,F1)) = 2*a", "coords": {} }, { "index": 3959, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3959.png", "answer": "2*sqrt(6)/3", "fact_expr": "C: Hyperbola;b: Number;a: Number;B: Point;A: Point;F: Point;l1: Line;l2: Line;e: Number;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Asymptote(C) = {l1, l2};RightFocus(C) = F;L: Line;PointOnCurve(F, L);IsPerpendicular(L, l1);FootPoint(L, l1) = A;Intersection(L, l2) = B;VectorOf(B, A) = 3*VectorOf(A, F);Eccentricity(C) = e", "coords": {} }, { "index": 3960, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3961, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3962, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3963, "success": false, "error": "\nk\\ge 0\n ^\nParseFatalException: Unknown symbol: \\ge, found '\\' (at char 1), (line:1, col:2)" }, { "index": 3964, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3964.png", "answer": "12*sqrt(3)", "fact_expr": "G: Hyperbola;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 - y^2/12 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 3965, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3965.png", "answer": "5/8", "fact_expr": "l: Line;G: Parabola;A: Point;O: Origin;B: Point;F: Point;Expression(G) = (y^2 = 2*x);Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B};Abs(LineSegmentOf(A, F)) = 4*Abs(LineSegmentOf(B, F))", "coords": {} }, { "index": 3966, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/train/hyperbola/problem_3966.png", "answer": "y=pm*(sqrt(3)/2)*x", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Parabola;H: Circle;P: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y^2 = -16*x);Expression(H) = (y^2 - 2*a*x + x^2 = 0);Coordinate(P) = (4, 3);IsTangent(Directrix(G),H);PointOnCurve(P, C)", "coords": { "P": [ 4.0, 3.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 3967, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3967.png", "answer": "sqrt(6)/3", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;A: Point;B: Point;F: Point;C: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (0, b/2);RightFocus(G)=F;PointOnCurve(A,H);IsPerpendicular(H,yAxis);Intersection(H,G)={B,C};AngleOf(B, F, C) = ApplyUnit(90, degree)", "coords": {} }, { "index": 3968, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3969, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3969.png", "answer": "2*x+pm*sqrt(2)*y-1=0", "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;P: Point;F: Point;Expression(G) = (y^2 = 2*x);Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A,B))=M;L:Line;PointOnCurve(M,L);IsPerpendicular(L,yAxis);Intersection(L,G)=P;Abs(LineSegmentOf(P, F)) = 3/2;M:Point", "coords": {} }, { "index": 3970, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3971, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3971.png", "answer": "8", "fact_expr": "G: Parabola;Expression(G) = (y^2 = a*x);a: Number;a>0;F:Point;Focus(G)=F;l: Line;PointOnCurve(F, l);Slope(l)=2;A:Point;Intersection(l,yAxis)=A;O: Origin;Area(TriangleOf(O, A, F))=4", "coords": {} }, { "index": 3972, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3973, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3973.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Ellipse;I: Line;A: Point;B: Point;F1: Point;F2: Point;Expression(G) = (x^2/16 + y^2/9 = 1);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, I);Intersection(I, G) = {A, B};Area(InscribedCircle(TriangleOf(A,B,F2)))=pi;x1:Number;x2:Number;y1:Number;y2:Number", "coords": {} }, { "index": 3974, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3975, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3975.png", "answer": "2*sqrt(6)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x);H: Line;Expression(H) = (-x + y + 1 = 0);A: Point;B: Point;Intersection(G, H) = {A, B};IsChordOf(LineSegmentOf(A, B), G)", "coords": {} }, { "index": 3976, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3976.png", "answer": "(-12, 0)", "fact_expr": "G: Hyperbola;k: Number;Expression(G) = (x^2/4 + y^2/k = 1);e: Number;Eccentricity(G) = e ;In(e, (1, 2))", "coords": {} }, { "index": 3977, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3977.png", "answer": "2", "fact_expr": "A: Point;F: Point;G: Parabola;O: Origin;M: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(A, G);AngleOf(O, F, A) = ApplyUnit(120, degree);MidPoint(LineSegmentOf(A, F)) = M", "coords": {} }, { "index": 3978, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3978.png", "answer": "x^2/4 - y^2/4 = 1", "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2 = a^2);a: Number;G: Parabola;Expression(G) = (y^2 = 16*x);A: Point;B: Point;O: Origin;Center(C) = O;PointOnCurve(Focus(C), xAxis);Intersection(C, Directrix(G)) = {A,B};Abs(LineSegmentOf(A, B)) = 4*sqrt(3)", "coords": {} }, { "index": 3979, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3979.png", "answer": "sqrt(3)/2", "fact_expr": "E: Ellipse;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;H: Point;Coordinate(H) = (4, 0);L: Line;PointOnCurve(H, L) = True;Intersection(L, E) = {A, B};A: Point;B: Point;Coordinate(MidPoint(LineSegmentOf(A,B))) = (2,-1)", "coords": { "H": [ 4.0, 0.0 ] } }, { "index": 3980, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3980.png", "answer": "9", "fact_expr": "G: Hyperbola;H: Circle;Z: Circle;P: Point;M: Point;N: Point;Expression(G) = (x^2/9 - y^2/16 = 1);Expression(H) = (y^2 + (x + 5)^2 = 4);Expression(Z) = (y^2 + (x - 5)^2 = 1);PointOnCurve(P, RightPart(G));PointOnCurve(M, H);PointOnCurve(N, Z)", "coords": {} }, { "index": 3981, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3981.png", "answer": "ApplyUnit(90, degree)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1))*Abs(LineSegmentOf(P, F2)) = 32", "coords": {} }, { "index": 3982, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3982.png", "answer": "y^2=x", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G) = True;F: Point;Focus(G) = F;PointOnCurve(F, LineSegmentOf(A, B)) = True;Abs(LineSegmentOf(A, F)) = 1;Abs(LineSegmentOf(B, F)) = 1/3", "coords": {} }, { "index": 3983, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3983.png", "answer": "4*sqrt(3)/3", "fact_expr": "C: Parabola;Expression(C) = (y = x^2/8);F: Point;Focus(C) = F;M: Point;l: Line;Directrix(C) = l;PointOnCurve(M, l);Intersection(LineSegmentOf(M,F), C) = N;N: Point;VectorOf(M,N) = (2/3)*VectorOf(M,F);O: Origin", "coords": {} }, { "index": 3984, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3984.png", "answer": "12", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 8*y);A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G);YCoordinate(MidPoint(LineSegmentOf(A, B))) = 4", "coords": {} }, { "index": 3985, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3985.png", "answer": "[1, sqrt(3)]", "fact_expr": "G: Parabola;A: Point;B: Point;P: Point;Expression(G) = (y^2 = 2*x);Coordinate(A) = (-1, 0);Coordinate(B) = (1, 0);PointOnCurve(P, G);m: Number;Abs(LineSegmentOf(P, A)) = m*Abs(LineSegmentOf(P, B))", "coords": { "A": [ -1.0, 0.0 ], "B": [ 1.0, 0.0 ] } }, { "index": 3986, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3986.png", "answer": "4*sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/4 + x^2/a^2 = 1);a: Number;Expression(OneOf(Asymptote(G))) = (y = x*(sqrt(3)/3))", "coords": {} }, { "index": 3987, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3987.png", "answer": "sqrt(6)/2", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (3, sqrt(10)/2);LeftFocus(C)=F1;RightFocus(C)=F2;PointOnCurve(P, RightPart(C));Abs(LineSegmentOf(P, F1)) - Abs(LineSegmentOf(P, F2)) = 4", "coords": { "P": [ 3.0, 1.5811388300841898 ] } }, { "index": 3988, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3988.png", "answer": "2", "fact_expr": "l: Line;C: Ellipse;a: Number;b: Number;Expression(C) = (x^2/25+y^2/16 = 1);Expression(l) = (a*x + b*y - 4*a + 2*b = 0)", "coords": {} }, { "index": 3989, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3990, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3990.png", "answer": "4/3", "fact_expr": "G: Parabola;H: Curve;a: Number;Expression(G) = (y^2 = 16*x);Expression(H) = (-y^2/7 + x^2/a^2 = 1);e:Number;Eccentricity(H)=e;RightFocus(H)=Focus(G)", "coords": {} }, { "index": 3991, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3991.png", "answer": "y^2/18 - x^2/27 = 1", "fact_expr": "G: Hyperbola;C:Hyperbola;A: Point;Expression(G) = (x^2/3 - y^2/2 = 1);Coordinate(A) = (sqrt(3), 2*sqrt(5));Asymptote(C) = Asymptote(G);PointOnCurve(A, C)", "coords": { "A": [ 1.7320508075688772, 4.47213595499958 ] } }, { "index": 3992, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_3992.png", "answer": "(\\sqrt{3} - 1)/2", "fact_expr": "F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;l: Line;PointOnCurve(F1, l);Intersection(l, C) = {A, B};A: Point;B: Point;Intersection(l, yAxis) = P;P: Point;VectorOf(F1, P) = VectorOf(A, F1);DotProduct(VectorOf(F2, A), VectorOf(F2, P)) = 0", "coords": {} }, { "index": 3993, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3993.png", "answer": "x^2/9 - y^2/16 = 1", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;Coordinate(F) = (5, 0);RightFocus(G) = F;H: Line;PointOnCurve(F, H);A: Point;B: Point;Intersection(H, G) = {A, B};M: Point;Coordinate(M) = (-45/7, -80/7);MidPoint(LineSegmentOf(A, B)) = M", "coords": { "F": [ 5.0, 0.0 ], "M": [ -6.428571428571429, -11.428571428571429 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 3994, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3994.png", "answer": "8", "fact_expr": "C: Parabola;Expression(C) = (x^2 = 2*(p*y));p: Number;p>0;F: Point;Focus(C) = F;M: Point;PointOnCurve(M, C);Distance(M, F) = 13;Distance(M, xAxis) = 9", "coords": {} }, { "index": 3995, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_3995.png", "answer": "8", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;A: Point;y1: Number;Coordinate(A) = (p/4, y1);PointOnCurve(A, C);Distance(A, Focus(C)) = 6", "coords": {} }, { "index": 3996, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 3997, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_ecc_mismatch", "params": { "a": 1.9257268905639648, "b": 1.7993911322594118, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 2.5576271215184017e-15, "converged": true, "iterations": 2 }, "output_path": "results/train/ellipse/problem_3997.png", "answer": "{4,9/4}", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2/3 + y^2/m = 1);Eccentricity(G) = 1/2", "coords": {}, "validation_reasons": [ "ellipse_ecc_mismatch" ] }, { "index": 3998, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_3998.png", "answer": "pi/2", "fact_expr": "l: Line;G: Parabola;P: Point;Q: Point;F: Point;Expression(G) = (y = -x^2/4);Coordinate(P) = (-4,-4);Focus(G) = F;Intersection(l,xAxis)=Q;TangentPoint(l,G)=P", "coords": { "P": [ -4.0, -4.0 ] } }, { "index": 3999, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_3999.png", "answer": "2*sqrt(5)", "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (x^2/4 - y^2/m = 1);FocalLength(G) = 6", "coords": {} }, { "index": 4000, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4000.png", "answer": "16/3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, G) = {A, B};O: Origin;Area(TriangleOf(A, O, F)) = 3*Area(TriangleOf(B, O ,F))", "coords": {} }, { "index": 4001, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4001.png", "answer": "pm*1", "fact_expr": "G: Hyperbola;H: Circle;l:Line;m: Number;B: Point;A: Point;Expression(G) = (x^2 - y^2/2 = 1);Expression(H) = (x^2 + y^2 = 5);Expression(l)=(x-y+m=0);Intersection(l,G)={A,B};Negation(A=B);PointOnCurve(MidPoint(LineSegmentOf(A,B)),H)", "coords": {} }, { "index": 4002, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 0.7071067811865476, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4002.png", "answer": "y=pm*(sqrt(2)/2)*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - 2*y^2 = 1)", "coords": {} }, { "index": 4003, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4003.png", "answer": "3/4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = x);A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G);Abs(LineSegmentOf(A, B)) = 2;M: Point;MidPoint(LineSegmentOf(A, B)) = M", "coords": {} }, { "index": 4004, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4004.png", "answer": "sqrt(5)/3", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;P: Point;PointOnCurve(P, G);F1: Point;F2: Point;Focus(G) = {F1, F2};DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;Tan(AngleOf(P, F1, F2)) = 1/2", "coords": {} }, { "index": 4005, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4005.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "E: Ellipse;b: Number;a: Number;A: Point;B: Point;F: Point;M:Point;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (-1, 0);OneOf(Focus(E))=F;PointOnCurve(F,LineSegmentOf(A,B));MidPoint(LineSegmentOf(A,B))=M;Coordinate(M)=(-2/3,1/3);IsChordOf(LineSegmentOf(A,B),E)", "coords": { "F": [ -1.0, 0.0 ], "M": [ -0.6666666666666666, 0.3333333333333333 ] } }, { "index": 4006, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4006.png", "answer": "sqrt(2)*3/4", "fact_expr": "G: Parabola;p: Number;A: Point;F: Point;B: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(A) = (0, 2);Focus(G) = F;MidPoint(LineSegmentOf(F, A)) = B;PointOnCurve(B, G)", "coords": { "A": [ 0.0, 2.0 ] } }, { "index": 4007, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4008, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4008.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "A: Point;a: Real;Coordinate(A) = (a, 0);G: Curve;Expression(G) = (y^2 = 2*x);P: Point;PointOnCurve(P, G)", "coords": {} }, { "index": 4009, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4009.png", "answer": "sqrt(5)/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G))) = (x - 2*y = 0)", "coords": {} }, { "index": 4010, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.8284271247461903, "b": 2.8284271247461903, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4010.png", "answer": "14+8*sqrt(2)", "fact_expr": "G: Hyperbola;P: Point;Q: Point;F1:Point;F2: Point;Expression(G) = (x^2 - y^2 = 8);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(F1,LineSegmentOf(P,Q));IsChordOf(LineSegmentOf(P,Q),G);Intersection(LineSegmentOf(P,Q),LeftPart(G))={P,Q};Abs(LineSegmentOf(P,Q))=7", "coords": {} }, { "index": 4011, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4012, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4012.png", "answer": "9", "fact_expr": "G: Hyperbola;H: Circle;C:Circle;P: Point;M: Point;N: Point;Expression(G) = (x^2/9 - y^2/16 = 1);Expression(H) = (10*x + x^2 + y^2 + 21 = 0);Expression(C)=(x^2+y^2-10*x+24=0);PointOnCurve(P, RightPart(G));PointOnCurve(M, H);PointOnCurve(N,C)", "coords": {} }, { "index": 4013, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4013.png", "answer": "3*sqrt(2)/4", "fact_expr": "F: Point;A: Point;G: Parabola;p: Number;B: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(A) = (0, 2);Focus(G) = F;MidPoint(LineSegmentOf(F, A)) = B;PointOnCurve(B, G)", "coords": { "A": [ 0.0, 2.0 ] } }, { "index": 4014, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4014.png", "answer": "2+4*sqrt(5)/5", "fact_expr": "P: Point;PointOnCurve(P, RightPart(G)) = True;G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1);H: Line;Expression(H) = (y = 2*x);F: Point;Coordinate(F) = (-2, 0)", "coords": { "F": [ -2.0, 0.0 ] } }, { "index": 4015, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.3300262689590454, "b": 2.30381441116333, "orientation": "horizontal" }, "optimization": { "final_loss": 6.658357051492203e-06, "converged": true, "iterations": 122 }, "output_path": "results/train/hyperbola/problem_4015.png", "answer": "(0, 1/2]", "fact_expr": "D: Hyperbola;Expression(D) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Expression(Asymptote(D)) = (y = pm*sqrt(3)*x);F1: Point;F2: Point;LeftFocus(D) = F1;RightFocus(D) = F2;P: Point;PointOnCurve(P, RightPart(D))", "coords": {} }, { "index": 4016, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4017, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_4017.png", "answer": "x^2/9 - y^2/27 = 1", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;D: Point;PointOnCurve(D, C) = True;P1: Point;P2: Point;Focus(C) = {P1, P2};Abs(Distance(D, P1) - Distance(D, P2)) = 6;Eccentricity(C) = 2", "coords": {} }, { "index": 4018, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4018.png", "answer": "x^2/24 + y^2/8 = 1", "fact_expr": "H: Ellipse;Expression(H) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;G: Parabola;Expression(G) = (y^2 = 16*x);RightFocus(H) = Focus(G);Eccentricity(H) = sqrt(6)/3", "coords": {} }, { "index": 4019, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4019.png", "answer": "sqrt(6)/3", "fact_expr": "G: Parabola;X: Circle;p: Number;H: Point;F: Point;O: Origin;M: Point;B: Point;x0: Number;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(M) = (x0, 2*sqrt(2));Focus(G) = F;PointOnCurve(M, G);Center(X) = M;PointOnCurve(O, X);IsTangent(Directrix(G), X);TangentPoint(Directrix(G), X)=H;Intersection(LineSegmentOf(H, F), G) = B", "coords": {} }, { "index": 4020, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.324555320336759, "b": 3.872983346207417, "major_axis": "x", "x_coef": 40.0, "y_coef": 15.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4020.png", "answer": "x^2/9 - y^2/16 = 1", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/40 + y^2/15 = 1);Focus(G)=Focus(H);Eccentricity(G)=5/3", "coords": {} }, { "index": 4021, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4021.png", "answer": "1/16", "fact_expr": "I: Point;Coordinate(I) = (0, -2);H: Line;PointOnCurve(I, H);G: Parabola;Expression(G) = (y^2 = 16*x);Intersection(H, G) = {A, B};A: Point;B: Point;Coordinate(A) = (x1, y1);x1: Number;y1: Number;Coordinate(B) = (x2, y2);x2: Number;y2: Number;y1^2 - y2^2 = 1;O: Origin", "coords": { "I": [ 0.0, -2.0 ] } }, { "index": 4022, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4022.png", "answer": "[1,2)", "fact_expr": "H: Line;Expression(H) = (y = k*x - 1);k: Number;PointOnCurve(Focus(G), xAxis) = True;G: Ellipse;Expression(G) = (x^2/2 + y^2/b = 1);IsIntersect(H, G) = True;b: Number", "coords": {} }, { "index": 4023, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4024, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.4641016151377544, "major_axis": "x", "x_coef": 16.0, "y_coef": 12.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4024.png", "answer": "(2*sqrt(3),sqrt(3))", "fact_expr": "G: Ellipse;A: Point;M: Point;F: Point;Expression(G) = (x^2/16 + y^2/12 = 1);Coordinate(A) = (-2, sqrt(3));RightFocus(G) = F;PointOnCurve(M, G);WhenMin(Abs(LineSegmentOf(M, A)) + 2*Abs(LineSegmentOf(M, F)))", "coords": { "A": [ -2.0, 1.7320508075688772 ] } }, { "index": 4025, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4025.png", "answer": "(\\sqrt{2}/2, 1)", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;c: Number;Coordinate(F) = (-c, 0);LeftFocus(G) = F;H: Circle;Center(H) = F;Radius(H) = c;NumIntersection(H, LeftDirectrix(G)) = 2", "coords": {} }, { "index": 4026, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4026.png", "answer": "a^2/b^2", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;P: Point;PointOnCurve(P, G);M: Point;IsPerpendicular(LineSegmentOf(P, M), xAxis);FootPoint(LineSegmentOf(P, M), xAxis) = M;Negation(M=P);A1: Point;A2: Point;MajorAxis(G) = LineSegmentOf(A1, A2);PointOnCurve(P, LineSegmentOf(P, M))", "coords": {} }, { "index": 4027, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 2.0, 0.0 ], "radius": 8.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_4027.png", "answer": "x^2/16+y^2/12=1", "fact_expr": "G: Circle;Expression(G) = (y^2 + (x + 2)^2 = 64);M: Point;Center(G) = M;A: Point;PointOnCurve(A, G);N: Point;Coordinate(N) = (2, 0);P: Point;Intersection(PerpendicularBisector(LineSegmentOf(A, N)), LineSegmentOf(M, A)) = P", "coords": {} }, { "index": 4028, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4028.png", "answer": "2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x);A: Point;B: Point;PointOnCurve(A, G);PointOnCurve(B, G);F: Point;Focus(G) = F;Distance(A, F) + Distance(B, F) = 5;M: Point;MidPoint(LineSegmentOf(A, B)) = M", "coords": {} }, { "index": 4029, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4030, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "y", "x_coef": 3.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4030.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 2*(p*y));p: Number;p>0;H: Ellipse;Expression(H) = (x^2/3 + y^2/4 = 1);Focus(G) = OneOf(Vertex(H))", "coords": {} }, { "index": 4031, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4031.png", "answer": "3*sqrt(5)", "fact_expr": "C1: Circle;C: Point;Center(C1) = C;G: Parabola;Expression(G) = (y^2 = 2*x);PointOnCurve(C, G) = True;TangentPoint(C1, yAxis) = A;A: Point;Intersection(C1, xAxis) = {P, Q};P: Point;Q: Point;O: Origin;DotProduct(VectorOf(O, C), VectorOf(O, A)) = 9", "coords": {} }, { "index": 4032, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 4.898979485566356, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4032.png", "answer": "24", "fact_expr": "G: Hyperbola;P: Point;H: Ellipse;F1: Point;F2: Point;Expression(G) = (x^2 - y^2/24 = 1);Expression(H) = (x^2/49 + y^2/24 = 1);Focus(G) = {F1, F2};OneOf(Intersection(G, H)) = P", "coords": {} }, { "index": 4033, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4033.png", "answer": "4", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2 - y^2/4 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, RightPart(G));Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 4034, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4034.png", "answer": "3", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1);F: Point;LeftFocus(G) = F;A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G);Inclination(LineSegmentOf(A, B)) = pi/6;PointOnCurve(F, LineSegmentOf(A, B))", "coords": {} }, { "index": 4035, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4035.png", "answer": "{15/4, 17/4}", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2/4 + y^2/m = 1);FocalLength(G) = 1", "coords": {} }, { "index": 4036, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4036.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "C: Parabola;p: Number;G: Line;P: Point;F: Point;Q: Point;A: Point;l: Line;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;Directrix(C) = l;PointOnCurve(F, G);Intersection(G, C) = {P, Q};Intersection(G, l) = A;VectorOf(P, F) = 3*VectorOf(F, Q)", "coords": {} }, { "index": 4037, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4037.png", "answer": "5/3", "fact_expr": "l: Line;C: Hyperbola;a: Number;b: Number;G: Line;Q: Point;P: Point;F1: Point;M: Point;F2: Point;e: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F2, G);Intersection(G, RightPart(C)) = {P, Q};DotProduct((VectorOf(F1, P) + VectorOf(F1, Q)), VectorOf(P, Q)) = 0;PointOnCurve(RightVertex(C), l);IsParallel(l, OneOf(Asymptote(C)));Intersection(l, LineSegmentOf(P, Q)) = M;Abs(LineSegmentOf(Q, M)) = 3*Abs(LineSegmentOf(P, M));Eccentricity(C) = e", "coords": {} }, { "index": 4038, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.535473169108841, "b": 3.535473169108841, "orientation": "horizontal" }, "optimization": { "final_loss": 4.3822798825688845e-07, "converged": true, "iterations": 93 }, "output_path": "results/train/hyperbola/problem_4038.png", "answer": "x^2/9 - y^2/16 = 1", "fact_expr": "G: Hyperbola;F1: Point;F2: Point;Coordinate(F1) = (5, 0);Coordinate(F2) = (-5, 0);Focus(G)={F1,F2};PointOnCurve(P,G);Abs(Distance(P,F1)-Distance(P,F2))=6;P:Point", "coords": { "F1": [ 5.0, 0.0 ], "F2": [ -5.0, 0.0 ] } }, { "index": 4039, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4039.png", "answer": "9*sqrt(3)", "fact_expr": "C: Parabola;p: Number;A: Point;F: Point;M: Point;l:Line;p>0;Expression(C) = (y^2=2*(p*x));Focus(C) = F;Directrix(C)=l;Expression(l)=(x=-3/2);PointOnCurve(M, C);PointOnCurve(A, l);IsPerpendicular(LineSegmentOf(M,A),l);Slope(LineOf(A,F))=k1;k1=-sqrt(3);k1:Number", "coords": {} }, { "index": 4040, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_4040.png", "answer": "3/16", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/n + x^2/m = 1);Eccentricity(G) = 2;m: Number;n: Number;H: Parabola;Expression(H) = (y^2 = 4*x);OneOf(Focus(G)) = Focus(H)", "coords": {} }, { "index": 4041, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4041.png", "answer": "4/5", "fact_expr": "C: Ellipse;Expression(C) = (y^2/9 + x^2/a^2 = 1);a: Number;G: Line;Expression(G) = (y = x - 4);PointOnCurve(OneOf(Focus(C)),G) = True", "coords": {} }, { "index": 4042, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.8284271247461903, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4042.png", "answer": "12*sqrt(6)", "fact_expr": "G: Hyperbola;A: Point;P: Point;F: Point;Expression(G) = (x^2 - y^2/8 = 1);Coordinate(A) = (0, 6*sqrt(6));RightFocus(G) = F;PointOnCurve(P, LeftPart(G));WhenMin(Perimeter(TriangleOf(A, P, F)))", "coords": { "A": [ 0.0, 14.696938456699067 ] } }, { "index": 4043, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4043.png", "answer": "2", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, F)) - Abs(LineSegmentOf(B, F)) = 3/2", "coords": {} }, { "index": 4044, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4045, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 1.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4045.png", "answer": "1+3*sqrt(6)/2", "fact_expr": "P: Point;H: Circle;Expression(H) = (x^2 + (y - 2)^2 = 1);PointOnCurve(P, H);G: Ellipse;Expression(G) = (x^2/9 + y^2 = 1);Q: Point;PointOnCurve(Q, G)", "coords": {} }, { "index": 4046, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4047, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4047.png", "answer": "(1, 2)", "fact_expr": "C: Parabola;P: Point;F: Point;K: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;Intersection(Directrix(C), xAxis) = K;PointOnCurve(P, C);Quadrant(P) = 1;WhenMin(Abs(LineSegmentOf(P, F))/Abs(LineSegmentOf(P, K)))", "coords": {} }, { "index": 4048, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4049, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4050, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4051, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4051.png", "answer": "32/15", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);A: Point;RightVertex(G)=A;F: Point;RightFocus(G)=F;H: Line;PointOnCurve(F,H);IsParallel(H,OneOf(Asymptote(G)));B: Point;Intersection(H,G)=B", "coords": {} }, { "index": 4052, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4053, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4053.png", "answer": "sqrt(3)/3", "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;P: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;L:Line;PointOnCurve(F1, L);IsPerpendicular(L,xAxis);Intersection(L, G) = P;AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 4054, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4055, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4055.png", "answer": "12", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;x1:Number;x2:Number;y2:Number;Expression(G) = (y^2 = 8*x);Coordinate(A) = (x1,x2);Coordinate(B) = (x2,y2);PointOnCurve(Focus(G),H);Intersection(H, G) = {A,B};Abs(LineSegmentOf(A,B))=16", "coords": {} }, { "index": 4056, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4057, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4057.png", "answer": "7/5", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;P: Point;F1: Point;F2: Point;Q: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, H);Intersection(H, RightPart(G)) = {P, Q};Abs(LineSegmentOf(P, F1)) = Abs(LineSegmentOf(F1, F2));3*Abs(LineSegmentOf(P, F2)) = 2*Abs(LineSegmentOf(Q, F2))", "coords": {} }, { "index": 4058, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4059, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4059.png", "answer": "4+2*sqrt(2)", "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2 = 1);F1: Point;F2: Point;Focus(C)={F1,F2};P: Point;PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) = 3*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 4060, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4060.png", "answer": "1", "fact_expr": "l: Line;C: Parabola;p: Number;A: Point;F: Point;B: Point;p>0;Expression(C) = (y^2 = 2*p*x);Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, F)) = 2/3;Abs(LineSegmentOf(B, F)) = 2", "coords": {} }, { "index": 4061, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4061.png", "answer": "4*x+9*y-13=0", "fact_expr": "G: Ellipse;A: Point;B: Point;M: Point;H:Line;Expression(G) = (x^2/9 + y^2/4 = 1);Coordinate(M) = (1, 1);PointOnCurve(M,H);Intersection(H,G)={A,B};IsChordOf(LineSegmentOf(A,B),G);MidPoint(LineSegmentOf(A, B)) = M", "coords": { "M": [ 1.0, 1.0 ] } }, { "index": 4062, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4062.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Ellipse;H: Line;P: Point;Q: Point;F2: Point;F1: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(G)={F1,F2};PointOnCurve(F1,H);Intersection(H,G)={P,Q}", "coords": {} }, { "index": 4063, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4063.png", "answer": "{2,18}", "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/9 + y^2/16 = 1);P: Point;PointOnCurve(P, G) = True;P1: Point;P2: Point;OneOf(Focus(G))=P1;OneOf(Focus(G))=P2;Negation(P1=P2);Distance(P, P1) = 10", "coords": {} }, { "index": 4064, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4064.png", "answer": "sqrt(13)/2", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);A: Point;RightVertex(G) = A;F: Point;RightFocus(G) = F;H: Circle;Center(H) = A;R: Number;Radius(H) = R;B: Point;C: Point;Intersection(H, G) = {B, C};PointOnCurve(F,LineOf(B,C)) = True", "coords": {} }, { "index": 4065, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4065.png", "answer": "1", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);F2: Point;F1: Point;LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G) = True;P: Point;A: Point;Coordinate(A) = ((-1)*4, 4)", "coords": { "A": [ -4.0, 4.0 ] } }, { "index": 4066, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4066.png", "answer": "2", "fact_expr": "G: Parabola;p: Number;t:Number;H: Point;I: Point;Expression(G) = (y^2 = 2*(p*x));Coordinate(H) = (2, t);Coordinate(I) = (p/2, 0);PointOnCurve(H, G);Distance(H, I) = 3", "coords": {} }, { "index": 4067, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4067.png", "answer": "3", "fact_expr": "G: Parabola;C: Circle;M: Point;A: Point;F: Point;Expression(G) = (y^2 = 4*x);Expression(C) = ((x - 3)^2 + (y - 1)^2 = 1);PointOnCurve(M, G);Focus(G) = F;PointOnCurve(A, C)", "coords": {} }, { "index": 4068, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4068.png", "answer": "(pm*(4*sqrt(3))/3,-1/3)", "fact_expr": "G: Ellipse;A: Point;P: Point;Expression(G) = (x^2 + 4*y^2 = 4);Coordinate(A) = (0, 1);PointOnCurve(A, G);PointOnCurve(P, G);IsChordOf(LineSegmentOf(A,P),G);WhenMax(Length(LineSegmentOf(A,P)))", "coords": { "A": [ 0.0, 1.0 ] } }, { "index": 4069, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4069.png", "answer": "x^2/4-y^2=1", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (2*x + y = 0);FocalLength(G)=2*sqrt(5) ;IsPerpendicular(OneOf(Asymptote(G)), H)", "coords": {} }, { "index": 4070, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4070.png", "answer": "4*sqrt(2)", "fact_expr": "l: Line;C: Parabola;A: Point;P: Point;Q: Point;F: Point;B: Point;D: Point;Expression(C) = (y^2 = 8*x);Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};O: Origin;MidPoint(LineSegmentOf(O, A)) = D;Projection(B, yAxis) = P;Projection(D, yAxis) = Q", "coords": {} }, { "index": 4071, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4071.png", "answer": "13", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);P: Point;PointOnCurve(P, G);H: Circle;Expression(H) = (y^2 + (x + 3)^2 = 4);M: Point;PointOnCurve(M, H);H1: Circle;Expression(H1) = (y^2 + (x - 3)^2 = 1);N: Point;PointOnCurve(N, H1)", "coords": {} }, { "index": 4072, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4073, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4073.png", "answer": "4", "fact_expr": "G: Parabola;p: Number;m: Number;A: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(A) = (1, m);PointOnCurve(A, G);Distance(A, Focus(G)) = 3", "coords": {} }, { "index": 4074, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4149149626832767, "b": 2.450642358201144, "orientation": "horizontal" }, "optimization": { "final_loss": 1.0319236343469396e-05, "converged": true, "iterations": 94 }, "output_path": "results/train/hyperbola/problem_4074.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/b^2 = 1);b: Number;b>0;Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x)", "coords": {} }, { "index": 4075, "success": false, "conic_type": "parabola", "error": "validation: parabola_point_off_curve", "params": { "p": 3.9731175899505615, "direction": "up" }, "optimization": { "final_loss": 0.00018243117665406317, "converged": true, "iterations": 127 }, "output_path": "results/train/parabola/problem_4075.png", "answer": "5", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 2*p*y);p: Number;p>0;H: Point;Coordinate(H) = (4, 1);PointOnCurve(H, G);Distance(H, Focus(G)) = d;d: Number", "coords": { "H": [ 4.0, 1.0 ] }, "validation_reasons": [ "parabola_point_off_curve" ] }, { "index": 4076, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4076.png", "answer": "(3+pm*sqrt(5))/4", "fact_expr": "G: Parabola;p: Number;H: Line;A: Point;B: Point;p>0;Expression(G) = (y^2 = 2*(p*x));PointOnCurve(Focus(G), H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 3;YCoordinate(MidPoint(LineSegmentOf(A, B))) = 1/2", "coords": {} }, { "index": 4077, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4078, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4078.png", "answer": "10", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);H: Line;PointOnCurve(Focus(G), H);A: Point;B: Point;x1: Number;x2: Number;y1: Number;y2: Number;Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);Intersection(H, G) = {A, B};x1+x2=8", "coords": {} }, { "index": 4079, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4079.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x);F: Point;Focus(G) = F;PointOnCurve(F, H) = True;Inclination(H) = ApplyUnit(45, degree);H: Line;Intersection(H, G) = {A, B};B: Point;A: Point", "coords": {} }, { "index": 4080, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4080.png", "answer": "3+sqrt(2)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;A: Point;M: Point;Coordinate(A) = (2, 1);Focus(G) = F;PointOnCurve(M, G);Negation(PointOnCurve(M,LineOf(A,F)))", "coords": { "A": [ 2.0, 1.0 ] } }, { "index": 4081, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4081.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (2, 0);Distance(P,OneOf(Asymptote(G)))=sqrt(2)", "coords": { "P": [ 2.0, 0.0 ] } }, { "index": 4082, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.449489742783178, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4082.png", "answer": "6/5", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/6 - y^2/3 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;M: Point;PointOnCurve(M, G);IsPerpendicular(LineSegmentOf(M, F1), xAxis)", "coords": {} }, { "index": 4083, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4083.png", "answer": "2*x**2-2*y**2-2*x-2*y-1=0", "fact_expr": "Q: Point;N: Point;G: Hyperbola;H: Line;L:Line;Expression(G) = (x^2 - y^2 = 1);PointOnCurve(Q,G);PointOnCurve(Q,L);IsPerpendicular(L,H);FootPoint(L,H)=N;Expression(H) = (x + y = 2);MidPoint(LineSegmentOf(Q, N)) = P;P:Point", "coords": {} }, { "index": 4084, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4084.png", "answer": "4", "fact_expr": "G: Hyperbola;a: Number;b: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(OneOf(Intersection(Asymptote(G), Directrix(G)))) = (1, sqrt(3))", "coords": {} }, { "index": 4085, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4086, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4086.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;H: Circle;Expression(H)=((x - 2)^2 + y^2 = 3);IsTangent(Asymptote(G), H)", "coords": {} }, { "index": 4087, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4087.png", "answer": "6", "fact_expr": "G: Ellipse;m: Real;Expression(G) = (x^2/5 + y^2/m = 1);Coordinate(OneOf(Focus(G))) = (0, 1)", "coords": {} }, { "index": 4088, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4088.png", "answer": "pm*sqrt(3)/3", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;k: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = k*x);Eccentricity(G) = sqrt(3)/3;XCoordinate(OneOf(Intersection(H,G))) = b", "coords": {} }, { "index": 4089, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4089.png", "answer": "[4, 5)", "fact_expr": "G: Ellipse;m: Real;H: Line;k: Number;Expression(G) = (x^2/5 + y^2/m = 1);Expression(H) = (y = k*x - 2);IsIntersect(H, G);PointOnCurve(Focus(G), xAxis)", "coords": {} }, { "index": 4090, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.6457513110645907, "major_axis": "x", "x_coef": 16.0, "y_coef": 7.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4090.png", "answer": "8", "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/16 + y^2/7 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C)", "coords": {} }, { "index": 4091, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4091.png", "answer": "y^2=2*x-2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);H: LineSegment;PointOnCurve(Focus(G), H);IsChordOf(H, G)", "coords": {} }, { "index": 4092, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4092.png", "answer": "sqrt(2)-1", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;M: Point;F2: Point;F1: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(F1,H);Inclination(H)=ApplyUnit(45,degree);OneOf(Intersection(H,G))=M;IsPerpendicular(LineSegmentOf(M,F2),xAxis)", "coords": {} }, { "index": 4093, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4093.png", "answer": "4*sqrt(3)/3", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);P: Point;PointOnCurve(P, G);B: Point;UpperVertex(G) = B", "coords": {} }, { "index": 4094, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4095, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4095.png", "answer": "6", "fact_expr": "G: Ellipse;P: Point;F1: Point;O: Origin;Q: Point;F2: Point;Expression(G) = (x^2/16 + y^2/9 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);MidPoint(LineSegmentOf(P, F1)) = Q;Abs(LineSegmentOf(O, Q)) = 1", "coords": {} }, { "index": 4096, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4096.png", "answer": "(1, \\sqrt{2}]", "fact_expr": "G: Ellipse;b: Number;a: Number;c: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);HalfFocalLength(G) = c", "coords": {} }, { "index": 4097, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4097.png", "answer": "-8/9", "fact_expr": "G: Ellipse;H: LineSegment;I: Point;Expression(G) = (x^2/9 + y^2/4 = 1);Coordinate(I) = (2, 1);IsChordOf(H,G);MidPoint(H)=I;Slope(OverlappingLine(H)) = k;k:Number", "coords": { "I": [ 2.0, 1.0 ] } }, { "index": 4098, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 7.0, "b": 4.898979485566356, "major_axis": "x", "x_coef": 49.0, "y_coef": 24.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4098.png", "answer": "4*pi", "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/49 + y^2/24 = 1);LeftFocus(C)=F1;RightFocus(C)=F2;PointOnCurve(P,C);DotProduct(VectorOf(P,F1),VectorOf(P,F2))=0", "coords": {} }, { "index": 4099, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4099.png", "answer": "5", "fact_expr": "G: Hyperbola;a: Number;a>0;Expression(G) = (-y^2/9 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G))) = (y = (3/5)*x)", "coords": {} }, { "index": 4100, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4100.png", "answer": "x^2/4 - y^2/2 = 1", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F: Point;Q: Point;O: Origin;G: Circle;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;Center(G) = F;Radius(G) = sqrt(3);Intersection(G, OneOf(Asymptote(C))) = {P, Q};VectorOf(P, Q) = 2*VectorOf(Q, O);IsPerpendicular(LineSegmentOf(P, F), xAxis)", "coords": {} }, { "index": 4101, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4102, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4103, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4103.png", "answer": "x^2/4 + y^2/2 = 1", "fact_expr": "B: Point;F: Point;M: Point;G: Ellipse;b: Number;a: Number;H: Line;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x + y - 3*sqrt(2) = 0);UpperVertex(G) = B;LeftFocus(G) = F;IsPerpendicular(LineOf(B,F),H);FootPoint(LineOf(B,F),H)=M;MidPoint(LineSegmentOf(M, F)) = B", "coords": {} }, { "index": 4104, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4104.png", "answer": "(-oo,(3*sqrt(2)-sqrt(10))/2)", "fact_expr": "G: Ellipse;H: Line;P: Point;A: Point;m: Real;Expression(G) = (x^2/4 + y^2 = 1);Expression(H) = (x + y - 3 = 0);PointOnCurve(A, H);PointOnCurve(P,G);LineSegmentOf(P,A)>m", "coords": {} }, { "index": 4105, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4106, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4106.png", "answer": "(4, 0)", "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;F: Point;P: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, B)) = 6;Intersection(PerpendicularBisector(LineSegmentOf(A, B)), xAxis) = P", "coords": {} }, { "index": 4107, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4108, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4108.png", "answer": "8", "fact_expr": "l: Line;C: Parabola;p: Number;M: Point;F: Point;N: Point;O: Origin;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {M, N};VectorOf(M, F) = 3*VectorOf(F, N);Area(TriangleOf(O, M, N)) = sqrt(3)*Abs(LineSegmentOf(M, N))", "coords": {} }, { "index": 4109, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4109.png", "answer": "9*pi/16", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;l: Line;PointOnCurve(F2, l);P: Point;Q: Point;Intersection(l, G) = {P, Q}", "coords": {} }, { "index": 4110, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.24967060983181, "direction": "right" }, "optimization": { "final_loss": 6.555423624376999e-06, "converged": true, "iterations": 116 }, "output_path": "results/train/parabola/problem_4110.png", "answer": "5/4", "fact_expr": "C: Parabola;p: Number;P: Point;F: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(P) = (1, 1);PointOnCurve(P, C);Focus(C) = F", "coords": { "P": [ 1.0, 1.0 ] } }, { "index": 4111, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4111.png", "answer": "sqrt(10)/2", "fact_expr": "A: Point;B: Point;C: Point;PointOnCurve(A, G) = True;PointOnCurve(B, G) = True;PointOnCurve(C, G) = True;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;O: Origin;PointOnCurve(O,LineOf(A,B)) = True;F: Point;RightFocus(G) = F;PointOnCurve(F,LineOf(A,C)) = True;IsPerpendicular(LineSegmentOf(B,F),LineSegmentOf(A,C));Abs(LineSegmentOf(B, F)) = Abs(LineSegmentOf(C, F))", "coords": {} }, { "index": 4112, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.4641016151377544, "major_axis": "x", "x_coef": 16.0, "y_coef": 12.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4112.png", "answer": "x^2 - y^2 / 3 = 1", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/16 + y^2/12 = 1);Eccentricity(G)=4*Eccentricity(H);PointOnCurve(Focus(G),xAxis)", "coords": {} }, { "index": 4113, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4113.png", "answer": "2+pm*sqrt(2)", "fact_expr": "P: Point;G: Parabola;Expression(G) = (y = x^2);PointOnCurve(P, G) ;Negation(P = O);O: Origin;IsPerpendicular(LineSegmentOf(P, Q), xAxis);FootPoint(LineSegmentOf(P, Q), xAxis) = Q;Q: Point;H: Line;PointOnCurve(MidPoint(LineSegmentOf(P, Q)), H);IsParallel(H, xAxis) ;Intersection(H, G) = M;M: Point;Intersection(LineOf(Q, M), yAxis) = N;N: Point", "coords": {} }, { "index": 4114, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4115, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4116, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4117, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4117.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(60, degree);Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 4118, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4118.png", "answer": "1/3", "fact_expr": "l: Line;G: Parabola;p: Number;A: Point;F: Point;B: Point;p>0;Expression(G) = (y^2 = 2*p*x);Focus(G) = F;PointOnCurve(F, l);Inclination(l) = ApplyUnit(120, degree);Intersection(l, G) = {A, B};Quadrant(A)=1;Quadrant(B)=4", "coords": {} }, { "index": 4119, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4120, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4120.png", "answer": "6", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1);H: Line;Slope(H) = 1;PointOnCurve(OneOf(Focus(G)), H);A: Point;B: Point;Intersection(H, G) = {A, B}", "coords": {} }, { "index": 4121, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4121.png", "answer": "9/2", "fact_expr": "C: Parabola;p: Number;A: Point;B: Point;F: Point;O: Origin;L:Line;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;PointOnCurve(F, L)=True;Intersection(C, L) = {A, B};VectorOf(A, B) = 3*VectorOf(F, B);Area(TriangleOf(O, A, B)) = (sqrt(2)/3)*Abs(LineSegmentOf(A, B))", "coords": {} }, { "index": 4122, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4122.png", "answer": "1", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;E: Point;PointOnCurve(E, Directrix(G));LineSegmentOf(E, F) = (2*sqrt(3))/3;H: Line;PointOnCurve(F, H);IsPerpendicular(H, LineSegmentOf(E, F));A: Point;B: Point;Intersection(H, G) = {A, B};VectorOf(A, F) = 3*VectorOf(F, B)", "coords": {} }, { "index": 4123, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4123.png", "answer": "sqrt(3)/3", "fact_expr": "E: Parabola;p: Number;P: Point;A: Point;F: Point;p>0;Expression(E) = (y^2 = 2*(p*x));Focus(E) = F;Intersection(Directrix(E),axis)=A;PointOnCurve(P, E);AngleOf(P, A, F) = ApplyUnit(30, degree)", "coords": {} }, { "index": 4124, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 2.0, -0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_4124.png", "answer": "x^2/3-y^2=1", "fact_expr": "G: Hyperbola;H: Circle;Expression(H) = (-4*x + x^2 + y^2 + 3 = 0);Center(H)=F;RightFocus(G)=F;IsTangent(Asymptote(G),H);F: Point", "coords": {} }, { "index": 4125, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4125.png", "answer": "sqrt(5)/5", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;H: Line;Expression(H) = (y = 2*x - 1);PointOnCurve(OneOf(Vertex(G)),H) = True;PointOnCurve(OneOf(Focus(G)),H)", "coords": {} }, { "index": 4126, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4127, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4128, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/train/hyperbola/problem_4128.png", "answer": "2", "fact_expr": "C: Hyperbola;a: Number;b: Number;G: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(G) = (2, 3);PointOnCurve(G, C);FocalLength(C) = 4", "coords": { "G": [ 2.0, 3.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 4129, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 10.0, "b": 8.0, "major_axis": "x", "x_coef": 100.0, "y_coef": 64.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4129.png", "answer": "100", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/100 + y^2/64 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G)", "coords": {} }, { "index": 4130, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4130.png", "answer": "{0,1,-1}", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);l: Line;H: Point;Coordinate(H) = (-1, 0);PointOnCurve(H, l) = True;NumIntersection(l, G) = 1", "coords": { "H": [ -1.0, 0.0 ] } }, { "index": 4131, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.0, "b": 0.7071067811865476, "major_axis": "y", "x_coef": 0.5, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4131.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Ellipse;P: Point;Coordinate(P) = (x1, y1);x1:Number;y1:Number;Expression(G)=(2*x^2+y^2=1)", "coords": {} }, { "index": 4132, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4132.png", "answer": "y^2 = -6*x", "fact_expr": "G: Hyperbola;H: Parabola;Expression(G) = (x^2/3 - y^2 = 1);O: Origin;Vertex(H) = O;Directrix(H) = RightDirectrix(G)", "coords": {} }, { "index": 4133, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4134, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4134.png", "answer": "pm*((2*sqrt(2))/3)", "fact_expr": "G: Line;Expression(G) = (y = k*(x + 2));k: Real;C: Parabola;Expression(C) = (y^2 = 8*x);A: Point;B: Point;Intersection(G, C) = {A, B};F: Point;Focus(C) = F;Abs(VectorOf(F, A)) = 2*Abs(VectorOf(F, B))", "coords": {} }, { "index": 4135, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4135.png", "answer": "7/4", "fact_expr": "G: Parabola;M: Point;Expression(G) = (y^2 = x);PointOnCurve(M, G);Distance(M, Focus(G)) = 2", "coords": {} }, { "index": 4136, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4136.png", "answer": "16/3", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);H: Circle;PointOnCurve(OneOf(Vertex(G)),H) = True;PointOnCurve(OneOf(Focus(G)),H) = True;PointOnCurve(Center(H),G) = True", "coords": {} }, { "index": 4137, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4137.png", "answer": "5/3", "fact_expr": "C: Hyperbola;a: Number;b: Number;G: Line;O: Origin;A: Point;F: Point;a>0;b>0;Expression(C) = (-x^2/b^2 + y^2/a^2 = 1);LowerFocus(C)=F;PointOnCurve(F,G);Slope(G)=2;Intersection(G,OneOf(Asymptote(C)))=A;Quadrant(A)=1;Abs(LineSegmentOf(O, A)) = Abs(LineSegmentOf(O, F))", "coords": {} }, { "index": 4138, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4138.png", "answer": "sqrt(3)/2", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;A: Point;B: Point;PointOnCurve(A, G);PointOnCurve(B, G);Z: Circle;Expression(Z) = (x^2 + y^2 = R^2);IsDiameter(LineSegmentOf(A, B),Z);R: Number;R>0;M: Point;PointOnCurve(M, G);Negation(M=A);Negation(M=B);Slope(LineOf(M, A))*Slope(LineOf(M, B)) = -1/4", "coords": {} }, { "index": 4139, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4140, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4140.png", "answer": "x^2/4+y^2/3=1", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;A: Point;UpperVertex(G) = A;Coordinate(A) = (0, sqrt(3));Area(InscribedCircle(TriangleOf(A, F1, F2))) = (1/3)*pi", "coords": { "A": [ 0.0, 1.7320508075688772 ] } }, { "index": 4141, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4141.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;O: Origin;P: Point;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x = -a);RightFocus(G) = F;PointOnCurve(P, H);AngleOf(O, P, F) = ApplyUnit(30, degree)", "coords": {} }, { "index": 4142, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4142.png", "answer": "4", "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;O: Origin;Expression(G) = (y^2 = 4*x);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);Intersection(l, G) = {A,B};DotProduct(VectorOf(O, A), VectorOf(O, B)) = -4;x1:Number;y1:Number;x2:Number;y2:Number", "coords": {} }, { "index": 4143, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4144, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4145, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4145.png", "answer": "(1/2,0)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x)", "coords": {} }, { "index": 4146, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4146.png", "answer": "8", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/16 + y^2/9 = 1);Focus(G)={F1,F2};PointOnCurve(P,G)", "coords": {} }, { "index": 4147, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4147.png", "answer": "sqrt(13)", "fact_expr": "G: Ellipse;A: Point;B: Point;Expression(G) = (x^2/9 + y^2/4 = 1);RightVertex(G)=A;UpperVertex(G)=B", "coords": {} }, { "index": 4148, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 0.3333333333333333, "b": 0.25, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4148.png", "answer": "5/6", "fact_expr": "G: Hyperbola;Expression(G) = (9*x^2 - 16*y^2 = 1)", "coords": {} }, { "index": 4149, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4149.png", "answer": "18*sqrt(2)/7", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;F: Point;RightFocus(C) = F;A: Point;B: Point;LeftVertex(C) = A;RightVertex(C) = B;Abs(LineSegmentOf(F,A)) = 3;Abs(LineSegmentOf(F,B)) = 1;G: Line;Expression(G) = (y = x + 1/2)", "coords": {} }, { "index": 4150, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4151, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4151.png", "answer": "5/2", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l: Line;Directrix(C) = l;P: Point;PointOnCurve(P, l);Q: Point;OneOf(Intersection(LineOf(P, F), C)) = Q;VectorOf(P, F) = 4*VectorOf(F, Q)", "coords": {} }, { "index": 4152, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4153, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4154, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4154.png", "answer": "x^2/5 + y^2/4 = 1", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (2*x - y + 2 = 0);PointOnCurve(OneOf(Vertex(G)),H);PointOnCurve(OneOf(Focus(G)),H)", "coords": {} }, { "index": 4155, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4155.png", "answer": "sqrt(2)/3", "fact_expr": "G:Ellipse;Expression(G)=(x^2/a^2+y^2/b^2=1);A:Point;B:Point;F:Point;a:Number;b:Number;a>b;b>0;H:Line;RightFocus(G)=F;PointOnCurve(F,H);Slope(H)=1;Intersection(H,G)={A,B};LineSegmentOf(A,F)=2*LineSegmentOf(F,B)", "coords": {} }, { "index": 4156, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4156.png", "answer": "5/3", "fact_expr": "G: Hyperbola;a: Number;b: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, RightPart(G));Abs(LineSegmentOf(P, F1)) = 4*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 4157, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4157.png", "answer": "7", "fact_expr": "G: Parabola;P: Point;A: Point;F: Point;Expression(G) = (y^2 = 8*x);Coordinate(A) = (5, 2);PointOnCurve(P, G);Focus(G) = F", "coords": { "A": [ 5.0, 2.0 ] } }, { "index": 4158, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4158.png", "answer": "x^2 - y^2 / 3 = 1", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(l)=(x+sqrt(3)*y=0);IsPerpendicular(OneOf(Asymptote(C)),l);Distance(OneOf(Focus(C)), l) = 1", "coords": {} }, { "index": 4159, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4159.png", "answer": "(0,-1)", "fact_expr": "G: Parabola;Expression(G) = (y = -x^2/4)", "coords": {} }, { "index": 4160, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4160.png", "answer": "x^2/5+y^2/4=1", "fact_expr": "A: Point;B: Point;G: Ellipse;b: Number;a: Number;H: Circle;P: Point;l1: Line;l2:Line;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2 + y^2 = 1);Coordinate(P) = (1, 1/2);PointOnCurve(Focus(G), xAxis);TangentOfPoint(P,H)={l1,l2};TangentPoint(l1,H)=A;TangentPoint(l2,H)=B;PointOnCurve(RightFocus(G),LineOf(A,B));PointOnCurve(UpperVertex(G),LineOf(A,B))", "coords": { "P": [ 1.0, 0.5 ] } }, { "index": 4161, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4161.png", "answer": "2*sqrt(2)", "fact_expr": "G: Parabola;P: Point;Q: Point;F: Point;l: Line;Expression(G) = (y^2 = 4*x);Focus(G) = F;Directrix(G) = l;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P,Q),xAxis);FootPoint(LineSegmentOf(P,Q),xAxis)=Q;LineSegmentOf(P,F)=3", "coords": {} }, { "index": 4162, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4163, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4163.png", "answer": "4", "fact_expr": "G: Ellipse;Expression(G) = (x^2 + y^2/m = 1);m: Number;PointOnCurve(Focus(G), yAxis);Length(MajorAxis(G)) = 2*Length(MinorAxis(G))", "coords": {} }, { "index": 4164, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4165, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4165.png", "answer": "x = -2", "fact_expr": "G: Parabola;p: Number;H: Line;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (2*x + y - 4 = 0);PointOnCurve(Focus(G), H)", "coords": {} }, { "index": 4166, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4166.png", "answer": "(1, \\sqrt{6}/2)", "fact_expr": "C: Hyperbola;P: Point;A: Point;B: Point;a: Number;b: Number;a>0;b>0;e: Number;Expression(C) = (x^2/a^2 - y^2/b^2 = 1);Coordinate(A) = (a, 0);Coordinate(B) = (2*a, 0);PointOnCurve(P,RightPart(C));Negation(P=A);VectorOf(P, A)^2 + VectorOf(P, B)^2 = a^2;Eccentricity(C) = e", "coords": {} }, { "index": 4167, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4167.png", "answer": "10", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 5*x);Q: Point;Coordinate(Q) = (m, n);m: Number;n: Number;PointOnCurve(Q, G);Distance(Q, Focus(G)) = 25/4", "coords": {} }, { "index": 4168, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4169, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4170, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 8.0, "b": 6.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4170.png", "answer": "16", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/64 - y^2/36 = 1);P: Point;PointOnCurve(P, G);Distance(P, RightFocus(G)) = 4", "coords": {} }, { "index": 4171, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4171.png", "answer": "16", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G) = True", "coords": {} }, { "index": 4172, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.453753471374512, "b": 4.020467281341553, "orientation": "horizontal" }, "optimization": { "final_loss": 4.0108716348186135e-07, "converged": true, "iterations": 129 }, "output_path": "results/train/hyperbola/problem_4172.png", "answer": "y=pm*2*sqrt(2)*x", "fact_expr": "C: Hyperbola;b: Number;a: Number;F1: Point;F2: Point;N: Point;M: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-6, 0);Coordinate(F2) = (6, 0);Coordinate(N) = (0, 4);LeftFocus(C)=F1;RightFocus(C)=F2;PointOnCurve(M, RightPart(C));Min(Perimeter(TriangleOf(M, N, F1))) = 4 + 4*sqrt(13)", "coords": { "F1": [ -6.0, 0.0 ], "F2": [ 6.0, 0.0 ], "N": [ 0.0, 4.0 ] } }, { "index": 4173, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4173.png", "answer": "4", "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;Expression(C) = (y^2 = 2*x);Expression(l) = (y = x - 1/2);Intersection(C, l) = {A, B}", "coords": {} }, { "index": 4174, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4174.png", "answer": "9", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;A: Point;B: Point;PointOnCurve(A, C);PointOnCurve(B, C);Negation(A=B);PointOnCurve(F, LineOf(A, B))", "coords": {} }, { "index": 4175, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4175.png", "answer": "4", "fact_expr": "G: Hyperbola;H: Parabola;p: Number;Expression(G) = (x^2/4 - y^2 = 1);Expression(H) = (y^2 = 2*(p*x));Focus(H) = RightVertex(G)", "coords": {} }, { "index": 4176, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4176.png", "answer": "y^2=4*sqrt(2)*x", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(C) = F;L: Line;Expression(L) = (y = 4);Intersection(L,C) = P;P: Point;Intersection(L,yAxis) = Q;Q: Point;Abs(LineSegmentOf(P, F)) = (3/2)*Abs(LineSegmentOf(P, Q))", "coords": {} }, { "index": 4177, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4178, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4178.png", "answer": "3/4", "fact_expr": "l: Line;C: Parabola;A: Point;M: Point;B: Point;Expression(C) = (x^2 = 4*y);Coordinate(M) = (3/2, -1);PointOnCurve(Focus(C), l);Intersection(l, C) = {A, B};IsTangent(LineSegmentOf(A, M), C)", "coords": { "M": [ 1.5, -1.0 ] } }, { "index": 4179, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4179.png", "answer": "1/3", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;Focus(C) = {F1, F2};L1: Line;L2: Line;Asymptote(C) = {L1, L2};IsPerpendicular(L1, L2);P: Point;PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) = 3*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 4180, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4181, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4181.png", "answer": "(0, +\\infty)", "fact_expr": "G: Hyperbola;O: Point;A: Point;P: Point;Expression(G) = (x^2 - y^2 = 1);Coordinate(O) = (0, 0);Coordinate(A) = (1, 1);PointOnCurve(P,RightPart(G))", "coords": { "O": [ 0.0, 0.0 ], "A": [ 1.0, 1.0 ] } }, { "index": 4182, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4182.png", "answer": "8/3", "fact_expr": "C: Parabola;F: Point;P: Point;Q: Point;l: Line;Expression(C) = (y^2 = 8*x);Focus(C) = F;Directrix(C) = l;PointOnCurve(P, l);OneOf(Intersection(LineOf(P,F), C))=Q;VectorOf(F, P) = 3*VectorOf(F, Q)", "coords": {} }, { "index": 4183, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4183.png", "answer": "{0}+[1,+oo)", "fact_expr": "G: Parabola;H: Line;k: Number;Expression(G) = (y^2 = 4*x);Expression(H) = (y = k*x + 1);NumIntersection(H, G) <= 1", "coords": {} }, { "index": 4184, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4185, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4185.png", "answer": "x^2=16*y", "fact_expr": "C1: Hyperbola;Expression(C1) = (x^2 - y^2/3 = 1);C2: Parabola;Expression(C2) = (x^2 = 2*p*y) ;p: Number;p>0;Distance(Focus(C2),Asymptote(C1)) = 2", "coords": {} }, { "index": 4186, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4186.png", "answer": "4", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;FocalLength(G) = 4*sqrt(2);l1: Line;l2: Line;Asymptote(G) = {l1, l2};IsPerpendicular(l1, l2) = True", "coords": {} }, { "index": 4187, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4187.png", "answer": "pi/3", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 - y^2/16 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);Abs(LineSegmentOf(P, F1))*Abs(LineSegmentOf(P, F2)) = 64", "coords": {} }, { "index": 4188, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4188.png", "answer": "2", "fact_expr": "C: Hyperbola;b: Number;a: Real;G: Circle;O: Origin;F: Point;A: Point;e: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(C) = e;e = sqrt(5)/2;RightFocus(C) = F;IsDiameter(LineSegmentOf(O, F), G);Intersection(G, OneOf(Asymptote(C))) = {O, A};Area(TriangleOf(A, O, F)) = 1", "coords": {} }, { "index": 4189, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4189.png", "answer": "3*x+4*y-7=0", "fact_expr": "G: Ellipse;H: Line;B: Point;A: Point;M: Point;Expression(G) = (x^2/4 + y^2/3 = 1);Coordinate(M) = (1, 1);PointOnCurve(M, H);Intersection(H, G) = {A, B};IsChordOf(LineSegmentOf(A,B),G);MidPoint(LineSegmentOf(A,B))=M", "coords": { "M": [ 1.0, 1.0 ] } }, { "index": 4190, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4190.png", "answer": "sqrt(5)/2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Focus(G) = F;F: Point;Coordinate(F) = (1, 0);Distance(F, Asymptote(G)) = sqrt(5)/5", "coords": { "F": [ 1.0, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 4191, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4191.png", "answer": "-(b^2/a^2)", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;A: Point;B: Point;Endpoint(MinorAxis(G)) = {A, B};P: Point;PointOnCurve(P, G);Negation(P = A);Negation(P = B)", "coords": {} }, { "index": 4192, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4192.png", "answer": "5", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;PointOnCurve(F, H);H: Line;Intersection(H, G) = {A, B};A: Point;B: Point;l1: Line;l2: Line;PointOnCurve(A, l1) ;PointOnCurve(B, l2) ;Directrix(G) = l;l: Line;IsPerpendicular(l1, l) ;IsPerpendicular(l2, l) ;FootPoint(l1, l) = C;FootPoint(l2, l) = D;C: Point;D: Point;Abs(LineSegmentOf(A, F)) = 4*Abs(LineSegmentOf(B, F))", "coords": {} }, { "index": 4193, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4194, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4195, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.23606797749979, "b": 1.0, "major_axis": "x", "x_coef": 5.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4195.png", "answer": "4*sqrt(5)/5", "fact_expr": "G: Ellipse;H: Circle;M: Point;N: Point;F: Point;Expression(G) = (x^2/5 + y^2 = 1);Expression(H) = (x^2 + y^2 = 1);RightFocus(G) = F;Quadrant(M) = 1;PointOnCurve(M, G);IsPerpendicular(LineSegmentOf(M,F),xAxis);TangentPoint(LineOf(M,N),H)=N", "coords": {} }, { "index": 4196, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4197, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4198, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4198.png", "answer": "5/2", "fact_expr": "A: Point;Coordinate(A) = (0, 2);l: Line;Expression(l) = (y = x + 3/2);C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;NumIntersection(l, C) = 1;P: Point;PointOnCurve(P, C)", "coords": { "A": [ 0.0, 2.0 ] } }, { "index": 4199, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.463192105293274, "b": 2.070901870727539, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4661259228887502e-06, "converged": true, "iterations": 106 }, "output_path": "results/train/hyperbola/problem_4199.png", "answer": "2*sqrt(6)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/4 + x^2/a^2 = 1);a: Number;a>0;Eccentricity(C) = sqrt(3)", "coords": {} }, { "index": 4200, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4200.png", "answer": "8", "fact_expr": "C: Parabola;p: Number;G: Line;A: Point;B: Point;F: Point;p>0;Expression(C) = (y^2 = 2*p*x);Expression(G) = (y = x - 1);Focus(C) = F;PointOnCurve(F,G);Intersection(G,C)={A,B}", "coords": {} }, { "index": 4201, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4201.png", "answer": "{x-2*y+1=0,y=1}", "fact_expr": "G: Parabola;H: Line;P:Point;Expression(G) = (y^2 = x);Coordinate(P) = (1, 1);NumIntersection(H,G)=1;PointOnCurve(P, H)", "coords": { "P": [ 1.0, 1.0 ] } }, { "index": 4202, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4202.png", "answer": "2*sqrt(2)", "fact_expr": "l: Line;C: Parabola;k: Number;k > 0;F: Point;A: Point;B: Point;A1: Point;B1: Point;Expression(C) = (y^2 = 6*x);Slope(l) = k;Focus(C) = F;PointOnCurve(F ,l);Intersection(l, C) = {A, B};l1: Line;l2: Line;PointOnCurve(A, l1);PointOnCurve(B, l2);IsPerpendicular(l1, xAxis);IsPerpendicular(l2, xAxis);FootPoint(l1, xAxis) = A1;FootPoint(l2, xAxis) = B1;Area(TriangleOf(A, B, B1))/Area(TriangleOf(A, B, A1)) = 2", "coords": {} }, { "index": 4203, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4203.png", "answer": "2", "fact_expr": "C: Hyperbola;G:Circle;a: Number;b: Number;A: Point;M: Point;N: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftVertex(C) = A;Center(G)=A;Radius(G)=b;Intersection(OneOf(Asymptote(C)), G) = {M, N};AngleOf(M, A, N) = ApplyUnit(120, degree)", "coords": {} }, { "index": 4204, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4205, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4206, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4206.png", "answer": "4", "fact_expr": "O: Origin;C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;A: Point;PointOnCurve(A, C);Distance(A, Focus(C)) = 4;M: Point;PointOnCurve(M, Directrix(C));Min(Abs(LineSegmentOf(M, A)) + Abs(LineSegmentOf(O, M))) = 2*sqrt(13)", "coords": {} }, { "index": 4207, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4207.png", "answer": "[\\sqrt{2}-1,1)", "fact_expr": "G: Ellipse;b: Number;a: Number;c:Number;F1: Point;F2: Point;Q: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);LeftFocus(G) =F1;RightFocus(G)=F2;Abs(VectorOf(F1, Q)) = 2*a;Area(TriangleOf(F1, Q, F2)) = b^2", "coords": {} }, { "index": 4208, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4209, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4210, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4210.png", "answer": "(0, 3)", "fact_expr": "G: Parabola;Expression(G) = (y = x^2/12)", "coords": {} }, { "index": 4211, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4211.png", "answer": "(1,5/3]", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P,Asymptote(C));Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 4212, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4212.png", "answer": "[4/5, 8/5]", "fact_expr": "l1: Line;l2: Line;C: Hyperbola;Expression(C) = (x^2/4 - y^2 = 1);Asymptote(C) = {l1, l2};P: Point;PointOnCurve(P, C);Range(Distance(P, l1)) = [1/2,1]", "coords": {} }, { "index": 4213, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5002657771110535, "direction": "right" }, "optimization": { "final_loss": 8.410525879298802e-06, "converged": true, "iterations": 102 }, "output_path": "results/train/parabola/problem_4213.png", "answer": "5/2", "fact_expr": "G: Parabola;p: Number;A: Point;Expression(G) = (y^2 = 2*p*x);Coordinate(A) = (2, 2);PointOnCurve(A, G)", "coords": { "A": [ 2.0, 2.0 ] } }, { "index": 4214, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4214.png", "answer": "8", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 4);F1: Point;LeftFocus(G) = F1;P1: Point;P2: Point;PointOnCurve(P1, RightPart(G));PointOnCurve(P2, RightPart(G))", "coords": {} }, { "index": 4215, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4215.png", "answer": "-4", "fact_expr": "G: Parabola;P: Point;Q: Point;A: Point;l1:Line;l2:Line;Expression(G) = (x^2 = 2*y);PointOnCurve(P,G);PointOnCurve(Q,G);XCoordinate(P)=4;XCoordinate(Q)=-2;TangentOfPoint(P,G)=l1;TangentOfPoint(Q,G)=l2;Intersection(l1,l2)=A", "coords": {} }, { "index": 4216, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4216.png", "answer": "7", "fact_expr": "G: Ellipse;H1: Circle;H2: Circle;P: Point;M: Point;N: Point;Expression(G) = (x^2/25 + y^2/16 = 1);Expression(H1) = (y^2 + (x + 3)^2 = 1);Expression(H2) = (y^2 + (x - 3)^2 = 4);PointOnCurve(P, G);PointOnCurve(M, H1);PointOnCurve(N, H2)", "coords": {} }, { "index": 4217, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4217.png", "answer": "-1/3", "fact_expr": "G: Ellipse;H: Line;a: Real;Expression(G) = (x^2/25 + y^2/16 = 1);Expression(H) = (a*x - y + 1 = 0);PointOnCurve(RightFocus(G),H)", "coords": {} }, { "index": 4218, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4218.png", "answer": "1/2", "fact_expr": "C: Ellipse;O: Origin;P: Point;F: Point;Expression(C) = (x^2/4 + y^2 = 1);RightFocus(C) = F;PointOnCurve(P, C);Abs(LineSegmentOf(O, P)) = Abs(LineSegmentOf(O, F))", "coords": {} }, { "index": 4219, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4219.png", "answer": "2", "fact_expr": "C: Parabola;G: Curve;k: Number;P: Point;F: Point;Expression(C) = (y = x^2/4);k>0;Expression(G) = (y = k/x);Focus(C) = F;Intersection(G, C) = P;IsPerpendicular(LineSegmentOf(P, F), yAxis)", "coords": {} }, { "index": 4220, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4220.png", "answer": "sqrt(5)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);M: Point;PointOnCurve(M, G) = True;Distance(M, Focus(G)) = 2", "coords": {} }, { "index": 4221, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4221.png", "answer": "4/3", "fact_expr": "G: Parabola;P: Point;A: Point;F: Point;O: Origin;l: Line;Expression(G) = (x^2 = 4*y);Focus(G) = F;Directrix(G) = l;PointOnCurve(P, G);PointOnCurve(P, LineSegmentOf(P,A));IsPerpendicular(LineSegmentOf(P,A),l);FootPoint(LineSegmentOf(P,A),l)=A;AngleOf(A, F, O) = ApplyUnit(30, degree)", "coords": {} }, { "index": 4222, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4222.png", "answer": "2*x+1=0", "fact_expr": "G: Parabola;Expression(G) = (x^2 = y)", "coords": {} }, { "index": 4223, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4223.png", "answer": "-1/2", "fact_expr": "G: Hyperbola;P: Point;Expression(G) = (x^2 - y^2 = 1);Coordinate(P) = (a, b);PointOnCurve(P, LeftPart(G));L:Line;OneOf(Asymptote(G))=L;Expression(L)=(y = x);Distance(P, L) = sqrt(2);a:Number;b:Number", "coords": {} }, { "index": 4224, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4224.png", "answer": "8", "fact_expr": "A: Point;B: Point;l: Line;G: Parabola;Expression(G) = (y^2 = 4*x);PointOnCurve(Focus(G), l);Intersection(l, G) = {A,B};XCoordinate(MidPoint(LineSegmentOf(A, B))) = 3", "coords": {} }, { "index": 4225, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4225.png", "answer": "6", "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/4 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};H: Line;PointOnCurve(F1, H);Intersection(H, G) = {A, B};A: Point;B: Point;Abs(LineSegmentOf(A, B)) + Abs(LineSegmentOf(F2, A)) = 10", "coords": {} }, { "index": 4226, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.463192105293274, "b": 2.070901870727539, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4661259228887502e-06, "converged": true, "iterations": 106 }, "output_path": "results/train/hyperbola/problem_4226.png", "answer": "y=pm*sqrt(2)*x", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/2 + x^2/a^2 = 1);a: Number;Eccentricity(G) = sqrt(3)", "coords": {} }, { "index": 4227, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4227.png", "answer": "sqrt(2)/2", "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/m^2 = 1);m: Number;m > 0;H: Line;Expression(H) = (y = x*sqrt(2)/2);OneOf(Intersection(H, G)) = M;M: Point;Projection(M, xAxis) = F;F: Point;RightFocus(G) = F", "coords": {} }, { "index": 4228, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4228.png", "answer": "x^2/6 + y^2/3 = 1", "fact_expr": "C: Ellipse;b: Number;a: Number;F1: Point;E: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;FocalLength(C) = 2*sqrt(3);PointOnCurve(E, C);IsPerpendicular(LineSegmentOf(E, F1), LineSegmentOf(E, F2));Slope(LineOf(E,F1))=b/c;c:Number;HalfFocalLength(C)=c", "coords": {} }, { "index": 4229, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 3.0, -0.0 ], "radius": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_4229.png", "answer": "14", "fact_expr": "G: Parabola;p: Number;H: Circle;p>0;Expression(G) = (y^2 = -2*p*x);Expression(H) = (-6*x + x^2 + y^2 - 7 = 0);IsTangent(Directrix(G),H)", "coords": {} }, { "index": 4230, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4230.png", "answer": "{(x^2=2*y),(x^2=4*y)}", "fact_expr": "M: Point;Coordinate(M) = (2, -2*p);p: Number;G: Parabola;Expression(G) = (x^2 = 2*p*y);p>0;l1: Line;l2: Line;TangentOfPoint(M, G) = {l1, l2};TangentPoint(l1, G) = A;TangentPoint(l2, G) = B;B: Point;A: Point;YCoordinate(MidPoint(LineSegmentOf(A, B))) = 6", "coords": {} }, { "index": 4231, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4231.png", "answer": "(x-5)^2+y^2=9", "fact_expr": "G: Hyperbola;H: Parabola;C:Circle;Expression(G) = (x^2/16 - y^2/9 = 1);Expression(H) = (y^2 = 20*x);Center(C)=Focus(H);IsTangent(Asymptote(G),C)", "coords": {} }, { "index": 4232, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4232.png", "answer": "pm*(5/2)", "fact_expr": "G: Hyperbola;H: Circle;m: Number;Expression(G) = (x^2/16 - y^2/9 = 1);Expression(H) = (x^2 + (-m + y)^2 = 4);IsTangent(Asymptote(G),H)", "coords": {} }, { "index": 4233, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4234, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4235, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4235.png", "answer": "4", "fact_expr": "C: Parabola;p: Number;M: Point;F: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(MidPoint(LineSegmentOf(M,F))) = (2, 2);PointOnCurve(M, C);Focus(C) = F", "coords": {} }, { "index": 4236, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.2352789464429823, "b": 2.2352789464429823, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4312591569844774e-06, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_4236.png", "answer": "x^2/9-y^2=1", "fact_expr": "G: Hyperbola;F1: Point;F2: Point;M: Point;Coordinate(F1) = (-sqrt(10), 0);Coordinate(F2) = (sqrt(10), 0);Focus(G) = {F1,F2};PointOnCurve(M, G);DotProduct(VectorOf(M, F1),VectorOf(M,F2))=0;Abs(VectorOf(M, F1))*Abs(VectorOf(M, F2)) = 2", "coords": { "F1": [ -3.1622776601683795, 0.0 ], "F2": [ 3.1622776601683795, 0.0 ] } }, { "index": 4237, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4237.png", "answer": "36", "fact_expr": "l: Line;G: Parabola;Expression(G) = (y^2 = 4*x);Intersection(l, G) = {A, B};A: Point;B: Point;O: Origin;Slope(LineOf(O,A)) * Slope(LineOf(O,B)) = -1;C: Circle;Center(C) = MidPoint(LineSegmentOf(A,B));Radius(C) = sqrt(2);Intersection(l,C) = {P,Q};P: Point;Q: Point", "coords": {} }, { "index": 4238, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4238.png", "answer": "2*sqrt(2)", "fact_expr": "G: Hyperbola;H: Parabola;p: Number;Expression(G) = (x^2 - y^2 = 1);p>0;Expression(H) = (y^2 = 2*p*x);PointOnCurve(OneOf(Focus(G)), Directrix(H))", "coords": {} }, { "index": 4239, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4239.png", "answer": "7/2", "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (y^2 = 2*x);Coordinate(A) = (3, 2);PointOnCurve(P, G);Focus(G) = F", "coords": { "A": [ 3.0, 2.0 ] } }, { "index": 4240, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4241, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4241.png", "answer": "11", "fact_expr": "C: Ellipse;H: Line;A: Point;B: Point;F: Point;G: Point;Expression(C) = (x^2/16 + y^2/9 = 1);Focus(C) = {F, G};PointOnCurve(G, H);Intersection(H, C) = {A, B};Abs(LineSegmentOf(A, B)) = 5", "coords": {} }, { "index": 4242, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4243, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4243.png", "answer": "x^2/16-y^2/9=1;3", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);FocalLength(G)=10;Slope(OneOf(Asymptote(G)))=3/4", "coords": {} }, { "index": 4244, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 6.0, "b": 8.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4244.png", "answer": "27", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/36 - y^2/64 = 1);PointOnCurve(P, G);Focus(G) = {F1, F2};Distance(P, F1) = 15", "coords": {} }, { "index": 4245, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 5.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4245.png", "answer": "{22,2}", "fact_expr": "G: Hyperbola;F1: Point;F2: Point;P:Point;Expression(G) = (x^2/25 - y^2/9 = 1);Focus(G)={F1,F2};PointOnCurve(P,G);Distance(P,F1)=12", "coords": {} }, { "index": 4246, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4247, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4247.png", "answer": "x^2/5 - y^2/20 = 1", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;l: Line;Expression(l) = (y = 2*x + 10);IsParallel(OneOf(Asymptote(G)),l) = True;PointOnCurve(OneOf(Focus(G)),l) = True", "coords": {} }, { "index": 4248, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4248.png", "answer": "sqrt(2)/4", "fact_expr": "l: Line;G: Ellipse;B: Point;A: Point;O: Origin;M: Point;N: Point;Expression(G) = (x^2/2 + y^2 = 1);Negation(PointOnCurve(O, l));Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A,B)) = M;Intersection(PerpendicularBisector(LineSegmentOf(A,B)), xAxis) = N", "coords": {} }, { "index": 4249, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4249.png", "answer": "sqrt(7)/4", "fact_expr": "l: Line;E: Ellipse;b: Number;a: Number;P: Point;F: Point;Q: Point;O:Origin;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(E) = F;PointOnCurve(O,l);Intersection(l, E) = {P, Q};Abs(LineSegmentOf(P, F)) = 3*Abs(LineSegmentOf(Q, F));AngleOf(P, F, Q) = ApplyUnit(120, degree)", "coords": {} }, { "index": 4250, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4250.png", "answer": "(2*sqrt(3))/3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;l: Line;Directrix(G) = l;H: Line;PointOnCurve(F,H);A: Point;B: Point;Intersection(H, G) = {A, B};L1: Line;L2: Line;C: Point;D: Point;PointOnCurve(A, L1);PointOnCurve(B, L2);IsPerpendicular(L1, l);IsPerpendicular(L2, l);FootPoint(L1, l) = C;FootPoint(L2, l) = D;Abs(LineSegmentOf(A, F)) = 2*Abs(LineSegmentOf(B, F));Area(TriangleOf(C, D, F)) = sqrt(2)", "coords": {} }, { "index": 4251, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4251.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(Asymptote(G)) = (y = pm*x)", "coords": {} }, { "index": 4252, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4253, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4253.png", "answer": "pi/2", "fact_expr": "C: Parabola;p: Number;P: Point;M: Point;Q:Point;L1: Line;L2: Line;p>0;Expression(C) = (y^2 = 2*(p*x));Intersection(Directrix(C), xAxis) = M;TangentOfPoint(M, C) = {L1, L2};TangentPoint(L1,C)=P;TangentPoint(L2,C)=Q", "coords": {} }, { "index": 4254, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4255, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4255.png", "answer": "-12", "fact_expr": "G: Parabola;H: Line;O: Origin;A: Point;B: Point;Expression(G) = (y^2 = 8*x);PointOnCurve(Focus(G), H);Intersection(G, H) = {A, B};Slope(H) = 1", "coords": {} }, { "index": 4256, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4256.png", "answer": "sqrt(6)", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;M: Point;N: Point;a>0;b>0;e:Number;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(C) = e;Expression(l)=(y=x);Intersection(l, C) = {M, N};Abs(LineSegmentOf(M, N)) = sqrt(2)*b", "coords": {} }, { "index": 4257, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4257.png", "answer": "x^2 - y^2/4 = 1", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Ellipse;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2/9 + y^2/4 = 1);Focus(C) = Focus(G);Expression(Asymptote(C)) = (y = pm*(2*x))", "coords": {} }, { "index": 4258, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4258.png", "answer": "x^2 - y^2/(1/2) = 1", "fact_expr": "P: Point;Coordinate(P) = (3, 2);PointOnCurve(P, Z);G: Hyperbola;Expression(G) = (x^2/4 - y^2/2 = 1);Z: Hyperbola;Expression(Asymptote(Z))=Expression(Asymptote(G))", "coords": { "P": [ 3.0, 2.0 ] } }, { "index": 4259, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4259.png", "answer": "2*sqrt(2)-6", "fact_expr": "C: Ellipse;Expression(C) = (x^2/9 + y^2/5 = 1);B: Point;Coordinate(B) = (-2, 0);A: Point;Coordinate(A) = (4, 2);M: Point;PointOnCurve(M, C)", "coords": { "B": [ -2.0, 0.0 ], "A": [ 4.0, 2.0 ] } }, { "index": 4260, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4260.png", "answer": "9", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2))", "coords": {} }, { "index": 4261, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "y", "x_coef": 9.0, "y_coef": 25.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4261.png", "answer": "y^2/4 - x^2/12 = 1", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/9 + y^2/25 = 1);FocalLength(G) = FocalLength(H);Eccentricity(G) + Eccentricity(H) = 14/5", "coords": {} }, { "index": 4262, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4262.png", "answer": "pm*8", "fact_expr": "G: Parabola;Expression(G) = (y^2 = m*x);m: Number;H: Ellipse;Expression(H) = (x^2/9 + y^2/5 = 1);Focus(G) = OneOf(Focus(H))", "coords": {} }, { "index": 4263, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/train/hyperbola/problem_4263.png", "answer": "x^2 - y^2 = 1", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>b;b>0;Expression(Asymptote(G)) = (y = pm*x);H: Point;Coordinate(H) = (sqrt(2), 1);PointOnCurve(H,G) = True", "coords": { "H": [ 1.4142135623730951, 1.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 4264, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4264.png", "answer": "-4", "fact_expr": "l: Line;C: Parabola;O: Origin;A: Point;B: Point;M: Point;l1: Line;l2: Line;Expression(C) = (y = x^2/4);Expression(l) = (y=-2);PointOnCurve(M, l);TangentOfPoint(M, C) = {l1, l2};TangentPoint(l1,C)=A;TangentPoint(l2,C)=B", "coords": {} }, { "index": 4265, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4266, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4266.png", "answer": "3", "fact_expr": "C: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Focus(C)={F1,F2};PointOnCurve(P, C);IsPerpendicular(VectorOf(P, F1),VectorOf(P,F2));Area(TriangleOf(P, F1, F2)) = 9", "coords": {} }, { "index": 4267, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4267.png", "answer": "5*sqrt(3)/3", "fact_expr": "E: Hyperbola;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(E) = F1;RightFocus(E) = F2;Abs(LineSegmentOf(F1, F2)) = 10;P: Point;PointOnCurve(P, RightPart(E));A: Point;Intersection(LineSegmentOf(P, F1), yAxis) = A;Q: Point;TangentPoint(LineSegmentOf(A, F2), InscribedCircle(TriangleOf(P, A, F2))) = Q;Abs(LineSegmentOf(A, Q)) = sqrt(3)", "coords": {} }, { "index": 4268, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4268.png", "answer": "8/3", "fact_expr": "C: Parabola;F: Point;A: Point;B: Point;l: Line;H: Line;Expression(C) = (y^2 = 4*x);Focus(C) = F;Directrix(C) = l;PointOnCurve(F, H);Intersection(H, l) = A;OneOf(Intersection(H, C)) = B;VectorOf(F, A) = 3*VectorOf(F, B)", "coords": {} }, { "index": 4269, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4269.png", "answer": "1", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1)", "coords": {} }, { "index": 4270, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4270.png", "answer": "x^2/2 - y^2/8 = 1", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/4 = 1);H: Hyperbola;Asymptote(G) = Asymptote(H)", "coords": {} }, { "index": 4271, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4271.png", "answer": "[3/4,3/2]", "fact_expr": "C:Hyperbola;P: Point;A2: Point;A1: Point;Expression(C) = (x^2/2 - y^2/3 = 1);LeftVertex(C)=A1;RightVertex(C)=A2;PointOnCurve(P,C);Range(Slope(LineOf(P,A2)))=[1,2]", "coords": {} }, { "index": 4272, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4272.png", "answer": "5*sqrt(3)/6", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/9 + y^2/5 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);AngleOf(F1, P, F2) = pi/3", "coords": {} }, { "index": 4273, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4274, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4274.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;a: Number;H: Circle;a>0;Expression(G) = (-y^2/4 + x^2/a^2 = 1);Expression(H) = (x^2 + (y - 2)^2 = 1);IsTangent(Asymptote(G),H)", "coords": {} }, { "index": 4275, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4275.png", "answer": "1/4", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);A: Point;RightVertex(G) = A;B: Point;UpperVertex(G) = B;l: Line;C: Point;D: Point;Intersection(l, G) = {C, D};IsParallel(l, LineOf(A, B));k1: Number;k2: Number;Slope(LineOf(A, C)) = k1;Slope(LineOf(B, D)) = k2", "coords": {} }, { "index": 4276, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4276.png", "answer": "sqrt(5)/2", "fact_expr": "G: Hyperbola;a: Number;Expression(G) = (-y^2/4 + x^2/a = 1);Expression(OneOf(Asymptote(G))) = (x - 2*y = 0)", "coords": {} }, { "index": 4277, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4277.png", "answer": "8", "fact_expr": "H: Ellipse;e: Number;Eccentricity(H) = e;e = 1/2;G: Hyperbola;Expression(G) = (x^2/3 - y^2 = 1);Focus(H) = Focus(G);P: Point;PointOnCurve(P, H);F1: Point;F2: Point;Focus(H) = {F1, F2}", "coords": {} }, { "index": 4278, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4278.png", "answer": "2*sqrt(3)/3", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;F: Point;A: Point;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (c, 0);RightFocus(C)=F;Center(G)=F;Radius(G)=a;Intersection(G,OneOf(Asymptote(C)))={A, B};Abs(LineSegmentOf(A, B)) = sqrt(2)*c;c:Number", "coords": {} }, { "index": 4279, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4279.png", "answer": "4*x - y -10 =0", "fact_expr": "l: Line;G: Parabola;Expression(G) = (y^2 = 16*x);A: Point;B: Point;Intersection(l, G) = {A, B};IsChordOf(LineSegmentOf(A, B), G);P: Point;Coordinate(P) = (3, 2);MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ 3.0, 2.0 ] } }, { "index": 4280, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 2.2628633975982666, "b": 1.8162860737044515, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 2.6997004191253788e-06, "converged": true, "iterations": 101 }, "output_path": "results/train/ellipse/problem_4280.png", "answer": "sqrt(5) + 2", "fact_expr": "C: Ellipse;a: Number;b: Number;A: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (1, 2);PointOnCurve(A, C)", "coords": { "A": [ 1.0, 2.0 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 4281, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4281.png", "answer": "x^2=6*y", "fact_expr": "C: Parabola;Expression(C) = (x^2 = 2*p*y);p: Number;p>0;F: Point;Focus(C) = F;l: Line;Directrix(C) = l;P: Point;PointOnCurve(P,C) = True;E: Point;l1: Line;PointOnCurve(P,l1) = True;IsPerpendicular(l,l1) = True;Intersection(l,l1) = E;AngleOf(P, F, E) = ApplyUnit(60, degree);Abs(LineSegmentOf(P, F)) = 6", "coords": {} }, { "index": 4282, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.4641016151377544, "b": 3.0, "major_axis": "x", "x_coef": 12.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4282.png", "answer": "sqrt(13)", "fact_expr": "G: Ellipse;A: Point;P: Point;Expression(G) = (x^2/12 + y^2/9 = 1);Coordinate(A) = (0, 1/2);PointOnCurve(P, G)", "coords": { "A": [ 0.0, 0.5 ] } }, { "index": 4283, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142164083257607, "b": 1.4142164083257607, "orientation": "horizontal" }, "optimization": { "final_loss": 2.216064987687002e-06, "converged": true, "iterations": 95 }, "output_path": "results/train/hyperbola/problem_4283.png", "answer": "6", "fact_expr": "F1: Point;F2: Point;Coordinate(F1) = (-2, 0);Coordinate(F2) = (2, 0);H: Ellipse;Expression(H) = (x^2 + 2*y^2 = 8);G: Hyperbola;Expression(G) = (x^2 - y^2 = 2);P: Point;OneOf(Intersection(H, G)) = P", "coords": { "F1": [ -2.0, 0.0 ], "F2": [ 2.0, 0.0 ] } }, { "index": 4284, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4284.png", "answer": "sqrt(3)-1", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;O: Origin;l: Line;PointOnCurve(O, l);Inclination(l) = ApplyUnit(60, degree);M: Point;OneOf(Intersection(l, C)) = M;Abs(VectorOf(M, F1) + VectorOf(M, F2)) = Abs(VectorOf(M, F1) - VectorOf(M, F2))", "coords": {} }, { "index": 4285, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4285.png", "answer": "{11, 29}", "fact_expr": "G: Ellipse;k: Number;Expression(G) = (x^2/20 + y^2/k = 1);FocalLength(G) = 6", "coords": {} }, { "index": 4286, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4286.png", "answer": "2*sqrt(2)", "fact_expr": "G: Parabola;H: Line;B: Point;O: Origin;F: Point;A: Point;Expression(G) = (y^2 = 8*x);Focus(G)=F;PointOnCurve(F, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, F)) = 6", "coords": {} }, { "index": 4287, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4288, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4289, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4290, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4291, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4291.png", "answer": "x^2/16+y^2/9=1", "fact_expr": "H: Circle;Expression(H) = (x^2 + y^2 = 1);L1: Line;IsTangent(L1, H) = True;G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);A: Point;B: Point;Intersection(L1, G) = {A, B};Intersection(TangentOnPoint(A, G), TangentOnPoint(B, G)) = P;P: Point", "coords": {} }, { "index": 4292, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4292.png", "answer": "2*sqrt(5)/5", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;H: Parabola;Expression(H) = (y^2 = 4*x);Length(InterceptChord(Directrix(H), G)) = b", "coords": {} }, { "index": 4293, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4293.png", "answer": "x^2=-(20/3)*y", "fact_expr": "C: Parabola;G: Line;O:Origin;Expression(G) = (2*x - 3*y - 5 = 0);Vertex(C)=O;SymmetryAxis(C)=yAxis;PointOnCurve(Focus(C),G)", "coords": {} }, { "index": 4294, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "major_axis": "x", "x_coef": 3.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4294.png", "answer": "4*sqrt(3)", "fact_expr": "G: Ellipse;B: Point;C: Point;A: Point;Expression(G) = (x^2/3 + y^2 = 1);PointOnCurve(F2, LineSegmentOf(B, C));PointOnCurve(B, G);PointOnCurve(C, G);F1: Point;F2: Point;OneOf(Focus(G)) = F1;OneOf(Focus(G)) = F2;Negation(F1 = F2);A = F1", "coords": {} }, { "index": 4295, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4296, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4296.png", "answer": "sqrt(2)/2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;H: Line;Expression(H) = (y = x);O: Origin;A: Point;Intersection(H, G) = {O, A};B: Point;Intersection(H, Directrix(G)) = B;C: Point;OneOf(Intersection(LineOf(A,F),G))=C", "coords": {} }, { "index": 4297, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4298, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4298.png", "answer": "10", "fact_expr": "C: Parabola;G: Line;A: Point;F: Point;P: Point;l:Line;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(P,C);Directrix(C)=l;PointOnCurve(P,G);IsPerpendicular(l,G);FootPoint(l,G)=A;Slope(LineOf(A,F))=-2", "coords": {} }, { "index": 4299, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4300, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4300.png", "answer": "1", "fact_expr": "G: Parabola;H: Line;A: Point;F: Point;B: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B}", "coords": {} }, { "index": 4301, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_4301.png", "answer": "(2,4]", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;O: Origin;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = 2;PointOnCurve(P, G);LeftFocus(G)=F1;RightFocus(G)=F2", "coords": {} }, { "index": 4302, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4303, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4304, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4305, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4305.png", "answer": "x=pm*(\\sqrt{5}/5)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/4 = 1)", "coords": {} }, { "index": 4306, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4306.png", "answer": "{(-oo,-sqrt(17))+(sqrt(17),+oo)}", "fact_expr": "G: Ellipse;H: Line;b: Number;Expression(G) = (x^2/4 + y^2 = 1);Expression(H) = (y = b + 2*x);NumIntersection(H,G)=0", "coords": {} }, { "index": 4307, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4307.png", "answer": "x^2/2 - y^2/2 = 1", "fact_expr": "C: Hyperbola;G: Parabola;O: Origin;Expression(G) = (y^2 = 8*x);Center(C) = O;SymmetryAxis(C)=axis;OneOf(Focus(C) )= Focus(G);Expression(OneOf(Asymptote(C))) = (x + y = 0)", "coords": {} }, { "index": 4308, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4308.png", "answer": "3", "fact_expr": "l: Line;C: Parabola;B: Point;A: Point;Expression(C) = (y^2 = 6*x);PointOnCurve(Focus(C), l);Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, B)) = 9", "coords": {} }, { "index": 4309, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4309.png", "answer": "sqrt(2) - 1", "fact_expr": "G: Ellipse;a: Number;b: Number;H: Circle;I: Line;M: Point;N: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(RightFocus(G), I);IsPerpendicular(I, xAxis);Intersection(I, G) = {M, N};IsDiameter(LineSegmentOf(M, N), H);PointOnCurve(LeftFocus(G), H)", "coords": {} }, { "index": 4310, "success": false, "conic_type": "parabola", "error": "validation: parabola_point_off_curve", "params": { "p": 0.9927864074707031, "direction": "right" }, "optimization": { "final_loss": 0.00015326803259085864, "converged": true, "iterations": 44 }, "output_path": "results/train/parabola/problem_4310.png", "answer": "5", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;P: Point;Coordinate(P) = (4, 4);PointOnCurve(P, G)", "coords": { "P": [ 4.0, 4.0 ] }, "validation_reasons": [ "parabola_point_off_curve" ] }, { "index": 4311, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4312, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_4312.png", "answer": "x^2 - y^2 / 3 = 1", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/n + x^2/m = 1);m: Number;n: Number;Eccentricity(G) = 2;H: Parabola;Expression(H) = (y^2 = 8*x);OneOf(Focus(G)) = Focus(H)", "coords": {} }, { "index": 4313, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.3166247903554, "b": 2.6457513110645907, "major_axis": "x", "x_coef": 11.0, "y_coef": 7.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4313.png", "answer": "2*sqrt(11)/11", "fact_expr": "G: Ellipse;Expression(G) = (x^2/11 + y^2/7 = 1)", "coords": {} }, { "index": 4314, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4315, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4315.png", "answer": "16", "fact_expr": "G: Parabola;Expression(G) = (y^2 = a*x);a: Number;a>0;V: Point;PointOnCurve(V, G) = True;XCoordinate(V) = 6;Distance(V, Focus(G)) = 10", "coords": {} }, { "index": 4316, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4317, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4317.png", "answer": "(1,0)", "fact_expr": "C: Parabola;p: Number;P: Point;F: Point;p>0;Expression(C) = (y^2 = 2*p*x);Focus(C)=F;PointOnCurve(P, C);Abs(LineSegmentOf(P, F)) = 4;Distance(P, yAxis) = 3", "coords": {} }, { "index": 4318, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.8284271247461903, "major_axis": "x", "x_coef": 9.0, "y_coef": 8.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4318.png", "answer": "{6}", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/8 = 1);F: Point;RightFocus(G) = F;H: Circle;Expression(H) = (x^2 + y^2 = 8);M: Point;Quadrant(M) = 1;Z: Line;TangentOnPoint(M, H) = Z;P: Point;Q: Point;Intersection(Z, G) = {P, Q}", "coords": {} }, { "index": 4319, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4319.png", "answer": "sqrt(2)-1", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = 2*x);HalfFocalLength(G) = c;XCoordinate(OneOf(Intersection(H,G)))=c;c:Number", "coords": {} }, { "index": 4320, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4320.png", "answer": "2", "fact_expr": "l1: Line;Expression(l1) = (3*x - 4*y - 9 = 0);l2: Line;Expression(l2) = (y = 1/4);G: Parabola;Expression(G) = (y = x^2);P: Point;PointOnCurve(P,G) = True", "coords": {} }, { "index": 4321, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4321.png", "answer": "(0,2*sqrt(5)/5)", "fact_expr": "l: Line;k:Number;Expression(l) = (y = k*x + 2);G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;B: Point;UpperVertex(G) = B;F: Point;LeftFocus(G) = F;PointOnCurve(B, l);PointOnCurve(F, l);H: Circle;Expression(H) = (x^2 + y^2 = 4);L: Number;Length(InterceptChord(l, H)) = L;L >= 4*sqrt(5)/5;e: Number;Eccentricity(G) = e", "coords": {} }, { "index": 4322, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4322.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Parabola;H: Line;b: Number;Expression(G) = (y^2 = 4*x);Expression(H) = (y = b + 2*x);Length(InterceptChord(H,G))=3*sqrt(5)", "coords": {} }, { "index": 4323, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4323.png", "answer": "5/2", "fact_expr": "A: Point;B: Point;G: Parabola;F: Point;Expression(G) = (y^2 = 2*x);Focus(G) = F;PointOnCurve(A, G);PointOnCurve(B, G);Abs(LineSegmentOf(A, F)) + Abs(LineSegmentOf(B, F)) = 6", "coords": {} }, { "index": 4324, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4324.png", "answer": "y = pm*(sqrt(2)/2)*x", "fact_expr": "G: Hyperbola;b: Number;a: Number;M: Point;F1: Point;F2: Point;l: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, l);IsPerpendicular(l, OneOf(Asymptote(G)));FootPoint(l, OneOf(Asymptote(G))) = M;Abs(VectorOf(M, F1)) = 3*Abs(VectorOf(M, F2))", "coords": {} }, { "index": 4325, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -1.0, 0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_4325.png", "answer": "x^2/4 + y^2/3 = 1", "fact_expr": "C: Circle;H:Circle;G:Circle;C1:Point;Expression(H)=((x+1)^2+y^2=1);Expression(G)=((x-1)^2+y^2=25);IsIntersect(C,H);IsTangent(C,G);Center(C)=C1", "coords": {} }, { "index": 4326, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.3300262689590454, "b": 2.30381441116333, "orientation": "horizontal" }, "optimization": { "final_loss": 6.658357051492203e-06, "converged": true, "iterations": 122 }, "output_path": "results/train/hyperbola/problem_4326.png", "answer": "x^2 - y^2 / 3 = 1", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 8*x);Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x);PointOnCurve(OneOf(Focus(G)), Directrix(H))", "coords": {} }, { "index": 4327, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "major_axis": "x", "x_coef": 3.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4327.png", "answer": "y=pm*x", "fact_expr": "C1: Ellipse;Expression(C1) = (x^2/3 + y^2 = 1);C2: Hyperbola;F1: Point;F2: Point;Focus(C1) = {F1, F2};Focus(C2) = {F1, F2};P: Point;Intersection(C1, C2) = P;O: Origin;LineSegmentOf(O, P) = LineSegmentOf(O, F1)", "coords": {} }, { "index": 4328, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4329, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4330, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4330.png", "answer": "sqrt(13)/2", "fact_expr": "G: Hyperbola;a: Number;H: Parabola;M: Point;F: Point;Expression(G) = (-y^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 4*x);OneOf(Intersection(H, G)) = M;Focus(H) = F;LineSegmentOf(M, F) = 3", "coords": {} }, { "index": 4331, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4331.png", "answer": "y=(-1/2)*x\ny=(1/2)*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1);L1:Line;L2:Line;Asymptote(G)={L1,L2}", "coords": {} }, { "index": 4332, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.4641016151377544, "major_axis": "x", "x_coef": 16.0, "y_coef": 12.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4332.png", "answer": "19", "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/12 = 1);P: Point;PointOnCurve(P, G);N: Circle;Expression(N) = (x^2 + (y - 1)^2 = 1);E: Point;F: Point;IsDiameter(LineSegmentOf(E, F), N) = True", "coords": {} }, { "index": 4333, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.9991092681884766, "direction": "right" }, "optimization": { "final_loss": 1.8838097730622394e-06, "converged": true, "iterations": 115 }, "output_path": "results/train/parabola/problem_4333.png", "answer": "-3/4", "fact_expr": "B: Point;Coordinate(B) = (8, 8);C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;PointOnCurve(B, C) = True;Intersection(TangentOnPoint(B,C),Directrix(C))=A;A: Point;F: Point;Focus(C) = F", "coords": { "B": [ 8.0, 8.0 ] } }, { "index": 4334, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4334.png", "answer": "3", "fact_expr": "G: Hyperbola;m: Real;H: Parabola;Expression(G) = (-y^2 + x^2/m = 1);Expression(H) = (y^2 = -8*x);Focus(H) = LeftFocus(G)", "coords": {} }, { "index": 4335, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.463192105293274, "b": 2.070901870727539, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4661259228887502e-06, "converged": true, "iterations": 106 }, "output_path": "results/train/hyperbola/problem_4335.png", "answer": "x^2/12 - y^2/24 = 1", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 8*x);Eccentricity(G) = sqrt(3);LeftDirectrix(G)=Directrix(H)", "coords": {} }, { "index": 4336, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.23606797749979, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 5.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4336.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "P: Point;Coordinate(P) = (2, 0);H: Line;Inclination(H) = ApplyUnit(60, degree);PointOnCurve(P, H);G: Ellipse;Expression(G) = (x^2/5 + y^2/3 = 1);A: Point;B: Point;Intersection(H, G) = {A, B}", "coords": { "P": [ 2.0, 0.0 ] } }, { "index": 4337, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4338, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4338.png", "answer": "-4/3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;P: Point;PointOnCurve(P, G) = True;A: Point;Coordinate(A) = (1, 1);WhenMin(Perimeter(TriangleOf(P,A,F))) = True", "coords": { "A": [ 1.0, 1.0 ] } }, { "index": 4339, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4339.png", "answer": "b^2/a^2", "fact_expr": "G: Hyperbola;b: Number;a: Number;A1: Point;P: Point;A2: Point;a>0;b>0;k1:Number;k2:Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, RightPart(G));Negation(P=Vertex(G));LeftVertex(G)=A1;RightVertex(G)=A2;Slope(LineOf(A1,P))=k1;Slope(LineOf(A2,P))=k2", "coords": {} }, { "index": 4340, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4340.png", "answer": "(1,2)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F2: Point;F1: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G) = True;AngleOf(P, F2, F1) = ApplyUnit(120, degree)", "coords": {} }, { "index": 4341, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4341.png", "answer": "0", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/2 - y^2/b^2 = 1);b: Number;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;Expression(OneOf(Asymptote(G))) = (y = x);P: Point;Coordinate(P) = (sqrt(3), y0);y0: Number;PointOnCurve(P, G)", "coords": {} }, { "index": 4342, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4343, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4343.png", "answer": "sqrt(2)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(C) = F;Z: Line;PointOnCurve(F, Z);IsPerpendicular(Z, OneOf(Asymptote(C)));M: Point;Intersection(Z, C) = M;N: Point;FootPoint(Z, OneOf(Asymptote(C))) = N;MidPoint(LineSegmentOf(F, N)) = M", "coords": {} }, { "index": 4344, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4344.png", "answer": "2", "fact_expr": "G: Parabola;M: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(M, G);Distance(M, Focus(G)) = 3", "coords": {} }, { "index": 4345, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4345.png", "answer": "34/3", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);M:Point;PointOnCurve(M,G);XCoordinate(M)=5", "coords": {} }, { "index": 4346, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4346.png", "answer": "4", "fact_expr": "G: Ellipse;O: Origin;N: Point;M: Point;F: Point;Expression(G) = (x^2/25 + y^2/9 = 1);PointOnCurve(M,G);OneOf(Focus(G))=F;Distance(M,F)=2;MidPoint(LineSegmentOf(M,F))=N", "coords": {} }, { "index": 4347, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4348, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 1.681220143106833, "b": 2.2412689930799665, "orientation": "horizontal" }, "optimization": { "final_loss": 4.985250663519028e-06, "converged": true, "iterations": 95 }, "output_path": "results/train/hyperbola/problem_4348.png", "answer": "x^2/36 - y^2/64 = 1", "fact_expr": "G: Hyperbola;Center(G) = O;O: Origin;F: Point;OneOf(Focus(G)) = F;Coordinate(F) = (10, 0);Expression(Asymptote(G)) = (y = pm*(4/3)*x)", "coords": { "F": [ 10.0, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 4349, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4349.png", "answer": "sqrt(3)", "fact_expr": "C: Parabola;G: Line;m: Number;k: Number;F: Point;M: Point;N: Point;Expression(C) = (y^2 = 4*x);k>0;Expression(G) = (y = k*x + m);Intersection(G,C)=M;Intersection(G, Directrix(C))=N;Focus(C) = F;2*VectorOf(F, M) = VectorOf(M, N)", "coords": {} }, { "index": 4350, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4350.png", "answer": "x^2 - y^2/3 = 1", "fact_expr": "G: Hyperbola;H: Ellipse;I: Point;Expression(H) = (x^2/9 + y^2/5 = 1);Focus(G) = Focus(H);Coordinate(I) = (sqrt(2), sqrt(3));PointOnCurve(I, G)", "coords": { "I": [ 1.4142135623730951, 1.7320508075688772 ] } }, { "index": 4351, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4352, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4352.png", "answer": "pm*sqrt(3)", "fact_expr": "l: Line;C: Parabola;A: Point;F: Point;B: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, F)) = 3*Abs(LineSegmentOf(B, F))", "coords": {} }, { "index": 4353, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4353.png", "answer": "5", "fact_expr": "l: Line;G: Parabola;M: Point;O: Origin;N: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {M, N};Area(TriangleOf(M, O, N)) = sqrt(5)", "coords": {} }, { "index": 4354, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4355, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4356, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.23606797749979, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4356.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/5 - y^2/4 = 1)", "coords": {} }, { "index": 4357, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4357.png", "answer": "3*sqrt(2)/4", "fact_expr": "G: Hyperbola;a: Number;b: Number;H: Line;P:Point;e: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = x*(b/(3*a)));Eccentricity(G) = e;Intersection(H,LeftPart(G))=P;LeftFocus(G)=F1;IsPerpendicular(LineSegmentOf(P,F1),xAxis);F1:Point", "coords": {} }, { "index": 4358, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4358.png", "answer": "x-y-3=0", "fact_expr": "G: Hyperbola;H: Line;B: Point;A: Point;P: Point;Expression(G) = (x^2 - 4*y^2 = 4);Coordinate(P) = (4, 1);Intersection(H, G) = {A, B};MidPoint(LineSegmentOf(A,B))=P", "coords": { "P": [ 4.0, 1.0 ] } }, { "index": 4359, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4360, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4360.png", "answer": "(\\sqrt{5}-1)/2", "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;F2: Point;A: Point;e: Number;c:Number;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;Abs(LineSegmentOf(F1, F2)) = 2*c;PointOnCurve(A, G);DotProduct(VectorOf(A, F1), VectorOf(F1, F2)) = 0;DotProduct(VectorOf(A, F1), VectorOf(A, F2)) = c^2;Eccentricity(G) = e", "coords": {} }, { "index": 4361, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4362, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4363, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 2.0, 0.0 ], "radius": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_4363.png", "answer": "{(x^2/9+y^2/5=1)&Negation(x=-3),(y=0)&Negation(x=-3)&Negation(x=-2)&Negation(x=2)}", "fact_expr": "P: Circle;M:Circle;N:Circle;Expression(M)=((x-2)^2+y^2=25);Expression(N)=((x+2)^2+y^2=1);IsTangent(P,M);IsTangent(P,N)", "coords": {} }, { "index": 4364, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4365, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4365.png", "answer": "sqrt(5)", "fact_expr": "C: Hyperbola;G: Circle;r: Number;Expression(C) = (-x^2/4 + y^2/5 = 1);r>0;Expression(G) = (y^2 + (x - 3)^2 = r^2);IsTangent(Asymptote(C),G)", "coords": {} }, { "index": 4366, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4367, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4367.png", "answer": "1", "fact_expr": "C: Parabola;Expression(C) = (x^2 = 4*y);P: Point;PointOnCurve(P, C);A: Point;Coordinate(A) = (0, 1);Distance(P, A) = 2", "coords": { "A": [ 0.0, 1.0 ] } }, { "index": 4368, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4368.png", "answer": "2", "fact_expr": "G: Ellipse;P: Point;Q: Point;H:Line;M: Point;S: Point;Expression(G) = (x^2/4 + y^2/3 = 1);Expression(H) = (x = 4);Coordinate(M) = (1, 0);PointOnCurve(M,LineSegmentOf(P,Q));IsChordOf(LineSegmentOf(P,Q),G);Intersection(LineOf(P,Q), H) = S", "coords": { "M": [ 1.0, 0.0 ] } }, { "index": 4369, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4369.png", "answer": "2", "fact_expr": "G: Parabola;A: Point;P: Point;Q: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (0, 2*sqrt(2));PointOnCurve(P, G);PointOnCurve(P, LineSegmentOf(P, Q));IsPerpendicular(LineSegmentOf(P, Q), yAxis);FootPoint(LineSegmentOf(P, Q), yAxis) = Q", "coords": { "A": [ 0.0, 2.8284271247461903 ] } }, { "index": 4370, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "major_axis": "x", "x_coef": 3.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4370.png", "answer": "1", "fact_expr": "G: Parabola;H: Ellipse;I: Circle;P: Point;Q: Point;O: Origin;Expression(G) = (x^2 = 2*(sqrt(6)*y));Expression(H) = (x^2/3 + y^2 = 1);PointOnCurve(P, H);PointOnCurve(Q, Directrix(G));IsDiameter(LineSegmentOf(P,Q),I);PointOnCurve(O, I)", "coords": {} }, { "index": 4371, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4371.png", "answer": "6", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/4 = 1)", "coords": {} }, { "index": 4372, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4372.png", "answer": "-3", "fact_expr": "G: Parabola;H: Line;O: Origin;A: Point;B: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(Focus(G),H);Intersection(G, H) = {A, B}", "coords": {} }, { "index": 4373, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4373.png", "answer": "sqrt(5)/3", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;M: Point;PointOnCurve(M, G);AngleOf(F1, M, F2) = ApplyUnit(90, degree);N: Point;Intersection(LineOf(M, F1), G) = N;4*Abs(LineSegmentOf(N, F2)) = 5*Abs(LineSegmentOf(M, F2))", "coords": {} }, { "index": 4374, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4374.png", "answer": "(1, 0)", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;Focus(G) = {F1, F2};e: Number;Eccentricity(G) = e;e = sqrt(2)/2;P: Point;PointOnCurve(P, G) = True;DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;Area(TriangleOf(P, F1, F2)) = 1;RightFocus(G) = F2", "coords": {} }, { "index": 4375, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4376, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4377, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4378, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4378.png", "answer": "x^2/3 - y^2/6 = 1", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;N: Point;M: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(MidPoint(LineSegmentOf(M,N))) = (3, 6);Coordinate(F) = (-3, 0);LeftFocus(G)=F;PointOnCurve(F, H);Intersection(H, G) = {M, N};F:Point", "coords": { "F": [ -3.0, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 4379, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4379.png", "answer": "3/2", "fact_expr": "G: Parabola;F: Point;P: Point;Q: Point;l: Line;Expression(G) = (y^2 = 4*x);Focus(G) = F;Directrix(G)=l;PointOnCurve(P, l);OneOf(Intersection(LineOf(P,F),G))=Q;VectorOf(F, P) = 4*VectorOf(F, Q)", "coords": {} }, { "index": 4380, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4380.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;M: Point;N: Point;H:Line;C:Circle;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(LeftFocus(G),H);IsPerpendicular(H,xAxis);Intersection(H,G) = {M, N};IsDiameter(LineSegmentOf(M,N),C);PointOnCurve(RightVertex(G),C)", "coords": {} }, { "index": 4381, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4381.png", "answer": "ApplyUnit(60,degree)", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2;Abs(LineSegmentOf(P, F1))*Abs(LineSegmentOf(P, F2)) = 4/3", "coords": {} }, { "index": 4382, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4382.png", "answer": "-8", "fact_expr": "r: Ellipse;Expression(r) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;Eccentricity(r) = sqrt(3)/2;A: Point;B: Point;C: Point;PointOnCurve(Vertex(TriangleOf(A, B, C)), r);D: Point;E: Point;M: Point;MidPoint(LineSegmentOf(A, B)) = D;MidPoint(LineSegmentOf(B, C)) = E;MidPoint(LineSegmentOf(A, C)) = M;k1: Number;k2: Number;k3: Number;Slope(OverlappingLine(LineSegmentOf(A, B))) = k1;Slope(OverlappingLine(LineSegmentOf(B, C))) = k2;Slope(OverlappingLine(LineSegmentOf(A, C))) = k3;Negation(k1=0);Negation(k2=0);Negation(k3=0);O: Origin;Slope(LineOf(O, D)) + Slope(LineOf(O, E)) + Slope(LineOf(O, M)) = 2", "coords": {} }, { "index": 4383, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_focus_mismatch", "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4383.png", "answer": "x^2/18+y^2/9=1", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;B: Point;A: Point;F: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (3, 0);RightFocus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};Coordinate(MidPoint(LineSegmentOf(A,B)))=(1,-1)", "coords": { "F": [ 3.0, 0.0 ] }, "validation_reasons": [ "ellipse_focus_mismatch" ] }, { "index": 4384, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4384.png", "answer": "4-2*sqrt(5)", "fact_expr": "G: Hyperbola;P: Point;O: Origin;F1: Point;x1:Number;y1:Number;Expression(G) = (x^2/4 - y^2 = 1);Coordinate(P) = (x1, y1);PointOnCurve(P, G);LeftFocus(G) = F1", "coords": {} }, { "index": 4385, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4385.png", "answer": "3", "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2/3 = 1);F1: Point;F2: Point;Focus(C) = {F1, F2};O: Origin;P: Point;PointOnCurve(P, C);Abs(LineSegmentOf(O, P)) = 2", "coords": {} }, { "index": 4386, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4386.png", "answer": "y^2 = 6*x", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(C) = F;l: Line;Directrix(C) = l;A: Point;PointOnCurve(A, C);G: Circle;Center(G) = F;Radius(G) = Abs(LineSegmentOf(F, A));B: Point;D: Point;Intersection(G, l) = {B, D};AngleOf(A, B, D) = ApplyUnit(90, degree);Area(TriangleOf(A, B, F)) = 9*sqrt(3)", "coords": {} }, { "index": 4387, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4387.png", "answer": "sqrt(6)/2", "fact_expr": "D: Hyperbola;C: Ellipse;P: Point;F1: Point;F2: Point;e: Number;Expression(C) = (x^2/4 + y^2 = 1);Focus(D) = {F1,F2};Focus(C) = {F1,F2};OneOf(Intersection(C,D))=P;IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2));Eccentricity(D) = e", "coords": {} }, { "index": 4388, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4389, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -0.0, 3.0 ], "radius": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_4389.png", "answer": "1/4, -1/28", "fact_expr": "G: Parabola;Expression(G) = (y = a*x^2);a: Number;H: Circle;Expression(H) = (-6*y + x^2 + y^2 - 7 = 0);IsTangent(Directrix(G), H) = True", "coords": {} }, { "index": 4390, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4390.png", "answer": "sqrt(3)/2", "fact_expr": "H: Ellipse;Expression(H) = (k^2 + x^2 = 3*k);k: Number;k>0;G: Parabola;Expression(G) = (y^2 = 12*x);OneOf(Focus(H)) = Focus(G)", "coords": {} }, { "index": 4391, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4391.png", "answer": "2", "fact_expr": "C: Parabola;A: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(A, C);Abs(LineSegmentOf(A, F)) = 3", "coords": {} }, { "index": 4392, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4392.png", "answer": "x^2 - y^2/4 = 1", "fact_expr": "G: Hyperbola;C: Parabola;P: Point;O: Origin;Expression(C) = (y^2 = 4*x);Coordinate(P) = (sqrt(5), 4);Center(G) = O;PointOnCurve(P, G);RightVertex(G) = Focus(C)", "coords": { "P": [ 2.23606797749979, 4.0 ] } }, { "index": 4393, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4393.png", "answer": "y^2 = 8*x", "fact_expr": "G: Parabola;F: Point;Coordinate(F) = (2, 0);Focus(G) = F", "coords": { "F": [ 2.0, 0.0 ] } }, { "index": 4394, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_4394.png", "answer": "(x-2)^2+y^2=1", "fact_expr": "H: Line;A: Point;Coordinate(A) = (4, 0);PointOnCurve(A, H);G: Circle;Expression(G) = (x^2 + y^2 = 4);B: Point;Intersection(H, G) = B;P: Point;MidPoint(LineSegmentOf(A, B)) = P", "coords": {} }, { "index": 4395, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4395.png", "answer": "8", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};H: Line;PointOnCurve(F1, H);A: Point;B: Point;Intersection(H, G) = {A, B};Abs(LineSegmentOf(F2, A)) + Abs(LineSegmentOf(F2, B)) = 12", "coords": {} }, { "index": 4396, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4397, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4397.png", "answer": "3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);H: Line;PointOnCurve(Focus(G), H) = True;Intersection(H, G) = {A, B};A: Point;B: Point;Abs(LineSegmentOf(A, B)) = 8", "coords": {} }, { "index": 4398, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4398.png", "answer": "(0, 1/2]", "fact_expr": "G: Ellipse;Z: Line;F2: Point;A: Point;F1: Point;B: Point;Expression(G) = (x^2/4 + y^2 = 1);Focus(G) = {F1,F2};PointOnCurve(F1, Z);Intersection(Z, G) = {A, B}", "coords": {} }, { "index": 4399, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4399.png", "answer": "3", "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (2, 2);Focus(G) = F;PointOnCurve(P, G)", "coords": { "A": [ 2.0, 2.0 ] } }, { "index": 4400, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4400.png", "answer": "2*sqrt(2)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/2 + y^2 = 1);Intersection(G, yAxis) = {M, N};M: Point;N: Point;H: Line;Expression(H) = (y = x);A1: Point;A2: Point;Intersection(H, G) = {A1, A2};P: Point;PointOnCurve(P, G) = True;Negation(P = A1);Negation(P = A2);Q: Point;IsPerpendicular(LineSegmentOf(Q, A1), LineSegmentOf(P, A1)) = True;IsPerpendicular(LineSegmentOf(Q, A2), LineSegmentOf(P, A2)) = True", "coords": {} }, { "index": 4401, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4401.png", "answer": "1", "fact_expr": "G: Parabola;H: Line;O: Origin;A: Point;B: Point;k1:Number;k2:Number;Expression(G) = (y^2 = 4*x);Expression(H) = (y = x - 1);Intersection(H, G) = {A, B};Slope(LineSegmentOf(O, A)) = k1;Slope(LineSegmentOf(O,B))=k2", "coords": {} }, { "index": 4402, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4402.png", "answer": "2", "fact_expr": "C: Hyperbola;b: Number;a: Number;M: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(M) = (a, b);LeftFocus(C) = F1;RightFocus(C) = F2;AngleOf(M, F1, F2) = ApplyUnit(30, degree)", "coords": {} }, { "index": 4403, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4403.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1)", "coords": {} }, { "index": 4404, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4404.png", "answer": "3/2+sqrt(2)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x);F: Point;Focus(G) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, G) = {A, B}", "coords": {} }, { "index": 4405, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4405.png", "answer": "x^2 - y^2 / 3 = 1", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;Coordinate(F) = (2, 0);OneOf(Focus(G)) = F;H: Circle;Expression(H) = (y^2 + (x - 2)^2 = 3);IsTangent(Asymptote(G), H)", "coords": { "F": [ 2.0, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 4406, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4406.png", "answer": "x^2/6 - y^2/8 = 1", "fact_expr": "G: Hyperbola;H: Point;C:Hyperbola;Expression(G) = (x^2/3 - y^2/4 = 1);Coordinate(H) = (3, 2);Asymptote(C) = Asymptote(G);PointOnCurve(H,C)", "coords": { "H": [ 3.0, 2.0 ] } }, { "index": 4407, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4407.png", "answer": "(-2, \\pm 2\\sqrt{2})", "fact_expr": "G: Parabola;Expression(G) = (y^2 = -4*x);M: Point;PointOnCurve(M, G);Distance(M, Focus(G)) = 3", "coords": {} }, { "index": 4408, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4408.png", "answer": "9*sqrt(13)/13", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;A: Point;RightVertex(G) = A;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;H: Line;PointOnCurve(F2, H);IsPerpendicular(H, xAxis);B: Point;OneOf(Intersection(H, G)) = B;Abs(LineSegmentOf(F1, F2)) = 2;Abs(LineSegmentOf(F2, B)) = 3/2", "coords": {} }, { "index": 4409, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4409.png", "answer": "3", "fact_expr": "H: Ellipse;Expression(H) = (x^2/4 + y^2/m = 1);m: Number;G: Hyperbola;Expression(G) = (x^2 - y^2/n = 1);n: Number;F1: Point;F2: Point;Focus(H) = {F1, F2};Focus(G) = {F1, F2};P: Point;OneOf(Intersection(H, G)) = P", "coords": {} }, { "index": 4410, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4411, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4411.png", "answer": "4", "fact_expr": "G: Hyperbola;F:Point;Expression(G) = (x^2/9 - y^2/16 = 1);RightFocus(G)=F", "coords": {} }, { "index": 4412, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4412.png", "answer": "y^2/4 + x^2 = 1", "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;a > b;b > 0;Expression(G) = (x^2/b^2 + y^2/a^2 = 1);Coordinate(A) = (1, 0);RightVertex(G) = A;H:LineSegment;IsChordOf(H,G);PointOnCurve(Focus(G),H);IsPerpendicular(H,MajorAxis(G));Length(H)=1", "coords": { "A": [ 1.0, 0.0 ] } }, { "index": 4413, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4413.png", "answer": "16/3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, G) = {A, B};y1: Number;Coordinate(A) = (3, y1);Abs(LineSegmentOf(A, F)) = 4", "coords": {} }, { "index": 4414, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4414.png", "answer": "12", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);F: Point;Focus(G) = F;A: Point;B: Point;PointOnCurve(A, G);PointOnCurve(B, G);VectorOf(A, F) = 2*VectorOf(F, B)", "coords": {} }, { "index": 4415, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4415.png", "answer": "[\\sqrt{3}/2, 1)", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G) ;AngleOf(F1, P, F2) = 2*pi/3", "coords": {} }, { "index": 4416, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4416.png", "answer": "sqrt(2)+1", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;N: Point;M: Point;O: Origin;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Vertex(H) = O;Focus(H) = F;LeftFocus(G)=F;Intersection(H,G)={M,N};MidPoint(LineSegmentOf(M,N)) = F", "coords": {} }, { "index": 4417, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4417.png", "answer": "34", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/5 = 1);F: Point;RightFocus(G) = F;P: Point;PointOnCurve(P, LeftPart(G));A: Point;Coordinate(A) = (0, 6*sqrt(6))", "coords": { "A": [ 0.0, 14.696938456699067 ] } }, { "index": 4418, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4418.png", "answer": "7/2", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 2*y);P: Point;PointOnCurve(P, G) = True;YCoordinate(P) = 3", "coords": {} }, { "index": 4419, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4419.png", "answer": "(\\sqrt{5}/5,1)", "fact_expr": "G: Ellipse;b: Number;a: Number;B: Point;A: Point;M: Point;e: Number;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(A,G);PointOnCurve(B,G);a > b;b > 0;Coordinate(M) = (a/5, 0);Intersection(PerpendicularBisector(LineSegmentOf(A,B)), xAxis) = M;Eccentricity(G) = e", "coords": {} }, { "index": 4420, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4420.png", "answer": "sqrt(3) + 1", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;O: Origin;L: Line;PointOnCurve(O, L);Inclination(L) = pi/3;P: Point;Q: Point;Intersection(L, LeftPart(G)) = P;Intersection(L, RightPart(G)) = Q;H: Circle;IsDiameter(LineSegmentOf(P, Q), H);PointOnCurve(F, H);RightFocus(G) = F;F:Point", "coords": {} }, { "index": 4421, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4421.png", "answer": "3", "fact_expr": "C: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Focus(C) = {F1, F2};PointOnCurve(P, C);IsPerpendicular(VectorOf(P, F2),VectorOf(P, F1));Area(TriangleOf(P, F1, F2)) = 9", "coords": {} }, { "index": 4422, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4422.png", "answer": "1", "fact_expr": "G: Hyperbola;E: Point;O: Origin;F: Point;Expression(G) = (x^2/2 - y^2/2 = 1);OneOf(Focus(G)) = F;L:Line;PointOnCurve(F,L);IsPerpendicular(OneOf(Asymptote(G)),L);FootPoint(OneOf(Asymptote(G)),L)=E", "coords": {} }, { "index": 4423, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4423.png", "answer": "(1,\\sqrt{2})", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;H: Line;PointOnCurve(RightFocus(G), H) = True;Inclination(H) = ApplyUnit(45, degree);NumIntersection(H, RightPart(G)) = 2;e: Number;Eccentricity(G) = e", "coords": {} }, { "index": 4424, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.4641016151377544, "major_axis": "x", "x_coef": 16.0, "y_coef": 12.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4424.png", "answer": "(2, -3)", "fact_expr": "G: Ellipse;H: Line;P: Point;Expression(G) = (x^2/16 + y^2/12 = 1);Expression(H) = (x - 2*y - 12 = 0);PointOnCurve(P, G);WhenMin(Distance(P,H))", "coords": {} }, { "index": 4425, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4425.png", "answer": "4", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;x0:Number;y0:Number;Expression(G) = (x^2/16 + y^2/9 = 1);Coordinate(P) = (x0, y0);PointOnCurve(P, G);Focus(G) = {F1, F2}", "coords": {} }, { "index": 4426, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4427, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4428, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4428.png", "answer": "3/4", "fact_expr": "F1: Point;F2: Point;C: Hyperbola;Expression(C) = (x^2 - y^2 = 2);LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C) = True;Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 4429, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4429.png", "answer": "sqrt(10)", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1:Point;F2: Point;O: Origin;a>0;b>0;L:Line;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(F2,L);IsPerpendicular(L,Asymptote(G));FootPoint(L,Asymptote(G))=P;LeftFocus(G)=F1;RightFocus(G)=F2;Tan(AngleOf(P, F2, O)) = 1/3", "coords": {} }, { "index": 4430, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_4430.png", "answer": "1", "fact_expr": "G: Hyperbola;a: Number;a>0;Expression(G) = (-y^2/3 + x^2/a^2 = 1);Eccentricity(G) = 2", "coords": {} }, { "index": 4431, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4431.png", "answer": "x**2/3-y**2=1", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;O: Origin;F: Point;A: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;IsDiameter(LineSegmentOf(O,F),G);Intersection(G,OneOf(Asymptote(C)))=A;sqrt(3)*Abs(LineSegmentOf(O, F)) = 2*Abs(LineSegmentOf(O, A));Distance(F,l) = 1;l: Line;OneOf(Asymptote(C)) = l", "coords": {} }, { "index": 4432, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4432.png", "answer": "1", "fact_expr": "C: Ellipse;A: Point;P: Point;F: Point;Expression(C) = (x^2/4 + y^2/3 = 1);RightFocus(C)=F;Coordinate(A) = (2, 4);PointOnCurve(P, C)", "coords": { "A": [ 2.0, 4.0 ] } }, { "index": 4433, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4433.png", "answer": "y=pm*4*x/3\ny^2=20*x", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/9 - y^2/16 = 1);G: Parabola;Focus(G) = RightFocus(C)", "coords": {} }, { "index": 4434, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4434.png", "answer": "(-oo, 0) + [1, +oo)", "fact_expr": "G: Parabola;H: Curve;a: Real;Expression(G) = (y^2 = 4*x);Expression(H) = And(x^2/a + y^2/4 = 1, y>=0);NumIntersection(Directrix(G), H) = 1", "coords": {} }, { "index": 4435, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4435.png", "answer": "sqrt(5)/5", "fact_expr": "G: Hyperbola;H: Point;Expression(G) = (x^2/4 - y^2 = 1);Coordinate(H) = (1, 0)", "coords": { "H": [ 1.0, 0.0 ] } }, { "index": 4436, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4437, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4437.png", "answer": "sqrt(5)", "fact_expr": "G: Parabola;H: Line;A: Point;P: Point;d: Number;Expression(G) = (x^2 = 4*y);Expression(H) = (y = -1);Coordinate(A) = (2, 0);PointOnCurve(P, G);Distance(P, H) = d", "coords": { "A": [ 2.0, 0.0 ] } }, { "index": 4438, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4438.png", "answer": "sqrt(2)+1", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;I: Point;lambda:Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, RightPart(G));LeftFocus(G) = F1;RightFocus(G) = F2;Abs(LineSegmentOf(F1, F2)) = b^2/a;Incenter(TriangleOf(P,F1,F2))=I;lambda*Area(TriangleOf(I,P,F1))=lambda*Area(TriangleOf(I,P,F2))+Area(TriangleOf(I,F1,F2))", "coords": {} }, { "index": 4439, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4440, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4440.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;B: Point;F: Point;A: Point;a>0;b>0;OneOf(Focus(G))=F;OneOf(Endpoint(ImageinaryAxis(G)))=B;Intersection(LineSegmentOf(B,F),OneOf(Asymptote(G))) = A;VectorOf(F, A) = 2*VectorOf(A, B)", "coords": {} }, { "index": 4441, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4441.png", "answer": "7;3*sqrt(15)/4-1", "fact_expr": "G: Ellipse;H: Circle;A: Point;B: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Expression(H) = (y^2 + (x - 1)^2 = 1);PointOnCurve(A, G);PointOnCurve(B, H)", "coords": {} }, { "index": 4442, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4442.png", "answer": "1", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0", "coords": {} }, { "index": 4443, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4444, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4444.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;a: Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Distance(OneOf(Focus(G)),OneOf(Asymptote(G)))=2*a", "coords": {} }, { "index": 4445, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 2.6457513110645907, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4445.png", "answer": "1", "fact_expr": "G: Hyperbola;A: Point;F: Point;Expression(G) = (x^2/9 - y^2/7 = 1);RightVertex(G) = A;RightFocus(G) = F", "coords": {} }, { "index": 4446, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4447, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4447.png", "answer": "sqrt(2)+1", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b<0;A: Point;RightVertex(C) = A;F: Point;RightFocus(C) = F;Intersection(LeftDirectrix(C), xAxis) = B;B: Point;MidPoint(LineSegmentOf(B, F)) = A", "coords": {} }, { "index": 4448, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4448.png", "answer": "4", "fact_expr": "C: Parabola;P: Point;A: Point;B: Point;Expression(C) = (x^2 = 4*y);PointOnCurve(P,Directrix(C));TangentOfPoint(P,C)={LineOf(P,A),LineOf(P,B)};TangentPoint(LineOf(P,A),C)=A;TangentPoint(LineOf(P,B),C)=B", "coords": {} }, { "index": 4449, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4449.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;b: Number;a: Number;L1: Line;L2: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Asymptote(G) = {L1, L2};IsPerpendicular(L1, L2)", "coords": {} }, { "index": 4450, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4450.png", "answer": "1", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2 = 1)", "coords": {} }, { "index": 4451, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4451.png", "answer": "sqrt(6)/2", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/2 - y^2 = 1)", "coords": {} }, { "index": 4452, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4452.png", "answer": "2", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;c: Number;G: Circle;P: Point;Q: Point;N: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;FocalLength(C) = 2*c;A: Point;RightVertex(C) = A;Center(G) = A;Radius(G) = (a + c)/2;PointOnCurve(F1, l);TangentPoint(G, l) = N;Intersection(l, C) = {P, Q};VectorOf(P, Q) = 2*VectorOf(P, N)", "coords": {} }, { "index": 4453, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -3.0, 0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_4453.png", "answer": "1/2", "fact_expr": "G: Parabola;Expression(G) = (y = a*x^2);a: Number;a>0;l: Line;Directrix(G) = l;C: Circle;Expression(C) = (y^2 + (x - 3)^2 = 1);Length(InterceptChord(l,C)) = sqrt(3)", "coords": {} }, { "index": 4454, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4455, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4455.png", "answer": "(2,4)", "fact_expr": "G: Parabola;H: Line;P: Point;Expression(G) = (y = x^2);Expression(H) = (y = 4*x - 5);IsParallel(TangentOnPoint(P, G), H)", "coords": {} }, { "index": 4456, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4457, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4457.png", "answer": "y^2 = 4*x", "fact_expr": "l: Line;G: Parabola;p: Number;H: Line;A: Point;F: Point;M: Point;B: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (x = -p/2);Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B};Intersection(l, H) = M;VectorOf(A, F) = VectorOf(F, M) ;Abs(LineSegmentOf(A, B)) = 16/3", "coords": {} }, { "index": 4458, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4458.png", "answer": "2", "fact_expr": "G: Line;Expression(G) = (y = b);C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;Intersection(G,OneOf(Asymptote(C))) = P;P: Point;LeftVertex(C) = A1;RightVertex(C) = A2;A1: Point;A2: Point;F: Point;LeftFocus(C) = F;Abs(LineSegmentOf(P, F)) = Abs(LineSegmentOf(A1, A2))", "coords": {} }, { "index": 4459, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4459.png", "answer": "17", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);F1: Point;LeftFocus(G) = F1;P: Point;PointOnCurve(P, G) = True;H: Circle;Expression(H) = (x^2 + (y - 2*sqrt(5))^2 = 1);M: Point;PointOnCurve(M, H) = True", "coords": {} }, { "index": 4460, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4460.png", "answer": "10/3", "fact_expr": "G: Hyperbola;H: Line;A: Point;F: Point;B: Point;L1:Line;L2:Line;Expression(G) = (x^2/9 - y^2/16 = 1);RightVertex(G)=A;RightFocus(G)=F;PointOnCurve(F, H);OneOf(Asymptote(G)) = L1;OneOf(Asymptote(G)) = L2;Negation(L1 = L2);IsParallel(L1,H);Intersection(H,L2) = B", "coords": {} }, { "index": 4461, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4461.png", "answer": "sqrt(2)", "fact_expr": "F: Point;A: Point;G: Parabola;p: Number;M: Point;l: Line;B: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(A) = (0, 2);Directrix(G)=l;Focus(G) = F;Intersection(LineSegmentOf(F, A), G) = B;L:Line;IsPerpendicular(l,L);PointOnCurve(B,L);FootPoint(l,L)=M;IsPerpendicular(LineSegmentOf(A, M), LineSegmentOf(M, F))", "coords": { "A": [ 0.0, 2.0 ] } }, { "index": 4462, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4463, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4464, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4464.png", "answer": "8*sqrt(3)/3", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 4);F: Point;RightFocus(G) = F;H: Line;PointOnCurve(F, H);Inclination(H) = ApplyUnit(105, degree);P: Point;Q: Point;Intersection(H, G) = {P, Q}", "coords": {} }, { "index": 4465, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4465.png", "answer": "25/8", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;C: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(Focus(G), H);Intersection(H, G) = {A, B};XCoordinate(A) = 4;L:Line;PointOnCurve(A,L);PointOnCurve(Vertex(G),L);Intersection(L,Directrix(G))=C", "coords": {} }, { "index": 4466, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4466.png", "answer": "2*x-y-15=0", "fact_expr": "P: Point;G: Hyperbola;A: Point;B: Point;H: Line;Expression(G) = (x^2 - 4*y^2 = 4);Coordinate(P) = (8, 1);PointOnCurve(P, H) = True;Intersection(H, G) = {A , B};MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ 8.0, 1.0 ] } }, { "index": 4467, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4467.png", "answer": "5", "fact_expr": "G: Parabola;F: Point;M: Point;A: Point;O:Origin;Coordinate(F) = (0, 1);Coordinate(M) = (-1, 4);Vertex(G)=O;Focus(G)=F;PointOnCurve(A,G)", "coords": { "F": [ 0.0, 1.0 ], "M": [ -1.0, 4.0 ] } }, { "index": 4468, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4468.png", "answer": "3", "fact_expr": "l: Line;C: Parabola;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;p: Number;A: Point;F: Point;D: Point;B: Point;p>0;Intersection(l,C)={A,B};Intersection(l,Directrix(C))=D;PointOnCurve(F, l);Abs(LineSegmentOf(A, F)) = 6;VectorOf(D, B) = 2*VectorOf(B, F)", "coords": {} }, { "index": 4469, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4469.png", "answer": "(\\sqrt{17}/4, +\\infty)", "fact_expr": "t: Number;P: Point;Coordinate(P) = (2, t);G: Parabola;Expression(G) = (x^2 = 4*y);PointOnCurve(P, G);M: Point;N: Point;PointOnCurve(M, G);PointOnCurve(N, G);Negation(M=P);Negation(N=P);Slope(LineOf(P, M))+Slope(LineOf(P, N)) = 3/2;Q: Point;MidPoint(LineSegmentOf(M, N)) = Q;d: Number;O: Origin;Distance(Q, O) = d", "coords": {} }, { "index": 4470, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.23606797749979, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4470.png", "answer": "{x^2/20-y^2/16=1, y^2/16-x^2/20=1}", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/5 - y^2/4 = 1);C: Hyperbola;Asymptote(C) = Asymptote(G);FocalLength(C) = 12", "coords": {} }, { "index": 4471, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4472, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4472.png", "answer": "{5*sqrt(5)/11,sqrt(5)}", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = ((x - 4)^2 + (y - 3)^2 = 5);IsTangent(OneOf(Asymptote(G)),H)", "coords": {} }, { "index": 4473, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4473.png", "answer": "(pm*sqrt(3), 0)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/2 = 1)", "coords": {} }, { "index": 4474, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2135674953460693, "b": 2.4270472526550293, "orientation": "horizontal" }, "optimization": { "final_loss": 4.453341261978494e-06, "converged": true, "iterations": 125 }, "output_path": "results/train/hyperbola/problem_4474.png", "answer": "sqrt(5)", "fact_expr": "C1: Hyperbola;Expression(C1) = (x^2/a^2 - y^2/b^2 = 1);a: Number;b: Number;a>0;b>0;C2: Hyperbola;Expression(C2) = (y^2/m^2 - x^2/n^2 = 1);m: Number;n: Number;m>0;n>0;Asymptote(C1) = Asymptote(C2);Eccentricity(C1) = sqrt(5)/2", "coords": {} }, { "index": 4475, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_4475.png", "answer": "(x^2/4+3*y^2/4=1)&Negation(x=pm*2)", "fact_expr": "P: Point;A: Point;B: Point;x1:Number;y1:Number;Coordinate(P) = (x1, y1);Coordinate(A) = (-2, 0);Coordinate(B) = (2, 0);Slope(LineSegmentOf(P,A))*Slope(LineSegmentOf(P,B))=-1/3", "coords": {} }, { "index": 4476, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4476.png", "answer": "4", "fact_expr": "G: Parabola;p: Number;F: Point;A: Point;B: Point;O: Origin;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;PointOnCurve(A, G);PointOnCurve(B, G);Negation(A = O);Negation(B = O);Negation(A = B);Abs(VectorOf(F, A)) + Abs(VectorOf(F, B)) + Abs(VectorOf(F, O)) = 12;VectorOf(F, A) + VectorOf(F, B) + VectorOf(F, O) = 0", "coords": {} }, { "index": 4477, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.463192105293274, "b": 2.070901870727539, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4661259228887502e-06, "converged": true, "iterations": 106 }, "output_path": "results/train/hyperbola/problem_4477.png", "answer": "{(-\u221e,-\u221a(2)], [\u221a(2),+\u221e)}", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Curve;k: Real;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y*(-k*x + y) = 0);Eccentricity(C) = sqrt(3);NumIntersection(G,C)=2", "coords": {} }, { "index": 4478, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4479, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4479.png", "answer": "-2", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;l: Line;PointOnCurve(Focus(C), l);A: Point;B: Point;Intersection(l, C) = {A, B};H: Circle;IsDiameter(LineSegmentOf(A, B), H);M: Point;Coordinate(M) = (-1, -1);Intersection(H, Directrix(C)) = M;k: Number;Slope(l) = k", "coords": { "M": [ -1.0, -1.0 ] } }, { "index": 4480, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4480.png", "answer": "(\\sqrt{5}-1)/2", "fact_expr": "l: Line;C: Ellipse;G: Circle;A: Point;B: Point;F: Point;O:Origin;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);OneOf(Focus(C)) = F;PointOnCurve(F, l);IsPerpendicular(l,xAxis);Intersection(l, C) = {A, B};IsDiameter(LineSegmentOf(A, B), G);PointOnCurve(O, G);a:Number;b:Number", "coords": {} }, { "index": 4481, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4482, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4482.png", "answer": "6*sqrt(3)", "fact_expr": "C: Parabola;G: Line;M: Point;N:Point;F: Point;P: Point;Q: Point;Expression(C) = (y^2 = 6*x);Focus(C)=F;PointOnCurve(F, G);Slope(G) = sqrt(3);Intersection(G, C) = {P, Q};Projection(P,Directrix(C))=M;Projection(Q,Directrix(C))=N", "coords": {} }, { "index": 4483, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4483.png", "answer": "13/2", "fact_expr": "G: Parabola;A: Point;B: Point;M: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(A, G);PointOnCurve(B, G);Abs(LineSegmentOf(A, B)) = 15;MidPoint(LineSegmentOf(A, B)) = M;XCoordinate(M)=x0;x0:Number", "coords": {} }, { "index": 4484, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4484.png", "answer": "2*sqrt(2)", "fact_expr": "G: Parabola;H: Line;P: Point;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(P) = (2, 0);PointOnCurve(P, H);Intersection(H, G) = {A, B};Focus(G) = F", "coords": { "P": [ 2.0, 0.0 ] } }, { "index": 4485, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4485.png", "answer": "1/3", "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/9 = 1);M: Point;PointOnCurve(M, G);H: Circle;Expression(H) = (x^2 + y^2 = 2);L1: Line;L2: Line;TangentOfPoint(M, H) = {L1, L2};A: Point;B: Point;TangentPoint(L1, H) = A;TangentPoint(L2, H) = B;l: Line;PointOnCurve(A, l);PointOnCurve(B, l);P: Point;Intersection(l, xAxis) = P;Q: Point;Intersection(l, yAxis) = Q;O: Origin", "coords": {} }, { "index": 4486, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4486.png", "answer": "2", "fact_expr": "G: Ellipse;A: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2/3 = 1);PointOnCurve(A, G);OneOf(Focus(G)) = F1;OneOf(Focus(G)) = F2;Negation(F1 = F2);Distance(A, F1) = 2", "coords": {} }, { "index": 4487, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "major_axis": "x", "x_coef": 3.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4487.png", "answer": "3*sqrt(2)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/3 + y^2 = 1);M: Point;PointOnCurve(M, G) = True;l: Line;Expression(l) = (x - y - 8 = 0)", "coords": {} }, { "index": 4488, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4488.png", "answer": "(2*sqrt(3)/3, +oo)", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y^2 + (x - 2)^2 = 1);NumIntersection(Asymptote(C), G)=0", "coords": {} }, { "index": 4489, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4489.png", "answer": "2*sqrt(5)", "fact_expr": "C: Parabola;A: Point;K: Point;F: Point;O: Origin;Expression(C) = (y^2 = 8*x);Focus(C) = F;Intersection(Directrix(C), xAxis) = K;PointOnCurve(A, C);Abs(LineSegmentOf(A, K)) = sqrt(2)*Abs(LineSegmentOf(A, F))", "coords": {} }, { "index": 4490, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.463192105293274, "b": 2.070901870727539, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4661259228887502e-06, "converged": true, "iterations": 106 }, "output_path": "results/train/hyperbola/problem_4490.png", "answer": "(\\sqrt{6}, 0)", "fact_expr": "G: Hyperbola;a: Number;a>0;Expression(G) = (-y^2/4 + x^2/a^2 = 1);Eccentricity(G) = sqrt(3)", "coords": {} }, { "index": 4491, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4491.png", "answer": "5/4", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);Asymptote(G) = Asymptote(H);PointOnCurve(Focus(H), yAxis);H: Hyperbola", "coords": {} }, { "index": 4492, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4492.png", "answer": "12/5", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P: Point;PointOnCurve(P, G);d1: Number;Distance(P, Directrix(G)) = d1;H: Line;Expression(H) = (3*x - 4*y + 9 = 0);d2: Number;Distance(P, H) = d2", "coords": {} }, { "index": 4493, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4494, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.8284271247461903, "b": 2.8284271247461903, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4494.png", "answer": "8", "fact_expr": "G: Hyperbola;H: Parabola;p: Number;Expression(G) = (x^2 - y^2 = 8);p>0;Expression(H) = (y^2 = 2*(p*x));Focus(H) = OneOf(Focus(G))", "coords": {} }, { "index": 4495, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4496, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4496.png", "answer": "8", "fact_expr": "P: Point;A: Point;B: Point;l: Line;F:Point;C: Ellipse;Expression(C) = (x^2/4 + y^2/2 = 1);RightFocus(C)=F;PointOnCurve(F,l);Slope(l)=1;Intersection(l,C)={A,B};Intersection(RightDirectrix(C),xAxis)=P;Slope(LineOf(P,A))=k1;Slope(LineOf(P,B))=k2;k1:Number;k2:Number", "coords": {} }, { "index": 4497, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4497.png", "answer": "y=pm*(sqrt(3)/3)*x", "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (y^2/16 - x^2/m = 1);Eccentricity(G) = e;e = 2;e:Number", "coords": {} }, { "index": 4498, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4499, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4499.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;Expression(H) = (x + 2*y - 1 = 0);IsPerpendicular(OneOf(Asymptote(G)), H);H: Line", "coords": {} }, { "index": 4500, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4500.png", "answer": "(5/4)*p^2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;K: Point;Intersection(Directrix(G), xAxis) = K;l: Line;PointOnCurve(K, l);A: Point;B: Point;Intersection(l, G) = {A, B};O: Origin", "coords": {} }, { "index": 4501, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4501.png", "answer": "sqrt(10)", "fact_expr": "C: Parabola;M: Point;F: Point;P: Point;N: Point;l: Line;Expression(C) = (y^2 = 4*x);Focus(C) = F;Directrix(C) = l;PointOnCurve(P, C);Quadrant(P)=1;IsPerpendicular(LineSegmentOf(P,M),l);FootPoint(LineSegmentOf(P,M),l)=M;Intersection(LineSegmentOf(M,F), C) = N;Slope(LineSegmentOf(P, F)) = 3/4", "coords": {} }, { "index": 4502, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4502.png", "answer": "(6-4*sqrt(2),6+4*sqrt(2))", "fact_expr": "C: Parabola;G: Line;k: Real;m:Number;A: Point;F: Point;B: Point;Expression(C) = (x^2 = 8*y);Expression(G) = (y = k*x + m);Focus(C) = F;Intersection(G, C) = {A, B};AngleOf(A,F,B)>ApplyUnit(90,degree)", "coords": {} }, { "index": 4503, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4504, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4505, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4506, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.75, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4506.png", "answer": "5", "fact_expr": "C: Parabola;M: Point;P: Point;N: Point;Expression(C) = (y^2 = 3*x);Coordinate(M) = (2, 0);Coordinate(N)=(3, 0);PointOnCurve(P, C)", "coords": { "M": [ 2.0, 0.0 ], "N": [ 3.0, 0.0 ] } }, { "index": 4507, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 0.7001950144767761, "b": 2.7117252349853516, "orientation": "horizontal" }, "optimization": { "final_loss": 1.1445769132478745e-06, "converged": true, "iterations": 350 }, "output_path": "results/train/hyperbola/problem_4507.png", "answer": "x^2 - y^2/15 = 1", "fact_expr": "C: Hyperbola;b: Number;a: Number;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(C) = 4", "coords": {} }, { "index": 4508, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4508.png", "answer": "8*sqrt(3)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;P: Point;PointOnCurve(P, G) = True;Q: Point;Coordinate(Q) = (9, 0);Abs(LineSegmentOf(Q, F)) = 2*Abs(LineSegmentOf(P, F))", "coords": { "Q": [ 9.0, 0.0 ] } }, { "index": 4509, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5002657771110535, "direction": "right" }, "optimization": { "final_loss": 8.410525879298802e-06, "converged": true, "iterations": 102 }, "output_path": "results/train/parabola/problem_4509.png", "answer": "5/2", "fact_expr": "G: Parabola;p: Number;M: Point;Expression(G) = (y^2 = 2*(p*x));Coordinate(M) = (2, 2);PointOnCurve(M, G)", "coords": { "M": [ 2.0, 2.0 ] } }, { "index": 4510, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4511, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4511.png", "answer": "sqrt(2)/2", "fact_expr": "G: Hyperbola;H: Parabola;p: Number;M: Point;P: Point;F: Point;Expression(G) = (x^2/2 - y^2 = 1/3);p>0;Expression(H) = (y^2 = 2*(p*x));Coordinate(M) = (-1, 0);Focus(H) = F;OneOf(Focus(G)) = F;Focus(H) = OneOf(Focus(G));PointOnCurve(P, H)", "coords": { "M": [ -1.0, 0.0 ] } }, { "index": 4512, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4513, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4514, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4514.png", "answer": "sqrt(2)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 8*x);F: Point;Focus(C) = F;P: Point;PointOnCurve(P, C);A: Point;Coordinate(A) = (-2, 0)", "coords": { "A": [ -2.0, 0.0 ] } }, { "index": 4515, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4515.png", "answer": "4", "fact_expr": "G: Parabola;p: Number;A: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(A) = ((-1)*2, 3);Distance(A, Focus(G))=5", "coords": { "A": [ -2.0, 3.0 ] } }, { "index": 4516, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4516.png", "answer": "3*sqrt(5)/5", "fact_expr": "C: Hyperbola;a: Number;b: Number;e: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);F:Point;RightFocus(C)=F;Distance(F,Asymptote(C))=2;Eccentricity(C) = e;O:Origin;Distance(F,O)=3", "coords": {} }, { "index": 4517, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4517.png", "answer": "pm*sqrt(3)/2", "fact_expr": "G: Parabola;M: Point;Expression(G) = (x^2 = y);PointOnCurve(M, G);Distance(M, Focus(G)) = 1", "coords": {} }, { "index": 4518, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4518.png", "answer": "2*sqrt(2)", "fact_expr": "l: Line;Expression(l) = (y = 2*x + b);b: Number;C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;Length(InterceptChord(l, C)) = 5;PointOnCurve(Focus(C), l);M: Point;PointOnCurve(M, C);N: Point;Coordinate(N) = (3, 0)", "coords": { "N": [ 3.0, 0.0 ] } }, { "index": 4519, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4519.png", "answer": "1/2", "fact_expr": "C: Ellipse;a:Number;b:Number;a>b;b>0;M: Point;F1: Point;N: Point;F2: Point;Expression(C)=(x^2/a^2 +y^2/b^2=1);LeftFocus(C) =F1;RightFocus(C)=F2;PointOnCurve(M,C);IsPerpendicular(LineSegmentOf(M,F2),xAxis);Intersection(LineOf(M,F1),C)={M,N};Slope(LineOf(M,N)) = 3/4", "coords": {} }, { "index": 4520, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4520.png", "answer": "4*sqrt(3)", "fact_expr": "G: Hyperbola;b: Number;a: Number;F1: Point;P: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;Distance(F2, Asymptote(G)) = 2;PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 4521, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4521.png", "answer": "sqrt(6)/2", "fact_expr": "l: Line;G: Hyperbola;b: Number;a: Number;A: Point;F: Point;B: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;PointOnCurve(F, l);Intersection(l, Asymptote(G)) = {A, B};IsPerpendicular(l,OneOf(Asymptote(G)));VectorOf(A, F) = 3*VectorOf(F, B)", "coords": {} }, { "index": 4522, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4522.png", "answer": "4", "fact_expr": "C: Hyperbola;F: Point;Expression(C) = (x^2/9 - y^2/16 = 1);OneOf(Focus(C)) = F", "coords": {} }, { "index": 4523, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4524, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.6457513110645907, "major_axis": "x", "x_coef": 16.0, "y_coef": 7.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4524.png", "answer": "8+sqrt(5)", "fact_expr": "F1: Point;F2: Point;G: Ellipse;Expression(G) = (x^2/16 + y^2/7 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;E: Point;PointOnCurve(E, G);N: Point;Coordinate(N) = (5, 1)", "coords": { "N": [ 5.0, 1.0 ] } }, { "index": 4525, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4526, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4526.png", "answer": "2*sqrt(13)", "fact_expr": "G: Parabola;A: Point;F: Point;P: Point;O: Origin;Expression(G) = (x^2 = 8*y);Focus(G) = F;PointOnCurve(P, Directrix(G));PointOnCurve(A, G);Abs(LineSegmentOf(A, F)) = 4", "coords": {} }, { "index": 4527, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4527.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/12 + y^2/4 = 1)", "coords": {} }, { "index": 4528, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4528.png", "answer": "y^2=8*x", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;A:Point;XCoordinate(A) = 1;PointOnCurve(A,G) = True;Distance(A,Vertex(G)) = Distance(A,Directrix(G))", "coords": {} }, { "index": 4529, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4529.png", "answer": "sqrt(17)/2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x);F: Point;Focus(G) = F;P: Point;PointOnCurve(P, G);A: Point;Coordinate(A) = (0, 2)", "coords": { "A": [ 0.0, 2.0 ] } }, { "index": 4530, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4531, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4531.png", "answer": "13", "fact_expr": "F1: Point;F2: Point;Focus(G) = {F1, F2};G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);M: Point;PointOnCurve(M, G);Distance(M, F1) = 7", "coords": {} }, { "index": 4532, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4532.png", "answer": "sqrt(10)-1", "fact_expr": "G: Parabola;A: Point;P:Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (2, 3);PointOnCurve(P, G)", "coords": { "A": [ 2.0, 3.0 ] } }, { "index": 4533, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4533.png", "answer": "x^2/16+y^2/25=1", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/5 - y^2/4 = -1);H: Ellipse;Focus(G) = Focus(H);Eccentricity(H) = 0.6", "coords": {} }, { "index": 4534, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4534.png", "answer": "x=-2", "fact_expr": "G: Parabola;p: Number;H: Ellipse;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (x^2/9 + y^2/5 = 1);Focus(G) = RightFocus(H)", "coords": {} }, { "index": 4535, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4535.png", "answer": "y=pm*(4/3)*x", "fact_expr": "C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Distance(RightFocus(C),OneOf(Asymptote(C)))=4;FocalLength(C) = 10", "coords": {} }, { "index": 4536, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4536.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(Asymptote(G)) = (y = pm*(2*x))", "coords": {} }, { "index": 4537, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4537.png", "answer": "3", "fact_expr": "P: Point;x0: Number;y0: Number;Coordinate(P) = (x0, y0);PointOnCurve(P, G);G: Parabola;Expression(G) = (y^2 = 4*x);H: Circle;Expression(H) = ((x + 2)^2 + (y - 4)^2 = 1);Q: Point;PointOnCurve(Q, H)", "coords": {} }, { "index": 4538, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4539, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4540, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4540.png", "answer": "sqrt(5)/3", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;LeftFocus(C) = F;l: Line;PointOnCurve(F, l);Slope(l) = 1/2;A: Point;B: Point;Intersection(l, C) = {A, B};O: Origin;Abs(LineSegmentOf(O, F)) = Abs(LineSegmentOf(O, A))", "coords": {} }, { "index": 4541, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4541.png", "answer": "10/3", "fact_expr": "G: Parabola;H: Line;B: Point;A: Point;F: Point;Expression(G) = (y^2 = 8*x);Focus(G) = F;PointOnCurve(F, H);Inclination(H) = ApplyUnit(60, degree);Intersection(H, G) = {A, B}", "coords": {} }, { "index": 4542, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 6.0, "b": 8.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4542.png", "answer": "27", "fact_expr": "P: Point;C: Hyperbola;Expression(C) = (x^2/36 - y^2/64 = 1);PointOnCurve(P, C) = True;A: Point;Coordinate(A) = (-10, 0);B: Point;Coordinate(B) = (10, 0);Abs(LineSegmentOf(P, A)) = 15", "coords": { "A": [ -10.0, 0.0 ], "B": [ 10.0, 0.0 ] } }, { "index": 4543, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4543.png", "answer": "x^2 - y^2 / 3 = 1 \ny = pm * sqrt(3) * x", "fact_expr": "H: Parabola;Expression(H) = (y^2 = 8*x);G: Hyperbola;RightFocus(G) = Focus(H);Eccentricity(G) = 2", "coords": {} }, { "index": 4544, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4544.png", "answer": "5", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 10*x)", "coords": {} }, { "index": 4545, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4546, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 6.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4546.png", "answer": "y=pm*x/2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/36 - y^2/9 = 1)", "coords": {} }, { "index": 4547, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4547.png", "answer": "sqrt(85)/5", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;P: Point;F: Point;Q: Point;O:Origin;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(O,G);Intersection(G, C) = {P, Q};Negation(P=Q);OneOf(Focus(C)) = F;Tan(AngleOf(P, F, Q)) = 4/3;Abs(LineSegmentOf(Q, F)) = 3*Abs(LineSegmentOf(P, F))", "coords": {} }, { "index": 4548, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 2.2628633975982666, "b": 1.8162860737044515, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 2.6997004191253788e-06, "converged": true, "iterations": 101 }, "output_path": "results/train/ellipse/problem_4548.png", "answer": "sqrt(5)+2", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;A: Point;Coordinate(A) = (1, 2);PointOnCurve(A,C) = True", "coords": { "A": [ 1.0, 2.0 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 4549, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4549.png", "answer": "3", "fact_expr": "C: Parabola;F: Point;P: Point;Q: Point;l: Line;Expression(C) = (y^2 = 8*x);Focus(C) = F;Directrix(C) = l;PointOnCurve(P, l);OneOf(Intersection(LineOf(P,F),C))=Q;VectorOf(F, P) = 4*VectorOf(F, Q)", "coords": {} }, { "index": 4550, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4550.png", "answer": "(7/4, pm*sqrt(7)/2)", "fact_expr": "G: Parabola;P: Point;Expression(G) = (y^2 = x);PointOnCurve(P, G);Distance(P, Focus(G)) = 2", "coords": {} }, { "index": 4551, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4551.png", "answer": "pi/2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, RightPart(G));Abs(LineSegmentOf(P, F1))*Abs(LineSegmentOf(P, F2)) = 32", "coords": {} }, { "index": 4552, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4552.png", "answer": "y^2/100 - x^2/25 = 1", "fact_expr": "C: Hyperbola;Expression(C) = (-x^2/b^2 + y^2/a^2 = 1);b: Number;a: Number;FocalLength(C) = 10*sqrt(5);P: Point;Coordinate(P) = (1, 2);PointOnCurve(P, Asymptote(C))", "coords": { "P": [ 1.0, 2.0 ] } }, { "index": 4553, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4553.png", "answer": "[2,+oo)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(G) = F;PointOnCurve(F, H) = True;Inclination(H) = ApplyUnit(60, degree);H: Line;NumIntersection(H, RightPart(G)) = 1", "coords": {} }, { "index": 4554, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4555, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4555.png", "answer": "x^2/4 + y^2 = 1", "fact_expr": "G: Ellipse;a: Number;b: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Distance(LeftFocus(G), RightDirectrix(G)) = 7*sqrt(3)/3;Distance(Center(G), Directrix(G)) = 4*sqrt(3)/3", "coords": {} }, { "index": 4556, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4556.png", "answer": "(1, \\sqrt{2}+1)", "fact_expr": "G: Hyperbola;b: Number;a: Number;F1: Point;F2: Point;P: Point;a>0;b>0;c:Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(P,G);Sin(AngleOf(P,F1,F2))/Sin(AngleOf(P,F2,F1))=a/c", "coords": {} }, { "index": 4557, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4557.png", "answer": "3", "fact_expr": "C: Hyperbola;M: Point;A: Point;B: Point;O: Origin;P: Point;Q: Point;Expression(C) = (x^2/4 - y^2/3 = 1);LeftVertex(C) = A;RightVertex(C) = B;PointOnCurve(M, C);Negation(M = A);Negation(M = B);Intersection(LineOf(A, M),yAxis)=P;Intersection(LineOf(B, M), yAxis) = Q", "coords": {} }, { "index": 4558, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.8284271247461903, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4558.png", "answer": "-1", "fact_expr": "G: Hyperbola;A: Point;B: Point;C: Point;O: Origin;D: Point;E: Point;F: Point;Expression(G) = (x^2 - y^2/8 = 1);PointOnCurve(A, G);PointOnCurve(B, G);PointOnCurve(C, G);MidPoint(LineSegmentOf(A, B)) = D;MidPoint(LineSegmentOf(B, C)) = E;MidPoint(LineSegmentOf(A, C)) = F;k1: Number;k2: Number;k3: Number;Slope(LineSegmentOf(O, F)) = k3;Slope(LineSegmentOf(O, D)) = k1;Slope(LineSegmentOf(O, E)) = k2;k1 + k2 + k3 = -8;Negation(k1 = 0);Negation(k2 = 0);Negation(k3 = 0);k4: Number;k5: Number;k6: Number;Slope(LineSegmentOf(A, B)) = k4;Slope(LineSegmentOf(B, C)) = k5;Slope(LineSegmentOf(A, C)) = k6", "coords": {} }, { "index": 4559, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4560, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.552875593585339, "b": 2.3303634978370953, "orientation": "horizontal" }, "optimization": { "final_loss": 1.3057248743840209e-06, "converged": true, "iterations": 98 }, "output_path": "results/train/hyperbola/problem_4560.png", "answer": "4", "fact_expr": "G: Hyperbola;PointOnCurve(Focus(G), xAxis);Expression(Asymptote(G)) = (y = pm*(3/2)*x);Length(ImageinaryAxis(G)) = 6", "coords": {} }, { "index": 4561, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4561.png", "answer": "sqrt(5)-1", "fact_expr": "C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);l1:Line;l2:Line;O:Origin;PointOnCurve(O,l1);PointOnCurve(O,l2);Slope(l1)>0;Slope(l2)>0;Inclination(l2)=2*Inclination(l1);OneOf(Asymptote(C))=l2;RightFocus(C)=F;L:Line;PointOnCurve(F,L);IsPerpendicular(l1,L);FootPoint(l1,L)=M;PointOnCurve(M,C);M:Point;F:Point", "coords": {} }, { "index": 4562, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4562.png", "answer": "{(y^2=16*x), (x^2=-12*y)}", "fact_expr": "G: Parabola;Vertex(G) = O;O: Origin;SymmetryAxis(G) = axis;H: Line;Expression(H) = (3*x - 4*y - 12 = 0);PointOnCurve(Focus(G), H)", "coords": {} }, { "index": 4563, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4564, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4565, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4565.png", "answer": "[3/8, 3/4]", "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2/3 = 1);A1: Point;A2: Point;LeftVertex(C) = A1;RightVertex(C) = A2;P: Point;PointOnCurve(P, C);Range(Slope(LineOf(P, A2))) = [-2, -1]", "coords": {} }, { "index": 4566, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4566.png", "answer": "3", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2));Area(TriangleOf(P, F1, F2)) = 9", "coords": {} }, { "index": 4567, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4568, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.1503682136535645, "b": 2.5736796855926514, "orientation": "horizontal" }, "optimization": { "final_loss": 5.191216132516274e-06, "converged": true, "iterations": 118 }, "output_path": "results/train/hyperbola/problem_4568.png", "answer": "2*sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/3 + x^2/a^2 = 1);a: Number;Eccentricity(G) = sqrt(6)/2;H: Circle;Expression(H) = (y^2 + (x - 6)^2 = r^2);r: Number;r>0;IsTangent(Asymptote(G), H)", "coords": {} }, { "index": 4569, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.449489742783178, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 6.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4569.png", "answer": "y^2 = 8*x", "fact_expr": "G: Parabola;H: Ellipse;Expression(H) = (x^2/6 + y^2/2 = 1);Focus(G) = RightFocus(H)", "coords": {} }, { "index": 4570, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -1.0, 0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_4570.png", "answer": "((x - 1/2)^2 + y^2 = 1/4)&(0 < x <= 1)", "fact_expr": "C: Circle;Expression(C) = (y^2 + (x - 1)^2 = 1);O: Origin;H: LineSegment;PointOnCurve(O,H);IsChordOf(H,C)", "coords": {} }, { "index": 4571, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -0.0, -3.0 ], "radius": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_4571.png", "answer": "{(y^2+x^2+3*y=0)&(5/3 b;b > 0;P: Point;Coordinate(P) = (0, sqrt(3));PointOnCurve(P,C);Length(MajorAxis(C)) = 2 * FocalLength(C)", "coords": { "P": [ 0.0, 1.7320508075688772 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 4576, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4577, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4577.png", "answer": "y=pm*4*x", "fact_expr": "C: Hyperbola;b: Number;a: Number;M: Point;O: Origin;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(M, C);Abs(LineSegmentOf(M, O)) = Abs(LineSegmentOf(O, F1));Area(TriangleOf(M, O, F1)) = 2*a*b", "coords": {} }, { "index": 4578, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4578.png", "answer": "3", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);F: Point;LeftFocus(G) = F;H: Line;Expression(H) = (x = m);m: Number;Intersection(H, G) = {A, B};A: Point;B: Point;WhenMax(Perimeter(TriangleOf(F, A, B)))", "coords": {} }, { "index": 4579, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4580, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "y", "x_coef": 9.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4580.png", "answer": "y^2 = pm*20*x", "fact_expr": "G: Parabola;H: Ellipse;O: Origin;Expression(H) = (x^2/9 + y^2/16 = 1);Vertex(G) = O;SymmetryAxis(G)=OverlappingLine(MinorAxis(H));Distance(Focus(G),Vertex(G))=5", "coords": {} }, { "index": 4581, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4581.png", "answer": "2", "fact_expr": "G: Parabola;M: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(M, G);Distance(M, Focus(G)) = 3;x1: Number;XCoordinate(M) = x1", "coords": {} }, { "index": 4582, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4582.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Curve;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2/b^2 + y^2/a^2 = 1);L:Line;PointOnCurve(OneOf(Focus(G)),L);IsPerpendicular(OneOf(Asymptote(G)),L);PointOnCurve(FootPoint(OneOf(Asymptote(G)),L),H)", "coords": {} }, { "index": 4583, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4584, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.125, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4584.png", "answer": "2", "fact_expr": "G: Parabola;Expression(G) = (y = 2*x^2);F: Point;Focus(G) = F;M: Point;N: Point;PointOnCurve(M, G);PointOnCurve(N, G);Abs(LineSegmentOf(M, F)) + Abs(LineSegmentOf(N, F)) = 17/4", "coords": {} }, { "index": 4585, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4585.png", "answer": "1/2", "fact_expr": "G: Ellipse;m: Number;F: Point;P: Point;m>3;Expression(G) = (y^2/3 + x^2/m = 1);OneOf(Focus(G)) = F;PointOnCurve(P, G);Max(Distance(P, F)) = 3", "coords": {} }, { "index": 4586, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4587, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4587.png", "answer": "x+y-1=0", "fact_expr": "l: Line;C: Ellipse;M: Point;A: Point;B: Point;F1: Point;Expression(C) = (x^2/4 + y^2/3 = 1);Coordinate(M) = (0, 1);PointOnCurve(M, l);Intersection(l, C) = {A, B};LeftFocus(C)=F1;WhenMax(Perimeter(TriangleOf(A,B,F1)))", "coords": { "M": [ 0.0, 1.0 ] } }, { "index": 4588, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4588.png", "answer": "sqrt(15)", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;O: Origin;M: Point;Expression(G) = (x^2/9 + y^2/5 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);MidPoint(LineSegmentOf(P,F1))=M;Abs(LineSegmentOf(O,F2))=Abs(LineSegmentOf(O,M))", "coords": {} }, { "index": 4589, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4589.png", "answer": "2*sqrt(5)", "fact_expr": "P: Point;G: Parabola;Expression(G) = (x^2 = 8*y);PointOnCurve(P, G) = True;A: Point;Coordinate(A) = (4, 0);H: Line;Expression(H) = (y = -2)", "coords": { "A": [ 4.0, 0.0 ] } }, { "index": 4590, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4591, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4591.png", "answer": "2", "fact_expr": "C: Parabola;Expression(C) = (x^2 = 4*y);F: Point;Focus(C) = F;l: Line;PointOnCurve(F, l);Intersection(l, C) = {A, B};A: Point;B: Point;Intersection(TangentOnPoint(A,C),xAxis) = M;Intersection(TangentOnPoint(A,C),yAxis) = N;M: Point;O: Origin;N: Point;Area(TriangleOf(M, O, N)) = 1/2", "coords": {} }, { "index": 4592, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4593, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4594, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4594.png", "answer": "5", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (y^2/4 + x^2/m = 1);PointOnCurve(Focus(G), xAxis);FocalLength(G) = 2", "coords": {} }, { "index": 4595, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4595.png", "answer": "8", "fact_expr": "G: Parabola;A: Point;B: Point;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;IsChordOf(LineSegmentOf(A, B), G);Inclination(LineSegmentOf(A, B)) = ApplyUnit(45, degree);PointOnCurve(F, LineSegmentOf(A, B))", "coords": {} }, { "index": 4596, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 9.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4596.png", "answer": "17", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/81 = 1);P: Point;PointOnCurve(P, G);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;Abs(LineSegmentOf(P, F1)) = 9", "coords": {} }, { "index": 4597, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4597.png", "answer": "2*sqrt(3)/3", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (1, 0);Distance(P, OneOf(Asymptote(C))) = 1/2", "coords": { "P": [ 1.0, 0.0 ] } }, { "index": 4598, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4599, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4599.png", "answer": "[2*sqrt(2), 4)", "fact_expr": "T: Parabola;C: Circle;l1: Line;l2: Line;P: Point;Q: Point;M: Point;Expression(T) = (y^2 = 4*x);Expression(C) = (y^2 + (x - 3)^2 = 4);PointOnCurve(M, T);TangentOfPoint(M, C) = {l1, l2};TangentPoint(l1, C) = P;TangentPoint(l2, C) = Q", "coords": {} }, { "index": 4600, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4600.png", "answer": "2*sqrt(2)", "fact_expr": "H: Line;Expression(H) = (y = x - 1);G: Parabola;Expression(G) = (y^2 = 4*x);Intersection(H,G) = {A,B};A: Point;B: Point;O: Origin", "coords": {} }, { "index": 4601, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4602, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4602.png", "answer": "3*sqrt(3)", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/25 + y^2/9 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 4603, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4603.png", "answer": "5", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(Focus(G), H);Slope(H)=-2;Intersection(H, G) = {A,B}", "coords": {} }, { "index": 4604, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4604.png", "answer": "2", "fact_expr": "A: Point;B: Point;G: Parabola;Expression(G) = (y^2 = 2*x);Focus(G) = F;F: Point;IsChordOf(LineSegmentOf(A, B), G) = True;PointOnCurve(F, LineSegmentOf(A, B)) = True;Abs(LineSegmentOf(A, B)) = 4;C: Point;MidPoint(LineSegmentOf(A, B)) = C;H: Line;Expression(H) = (x + 1/2 = 0)", "coords": {} }, { "index": 4605, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4605.png", "answer": "16/5", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);P: Point;PointOnCurve(P, G);d1: Number;d2: Number;H: Line;Expression(H) = (4*x + 3*y + 8 = 0);Distance(P, Directrix(G)) = d1;Distance(P, H) = d2", "coords": {} }, { "index": 4606, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_4606.png", "answer": "sqrt(3)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Eccentricity(C) = 2;G: Parabola;Expression(G) = (y^2 = 16*x);RightFocus(C) = Focus(G)", "coords": {} }, { "index": 4607, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4607.png", "answer": "8", "fact_expr": "H: Parabola;Expression(H) = (y^2 = 2*(p*x));p: Number;p>0;l: Line;Directrix(H) = l;G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1);Z1: Line;Z2: Line;Asymptote(G) = {Z1, Z2};A: Point;Intersection(l, Z1) = A;B: Point;Intersection(l, Z2) = B;Abs(LineSegmentOf(A, B)) = 4", "coords": {} }, { "index": 4608, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4608.png", "answer": "pm*1", "fact_expr": "C: Parabola;P: Point;A: Point;F: Point;t:Number;Expression(C) = (x^2 = 8*y);Intersection(Directrix(C), yAxis) = A;Focus(C) = F;PointOnCurve(P, C);t=Abs(LineSegmentOf(P,A))/Abs(LineSegmentOf(P,F));WhenMax(t)", "coords": {} }, { "index": 4609, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4609.png", "answer": "16.5", "fact_expr": "G: Hyperbola;H: Point;P: Point;D:Point;Expression(G) = (x^2/16 - y^2/9 = 1);Coordinate(H) = (5, 0);Coordinate(D)=(-5,0);PointOnCurve(P, G);Distance(P, H) = 8.5", "coords": { "H": [ 5.0, 0.0 ], "D": [ -5.0, 0.0 ] } }, { "index": 4610, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4610.png", "answer": "pm*sqrt(3)", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;k:Number;Expression(G) = (y^2 = 4*x);Expression(H) = (y = k*(x - 1));Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 16/3", "coords": {} }, { "index": 4611, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4611.png", "answer": "sqrt(3)/2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);Intersection(Directrix(G), xAxis) = P;P: Point;PointOnCurve(P, H);Slope(H) = k;k: Number;k>0;H: Line;Intersection(H, G) = {A, B};B: Point;A: Point;Focus(G) = F;F: Point;Abs(LineSegmentOf(F, A)) = 3*Abs(LineSegmentOf(F, B));Slope(LineOf(A, B)) = k", "coords": {} }, { "index": 4612, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4612.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;b: Number;H: Parabola;b>0;Expression(G) = (x^2 - y^2/b^2 = 1);Expression(H) = (y^2 = 8*x);Focus(H) = RightFocus(G)", "coords": {} }, { "index": 4613, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4613.png", "answer": "y^2 = - 20*x", "fact_expr": "G: Hyperbola;H: Parabola;Expression(G) = (x^2/16 - y^2/9 = 1);Center(G) = Vertex(H);LeftFocus(G) = Focus(H)", "coords": {} }, { "index": 4614, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4614.png", "answer": "sqrt(85)/5", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;A: Point;B: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F2;PointOnCurve(F2, l);Intersection(l, RightPart(C)) = {A, B};IsChordOf(LineSegmentOf(A, B), C);Abs(LineSegmentOf(A, B)) = 5*a;2*Abs(LineSegmentOf(B, F2)) = 3*Abs(LineSegmentOf(A, F2))", "coords": {} }, { "index": 4615, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 2.8284271247461903, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4615.png", "answer": "64*pi", "fact_expr": "G: Hyperbola;H: Circle;O: Origin;N: Point;F1: Point;P: Point;M: Point;F2: Point;lambda:Number;lambda>0;mu:Number;Expression(G) = (x^2/16 - y^2/8 = 1);PointOnCurve(P, RightPart(G));LeftFocus(G) = F1;RightFocus(G) = F2;VectorOf(F1, P) = lambda*VectorOf(P, M);VectorOf(P,N)=mu*(VectorOf(P,M)/Abs(VectorOf(P,M))+VectorOf(P,F2)/Abs(VectorOf(P,F2)));Abs(VectorOf(P, F2)) = 4;DotProduct(VectorOf(P,N),VectorOf(F2,N))=0;Center(H)=O;Radius(H)=LineSegmentOf(O,N)", "coords": {} }, { "index": 4616, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4616.png", "answer": "x^2/2 - y^2/4 = 1", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Point;a>0;b>0;l:Line;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(H) = (1, 0);FocalLength(G) = 2*sqrt(6);OneOf(Asymptote(G)) = l;Distance(H, l) = sqrt(6)/3", "coords": { "H": [ 1.0, 0.0 ] } }, { "index": 4617, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4617.png", "answer": "8/3", "fact_expr": "G: Ellipse;M: Point;N: Point;P: Point;Expression(G) = (x^2/9 + y^2/5 = 1);Coordinate(M) = (2, 0);Coordinate(N) = (-2, 0);PointOnCurve(P, G)", "coords": { "M": [ 2.0, 0.0 ], "N": [ -2.0, 0.0 ] } }, { "index": 4618, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4618.png", "answer": "9*x - 32*y + 73 = 0", "fact_expr": "G: Ellipse;H: LineSegment;M: Point;Expression(G) = (x^2/16 + y^2/9 = 1);Coordinate(M) = (-1, 2);IsChordOf(H, G);MidPoint(H) = M", "coords": { "M": [ -1.0, 2.0 ] } }, { "index": 4619, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4619.png", "answer": "pi/3, 2*pi/3", "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;F: Point;Expression(C) = (y^2 = 12*x);Focus(C) = F;PointOnCurve(F, l);Intersection(l,C)={A,B};IsChordOf(LineSegmentOf(A,B),C);Length(LineSegmentOf(A, B)) = 16", "coords": {} }, { "index": 4620, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4620.png", "answer": "(\\sqrt{5} - 1)/2", "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;F: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(A, G);OneOf(Focus(G)) = F;IsPerpendicular(LineSegmentOf(A,F), xAxis);c: Number;Abs(LineSegmentOf(A, F)) = c;HalfFocalLength(G) = c", "coords": {} }, { "index": 4621, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4621.png", "answer": "y=2*x+1", "fact_expr": "G: Parabola;H: LineSegment;M: Point;Expression(G) = (y^2 = -12*x);Coordinate(M) = (-2, -3);MidPoint(H)=M;IsChordOf(H,G)", "coords": { "M": [ -2.0, -3.0 ] } }, { "index": 4622, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4623, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4623.png", "answer": "y = pm*sqrt(3)*x", "fact_expr": "C: Hyperbola;b: Number;a: Number;B: Point;F: Point;A: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftVertex(C) = A;RightFocus(C) = F;PointOnCurve(B, C);IsPerpendicular(LineSegmentOf(B, F), LineSegmentOf(A, F));Abs(LineSegmentOf(A, F)) = Abs(LineSegmentOf(B, F))", "coords": {} }, { "index": 4624, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4624.png", "answer": "4/9", "fact_expr": "C: Ellipse;b: Number;a: Number;G: Line;P: Point;F1: Point;F2: Point;A: Point;I: Point;B: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);Incenter(TriangleOf(P,F1,F2))=I;PointOnCurve(I,G);IsParallel(G,xAxis);Intersection(G,LineSegmentOf(P,F1))=A;Intersection(G,LineSegmentOf(P,F2))=B;Eccentricity(C)=e;e:Number;e=1/2", "coords": {} }, { "index": 4625, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4625.png", "answer": "(\\sqrt{5} - 1)/2", "fact_expr": "A: Point;B: Point;G: Ellipse;b: Number;a: Number;e: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (a, 0);Coordinate(B) = (0, b);Eccentricity(G) = e;O: Origin;Distance(O, LineOf(A, B)) = c ;c: Number;HalfFocalLength(G) = c", "coords": {} }, { "index": 4626, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4626.png", "answer": "y = (pm*sqrt(3)/3)*x", "fact_expr": "G: Hyperbola;a: Number;b: Number;E: Ellipse;a1: Number;b1: Number;a1 > 0;b1 > 0;Expression(G) = (x^2/a1^2 - y^2/b1^2 = 1/2);a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);Focus(G) = Focus(E)", "coords": {} }, { "index": 4627, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.8284271247461903, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4627.png", "answer": "4*sqrt(3)", "fact_expr": "G: Hyperbola;H: Parabola;O: Origin;Expression(G) = (x^2 - 2*y^2 = 8);Vertex(H) = O;Focus(H) = OneOf(Focus(G))", "coords": {} }, { "index": 4628, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4628.png", "answer": "4*sqrt(6)", "fact_expr": "l: Line;M: Point;Coordinate(M) = (2, 0);PointOnCurve(M, l);Slope(l) = 1;G: Parabola;Expression(G) = (y^2 = 4*x);A: Point;B: Point;Intersection(l, G) = {A, B}", "coords": { "M": [ 2.0, 0.0 ] } }, { "index": 4629, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4630, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4630.png", "answer": "{2, 2*sqrt(3)}", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1);F: Point;RightFocus(G) = F;P: Point;PointOnCurve(P, Asymptote(G));O: Origin;Abs(LineSegmentOf(O, P)) = 2", "coords": {} }, { "index": 4631, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 2.0, "major_axis": "x", "x_coef": 8.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4631.png", "answer": "-1/4", "fact_expr": "G: Ellipse;A: Point;P: Point;B:Point;k1:Number;k2:Number;Expression(G) = (x^2/8 + y^2/4 = 1);LeftVertex(G)=A;RightVertex(G)=B;PointOnCurve(P,G);Slope(LineOf(P,A))=k1;Slope(LineOf(P,B))=k2;k1=2", "coords": {} }, { "index": 4632, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4632.png", "answer": "y^2=2*x-1", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;J: Point;Coordinate(J) = (1/2, 0);PointOnCurve(J, H) ;H: Line;Intersection(H, G) = {P, Q};Q: Point;P: Point;MidPoint(LineSegmentOf(P, Q)) = M;M: Point;Negation(P=Q)", "coords": { "J": [ 0.5, 0.0 ] } }, { "index": 4633, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4634, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4634.png", "answer": "1/4", "fact_expr": "C: Hyperbola;P: Point;F1: Point;F2: Point;Expression(C) = (x^2 - y^2/3 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 4635, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4635.png", "answer": "2", "fact_expr": "l: Line;E: Parabola;p: Number;A: Point;O: Origin;B: Point;p>0;Expression(E)=(y^2=2*p*x);Focus(E)=F;PointOnCurve(F,l);Intersection(l,E)={A,B};Quadrant(A)=1;Area(TriangleOf(O,A,B))=(-3/2)*Tan(AngleOf(A,O,B));F:Point", "coords": {} }, { "index": 4636, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4636.png", "answer": "3*sqrt(6)", "fact_expr": "G: Parabola;p: Number;H: Line;A: Point;F: Point;B: Point;O: Origin;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G)=F;PointOnCurve(F,H);Intersection(H, G) = {A, B};VectorOf(A, F) = 2*VectorOf(F, B);Area(TriangleOf(O,A,B))= sqrt(3)*Abs(LineSegmentOf(A, B))", "coords": {} }, { "index": 4637, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 10.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_4637.png", "answer": "{x^2/36-y^2/64=1,y^2/64-x^2/36=1}", "fact_expr": "G: Hyperbola;H: Circle;Expression(Asymptote(G)) = (y = pm*((4/3)*x));Expression(H) = (x^2 + y^2 = 100);PointOnCurve(Focus(G), H) = True", "coords": {} }, { "index": 4638, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4638.png", "answer": "(2,0)", "fact_expr": "G: Ellipse;H: Circle;b: Number;a: Number;Expression(G) = (x^2/9 + y^2/5 = 1);Expression(H) = ((-a + x)^2 + (-b + y)^2 = 1);Center(H)=RightFocus(G)", "coords": {} }, { "index": 4639, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4639.png", "answer": "0", "fact_expr": "G: Parabola;H: Line;O: Origin;A: Point;B: Point;Expression(G) = (y^2 = 2*x);Expression(H) = (y = x - 2);Intersection(H, G) = {A, B}", "coords": {} }, { "index": 4640, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4641, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4641.png", "answer": "3", "fact_expr": "G: Parabola;H: Point;Expression(G) = (y^2 = 4*x);Coordinate(H) = (2, 2*sqrt(2));PointOnCurve(H, G)", "coords": { "H": [ 2.0, 2.8284271247461903 ] } }, { "index": 4642, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4642.png", "answer": "48*x-y-95=0", "fact_expr": "C: Parabola;p: Number;A: Point;B: Point;P: Point;F: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(P) = (6, m);Coordinate(M) = (2, 1);PointOnCurve(P, C);Focus(C) = F;Distance(P, F) = 7;IsChordOf(LineSegmentOf(A,B),C);MidPoint(LineSegmentOf(A,B))=M;m:Number;M:Point", "coords": { "M": [ 2.0, 1.0 ] } }, { "index": 4643, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4644, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4645, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4646, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4646.png", "answer": "8", "fact_expr": "G: Ellipse;N: Circle;E: Point;F: Point;P: Point;Expression(G) = (x^2/4 + y^2/3 = 1);Expression(N) = (y^2 + (x - 1)^2 = 1);PointOnCurve(P, G);IsDiameter(LineSegmentOf(E, F), N)", "coords": {} }, { "index": 4647, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4647.png", "answer": "1/2", "fact_expr": "C: Parabola;G: Line;E: Point;P: Point;Q: Point;k: Number;Expression(C) = (x^2 = 8*y);Coordinate(E) = (2, -2);PointOnCurve(Focus(C), G);Slope(G) = k;Intersection(G, C) = {P, Q};AngleOf(P, E, Q) = ApplyUnit(90, degree)", "coords": { "E": [ 2.0, -2.0 ] } }, { "index": 4648, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 5.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4648.png", "answer": "3*sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/25 - y^2/9 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(120, degree)", "coords": {} }, { "index": 4649, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4649.png", "answer": "4", "fact_expr": "G: Parabola;H: Line;P: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (x = -2);PointOnCurve(P, G);Distance(P, H) = 5", "coords": {} }, { "index": 4650, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4650.png", "answer": "x=-1", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x)", "coords": {} }, { "index": 4651, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2135674953460693, "b": 2.4270472526550293, "orientation": "horizontal" }, "optimization": { "final_loss": 4.453341261978494e-06, "converged": true, "iterations": 125 }, "output_path": "results/train/hyperbola/problem_4651.png", "answer": "x^2/5 - y^2/20 = 1", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 20*x);Eccentricity(G)=sqrt(5);OneOf(Focus(G))=Focus(H)", "coords": {} }, { "index": 4652, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4652.png", "answer": "[1/8,1/4)", "fact_expr": "C: Ellipse;Expression(C) = (x^2/2 + y^2 = 1);l: Line;F: Point;LeftFocus(C) = F;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, C) = {A, B};M: Point;Intersection(PerpendicularBisector(LineSegmentOf(A, B)), xAxis) = M", "coords": {} }, { "index": 4653, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4653.png", "answer": "13", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 - y^2/16 = 1);PointOnCurve(P, G);LeftFocus(G) =F1;RightFocus(G)=F2;Abs(LineSegmentOf(P, F1)) = 7", "coords": {} }, { "index": 4654, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.0, "b": 0.5, "major_axis": "y", "x_coef": 0.25, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4654.png", "answer": "y=x", "fact_expr": "G: Ellipse;Expression(G) = (4*x^2 + y^2 = 1);H: Line;Expression(H) = (y = m + x);m: Number;WhenMax(Length(InterceptChord(H, G)))", "coords": {} }, { "index": 4655, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4655.png", "answer": "(2,2)", "fact_expr": "G: Parabola;A: Point;M: Point;F: Point;S:Number;Expression(G) = (y^2 = 2*x);Coordinate(A) = (3, 2);Focus(G)=F;PointOnCurve(M,G);Distance(A,F)+Distance(A,M)=Abs(LineSegmentOf(M,A))+Abs(LineSegmentOf(M,F));Abs(LineSegmentOf(M,A))+Abs(LineSegmentOf(M,F))=S;WhenMin(S)", "coords": { "A": [ 3.0, 2.0 ] } }, { "index": 4656, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 4.0, "major_axis": "x", "x_coef": 36.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4656.png", "answer": "16", "fact_expr": "G: Ellipse;Expression(G) = (x^2/36 + y^2/16 = 1);P: Point;PointOnCurve(P, G);F1: Point;F2: Point;Focus(G) = {F1, F2};IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2))", "coords": {} }, { "index": 4657, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4658, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4658.png", "answer": "x^2/6 - y^2/3 = 1", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/2 - y^2 = 1);Coordinate(Focus(Z)) = (3, 0);Z: Hyperbola;Asymptote(G) = Asymptote(Z)", "coords": {} }, { "index": 4659, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4659.png", "answer": "sqrt(2)", "fact_expr": "G: Parabola;H: Point;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(H) = (0, -1);PointOnCurve(P, G)", "coords": { "H": [ 0.0, -1.0 ] } }, { "index": 4660, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4660.png", "answer": "x = pm*sqrt(5)/5", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/4 = 1)", "coords": {} }, { "index": 4661, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4661.png", "answer": "(1, 2)", "fact_expr": "G: Hyperbola;a: Number;Expression(G) = (-y^2/3 + x^2/a^2 = 1);a > 1", "coords": {} }, { "index": 4662, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4662.png", "answer": "x = -1", "fact_expr": "l: Line;G: Parabola;p: Number;A: Point;B: Point;P: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(P) = (1, 1);Slope(l) = 2;Intersection(l, G) = {A, B};IsChordOf(LineSegmentOf(A, B), G);MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ 1.0, 1.0 ] } }, { "index": 4663, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4663.png", "answer": "sqrt(5)+2", "fact_expr": "l: Line;E: Parabola;A: Point;F: Point;B: Point;C: Point;Expression(E) = (y^2 = 4*x);Focus(E) = F;Expression(l)=(y = k*(x - 1));Intersection(l, E) = {A, B};Intersection(l,yAxis)=C;Abs(LineSegmentOf(A, F)) = Abs(LineSegmentOf(B, C));k:Number", "coords": {} }, { "index": 4664, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4665, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4665.png", "answer": "x^2+(3/2)*y^2=1", "fact_expr": "E: Ellipse;b: Number;G: Line;A: Point;F1: Point;B: Point;F2: Point;0 < b;b < 1;Expression(E) = (x^2 + y^2/b^2 = 1);LeftFocus(E) = F1;RightFocus(E) = F2;PointOnCurve(F1, G);Intersection(G, E) = {A, B};Abs(LineSegmentOf(A, F1)) = 3*Abs(LineSegmentOf(B, F1));IsPerpendicular(LineSegmentOf(A,F2),xAxis)", "coords": {} }, { "index": 4666, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4666.png", "answer": "4/7", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F2), xAxis)", "coords": {} }, { "index": 4667, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4667.png", "answer": "24", "fact_expr": "G: Hyperbola;O: Origin;A: Point;B: Point;Expression(G) = (x^2/3 - y^2/4 = 1);PointOnCurve(A,G);PointOnCurve(B,G);Negation(A=B);IsPerpendicular(LineSegmentOf(O,A),LineSegmentOf(O,B))", "coords": {} }, { "index": 4668, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4669, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4670, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4670.png", "answer": "sqrt(5)/2", "fact_expr": "C: Hyperbola;a: Number;G: Parabola;l: Line;e: Number;Expression(C) = (-y^2 + x^2/a^2 = 1);Expression(G) = (y^2 = 8*x);Directrix(G) = l;IsTangent(l, C);Eccentricity(C) = e", "coords": {} }, { "index": 4671, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4672, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_4672.png", "answer": "x^2 - y^2/3 = 1", "fact_expr": "C: Hyperbola;m: Number;n: Number;Negation(m*n=0);Expression(C) = (x^2/m^2 - y^2/n^2 = 1);E: Parabola;Expression(E) = (y^2 = 8*x);RightFocus(C) = Focus(E);Eccentricity(C) = 2", "coords": {} }, { "index": 4673, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4673.png", "answer": "y^2=12*(x+3)", "fact_expr": "G: Hyperbola;H: Parabola;Expression(G) = (x^2/4 - y^2/5 = 1);Focus(H)= Center(G);LeftFocus(G)=Vertex(H)", "coords": {} }, { "index": 4674, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4674.png", "answer": "11", "fact_expr": "G: Hyperbola;A: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 - y^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(A, LeftPart(G));Abs(LineSegmentOf(A, F1)) = 5", "coords": {} }, { "index": 4675, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4675.png", "answer": "2*sqrt(2)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(C) = F;l: Line;Directrix(C) = l;A: Point;Intersection(l, xAxis) = A;M: Point;PointOnCurve(M, C);IsPerpendicular(LineSegmentOf(M, F), xAxis);G: Circle;IsDiameter(LineSegmentOf(A, F), G);Length(InterceptChord(LineOf(A, M), G)) = 2", "coords": {} }, { "index": 4676, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4676.png", "answer": "4*sqrt(3)/3", "fact_expr": "G: Ellipse;A: Point;P: Point;Expression(G) = (x^2 + 4*y^2 = 4);Coordinate(A) = (0, 1);PointOnCurve(A, G);PointOnCurve(P, G);Negation(P = A)", "coords": { "A": [ 0.0, 1.0 ] } }, { "index": 4677, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4677.png", "answer": "-4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;H: Line;PointOnCurve(F, H);A: Point;B: Point;Intersection(H, G) = {A, B};O: Origin;k1: Number;k2: Number;Slope(LineOf(O, A)) = k1;Slope(LineOf(O, B)) = k2", "coords": {} }, { "index": 4678, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4679, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4680, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4680.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;b: Number;a: Number;O: Origin;A: Point;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;Quadrant(A) = 1;PointOnCurve(A, G);DotProduct(VectorOf(O, A), VectorOf(O, F)) = Abs(VectorOf(O, F))^2;Expression(LineOf(O,A)) = (y = x*(2*sqrt(3)/3))", "coords": {} }, { "index": 4681, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4681.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;F: Point;LeftFocus(G) = F;O: Circle;Expression(O) = (x^2 + y^2 = a^2);L1: Line;L2: Line;TangentOfPoint(F, O) = {L1, L2};A: Point;C: Point;B: Point;TangentPoint(L1, O) = A;TangentPoint(L2, O) = B;LeftVertex(G) = C;AngleOf(A, C, B) = ApplyUnit(120, degree)", "coords": {} }, { "index": 4682, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4682.png", "answer": "(1,3)+(3,+oo)", "fact_expr": "G: Ellipse;m: Number;H: Line;Expression(G) = (y^2/3 + x^2/m = 1);Expression(H) = (y = x + 2);NumIntersection(H, G) = 2", "coords": {} }, { "index": 4683, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4683.png", "answer": "(3/2,-6/5)", "fact_expr": "J: Point;Coordinate(J) = (3, 0);PointOnCurve(J,L) = True;L: Line;Slope(L) = 4/5;G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1)", "coords": { "J": [ 3.0, 0.0 ] } }, { "index": 4684, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4685, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4685.png", "answer": "x - 2*y = 0", "fact_expr": "G: Parabola;H: LineSegment;A: Point;Expression(G) = (y^2 = 2*x);Coordinate(A) = (4, 2);MidPoint(H) = A;IsChordOf(H, G)", "coords": { "A": [ 4.0, 2.0 ] } }, { "index": 4686, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4686.png", "answer": "(1,3]", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;Focus(G)={F1,F2};P: Point;PointOnCurve(P,G);Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 4687, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4688, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4688.png", "answer": "sqrt(2)/6", "fact_expr": "G: Parabola;p: Number;H: Circle;p>0;Z: Point;X: Point;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (x^2 + (y - 1)^2 = 1);Intersection(G, H) = {Z, X};Distance(Z, X) = 2*sqrt(3)/3", "coords": {} }, { "index": 4689, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4689.png", "answer": "sqrt(7)/2", "fact_expr": "G: Hyperbola;e: Number;Expression(G) = (x^2/4 - y^2/3 = 1);Eccentricity(G) = e", "coords": {} }, { "index": 4690, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4690.png", "answer": "(0, 1/2]", "fact_expr": "C: Ellipse;b: Number;a: Number;G: Line;A: Point;F: Point;B: Point;O:Origin;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;PointOnCurve(O,G);Intersection(G,C) = {A, B};AngleOf(A,F,B)>=ApplyUnit(120,degree)", "coords": {} }, { "index": 4691, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4691.png", "answer": "6", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1)", "coords": {} }, { "index": 4692, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4693, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4693.png", "answer": "2", "fact_expr": "G: Parabola;H: Line;O: Origin;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, F)) = 2", "coords": {} }, { "index": 4694, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4694.png", "answer": "y=pm*(4/3)*x", "fact_expr": "G: Hyperbola;a: Number;H: Parabola;a>0;Expression(G) = (-y^2/9 + x^2/a^2 = 1);Expression(H) = (y^2 = 20*x);Focus(H) = RightFocus(G)", "coords": {} }, { "index": 4695, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4696, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4696.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "C: Parabola;p: Number;A: Point;O: Origin;B: Point;F: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;G: Line;PointOnCurve(F, G);Intersection(G, C) = {A, B};Abs(LineSegmentOf(A, F)) = 2*Abs(LineSegmentOf(B, F));Area(TriangleOf(A, O, B)) = 6*sqrt(2)", "coords": {} }, { "index": 4697, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4697.png", "answer": "y = -1", "fact_expr": "G: Parabola;p: Number;Expression(G) = (x^2 = 2*p*y);Coordinate(Focus(G)) = (0, 1)", "coords": {} }, { "index": 4698, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4698.png", "answer": "x^2=(16/5)*y", "fact_expr": "C: Parabola;p: Number;A: Point;P: Point;H: Point;0 < p;p < 4;Expression(C) = (x^2 = 2*(p*y));Coordinate(A) = (0, 4);Directrix(C)=l;PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P,H),l);FootPoint(LineSegmentOf(P,H),l)=H;Abs(LineSegmentOf(P, H)) = Abs(LineSegmentOf(P, A));AngleOf(A, P, H) = ApplyUnit(120, degree);l:Line", "coords": { "A": [ 0.0, 4.0 ] } }, { "index": 4699, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4699.png", "answer": "(1/2)*(a-p)", "fact_expr": "A: Point;B: Point;a: Number;Length(LineSegmentOf(A, B)) = a;Endpoint(LineSegmentOf(A, B)) = {A, B};G: Parabola;Expression(G) = (y^2=2*p*x);p>0;a>2*p;PointOnCurve(A, G) = True;PointOnCurve(B, G) = True;M: Point;MidPoint(LineSegmentOf(A, B)) = M;p: Number", "coords": {} }, { "index": 4700, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4700.png", "answer": "3", "fact_expr": "C: Ellipse;a: Number;b: Number;F1: Point;P: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, C);LeftFocus(C) =F1;RightFocus(C)=F2;AngleOf(F1, P, F2) = ApplyUnit(90, degree);Area(TriangleOf(P,F1,F2)) = 9", "coords": {} }, { "index": 4701, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4702, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4702.png", "answer": "(0, 6)", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 2*(p*y));p: Number;p>0;H: Circle;Expression(H) = (6*y + x^2 + y^2 - 16 = 0);A: Point;B: Point;Intersection(Directrix(G), H) = {A, B};LineSegmentOf(A, B) = 8", "coords": {} }, { "index": 4703, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4703.png", "answer": "sqrt(3)/3", "fact_expr": "A: Point;T: Point;B: Point;F: Point;C: Ellipse;Coordinate(A) = (0, b);Coordinate(B) = (0, -b);Coordinate(F) = (c, 0);Coordinate(T) = (3*c, 0);Expression(C) = (y^2/b^2 + x^2/a^2 = 1);a:Number;b:Number;c:Number;a>b;b>0;c = sqrt(a^2 - b^2);PointOnCurve(Intersection(LineOf(A,T),LineOf(B,F)), C)", "coords": {} }, { "index": 4704, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4704.png", "answer": "(1, 2)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/12 = 1);F: Point;RightFocus(G) = F;E: Hyperbola;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;Distance(F, Asymptote(E))0;b>0;RightFocus(M) = F;P: Point;PointOnCurve(P, M);O: Origin;Abs(LineSegmentOf(O, P)) = 2*b;AngleOf(P, O, F) = pi/6", "coords": {} }, { "index": 4710, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4710.png", "answer": "x^2/5+y^2=1\n2*sqrt(5)/5", "fact_expr": "H: Line;Expression(H) = (x - 2*y + 2 = 0);G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;PointOnCurve(OneOf(Vertex(G)), H) = True;PointOnCurve(OneOf(Focus(G)), H) = True", "coords": {} }, { "index": 4711, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4711.png", "answer": "sqrt(3)/2", "fact_expr": "P: Point;B1: Point;G: Ellipse;a: Number;b: Number;B2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);UpperVertex(G) = B2;LowerVertex(G) = B1;PointOnCurve(P, G);Slope(LineOf(P, B1)) = 1/4;Slope(LineOf(P, B2)) = -1", "coords": {} }, { "index": 4712, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4712.png", "answer": "3*sqrt(5) - 1", "fact_expr": "G: Parabola;A: Point;P: Point;M: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (4, 6);PointOnCurve(P, G);Projection(P, yAxis) = M", "coords": { "A": [ 4.0, 6.0 ] } }, { "index": 4713, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4713.png", "answer": "8", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);F: Point;Focus(G) = F;l: Line;Directrix(G) = l;P: Point;PointOnCurve(P,G) = True;IsPerpendicular(LineSegmentOf(P,A),l) = True;A: Point;FootPoint(LineSegmentOf(P,A),l) = A;Slope(LineOf(A,F)) = -sqrt(3)", "coords": {} }, { "index": 4714, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4714.png", "answer": "6", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2/5 + y^2/m = 1);Coordinate(OneOf(Focus(G))) = (0, 1)", "coords": {} }, { "index": 4715, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4715.png", "answer": "4", "fact_expr": "G: Parabola;a: Number;M: Point;l: Line;a>0;m:Number;Expression(G) = (x^2 = a*y);Coordinate(M) = (m, 1);PointOnCurve(M, G);Directrix(G) = l;Distance(M, l) = 2", "coords": {} }, { "index": 4716, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4716.png", "answer": "sqrt(5)/2", "fact_expr": "E: Hyperbola;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;F: Point;RightFocus(E) = F;l1: Line;l2: Line;OneOf(Asymptote(E)) = l1;OneOf(Asymptote(E)) = l2;Negation(l1 = l2);H: Line;PointOnCurve(F, H);IsPerpendicular(l1,H);FootPoint(l1,H) = P;P: Point;Intersection(H,l2) = Q;Q: Point;5*VectorOf(P, F) = 3*VectorOf(F, Q)", "coords": {} }, { "index": 4717, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4717.png", "answer": "2*p^2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;C: Point;Intersection(Directrix(G), xAxis) = C;H: Line;PointOnCurve(F, H);A: Point;B: Point;Intersection(H, G) = {A, B};DotProduct(VectorOf(B, C), VectorOf(B, F)) = 0", "coords": {} }, { "index": 4718, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4718.png", "answer": "1", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/3 - y^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) + Abs(LineSegmentOf(P, F2)) = 2*sqrt(5)", "coords": {} }, { "index": 4719, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4720, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4720.png", "answer": "y=pm*sqrt(2)*x/2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/2 = 1)", "coords": {} }, { "index": 4721, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4721.png", "answer": "sqrt(21)/3", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;l: Line;PointOnCurve(F2, l) = True;Intersection(l, RightPart(C)) = {P, Q};P: Point;Q: Point;Abs(LineSegmentOf(P, F1)) = 3*Abs(LineSegmentOf(P, F2));Abs(LineSegmentOf(P, Q)) = 3*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 4722, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4722.png", "answer": "pm*sqrt(3)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;O: Origin;L1: Line;L2: Line;OneOf(Asymptote(C)) = L1;OneOf(Asymptote(C)) = L2;Negation(L1=L2);L: Line;PointOnCurve(F2, L);IsPerpendicular(L, L1);P: Point;FootPoint(L, L1) = P;Q: Point;Intersection(LineSegmentOf(P, F1), L2) = Q;3*VectorOf(F1, Q) = 2*VectorOf(F1, P)", "coords": {} }, { "index": 4723, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4723.png", "answer": "4", "fact_expr": "C: Ellipse;P: Point;Q: Point;Expression(C) = (x^2/4 + y^2/3 = 1);PointOnCurve(P, C);PointOnCurve(Q, C)", "coords": {} }, { "index": 4724, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4724.png", "answer": "1/2", "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;I: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2;Incenter(TriangleOf(P, F1, F2)) = I;Area(TriangleOf(P, I, F1)) + Area(TriangleOf(P, I, F2)) = 2*Area(TriangleOf(I, F1, F2))", "coords": {} }, { "index": 4725, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.4641016151377544, "major_axis": "x", "x_coef": 16.0, "y_coef": 12.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4725.png", "answer": "3", "fact_expr": "C: Hyperbola;G: Ellipse;P: Point;F2: Point;M: Point;Expression(G) = (x^2/16 + y^2/12 = 1);InterReciprocal(Eccentricity(C), Eccentricity(G));PointOnCurve(P, RightPart(C));Focus(C) = {F1,F2};Distance(P, F2) = 4;MidPoint(LineSegmentOf(P, F2)) = M;O:Origin;F1:Point;Focus(G)={F1,F2};RightFocus(C)=F2", "coords": {} }, { "index": 4726, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4727, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4727.png", "answer": "{-sqrt(5)/2, -1, 1, sqrt(5)/2}", "fact_expr": "G: Hyperbola;H: Line;k: Real;Expression(G) = (x^2 - y^2 = 4);Expression(H) = (y = k*x - 1);NumIntersection(H, G)=1", "coords": {} }, { "index": 4728, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4728.png", "answer": "1", "fact_expr": "C: Parabola;A: Point;Expression(C) = (x^2 = 4*y);PointOnCurve(A, C);Distance(A, Focus(C)) = 2;yA: Number;YCoordinate(A) = yA", "coords": {} }, { "index": 4729, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4729.png", "answer": "sqrt(17)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 8*x);F: Point;Focus(C) = F;G: Line;M: Point;Coordinate(M) = (-2, 0);PointOnCurve(M, G);A: Point;B: Point;Intersection(G, C) = {A, B};O: Origin;IsParallel(LineSegmentOf(O, A), LineSegmentOf(B, F))", "coords": { "M": [ -2.0, 0.0 ] } }, { "index": 4730, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4730.png", "answer": "4", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1);P: Point;PointOnCurve(P, LeftPart(G)) = True;Distance(P, RightFocus(G)) = 13", "coords": {} }, { "index": 4731, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4732, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4733, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4734, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4735, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4735.png", "answer": "sqrt(2)", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;O:Origin;A: Point;B: Point;F: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;PointOnCurve(O,G);Intersection(G,C)={A,B};Abs(LineSegmentOf(A, B)) = sqrt(10);Abs(LineSegmentOf(A, F)) = sqrt(2);Cos(AngleOf(A, B, F)) = (2*sqrt(5))/5;Length(RealAxis(C)) = 2*a", "coords": {} }, { "index": 4736, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4736.png", "answer": "20", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);H: Line;PointOnCurve(Focus(G), H);A: Point;B: Point;x1: Number;x2: Number;y1: Number;y2: Number;Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);Intersection(H, G) = {A, B};x1 + x2 = 16", "coords": {} }, { "index": 4737, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "major_axis": "x", "x_coef": 3.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4737.png", "answer": "(0,pm*1)", "fact_expr": "G: Ellipse;F1: Point;A: Point;F2: Point;B: Point;Expression(G) = (x^2/3 + y^2 = 1);Focus(G) = {F1, F2};PointOnCurve(A, G);PointOnCurve(B, G);VectorOf(F1, A) = 5*VectorOf(F2, B)", "coords": {} }, { "index": 4738, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4739, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4739.png", "answer": "2*sqrt(3)/3", "fact_expr": "C: Hyperbola;b: Number;a: Number;M: Point;F: Point;N: Point;l1: Line;l2: Line;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;OneOf(Asymptote(G))=l1;OneOf(Asymptote(G))=l2;Negation(l1=l2);L:Line;PointOnCurve(F, L);IsPerpendicular(l1, L);FootPoint(l1,L)=M;Intersection(l2,L)=N;2*VectorOf(M, F) = VectorOf(F, N)", "coords": {} }, { "index": 4740, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4740.png", "answer": "2", "fact_expr": "G: Parabola;H: Point;Expression(G) = (y^2 = 4*x);Coordinate(H) = (m, 2);PointOnCurve(H,G);m:Number", "coords": {} }, { "index": 4741, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4742, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4742.png", "answer": "48/5", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);A: Point;B: Point;C: Point;F: Point;Coordinate(F) = (3, 0);PointOnCurve(A, G);PointOnCurve(B, G);PointOnCurve(C, G);VectorOf(F, A) + VectorOf(F, B) + VectorOf(F, C) = 0", "coords": { "F": [ 3.0, 0.0 ] } }, { "index": 4743, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 5.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4743.png", "answer": "y=-5", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 20*y)", "coords": {} }, { "index": 4744, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4745, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4745.png", "answer": "3", "fact_expr": "C: Ellipse;a: Number;b: Number;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);E: Parabola;p: Number;p>0;Expression(E) = (x^2 = 2*(p*y));F: Point;Focus(E) = F;UpperVertex(C) = F;P: Point;OneOf(Intersection(C, E)) = P;Eccentricity(C) = sqrt(3)/3;Abs(LineSegmentOf(P, F)) = 2", "coords": {} }, { "index": 4746, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4746.png", "answer": "(x-5)^2+y^2=16", "fact_expr": "H: Ellipse;Expression(H) = (x^2/169 + y^2/144 = 1);Z: Circle;Center(Z) = RightFocus(H);G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);IsTangent(Asymptote(G), Z)", "coords": {} }, { "index": 4747, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4748, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4748.png", "answer": "sqrt(2)", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;A: Point;O: Origin;B: Point;l1: Line;l2: Line;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2 + y^2 = a^2);TangentOfPoint(OneOf(Focus(C)), G) = {l1, l2};TangentPoint(l1, G) = A;TangentPoint(l2, G) = B;AngleOf(A, O, B) = ApplyUnit(90, degree)", "coords": {} }, { "index": 4749, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4749.png", "answer": "2*sqrt(14)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/5 = 1)", "coords": {} }, { "index": 4750, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4751, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4752, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4752.png", "answer": "3", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;PointOnCurve(P, C);IsPerpendicular(VectorOf(P, F1), VectorOf(P, F2));Area(TriangleOf(P, F1, F2)) = 9", "coords": {} }, { "index": 4753, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4754, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4754.png", "answer": "(2,2)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x);F: Point;Focus(G) = F;P: Point;PointOnCurve(P, G) = True;A: Point;Coordinate(A) = (3, 2);WhenMin(Abs(LineSegmentOf(P, A)) + Abs(LineSegmentOf(P, F)))", "coords": { "A": [ 3.0, 2.0 ] } }, { "index": 4755, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4755.png", "answer": "4/3", "fact_expr": "F: Point;C: Parabola;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F, l) = True;l: Line;Intersection(l, C) = {A, B};A: Point;B: Point;Intersection(l, Directrix(C)) = D;D: Point;MidPoint(LineSegmentOf(A, D)) = F", "coords": {} }, { "index": 4756, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.449489742783178, "b": 2.449489742783178, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4756.png", "answer": "(-sqrt(15)/3,-1)", "fact_expr": "H: Line;Expression(H) = (y = k*x + 2);k: Number;G: Hyperbola;Expression(G) = (x^2 - y^2 = 6);NumIntersection(H, RightPart(G)) = 2", "coords": {} }, { "index": 4757, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4757.png", "answer": "8", "fact_expr": "C: Ellipse;Expression(C)=(x^2/a^2+y^2/b^2=1);a: Number;b: Number;a > b;b > 0;F1:Point;F2:Point;LeftFocus(C)=F1;RightFocus(C)=F2;Abs(LineSegmentOf(F1,F2))=2*sqrt(2);O:Origin;B:Point;OneOf(Endpoint(MinorAxis(C)))=B;Area(TriangleOf(B,F1,O))=sqrt(7)", "coords": {} }, { "index": 4758, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4758.png", "answer": "(2,0)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 8*x)", "coords": {} }, { "index": 4759, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4759.png", "answer": "5", "fact_expr": "G: Parabola;P: Point;F: Point;A: Point;Expression(G) = (y^2 = 8*x);PointOnCurve(P, G);Coordinate(A)=(3, 1);Focus(G) = F", "coords": { "A": [ 3.0, 1.0 ] } }, { "index": 4760, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4760.png", "answer": "x+2*y-2=0", "fact_expr": "l: Line;C: Ellipse;A: Point;B: Point;P: Point;Coordinate(P) = (1, 1/2);Expression(C) = (x^2/4 + y^2 = 1);PointOnCurve(P, l);Intersection(l, C) = {A, B};MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ 1.0, 0.5 ] } }, { "index": 4761, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4762, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4762.png", "answer": "5", "fact_expr": "G: Ellipse;H: Line;A: Point;B: Point;F2: Point;F1: Point;Expression(G) = (x^2/9 + y^2/5 = 1);Focus(G) = {F1, F2};PointOnCurve(F1, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 5", "coords": {} }, { "index": 4763, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4763.png", "answer": "y = pm*(2/3)*x", "fact_expr": "G: Hyperbola;a: Number;Expression(G) = (x^2/9 - y^2/a = 1);Coordinate(RightFocus(G)) = (sqrt(13), 0)", "coords": {} }, { "index": 4764, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.8284271247461903, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4764.png", "answer": "y = pm*(sqrt(2)/2)*x", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/8 - y^2/4 = 1)", "coords": {} }, { "index": 4765, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4765.png", "answer": "8", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);F: Point;Focus(G) = F;H: Line;PointOnCurve(F, H);A: Point;B: Point;Intersection(H, G) = {A, B};M: Point;MidPoint(LineSegmentOf(A, B)) = M;Z: Line;PointOnCurve(M, Z);IsPerpendicular(Z, yAxis);N: Point;Intersection(Z, yAxis) = N;Abs(LineSegmentOf(M, N)) = 2", "coords": {} }, { "index": 4766, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4767, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_4767.png", "answer": "1/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;F: Point;A:Point;d1:Number;d2:Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (-a*y + b*x = 0);Eccentricity(G)=2;RightVertex(G)=A;RightFocus(G)=F;Distance(A, H) = d1;Distance(F,H)=d2", "coords": {} }, { "index": 4768, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4768.png", "answer": "11", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;Expression(G) = (y^2 = 6*x);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);PointOnCurve(Focus(G), H);Intersection(H, G) = {A,B};x1+x2=8;x1:Number;y1:Number;x2:Number;y2:Number", "coords": {} }, { "index": 4769, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4769.png", "answer": "18", "fact_expr": "G: Ellipse;H: Line;P: Point;Q: Point;F: Point;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(Center(G), H);Intersection(H, G) = {P, Q};OneOf(Focus(G)) = F", "coords": {} }, { "index": 4770, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4770.png", "answer": "6", "fact_expr": "G: Parabola;H: Line;k: Number;B: Point;A: Point;Expression(G) = (y^2 = 8*x);Expression(H) = (y = k*x - 2);Intersection(H, G) = {A, B};XCoordinate(MidPoint(LineSegmentOf(A, B))) = 3", "coords": {} }, { "index": 4771, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 2.8284271247461903 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_4771.png", "answer": "x^2/8 + y^2/2 = 1", "fact_expr": "G: Circle;P: Point;D: Point;M: Point;Expression(G) = (x^2 + y^2 = 8);PointOnCurve(P, G);PointOnCurve(P, LineSegmentOf(P, D));IsPerpendicular(LineSegmentOf(P, D), xAxis);FootPoint(LineSegmentOf(P, D), xAxis) = D;MidPoint(LineSegmentOf(P, D)) = M", "coords": {} }, { "index": 4772, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4772.png", "answer": "1", "fact_expr": "G: Hyperbola;H: Line;k: Number;Expression(G) = (x^2 - y^2 = 1);k > 0;Expression(H) = (y = k*x);OneOf(Asymptote(G)) = H", "coords": {} }, { "index": 4773, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4773.png", "answer": "4*x-3*y-4=0", "fact_expr": "G: Parabola;B: Point;A: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(A, G);PointOnCurve(B, G);VectorOf(A, F) = 4*VectorOf(F, B);Quadrant(A)=1", "coords": {} }, { "index": 4774, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4774.png", "answer": "-4/3", "fact_expr": "G: Ellipse;H:Line;A: Point;B: Point;M: Point;Expression(G) = (x^2/4 + y^2/3 = 1);Coordinate(M) = (m, 1);Slope(H)=1;Intersection(H, G) = {A, B};MidPoint(LineSegmentOf(A, B)) = M;m:Number", "coords": {} }, { "index": 4775, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4775.png", "answer": "4", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/4 - y^2/3 = 1);M: Point;N: Point;PointOnCurve(M, C);PointOnCurve(N, C);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2", "coords": {} }, { "index": 4776, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/train/hyperbola/problem_4776.png", "answer": "y=pm*(\\sqrt{3})*x", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;FocalLength(C) = 4;A: Point;Coordinate(A) = (2, 3);PointOnCurve(A, C)", "coords": { "A": [ 2.0, 3.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 4777, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4777.png", "answer": "(-1/4, 1)", "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (y^2 = -4*x);Coordinate(A) = (-2,1);Focus(G) = F;PointOnCurve(P, G);WhenMin(Abs(LineSegmentOf(P,F))+Abs(LineSegmentOf(P,A)))", "coords": { "A": [ -2.0, 1.0 ] } }, { "index": 4778, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4778.png", "answer": "x^2+(y-4)^2=64", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 16*y);H: Circle;Center(H) = Focus(G);IsTangent(Directrix(G), H)", "coords": {} }, { "index": 4779, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.4641016151377544, "major_axis": "x", "x_coef": 16.0, "y_coef": 12.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4779.png", "answer": "6", "fact_expr": "G: Ellipse;M: Point;F1: Point;F2: Point;O: Origin;N: Point;Expression(G) = (x^2/16 + y^2/12 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(M, G);MidPoint(LineSegmentOf(M, F1)) = N;LineSegmentOf(O, N) = 1", "coords": {} }, { "index": 4780, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4781, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.75, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4781.png", "answer": "12*x-8*y-7=0", "fact_expr": "l: Line;C: Parabola;A: Point;F: Point;B: Point;Expression(C) = (y^2 = 3*x);Focus(C) = F;Slope(l)=3/2;Intersection(l, C) = {A, B};Intersection(l,xAxis)=P;Abs(LineSegmentOf(A,F))+Abs(LineSegmentOf(B,F))=4;P:Point", "coords": {} }, { "index": 4782, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4783, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4783.png", "answer": "sqrt(5)-sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2/5 = 1);F: Point;LeftFocus(G) = F;H: Circle;Expression(H) = (x^2 + y^2 = 3);P: Point;T: Point;TangentOfPoint(F, H) = LineOf(F,P);Intersection(LineSegmentOf(F, P), RightPart(G)) = P;TangentPoint(LineOf(F, P), H) = T;M: Point;MidPoint(LineSegmentOf(F, P)) = M;O: Origin", "coords": {} }, { "index": 4784, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4785, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4785.png", "answer": "3+sqrt(3)", "fact_expr": "C: Ellipse;b: Number;a: Number;G: Line;A: Point;F: Point;B: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y = sqrt(3)*x);Intersection(G, C) = {A, B};Abs(LineSegmentOf(A, B)) = 4*sqrt(3);RightFocus(C) = F;IsPerpendicular(LineSegmentOf(A, F), LineSegmentOf(B, F))", "coords": {} }, { "index": 4786, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4787, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4787.png", "answer": "2*sqrt(3)", "fact_expr": "G: Parabola;p: Number;H: Line;B: Point;A: Point;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;PointOnCurve(F, H);Slope(H) = 1;Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, F)) - Abs(LineSegmentOf(B, F)) = sqrt(6)", "coords": {} }, { "index": 4788, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4789, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4790, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4790.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 8*y);F: Point;Focus(G) = F;H: Line;PointOnCurve(F, H);A: Point;B: Point;Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, F)) = 4", "coords": {} }, { "index": 4791, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 3.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4791.png", "answer": "2*x - 3*y + 5 = 0", "fact_expr": "l: Line;G: Ellipse;B: Point;A: Point;M: Point;Expression(G) = (x^2/3 + y^2/2 = 1);Coordinate(M) = (-1, 1);PointOnCurve(M, l);Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A,B)) = M", "coords": { "M": [ -1.0, 1.0 ] } }, { "index": 4792, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4793, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4793.png", "answer": "6*sqrt(3)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);A: Point;B: Point;LeftVertex(G) = A;RightVertex(G) = B;P: Point;PointOnCurve(P, G);Negation(P = A);Negation(P = B);H: Line;Expression(H) = (x = 10);M: Point;Intersection(LineOf(A,P),H)=M;N: Point;Intersection(LineOf(B,P),H)=N", "coords": {} }, { "index": 4794, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.0, "major_axis": "y", "x_coef": 4.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4794.png", "answer": "2*x+y-2=0", "fact_expr": "l: Line;G: Ellipse;Expression(G) = (x^2/4 + y^2/16 = 1);Coordinate(MidPoint(InterceptChord(l,G))) = (1/2, 1)", "coords": {} }, { "index": 4795, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4795.png", "answer": "4*sqrt(3)/3", "fact_expr": "G: Parabola;p: Number;A: Point;F: Point;P: Point;p>0;Expression(G) = (y^2 = 2*p*x);Coordinate(A) = (3, b);Quadrant(A)=1;Focus(G)=F;Distance(A,F)=4;PointOnCurve(P, Directrix(G));WhenMin(Perimeter(TriangleOf(P,A,F)));b:Number", "coords": {} }, { "index": 4796, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4796.png", "answer": "2", "fact_expr": "A: Point;B: Point;G: Parabola;Expression(G) = (x^2 = 4*y);PointOnCurve(A, G);PointOnCurve(B, G);Length(LineSegmentOf(A,B)) = 6;MidPoint(LineSegmentOf(A,B)) = M;M: Point", "coords": {} }, { "index": 4797, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2135674953460693, "b": 2.4270472526550293, "orientation": "horizontal" }, "optimization": { "final_loss": 4.453341261978494e-06, "converged": true, "iterations": 125 }, "output_path": "results/train/hyperbola/problem_4797.png", "answer": "5/2", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Eccentricity(C) = sqrt(5);Distance(Focus(C), Asymptote(C)) = 5", "coords": {} }, { "index": 4798, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.125, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4798.png", "answer": "-1/2", "fact_expr": "L: Line;Expression(L) = (m + 2*x + 4*y = 0);m: Number;G: Parabola;Expression(G) = (y = 2*x^2);PointOnCurve(Focus(G), L) = True", "coords": {} }, { "index": 4799, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4799.png", "answer": "sqrt(6)/3", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;l: Line;PointOnCurve(RightFocus(G), l);A: Point;B: Point;Intersection(l, G) = {A, B};M: Point;Intersection(l, yAxis) = M;VectorOf(A, F2) = 3*VectorOf(F2, B);VectorOf(M, A) = VectorOf(A, F2)", "coords": {} }, { "index": 4800, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4801, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4802, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4802.png", "answer": "(1/3)*sqrt(3)", "fact_expr": "G: Ellipse;a: Number;b: Number;H: Line;F2: Point;A: Point;B: Point;C: Point;F1: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, H);IsPerpendicular(H, xAxis);Intersection(H, G) = {A, B};Intersection(LineOf(A,F2),G)={A,C};Area(TriangleOf(A, B, C)) = 4*Area(TriangleOf(B, C, F2));Negation(A=C)", "coords": {} }, { "index": 4803, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4803.png", "answer": "16", "fact_expr": "G: Hyperbola;m: Number;F: Point;Expression(G) = (-x^2/9 + y^2/m = 1);Coordinate(F) = (0, 5);OneOf(Focus(G)) = F", "coords": { "F": [ 0.0, 5.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 4804, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.4641016151377544, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 12.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4804.png", "answer": "6", "fact_expr": "G: Parabola;p: Number;H: Ellipse;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (x^2/12 + y^2/3 = 1);Focus(G) = RightFocus(H)", "coords": {} }, { "index": 4805, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4805.png", "answer": "16/3", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, C) = {A, B};Distance(A, Directrix(C)) = 4", "coords": {} }, { "index": 4806, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_ecc_mismatch;ellipse_focus_mismatch", "params": { "a": 1.9257268905639648, "b": 1.7993911322594118, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 2.5576271215184017e-15, "converged": true, "iterations": 2 }, "output_path": "results/train/ellipse/problem_4806.png", "answer": "x^2/16 + y^2/12 = 1", "fact_expr": "G: Ellipse;m: Number;n: Number;F: Point;m>0;n>0;Expression(G) = (y^2/n^2 + x^2/m^2 = 1);Coordinate(F) = (2, 0);RightFocus(G) = F;Eccentricity(G) = 1/2", "coords": { "F": [ 2.0, 0.0 ] }, "validation_reasons": [ "ellipse_ecc_mismatch", "ellipse_focus_mismatch" ] }, { "index": 4807, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4807.png", "answer": "pm*2*sqrt(2)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);H: Line;A: Point;F: Point;B: Point;Focus(G) = F;PointOnCurve(F,H);Intersection(H, G) = {A, B};WhenMin(Abs(LineSegmentOf(A, F)) + 4*Abs(LineSegmentOf(B, F)))", "coords": {} }, { "index": 4808, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4808.png", "answer": "12", "fact_expr": "C: Parabola;F: Point;M: Point;N: Point;Expression(C) = (y^2 = 16*x);Focus(C) = F;PointOnCurve(M, C);Intersection(OverlappingLine(LineSegmentOf(F,M)),yAxis)=N;MidPoint(LineSegmentOf(F, N)) = M", "coords": {} }, { "index": 4809, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4809.png", "answer": "Answer missing", "fact_expr": "F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;PointOnCurve(F1,LineOf(A,B)) = True;Intersection(LineOf(A,B),G) = {A,B};A: Point;B: Point;2*Abs(LineSegmentOf(A, F1)) = 3*Abs(LineSegmentOf(F1, B));Abs(LineSegmentOf(B, F2)) = 2*Abs(LineSegmentOf(B, F1))", "coords": {} }, { "index": 4810, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4810.png", "answer": "12", "fact_expr": "E: Ellipse;M: Circle;P: Point;F1: Point;F2: Point;Q: Point;Expression(E) = (x^2/9 + y^2/5 = 1);Expression(M) = (-8*y - 10*x + x^2 + y^2 + 40 = 0);PointOnCurve(Q, M);LeftFocus(E) = F1;RightFocus(E) = F2;PointOnCurve(P, E)", "coords": {} }, { "index": 4811, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4811.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Ellipse;m: Number;n: Number;m > 0;n > 0;Expression(G) = (y^2/n^2 + x^2/m^2 = 1);2/n + 1/m = 1;WhenMin(m*n)", "coords": {} }, { "index": 4812, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4812.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G))) = (y = -2*x)", "coords": {} }, { "index": 4813, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4813.png", "answer": "x^2/4 - y^2/8 = 1", "fact_expr": "l: Line;Expression(l) = (y = sqrt(2)*(x + 2*sqrt(3)));C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;F: Point;LeftFocus(C) = F;PointOnCurve(F, l) = True;NumIntersection(l, C)=1", "coords": {} }, { "index": 4814, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4815, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4816, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4816.png", "answer": "1/2", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;M: Point;N: Point;Intersection(LineOf(M, N), LeftPart(C)) = M;Intersection(LineOf(M, N), RightPart(C)) = N;P: Point;Intersection(LineOf(M, N), RightDirectrix(C)) = P;F: Point;RightFocus(C) = F;lambda: Real;00;Expression(G) = (x^2 = 2*(p*y));a > b;b > 0;Expression(H)=(x^2/b^2+y^2/a^2=1);Focus(G) = F;OneOf(Focus(H)) = F;Intersection(G,H)={P,Q};PointOnCurve(F,LineOf(P,Q))", "coords": {} }, { "index": 4819, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4819.png", "answer": "2*sqrt(2)", "fact_expr": "Gamma: Parabola;Expression(Gamma) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(Gamma) = F;l: Line;Slope(l) = 1;A: Point;B: Point;Intersection(l, Gamma) = {A, B};Abs(LineSegmentOf(A, F)) = 3;Abs(LineSegmentOf(B, F)) = 5", "coords": {} }, { "index": 4820, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4820.png", "answer": "12*sqrt(7)/7", "fact_expr": "G: Ellipse;P: Point;F2: Point;Expression(G) = (x^2/16 + y^2/9 = 1);PointOnCurve(P, G);RightFocus(G) = F2;Distance(P, F2) = 5", "coords": {} }, { "index": 4821, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.02500000037252903, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4821.png", "answer": "1/20", "fact_expr": "G: Parabola;Expression(G) = (y = 10*x^2)", "coords": {} }, { "index": 4822, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.23606797749979, "b": 2.0, "major_axis": "x", "x_coef": 5.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4822.png", "answer": "4", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/5 + y^2/4 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G)", "coords": {} }, { "index": 4823, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4824, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4824.png", "answer": "5/3", "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/16 + y^2/9 = 1)", "coords": {} }, { "index": 4825, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4825.png", "answer": "x=-2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;F: Point;Focus(G) = F;H: Line;PointOnCurve(F, H);Inclination(H) = ApplyUnit(90,degree);A: Point;B: Point;Intersection(H, G) = {A, B};Length(LineSegmentOf(A, B)) = 8", "coords": {} }, { "index": 4826, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4826.png", "answer": "[2,+oo)", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F: Point;O: Origin;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;PointOnCurve(P, RightPart(C));Abs(LineSegmentOf(P, F)) - Abs(LineSegmentOf(P, O)) = 2*a", "coords": {} }, { "index": 4827, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4828, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4828.png", "answer": "10", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P: Point;PointOnCurve(P,G) = True;L: Line;PointOnCurve(P,L) = True;IsPerpendicular(L,Directrix(G)) = True;FootPoint(L,Directrix(G)) = M;M: Point;Focus(G) = F;F: Point;Abs(LineSegmentOf(P, F)) = 5", "coords": {} }, { "index": 4829, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4830, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4831, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4831.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "P: Point;A: Point;B: Point;G: Ellipse;Expression(G) = (y^2/n^2 + x^2/m^2 = 1);m: Number;n: Number;m>0;n>0;PointOnCurve(A, G);PointOnCurve(B, G);PointOnCurve(P, G);Negation(A = B);Negation(A = P);Negation(B = P);O: Origin;PointOnCurve(O, LineSegmentOf(A, B)) ;k1: Number;k2: Number;Slope(LineOf(P, A)) = k1;Slope(LineOf(P, B)) = k2;k1*k2 = -2", "coords": {} }, { "index": 4832, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4833, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4833.png", "answer": "7/2", "fact_expr": "A: Point;Coordinate(A) = (3, 2);F: Point;Focus(G) = F;G: Parabola;Expression(G) = (y^2 = 2*x);P: Point;PointOnCurve(P, G) = True", "coords": { "A": [ 3.0, 2.0 ] } }, { "index": 4834, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.1622776601683795, "b": 2.449489742783178, "major_axis": "y", "x_coef": 6.0, "y_coef": 10.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4834.png", "answer": "x^2/12 + y^2/16 = 1", "fact_expr": "G: Ellipse;Z: Ellipse;P: Point;Expression(G) = (x^2/6 + y^2/10 = 1);Coordinate(P) = (-3, 2);PointOnCurve(P, Z);Focus(G) = Focus(Z)", "coords": { "P": [ -3.0, 2.0 ] } }, { "index": 4835, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4835.png", "answer": "-3", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);R: Point;PointOnCurve(R, Directrix(G)) ;L1: Line;L2: Line;TangentOfPoint(R, G) = {L1, L2};TangentPoint(L1, G) = M;TangentPoint(L2, G) = N;M: Point;N: Point;O: Origin", "coords": {} }, { "index": 4836, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4836.png", "answer": "sqrt(3)/2\nx^2/9-y^2/3=1", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H)=(x^2+4*y^2=16);Focus(G)=Focus(H);Expression(OneOf(Asymptote(G)))=(x + sqrt(3)*y = 0)", "coords": {} }, { "index": 4837, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4837.png", "answer": "sqrt(52)", "fact_expr": "G: Curve;P: Point;x1: Number;y1: Number;Expression(G) = (x^2/16 + y^2/9 = 1);Coordinate(P) = (x1, y1);PointOnCurve(P, G)", "coords": {} }, { "index": 4838, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4838.png", "answer": "[(-4-sqrt(7))/12, (-4+sqrt(7))/12]", "fact_expr": "C: Circle;Expression(C) = (-8*y - 8*x + x^2 + y^2 + 28 = 0);P: Point;PointOnCurve(P, C);N: Curve;Expression(N) = (x^2 + 4*y^2 = 4);A: Point;B: Point;Intersection(N, xAxis) = {A, B};O: Origin;M: Point;Intersection(LineOf(O, P), N) = M;k1: Number;k2: Number;k3: Number;Slope(LineOf(M, A)) = k1;Slope(LineOf(M, B)) = k2;Slope(LineOf(O, P)) = k3", "coords": {} }, { "index": 4839, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4839.png", "answer": "3", "fact_expr": "F: Point;C: Parabola;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F, G) = True;Slope(G) = sqrt(3);G: Line;Intersection(G, C) = {A, B};A: Point;B: Point;Abs(LineSegmentOf(F, A)) > Abs(LineSegmentOf(F, B))", "coords": {} }, { "index": 4840, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4840.png", "answer": "(1/2, 1)", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;I: Point;M: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(I) = (c/2, 0);Coordinate(M) = (x0, y0);Quadrant(M)=1;PointOnCurve(M,G);Expression(TangentOnPoint(M,G))=(x0*x/a^2+y0*y/b^2=1);PointOnCurve(M,H);IsPerpendicular(TangentOnPoint(M,G),H);PointOnCurve(I,H);HalfFocalLength(G)=c;c:Number;x0:Number;y0:Number", "coords": {} }, { "index": 4841, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4841.png", "answer": "2", "fact_expr": "C: Parabola;p: Number;F: Point;Expression(C) = (y^2 = 2*(p*x));Coordinate(F) = (1, 0);Focus(C) = F", "coords": { "F": [ 1.0, 0.0 ] } }, { "index": 4842, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4842.png", "answer": "4+sqrt(2)", "fact_expr": "G: Parabola;N: Point;M: Point;F: Point;Expression(G) = (y^2 = 6*x);Coordinate(N) = (5/2, 1);Focus(G) = F;PointOnCurve(M, G)", "coords": { "N": [ 2.5, 1.0 ] } }, { "index": 4843, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4843.png", "answer": "sqrt(2)/2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;Intersection(Directrix(G), xAxis) = K;K: Point;P: Point;PointOnCurve(P, G) = True", "coords": {} }, { "index": 4844, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4844.png", "answer": "2", "fact_expr": "l: Line;C: Parabola;p: Number;B: Point;A: Point;F: Point;M:Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;Inclination(l)=ApplyUnit(30,degree);PointOnCurve(F, l);Intersection(l, C) = {A, B};MidPoint(LineSegmentOf(A,B))=M;XCoordinate(M)=7", "coords": {} }, { "index": 4845, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4845.png", "answer": "3", "fact_expr": "G: Hyperbola;F: Point;Expression(G) = (x^2/4 - y^2/9 = 1);RightFocus(G) = F", "coords": {} }, { "index": 4846, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4846.png", "answer": "[\\sqrt{2}/2,1)", "fact_expr": "G: Ellipse;a: Number;b: Number;a > b;b > 0;P:Point;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P,G);Max(Distance(P,UpperVertex(G)))=Distance(Center(G),Directrix(G))", "coords": {} }, { "index": 4847, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4848, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4849, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4849.png", "answer": "16", "fact_expr": "F: Point;Focus(C) = F;C: Parabola;Expression(C) = (y^2 = 8*x);l: Line;PointOnCurve(F, l) = True;Intersection(l, C) = {A, B};A: Point;B: Point;YCoordinate(MidPoint(LineSegmentOf(A,B))) = 4", "coords": {} }, { "index": 4850, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4850.png", "answer": "-4/9", "fact_expr": "G: Ellipse;H:Line;B: Point;A: Point;O: Origin;P: Point;Expression(G) = (x^2/9 + y^2/4 = 1);Intersection(H, G) = {A, B};MidPoint(LineSegmentOf(A,B)) = P;Slope(H) = k1;Slope(LineOf(O, P)) = k2;k1:Number;k2:Number", "coords": {} }, { "index": 4851, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4851.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(H) = (1, 2);PointOnCurve(H,OneOf(Asymptote(G)))", "coords": { "H": [ 1.0, 2.0 ] } }, { "index": 4852, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4852.png", "answer": "2*sqrt(3)/3\n1", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2 = 1)", "coords": {} }, { "index": 4853, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4853.png", "answer": "(-2*sqrt(6)/3,2*sqrt(6)/3)", "fact_expr": "G: Ellipse;H: Line;A1: Point;A2: Point;P: Point;F1: Point;F2: Point;M: Point;Expression(G) = (x^2/4 + y^2 = 1);Focus(G) = {F1,F2};PointOnCurve(M, LineSegmentOf(A1, A2));IsPerpendicular(LineSegmentOf(A1, A2), H);Intersection(H, G) = P;DotProduct(VectorOf(P,F1),VectorOf(P,F2))<0", "coords": {} }, { "index": 4854, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4854.png", "answer": "sqrt(6)/3", "fact_expr": "l: Line;C: Ellipse;a: Number;b: Number;M: Point;a > b;b > 0;Slope(l) = -1/3;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);MidPoint(InterceptChord(l, C)) = M;Coordinate(M) = (1, 1)", "coords": { "M": [ 1.0, 1.0 ] } }, { "index": 4855, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4855.png", "answer": "2", "fact_expr": "C: Parabola;p: Number;G: Circle;H: Point;M: Point;F: Point;p>0;Expression(C) = (y^2 = 2*p*x);Coordinate(H) = (0, 1);Focus(C) = F;PointOnCurve(M, C);Abs(LineSegmentOf(M, F)) = 2;IsDiameter(LineSegmentOf(M,F),G);PointOnCurve(H,G)", "coords": { "H": [ 0.0, 1.0 ] } }, { "index": 4856, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 2.0, "major_axis": "y", "x_coef": 4.0, "y_coef": 8.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4856.png", "answer": "8", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/8 = 1);K: Point;PointOnCurve(K, G);F1: Point;F2: Point;Focus(G) = {F1, F2};M: Number;Distance(K, F1)*Distance(K, F2) = M", "coords": {} }, { "index": 4857, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4857.png", "answer": "4*sqrt(5)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);M: Point;Coordinate(M) = (4, -4);PointOnCurve(A, C) = True;PointOnCurve(B, C) = True;A: Point;B: Point;DotProduct(VectorOf(M, A), VectorOf(M, B)) = 0;PointOnCurve(M, C)", "coords": { "M": [ 4.0, -4.0 ] } }, { "index": 4858, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.8284271247461903, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4858.png", "answer": "y=x+pm*1", "fact_expr": "l: Line;Slope(l) = 1;G: Hyperbola;Expression(G) = (x^2/4 - y^2/8 = 1);Intersection(l,G) = {A,B};A: Point;B: Point;Abs(LineSegmentOf(A, B)) = 4*sqrt(5)", "coords": {} }, { "index": 4859, "success": false, "error": "Circle from diameter: missing point coordinates" }, { "index": 4860, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4860.png", "answer": "(0,2)+(8,+oo)", "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2/m = 1);m: Real;F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;PointOnCurve(P, C) = True;DotProduct(VectorOf(P, F1), VectorOf(P, F2)) < 0", "coords": {} }, { "index": 4861, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4862, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4862.png", "answer": "4*sqrt(6)", "fact_expr": "H: Line;Expression(H) = (y = x + 1);G: Hyperbola;Expression(G) = (x^2/2 - y^2/3 = 1);A: Point;B: Point;Intersection(H, G) = {A, B}", "coords": {} }, { "index": 4863, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4863.png", "answer": "2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x)", "coords": {} }, { "index": 4864, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4865, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4866, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4866.png", "answer": "6*sqrt(2)/7", "fact_expr": "l: Line;G: Ellipse;H: Point;A: Point;O: Origin;B: Point;Expression(G) = (x^2/4 + y^2/3 = 1);Coordinate(H) = (1, 0);PointOnCurve(H, l);Inclination(l) = ApplyUnit(45, degree);Intersection(l, G) = {A, B}", "coords": { "H": [ 1.0, 0.0 ] } }, { "index": 4867, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4867.png", "answer": "2*sqrt(5)-1", "fact_expr": "C: Parabola;p: Number;G: Circle;H: Point;P: Point;Q: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Expression(G) = (y^2 + (x - 6)^2 = 1);Coordinate(H) = (5, t);PointOnCurve(H, C);PointOnCurve(P,C);PointOnCurve(Q,G);Distance(H, Focus(C)) = 6;t:Number", "coords": {} }, { "index": 4868, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4869, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4869.png", "answer": "6", "fact_expr": "G: Ellipse;O: Point;P: Point;F: Point;Expression(G) = (x^2/4 + y^2/3 = 1);PointOnCurve(P, G);Center(G)=O;LeftFocus(G)=F", "coords": {} }, { "index": 4870, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.1622776601683795, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4870.png", "answer": "4*sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/10 - y^2/2 = 1)", "coords": {} }, { "index": 4871, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4872, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4872.png", "answer": "2*sqrt(3)", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2 - y^2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2))", "coords": {} }, { "index": 4873, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4874, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4874.png", "answer": "(\\sqrt{5}+1)/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;A: Point;B: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);L:Line;PointOnCurve(OneOf(Focus(G)),L);IsPerpendicular(L,RealAxis(G));Intersection(L,G)={A,B};Length(LineSegmentOf(A,B))=FocalLength(G)", "coords": {} }, { "index": 4875, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4875.png", "answer": "sqrt(5)", "fact_expr": "C: Hyperbola;b: Number;a: Number;H: Circle;P: Point;F1:Point;F2: Point;G: Point;e: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2 + y^2 = b^2);LeftFocus(C) =F1;RightFocus(C)=F2;PointOnCurve(P,RightPart(C));TangentPoint(LineSegmentOf(P,F2),H)=G;MidPoint(LineSegmentOf(P, F2)) = G;Eccentricity(G) = e", "coords": {} }, { "index": 4876, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4876.png", "answer": "(5/4,-3/4)", "fact_expr": "G: Hyperbola;H: Line;Expression(G) = (x^2 - y^2 = 1);Expression(H) = (y = x);P:Point;Distance(P,H)=sqrt(2);PointOnCurve(P,RightPart(G))", "coords": {} }, { "index": 4877, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4877.png", "answer": "sqrt(3)/3", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 4878, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4879, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 1.6666666666666667, "major_axis": "x", "x_coef": 25.0, "y_coef": 2.7777777777777777 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4879.png", "answer": "10\n10/3\n(pm*(10/3)*sqrt(2),0)\n{(pm*5,0),(0,pm*5/3)}", "fact_expr": "G: Ellipse;Expression(G) = (x^2 + 9*y^2 = 25)", "coords": {} }, { "index": 4880, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4880.png", "answer": "y^2/16 + x^2/9 = 1", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);H: Ellipse;Vertex(H) = {Focus(G), Endpoint(MinorAxis(G))}", "coords": {} }, { "index": 4881, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4881.png", "answer": "x^2/16 - y^2/9 = 1", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G)))=(y=(3/4)*x);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2;IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2));Abs(LineSegmentOf(P, F1))*Abs(LineSegmentOf(P, F2)) = 18", "coords": {} }, { "index": 4882, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4882.png", "answer": "8", "fact_expr": "C: Parabola;p: Number;O: Origin;F: Point;M: Point;p>0;Expression(C) = (y^2 = 2*p*x);Focus(C)=F;PointOnCurve(M,C);Area(CircumCircle(TriangleOf(O,F,M)))=36*pi;IsTangent(CircumCircle(TriangleOf(O,F,M)),Directrix(C))", "coords": {} }, { "index": 4883, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4883.png", "answer": "sqrt(5)", "fact_expr": "G: Ellipse;Expression(G) = (y^2/4 + x^2/a^2 = 1);a: Number;a>2;Eccentricity(G) = sqrt(5)/5", "coords": {} }, { "index": 4884, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4884.png", "answer": "(-oo,0)", "fact_expr": "G: ConicSection;H: ConicSection;b: Number;a: Number;a>b;b>0;e1:Number;e2:Number;Expression(H) = (y^2/b^2 + x^2/a^2 = 1);Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(H)=e1;Eccentricity(G)=e2;m:Number;m=lg(e1)+lg(e2)", "coords": {} }, { "index": 4885, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4886, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4887, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4887.png", "answer": "(3, 4)", "fact_expr": "G: Ellipse;t: Number;Expression(G) = (x^2/4 + y^2/t = 1);PointOnCurve(Focus(G), xAxis);P: Point;PointOnCurve(P, G);Distance(P, OneOf(Focus(G))) > 1", "coords": {} }, { "index": 4888, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4888.png", "answer": "5", "fact_expr": "C: Ellipse;b: Number;a: Number;M: Point;F1: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Focus(C) = {F1, F2};PointOnCurve(M, C);Max(Abs(LineSegmentOf(M, F1))*Abs(LineSegmentOf(M, F2))) = 25", "coords": {} }, { "index": 4889, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4889.png", "answer": "(0, \\sqrt{3}/2]", "fact_expr": "l: Line;E: Ellipse;b: Number;F: Point;M: Point;0 < b;b < 2;Expression(E) = (x^2/4 + y^2/b^2 = 1);Expression(l)=(3*x-4*y=0);RightFocus(E) = F;OneOf(Endpoint(MinorAxis(E)))=M;Negation(Distance(M, l) < 4/5)", "coords": {} }, { "index": 4890, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4890.png", "answer": "pm*(3/2)", "fact_expr": "H: Line;Expression(H) = (y = k*x);k: Number;G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);A: Point;B: Point;Intersection(H, G) = {A, B};L1: Line;L2: Line;PointOnCurve(A, L1);PointOnCurve(B, L2);IsPerpendicular(L1, xAxis);IsPerpendicular(L2, xAxis);F1: Point;F2: Point;Focus(G) = {F1, F2};FootPoint(L1, xAxis) = F1;FootPoint(L2, xAxis) = F2", "coords": {} }, { "index": 4891, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4892, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4893, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4893.png", "answer": "sqrt(13)/2", "fact_expr": "G: Hyperbola;e: Number;Expression(G) = (x^2/4 - y^2/9 = 1);Eccentricity(G) = e", "coords": {} }, { "index": 4894, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 2.0, "major_axis": "x", "x_coef": 8.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4894.png", "answer": "x=4", "fact_expr": "G: Ellipse;Expression(G) = (x^2/8 + y^2/4 = 1)", "coords": {} }, { "index": 4895, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4895.png", "answer": "(-1,-3/2)", "fact_expr": "G: Ellipse;A: Point;F: Point;O: Origin;B: Point;C: Point;Expression(G) = (x^2/4 + y^2/3 = 1);RightFocus(G) = F;PointOnCurve(A,G);Quadrant(A) = 1;Intersection(OverlappingLine(LineSegmentOf(A, F)), G) = B;Intersection(OverlappingLine(LineSegmentOf(A, O)), G) = C;Area(TriangleOf(A, B, C)) = 3", "coords": {} }, { "index": 4896, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4896.png", "answer": "2", "fact_expr": "G: Parabola;H: Line;k: Number;A: Point;B: Point;Expression(G) = (y^2 = 8*x);Expression(H) = (y = k*x - 2);Intersection(H, G) = {A, B};Negation(A=B);YCoordinate(MidPoint(LineSegmentOf(A, B))) = 2", "coords": {} }, { "index": 4897, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4897.png", "answer": "1", "fact_expr": "C: Hyperbola;F: Point;Expression(C) = (x^2 - y^2 = 1);RightFocus(C) = F", "coords": {} }, { "index": 4898, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4898.png", "answer": "16/9", "fact_expr": "G: Parabola;A: Point;B: Point;F: Point;M: Point;Q: Point;P: Point;l: Line;Expression(G) = (y^2 = 2*x);Focus(G) = F;Directrix(G) = l;IsChordOf(LineSegmentOf(A,B),G);PointOnCurve(F,LineSegmentOf(A,B));MidPoint(LineSegmentOf(A,B))=M;L1:Line;L2:Line;PointOnCurve(F,L1);PointOnCurve(M,L2);IsPerpendicular(L1,LineSegmentOf(A,B));IsPerpendicular(L2,LineSegmentOf(A,B));Intersection(L1,l)=P;Intersection(L2,l)=Q;Abs(LineSegmentOf(A,F))=3*Abs(LineSegmentOf(B,F))", "coords": {} }, { "index": 4899, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4899.png", "answer": "sqrt(7)/2", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;l: Line;OneOf(Asymptote(C)) = l;G: Circle;Expression(G) = (y^2 + (-c + x)^2 = a^2);c: Number;A: Point;B: Point;Intersection(l, G) = {A, B};c^2 = a^2 + b^2;Abs(LineSegmentOf(A, B)) = a", "coords": {} }, { "index": 4900, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4900.png", "answer": "3*sqrt(3)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 6*x);F: Point;Focus(C) = F;G: Line;A: Point;B: Point;PointOnCurve(F, G);Inclination(G) = ApplyUnit(30, degree);Intersection(G, C) = {A, B}", "coords": {} }, { "index": 4901, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4902, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4902.png", "answer": "(1,0)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x)", "coords": {} }, { "index": 4903, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4904, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4904.png", "answer": "3*sqrt(2)/4", "fact_expr": "H: Line;Expression(H) = (y = x*b/(3*a));G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Intersection(H,LeftPart(G)) = P;P: Point;F1: Point;LeftFocus(G) = F1;IsPerpendicular(LineSegmentOf(P,F1),xAxis) = True;e: Number;Eccentricity(G) = e", "coords": {} }, { "index": 4905, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4906, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4906.png", "answer": "x^2/2-y^2=1", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/2 = 1);Z: Hyperbola;Asymptote(Z) = Asymptote(G);P: Point;Coordinate(P) = (2, 1);PointOnCurve(P, Z)", "coords": { "P": [ 2.0, 1.0 ] } }, { "index": 4907, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4907.png", "answer": "y = x", "fact_expr": "G: Parabola;H: Line;P: Point;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Coordinate(P) = (2, 2);PointOnCurve(P,H);Intersection(H, G) = {A, B};VectorOf(P, A) + VectorOf(P, B) = 0", "coords": { "P": [ 2.0, 2.0 ] } }, { "index": 4908, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4909, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4909.png", "answer": "[2, 3]", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);x1: Number;y1: Number;P: Point;Coordinate(P) = (x1, y1);PointOnCurve(P, G);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2", "coords": {} }, { "index": 4910, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4910.png", "answer": "(x-5)^2+y^2=16", "fact_expr": "G: Hyperbola;H: Circle;Expression(G) = (x^2/9 - y^2/16 = 1);RightFocus(G) = Center(H);IsTangent(Asymptote(G),H)", "coords": {} }, { "index": 4911, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4911.png", "answer": "pm*(sqrt(3)/3)", "fact_expr": "l: Line;G: Parabola;M: Circle;B:Point;A:Point;Expression(G) = (y^2 = 2*x);Intersection(l,G)={A,B};IsDiameter(LineSegmentOf(A,B),M);IsTangent(Directrix(G),M);Area(M)=16*pi", "coords": {} }, { "index": 4912, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4912.png", "answer": "y^2=4*x", "fact_expr": "C: Parabola;G: Line;B: Point;A: Point;P: Point;O: Origin;Expression(G) = (y = x);Coordinate(P) = (2, 2);Vertex(C) = O;PointOnCurve(Focus(C), xAxis);Intersection(G, C) = {A, B};MidPoint(LineSegmentOf(A,B)) = P", "coords": { "P": [ 2.0, 2.0 ] } }, { "index": 4913, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4914, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4915, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4915.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1)", "coords": {} }, { "index": 4916, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4916.png", "answer": "sqrt(5)/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = 2*x);IsPerpendicular(OneOf(Asymptote(G)),H)", "coords": {} }, { "index": 4917, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4918, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4918.png", "answer": "3/5", "fact_expr": "G: Hyperbola;H: Parabola;Expression(G) = (x^2/16 - y^2/9 = 1);Expression(H) = (y^2 = 4*x)", "coords": {} }, { "index": 4919, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4919.png", "answer": "2", "fact_expr": "C: Hyperbola;a: Number;b: Number;B: Point;F: Point;A: Point;a>0;b>0;Expression(C) = (-x^2/b^2 + y^2/a^2 = 1);UpperFocus(C) = F;UpperVertex(C) = A ;PointOnCurve(B, C);IsPerpendicular(LineSegmentOf(B, F), yAxis);Slope(LineSegmentOf(A, B)) = 1/3", "coords": {} }, { "index": 4920, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4920.png", "answer": "-2*p", "fact_expr": "G: Parabola;p: Number;H: Line;O: Origin;A: Point;M: Point;B: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(M) = (1, 0);PointOnCurve(M,SymmetryAxis(G));PointOnCurve(M,H);Intersection(H, G) = {A, B}", "coords": { "M": [ 1.0, 0.0 ] } }, { "index": 4921, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4921.png", "answer": "[1, 3]", "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/4 + y^2/3 = 1);Focus(C) = {F1, F2};PointOnCurve(P, C)", "coords": {} }, { "index": 4922, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4922.png", "answer": "x^2 = pm*8*y", "fact_expr": "G: Hyperbola;H: Parabola;Expression(G) = (-x^2/3 + y^2/6 = 1);Center(G) = Vertex(H);Directrix(H) = Directrix(G)", "coords": {} }, { "index": 4923, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4923.png", "answer": "y=pm*3*x", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;t: Number;Negation(t=0);P: Point;A: Point;B: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = t + 2*x);Coordinate(P) = (4*t, 0);L1:Line;L2:Line;Asymptote(G)={L1,L2};Intersection(H, L1) = A;Intersection(H,L2)=B;Abs(LineSegmentOf(P, A)) = Abs(LineSegmentOf(P, B))", "coords": {} }, { "index": 4924, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4925, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 9.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4925.png", "answer": "2\n2*pi/3", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 + y^2/2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 4", "coords": {} }, { "index": 4926, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4926.png", "answer": "56/5", "fact_expr": "C: Ellipse;A: Point;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/25 + y^2/9 = 1);Coordinate(A) = (4, 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);WhenMax(Perimeter(TriangleOf(A, P, F1)))", "coords": { "A": [ 4.0, 1.0 ] } }, { "index": 4927, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4927.png", "answer": "(0, 5)", "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/9 + y^2/16 = 1)", "coords": {} }, { "index": 4928, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4928.png", "answer": "2", "fact_expr": "C: Parabola;p: Number;G: Line;M: Point;F: Point;p>0;Expression(C) = (x^2 = 2*(p*y));Expression(G) = (-x + y + 1 = 0);Focus(C) = F;Intersection(G, C) = M;IsPerpendicular(LineSegmentOf(M,F),SymmetryAxis(C))", "coords": {} }, { "index": 4929, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4929.png", "answer": "6", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2/3 = 1);PointOnCurve(P, G);Focus(G) = {F1, F2}", "coords": {} }, { "index": 4930, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4930.png", "answer": "sqrt(10)/2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;LeftFocus(G) = F;H: Circle;Expression(H) = (x^2 + y^2 = a^2/4);L: Line;TangentOfPoint(F, H) = L;TangentPoint(L, H) = E;E: Point;Intersection(OverlappingLine(LineSegmentOf(F, E)), RightPart(G)) = P;P: Point;MidPoint(LineSegmentOf(P, F)) = E", "coords": {} }, { "index": 4931, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4931.png", "answer": "x^2/5+y^2/4=1", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;e: Number;Eccentricity(G) = e ;e = sqrt(5)/5;Coordinate(OneOf(Vertex(G))) = (0, 2)", "coords": {} }, { "index": 4932, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.242349737599891, "b": 4.242349737599891, "orientation": "horizontal" }, "optimization": { "final_loss": 1.0299271839804645e-06, "converged": true, "iterations": 86 }, "output_path": "results/train/hyperbola/problem_4932.png", "answer": "-11", "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (x^2/25 + y^2/m = 1);F1: Point;F2: Point;Coordinate(F1) = (-6, 0);Coordinate(F2) = (6, 0);Focus(G) = {F1, F2}", "coords": { "F1": [ -6.0, 0.0 ], "F2": [ 6.0, 0.0 ] } }, { "index": 4933, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4933.png", "answer": "x^2/12 - y^2/4 = 1", "fact_expr": "G: Hyperbola;H: Parabola;Expression(OneOf(Asymptote(G))) = (x + sqrt(3)*y = 0);OneOf(Focus(G)) = Focus(H);Expression(H) = (y^2 = 16*x)", "coords": {} }, { "index": 4934, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4934.png", "answer": "sqrt(2)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x);F: Point;Focus(G) = F;A: Point;PointOnCurve(A, G);B: Point;Coordinate(B) = (-1/2, 0);WhenMax(Abs(LineSegmentOf(A, B))/Abs(LineSegmentOf(A, F))) = True", "coords": { "B": [ -0.5, 0.0 ] } }, { "index": 4935, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4935.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/12 = 1)", "coords": {} }, { "index": 4936, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.125, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4936.png", "answer": "1/2", "fact_expr": "C: Parabola;G:Circle;F: Point;l: Line;A: Point;B: Point;Expression(C) = (y = 2*x^2);Focus(C) = F;Directrix(C) = l;Center(G)=F;IsTangent(l,G);Intersection(G, C) = {A, B}", "coords": {} }, { "index": 4937, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4938, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4939, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_4939.png", "answer": "pi/6", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (-3*a*y + b*x = 0);Eccentricity(G) = 2", "coords": {} }, { "index": 4940, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4940.png", "answer": "y^2 = 8*x", "fact_expr": "G: Parabola;p: Number;M: Point;F: Point;O: Origin;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;PointOnCurve(M, G);Abs(LineSegmentOf(M, F)) = 4*Abs(LineSegmentOf(O, F));Area(TriangleOf(M, F, O)) = 4*sqrt(3)", "coords": {} }, { "index": 4941, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4941.png", "answer": "-2", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/2 + y^2/m = 1);m: Number;e: Number;Eccentricity(C) = e;M: Circle;Expression(M) = (y^2 + (x - 2)^2 = e^2);IsTangent(Asymptote(C), M) = True", "coords": {} }, { "index": 4942, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4942.png", "answer": "7", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);P: Point;Q: Point;PointOnCurve(P, G);PointOnCurve(Q, G);Negation(P=Endpoint(MajorAxis(G)));Negation(Q=Endpoint(MajorAxis(G)));O: Origin;K1: Number;K2: Number;Slope(LineOf(O, P)) = K1;Slope(LineOf(O, Q)) = K2;K1*K2 = -3/4", "coords": {} }, { "index": 4943, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4944, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4945, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_ecc_mismatch", "params": { "a": 1.9257268905639648, "b": 1.7993911322594118, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 2.5576271215184017e-15, "converged": true, "iterations": 2 }, "output_path": "results/train/ellipse/problem_4945.png", "answer": "2*sqrt(3)", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;Eccentricity(G) = 1/2;H: Line;PointOnCurve(F2, H);Intersection(H, G) = {A, B};A: Point;B: Point;Perimeter(TriangleOf(A, B, F1)) = 8", "coords": {}, "validation_reasons": [ "ellipse_ecc_mismatch" ] }, { "index": 4946, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4947, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4947.png", "answer": "sqrt(10)/3", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = -4*sqrt(2)*x);Distance(Focus(H), OneOf(Asymptote(G))) = sqrt(5)/5", "coords": {} }, { "index": 4948, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 2.0, -0.0 ], "radius": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_4948.png", "answer": "{(y^2 = 8*x)&(x > 0), (y = 0)&(x < 0)}", "fact_expr": "G: Circle;Expression(G) = (-4*x + x^2 + y^2 = 0);Z:Circle;IsOutTangent(Z, G);IsTangent(Z, yAxis)", "coords": {} }, { "index": 4949, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4949.png", "answer": "2*sqrt(3+2*sqrt(3))", "fact_expr": "C: Ellipse;Expression(C) = (y^2/3 + x^2/a^2 = 1);a: Number;a>sqrt(3);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;O: Origin;G: Line;PointOnCurve(O, G);Inclination(G) = ApplyUnit(60, degree);M: Point;OneOf(Intersection(G, C)) = M;Abs(VectorOf(M, F1) + VectorOf(M, F2)) = Abs(VectorOf(M, F1) - VectorOf(M, F2))", "coords": {} }, { "index": 4950, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4951, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 4.47213595499958, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4951.png", "answer": "17", "fact_expr": "G: Hyperbola;F1: Point;F2: Point;P: Point;Expression(G) = (x^2/16 - y^2/20 = 1);Focus(G)={F1,F2};PointOnCurve(P, G);Distance(P,F1) = 9", "coords": {} }, { "index": 4952, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 4.0, -0.0 ], "radius": 10.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_4952.png", "answer": "x^2/25+y^2/9=1", "fact_expr": "G: Circle;C:Circle;A: Point;Expression(G) = (-8*x + x^2 + y^2 - 84 = 0);Coordinate(A) = (-4, 0);PointOnCurve(A, C);IsTangent(G,C);P:Point;Center(C)=P", "coords": {} }, { "index": 4953, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4953.png", "answer": "(p*k^2 + p, -p*k)", "fact_expr": "G: Parabola;p: Number;l1: Line;l2: Line;P: Point;Q: Point;M: Point;N: Point;T: Point;k: Number;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(X) = (p + p/k^2, p/k);Coordinate(T) = (p, 0);PointOnCurve(T, l1);PointOnCurve(T, l2);IsPerpendicular(l1, l2);Intersection(l1, G) = {P, Q};Intersection(l2, G) = {M, N};Slope(l1) = k;IsChordOf(LineSegmentOf(P, Q), G);IsChordOf(LineSegmentOf(M, N), G);X: Point;MidPoint(LineSegmentOf(P, Q)) = X", "coords": {} }, { "index": 4954, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4954.png", "answer": "x \\pm y - 1 = 0", "fact_expr": "l: Line;G: Parabola;a: Number;F: Point;A: Point;B: Point;Expression(G) = (y^2 = a*x);Coordinate(F) = (1, 0);Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B};LineSegmentOf(A, B) = 8", "coords": { "F": [ 1.0, 0.0 ] } }, { "index": 4955, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4955.png", "answer": "18", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1);P: Point;PointOnCurve(P, LeftPart(G));Distance(P, LeftFocus(G)) = 10", "coords": {} }, { "index": 4956, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4956.png", "answer": "8", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;x1:Number;y1:Number;x2:Number;y2:Number;Expression(G) = (y^2 = 4*x);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);PointOnCurve(Focus(G), H);Intersection(H, G) = {A,B};x1 + x2 = 6", "coords": {} }, { "index": 4957, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4957.png", "answer": "3/2", "fact_expr": "C: Hyperbola;P: Point;F: Point;A: Point;Expression(C) = (x^2 - y^2/3 = 1);Coordinate(A) = (1, 3);RightFocus(C)=F;PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P,F),xAxis)", "coords": { "A": [ 1.0, 3.0 ] } }, { "index": 4958, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4958.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "H: Line;Expression(H) = (y = k*x + 1);k: Number;G: Ellipse;Expression(G) = (x^2/5 + y^2/m = 1);m: Number;IsIntersect(H, G)", "coords": {} }, { "index": 4959, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4960, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 2.560132719491411, "b": 1.280571558186289, "orientation": "horizontal" }, "optimization": { "final_loss": 5.001754676787936e-06, "converged": true, "iterations": 73 }, "output_path": "results/train/hyperbola/problem_4960.png", "answer": "x^2/4-y^2=1", "fact_expr": "G: Hyperbola;H: Point;Coordinate(H) = (4, sqrt(3));PointOnCurve(H,G) = True;Expression(Asymptote(G)) = (y=pm*(1/2)*x)", "coords": { "H": [ 4.0, 1.7320508075688772 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 4961, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4962, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4962.png", "answer": "1/4", "fact_expr": "G: Ellipse;a: Number;b: Number;F: Point;A: Point;O:Origin;H: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Center(G) = O;RightFocus(G)=F;RightVertex(G)=A;Intersection(RightDirectrix(G),xAxis)=H", "coords": {} }, { "index": 4963, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4963.png", "answer": "1", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x);P: Point;Coordinate(P) = (2, 2);l: Line;TangentOfPoint(P, G) = l", "coords": { "P": [ 2.0, 2.0 ] } }, { "index": 4964, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4965, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4965.png", "answer": "1", "fact_expr": "G: Hyperbola;b: Number;b>0;Expression(G) = (x^2/4 - y^2/b^2 = 1);Expression(Asymptote(G)) = (y = pm*(x/2))", "coords": {} }, { "index": 4966, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4966.png", "answer": "7", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;G: Point;Coordinate(G) = (-2, 0);l: Line;Slope(l) = 2/3;PointOnCurve(G, l);A: Point;B: Point;Intersection(l, C) = {A, B}", "coords": { "G": [ -2.0, 0.0 ] } }, { "index": 4967, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4967.png", "answer": "5", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P: Point;PointOnCurve(P, G);F: Point;Focus(G) = F;B: Point;Coordinate(B) = (4, 2)", "coords": { "B": [ 4.0, 2.0 ] } }, { "index": 4968, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4969, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4970, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4970.png", "answer": "y^2=12*x", "fact_expr": "G: Hyperbola;H: Parabola;Expression(G) = (x^2/4 - y^2/5 = 1);Vertex(H) = Center(G);RightFocus(G)=Focus(H)", "coords": {} }, { "index": 4971, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4972, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4972.png", "answer": "2*sqrt(3)/3", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y^2 + (x - 2)^2 = 4);Length(InterceptChord(OneOf(Asymptote(C)),G))=2*sqrt(3)", "coords": {} }, { "index": 4973, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4973.png", "answer": "2", "fact_expr": "C: Ellipse;A: Point;B: Point;P: Point;b:Real;b>0;b<5;Coordinate(A) = (-4, 0);Coordinate(B) = (-1, 0);Expression(C) = (x^2/25 + y^2/b^2 = 1);PointOnCurve(P,C);Abs(LineSegmentOf(P,A))=2*Abs(LineSegmentOf(P,B))", "coords": { "A": [ -4.0, 0.0 ], "B": [ -1.0, 0.0 ] } }, { "index": 4974, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4974.png", "answer": "1", "fact_expr": "C: Parabola;Expression(C) = (y^2 = x);F: Point;Focus(C) = F;A: Point;x0: Number;y0: Number;Coordinate(A) = (x0, y0);PointOnCurve(A, C);Abs(LineSegmentOf(A, F)) = (5/4)*x0", "coords": {} }, { "index": 4975, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4975.png", "answer": "y=2", "fact_expr": "G: Parabola;Expression(G) = (y = -x^2/8)", "coords": {} }, { "index": 4976, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4976.png", "answer": "sqrt(2)", "fact_expr": "P: Point;Coordinate(P) = (sqrt(2)/2, 1);M: Point;N: Point;PointOnCurve(M, G);PointOnCurve(N, G);G: Ellipse;Expression(G) = (x^2 + y^2/2 = 1);k1: Number;k2: Number;k: Number;Slope(LineOf(P, M)) = k1;Slope(LineOf(P, N)) = k2;Slope(LineOf(M, N)) = k;k1 + k2 = 0", "coords": { "P": [ 0.7071067811865476, 1.0 ] } }, { "index": 4977, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4977.png", "answer": "12/25", "fact_expr": "F1: Point;F2: Point;E: Ellipse;LeftFocus(E) = F1;RightFocus(E) = F2;a: Number;b: Number;a > b;b > 0;Expression(E) = (x^2/a^2 + y^2/b^2 = 1);B: Point;C: Point;UpperVertex(E) = B;LowerVertex(E) = C;D: Point;Intersection(LineOf(B,F2),E)={B,D};Cos(AngleOf(F1, B, F2)) = 7/25;Negation(B=D)", "coords": {} }, { "index": 4978, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4978.png", "answer": "[1, 7)", "fact_expr": "H: Line;Expression(H) = (y = k*x + 1);k: Real;G: Ellipse;Expression(G) = (x^2/7 + y^2/a = 1);a: Real;PointOnCurve(Focus(G), xAxis);IsIntersect(H, G)", "coords": {} }, { "index": 4979, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4980, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4980.png", "answer": "3", "fact_expr": "l: Line;G: Parabola;O: Origin;M: Point;A: Point;B: Point;P: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B};VectorOf(O, M) = (VectorOf(O, A) + VectorOf(O, B))/2;Z: Line;PointOnCurve(M, Z);IsPerpendicular(yAxis, Z);Intersection(Z, G) = P;Abs(LineSegmentOf(P, F)) = 2", "coords": {} }, { "index": 4981, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4981.png", "answer": "y=pm*x", "fact_expr": "H: Parabola;Expression(H) = (y^2 = 2*(p*x));p: Number;p>0;G: Hyperbola;Expression(G) = (x^2/8 - y^2/p = 1);Focus(H) = OneOf(Focus(G))", "coords": {} }, { "index": 4982, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4982.png", "answer": "1\n2", "fact_expr": "C1: Hyperbola;C2: Hyperbola;a:Number;b:Number;F: Point;Expression(C1) = (x^2/a^2-y^2/b^2=1);Expression(C2) = (x^2/4-y^2/16=1);Coordinate(F) = (sqrt(5), 0);RightFocus(C1) = F;a>0;b>0;Asymptote(C1)=Asymptote(C2)", "coords": { "F": [ 2.23606797749979, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 4983, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 5.656854249492381, "major_axis": "x", "x_coef": 36.0, "y_coef": 32.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4983.png", "answer": "[21/2, 27/2]", "fact_expr": "G: Ellipse;H: Circle;C:Circle;P: Point;M: Point;N: Point;Expression(G) = (x^2/36 + y^2/32 = 1);Expression(H) = (y^2 + (x - 2)^2 = 1);Expression(C) = (y^2 + (x + 2)^2 = 1/4);PointOnCurve(P, G);PointOnCurve(M, H);PointOnCurve(N, C)", "coords": {} }, { "index": 4984, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4984.png", "answer": "sqrt(6)/2", "fact_expr": "l: Line;C: Ellipse;P: Point;Expression(C) = (x^2/2 + y^2 = 1);PointOnCurve(P, C);Expression(l)=(x - y + 2*sqrt(3)=0)", "coords": {} }, { "index": 4985, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 2.6457513110645907, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4985.png", "answer": "(pm*3, 0)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/7 = 1)", "coords": {} }, { "index": 4986, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 3.872983346207417, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4986.png", "answer": "6", "fact_expr": "P: Point;G: Hyperbola;Expression(G) = (x^2 - y^2/15 = 1);PointOnCurve(P, RightPart(G)) = True;M: Point;N: Point;H1: Circle;Expression(H1) = (y^2 + (x + 4)^2 = 4);H2: Circle;Expression(H2) = (y^2 + (x - 4)^2 = 4);PointOnCurve(M, H1) = True;PointOnCurve(N, H2) = True", "coords": {} }, { "index": 4987, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.0, "b": 0.5773502691896257, "major_axis": "x", "x_coef": 1.0, "y_coef": 0.3333333333333333 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4987.png", "answer": "2*sqrt(3)/3", "fact_expr": "G: Ellipse;Expression(G) = (x^2 + 3*y^2 = 1)", "coords": {} }, { "index": 4988, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4988.png", "answer": "(4/3)*sqrt(6)", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;A: Point;Coordinate(A) = (2, 0);OneOf(Endpoint(MajorAxis(G))) = A;B: Point;C: Point;IsChordOf(LineSegmentOf(B, C), G);O: Point;Center(G) = O;PointOnCurve(O, LineSegmentOf(B, C));DotProduct(VectorOf(A, C), VectorOf(B, C)) = 0;Abs(-VectorOf(O, B) + VectorOf(O, C)) = 2*Abs(-VectorOf(B, A) + VectorOf(B, C))", "coords": { "A": [ 2.0, 0.0 ] } }, { "index": 4989, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.3300262689590454, "b": 2.30381441116333, "orientation": "horizontal" }, "optimization": { "final_loss": 6.658357051492203e-06, "converged": true, "iterations": 122 }, "output_path": "results/train/hyperbola/problem_4989.png", "answer": "3", "fact_expr": "G: Hyperbola;H:Parabola;a: Number;M: Point;y0:Number;Expression(G) = (-y^2/3 + x^2/a = 1);Expression(H) = (y^2 = 4*(a*x));Coordinate(M) = (2, y0);Expression(OneOf(Asymptote(G)))= (y = sqrt(3)*x);PointOnCurve(M, H);Focus(H) = F;F:Point", "coords": {} }, { "index": 4990, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4990.png", "answer": "[sqrt(3)/3, sqrt(2)/2]", "fact_expr": "M: Ellipse;a: Number;b: Number;c: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(M) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(M) = F1;RightFocus(M) = F2;PointOnCurve(P, M);c = sqrt(a^2 - b^2);Range(Max(Abs(LineSegmentOf(P, F1))*Abs(LineSegmentOf(P, F2)))) = [2*c^2, 3*c^2]", "coords": {} }, { "index": 4991, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4992, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.23606797749979, "b": 2.0, "major_axis": "x", "x_coef": 5.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4992.png", "answer": "5", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/5 + y^2/4 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G)", "coords": {} }, { "index": 4993, "success": false, "conic_type": "parabola", "error": "validation: parabola_focus_mismatch", "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_4993.png", "answer": "4\n5/2", "fact_expr": "G: Parabola;p: Number;F: Point;A: Point;E: Point;p>0;Expression(G) = (y^2 = 2*p*x);Coordinate(F) = (2, 0);Coordinate(A) = (3, 2);Focus(G) = F;l:Line;PointOnCurve(A,l);IsPerpendicular(l,Directrix(G));Intersection(l,G)=E", "coords": { "F": [ 2.0, 0.0 ], "A": [ 3.0, 2.0 ] }, "validation_reasons": [ "parabola_focus_mismatch" ] }, { "index": 4994, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4995, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.6457513110645907, "major_axis": "x", "x_coef": 16.0, "y_coef": 7.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_4995.png", "answer": "y=pm*(sqrt(7)/3)*x", "fact_expr": "G: Ellipse;Expression(G)=(x^2/16+y^2/7=1);C:Hyperbola;Focus(C)=Endpoint(MajorAxis(G));Endpoint(RealAxis(C))=Focus(G)", "coords": {} }, { "index": 4996, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.02083333395421505, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_4996.png", "answer": "1/24", "fact_expr": "G: Parabola;Expression(G) = (y = 12*x^2)", "coords": {} }, { "index": 4997, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4998, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 4999, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_4999.png", "answer": "x^2 - y^2/(1/2) = 1", "fact_expr": "E: Hyperbola;P: Point;Expression(E) = (x^2/4 - y^2/2 = 1);Coordinate(P) = (3, 2);C: Hyperbola;PointOnCurve(P, C);Expression(Asymptote(C)) = Expression(Asymptote(E))", "coords": { "P": [ 3.0, 2.0 ] } }, { "index": 5000, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5000.png", "answer": "13", "fact_expr": "G: Parabola;M: Point;Expression(G) = (y^2 = 16*x);PointOnCurve(M, G);Distance(M, xAxis) = 12", "coords": {} }, { "index": 5001, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5001.png", "answer": "2*sqrt(2)", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;H: Line;P: Point;F2: Point;O: Origin;M: Point;Area(TriangleOf(M,F2,O))/Area(TriangleOf(P,M,O))=7;F1: Point;y0:Number;y0>0;a > b;b > 0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = x*(b/a));Coordinate(P) = (0, y0);LeftFocus(C) = F1;RightFocus(C) = F2;IsDiameter(LineSegmentOf(O, P), G);Intersection(G, H) = {O, M};PointOnCurve(M, LineSegmentOf(P,F2))", "coords": {} }, { "index": 5002, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5002.png", "answer": "x^2+(y+2)^2=20", "fact_expr": "G: Ellipse;A: Point;D: Point;B: Point;P: Point;Expression(G) = (x^2 + y^2/5 = 1);Coordinate(A) = (0, -2);Coordinate(B) = (0, 2);PointOnCurve(D, G);Abs(LineSegmentOf(P, D)) = Abs(LineSegmentOf(B, D));PointOnCurve(P,OverlappingLine(LineSegmentOf(A,D)))", "coords": { "A": [ 0.0, -2.0 ], "B": [ 0.0, 2.0 ] } }, { "index": 5003, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5004, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5004.png", "answer": "x+pm*y=0;pm*3", "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2 = 1);A: Point;RightVertex(C) = A;l: Line;PointOnCurve(A, l) = True;J1: Line;J2: Line;Asymptote(C) = {J1, J2};Intersection(l, J1) = P;Intersection(l, J2) = Q;P: Point;Q: Point;LineSegmentOf(P, A) = 2*LineSegmentOf(A, Q)", "coords": {} }, { "index": 5005, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5005.png", "answer": "x=-1", "fact_expr": "l: Line;C: Parabola;Expression(C) = (y^2 = 4*x);PointOnCurve(Focus(C), l);A: Point;B: Point;Intersection(l, C) = {A, B};Z1: Line;Z2: Line;TangentOfPoint(A, C) = Z1;TangentOfPoint(B, C) = Z2;P: Point;Intersection(Z1, Z2) = P", "coords": {} }, { "index": 5006, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 3.0, "major_axis": "x", "x_coef": 36.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5006.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "l: Line;G: Ellipse;I: Point;Expression(G) = (x^2/36 + y^2/9 = 1);Coordinate(I) = (4, 2);MidPoint(InterceptChord(l, G)) = I", "coords": { "I": [ 4.0, 2.0 ] } }, { "index": 5007, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5008, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5009, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5010, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5010.png", "answer": "-1/3", "fact_expr": "L: Line;Slope(L) = 1/2;G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);Intersection(L,G) = {A,B};A: Point;B: Point;M: Point;MidPoint(LineSegmentOf(A, B)) = M;Coordinate(M) = (m, 1/2);m: Number", "coords": {} }, { "index": 5011, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5011.png", "answer": "(-1, 0)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/2 + y^2 = 1)", "coords": {} }, { "index": 5012, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5012.png", "answer": "8/3", "fact_expr": "l: Line;C: Parabola;F: Point;M: Point;P: Point;Q: Point;Expression(C) = (y^2 = 8*x);Focus(C)=F;PointOnCurve(F,l);Intersection(l, C) = {P, Q};Intersection(l,Directrix(C))=M;VectorOf(F, M) = 3*VectorOf(F, P)", "coords": {} }, { "index": 5013, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5013.png", "answer": "12", "fact_expr": "G: Hyperbola;P: Point;F1:Point;F2: Point;Q: Point;Expression(G) = (x^2 - y^2/4 = 1);LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(F1, l);Intersection(l, LeftPart(G)) = {P, Q};Abs(LineSegmentOf(P, Q)) = 4;l:Line", "coords": {} }, { "index": 5014, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_5014.png", "answer": "6", "fact_expr": "G: Hyperbola;m: Real;Expression(G) = (x^2/2 - y^2/m = 1);Eccentricity(G) = 2", "coords": {} }, { "index": 5015, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5016, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.6457513110645907, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5016.png", "answer": "32", "fact_expr": "H: Parabola;Expression(H) = (y^2 = 2*(p*x));p: Number;F: Point;Focus(H) = F;G: Hyperbola;Expression(G) = (x^2/7 - y^2/9 = 1);RightFocus(G) = F;Intersection(Directrix(H), xAxis) = K;K: Point;A: Point;PointOnCurve(A, H) = True;Abs(LineSegmentOf(A, K)) = sqrt(2)*Abs(LineSegmentOf(A, F))", "coords": {} }, { "index": 5017, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5017.png", "answer": "(pm*1)+[-sqrt(5)/2,]", "fact_expr": "G: Hyperbola;H: Line;k: Number;Expression(G) = (x^2 - y^2 = 4);Expression(H) = (y = k*x - 1);IsIntersect(H, G)", "coords": {} }, { "index": 5018, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5018.png", "answer": "(0, pm*2)", "fact_expr": "G: Curve;Expression(G) = (x^2 + y^2/5 = 1)", "coords": {} }, { "index": 5019, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5019.png", "answer": "(2, 2)", "fact_expr": "G: Parabola;P: Point;A: Point;F: Point;Expression(G) = (y^2 = 2*x);Coordinate(A) = (3, 2);Focus(G) = F;PointOnCurve(P, G);WhenMin(LineSegmentOf(P, A) + LineSegmentOf(P, F))", "coords": { "A": [ 3.0, 2.0 ] } }, { "index": 5020, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5021, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.3300262689590454, "b": 2.30381441116333, "orientation": "horizontal" }, "optimization": { "final_loss": 6.658357051492203e-06, "converged": true, "iterations": 122 }, "output_path": "results/train/hyperbola/problem_5021.png", "answer": "x^2/16 - y^2/48 = 1", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x);H: Parabola;Expression(H) = (y^2 = 32*x);OneOf(Focus(G)) = Focus(H)", "coords": {} }, { "index": 5022, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5022.png", "answer": "5/4", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1)", "coords": {} }, { "index": 5023, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5024, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5024.png", "answer": "32", "fact_expr": "G: Parabola;H: Line;P: Point;A: Point;B: Point;x1:Number;x2:Number;y1:Number;y2:Number;Expression(G) = (y^2 = 4*x);Coordinate(P) = (4, 0);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);PointOnCurve(P, H);Intersection(H, G) = {A,B}", "coords": { "P": [ 4.0, 0.0 ] } }, { "index": 5025, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5025.png", "answer": "12", "fact_expr": "G: Parabola;p: Number;H: Line;l: Line;A: Point;B: Point;C: Point;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};Directrix(G) = l;Intersection(H, l) = C;VectorOf(C, F) = 2*VectorOf(F, A);LineSegmentOf(A, F) = 8", "coords": {} }, { "index": 5026, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.449489742783178, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 6.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5026.png", "answer": "4", "fact_expr": "G: Ellipse;Expression(G) = (x^2 + 3*y^2 = 6)", "coords": {} }, { "index": 5027, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5028, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5028.png", "answer": "2", "fact_expr": "G: Parabola;p: Number;Expression(G) = (y^2 = 2*(p*x));Distance(Focus(G),Directrix(G))=1;H:LineSegment;PointOnCurve(Focus(G),H);IsChordOf(H,G)", "coords": {} }, { "index": 5029, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5030, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5030.png", "answer": "2*sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2/4 = 1)", "coords": {} }, { "index": 5031, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5031.png", "answer": "x^2/8-y^2=1", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;RightFocus(C) = F;F: Point;Coordinate(F) = (3, 0);Distance(F,OneOf(Asymptote(C))) = 1", "coords": { "F": [ 3.0, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 5032, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 10.0, "b": 8.0, "major_axis": "x", "x_coef": 100.0, "y_coef": 64.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5032.png", "answer": "13", "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/100 + y^2/64 = 1);PointOnCurve(P, G);Distance(P, LeftFocus(G)) = 7", "coords": {} }, { "index": 5033, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5033.png", "answer": "(\\sqrt{5} - 1)/2", "fact_expr": "G: Ellipse;b: Number;a: Number;H: LineSegment;PointOnCurve(Focus(G),H);a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);IsPerpendicular(MajorAxis(G), H);IsChordOf(H, G) ;FocalLength(G) = Length(H)", "coords": {} }, { "index": 5034, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5034.png", "answer": "2*sqrt(3)/3", "fact_expr": "E: Hyperbola;a: Number;b: Number;C: Circle;a>0;b>0;Expression(E) = (-x^2/b^2 + y^2/a^2 = 1);Expression(C) = (-16*y + x^2 + y^2 + 48 = 0);IsTangent(C, Asymptote(E))", "coords": {} }, { "index": 5035, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5035.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (sqrt(3)*x - y + sqrt(3) = 0);IsParallel(OneOf(Asymptote(G)),H)", "coords": {} }, { "index": 5036, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5036.png", "answer": "sqrt(21)/3", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, Asymptote(G));Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2));AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 5037, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5037.png", "answer": "sqrt(5)/3", "fact_expr": "C: Ellipse;b: Number;a: Number;P: Point;F2: Point;F1: Point;O: Origin;N: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) =F1;RightFocus(C)=F2;PointOnCurve(P,C);Quadrant(P)=2;Intersection(LineSegmentOf(P, F2), yAxis) = N;DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;Abs(LineSegmentOf(F1, F2)) = 4*Abs(LineSegmentOf(O, N))", "coords": {} }, { "index": 5038, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5038.png", "answer": "y^2=8*x", "fact_expr": "G: Parabola;p: Number;P: Point;m:Number;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(P) = (2, m);PointOnCurve(P,G);Distance(P,yAxis)=Distance(P,Focus(G))/2", "coords": {} }, { "index": 5039, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5039.png", "answer": "2", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;O: Origin;M: Point;F: Point;N: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;PointOnCurve(F,G);L1:Line;L2:Line;Asymptote(C)={L1,L2};Intersection(G,L1)=M;Intersection(G,L2)=N;DotProduct(VectorOf(O, M), VectorOf(M, F)) = 0;Abs(LineSegmentOf(M, N)) = b", "coords": {} }, { "index": 5040, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5040.png", "answer": "2*sqrt(5)/5", "fact_expr": "F: Point;C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;RightFocus(C) = F;G: Line;Expression(G) = (y = x*b/a);Intersection(G, C) = {A, B};A: Point;B: Point;Cos(AngleOf(A, F, B)) = 1/3", "coords": {} }, { "index": 5041, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5041.png", "answer": "(x-\\sqrt{5})^2+y^2=1", "fact_expr": "G: Hyperbola;H:Circle;Expression(G) = (x^2/4 - y^2 = 1);RightFocus(G)=Center(H);IsTangent(Asymptote(G),H)", "coords": {} }, { "index": 5042, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5042.png", "answer": "8", "fact_expr": "G: Parabola;H: Line;Expression(G) = (y = x^2/4);Slope(H)=1;PointOnCurve(Focus(G),H)", "coords": {} }, { "index": 5043, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5043.png", "answer": "6", "fact_expr": "Gamma: Ellipse;O: Origin;P: Point;F: Point;Expression(Gamma) = (x^2/4 + y^2/3 = 1);LeftFocus(Gamma) = F;PointOnCurve(P, Gamma)", "coords": {} }, { "index": 5044, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5044.png", "answer": "2", "fact_expr": "C: Parabola;p: Number;G: Line;A: Point;E: Point;B: Point;D: Point;Q: Point;F: Point;l: Line;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(Q) = (-p/2, 0);Focus(C) = F;Directrix(C) = l;PointOnCurve(Q, G);Intersection(G, C) = {A, B};L1:Line;L2:Line;PointOnCurve(A, L1);PointOnCurve(B,L2);Intersection(l,L1)=D;Intersection(l, L2)=E;Slope(LineSegmentOf(A, E))=k1;Slope(LineSegmentOf(B, D)) = k2;k1:Number;k2:Number;IsPerpendicular(l,L1);IsPerpendicular(l,L2)", "coords": {} }, { "index": 5045, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5045.png", "answer": "4", "fact_expr": "G: Parabola;B: Point;P: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(B) = (3, 2);PointOnCurve(P, G);Focus(G) = F", "coords": { "B": [ 3.0, 2.0 ] } }, { "index": 5046, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5046.png", "answer": "4", "fact_expr": "G: Hyperbola;a: Number;b: Number;c: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = (2*sqrt(3))/3;FocalLength(G) = 2*c;2*a^2 = 3*c;PointOnCurve(P, G);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 2;LeftFocus(G) = F1;RightFocus(G) = F2", "coords": {} }, { "index": 5047, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5048, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5048.png", "answer": "2*sqrt(5)", "fact_expr": "C: Parabola;G: Circle;A: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(A, C);Abs(LineSegmentOf(A, F)) = 3;Center(G)=A;Radius(G)=LineSegmentOf(A,F)", "coords": {} }, { "index": 5049, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5049.png", "answer": "x^2 - y^2 / 3 = 1", "fact_expr": "G: Hyperbola;H: Parabola;O: Origin;A: Point;B: Point;Expression(H) = (y^2 = 8*x);Focus(H)=F;OneOf(Focus(G))=F;Intersection(Directrix(H), G) = {A, B};Area(TriangleOf(O,A,B))=6;F:Point", "coords": {} }, { "index": 5050, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5050.png", "answer": "4/3", "fact_expr": "G: Parabola;P: Point;F: Point;A: Point;O: Origin;l: Line;Expression(G) = (x^2 = 4*y);Focus(G) = F;Directrix(G)=l;PointOnCurve(P,G);PointOnCurve(P,LineSegmentOf(P,A));IsPerpendicular(LineSegmentOf(P,A),l);FootPoint(LineSegmentOf(P,A),l) = A;AngleOf(A, F, O) = ApplyUnit(30, degree)", "coords": {} }, { "index": 5051, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5051.png", "answer": "x^2/12 - y^2/4 = 1", "fact_expr": "G: Hyperbola;H: Parabola;Expression(OneOf(Asymptote(G)))=(x + sqrt(3)*y = 0);Expression(H)=(y^2=16*x);OneOf(Focus(G))=Focus(H)", "coords": {} }, { "index": 5052, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5052.png", "answer": "6*sqrt(2)", "fact_expr": "G: Parabola;M: Point;Expression(G) = (y^2 = 12*x);PointOnCurve(M, G);Distance(M, Focus(G)) = 9", "coords": {} }, { "index": 5053, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5053.png", "answer": "-1/2", "fact_expr": "C: Ellipse;A: Point;M: Point;B: Point;Expression(C) = (x^2/4 + y^2/2 = 1);Intersection(C, xAxis) = {A, B};PointOnCurve(M, C);Negation(M = A);Negation(M = B);Slope(LineOf(M,A))=k1;Slope(LineOf(M,B))=k2;k1:Number;k2:Number", "coords": {} }, { "index": 5054, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5054.png", "answer": "sqrt(2)", "fact_expr": "C: Hyperbola;b: Number;a: Number;H: Parabola;A: Point;B: Point;e: Number;l:Line;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (4*x^2=y);Directrix(H)=l;Intersection(l,Asymptote(C))={A, B};Abs(LineSegmentOf(A, B)) = 1/8;Eccentricity(C)=e", "coords": {} }, { "index": 5055, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5055.png", "answer": "2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;H: Line;PointOnCurve(F, H);A: Point;B: Point;Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, F)) = 2", "coords": {} }, { "index": 5056, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5056.png", "answer": "(-1/4,1)", "fact_expr": "G: Parabola;A: Point;P: Point;Expression(G) = (y^2 = -4*x);Coordinate(A) = (-2, 1);PointOnCurve(P, G);WhenMin(Distance(P,F)+Distance(P,A));Focus(G)=F;F:Point", "coords": { "A": [ -2.0, 1.0 ] } }, { "index": 5057, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5057.png", "answer": "1", "fact_expr": "F1: Point;F2: Point;G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1);Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G) = True;AngleOf(F1, P, F2) = ApplyUnit(90, degree)", "coords": {} }, { "index": 5058, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5059, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5059.png", "answer": "4/5", "fact_expr": "F2: Point;F1: Point;LeftFocus(G) = F1;RightFocus(G) = F2;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;P: Point;PointOnCurve(P,RightPart(G)) = True;Abs(LineSegmentOf(P, F2)) = Abs(LineSegmentOf(F1, F2));Expression(Asymptote(G)) = (4*x+pm*3*y = 0)", "coords": {} }, { "index": 5060, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5060.png", "answer": "6", "fact_expr": "G: Ellipse;H: Line;F2: Point;A: Point;B: Point;F1: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(G) = {F1, F2};PointOnCurve(F1, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(F2, A)) + Abs(LineSegmentOf(F2, B)) = 14", "coords": {} }, { "index": 5061, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5061.png", "answer": "2*sqrt(3)", "fact_expr": "H: Circle;Expression(H) = (4*y + x^2 + y^2 = 0);G: Parabola;Expression(G) = (x^2 = 4*y)", "coords": {} }, { "index": 5062, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5062.png", "answer": "ApplyUnit(60,degree)", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/16 + y^2/9 = 1);PointOnCurve(P, G);LeftFocus(G) = F1;RightFocus(G) = F2;Abs(LineSegmentOf(P, F1))*Abs(LineSegmentOf(P, F2)) = 12", "coords": {} }, { "index": 5063, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5063.png", "answer": "8", "fact_expr": "l: Line;G: Parabola;B: Point;A: Point;Expression(G) = (y^2 = 4*x);Intersection(l, G) = {A, B};Coordinate(MidPoint(LineSegmentOf(A,B)))=(3,2)", "coords": {} }, { "index": 5064, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5064.png", "answer": "sqrt(3)/3", "fact_expr": "C: Ellipse;b: Number;a: Number;B: Point;F: Point;D: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;OneOf(Endpoint(MinorAxis(C)))=B;Intersection(OverlappingLine(LineSegmentOf(B,F)), C) = D;VectorOf(B, F) = 2*VectorOf(F, D)", "coords": {} }, { "index": 5065, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5065.png", "answer": "{(3,6),(3,-6)}", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 12*x);P:Point;Distance(P,Focus(G))=6;PointOnCurve(P,G)", "coords": {} }, { "index": 5066, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 5.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5066.png", "answer": "-1", "fact_expr": "G: Hyperbola;H: Circle;P: Point;F: Point;M: Point;N: Point;O: Origin;Expression(G) = (x^2/16 - y^2/25 = 1);Expression(H) = (x^2 + y^2 = 16);LeftFocus(G) = F;PointOnCurve(P, RightPart(G));TangentPoint(LineSegmentOf(P, F), H) = N;MidPoint(LineSegmentOf(P, F)) = M", "coords": {} }, { "index": 5067, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5067.png", "answer": "y=pm*sqrt(3)*x", "fact_expr": "G: Hyperbola;b: Number;a: Number;F: Point;N: Point;M: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (-2, 0);LeftFocus(G)=F;L:Line;PointOnCurve(F,L);IsPerpendicular(L,L1);FootPoint(L,L1)=N;VectorOf(M, N) = VectorOf(N, F);L1:Line;L2:Line;Intersection(L,L2)=M;OneOf(Asymptote(G))=L1;OneOf(Asymptote(G))=L2;Negation(L1=L2)", "coords": { "F": [ -2.0, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 5068, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5068.png", "answer": "(2*a^2*Abs(k)/(1 + a^2*k^2))*sqrt(1 + k^2)", "fact_expr": "G: Ellipse;a: Number;H: Line;k: Number;a>1;Expression(G) = (y^2 + x^2/a^2 = 1);Expression(H) = (y = k*x + 1)", "coords": {} }, { "index": 5069, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -1.0, 0.0 ], "radius": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_5069.png", "answer": "x^2/9+y^2/8=1", "fact_expr": "M: Circle;C1:Circle;Expression(C1)=((x+1)^2+y^2=25);C2:Circle;Expression(C2)=((x-1)^2+y^2=1);IsInTangent(M,C1);IsOutTangent(M,C2);M1:Point;Center(M)=M1", "coords": {} }, { "index": 5070, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5070.png", "answer": "5/3", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;G: Circle;T: Point;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2 + y^2 = a^2);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, l);TangentPoint(l, G) = T;Intersection(l, RightPart(C)) = P;VectorOf(T, P) = 3*VectorOf(F1, T)", "coords": {} }, { "index": 5071, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5071.png", "answer": "3/2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x);A: Point;B: Point;PointOnCurve(A, G);PointOnCurve(B, G);IsPerpendicular(LineSegmentOf(A, B), xAxis);Abs(LineSegmentOf(A, B)) = 2*sqrt(2)", "coords": {} }, { "index": 5072, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5072.png", "answer": "-2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = a*x);a: Number;Expression(Directrix(G)) = (x = 1/2)", "coords": {} }, { "index": 5073, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.449489742783178, "major_axis": "x", "x_coef": 16.0, "y_coef": 6.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5073.png", "answer": "x^2/9-y^2=1", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/16 + y^2/6 = 1);Focus(G) = Focus(H);Expression(Asymptote(G)) = (y = pm*(x/3))", "coords": {} }, { "index": 5074, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5075, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5075.png", "answer": "[sqrt(2)/2,sqrt(5)/3]", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G) = True;lambda: Number;Abs(LineSegmentOf(P, F1)) = lambda*Abs(LineSegmentOf(P, F2));1/2 <= lambda;lambda <= 2;AngleOf(F1, P, F2) = pi/2", "coords": {} }, { "index": 5076, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5077, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2631799883090584, "b": 2.5275303393962245, "orientation": "horizontal" }, "optimization": { "final_loss": 4.481942266905428e-06, "converged": true, "iterations": 95 }, "output_path": "results/train/hyperbola/problem_5077.png", "answer": "x^2 - y^2/4 = 1", "fact_expr": "G: Hyperbola;Expression(OneOf(Asymptote(G))) = (y = 2*x);H: Parabola;Expression(H) = (y^2 = 4*x);PointOnCurve(Focus(H), G)", "coords": {} }, { "index": 5078, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5078.png", "answer": "7/4", "fact_expr": "M: Point;N: Point;Length(LineSegmentOf(M, N)) = 4;Endpoint(LineSegmentOf(M, N)) = {M, N};G: Parabola;Expression(G) = (y^2 = x);PointOnCurve(M, G);PointOnCurve(N, G);P: Point;MidPoint(LineSegmentOf(M, N)) = P", "coords": {} }, { "index": 5079, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.6457513110645907, "major_axis": "x", "x_coef": 16.0, "y_coef": 7.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5079.png", "answer": "16", "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/7 = 1);F2: Point;F1: Point;LeftFocus(G) = F1;RightFocus(G) = F2;H: Line;A: Point;B: Point;PointOnCurve(F1, H);Intersection(H, G) = {A,B}", "coords": {} }, { "index": 5080, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5080.png", "answer": "2/5", "fact_expr": "C: Ellipse;Expression(C) = (x^2/25 + y^2/16 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;M: Point;PointOnCurve(M, C)", "coords": {} }, { "index": 5081, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5081.png", "answer": "sqrt(3)-1", "fact_expr": "G: Ellipse;H: Line;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);Expression(H) = (x - sqrt(3)*y + 4 + sqrt(3) = 0);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, H);WhenMax(AngleOf(F1,P,F2))", "coords": {} }, { "index": 5082, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5082.png", "answer": "x + 2*y-2 = 0", "fact_expr": "M: Ellipse;Expression(M) = (x^2/4 + y^2 = 1);l: Line;Intersection(l, M) = {A, B};A: Point;B: Point;D: Point;Coordinate(D) = (1, 1/2);IsChordOf(LineSegmentOf(A,B),M) ;MidPoint(LineSegmentOf(A, B)) = D", "coords": { "D": [ 1.0, 0.5 ] } }, { "index": 5083, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5083.png", "answer": "sqrt(10)/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;P: Point;F1: Point;F2: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2 + y^2 = a^2/4);PointOnCurve(P,RightPart(G));LeftFocus(G)=F1;RightFocus(G)=F2;DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;NumIntersection(LineOf(P,F1), H) = 1", "coords": {} }, { "index": 5084, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5084.png", "answer": "1", "fact_expr": "C: Ellipse;b: Number;a: Number;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);FocalLength(C) = 4;P:Point;PointOnCurve(P,C);Max(Distance(P,OneOf(Focus(C))))=5", "coords": {} }, { "index": 5085, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5085.png", "answer": "sqrt(34)-1", "fact_expr": "P: Point;G: Parabola;Expression(G) = (y^2 = 4*x);PointOnCurve(P, G) = True;Projection(P, yAxis) = Q;Q: Point;M: Point;Coordinate(M) = (4, 5)", "coords": { "M": [ 4.0, 5.0 ] } }, { "index": 5086, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5086.png", "answer": "5", "fact_expr": "P: Point;G: Parabola;Expression(G) = (y^2 = 16*x);PointOnCurve(P, G);Distance(P, Directrix(G)) = d;d: Number;Q: Point;Coordinate(Q) = (0, -3)", "coords": { "Q": [ 0.0, -3.0 ] } }, { "index": 5087, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5087.png", "answer": "16/3", "fact_expr": "C: Circle;G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);PointOnCurve(OneOf(Vertex(G)), C);PointOnCurve(OneOf(Focus(G)), C);PointOnCurve(Center(C), G)", "coords": {} }, { "index": 5088, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5089, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 6.0, "b": 6.708203932499369, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5089.png", "answer": "25", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2:Point;Expression(G) = (x^2/36 - y^2/45 = 1);PointOnCurve(P, G);LeftFocus(G) =F1;RightFocus(G)=F2;Distance(P, F1) = 13", "coords": {} }, { "index": 5090, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5090.png", "answer": "sqrt(3)/3", "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;P: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P,F2),xAxis);Slope(LineOf(P,F1)) = sqrt(3)/3", "coords": {} }, { "index": 5091, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5091.png", "answer": "(pm*sqrt(5),0)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/4 = 1)", "coords": {} }, { "index": 5092, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5093, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5094, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5094.png", "answer": "sqrt(2)/2", "fact_expr": "l: Line;G: Ellipse;b: Number;a: Number;P: Point;A: Point;F1: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;RightVertex(G)=A;PointOnCurve(P,l);PointOnCurve(A,l);IsPerpendicular(l,xAxis);WhenMin(Area(CircumCircle(TriangleOf(F1,F2,P))));Length(LineSegmentOf(P,A))=HalfFocalLength(G)", "coords": {} }, { "index": 5095, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5095.png", "answer": "2*sqrt(3)", "fact_expr": "G: Ellipse;Expression(G) = (x^2 + y^2/4 = 1)", "coords": {} }, { "index": 5096, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5096.png", "answer": "8", "fact_expr": "G: Hyperbola;M: Parabola;A: Point;B: Point;F: Point;Expression(G) = (x^2 - y^2 = 1);Expression(M) = (y^2 = 8*x);Focus(M) = F;Intersection(OneOf(Asymptote(G)), M) = {A, B}", "coords": {} }, { "index": 5097, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 2.0, 1.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_5097.png", "answer": "pm*(sqrt(3)/2)", "fact_expr": "G: Circle;Q: Curve;a: Number;A: Point;B: Point;Expression(Q) = (x^2/(2*a^2) - y^2/a^2 = 1)&(x>0);Expression(G)=((x-2)^2+(y-1)^2=1);PointOnCurve(A, Q);PointOnCurve(B, Q);IsDiameter(LineSegmentOf(A,B),G)", "coords": {} }, { "index": 5098, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5098.png", "answer": "y=pm*((sqrt(5)+1)/2)*x", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;P: Point;Q: Point;F2: Point;F1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1,l);IsPerpendicular(l,OneOf(Asymptote(C)));Intersection(l,LeftPart(C))=Q;Intersection(l,RightPart(C))=P;-Abs(LineSegmentOf(P, F2)) + Abs(LineSegmentOf(P, Q)) = a", "coords": {} }, { "index": 5099, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5099.png", "answer": "y = pm*x*sqrt(5)/2", "fact_expr": "G: Hyperbola;b: Number;H: Point;Expression(G) = (x^2/4 - y^2/b = 1);Coordinate(H) = (3, 0);RightFocus(G) = H", "coords": { "H": [ 3.0, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 5100, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 2.0, 0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_5100.png", "answer": "(x^2/(1/4)-y^2/(15/4)=1)&(x<0)", "fact_expr": "G: Circle;Expression(G) = (y^2 + (x + 2)^2 = 1);C:Circle;Expression(C)=((x-2)^2+y^2=4);H:Circle;IsOutTangent(H,G);IsOutTangent(H,C)", "coords": {} }, { "index": 5101, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5101.png", "answer": "sqrt(5)/5", "fact_expr": "G: Ellipse;a: Number;b: Number;H: Line;A: Point;F2: Point;B: Point;C: Point;F1: Point;a > b;b > 0;Expression(G) = (x^2/b^2 + y^2/a^2 = 1);UpperFocus(G) = F1;LowerFocus(G) = F2;PointOnCurve(F1, H);IsPerpendicular(H, yAxis);Intersection(H, G) = {A, B};Intersection(LineOf(A,F2),G)={A,C};Area(TriangleOf(A, B, C)) = 3*Area(TriangleOf(B, C, F2));Negation(A=C)", "coords": {} }, { "index": 5102, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5102.png", "answer": "y^2=12*x", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(C) = F;G: Ellipse;Expression(G) = (x^2/9 + y^2/4 = 1);Distance(F, Directrix(C)) = Length(MajorAxis(G))", "coords": {} }, { "index": 5103, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5103.png", "answer": "2", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);M: Point;Coordinate(M) = (-1, 1);L: Line;PointOnCurve(Focus(C), L) = True;Slope(L) = k;k: Real;Intersection(L, C) = {A, B};A: Point;B: Point;DotProduct(VectorOf(M, A), VectorOf(M, B)) = 0", "coords": { "M": [ -1.0, 1.0 ] } }, { "index": 5104, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5104.png", "answer": "x^2=-24*y", "fact_expr": "G: Parabola;O: Origin;Vertex(G) = O;SymmetryAxis(G) = yAxis;H: Line;Expression(H) = (3*x - 4*y - 24 = 0);PointOnCurve(Focus(G), H)", "coords": {} }, { "index": 5105, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5106, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5106.png", "answer": "2*sqrt(3)/3", "fact_expr": "G: Hyperbola;a: Number;b: Number;e: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(Asymptote(G)) = (x + pm*sqrt(3)*y = 0);Eccentricity(G) = e", "coords": {} }, { "index": 5107, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5108, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5108.png", "answer": "sqrt(2)", "fact_expr": "G: Parabola;p: Number;Q: Point;O: Origin;F: Point;P: Point;p>0;Expression(G) = (y^2 = 2*(p*x));PointOnCurve(P, G);Focus(G)=F;PointOnCurve(Q, LineSegmentOf(P,F));2*Abs(VectorOf(P, Q)) = Abs(VectorOf(Q, F))", "coords": {} }, { "index": 5109, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5110, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5110.png", "answer": "24", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 2*(p*y));p: Number;p>0;H: Point;Coordinate(H) = (m, 3);m: Number;PointOnCurve(H, G);Distance(H, Focus(G)) = 5", "coords": {} }, { "index": 5111, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5111.png", "answer": "3*sqrt(2)/4", "fact_expr": "G: Line;Expression(G) = (y = k*x - 2);k: Number;k>0;C: Parabola;Expression(C) = (x^2 = 8*y);A: Point;B: Point;Intersection(G, C) = {A, B};F: Point;Focus(C) = F;Abs(LineSegmentOf(F, A)) = 2*Abs(LineSegmentOf(F, B))", "coords": {} }, { "index": 5112, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 1.0, -2.0 ], "radius": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_5112.png", "answer": "3/4", "fact_expr": "C1: Circle;C:Hyperbola;a: Number;a>0;Expression(C) = (-y^2 + x^2/a^2 = 1);Expression(C1) = ((x - 1)^2 + (y + 2)^2 = 4);IsTangent(OneOf(Asymptote(C)),C1);Quadrant(TangentPoint(OneOf(Asymptote(C)),C1))=3", "coords": {} }, { "index": 5113, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5113.png", "answer": "y^2/2 - x^2/4 = 1", "fact_expr": "G: Hyperbola;C: Hyperbola;A: Point;Expression(G) = (x^2/2 - y^2 = 1);Coordinate(A) = (-2, 2);PointOnCurve(A, C);Asymptote(G) = Asymptote(C)", "coords": { "A": [ -2.0, 2.0 ] } }, { "index": 5114, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.23606797749979, "b": 2.0, "major_axis": "x", "x_coef": 5.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5114.png", "answer": "2", "fact_expr": "G: Ellipse;Expression(G) = (x^2/5 + y^2/4 = 1)", "coords": {} }, { "index": 5115, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5115.png", "answer": "8/3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;A: Point;B: Point;PointOnCurve(A, G);PointOnCurve(B, G);VectorOf(A, F) = 3*VectorOf(F, B);IsChordOf(LineSegmentOf(A, B), G)", "coords": {} }, { "index": 5116, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5117, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5118, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5118.png", "answer": "pm*p/2", "fact_expr": "G: Parabola;p: Number;A: Point;B: Point;M: Point;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(M) = (xm, ym);xm:Number;ym:Number;Focus(G) = F;IsChordOf(LineSegmentOf(A, B), G);PointOnCurve(F, LineSegmentOf(A, B));MidPoint(LineSegmentOf(A, B)) = M;xm = (3/4)*p", "coords": {} }, { "index": 5119, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5119.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;Asymptote(G) = {l1, l2};l1: Line;l2: Line;PointOnCurve(F2, V) = True;IsPerpendicular(V, l1) = True;V: Line;Intersection(V, l1) = P;Intersection(V, l2) = Q;P: Point;Q: Point;O: Origin;VectorOf(O, F2) + VectorOf(O, Q) = 2*VectorOf(O, P)", "coords": {} }, { "index": 5120, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5120.png", "answer": "{1/3, 3}", "fact_expr": "C: Ellipse;G: Line;A: Point;B: Point;F1: Point;F2: Point;Expression(C) = (x^2/2 + y^2 = 1);LeftFocus(C) =F1;RightFocus(C)=F2;PointOnCurve(F2, G);Intersection(G, C) = {A, B};Area(TriangleOf(A, B, F1)) = 4/3", "coords": {} }, { "index": 5121, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5121.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;F1: Point;F2: Point;P: Point;Expression(G) = (x^2/2 - y^2 = 1);PointOnCurve(P, G);Focus(G) = {F1, F2};Abs(LineSegmentOf(P, F1)) + Abs(LineSegmentOf(P, F2)) = 4*sqrt(2)", "coords": {} }, { "index": 5122, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5122.png", "answer": "sqrt(17)/3", "fact_expr": "G: Hyperbola;b: Number;a: Number;A: Point;B: Point;F: Point;C: Point;O:Origin;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(A, G);PointOnCurve(B, G);PointOnCurve(C, G);PointOnCurve(O,LineSegmentOf(A,B));RightFocus(G)=F;PointOnCurve(F,LineSegmentOf(A,C));IsPerpendicular(LineSegmentOf(B, F), LineSegmentOf(A, C));3*Abs(LineSegmentOf(A, F)) = Abs(LineSegmentOf(A, C))", "coords": {} }, { "index": 5123, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5124, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5124.png", "answer": "sqrt(5)", "fact_expr": "C: Hyperbola;a: Number;a>0;Expression(C) = (-y^2/5 + x^2/a^2 = 1)", "coords": {} }, { "index": 5125, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5125.png", "answer": "sqrt(2)/2", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;H: Line;Expression(H) = (x = -a^2/c);c: Number;PointOnCurve(P, H);Negation(PointOnCurve(P, axis));Max(AngleOf(F1, P, F2)) = pi/6", "coords": {} }, { "index": 5126, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5126.png", "answer": "2*sqrt(3)", "fact_expr": "C: Hyperbola;F1: Point;P: Point;F2: Point;Expression(C) = (x^2 - y^2 = 2);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 5127, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5128, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5129, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5129.png", "answer": "8", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);F: Point;Focus(G) = F;C: Point;Intersection(Directrix(G), xAxis) = C;H: Line;PointOnCurve(F, H);A: Point;B: Point;Intersection(H, G) = {A, B};AngleOf(C, B, F) = ApplyUnit(90, degree)", "coords": {} }, { "index": 5130, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5130.png", "answer": "sqrt(2)", "fact_expr": "C: Ellipse;Expression(C) = (x^2/a^2+y^2/b^2=1);a: Number;b: Number;a>b;b>0;Eccentricity(C) = sqrt(3)/2;F: Point;RightFocus(C) = F;G: Line;PointOnCurve(F, G);k: Number;k>0;Slope(G) = k;A: Point;B: Point;Intersection(G, C) = {A, B};VectorOf(A, F) = 3*VectorOf(F, B)", "coords": {} }, { "index": 5131, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5131.png", "answer": "5/4", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);F: Point;Focus(G) = F;A: Point;Quadrant(A) = 1;PointOnCurve(A, G);Abs(LineSegmentOf(A, F)) = 2;P: Point;PointOnCurve(P, Directrix(G));WhenMin(Abs(LineSegmentOf(P, A))+ Abs(LineSegmentOf(P, F)))", "coords": {} }, { "index": 5132, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5133, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5134, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5134.png", "answer": "4", "fact_expr": "C: Parabola;A: Point;F: Point;B: Point;l: Line;Expression(C) = (y^2 = 4*x);Focus(C) = F;Directrix(C) = l;In(A, l);Intersection(LineSegmentOf(A,F), C) = B;VectorOf(F, A) = 3*VectorOf(F, B)", "coords": {} }, { "index": 5135, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5136, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5137, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5137.png", "answer": "(1,\\sqrt{5}/2]", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;P:Point;PointOnCurve(P, RightPart(C)) = True;L1: Line;L2: Line;PointOnCurve(P, L1) = True;PointOnCurve(P, L2) = True;J1: Line;J2: Line;Asymptote(C) = {J1, J2};IsParallel(L1, J1) = True;IsParallel(L2, J2) = True;Intersection(L1, J2) = M;Intersection(L2, J1) = N;M: Point;N: Point;O: Origin;Area(TriangleOf(O, M, N)) = S;S: Number;S >= b^2/2", "coords": {} }, { "index": 5138, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5139, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5140, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5141, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5141.png", "answer": "4", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);A: Point;LeftVertex(G) = A;I: Point;Coordinate(I) = (-6/5, 0);M: Point;N: Point;PointOnCurve(I, LineOf(M, N));Intersection(LineOf(M, N), G) = {M, N};Negation(A=M);Negation(A=N);k1: Number;Slope(LineOf(A, M)) = k1;k2: Number;Slope(LineOf(A, N)) = k2", "coords": { "I": [ -1.2, 0.0 ] } }, { "index": 5142, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5143, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5143.png", "answer": "sqrt(3)", "fact_expr": "l: Line;C: Parabola;G: Circle;B: Point;A: Point;F: Point;Expression(C) = (y^2 = 2*x);Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, F)) = Abs(LineSegmentOf(B, F));IsDiameter(LineSegmentOf(A,B),G)", "coords": {} }, { "index": 5144, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5145, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5145.png", "answer": "{y^2=12*x, y^2=-4*x}", "fact_expr": "G: Parabola;O: Origin;Vertex(G) = O;PointOnCurve(Focus(G), xAxis);H: Line;Expression(H) = (2*x - y + 1 = 0);Length(InterceptChord(H, G)) = sqrt(15)", "coords": {} }, { "index": 5146, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5146.png", "answer": "8*x+9*y-25=0", "fact_expr": "C: Ellipse;Expression(C) = (x^2/9 + y^2/4 = 1);P: Point;Coordinate(P) = (2, 1);PointOnCurve(P, l) = True;l: Line;Intersection(l, C) = {A, B};A: Point;B: Point;MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ 2.0, 1.0 ] } }, { "index": 5147, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5147.png", "answer": "5", "fact_expr": "G: Parabola;M: Point;N: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(M, G);IsPerpendicular(LineSegmentOf(M, N), xAxis);Intersection(LineSegmentOf(M, N), xAxis) = N;Abs(LineSegmentOf(M, F)) = 6", "coords": {} }, { "index": 5148, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5148.png", "answer": "-2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1);A1: Point;LeftVertex(G) = A1;F2: Point;RightFocus(G) = F2;P: Point;PointOnCurve(P, RightPart(G))", "coords": {} }, { "index": 5149, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5149.png", "answer": "{x^2 = 2*y, x^2 = 4*y}", "fact_expr": "G: Parabola;p: Number;A: Point;B: Point;M: Point;p>0;Expression(G) = (x^2 = 2*(p*y));Coordinate(M) = (2, -2*p);l1: Line;l2: Line;TangentOfPoint(M, G) = {l1, l2};TangentPoint(l1, G) = A;TangentPoint(l2, G) = B;YCoordinate(MidPoint(LineSegmentOf(A, B))) = 6", "coords": {} }, { "index": 5150, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5150.png", "answer": "y^2 = -6x", "fact_expr": "G: Parabola;C:Curve;O:Origin;Expression(C)=(2*x-4*y+3=0);SymmetryAxis(G)=xAxis;Vertex(G)=O;PointOnCurve(Focus(G),C)", "coords": {} }, { "index": 5151, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5151.png", "answer": "2*sqrt(3)/3", "fact_expr": "H: Circle;Expression(H) = ((x - sqrt(3))^2 + (y - 1)^2 = 9);G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Intersection(H,Asymptote(G)) = {A,B};Quadrant(A) = 2;Quadrant(B) = 4;A: Point;B: Point;Abs(LineSegmentOf(A, B)) = 2*sqrt(6)", "coords": {} }, { "index": 5152, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5152.png", "answer": "25", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G)", "coords": {} }, { "index": 5153, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5153.png", "answer": "8*sqrt(2)/7", "fact_expr": "G: Ellipse;Expression(G) = (x^2/2 + y^2 = 1);F1: Point;LeftFocus(G) = F1;l: Line;PointOnCurve(F1, l) = True;Inclination(l) = ApplyUnit(60, degree);Intersection(l, G) = {A, B};A: Point;B: Point", "coords": {} }, { "index": 5154, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.47213595499958, "b": 2.0, "major_axis": "x", "x_coef": 20.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5154.png", "answer": "4", "fact_expr": "F: Point;C: Ellipse;Expression(C) = (x^2/20 + y^2/4 = 1);RightFocus(C) = F;P: Point;PointOnCurve(P, C) = True;A: Point;Coordinate(A) = (-2, 1);WhenMin(Perimeter(TriangleOf(A, P, F))) = True", "coords": { "A": [ -2.0, 1.0 ] } }, { "index": 5155, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5156, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5157, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5157.png", "answer": "2", "fact_expr": "G: Parabola;H: Point;x0:Number;y0:Number;Expression(G) = (y^2 = 8*x);Coordinate(H) = (x0, y0);PointOnCurve(H, G);Distance(H, Focus(G)) = x0^2", "coords": {} }, { "index": 5158, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5158.png", "answer": "1", "fact_expr": "F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);M: Point;PointOnCurve(M, G) = True", "coords": {} }, { "index": 5159, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5160, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5161, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5161.png", "answer": "2*sqrt(2)", "fact_expr": "C: Parabola;Expression(C) = (x^2 = 4*y);F: Point;Focus(C) = F;P: Point;G: Line;Expression(G) = (x - y - 2 = 0);PointOnCurve(P, G);TangentOfPoint(P, C) = {LineOf(P, A), LineOf(P, B)};TangentPoint(LineOf(P, A), C) = A;TangentPoint(LineOf(P, B), C) = B;B: Point;A: Point;O: Origin", "coords": {} }, { "index": 5162, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5163, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5164, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5164.png", "answer": "2", "fact_expr": "G: Hyperbola;C:Hyperbola;H: Point;Expression(G) = (x^2/9 - y^2/16 = 1);Coordinate(H) = (-3, 2*sqrt(3));Asymptote(G) = Asymptote(C);PointOnCurve(H,C)", "coords": { "H": [ -3.0, 3.4641016151377544 ] } }, { "index": 5165, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5165.png", "answer": "x-y-2=0", "fact_expr": "G: Parabola;H: LineSegment;A: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (4, 2);MidPoint(H)=A;IsChordOf(H,G)", "coords": { "A": [ 4.0, 2.0 ] } }, { "index": 5166, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_5166.png", "answer": "12", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/n + x^2/m = 1);m: Number;n: Number;m>0;n>0;Eccentricity(G) = 2;H: Parabola;Expression(H) = (y^2 = 4*(m*x));OneOf(Focus(G)) = Focus(H)", "coords": {} }, { "index": 5167, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_5167.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;Eccentricity(G) = 2", "coords": {} }, { "index": 5168, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5168.png", "answer": "2*sqrt(3) - 3", "fact_expr": "G: Hyperbola;H: Circle;M: Point;T: Point;F: Point;P: Point;O: Origin;Expression(G) = (x^2/9 - y^2/12 = 1);Expression(H) = (x^2 + y^2 = 9);LeftFocus(G)=F;IsTangent(LineOf(M, T), H);PointOnCurve(F, LineOf(M, T));TangentPoint(LineOf(M,T),H)=T;Intersection(LineOf(M,T),RightPart(G))=M;MidPoint(LineSegmentOf(M, F)) = P", "coords": {} }, { "index": 5169, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5170, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5170.png", "answer": "sqrt(6)/2", "fact_expr": "l: Line;G: Hyperbola;b: Number;a: Number;l1:Line;F: Point;B: Point;A: Point;l2: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G)=F;Asymptote(G)={l1,l2};PointOnCurve(F,l);IsPerpendicular(l,l1);Intersection(l,l1)=A;Intersection(l,l2)=B;VectorOf(F, B) = 3*VectorOf(A, F)", "coords": {} }, { "index": 5171, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5171.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;A: Point;F2: Point;F1: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(A, G);IsPerpendicular(LineSegmentOf(A, F2), xAxis);Abs(LineSegmentOf(A, F1))/Abs(LineSegmentOf(A, F2)) = 5/3", "coords": {} }, { "index": 5172, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5172.png", "answer": "{2,18}", "fact_expr": "l: Line;G: Parabola;p: Number;H: Circle;p>0;Expression(G) = (y^2 = 2*p*x);Slope(l)=1;Focus(G)=F;Expression(H) = (y^2 + (x - 5)^2 = 8);IsTangent(l,H);F:Point;PointOnCurve(F,l)", "coords": {} }, { "index": 5173, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5173.png", "answer": "(1,\\sqrt{2}]", "fact_expr": "l: Line;G: Hyperbola;b: Number;a: Number;C: Parabola;p: Number;A: Point;B: Point;F: Point;k:Number;k>=sqrt(2);a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);p>0;Expression(C) = (y^2 = 2*p*x);Focus(C)=F;PointOnCurve(F,l);Intersection(l,Directrix(C))=A;OneOf(Intersection(l,C))=B;VectorOf(A, B) = k*VectorOf(B, F);IsPerpendicular(l,OneOf(Asymptote(G)))", "coords": {} }, { "index": 5174, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5174.png", "answer": "3/4", "fact_expr": "G: Parabola;H: Line;M: Point;N: Point;F:Point;x1:Number;y1:Number;x2:Number;y2:Number;Expression(G)=(y^2 = x);Coordinate(M)=(x1,y1);Coordinate(N)=(x2,y2);Focus(G)=F;PointOnCurve(F,H);Intersection(H,G)={M,N};x1+x2=1", "coords": {} }, { "index": 5175, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5175.png", "answer": "2", "fact_expr": "G: Parabola;p: Number;p>0;Expression(G) = (y^2 = 2*(p*x));P: Point;XCoordinate(P) = 4;PointOnCurve(P, G);Distance(P, Focus(G)) = 5", "coords": {} }, { "index": 5176, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5177, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5177.png", "answer": "1/2", "fact_expr": "G: Hyperbola;H: Ellipse;b: Number;a: Number;a > b;b > 0;Expression(H) = (y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = 2;Center(G)=O;O:Origin;Vertex(G)=Focus(H);Focus(G)=Vertex(H)", "coords": {} }, { "index": 5178, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5178.png", "answer": "x^2/11 - y^2/(99/16) = 1", "fact_expr": "G: Hyperbola;C:Hyperbola;H: Point;Expression(G) = (x^2/16 - y^2/9 = 1);Coordinate(H) = (3*sqrt(3), -3);Asymptote(C) = Asymptote(G);PointOnCurve(H, C)", "coords": { "H": [ 5.196152422706632, -3.0 ] } }, { "index": 5179, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5179.png", "answer": "(\\sqrt{5}-1)/2", "fact_expr": "C: Hyperbola;b: Number;a: Number;M: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(M, Asymptote(C));IsPerpendicular(LineSegmentOf(M, F1), LineSegmentOf(M, F2));Abs(LineSegmentOf(M, F1)) = 2*b + Abs(LineSegmentOf(M, F2))", "coords": {} }, { "index": 5180, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5181, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5181.png", "answer": "4", "fact_expr": "l: Line;C: Parabola;p: Number;M: Point;F: Point;A: Point;B: Point;N: Point;p>0;Expression(C) = (y^2 = 2*p*x);Focus(C)=F;PointOnCurve(F, l);Intersection(l, C) = {A, B};Projection(A,Directrix(C))=M;Projection(B, Directrix(C)) = N;Area(TriangleOf(M,F,N))/Area(TriangleOf(A,F,M))=lambda;Area(TriangleOf(B,F,N))/Area(TriangleOf(M,F,N))=mu;lambda:Number;mu:Number", "coords": {} }, { "index": 5182, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5182.png", "answer": "[sqrt(2)/2, 1]", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;P: Point;Coordinate(P) = (x1, y1);x1: Number;y1: Number;PointOnCurve(P, G) = True;A: Point;Coordinate(A) = (-1, 0)", "coords": { "A": [ -1.0, 0.0 ] } }, { "index": 5183, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5183.png", "answer": "2*sqrt(3)", "fact_expr": "C: Ellipse;C1:Hyperbola;b: Number;b>0;Expression(C1) = (x^2/4 - y^2/b^2 = 1);Expression(C) = (x^2/16 + y^2/12 = 1);InterReciprocal(Eccentricity(C1), Eccentricity(C))", "coords": {} }, { "index": 5184, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5184.png", "answer": "sqrt(3)-1", "fact_expr": "C1: Ellipse;Expression(C1) = (x^2/a^2 + y^2/b^2 = 1);b: Number;a: Number;a > b;b > 0;H: Circle;Expression(H) = (x^2 + y^2 = a^2 - b^2);P: Point;OneOf(Intersection(C1,H))=P;F2: Point;F1: Point;LeftFocus(C1) = F1;RightFocus(C1) = F2;2*AngleOf(P, F2, F1) = AngleOf(P, F1, F2)", "coords": {} }, { "index": 5185, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.449489742783178, "b": 3.1622776601683795, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5185.png", "answer": "8", "fact_expr": "G: Hyperbola;H: Parabola;p: Real;Expression(G) = (x^2/6 - y^2/10 = 1);p>0;Expression(H) = (y^2 = 2*(p*x));Focus(H) = RightFocus(G)", "coords": {} }, { "index": 5186, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5186.png", "answer": "12", "fact_expr": "G: Parabola;A: Point;B: Point;Expression(G) = (y^2 = 8*x);IsChordOf(LineSegmentOf(A, B), G);a: Number;Coordinate(MidPoint(LineSegmentOf(A, B))) = (4, a);PointOnCurve(Focus(G), LineSegmentOf(A, B))", "coords": {} }, { "index": 5187, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5187.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/4 + y^2/2 = 1)", "coords": {} }, { "index": 5188, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5188.png", "answer": "4*sqrt(10)-8", "fact_expr": "G: Circle;H:Curve;M: Point;P: Point;Q: Point;Expression(G) = (x^2 + (y - 2)^2 = 1);Expression(H) = (x^2 = 8*y);Coordinate(M) = (0, 5);PointOnCurve(M,H);PointOnCurve(Q, G);PointOnCurve(P,H)", "coords": { "M": [ 0.0, 5.0 ] } }, { "index": 5189, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5189.png", "answer": "x^2/25 + y^2/16 = 1", "fact_expr": "G: Hyperbola;C: Ellipse;H: Point;Expression(G) = (x^2/4 - y^2/5 = 1);Coordinate(H) = (5, 0);Center(G) = Center(C);Focus(G) = Focus(C);PointOnCurve(H,C)", "coords": { "H": [ 5.0, 0.0 ] } }, { "index": 5190, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_5190.png", "answer": "(-\\infty,-1/3)+(1/3,+\\infty)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/(4*k^2) + x^2/(9*k^2) = 1);k: Real;H: Circle;Expression(H) = (x^2 + y^2 = 1);NumIntersection(G,H)=0", "coords": {} }, { "index": 5191, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5191.png", "answer": "(-2, 0) \nx = 2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = -8*x)", "coords": {} }, { "index": 5192, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5192.png", "answer": "6", "fact_expr": "C: Parabola;p: Number;G: Line;A: Point;A1: Point;F: Point;B: Point;B1: Point;l1: Line;l: Line;l2: Line;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;PointOnCurve(F, G);Intersection(G, C) = {A, B};PointOnCurve(A, l1);PointOnCurve(B, l2);IsPerpendicular(l1, l);IsPerpendicular(l2, l);FootPoint(l1, l) = A1;FootPoint(l2, l) = B1;Area(TriangleOf(A, A1, F)) = 9;Area(TriangleOf(B, B1, F)) = 1;Directrix(C) = l", "coords": {} }, { "index": 5193, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5193.png", "answer": "2*sqrt(3)/3", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);AngleOf(F1, P, F2) = pi/3", "coords": {} }, { "index": 5194, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5194.png", "answer": "6", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);Focus(C) = F;F: Point;l: Line;Intersection(l,C) = {A,B};A: Point;B: Point;Intersection(PerpendicularBisector(LineSegmentOf(A,B)),xAxis) = P;P: Point;Coordinate(P) = (4, 0)", "coords": { "P": [ 4.0, 0.0 ] } }, { "index": 5195, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5195.png", "answer": "2", "fact_expr": "G: Hyperbola;A1: Point;P: Point;A2: Point;Expression(G) = (x^2 - y^2 = 1);Vertex(G)={A1,A2};PointOnCurve(P, G);Slope(LineOf(P,A1)) = 1/2", "coords": {} }, { "index": 5196, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5196.png", "answer": "6", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/b^2 = 1);b: Number;b>0;Expression(OneOf(Asymptote(G))) = (3*x + 2*y = 0)", "coords": {} }, { "index": 5197, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5197.png", "answer": "[2,2*sqrt(2)]", "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/2 + y^2 = 1);Coordinate(P) = (x0, y0);Focus(C) = {F1,F2};00;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;PointOnCurve(A,C);PointOnCurve(B,C);AngleOf(A, F, B) = ApplyUnit(60, degree);IsChordOf(LineSegmentOf(A, B), C);MidPoint(LineSegmentOf(A, B)) = M;PointOnCurve(M,LineSegmentOf(M,N));IsPerpendicular(Directrix(C),LineSegmentOf(M,N));FootPoint(Directrix(C),LineSegmentOf(M,N))=N", "coords": {} }, { "index": 5200, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5200.png", "answer": "{0,1/2}", "fact_expr": "G: Parabola;H: Line;k: Real;Expression(G) = (y^2 = 4*x);Expression(H) = (y = k*x + 2);NumIntersection(H, G)=1", "coords": {} }, { "index": 5201, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5201.png", "answer": "(1/2,1)", "fact_expr": "G: Parabola;A: Point;M: Point;F: Point;Expression(G) = (y^2 = 2*x);Coordinate(A) = (3, 1);Focus(G) = F;PointOnCurve(M, G);WhenMin(Abs(LineSegmentOf(M, A)) + Abs(LineSegmentOf(M, F)))", "coords": { "A": [ 3.0, 1.0 ] } }, { "index": 5202, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5202.png", "answer": "x \\pm \\sqrt{3} y = 0", "fact_expr": "G: Hyperbola;a: Number;b: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Distance(OneOf(Focus(G)),OneOf(Asymptote(G)))=(1/4)*FocalLength(G)", "coords": {} }, { "index": 5203, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5203.png", "answer": "y=pm*sqrt(3)*x/3\n4", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/3 - y^2 = 1);G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;Focus(G) = OneOf(Focus(C))", "coords": {} }, { "index": 5204, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5204.png", "answer": "sqrt(2)/2", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;A: Point;B: Point;RightVertex(G) = A;UpperVertex(G) = B;LineSegmentOf(A, B) = (sqrt(3)/2)*LineSegmentOf(F1, F2)", "coords": {} }, { "index": 5205, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5205.png", "answer": "sqrt(10)/3", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;O: Origin;F: Point;P: Point;a>0;b>0;l:Line;RightFocus(G) = F;PointOnCurve(F,l);Slope(l)=-1;Intersection(l,Asymptote(G))=P;Quadrant(P)=1;Area(TriangleOf(O, F, P)) = (a^2 + b^2)/8", "coords": {} }, { "index": 5206, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.3300262689590454, "b": 2.30381441116333, "orientation": "horizontal" }, "optimization": { "final_loss": 6.658357051492203e-06, "converged": true, "iterations": 122 }, "output_path": "results/train/hyperbola/problem_5206.png", "answer": "12", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;I: Circle;Expression(I) = (y^2 + (x - 3)^2 = 225);IsTangent(Directrix(G), I);C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;Expression(OneOf(Asymptote(C))) = (y = sqrt(3)*x);OneOf(Focus(C)) = Focus(G)", "coords": {} }, { "index": 5207, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 0.7071067811865476, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5207.png", "answer": "(-sqrt(6)/2,0)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - 2*y^2 = 1)", "coords": {} }, { "index": 5208, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5209, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5209.png", "answer": "y=pm*sqrt(3)*x", "fact_expr": "G: Hyperbola;a: Number;H: Parabola;Expression(G) = (-y^2/3 + x^2/a^2 = 1);Expression(H) = (y^2 = 8*x);PointOnCurve(OneOf(Focus(G)), Directrix(H))", "coords": {} }, { "index": 5210, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5211, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5212, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5212.png", "answer": "x=-1", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x)", "coords": {} }, { "index": 5213, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.23606797749979, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5213.png", "answer": "x^2/25 + y^2/16 = 1", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/5 - y^2/4 = 1);Focus(G) = Focus(H);H: Ellipse;Eccentricity(H) = 3/5", "coords": {} }, { "index": 5214, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5214.png", "answer": "5/3", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G);PointOnCurve(F1, LineSegmentOf(A, B));Perimeter(InscribedCircle(TriangleOf(A, B, F2))) = pi;x1: Number;x2: Number;y1: Number;y2: Number;Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2)", "coords": {} }, { "index": 5215, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5215.png", "answer": "3/5", "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;m:Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (m, 4);PointOnCurve(P, G);Focus(G) = {F1, F2};Radius(InscribedCircle(TriangleOf(P,F1,F2)))=3/2", "coords": {} }, { "index": 5216, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5217, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5218, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5218.png", "answer": "sqrt(41)/5", "fact_expr": "G: Hyperbola;b: Number;a: Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G))) = (y = (4/5)*x)", "coords": {} }, { "index": 5219, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.242640687119285, "b": 3.0, "major_axis": "x", "x_coef": 18.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5219.png", "answer": "2*sqrt(5) + 1", "fact_expr": "G: Ellipse;H: Circle;M: Point;N: Point;Expression(G) = (x^2/18 + y^2/9 = 1);Expression(H) = (x^2 + (y - 1)^2 = 1);PointOnCurve(M, G);PointOnCurve(N, H)", "coords": {} }, { "index": 5220, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5220.png", "answer": "5/4", "fact_expr": "G: Ellipse;a: Number;a>0;Expression(G) = (y^2 + x^2/a^2 = 1);Distance(LeftFocus(G),RightVertex(G))=2", "coords": {} }, { "index": 5221, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5221.png", "answer": "2", "fact_expr": "G: Circle;Expression(G) = (y^2 + (x - 4)^2 = 4);C: Hyperbola;Expression(C) = (-x^2/b^2 + y^2/a^2 = 1);b: Number;a: Number;a>0;b>0;IsTangent(G, Asymptote(C))", "coords": {} }, { "index": 5222, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5222.png", "answer": "10*x+9*y-29=0", "fact_expr": "l: Line;C: Ellipse;A: Point;B: Point;M: Point;Expression(C) = (x^2/9 + y^2/5 = 1);Coordinate(M) = (2, 1);PointOnCurve(M, l);Intersection(l, C) = {A, B};MidPoint(LineSegmentOf(A, B)) = M", "coords": { "M": [ 2.0, 1.0 ] } }, { "index": 5223, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.463192105293274, "b": 2.070901870727539, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4661259228887502e-06, "converged": true, "iterations": 106 }, "output_path": "results/train/hyperbola/problem_5223.png", "answer": "2*sqrt(2)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/b^2 = 1);b: Number;b>0;Eccentricity(G) = sqrt(3)", "coords": {} }, { "index": 5224, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5225, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5225.png", "answer": "2*sqrt(3)-1", "fact_expr": "G: Parabola;H: Circle;P: Point;Q: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (y^2 + (x - 4)^2 = 1);PointOnCurve(P, G);PointOnCurve(Q, H)", "coords": {} }, { "index": 5226, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5227, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5227.png", "answer": "(3, 2*sqrt(6))", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 8*x);F: Point;Focus(C) = F;Q: Point;PointOnCurve(Q, C);Quadrant(Q) = 1;Abs(LineSegmentOf(Q, F)) = 5", "coords": {} }, { "index": 5228, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5228.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Hyperbola;H: Ellipse;A: Point;Expression(H) = (x^2/16 + y^2/9 = 1);Coordinate(A) = (4, -5);PointOnCurve(A, G);Vertex(MinorAxis(H))=Focus(G)", "coords": { "A": [ 4.0, -5.0 ] } }, { "index": 5229, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5230, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5230.png", "answer": "sqrt(2)", "fact_expr": "l: Line;G: Hyperbola;a: Number;C: Circle;a>0;Expression(G) = (-y^2/2 + x^2/a^2 = 1);Center(C)=RightFocus(G);IsTangent(C, Asymptote(G));Expression(l)=(x-sqrt(3)*y=0);Length(InterceptChord(l,C))=2", "coords": {} }, { "index": 5231, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5232, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5232.png", "answer": "sqrt(5)", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);L: Line;Slope(L) = -1/2;A: Point;B: Point;Intersection(L, G) = {A, B};H: Circle;IsDiameter(LineSegmentOf(A, B), H);P: Point;TangentPoint(H, Directrix(G)) = P", "coords": {} }, { "index": 5233, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5233.png", "answer": "2", "fact_expr": "G: Hyperbola;F: Point;Expression(G) = (x^2 - y^2/4 = 1);RightFocus(G) = F", "coords": {} }, { "index": 5234, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5235, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5235.png", "answer": "18", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - 4*y^2 = 4);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;L: Line;PointOnCurve(F1,L) = True;A: Point;B: Point;Intersection(L, LeftPart(G)) = {A, B};Abs(LineSegmentOf(A, B)) = 5", "coords": {} }, { "index": 5236, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.0, "major_axis": "y", "x_coef": 4.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5236.png", "answer": "3", "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/4 + y^2/16 = 1);PointOnCurve(P, G);F1:Point;F2:Point;OneOf(Focus(G))=F1;OneOf(Focus(G))=F2;Negation(F1=F2);Distance(P, F1) = 5", "coords": {} }, { "index": 5237, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5237.png", "answer": "sqrt(3)/3", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);PointOnCurve(P, G);Focus(G) = {F1, F2};AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 5238, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5239, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5240, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5240.png", "answer": "sqrt(6)/3", "fact_expr": "H: Line;Expression(H) = (x - y - 4 = 0);G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Intersection(H, G) = {A, B};A: Point;B: Point;XCoordinate(MidPoint(LineSegmentOf(A, B))) = 3", "coords": {} }, { "index": 5241, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5242, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5242.png", "answer": "sqrt(17)/2", "fact_expr": "P: Point;G: Parabola;Expression(G) = (y^2 = -2*x);PointOnCurve(P, G) = True;M: Point;Coordinate(M) = (0, 2)", "coords": { "M": [ 0.0, 2.0 ] } }, { "index": 5243, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5243.png", "answer": "2", "fact_expr": "G: Parabola;p: Number;H: Ellipse;Expression(G) = (y^2 = 2*p*x);Expression(H) = (x^2/5 - y^2/p^2 = 1);PointOnCurve(LeftFocus(H),Directrix(G))", "coords": {} }, { "index": 5244, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5244.png", "answer": "1", "fact_expr": "G: Parabola;p: Number;A: Point;B: Point;C: Point;D: Point;F: Point;p>0;Expression(G) = (x^2 = 2*(p*y));Focus(G) = F;PointOnCurve(A,G);PointOnCurve(B,G);AngleOf(A, F, B) = ApplyUnit(60, degree);IsChordOf(LineSegmentOf(A,B),G);MidPoint(LineSegmentOf(A,B))=C;PointOnCurve(C,LineOf(C,D));IsPerpendicular(Directrix(G),LineOf(C,D));FootPoint(Directrix(G),LineOf(C,D))=D", "coords": {} }, { "index": 5245, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.4970548748970032, "direction": "up" }, "optimization": { "final_loss": 0.00010081290383823216, "converged": true, "iterations": 80 }, "output_path": "results/train/parabola/problem_5245.png", "answer": "3/2", "fact_expr": "G: Parabola;p: Number;A: Point;p>0;Expression(G) = (x^2 = 2*(p*y));Coordinate(A) = (sqrt(2), 1);PointOnCurve(A, G);F: Point;Focus(G) = F", "coords": { "A": [ 1.4142135623730951, 1.0 ] } }, { "index": 5246, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5247, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5247.png", "answer": "sqrt(2)/2", "fact_expr": "G: Parabola;p: Number;H: Line;P: Point;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (y = x + 3);Expression(Directrix(G)) = (x = -2);PointOnCurve(P, G)", "coords": {} }, { "index": 5248, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5248.png", "answer": "6;3", "fact_expr": "G: Ellipse;Expression(G) = (x^2/a^2+y^2/b^2 = 1);a: Number;b: Number;a > b;b > 0;Eccentricity(G) = sqrt(3)/2;P: Point;PointOnCurve(P, G);F1: Point;F2: Point;Focus(G) = {F1, F2};Distance(P, F1) + Distance(P, F2) = 12", "coords": {} }, { "index": 5249, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5249.png", "answer": "x^2/4 - y^2/12 = 1", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;l: Line;Expression(l) = (x + sqrt(3)*y = 0);IsPerpendicular(OneOf(Asymptote(C)), l);Distance(OneOf(Focus(C)), l) = 2", "coords": {} }, { "index": 5250, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5250.png", "answer": "8*sqrt(3)/5", "fact_expr": "A: Point;F: Point;LowerVertex(C) = A;LeftFocus(C) = F;C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;l: Line;PointOnCurve(A,l) = True;Inclination(l) = ApplyUnit(60,degree);Intersection(l,xAxis) = M;Intersection(l,C) = N;M: Point;N: Point;YCoordinate(N) = 3*b/5;Perimeter(TriangleOf(F, M, N)) = 6", "coords": {} }, { "index": 5251, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.449489742783178, "major_axis": "y", "x_coef": 6.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5251.png", "answer": "2*sqrt(6)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/6 + y^2/9 = 1)", "coords": {} }, { "index": 5252, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5252.png", "answer": "8*sqrt(2)/5", "fact_expr": "H: Line;Expression(H) = (y = x - 1);G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1)", "coords": {} }, { "index": 5253, "success": false, "error": "Circle from diameter: cannot parse diameter pattern" }, { "index": 5254, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5255, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5255.png", "answer": "Answer missing", "fact_expr": "H: Ellipse;Expression(H) = (x^2/6 + y^2/2 = 1);G: Hyperbola;Expression(G) = (x^2/2 - y^2/2 = 1);F1: Point;F2: Point;Focus(H) = {F1, F2};P: Point;OneOf(Intersection(H, G)) = P;Focus(G) = {F1, F2}", "coords": {} }, { "index": 5256, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5257, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0010501146316528, "direction": "right" }, "optimization": { "final_loss": 3.455598402979376e-07, "converged": true, "iterations": 81 }, "output_path": "results/train/parabola/problem_5257.png", "answer": "2\n(1,0)", "fact_expr": "G: Parabola;p: Number;H:Point;F:Point;p>0;Focus(G)=F;Expression(G) = (y^2 = 2*(p*x));Coordinate(H) = (1, 2);PointOnCurve(H, G)", "coords": { "H": [ 1.0, 2.0 ] } }, { "index": 5258, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5258.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "E: Ellipse;Expression(E) = (x^2 + y^2/b^2 = 1);b: Number;0 < b;b < 1;F1: Point;F2: Point;LeftFocus(E) = F1;RightFocus(E) = F2;G: Line;PointOnCurve(F1, G);A: Point;B: Point;Intersection(G, E) = {A, B};Abs(LineSegmentOf(A, F1)) = 3*Abs(LineSegmentOf(B, F1));IsPerpendicular(LineSegmentOf(A, F2), xAxis)", "coords": {} }, { "index": 5259, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5259.png", "answer": "pm*sqrt(2)/2", "fact_expr": "H: Line;Expression(H) = (y = k*x - 3);k: Real;G: Hyperbola;Expression(G) = (x^2/2 - y^2 = 1);NumIntersection(H, G) = 1;Negation(IsTangent(H, G))", "coords": {} }, { "index": 5260, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5261, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5262, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 10.0, "b": 8.0, "major_axis": "x", "x_coef": 100.0, "y_coef": 64.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5262.png", "answer": "64", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/100 + y^2/64 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2))", "coords": {} }, { "index": 5263, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5263.png", "answer": "24/7", "fact_expr": "G: Ellipse;H: Line;B: Point;A: Point;Expression(G) = (x^2/4 + y^2/3 = 1);Expression(H) = (y = x - 1);Intersection(H, G) = {A, B}", "coords": {} }, { "index": 5264, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5264.png", "answer": "pm*4/3", "fact_expr": "l: Line;C: Parabola;p: Number;M: Point;F: Point;N: Point;p>0;Expression(C) = (y^2 = 2*p*x);Focus(C)=F;PointOnCurve(F, l);Intersection(l, C) = {M, N};VectorOf(M, F) = 4*VectorOf(F, N)", "coords": {} }, { "index": 5265, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5266, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 44.93328387732194, "b": 44.93328387732194, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5266.png", "answer": "ApplyUnit(40,degree)", "fact_expr": "G: Hyperbola;A1: Point;P: Point;A2: Point;Expression(G) = (x^2 - y^2 = 2019);LeftVertex(G)=A1;RightVertex(G)=A2;PointOnCurve(P,RightPart(G));AngleOf(A1, P, A2) = ApplyUnit(10, degree)", "coords": {} }, { "index": 5267, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 1.0, 2.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_5267.png", "answer": "(1,0)", "fact_expr": "C: Parabola;a: Number;M: Circle;Expression(C) = (a*y^2 + 4*x = 0);Expression(M) = ((x - 1)^2 + (y - 2)^2 = 1);PointOnCurve(Center(M),C)", "coords": {} }, { "index": 5268, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5268.png", "answer": "8", "fact_expr": "C: Ellipse;G: Line;M: Point;N: Point;F2: Point;F1: Point;Expression(C) = (x^2/4 + y^2 = 1);Focus(C) = {F1, F2};PointOnCurve(F1, G);Negation(IsParallel(G, axis));Intersection(G, C) = {M, N}", "coords": {} }, { "index": 5269, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5269.png", "answer": "(3, 2)", "fact_expr": "C: Parabola;O: Origin;Vertex(C) = O;F: Point;Coordinate(F) = (1, 0);Focus(C) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, C) = {A, B};Inclination(l) = ApplyUnit(45, degree);IsChordOf(LineSegmentOf(A, B), C)", "coords": { "F": [ 1.0, 0.0 ] } }, { "index": 5270, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5270.png", "answer": "(-17/8, 15/8)", "fact_expr": "G: Hyperbola;H: Line;Expression(G) = (x^2 - y^2 = 1);Expression(H) = (y = x + 4)", "coords": {} }, { "index": 5271, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5271.png", "answer": "8", "fact_expr": "C: Parabola;G: Line;k: Number;A: Point;N: Point;M: Point;F: Point;Expression(C) = (y^2 = 4*x);k>0;Expression(G) = (x + 2 = k*y)&(k>0);Coordinate(A) = (-2, 0);Focus(C) = F;Intersection(G, C) = {M, N};VectorOf(A, N) = 2*VectorOf(A, M)", "coords": { "A": [ -2.0, 0.0 ] } }, { "index": 5272, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/train/hyperbola/problem_5272.png", "answer": "x^{-y^2/3}=1", "fact_expr": "P: Point;Coordinate(P) = (2, -3);PointOnCurve(P, G);G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;F1: Point;F2: Point;Focus(G) = {F1, F2};Distance(F1, F2) = 4", "coords": { "P": [ 2.0, -3.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 5273, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5274, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5274.png", "answer": "2", "fact_expr": "G: Parabola;A: Point;B: Point;M: Point;Expression(G) = (y = -x^2/4);PointOnCurve(A,G);PointOnCurve(B, G);Abs(LineSegmentOf(A, B)) = 6;MidPoint(LineSegmentOf(A, B)) = M", "coords": {} }, { "index": 5275, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -2.0, -0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_5275.png", "answer": "sqrt(5)", "fact_expr": "G: Circle;C:Circle;T:Circle;H: Line;M: Curve;P0: Point;Expression(G) = (4*x + x^2 + y^2 + 3 = 0);Expression(T) = (-4*x + x^2 + y^2 - 45 = 0);Expression(H) = (x - 2*y - 1 = 0);IsOutTangent(C,G);IsInTangent(C,T);C1:Point;Center(C)=C1;Locus(C1)=M;PointOnCurve(P0, M)", "coords": {} }, { "index": 5276, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5276.png", "answer": "sqrt(7)/4", "fact_expr": "C: Ellipse;Expression(C) = (y^2/9 + x^2/a^2 = 1);a: Number;a>0;F: Point;RightFocus(C) = F;M: Point;PointOnCurve(M, C) = True;N: Point;MidPoint(LineSegmentOf(M, F)) = N;O: Origin;Abs(LineSegmentOf(M, F)) = 2*Abs(LineSegmentOf(O, N));2*Abs(LineSegmentOf(O, N)) = 4", "coords": {} }, { "index": 5277, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5277.png", "answer": "2", "fact_expr": "C: Parabola;A: Point;M: Point;F: Point;l: Line;Expression(C) = (y^2 = 4*x);Focus(C) = F;Directrix(C) = l;Intersection(l, xAxis) = A;PointOnCurve(M, C);Abs(LineSegmentOf(M, A))/Abs(LineSegmentOf(M, F)) = sqrt(2)", "coords": {} }, { "index": 5278, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2135674953460693, "b": 2.4270472526550293, "orientation": "horizontal" }, "optimization": { "final_loss": 4.453341261978494e-06, "converged": true, "iterations": 125 }, "output_path": "results/train/hyperbola/problem_5278.png", "answer": "x^2 - y^2/4 = 1", "fact_expr": "C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(C) = sqrt(5);Length(ImageinaryAxis(C))>1", "coords": {} }, { "index": 5279, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5280, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5281, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5282, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.8284271247461903, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5282.png", "answer": "32", "fact_expr": "C: Hyperbola;A: Point;P: Point;F: Point;Expression(C) = (x^2 - y^2/8 = 1);Coordinate(A) = (0, 6*sqrt(6));RightFocus(C) = F;PointOnCurve(P, LeftPart(C))", "coords": { "A": [ 0.0, 14.696938456699067 ] } }, { "index": 5283, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5283.png", "answer": "y=pm*3*x", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Distance(OneOf(Focus(G)),OneOf(Asymptote(G)))=3*a", "coords": {} }, { "index": 5284, "success": false, "error": "Circle from diameter: missing point coordinates" }, { "index": 5285, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 8.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5285.png", "answer": "12", "fact_expr": "G: Ellipse;H: Line;P: Point;Q: Point;x1:Number;x2:Number;y1:Number;y2:Number;Expression(G) = (x^2/8 + y^2/2 = 1);Expression(H) = (x + 2*y - 8 = 0);Coordinate(P) = (x1,y1);Coordinate(Q) = (x2,y2);PointOnCurve(P, G);PointOnCurve(Q, H)", "coords": {} }, { "index": 5286, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5286.png", "answer": "2*sqrt(3)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1)", "coords": {} }, { "index": 5287, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5287.png", "answer": "3", "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/4 + y^2/3 = 1);PointOnCurve(P, G);Distance(P, LeftFocus(G)) = 5/2", "coords": {} }, { "index": 5288, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5288.png", "answer": "4/7", "fact_expr": "l: Line;G: Parabola;p: Number;A: Point;B: Point;N: Point;M:Point;p>0;Expression(G) = (y^2 = 2*(p*x));Expression(l) = (-x + y + 2 = 0);Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A, B))=M;L:Line;PointOnCurve(M, L);IsPerpendicular(L, yAxis);Intersection(L, G) = N;AngleOf(A, N, B) = pi/2", "coords": {} }, { "index": 5289, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5289.png", "answer": "y=-3", "fact_expr": "G: Parabola;Expression(G) = (y = a*x^2);a: Number;a>0;H: Ellipse;Expression(H) = (x^2 + y^2/10 = 1);Focus(G)=OneOf(Focus(H))", "coords": {} }, { "index": 5290, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5291, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5292, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5293, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5293.png", "answer": "sqrt(5)", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y = 2*x);OneOf(Asymptote(C))=G", "coords": {} }, { "index": 5294, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.8284271247461903, "major_axis": "x", "x_coef": 9.0, "y_coef": 8.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5294.png", "answer": "7/2", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 + y^2/8 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P,G);Abs(LineSegmentOf(P,F1))=5/2", "coords": {} }, { "index": 5295, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5295.png", "answer": "2", "fact_expr": "G: Ellipse;H: Circle;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2/3 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(P,H);IsDiameter(MinorAxis(G),H)", "coords": {} }, { "index": 5296, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5296.png", "answer": "sqrt(21)/3", "fact_expr": "C: Hyperbola;b: Number;a: Number;O: Origin;A: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) =F1;RightFocus(C)=F2;PointOnCurve(A,OneOf(Asymptote(C)));IsPerpendicular(LineSegmentOf(O, A), LineSegmentOf(F1, A));Abs(LineSegmentOf(A, F2)) = 2*Abs(LineSegmentOf(A, F1))", "coords": {} }, { "index": 5297, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5297.png", "answer": "(1,\\sqrt{2})", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 + x^2 - 4*x + 2 = 0);IsIntersect(Asymptote(G),H)", "coords": {} }, { "index": 5298, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_5298.png", "answer": "1", "fact_expr": "G: Hyperbola;a: Number;a>0;Expression(G) = (-y^2/3 + x^2/a^2 = 1);Eccentricity(G) = 2", "coords": {} }, { "index": 5299, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5299.png", "answer": "y=x", "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;F: Point;O: Origin;Focus(C)=F;Coordinate(F) = (1, 0);Vertex(C) = O;Intersection(l, C) = {A, B};Coordinate(MidPoint(LineSegmentOf(A, B))) = (2,2)", "coords": { "F": [ 1.0, 0.0 ] } }, { "index": 5300, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.8284271247461903, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5300.png", "answer": "6*sqrt(5)+2", "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2/8 = 1);F: Point;RightFocus(C) = F;N: Point;Coordinate(N) = (0, 6);M: Point;PointOnCurve(M, LeftPart(C))", "coords": { "N": [ 0.0, 6.0 ] } }, { "index": 5301, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5301.png", "answer": "(4*sqrt(5))/3", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/5 = 1);H: Circle;Expression(H) = (x^2 + y^2 - 8*x + 15 = 0);C: Point;Center(H) = C", "coords": {} }, { "index": 5302, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5302.png", "answer": "4", "fact_expr": "C: Parabola;p: Number;G: Line;A: Point;F: Point;B: Point;p>0;Expression(C) = (y^2 = 2*p*x);PointOnCurve(F, G);Intersection(G, C) = {A, B};Abs(LineSegmentOf(A, F)) = p + 2;Abs(LineSegmentOf(B, F)) = p - 1;Focus(C)=F", "coords": {} }, { "index": 5303, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5303.png", "answer": "10", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(P, G);Focus(G)={F1,F2}", "coords": {} }, { "index": 5304, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5305, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5306, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5306.png", "answer": "(2,4)", "fact_expr": "G: Parabola;A: Point;M: Point;F: Point;Expression(G) = (y^2 = 8*x);Coordinate(A) = (3, 4);Focus(G)=F;PointOnCurve(M, G);WhenMin(Abs(LineSegmentOf(A,M))+Abs(LineSegmentOf(M,F)))", "coords": { "A": [ 3.0, 4.0 ] } }, { "index": 5307, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5307.png", "answer": "2", "fact_expr": "C: Ellipse;Expression(C) = (x^2/16 + y^2/4 = 1);F: Point;LeftFocus(C) = F;L: Line;PointOnCurve(F, L);IsPerpendicular(L, xAxis);A: Point;B: Point;Intersection(L, C) = {A, B}", "coords": {} }, { "index": 5308, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5308.png", "answer": "2*sqrt(2)", "fact_expr": "O: Origin;G: Line;Expression(G) = (x = a);C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;L1: Line;L2: Line;Asymptote(C) = {L1, L2};Intersection(G, L1) = D;Intersection(G, L2) = E;D: Point;E: Point;Area(TriangleOf(O, D, E)) = 1", "coords": {} }, { "index": 5309, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5309.png", "answer": "(\\sqrt{13} - 2)/3", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;H: Line;PointOnCurve(F1, H);A: Point;B: Point;Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, F1)) = 2*Abs(LineSegmentOf(B, F1));2*Abs(LineSegmentOf(B, F1)) = Abs(LineSegmentOf(F1, F2))", "coords": {} }, { "index": 5310, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5311, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5311.png", "answer": "4*sqrt(5)/5", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*p*x);Expression(Directrix(C)) = (x = -2);p: Number;p>0;F: Point;Focus(C) = F;A: Point;Intersection(Directrix(C), xAxis) = A;B: Point;PointOnCurve(B, C);sqrt(5)*Abs(LineSegmentOf(B, F)) = 2*Abs(LineSegmentOf(A, B))", "coords": {} }, { "index": 5312, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5312.png", "answer": "4", "fact_expr": "E: Hyperbola;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;P: Point;A: Point;F: Point;a>0;b>0;LeftFocus(E) = F;RightVertex(E)=A;PointOnCurve(P,LeftPart(E));AngleOf(P, A, F) = ApplyUnit(60, degree);Abs(LineSegmentOf(P,A))=Abs(LineSegmentOf(A,F))", "coords": {} }, { "index": 5313, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5314, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5314.png", "answer": "8", "fact_expr": "M: Point;Coordinate(M) = (sqrt(3), 0);H: Line;Expression(H) = (y = k*(x + sqrt(3)));G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);A: Point;B: Point;Intersection(H, G) = {A, B};k: Number", "coords": { "M": [ 1.7320508075688772, 0.0 ] } }, { "index": 5315, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 8.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5315.png", "answer": "15", "fact_expr": "G: Hyperbola;P: Point;Expression(G) = (x^2/16 - y^2/64 = 1);PointOnCurve(P, G);F1:Point;F2:Point;OneOf(Focus(G))=F1;OneOf(Focus(G))=F2;Negation(F1=F2);Distance(P, F1) = 7", "coords": {} }, { "index": 5316, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5316.png", "answer": "y^2=x", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;Distance(Focus(G),Directrix(G))=1/2", "coords": {} }, { "index": 5317, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5317.png", "answer": "5/4", "fact_expr": "G: Hyperbola;a: Number;b: Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(Asymptote(G)) = (y = pm*((3/4)*x))", "coords": {} }, { "index": 5318, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5318.png", "answer": "4", "fact_expr": "G: Parabola;A: Point;F: Point;P: Point;l: Line;Expression(G) = (y^2 = 4*x);Focus(G) = F;Directrix(G)=l;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P,A),l);FootPoint(LineSegmentOf(P,A),l)=A;Inclination(LineSegmentOf(A, F)) = (2*pi)/3", "coords": {} }, { "index": 5319, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 2.0, -0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_5319.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2 + x^2/m = 1);m: Number;H: Circle;Expression(H) = (-4*x + x^2 + y^2 + 3 = 0);IsTangent(OneOf(Asymptote(G)), H)", "coords": {} }, { "index": 5320, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5320.png", "answer": "x^2/9 - y^2/3 = 1", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2 + 4*y^2 = 16);Focus(G)=Focus(H);Expression(OneOf(Asymptote(G))) = (x + sqrt(3)*y = 0)", "coords": {} }, { "index": 5321, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5321.png", "answer": "1/3", "fact_expr": "G: Hyperbola;H: Ellipse;P:Point;F1:Point;F2:Point;Expression(G) = (x^2/3 - y^2 = 1);Expression(H) = (x^2/6 + y^2/2 = 1);Focus(G)={F1,F2};Focus(H)={F1,F2};OneOf(Intersection(G,H))=P", "coords": {} }, { "index": 5322, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5322.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F2: Point;F1: Point;l1: Line;l2: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;Asymptote(G)={l1,l2};Quadrant(P) = 1;PointOnCurve(P, l1);IsParallel(l2,LineSegmentOf(P,F2))", "coords": {} }, { "index": 5323, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.449489742783178, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 6.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5323.png", "answer": "{1,4}", "fact_expr": "G: Ellipse;Expression(G) = (y^2/2 + x^2/m = 1);m: Number;G1: Ellipse;Expression(G1) = (x^2/6 + y^2/3 = 1);Eccentricity(G) = Eccentricity(G1)", "coords": {} }, { "index": 5324, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5324.png", "answer": "pm*sqrt(2)", "fact_expr": "l: Line;C: Parabola;Expression(C) = (y^2 = 4*x);A: Point;B: Point;Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, B)) = 6;WhenMin(XCoordinate(MidPoint(LineSegmentOf(A, B)))) = True", "coords": {} }, { "index": 5325, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5325.png", "answer": "sqrt(15)/2", "fact_expr": "C: Ellipse;B: Point;A: Point;F: Point;Expression(C) = (x^2/4 + y^2/3 = 1);LeftFocus(C) = F;PointOnCurve(A, C);PointOnCurve(B, C);Negation(A=B);XCoordinate(MidPoint(LineSegmentOf(A, B))) = 1", "coords": {} }, { "index": 5326, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.125, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5326.png", "answer": "7/8", "fact_expr": "G: Parabola;M: Point;Expression(G) = (y = 2*x^2);PointOnCurve(M, G);Distance(M, Focus(G)) = 1", "coords": {} }, { "index": 5327, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5328, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5328.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;B: Point;Intersection(LeftDirectrix(G), xAxis) = B;A: Point;Coordinate(A) = (0, b);P: Point;LineSegmentOf(A, P) = 2*LineSegmentOf(A, B);PointOnCurve(P, G)", "coords": {} }, { "index": 5329, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5329.png", "answer": "1/2", "fact_expr": "G: Ellipse;b: Number;a: Number;P:Point;F2: Point;F1: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (a, b);LeftFocus(G) = F1;RightFocus(G) = F2;Abs(LineSegmentOf(P, F2)) = Abs(LineSegmentOf(F1, F2))", "coords": {} }, { "index": 5330, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/train/hyperbola/problem_5330.png", "answer": "9", "fact_expr": "G: Hyperbola;m: Number;m>0;Expression(G) = (x^2/16 - y^2/m = 1);Eccentricity(G) = 5/4", "coords": {} }, { "index": 5331, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5331.png", "answer": "y^2 = -36*(x - 4)", "fact_expr": "G: Hyperbola;H: Parabola;Expression(G) = (x^2/16 - y^2/9 = 1);LeftVertex(G) = Vertex(H);Focus(H) = RightFocus(G)", "coords": {} }, { "index": 5332, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5333, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5333.png", "answer": "(18,pm*12)", "fact_expr": "G: Parabola;P: Point;Expression(G) = (y^2=8*x);PointOnCurve(P,G);Distance(P,Focus(G))=20", "coords": {} }, { "index": 5334, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.0833333358168602, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5334.png", "answer": "(0, 1/12)", "fact_expr": "G: Parabola;Expression(G) = (y = 3*x^2)", "coords": {} }, { "index": 5335, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5335.png", "answer": "11/3", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/4 - y^2/3 = 1);F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;PointOnCurve(P, C);PointOnCurve(MidPoint(LineSegmentOf(P, F1)), yAxis)", "coords": {} }, { "index": 5336, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5336.png", "answer": "(1, 3]", "fact_expr": "F1: Point;F2: Point;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, LeftPart(G)) = True;Abs(LineSegmentOf(P, F2))^2/Abs(LineSegmentOf(P, F1)) = 8*a", "coords": {} }, { "index": 5337, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5337.png", "answer": "(\\sqrt{5}, +\\infty)", "fact_expr": "C: Hyperbola;b: Number;a: Number;F2: Point;P: Point;O: Origin;F1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P,RightPart(C));Abs(LineSegmentOf(O, P)) = Abs(LineSegmentOf(O, F2));IsIntersect(LineOf(P,F2),LeftPart(C))", "coords": {} }, { "index": 5338, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5338.png", "answer": "2*sqrt(5)/5", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/4 - y^2 = 1)", "coords": {} }, { "index": 5339, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5339.png", "answer": "10", "fact_expr": "G: Parabola;p: Real;O:Origin;p>0;Expression(G) = (y^2 = 2*(p*x));Distance(Focus(G),O)=5", "coords": {} }, { "index": 5340, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5340.png", "answer": "(x-2)^2+y^2=2", "fact_expr": "G: Hyperbola;E: Circle;Expression(G) = (x^2 - y^2 = 2);Center(E) = RightFocus(G);IsTangent(Asymptote(G), E) = True", "coords": {} }, { "index": 5341, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5341.png", "answer": "sqrt(2)/2", "fact_expr": "l: Line;C: Ellipse;b: Number;a: Number;A: Point;F1: Point;B: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(Focus(C),xAxis);LeftFocus(C)=F1;RightFocus(C)=F2;PointOnCurve(F1, l);Intersection(l,C)={A,B};VectorOf(A, F1) = 3*VectorOf(F1, B);3*Abs(VectorOf(B, F2)) = 5*Abs(VectorOf(A, F2))", "coords": {} }, { "index": 5342, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5343, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5344, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5344.png", "answer": "20", "fact_expr": "C: Hyperbola;m: Number;G: Parabola;A: Point;B: Point;m>0;Expression(C) = (2*x^2 - y^2 = m);Expression(G) = (y^2 = 16*x);Intersection(C, Directrix(G)) = {A,B};Abs(LineSegmentOf(A, B)) = 4*sqrt(3)", "coords": {} }, { "index": 5345, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5346, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5346.png", "answer": "2*sqrt(5)/5", "fact_expr": "G: Parabola;P: Point;F: Point;A: Point;Expression(G) = (x^2 = 4*y);Coordinate(A) = (3/2, -1);Focus(G) = F;PointOnCurve(P, G)", "coords": { "A": [ 1.5, -1.0 ] } }, { "index": 5347, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5347.png", "answer": "x^2 + y^2 = 2", "fact_expr": "C: Ellipse;Expression(C) = (x^2/2 + y^2 = 1);M: Point;PointOnCurve(M, C);l: Line;PointOnCurve(M, l);IsPerpendicular(l,xAxis);N: Point;FootPoint(l,xAxis) = N;P: Point;VectorOf(N, P) = sqrt(2)*VectorOf(N, M)", "coords": {} }, { "index": 5348, "success": false, "error": "Circle from diameter: missing point coordinates" }, { "index": 5349, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5349.png", "answer": "pi/6", "fact_expr": "P: Point;Coordinate(P) = (3*sqrt(3), 4);l: Line;PointOnCurve(P, l);C: Parabola;Expression(C) = (x^2 = 18*y);A: Point;B: Point;Intersection(l, C) = {A, B};IsChordOf(LineSegmentOf(A, B), C);MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ 5.196152422706632, 4.0 ] } }, { "index": 5350, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5350.png", "answer": "4", "fact_expr": "G: Ellipse;O: Circle;P: Point;F2: Point;Q: Point;M: Point;Expression(G) = (x^2/4 + y^2/3 = 1);RightFocus(G) = F2;Expression(O) = (x^2 + y^2 = 3);PointOnCurve(M, O) = True;Quadrant(M) = 1;Intersection(TangentOfPoint(M,O), G) = {P, Q}", "coords": {} }, { "index": 5351, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5351.png", "answer": "sqrt(2)", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/a^2 - y^2/b^2 = 1);a: Number;b: Number;a>b;b>0;F: Point;RightFocus(C) = F;A1: Point;A2: Point;LeftVertex(C) = A1;RightVertex(C) = A2;L1: Line;L2: Line;OneOf(Asymptote(C)) = L1;OneOf(Asymptote(C)) = L2;Negation(L1 = L2);H: Line;PointOnCurve(F,H);IsParallel(L1,H);P: Point;Intersection(H,L2) = P;G: Circle;IsDiameter(LineSegmentOf(A1,A2),G);PointOnCurve(P,G)", "coords": {} }, { "index": 5352, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5352.png", "answer": "(3,0)\ny^2=12*x", "fact_expr": "C: Parabola;G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);RightFocus(G)=F;Vertex(C)=O;O:Origin;Focus(C)=F;F:Point", "coords": {} }, { "index": 5353, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5353.png", "answer": "6", "fact_expr": "C: Parabola;F: Point;M: Point;N: Point;Expression(C) = (y^2 = 8*x);Focus(C) = F;PointOnCurve(M, C);Intersection(OverlappingLine(LineSegmentOf(F, M)), yAxis) = N;MidPoint(LineSegmentOf(F, N)) = M", "coords": {} }, { "index": 5354, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5355, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5355.png", "answer": "3/4", "fact_expr": "C: Parabola;p: Number;G: Line;A: Point;F: Point;B: Point;O: Origin;p>0;Expression(C) = (x^2 = 2*p*y);Focus(C) = F;PointOnCurve(F, G);Intersection(G, C) = {A, B};3*Abs(LineSegmentOf(A, F)) = Abs(LineSegmentOf(B, F))", "coords": {} }, { "index": 5356, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5356.png", "answer": "4*sqrt(10)/5", "fact_expr": "l: Line;Slope(l) = 1;G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);Intersection(l, G) = {A, B};A: Point;B: Point", "coords": {} }, { "index": 5357, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5357.png", "answer": "sqrt(3)", "fact_expr": "G: Ellipse;P: Point;A: Point;M: Point;x1:Number;y1:Number;Expression(G) = (x^2/25 + y^2/16 = 1);Coordinate(P) = (x1, y1);Coordinate(A) = (3, 0);PointOnCurve(P, G);Abs(LineSegmentOf(A, M)) = 1;DotProduct(VectorOf(P, M), VectorOf(A, M)) = 0", "coords": { "A": [ 3.0, 0.0 ] } }, { "index": 5358, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5358.png", "answer": "sqrt(3) - 1", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;Z: Line;PointOnCurve(F2,Z);Inclination(Z) = ApplyUnit(120,degree);OneOf(Intersection(Z,G)) = M;M: Point;IsPerpendicular(LineSegmentOf(M,F1),LineSegmentOf(M,F2))", "coords": {} }, { "index": 5359, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5359.png", "answer": "3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;Endpoint(Z) = F;A: Point;Intersection(Z, G) = A;B: Point;Intersection(Z, yAxis) = B;Abs(LineSegmentOf(A, F)) = 2;Abs(LineSegmentOf(B, F)) = 3;Z: Ray", "coords": {} }, { "index": 5360, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5360.png", "answer": "sqrt(17)", "fact_expr": "l: Line;C: Parabola;P: Point;B: Point;M: Point;F: Point;N: Point;Q: Point;Expression(C) = (x^2 = 8*y);Coordinate(P) = (0, -2);Coordinate(B) = (0, 3);Focus(C) = F;PointOnCurve(P, l);Intersection(l, C) = {M, N};Abs(LineSegmentOf(M, F)) + Abs(LineSegmentOf(N, F)) = 32;PointOnCurve(Q, l)", "coords": { "P": [ 0.0, -2.0 ], "B": [ 0.0, 3.0 ] } }, { "index": 5361, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5361.png", "answer": "3", "fact_expr": "G: Parabola;P: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(P, G);Distance(P, Focus(G)) = 4", "coords": {} }, { "index": 5362, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142164151010692, "b": 1.4142164151010692, "orientation": "horizontal" }, "optimization": { "final_loss": 2.2160979279094996e-06, "converged": true, "iterations": 95 }, "output_path": "results/train/hyperbola/problem_5362.png", "answer": "12", "fact_expr": "G: Hyperbola;a: Number;F1: Point;F2: Point;P: Point;a>0;Expression(G) = (x^2 - y^2/a^2 = 1);Coordinate(F1) = (-2, 0);Coordinate(F2) = (2, 0);Focus(G) = {F1,F2};PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": { "F1": [ -2.0, 0.0 ], "F2": [ 2.0, 0.0 ] } }, { "index": 5363, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5363.png", "answer": "2*sqrt(3)/3", "fact_expr": "E: Hyperbola;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;A: Point;Coordinate(A) = (2, 0);RightVertex(E) = A;Distance(A,OneOf(Asymptote(E))) = 1", "coords": { "A": [ 2.0, 0.0 ] } }, { "index": 5364, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5364.png", "answer": "6", "fact_expr": "G: Parabola;H: Line;F: Point;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A,B};Abs(LineSegmentOf(F, A))*Abs(LineSegmentOf(F, B)) = 6", "coords": {} }, { "index": 5365, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5365.png", "answer": "8", "fact_expr": "A: Point;B: Point;l: Line;G: Parabola;F: Point;Expression(G) = (y^2 = 8*x);Focus(G) = F;PointOnCurve(F, l);Intersection(l,G)={A,B};Distance(MidPoint(LineSegmentOf(A,B)), yAxis) = 2", "coords": {} }, { "index": 5366, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5366.png", "answer": "5", "fact_expr": "G: Hyperbola;b: Number;a: Number;O: Origin;P: Point;F2: Point;F1: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, RightPart(G));DotProduct((VectorOf(O,P)+VectorOf(O,F2)),VectorOf(P,F2))=0;3*Abs(LineSegmentOf(P, F1)) = 4*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 5367, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5367.png", "answer": "sqrt(10)/2", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;LeftFocus(C) = F1;RightFocus(C) = F2;F1: Point;F2: Point;PointOnCurve(F2,L) = True;L: Line;Intersection(L,RightPart(C)) = {P,Q};P: Point;Q: Point;IsPerpendicular(LineSegmentOf(P, Q), LineSegmentOf(P, F1)) = True;3*Abs(LineSegmentOf(P, Q)) = 4*Abs(LineSegmentOf(P, F1))", "coords": {} }, { "index": 5368, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5368.png", "answer": "10", "fact_expr": "G: Ellipse;P: Point;F: Point;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(P, G);LeftFocus(G) = F;Distance(P, F) = 6", "coords": {} }, { "index": 5369, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5370, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5370.png", "answer": "2*sqrt(3)", "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;O: Origin;F: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;PointOnCurve(P, G);Distance(P,O)=HalfFocalLength(G);Area(TriangleOf(P, O, F)) = 6", "coords": {} }, { "index": 5371, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5371.png", "answer": "8", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 12*y);F: Point;Focus(G) = F;P: Point;Coordinate(P) = (2, 1);l: Line;PointOnCurve(P, l) ;Intersection(l, G) = {A, B};A: Point;B: Point;MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ 2.0, 1.0 ] } }, { "index": 5372, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5373, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/train/hyperbola/problem_5373.png", "answer": "x^2/(8/5)-y^2/4=1", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;P: Point;Coordinate(P) = (sqrt(2), 1);PointOnCurve(P, C);Distance(OneOf(Focus(C)), OneOf(Asymptote(C))) = 2", "coords": { "P": [ 1.4142135623730951, 1.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 5374, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_focus_mismatch", "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5374.png", "answer": "x^2/8+y^2/4=1", "fact_expr": "C: Ellipse;b: Number;a: Number;B: Point;F: Point;M: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (2, 0);Coordinate(M) = (-sqrt(2)*a/2, 0);UpperVertex(C) = B;RightFocus(C) = F;IsPerpendicular(LineSegmentOf(B, F), LineSegmentOf(B, M))", "coords": { "F": [ 2.0, 0.0 ] }, "validation_reasons": [ "ellipse_focus_mismatch" ] }, { "index": 5375, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5375.png", "answer": "24/7", "fact_expr": "C: Ellipse;G: Line;Expression(C) = (x^2/4 + y^2/3 = 1);PointOnCurve(LeftFocus(C), G);Slope(G)=1", "coords": {} }, { "index": 5376, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5376.png", "answer": "2+4*sqrt(2)", "fact_expr": "C: Hyperbola;M: Point;P: Point;F: Point;Expression(C) = (x^2 - y^2/3 = 1);Coordinate(M) = (0, 2);RightFocus(C)=F;PointOnCurve(P,LeftPart(C))", "coords": { "M": [ 0.0, 2.0 ] } }, { "index": 5377, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5377.png", "answer": "2*pi/3", "fact_expr": "P: Point;Coordinate(P) = (x1, y1);x1: Number;y1: Number;Q: Point;Coordinate(Q) = (x2, y2);x2: Number;y2: Number;G: Parabola;Expression(G) = (y^2 = 4*x);PointOnCurve(P, G) = True;PointOnCurve(Q, G) = True;x1 + x2 + 2 = (2/3)*sqrt(3)*Abs(LineSegmentOf(P, Q));F: Point;Focus(G) = F", "coords": {} }, { "index": 5378, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5378.png", "answer": "(1/2,-1)", "fact_expr": "G: Parabola;H: Line;Expression(G) = (y^2 = 2*x);Expression(H) = (x - y + 3 = 0);P:Point;PointOnCurve(P,G);Q:Point;PointOnCurve(Q,H);WhenMin(Distance(P,Q))", "coords": {} }, { "index": 5379, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.23606797749979, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5379.png", "answer": "ApplyUnit(60,degree)", "fact_expr": "G: Hyperbola;F1: Point;P: Point;F2: Point;Expression(G) = (3*x^2 - 5*y^2 = 15);PointOnCurve(P, G);Focus(G)={F1,F2};Area(TriangleOf(F1, P, F2)) = 3*sqrt(3)", "coords": {} }, { "index": 5380, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5380.png", "answer": "5/3", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;e: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, RightPart(G));Abs(LineSegmentOf(P, F1)) = 4*Abs(LineSegmentOf(P, F2));Eccentricity(G) = e", "coords": {} }, { "index": 5381, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5381.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);A: Point;B: Point;PointOnCurve(A,G) = True;PointOnCurve(B,G) = True;O: Origin;DotProduct(VectorOf(O, A), VectorOf(O, B)) = 0;L: Line;PointOnCurve(O,L) = True;IsPerpendicular(L,LineOf(A,B)) = True;FootPoint(L,LineOf(A,B)) = P;P: Point;F: Point;Focus(G) = F", "coords": {} }, { "index": 5382, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5382.png", "answer": "sqrt(10)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Z: Curve;Expression(Z) = (3*x + y - 4 = 0);IsParallel(OneOf(Asymptote(C)), Z)", "coords": {} }, { "index": 5383, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5383.png", "answer": "16", "fact_expr": "G: Hyperbola;P: Point;Q: Point;F2: Point;F1: Point;Expression(G) = (x^2/16 - y^2/9 = 1);Focus(G) = {F1, F2};PointOnCurve(F1,LineSegmentOf(P,Q))", "coords": {} }, { "index": 5384, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5385, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5386, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.6457513110645907, "major_axis": "x", "x_coef": 16.0, "y_coef": 7.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5386.png", "answer": "4/3", "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/7 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G);PointOnCurve(F1, LineSegmentOf(A, B));Perimeter(InscribedCircle(TriangleOf(A, B, F2))) = pi;x1: Number;x2: Number;y1: Number;y2: Number;Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2)", "coords": {} }, { "index": 5387, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 2.0, "major_axis": "x", "x_coef": 8.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5387.png", "answer": "4", "fact_expr": "C: Ellipse;G: Line;A: Point;B: Point;F: Point;O: Origin;Expression(C) = (x^2/8 + y^2/4 = 1);RightFocus(C) = F;PointOnCurve(O, G);Intersection(G, C) = {A, B}", "coords": {} }, { "index": 5388, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5388.png", "answer": "{(4,4),(4,-4)}", "fact_expr": "C: Parabola;P: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(P, C);Abs(LineSegmentOf(P, F)) = 5", "coords": {} }, { "index": 5389, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5389.png", "answer": "3", "fact_expr": "C: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Focus(C) = {F1, F2};PointOnCurve(P, C);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;Area(TriangleOf(P, F1, F2)) = 9", "coords": {} }, { "index": 5390, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.3907806873321533, "b": 2.9160258769989014, "orientation": "horizontal" }, "optimization": { "final_loss": 1.1408701539039612e-07, "converged": true, "iterations": 124 }, "output_path": "results/train/hyperbola/problem_5390.png", "answer": "8", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;F1: Point;Coordinate(F1) = (-2*sqrt(5), 0);LeftFocus(C) = F1;F2: Point;Coordinate(F2) = (2*sqrt(5), 0);RightFocus(C) = F2;e: Number;Eccentricity(C)=e;e = sqrt(5)/2;P: Point;PointOnCurve(P, RightPart(C))", "coords": { "F1": [ -4.47213595499958, 0.0 ], "F2": [ 4.47213595499958, 0.0 ] } }, { "index": 5391, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5391.png", "answer": "9*sqrt(2)/4", "fact_expr": "G: Parabola;H: Line;P: Point;Q: Point;Expression(G) = (y^2 = 2*x);Expression(H) = (x + y + 5 = 0);PointOnCurve(P, H);PointOnCurve(Q, G)", "coords": {} }, { "index": 5392, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5392.png", "answer": "3", "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(P, G);F1:Point;F2:Point;OneOf(Focus(G))=F1;OneOf(Focus(G))=F2;Negation(F1=F2);Distance(P, F1) = 7", "coords": {} }, { "index": 5393, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.6057229042053223, "b": 1.3912456035614014, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 1.918481530083227e-06, "converged": true, "iterations": 111 }, "output_path": "results/train/ellipse/problem_5393.png", "answer": "4/3", "fact_expr": "G: Ellipse;C:Curve;Expression(C) = (x^2+y^2/m=1);C=G;PointOnCurve(Focus(G), yAxis);Eccentricity(G) = 1/2;m: Number", "coords": {} }, { "index": 5394, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5394.png", "answer": "[\\sqrt{2}/2,1)", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;LeftFocus(G) = F;O: Origin;H: Line;PointOnCurve(O,H);A: Point;B: Point;Intersection(H, G) = {A, B};IsPerpendicular(LineSegmentOf(A, F), LineSegmentOf(B, F))", "coords": {} }, { "index": 5395, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5396, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5397, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 3.0, "major_axis": "x", "x_coef": 36.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5397.png", "answer": "y = -0.5*x+4", "fact_expr": "G: Ellipse;Expression(G) = (x^2/36 + y^2/9 = 1);H: LineSegment;IsChordOf(H,G) = True;Coordinate(MidPoint(H)) = (4,2)", "coords": {} }, { "index": 5398, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5399, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5399.png", "answer": "4", "fact_expr": "G: Ellipse;M: Point;F1: Point;O: Origin;N: Point;Expression(G) = (x^2/25 + y^2/9 = 1);PointOnCurve(M, G);LeftFocus(G)=F1;Distance(M, F1) = 2;MidPoint(LineSegmentOf(M, F1)) = N", "coords": {} }, { "index": 5400, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5400.png", "answer": "[0, 2]", "fact_expr": "E: Ellipse;Expression(E) = (x^2/4 + y^2/2 = 1);F1: Point;F2: Point;Focus(E) = {F1, F2};P: Point;PointOnCurve(P, E)", "coords": {} }, { "index": 5401, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5401.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = sqrt(3)*x);OneOf(Asymptote(G)) = H", "coords": {} }, { "index": 5402, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5403, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5403.png", "answer": "x^2/18 - y^2/9 = 1", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - 2*y^2 = 2);M: Point;Coordinate(M) = (-6, 3);Z: Hyperbola;PointOnCurve(M, Z);Asymptote(G) = Asymptote(Z)", "coords": { "M": [ -6.0, 3.0 ] } }, { "index": 5404, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5404.png", "answer": "sqrt(5)/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;m: Number;Negation(m=0);P: Point;A: Point;B: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (m + x - 3*y = 0);Coordinate(P) = (m, 0);L1:Line;L2:Line;Asymptote(G)={L1,L2};Intersection(H,L1)=A;Intersection(H,L2)=B;Abs(LineSegmentOf(P, A)) = Abs(LineSegmentOf(P, B))", "coords": {} }, { "index": 5405, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5405.png", "answer": "y^2/4 - x^2/16 = 1", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1);Asymptote(G) = Asymptote(G1);H: Point;Coordinate(H) = (2, sqrt(5));PointOnCurve(H, G1) = True;G1: Hyperbola", "coords": { "H": [ 2.0, 2.23606797749979 ] } }, { "index": 5406, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "y", "x_coef": 9.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5406.png", "answer": "6*sqrt(2)", "fact_expr": "M: Point;PointOnCurve(M, G);G: Ellipse;Expression(G) = (x^2/9 + y^2/16 = 1);H: Line;Expression(H) = (x + y - 7 = 0)", "coords": {} }, { "index": 5407, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5407.png", "answer": "(0,pm*2)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/4 = 1);P: Point;PointOnCurve(P, G);F1: Point;F2: Point;Focus(G) = {F1, F2};Area(TriangleOf(P, F1, F2)) = 2*sqrt(5)", "coords": {} }, { "index": 5408, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5408.png", "answer": "(1, 0)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);D: Point;PointOnCurve(D, G) = True;l: Line;Expression(l) = (x = -1);Distance(D, l) = Distance(D, F);F: Point", "coords": {} }, { "index": 5409, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5409.png", "answer": "3", "fact_expr": "G: Hyperbola;m: Number;H: Parabola;Expression(G) = (x^2 - y^2/m = 1);Expression(H) = (y^2 = 8*x);OneOf(Focus(G)) = Focus(H)", "coords": {} }, { "index": 5410, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5411, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5411.png", "answer": "sqrt(14)/4", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Circle;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2 + y^2 = 3*b^2);OneOf(Intersection(G, H))=P;LeftFocus(G) = F1;RightFocus(G) = F2;Abs(LineSegmentOf(P, F1)) = 3*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 5412, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5412.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/9 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);PointOnCurve(MidPoint(LineSegmentOf(P, F1)), yAxis) = True", "coords": {} }, { "index": 5413, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5414, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5415, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5415.png", "answer": "4", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2/3 + y^2/m = 1);PointOnCurve(Focus(G), yAxis);Eccentricity(G) = (1/2)", "coords": {} }, { "index": 5416, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5417, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5417.png", "answer": "(1,\\sqrt{2})", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(RightFocus(G), H);Inclination(H) = ApplyUnit(45, degree);NumIntersection(H, RightPart(G)) = 2", "coords": {} }, { "index": 5418, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5418.png", "answer": "2+2*sqrt(2)", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;Expression(G) = (y = x^2/4);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);x1:Number;x2:Number;y1:Number;y2:Number;PointOnCurve(Focus(G), H);Intersection(H, G) = {A,B};y1+y2=2*sqrt(2);IsChordOf(LineSegmentOf(A,B),G)", "coords": {} }, { "index": 5419, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5420, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5420.png", "answer": "(1, 3/2)", "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2/3 = 1);P: Point;Coordinate(P) = (2, 1);l: Line;PointOnCurve(P, l);M: Point;TangentPoint(l, C) = M;Quadrant(M) = 1", "coords": { "P": [ 2.0, 1.0 ] } }, { "index": 5421, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5422, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5422.png", "answer": "1/2", "fact_expr": "G: Hyperbola;H: Line;P: Point;a: Number;b: Number;Expression(G) = (x^2 - y^2 = 1);Expression(H) = (y = x);Coordinate(P) = (a, b);PointOnCurve(P, RightPart(G));Distance(P, H) = sqrt(2)", "coords": {} }, { "index": 5423, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5423.png", "answer": "3*sqrt(5)/20", "fact_expr": "G: Parabola;H: Line;P: Point;Expression(G) = (y^2 = 2*x);Expression(H) = (y = 2*x + 1);PointOnCurve(P,G)", "coords": {} }, { "index": 5424, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5425, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5425.png", "answer": "3/4", "fact_expr": "C: Hyperbola;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/2 - y^2/2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 5426, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.8284271247461903, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5426.png", "answer": "4", "fact_expr": "G: Hyperbola;H: Ellipse;b: Number;Expression(G) = (x^2/8 - y^2 = 1);b>0;Expression(H) = (x^2/25 + y^2/b^2 = 1);Focus(H) = Focus(G)", "coords": {} }, { "index": 5427, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5427.png", "answer": "3", "fact_expr": "C: Parabola;p: Number;G: Line;M: Point;A: Point;B: Point;F: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(M) = (-p/2, 0);Focus(C) = F;PointOnCurve(F, G);Intersection(G, C) = {A, B};Abs(LineSegmentOf(A, B)) = 12 ;Tan(AngleOf(A, M, B)) = 2*sqrt(2)", "coords": {} }, { "index": 5428, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5428.png", "answer": "1/2", "fact_expr": "C: Parabola;Expression(C) = (x^2 = 2*(p*y));p: Number;p>0;F: Point;Focus(C) = F;x0: Number;P: Point;Coordinate(P) = (x0, 1/2);PointOnCurve(P, C);Abs(LineSegmentOf(P, F)) = 3/4", "coords": {} }, { "index": 5429, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5429.png", "answer": "3/2", "fact_expr": "G: Parabola;p: Number;H: Line;A: Point;B: Point;F: Point;p>0;Expression(G) = (y^2 = 2*p*x);Focus(G) = F;PointOnCurve(F, H);IsPerpendicular(H,xAxis);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 3", "coords": {} }, { "index": 5430, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5430.png", "answer": "4", "fact_expr": "F1: Point;F2: Point;G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G) = True;O: Origin;M: Point;MidPoint(LineSegmentOf(F1, P)) = M;Abs(LineSegmentOf(O, M)) = 3", "coords": {} }, { "index": 5431, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5431.png", "answer": "sqrt(2)/2", "fact_expr": "G: Ellipse;Expression(G) = (x^2 + y^2/b^2 = 1);b: Number;0 < b;b < 1;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;H: Line;Slope(H) = 1;PointOnCurve(F1, H);A: Point;B: Point;Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 4/3", "coords": {} }, { "index": 5432, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5432.png", "answer": "y=pm*(sqrt(10)/4)*x", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;p: Number;F: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);p>0;Expression(H) = (y^2 = 2*p*x);Focus(H)=F;RightFocus(G)=F;P:Point;Coordinate(P)=(3*a^2/p,2*b^2/p);PointOnCurve(P,G)", "coords": {} }, { "index": 5433, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5433.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(F1, F2),LineSegmentOf(P, F2));AngleOf(P,F1,F2)=ApplyUnit(30,degree)", "coords": {} }, { "index": 5434, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5434.png", "answer": "-1", "fact_expr": "P: Point;Coordinate(P) = (-3, 3);M: Point;Coordinate(M) = (3, 0);L: Line;PointOnCurve(M,L) = True;G: Parabola;Expression(G) = (y^2 = 4*x);Intersection(L,G) = {A,B};A: Point;B: Point;Slope(LineOf(P,A)) = k1;k1: Number;Slope(LineOf(P,B)) = k2;k2: Number", "coords": { "P": [ -3.0, 3.0 ], "M": [ 3.0, 0.0 ] } }, { "index": 5435, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.463192105293274, "b": 2.070901870727539, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4661259228887502e-06, "converged": true, "iterations": 106 }, "output_path": "results/train/hyperbola/problem_5435.png", "answer": "x \\pm \\sqrt{2} \\cdot y = 0", "fact_expr": "a: Number;b: Number;a > b;b > 0;C1: Ellipse;Expression(C1) = (y^2/b^2 + x^2/a^2 = 1);C2: Hyperbola;Expression(C2) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(C1)*Eccentricity(C2) = sqrt(3)/2", "coords": {} }, { "index": 5436, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5436.png", "answer": "8*sqrt(6)/9", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;Coordinate(F) = (1, 0);l: Line;Expression(l) = (y = m + x);A: Point;B: Point;Intersection(l, G) = {A, B};m: Number;m >= 0;m < 1;Negation(A=B)", "coords": { "F": [ 1.0, 0.0 ] } }, { "index": 5437, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5437.png", "answer": "3/5", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1)", "coords": {} }, { "index": 5438, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5439, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5439.png", "answer": "13/6", "fact_expr": "G: Ellipse;b: Number;a: Number;a > b;b > 0;e:Number;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);a <= sqrt(3);Eccentricity(G)=e", "coords": {} }, { "index": 5440, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5440.png", "answer": "sqrt(2)+1", "fact_expr": "C: Hyperbola;b: Number;a: Number;F: Point;B: Point;A: Point;a>0;b<0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightVertex(C) = A;RightFocus(C) = F;Intersection(LeftDirectrix(C), xAxis) = B;MidPoint(LineSegmentOf(B,F)) = A", "coords": {} }, { "index": 5441, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5441.png", "answer": "{2, 1/2}", "fact_expr": "P: Point;a: Number;b: Number;G: Hyperbola;Coordinate(P) = (a, b);Expression(G) = (x^2 - y^2 = 1);PointOnCurve(P, RightPart(G));Distance(P, Asymptote(G)) = sqrt(2)", "coords": {} }, { "index": 5442, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5443, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.0, "b": 0.5, "major_axis": "y", "x_coef": 0.25, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5443.png", "answer": "pm*1", "fact_expr": "G: Ellipse;H: Line;m: Number;Expression(G) = (4*x^2 + y^2 = 1);Expression(H) = (y = m + x);Length(InterceptChord(H,G))=2*sqrt(2)/5", "coords": {} }, { "index": 5444, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5445, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5445.png", "answer": "2", "fact_expr": "G: Parabola;Q: Point;P: Point;x0: Number;y0: Number;Expression(G) = (x^2 = -4*y);Coordinate(Q) = (-2*sqrt(2), 0);Coordinate(P) = (x0, y0);PointOnCurve(P, G)", "coords": { "Q": [ -2.8284271247461903, 0.0 ] } }, { "index": 5446, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5446.png", "answer": "16", "fact_expr": "l: Line;G: Parabola;a: Number;O: Origin;A: Point;F: Point;a>0;Expression(G) = (y^2 = a*x);Slope(l) = 1;Focus(G) = F;PointOnCurve(F, l);Intersection(l, yAxis) = A;Area(TriangleOf(O, A, F)) = 8", "coords": {} }, { "index": 5447, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5447.png", "answer": "sqrt(10)/2", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;B: Point;F1: Point;F2: Point;A: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, l);Intersection(l, LeftPart(C)) = {A, B};Area(TriangleOf(B, F1, F2)) = 3*Area(TriangleOf(A, F1, F2));AngleOf(F1, A, F2) = ApplyUnit(90, degree)", "coords": {} }, { "index": 5448, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5449, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5449.png", "answer": "(x+pm*1)^2+(y-(1/2))^2=1", "fact_expr": "H: Circle;G: Parabola;Expression(G) = (y = x^2/2);PointOnCurve(Center(H), G);IsTangent(Directrix(G), H);IsTangent(yAxis, H)", "coords": {} }, { "index": 5450, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5450.png", "answer": "x^2+(y-1)^2=1/5", "fact_expr": "G: Hyperbola;H: Parabola;C:Circle;Expression(G) = (-x^2 + y^2/4 = 1);Expression(H) = (y = x^2/4);Center(C)=Focus(H);IsTangent(OneOf(Asymptote(G)),C)", "coords": {} }, { "index": 5451, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0707900524139404, "b": 1.5409371596115455, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 3.1717604031648706e-06, "converged": true, "iterations": 78 }, "output_path": "results/train/ellipse/problem_5451.png", "answer": "{20/9, 36/5}", "fact_expr": "G: Ellipse;k: Real;Expression(G) = (x^2/4 + y^2/k = 1);Eccentricity(G) = 2/3", "coords": {} }, { "index": 5452, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5452.png", "answer": "1/2", "fact_expr": "G: Ellipse;a: Number;b: Number;F2: Point;M: Point;P: Point;O: Origin;F1: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(M) = (0, b*(sqrt(15)/2));LeftFocus(G) = F1;RightFocus(G) = F2;Intersection(LineSegmentOf(M,F2),G)=P;Abs(LineSegmentOf(P,O))+Abs(LineSegmentOf(P,F1))=2*a", "coords": {} }, { "index": 5453, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5453.png", "answer": "1", "fact_expr": "F1: Point;F2: Point;Focus(G) = {F1, F2};G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1);P: Point;PointOnCurve(P, G);AngleOf(F1,P,F2) = ApplyUnit(90,degree)", "coords": {} }, { "index": 5454, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5454.png", "answer": "2", "fact_expr": "G: Ellipse;O: Circle;A: Point;B: Point;P: Point;Expression(G) = (x^2/4 + y^2/3 = 1);Expression(O) = (x^2 + y^2 = 1);IsDiameter(LineSegmentOf(A, B), O);PointOnCurve(P, G)", "coords": {} }, { "index": 5455, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5455.png", "answer": "8", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);Focus(G) = F;F: Point;H: Line;PointOnCurve(F, H) = True;Intersection(H, G) = {A, B};A: Point;B: Point;XCoordinate(MidPoint(LineSegmentOf(A,B))) = 2", "coords": {} }, { "index": 5456, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5456.png", "answer": "4*sqrt(7)/7", "fact_expr": "C: Hyperbola;G:Circle;b: Number;a: Number;A: Point;O: Origin;B: Point;F: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;Center(G)=F;Radius(G)=HalfFocalLength(C);Intersection(G,Asymptote(C))={O,A,B};Perimeter(TriangleOf(A, O, B)) = 7*a", "coords": {} }, { "index": 5457, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5458, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5458.png", "answer": "2", "fact_expr": "G: Parabola;A: Point;B: Point;F: Point;Expression(G) = (x^2 = 4*y);Focus(G) = F;PointOnCurve(F, LineSegmentOf(A, B));IsChordOf(LineSegmentOf(A, B), G);Length(LineSegmentOf(A, B)) = 6;MidPoint(LineSegmentOf(A, B)) = M;M: Point", "coords": {} }, { "index": 5459, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5459.png", "answer": "9/4", "fact_expr": "C: Parabola;B: Point;A: Point;O: Origin;F: Point;Expression(C) = (x^2 = 4*y);Focus(C) = F;Intersection(LineOf(A, B), C) = {A, B};2*VectorOf(O, A) + VectorOf(O, B) - 3*VectorOf(O, F) = 0;IsChordOf(LineSegmentOf(A, B), C)", "coords": {} }, { "index": 5460, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5460.png", "answer": "5", "fact_expr": "G:Hyperbola;Expression(G) = (-y^2/9 + x^2/a^2 = 1);a: Number;P: Point;PointOnCurve(P, RightPart(G));Expression(OneOf(Asymptote(G))) = (3*x - y = 0);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;Abs(LineSegmentOf(P, F2)) = 3", "coords": {} }, { "index": 5461, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 1.4142135623730951, "major_axis": "y", "x_coef": 2.0, "y_coef": 8.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5461.png", "answer": "[-1, 3]", "fact_expr": "C: Ellipse;A: Point;B: Point;a: Real;Expression(C) = (x^2/2 + y^2/8 = 1);Coordinate(A) = (a, 2*a);Coordinate(B) = (a - 2, 2*a - 4);IsIntersect(LineSegmentOf(A,B), C)", "coords": {} }, { "index": 5462, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5463, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5464, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5464.png", "answer": "8", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;A: Point;B: Point;C: Point;Negation(A=B);Negation(A=C);Negation(B=C);PointOnCurve(A, G);PointOnCurve(B, G);PointOnCurve(C, G);F: Point;Focus(G) = F;VectorOf(F, A) + VectorOf(F, B) + VectorOf(F, C) = 0;Expression(OverlappingLine(LineSegmentOf(B, C))) = (4*x + y - 20 = 0)", "coords": {} }, { "index": 5465, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5466, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5466.png", "answer": "10 + sqrt(2)", "fact_expr": "G: Ellipse;A: Point;B: Point;P: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Coordinate(A) = (3, 1);Coordinate(B) = (-4, 0);PointOnCurve(P, G)", "coords": { "A": [ 3.0, 1.0 ], "B": [ -4.0, 0.0 ] } }, { "index": 5467, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.0625, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5467.png", "answer": "1/8", "fact_expr": "G: Parabola;Expression(G) = (y = 4*x^2)", "coords": {} }, { "index": 5468, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5468.png", "answer": "16", "fact_expr": "G: Parabola;M: Point;O: Origin;N: Point;Expression(G) = (y^2 = 8*x);PointOnCurve(M,G);PointOnCurve(N,G);AngleOf(M, O, N) = ApplyUnit(90, degree)", "coords": {} }, { "index": 5469, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5470, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5470.png", "answer": "y^2/18 - x^2/9 = 1", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/2 = 1);Z: Hyperbola;H: Point;Coordinate(H) = (3, -6);PointOnCurve(H, Z);Asymptote(G) = Asymptote(Z)", "coords": { "H": [ 3.0, -6.0 ] } }, { "index": 5471, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5472, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5473, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5474, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5474.png", "answer": "(1, 0)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;H: Circle;Expression(H) = (-2*y + x^2 + y^2 - 1 = 0);Length(InterceptChord(Directrix(G), H)) = 2", "coords": {} }, { "index": 5475, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5476, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5476.png", "answer": "(0, pm*2)", "fact_expr": "P: Point;F1: Point;F2: Point;Focus(G) = {F1, F2};G: Ellipse;Expression(G) = (x^2/9 + y^2/4 = 1);PointOnCurve(P, G);Vertex(TriangleOf(P, F1, F2)) = {P, F1, F2};Area(TriangleOf(P, F1, F2)) = 2*sqrt(5)", "coords": {} }, { "index": 5477, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5477.png", "answer": "2", "fact_expr": "G: Hyperbola;a: Number;Expression(G) = (-y^2/9 + x^2/a^2 = 1);a>0;Expression(OneOf(Asymptote(G))) = (3*x - 2*y = 0)", "coords": {} }, { "index": 5478, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5478.png", "answer": "y^2/4 - x^2/16 = 1", "fact_expr": "G: Hyperbola;C: Hyperbola;H: Point;Expression(G) = (x^2 - 4*y^2 = 4);Coordinate(H) = (2, sqrt(5));Asymptote(G) = Asymptote(C);PointOnCurve(H, C)", "coords": { "H": [ 2.0, 2.23606797749979 ] } }, { "index": 5479, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.674137386787269, "b": 1.1170897804012059, "orientation": "horizontal" }, "optimization": { "final_loss": 2.78399227733666e-06, "converged": true, "iterations": 61 }, "output_path": "results/train/hyperbola/problem_5479.png", "answer": "{13/12, 13/5}", "fact_expr": "G: Hyperbola;Expression(Asymptote(G)) = (y = pm*(5/12)*x)", "coords": {} }, { "index": 5480, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.6259966743174588, "b": 2.298015331604694, "orientation": "horizontal" }, "optimization": { "final_loss": 1.93011531933813e-06, "converged": true, "iterations": 91 }, "output_path": "results/train/hyperbola/problem_5480.png", "answer": "sqrt(6)/2", "fact_expr": "G: Hyperbola;a: Number;Expression(G) = (-x^2 + y^2/a^2 = 1);Expression(Asymptote(G)) = (y = pm*sqrt(2)*x)", "coords": {} }, { "index": 5481, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5482, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5483, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.125, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5483.png", "answer": "y = -1/8", "fact_expr": "G: Parabola;Expression(G) = (y = 2*x^2)", "coords": {} }, { "index": 5484, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_ecc_mismatch", "params": { "a": 1.9257268905639648, "b": 1.7993911322594118, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 2.5576271215184017e-15, "converged": true, "iterations": 2 }, "output_path": "results/train/ellipse/problem_5484.png", "answer": "3", "fact_expr": "G: Ellipse;m: Number;m<4;Expression(G) = (x^2/4 + y^2/m = 1);Eccentricity(G) = 1/2", "coords": {}, "validation_reasons": [ "ellipse_ecc_mismatch" ] }, { "index": 5485, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5486, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5486.png", "answer": "(2/3)*sqrt(21)", "fact_expr": "G: Ellipse;A: Point;P: Point;Expression(G) = (x^2/4 + y^2 = 1);Coordinate(A) = (0, 2);PointOnCurve(P, G)", "coords": { "A": [ 0.0, 2.0 ] } }, { "index": 5487, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5488, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5488.png", "answer": "3", "fact_expr": "A: Point;B: Point;G: Parabola;H: Line;Expression(G) = (y^2 = 4*x);PointOnCurve(Focus(G), H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 8", "coords": {} }, { "index": 5489, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5489.png", "answer": "2*sqrt(5)", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;F1: Point;F2: Point;A: Point;B: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;Abs(LineSegmentOf(F1, F2)) = 2;PointOnCurve(LeftFocus(G), H);Slope(H) = 2;Intersection(H, G) = {A, B};Area(TriangleOf(A,B,F2))= 4;IsChordOf(LineSegmentOf(A,B),G)", "coords": {} }, { "index": 5490, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5490.png", "answer": "x = 4", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1)", "coords": {} }, { "index": 5491, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5491.png", "answer": "(\\sqrt{5}-1)/2", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;FocalLength(C) = 2*c;c: Number;M: Circle;Expression(M) = (-2*c*y + x^2 + y^2 = 0);Intersection(M, C) = {A, B};A: Point;B: Point;IsPerpendicular(LineSegmentOf(O, A), LineSegmentOf(O, B)) = True;O: Origin", "coords": {} }, { "index": 5492, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5493, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 4.0, "major_axis": "x", "x_coef": 36.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5493.png", "answer": "x^2/15 - y^2/5 = 1", "fact_expr": "C: Hyperbola;G: Ellipse;Expression(G) = (x^2/36 + y^2/16 = 1);Focus(C) = Focus(G);Expression(Asymptote(C))=(pm*sqrt(3)*y+x=0)", "coords": {} }, { "index": 5494, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5494.png", "answer": "(\\sqrt{5}, +\\infty)", "fact_expr": "G: Hyperbola;a: Number;b: Number;H: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = 2*x);IsIntersect(G, H);Eccentricity(G)=e;e:Number", "coords": {} }, { "index": 5495, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5495.png", "answer": "x^2 - y^2 / 3 = 1", "fact_expr": "C: Hyperbola;b: Number;a: Number;F: Circle;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(F) = (y^2 + (x - 2)^2 = 3);IsTangent(Asymptote(C),F);OneOf(Focus(C))=Center(F)", "coords": {} }, { "index": 5496, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.0654510707333538, "b": 2.6632060182000887, "orientation": "horizontal" }, "optimization": { "final_loss": 3.6995812463854516e-05, "converged": true, "iterations": 75 }, "output_path": "results/train/hyperbola/problem_5496.png", "answer": "4*y^2/31 - 25*x^2/31 = 1", "fact_expr": "C: Hyperbola;G: Point;Coordinate(G) = (3, 8);Expression(Asymptote(C)) = (y = pm*(5/2)*x);PointOnCurve(G,C)", "coords": { "G": [ 3.0, 8.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 5497, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5497.png", "answer": "2*sqrt(3)/3", "fact_expr": "C: Hyperbola;b: Number;a: Number;A: Circle;M: Point;N: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightVertex(C) = A1;A1:Point;Center(A)=A1;Radius(A)=b;Intersection(A, OneOf(Asymptote(C))) = {M, N};Abs(LineSegmentOf(M, N)) = b", "coords": {} }, { "index": 5498, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5498.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/b^2 = 1);b: Number;P: Point;PointOnCurve(P, G);Expression(OneOf(Asymptote(G))) = (3*x - 2*y = 0);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;Abs(LineSegmentOf(P, F1)) = 5", "coords": {} }, { "index": 5499, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5499.png", "answer": "2", "fact_expr": "A: Point;B: Point;PointOnCurve(A, G) = True;PointOnCurve(B, G) = True;Negation(A = B);G: Parabola;Expression(G) = (y^2 = 4*x);IsChordOf(LineSegmentOf(A, B), G) = True;Negation(IsParallel(LineSegmentOf(A, B), yAxis));Intersection(PerpendicularBisector(LineSegmentOf(A, B)), xAxis) = P;P: Point;Coordinate(P) = (4, 0)", "coords": { "P": [ 4.0, 0.0 ] } }, { "index": 5500, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.4641016151377544, "major_axis": "x", "x_coef": 16.0, "y_coef": 12.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5500.png", "answer": "3*x-8*y+19=0", "fact_expr": "G: Ellipse;H: LineSegment;M: Point;Expression(G) = (x^2/16 + y^2/12 = 1);Coordinate(M) = (-1, 2);IsChordOf(H, G);MidPoint(H) = M", "coords": { "M": [ -1.0, 2.0 ] } }, { "index": 5501, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5502, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5502.png", "answer": "4*sqrt(2)", "fact_expr": "C: Parabola;P: Point;A: Point;O: Origin;B: Point;a:Number;Expression(C) = (x^2 = 4*y);Coordinate(P) = (a,-2);L1:Line;L2:Line;PointOnCurve(P,L1);PointOnCurve(P,L2);TangentPoint(L1,C)=A;TangentPoint(L2,C)=B", "coords": {} }, { "index": 5503, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5504, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5504.png", "answer": "sqrt(3)/2", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) - Abs(LineSegmentOf(P, F2)) = 2*b;Abs(LineSegmentOf(P, F1))*Abs(LineSegmentOf(P, F2)) = (3/2)*(a*b)", "coords": {} }, { "index": 5505, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5506, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5507, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5507.png", "answer": "1/2", "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F: Point;A: Point;Q: Point;e: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);RightFocus(G)=F;LeftVertex(G)=A;PointOnCurve(P, G);PointOnCurve(Q, G);IsPerpendicular(LineSegmentOf(P, F), LineSegmentOf(A, F));Tan(AngleOf(P, A, F)) = 1/2;Eccentricity(G) = e", "coords": {} }, { "index": 5508, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5508.png", "answer": "4", "fact_expr": "G: Hyperbola;H: Parabola;p: Real;Expression(G) = (x^2 - y^2/3 = 1);p>0;Expression(H) = (y^2 = 2*(p*x));PointOnCurve(LeftFocus(G), Directrix(H))", "coords": {} }, { "index": 5509, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5509.png", "answer": "sqrt(2)/2", "fact_expr": "l: Line;C: Ellipse;b: Number;a: Number;A: Point;F: Point;B: Point;e: Number;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Expression(l)=(y=x+1);Intersection(l,C)={A,B};Intersection(l,xAxis)=F;LeftFocus(C)=F;VectorOf(A, F) = 3*VectorOf(F, B);Eccentricity(C) = e", "coords": {} }, { "index": 5510, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5511, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5511.png", "answer": "27/4", "fact_expr": "G: Parabola;H: Line;A: Point;O: Origin;B: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};AngleOf(A, O, B) = ApplyUnit(120, degree)", "coords": {} }, { "index": 5512, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5512.png", "answer": "(-2,0) \nx=9/2", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/5 = 1)", "coords": {} }, { "index": 5513, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5513.png", "answer": "sqrt(5)/3", "fact_expr": "C: Ellipse;b: Number;a: Number;G: Circle;F2: Point;M: Point;F1: Point;E: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2 + y^2 = b^2);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(M, C);TangentPoint(LineSegmentOf(M,F2), G) = E;MidPoint(LineSegmentOf(M,F2)) = E", "coords": {} }, { "index": 5514, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5514.png", "answer": "8/3", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 8*x);F: Point;Focus(C) = F;l: Line;Directrix(C) = l;P: Point;PointOnCurve(P, l);Q: Point;OneOf(Intersection(LineOf(P, F), C)) = Q;VectorOf(P, F) = 3*VectorOf(Q, F)", "coords": {} }, { "index": 5515, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5515.png", "answer": "2*p", "fact_expr": "G: Parabola;p: Number;A: Point;B: Point;C: Point;F: Point;p>0;Expression(G) = (y^2 = 2*p*x);Focus(G)=F;Intersection(Directrix(G), xAxis) = C;IsChordOf(LineSegmentOf(A,B),G);PointOnCurve(F,LineSegmentOf(A,B));AngleOf(C, B, F) = ApplyUnit(90, degree)", "coords": {} }, { "index": 5516, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_focus_mismatch", "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5516.png", "answer": "x^2/16+y^2/12=1", "fact_expr": "C: Ellipse;b: Number;a: Number;F1: Point;P: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F2) = (2, 0);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);Tan(AngleOf(F1, P, F2)) = 4/3;Area(TriangleOf(P,F1,F2)) = 6", "coords": { "F2": [ 2.0, 0.0 ] }, "validation_reasons": [ "ellipse_focus_mismatch" ] }, { "index": 5517, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5517.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;H: Parabola;Expression(H) = (y = x^2/2 + 2);IsTangent(Asymptote(G), H)", "coords": {} }, { "index": 5518, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5519, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5519.png", "answer": "x^2/3 - y^2 = 1", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/9 + y^2/5 = 1);Focus(G)=Focus(H);Distance(Focus(G),Asymptote(G))=1", "coords": {} }, { "index": 5520, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5520.png", "answer": "8*sqrt(2)/3", "fact_expr": "G: Ellipse;H: Line;F1: Point;A: Point;B: Point;Expression(G) = (x^2/2 + y^2 = 1);Expression(H) = (y = x - 1);LeftFocus(G) = F1;Intersection(H, G) = {A, B}", "coords": {} }, { "index": 5521, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 4.47213595499958, "major_axis": "x", "x_coef": 36.0, "y_coef": 20.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5521.png", "answer": "20", "fact_expr": "F1: Point;F2: Point;Focus(G) = {F1, F2};G: Ellipse;Expression(G) = (x^2/36 + y^2/20 = 1);M: Point;PointOnCurve(M, G) = True;IsPerpendicular(LineSegmentOf(M, F1), LineSegmentOf(M, F2)) = True", "coords": {} }, { "index": 5522, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5522.png", "answer": "y=pm*sqrt(3)*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/12 = 1)", "coords": {} }, { "index": 5523, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5523.png", "answer": "2", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);P: Point;PointOnCurve(P, G);Distance(P, LeftDirectrix(G)) = 10;F: Point;LeftFocus(G) = F;M: Point;O: Origin;VectorOf(O, M) = (1/2)*(VectorOf(O, P) + VectorOf(O, F))", "coords": {} }, { "index": 5524, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5524.png", "answer": "2*sqrt(3)/3", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Distance(OneOf(Focus(G)),OneOf(Asymptote(G)))=(1/4)*FocalLength(G)", "coords": {} }, { "index": 5525, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5526, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5526.png", "answer": "pm*sqrt(3)", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;A: Point;F: Point;B: Point;k: Real;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(l) = (-3*a*k + 2*(k*x) - 2*y = 0);RightFocus(C) = F;PointOnCurve(F,l);Intersection(l, C) = {A, B};VectorOf(A, F) = 7*VectorOf(F, B)", "coords": {} }, { "index": 5527, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5527.png", "answer": "[\\sqrt{3}/3,1)", "fact_expr": "P: Point;F1: Point;F2:Point;G: Ellipse;b: Number;a: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(P, RightDirectrix(G));PointOnCurve(F2,PerpendicularBisector(LineSegmentOf(P,F1)))", "coords": {} }, { "index": 5528, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5528.png", "answer": "4", "fact_expr": "H: Line;Inclination(H) = pi/4;G: Parabola;Expression(G) = (y^2 = 2*x);F: Point;Focus(G) = F;A: Point;B: Point;Intersection(H, G) = {A, B};PointOnCurve(F, H)", "coords": {} }, { "index": 5529, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5530, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 9.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5530.png", "answer": "Answer is incorrect\nApplyUnit(120, degree)", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 + y^2/2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 4", "coords": {} }, { "index": 5531, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/train/hyperbola/problem_5531.png", "answer": "x^2 - y^2 = 1", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;e: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (sqrt(2), 1);RightFocus(C) = F2;LeftFocus(C) = F1;PointOnCurve(P, C);Eccentricity(C) = e;e = sqrt(2)", "coords": { "P": [ 1.4142135623730951, 1.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 5532, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5533, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2631799883090584, "b": 2.5275303393962245, "orientation": "horizontal" }, "optimization": { "final_loss": 4.481942266905428e-06, "converged": true, "iterations": 95 }, "output_path": "results/train/hyperbola/problem_5533.png", "answer": "2/3", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/(m + 2) + x^2/m = 1);m: Number;m>0;Expression(OneOf(Asymptote(G))) = (y = 2*x)", "coords": {} }, { "index": 5534, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5535, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.9970735311508179, "direction": "right" }, "optimization": { "final_loss": 2.85904548036342e-06, "converged": true, "iterations": 88 }, "output_path": "results/train/parabola/problem_5535.png", "answer": "16*sqrt(2)", "fact_expr": "P: Point;Coordinate(P) = (2, 4);PointOnCurve(P, C);C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;F: Point;Focus(C) = F;l: Line;Inclination(l) = ApplyUnit(45, degree);PointOnCurve(F, l);M: Point;N: Point;Intersection(l, C) = {M, N}", "coords": { "P": [ 2.0, 4.0 ] } }, { "index": 5536, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5536.png", "answer": "4", "fact_expr": "O: Origin;G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>=0;A: Point;B: Point;PointOnCurve(A, G);PointOnCurve(B, G);P: Point;Negation(P=A);Negation(P=B);AngleOf(A, O, B) = ApplyUnit(90, degree);Max(Distance(O, LineOf(A, B))) = 8", "coords": {} }, { "index": 5537, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.3300262689590454, "b": 2.30381441116333, "orientation": "horizontal" }, "optimization": { "final_loss": 6.658357051492203e-06, "converged": true, "iterations": 122 }, "output_path": "results/train/hyperbola/problem_5537.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;Expression(Asymptote(G)) = (y = pm*(sqrt(3)*x))", "coords": {} }, { "index": 5538, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5538.png", "answer": "(\\sqrt{3}+1)/2", "fact_expr": "F: Point;RightFocus(C) = F;C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;M: Point;PointOnCurve(M,RightPart(C)) = True;O: Origin;Abs(LineSegmentOf(F, M)) = 2*Abs(LineSegmentOf(O, F));AngleOf(O, F, M) = ApplyUnit(120, degree)", "coords": {} }, { "index": 5539, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.9959803819656372, "direction": "right" }, "optimization": { "final_loss": 3.6780886603082763e-07, "converged": true, "iterations": 56 }, "output_path": "results/train/parabola/problem_5539.png", "answer": "5/4", "fact_expr": "G: Parabola;a: Number;A: Point;Expression(G) = (y^2 = a*x);Coordinate(A) = (1/4, 1);PointOnCurve(A, G)", "coords": { "A": [ 0.25, 1.0 ] } }, { "index": 5540, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5540.png", "answer": "(2*sqrt(3))/3", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);A: Point;B: Point;Endpoint(LineSegmentOf(A, B)) = {A, B};PointOnCurve(A, G);PointOnCurve(B, G);Abs(LineSegmentOf(A, B)) = 2;x0: Number;y0: Number;M: Point;Coordinate(M) = (x0, y0);MidPoint(LineSegmentOf(A, B)) = M", "coords": {} }, { "index": 5541, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5541.png", "answer": "x^2/2 - y^2/6 = 1", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;a>0;b>0;l:Line;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 8*sqrt(2)*x);Expression(l)=(y=sqrt(3)*x-1);IsParallel(l,OneOf(Asymptote(G)));PointOnCurve(OneOf(Focus(G)),Directrix(H))", "coords": {} }, { "index": 5542, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5542.png", "answer": "9/2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, G) = {A, B};C: Point;Intersection(l, Directrix(G)) = C;VectorOf(F, C) = 4*VectorOf(F, B)", "coords": {} }, { "index": 5543, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5543.png", "answer": "2*sqrt(5)/3", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/5 = 1)", "coords": {} }, { "index": 5544, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5545, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5545.png", "answer": "sqrt(3)/2", "fact_expr": "G: Ellipse;M: Point;P: Point;N: Point;a:Number;b:Number;a>b;b>0;k1:Number;k2:Number;Expression(G)=(x^2/a^2+y^2/b^2=1);LeftVertex(G)=M;RightVertex(G)=N;PointOnCurve(P, G);Slope(LineOf(P, M)) = k1;Slope(LineOf(P,N))=k2;Negation(k1*k2=0);Min(Abs(k1) + Abs(k2)) = 1", "coords": {} }, { "index": 5546, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5547, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5547.png", "answer": "sqrt(14)/4", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;F: Point;RightFocus(C)=F;G: Circle;H: Line;Expression(H) = (y = b/3);A: Point;B: Point;Intersection(H, C) = {A, B};IsDiameter(LineSegmentOf(A,B),G);PointOnCurve(F, G)", "coords": {} }, { "index": 5548, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.463192105293274, "b": 2.070901870727539, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4661259228887502e-06, "converged": true, "iterations": 106 }, "output_path": "results/train/hyperbola/problem_5548.png", "answer": "y=pm*(sqrt(2)*x)", "fact_expr": "G: Hyperbola;m: Number;m>0;Expression(G) = (-y^2/4 + x^2/m = 1);Eccentricity(G) = sqrt(3)", "coords": {} }, { "index": 5549, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5549.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;A: Point;B: Point;C: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightVertex(G)=A;PointOnCurve(A,H);Slope(H)=-1;L1:Line;L2:Line;Asymptote(G)={L1,L2};Intersection(H,L1)=B;Intersection(H,L2)=C;VectorOf(A, B) = VectorOf(B, C)/2", "coords": {} }, { "index": 5550, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5550.png", "answer": "sqrt(2)", "fact_expr": "C: Ellipse;A: Point;F: Point;B: Point;l: Line;Expression(C) = (x^2/2 + y^2 = 1);LeftFocus(C) = F;LeftDirectrix(C) = l;PointOnCurve(A,l);VectorOf(F, A) = 3*VectorOf(F, B);Intersection(LineSegmentOf(A,F), C) = B", "coords": {} }, { "index": 5551, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5551.png", "answer": "x = 1", "fact_expr": "G: Parabola;Expression(G) = (y^2 = -4*x)", "coords": {} }, { "index": 5552, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5552.png", "answer": "sqrt(3)*x+pm*2*y=0", "fact_expr": "C: Hyperbola;a: Number;b: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, RightPart(C));Abs(LineSegmentOf(P, F1)) + Abs(LineSegmentOf(P, F2)) = 4*a;AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 5553, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5553.png", "answer": "14", "fact_expr": "E: Parabola;Expression(E) = (y^2 = 8*x);F: Point;Focus(E) = F;PointOnCurve(F, G) = True;G: Line;Intersection(G, E) = {A, B};A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), E) = True;MidPoint(LineSegmentOf(A, B)) = M;M: Point;L: Line;PointOnCurve(M, L) = True;IsPerpendicular(L, Directrix(E)) = True;Intersection(L, E) = P;P: Point;Abs(LineSegmentOf(P, F)) = 7/2", "coords": {} }, { "index": 5554, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5554.png", "answer": "3", "fact_expr": "C: Parabola;l: Line;P: Point;F: Point;Q: Point;O: Origin;Expression(C) = (y^2 = 8*x);Focus(C) = F;Directrix(C) = l;PointOnCurve(P, l);OneOf(Intersection(LineOf(P, F), C)) = Q;VectorOf(F, P) = 4*VectorOf(F, Q)", "coords": {} }, { "index": 5555, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5556, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5557, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5557.png", "answer": "-2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);H: Circle;m: Number;Expression(H) = (m*x + x^2 + y^2 - 4 = 0);Focus(G) = Center(H)", "coords": {} }, { "index": 5558, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.75, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5558.png", "answer": "15/4", "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (y^2 = 3*x);Coordinate(A) = (3, 2);Focus(G) = F;PointOnCurve(P, G)", "coords": { "A": [ 3.0, 2.0 ] } }, { "index": 5559, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5559.png", "answer": "(3*sqrt(3))/2", "fact_expr": "P: Point;G: Curve;Expression(G) = (y = x^2);PointOnCurve(P, G);l: Line;TangentOnPoint(P, G) = l;H: Line;PointOnCurve(P, H);IsPerpendicular(H, l);Q: Point;Intersection(H, G) = {P, Q}", "coords": {} }, { "index": 5560, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5561, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5561.png", "answer": "20", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);F2: Point;F1: Point;Focus(G) = {F1, F2};H: Line;PointOnCurve(F1, H) = True;Intersection(H, G) = {A, B};A: Point;B: Point", "coords": {} }, { "index": 5562, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5562.png", "answer": "pm*1", "fact_expr": "Z: Line;Expression(Z) = (y = k*x + 1);k: Number;Negation(k=0);G: Parabola;Expression(G) = (x^2 = 4*y);E: Point;F: Point;Intersection(Z, G) = {E, F};H: Circle;IsDiameter(LineSegmentOf(E, F), H);Length(InterceptChord(xAxis, H)) = 2*sqrt(7)", "coords": {} }, { "index": 5563, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5564, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_focus_mismatch", "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5564.png", "answer": "1/2", "fact_expr": "Gamma: Ellipse;A: Point;F: Point;B: Point;M: Point;N: Point;Expression(Gamma) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (3, 0);Coordinate(N) = (12, 0);RightFocus(Gamma) = F;UpperVertex(Gamma) = A;LowerVertex(Gamma) = B;Intersection(LineOf(A,F), Gamma) = {A,M};Intersection(LineOf(B,M), xAxis) = N;a:Number;b:Number;a>b;b>0", "coords": { "F": [ 3.0, 0.0 ], "N": [ 12.0, 0.0 ] }, "validation_reasons": [ "ellipse_focus_mismatch" ] }, { "index": 5565, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5566, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5566.png", "answer": "sqrt(3)+1", "fact_expr": "F1: Point;F2: Point;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;LeftFocus(G) = F1;RightFocus(G) = F2;IsDiameter(LineSegmentOf(F1, F2), H) = True;H: Circle;OneOf(Intersection(G, H)) = P;P: Point;AngleOf(P, F1, F2) = 2*AngleOf(P, F2, F1)", "coords": {} }, { "index": 5567, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5567.png", "answer": "2", "fact_expr": "C1: Ellipse;C2: Hyperbola;a, b: Number;a_, b_: Number;a > b;b > 0;Expression(C1) = (x^2/a^2 + y^2/b^2 = 1);a_ > 0;b_ > 0;Expression(C2) = (x^2/a_^2 - y^2/b_^2 = 1);F1: Point;F2: Point;Focus(C1) = {F1, F2};Focus(C2) = {F1, F2};P: Point;OneOf(Intersection(C1, C2)) = P;AngleOf(F1, P, F2) = ApplyUnit(90, degree);e1: Number;e2: Number;Eccentricity(C1) = e1;Eccentricity(C2) = e2", "coords": {} }, { "index": 5568, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 3.8253217034665385, "b": 1.8623655158696077, "major_axis": "x", "x_coef": 16.0, "y_coef": 4.0 }, "optimization": { "final_loss": 3.1260700230116838e-06, "converged": true, "iterations": 93 }, "output_path": "results/train/ellipse/problem_5568.png", "answer": "sqrt(3)/2", "fact_expr": "G: Ellipse;H: Point;I: Point;O: Origin;Coordinate(H) = (4, 0);Coordinate(I) = (0, 2);Center(G) = O;SymmetryAxis(G)=axis;PointOnCurve(H,G);PointOnCurve(I,G)", "coords": { "H": [ 4.0, 0.0 ], "I": [ 0.0, 2.0 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 5569, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -2.0, 0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_5569.png", "answer": "{x^2/25+y^2/21=1,x^2/16+y^2/12=1}", "fact_expr": "M: Circle;C1:Circle;C2:Circle;M1:Point;Expression(C1)=((x+2)^2+y^2=1);Expression(C2)=((x-2)^2+y^2=81);IsTangent(M,C1);IsTangent(M,C2);Center(M)=M1", "coords": {} }, { "index": 5570, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5570.png", "answer": "8", "fact_expr": "G: Parabola;Expression(G) = (x^2 = -4*y);l: Line;PointOnCurve(Focus(G), l);P: Point;Q: Point;x1: Number;x2: Number;y1: Number;y2: Number;Coordinate(P) = (x1, y1);Coordinate(Q) = (x2, y2);Intersection(l, G) = {P, Q};y1 + y2 = -6", "coords": {} }, { "index": 5571, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5571.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Hyperbola;A: Point;P: Point;F: Point;Expression(G) = (x^2/3 - y^2 = 1);Coordinate(A) = (3, 1);PointOnCurve(P, RightPart(G));RightFocus(G) = F", "coords": { "A": [ 3.0, 1.0 ] } }, { "index": 5572, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.4641016151377544, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 12.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5572.png", "answer": "1/7", "fact_expr": "G: Ellipse;Expression(G) = (x^2/12 + y^2/3 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G) = True;PointOnCurve(MidPoint(LineSegmentOf(P, F1)), yAxis) = True", "coords": {} }, { "index": 5573, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5573.png", "answer": "1/2", "fact_expr": "G: Parabola;H: Line;O: Origin;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (y = 2*x - 3);Intersection(H, G) = {A, B};Slope(LineOf(O, A)) = k1;Slope(LineOf(O, B)) = k2;k1:Number;k2:Number", "coords": {} }, { "index": 5574, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5574.png", "answer": "4", "fact_expr": "G: Parabola;P: Point;F: Point;Expression(G) = (x^2 = 4*y);PointOnCurve(P, G);Focus(G) = F;Abs(LineSegmentOf(P, F)) = 5", "coords": {} }, { "index": 5575, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5575.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;M: Point;F2: Point;F1: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, H);Inclination(H) = ApplyUnit(30, degree);Intersection(H, RightPart(G)) = M;IsPerpendicular(LineSegmentOf(M, F2), xAxis)", "coords": {} }, { "index": 5576, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5576.png", "answer": "(x-1)^2+y^2=36", "fact_expr": "G: Parabola;H: Circle;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(P) = (5, -2*sqrt(5));Focus(G)=Center(H);PointOnCurve(P,H)", "coords": { "P": [ 5.0, -4.47213595499958 ] } }, { "index": 5577, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5577.png", "answer": "sqrt(5)", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;P: Point;F2: Point;F1: Point;O:Origin;B: Point;e:Number;b > a;a > 0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2 + y^2 = b^2);LeftFocus(C) = F1;RightFocus(C) = F2;Eccentricity(C)=e;PointOnCurve(P, RightPart(C));IsPerpendicular(LineSegmentOf(P, F2), LineSegmentOf(F1, F2));PointOnCurve(B, G);VectorOf(O, B) = (2/sqrt(21))*VectorOf(O, P)", "coords": {} }, { "index": 5578, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5578.png", "answer": "34", "fact_expr": "G: Hyperbola;a: Number;P: Point;F1: Point;F2: Point;a>0;Expression(G) = (-y^2/9 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);Abs(LineSegmentOf(P,F1))=2*Abs(LineSegmentOf(P,F2));Abs(LineSegmentOf(P,F1))=16", "coords": {} }, { "index": 5579, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5579.png", "answer": "3*sqrt(2)/4", "fact_expr": "C2: Hyperbola;C1: Ellipse;F1: Point;A: Point;F2: Point;Expression(C1) = (x^2/4 + y^2 = 1);Focus(C1) = {F1, F2};Focus(C2) = {F1, F2};Intersection(C1, C2) = A;Quadrant(A) = 2;AngleOf(F1, A, F2) = pi/3", "coords": {} }, { "index": 5580, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_5580.png", "answer": "y = pm*sqrt(3)*x", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;Eccentricity(G) = 2", "coords": {} }, { "index": 5581, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5582, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5582.png", "answer": "9*sqrt(5)/2", "fact_expr": "G: Parabola;H: Line;I: Point;A: Point;B: Point;C: Point;Expression(G) = (y^2 = 6*x);Coordinate(I) = (3, 0);PointOnCurve(I, H);Intersection(H, G) = {A, B};Intersection(H,yAxis)=C;VectorOf(A, B) = 3*VectorOf(B, C)", "coords": { "I": [ 3.0, 0.0 ] } }, { "index": 5583, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5583.png", "answer": "8", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l: Line;Inclination(l) = ApplyUnit(45, degree);PointOnCurve(F, l);A: Point;B: Point;Intersection(l, C) = {A, B}", "coords": {} }, { "index": 5584, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5584.png", "answer": "sqrt(5)*2", "fact_expr": "C: Parabola;A: Point;K: Point;F: Point;O: Origin;Expression(C) = (y^2 = 8*x);Focus(C) = F;Intersection(Directrix(C), xAxis) = K;PointOnCurve(A, C);Abs(LineSegmentOf(A, K)) = sqrt(2)*Abs(LineSegmentOf(A, F))", "coords": {} }, { "index": 5585, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5585.png", "answer": "(0, \\sqrt{2}/2]", "fact_expr": "G: Ellipse;b: Number;a: Number;p: Point;F1: Point;F2: Point;a>b;b>0;x1:Number;y1:Number;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(p) = (x1, y1);PointOnCurve(p, G);Focus(G) = {F1, F2};AngleOf(F1,p,F2)<=ApplyUnit(90,degree)", "coords": {} }, { "index": 5586, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5586.png", "answer": "sqrt(2)/2", "fact_expr": "H: Line;Expression(H) = (x + 2*y - 3 = 0);G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;A: Point;B: Point;Intersection(H, G) = {A, B};Z: Line;Expression(Z) = (3*x - 4*y + 1 = 0);PointOnCurve(MidPoint(LineSegmentOf(A, B)), Z)", "coords": {} }, { "index": 5587, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5588, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5589, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.4149149626832767, "b": 2.450642358201144, "orientation": "horizontal" }, "optimization": { "final_loss": 1.0319236343469396e-05, "converged": true, "iterations": 94 }, "output_path": "results/train/hyperbola/problem_5589.png", "answer": "x^2 - y^2 / 3 = 1", "fact_expr": "O: Origin;Center(G) = O;G: Hyperbola;Expression(Asymptote(G)) = (y = pm*(sqrt(3)*x));H: Point;Coordinate(H) = (sqrt(2), sqrt(3));PointOnCurve(H, G)", "coords": { "H": [ 1.4142135623730951, 1.7320508075688772 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 5590, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5591, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 0.929953396320343, "b": 2.629857301712036, "orientation": "horizontal" }, "optimization": { "final_loss": 4.594453912432073e-06, "converged": true, "iterations": 175 }, "output_path": "results/train/hyperbola/problem_5591.png", "answer": "8", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Eccentricity(G) = 3;H: Circle;Expression(H) = (m - 6*y + x^2 + y^2 = 0);m: Number;IsTangent(Asymptote(G), H) = True", "coords": {} }, { "index": 5592, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5592.png", "answer": "y=pm*sqrt(3)*x", "fact_expr": "C: Hyperbola;b: Number;a: Number;D: Ellipse;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(D) = (x^2/4 + y^2/3 = 1);InterReciprocal(Eccentricity(C),Eccentricity(D))", "coords": {} }, { "index": 5593, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5593.png", "answer": "(4, 4)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);Q: Point;PointOnCurve(Q, G);A: Point;Coordinate(A) = (7, 8);WhenMin(Distance(Q, A) + Distance(Q, Directrix(G)))", "coords": { "A": [ 7.0, 8.0 ] } }, { "index": 5594, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5594.png", "answer": "2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;M: Point;PointOnCurve(M, G);XCoordinate(M) = 4;F: Point;Focus(G) = F;Distance(M, F) = 5", "coords": {} }, { "index": 5595, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5595.png", "answer": "[sqrt(2)/2, 1]", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(G) = F;L: Line;PointOnCurve(F, L);IsPerpendicular(L, OneOf(Asymptote(G)));A: Point;FootPoint(L, OneOf(Asymptote(G))) = A;O: Origin;Area(TriangleOf(O, A, F)) = 2;e: Number;Eccentricity(G) = e;In(e, [sqrt(17), sqrt(65)])", "coords": {} }, { "index": 5596, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5596.png", "answer": "x^2 - y^2 / 3 = 1", "fact_expr": "H: Parabola;Expression(H) = (y^2 = 4*x);G: Hyperbola;Focus(H) = Vertex(G);Vertex(H) = Center(G);Eccentricity(G) = 2", "coords": {} }, { "index": 5597, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5597.png", "answer": "pm*1", "fact_expr": "G: Parabola;B: Point;A: Point;Expression(G) = (x^2 = 4*y);IsChordOf(LineSegmentOf(A, B),G);Length(LineSegmentOf(A, B)) = 8;WhenMin(Distance(MidPoint(LineSegmentOf(A,B)),xAxis))", "coords": {} }, { "index": 5598, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5598.png", "answer": "5/3", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Expression(Asymptote(G)) = (y = pm*x*4/3);e: Number;Eccentricity(G) = e", "coords": {} }, { "index": 5599, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5599.png", "answer": "2*sqrt(5)", "fact_expr": "C: Parabola;Expression(C) = (x^2 = 2*p*y);p: Number;l: Line;Directrix(C) = l;Distance(Focus(C), l) = 4;G: Circle;Expression(G) = (-6*x + x^2 + y^2 = 0)", "coords": {} }, { "index": 5600, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5600.png", "answer": "8/3", "fact_expr": "C: Parabola;G: Line;A: Point;B: Point;F: Point;E: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F, G);Intersection(G, C) = {A, B};VectorOf(A, F) = 3*VectorOf(F, B);MidPoint(LineSegmentOf(A, B)) = E", "coords": {} }, { "index": 5601, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5602, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5602.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(G) = (x^2/16 - y^2/9 = 1);Focus(H)=Vertex(G);InterReciprocal(Eccentricity(G),Eccentricity(H))", "coords": {} }, { "index": 5603, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5604, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/train/hyperbola/problem_5604.png", "answer": "3*x+pm*4*y=0", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Eccentricity(G) = 5/4", "coords": {} }, { "index": 5605, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5605.png", "answer": "1", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);P:Point;PointOnCurve(P,G)", "coords": {} }, { "index": 5606, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.5030499696731567, "b": 2.00545072555542, "orientation": "horizontal" }, "optimization": { "final_loss": 1.3176339734854992e-06, "converged": true, "iterations": 109 }, "output_path": "results/train/hyperbola/problem_5606.png", "answer": "3", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;G: Circle;P: Point;T: Point;F: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2 + y^2 = a^2);LeftFocus(C) = F;PointOnCurve(F, l);TangentPoint(l, G) = T;Intersection(l, RightPart(C)) = P;Eccentricity(C) = 5/3", "coords": {} }, { "index": 5607, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/train/hyperbola/problem_5607.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Hyperbola;a: Number;H: Point;a>0;Expression(G) = (-y^2/16 + x^2/a^2 = 1);Coordinate(H) = (3, 0);PointOnCurve(H, G)", "coords": { "H": [ 3.0, 0.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 5608, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5609, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5609.png", "answer": "y=pm*sqrt(3)*x", "fact_expr": "C: Hyperbola;b: Number;a: Number;O: Origin;F: Point;A: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;RightVertex(C)=A;Abs(LineSegmentOf(O, F)) = 2*Abs(LineSegmentOf(O, A))", "coords": {} }, { "index": 5610, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 1.0, 2.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_5610.png", "answer": "(0,1/8)", "fact_expr": "C: Parabola;m: Number;M: Circle;Expression(C) = (m*y + 4*x^2 = 0);Expression(M) = ((x - 1)^2 + (y - 2)^2 = 1);PointOnCurve(Center(M),C)", "coords": {} }, { "index": 5611, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5612, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5613, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5613.png", "answer": "sqrt(6)/3", "fact_expr": "C: Ellipse;b: Number;a: Number;G: Circle;F: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2 + y^2 = b^2);RightFocus(C) = F;L1:Line;L2:Line;TangentOfPoint(F,G)={L1,L2};IsPerpendicular(L1,L2)", "coords": {} }, { "index": 5614, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5614.png", "answer": "8/5", "fact_expr": "G: Ellipse;H: Line;A: Point;B: Point;Expression(G) = (x^2/4 + y^2 = 1);PointOnCurve(RightFocus(G), H);Slope(H) = 1;Intersection(H, G) = {A, B};IsChordOf(LineSegmentOf(A, B), G)", "coords": {} }, { "index": 5615, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5616, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.125, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5616.png", "answer": "(1/2,1/2)", "fact_expr": "G: Parabola;H: Line;Expression(G) = (y = 2*x^2);Expression(H) = (2*x - y - 4 = 0);PointOnCurve(P, G);WhenMin(Distance(P,H));P:Point", "coords": {} }, { "index": 5617, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5617.png", "answer": "(-oo, 0)", "fact_expr": "a: Number;b: Number;e1: Number;e2: Number;m: Number;a>b;b>0;C1: ConicSection;C2: ConicSection;Expression(C1) = (x^2/a^2+y^2/b^2=1);Expression(C2) = (x^2/a^2-y^2/b^2=1);Eccentricity(C1) = e1;Eccentricity(C2) = e2;m = lg(e1) + lg(e2)", "coords": {} }, { "index": 5618, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5618.png", "answer": "3", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;PointOnCurve(P, C);AngleOf(F1, P, F2) = ApplyUnit(60, degree);Area(TriangleOf(P, F1, F2)) = 3*sqrt(3)", "coords": {} }, { "index": 5619, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5619.png", "answer": "5/3", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;F1: Point;F2: Point;B: Point;A: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F2, l);Intersection(l, RightPart(C)) = {A, B};Abs(LineSegmentOf(F1, F2)) = Abs(LineSegmentOf(F1, B));VectorOf(A, F2) = 2*VectorOf(F2, B)", "coords": {} }, { "index": 5620, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5620.png", "answer": "16", "fact_expr": "F2: Point;F1: Point;LeftFocus(G) = F1;RightFocus(G) = F2;G: Hyperbola;Expression(G) = (x^2/4 - y^2/3 = 1);P: Point;PointOnCurve(P, LeftPart(G))", "coords": {} }, { "index": 5621, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5622, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5623, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5623.png", "answer": "2", "fact_expr": "G: Parabola;l1:Line;l2:Line;Expression(l1)=(3*x - 4*y - 9 = 0);Expression(l2)=(y = -1/4);Expression(G)=(y=x^2);PointOnCurve(P,G);P:Point", "coords": {} }, { "index": 5624, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5624.png", "answer": "pi/4", "fact_expr": "E: Parabola;p: Number;B: Point;A: Point;F: Point;p>0;Expression(E) = (y^2 = 2*(p*x));Focus(E) = F;PointOnCurve(A,E);PointOnCurve(B,E);Quadrant(A)=1;Quadrant(B)=1;Abs(LineSegmentOf(F, A)) = 3;Abs(LineSegmentOf(F, B)) = 5;Abs(LineSegmentOf(A, B)) = 2*sqrt(2)", "coords": {} }, { "index": 5625, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5625.png", "answer": "sqrt(2)/2", "fact_expr": "G: Parabola;P: Point;A: Point;F: Point;x1:Number;y1:Number;Expression(G) = (y^2 = 4*x);Coordinate(P) = (x1, y1);Coordinate(A) = (-1, 0);Focus(G) = F;PointOnCurve(P, G)", "coords": { "A": [ -1.0, 0.0 ] } }, { "index": 5626, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5627, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5627.png", "answer": "sqrt(2)+1", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;I: Point;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) =F1;RightFocus(C)=F2;PointOnCurve(P, C);Center(InscribedCircle(TriangleOf(P,F1,F2)))=I;DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = DotProduct(VectorOf(P, I), VectorOf(P, F1));IsPerpendicular(LineSegmentOf(P,F2),LineSegmentOf(F1,F2))", "coords": {} }, { "index": 5628, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5628.png", "answer": "6", "fact_expr": "C: Parabola;A: Point;B: Point;P: Point;Expression(C) = (y^2 = 4*x);IsChordOf(LineSegmentOf(A,B),C);MidPoint(LineSegmentOf(A,B))=P;Distance(P, yAxis) = 2", "coords": {} }, { "index": 5629, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5630, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5630.png", "answer": "15", "fact_expr": "G: Ellipse;P: Point;M: Point;F1: Point;F2: Point;Expression(G) = (x^2/25 + y^2/16 = 1);Coordinate(M) = (6, 4);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G)", "coords": { "M": [ 6.0, 4.0 ] } }, { "index": 5631, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5632, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5633, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5633.png", "answer": "6", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);F: Point;Focus(G) = F;l: Line;Directrix(G) = l;P: Point;PointOnCurve(P,G) = True;A: Point;IsPerpendicular(LineSegmentOf(P, A), l) = True;FootPoint(LineSegmentOf(P, A), l) = A;Slope(LineOf(A, F)) = -sqrt(2)", "coords": {} }, { "index": 5634, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5635, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5635.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;a: Number;H: Parabola;Expression(G) = (-y^2/8 + x^2/a^2 = 1);Expression(H) = (y^2 = 16*x);PointOnCurve(OneOf(Focus(G)), Directrix(H))", "coords": {} }, { "index": 5636, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5636.png", "answer": "22", "fact_expr": "G: Hyperbola;P: Point;Q: Point;F2: Point;F1:Point;Expression(G) = (x^2 - y^2 = 4);Abs(LineSegmentOf(P, Q)) = 7;RightFocus(G) = F2;LeftFocus(G)=F1;PointOnCurve(F1,LineSegmentOf(P,Q));IsChordOf(LineSegmentOf(P,Q),LeftPart(G))", "coords": {} }, { "index": 5637, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "y", "x_coef": 4.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5637.png", "answer": "6", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/9 = 1);F1:Point;F2:Point;P:Point;Focus(G)={F1,F2};PointOnCurve(P,G)", "coords": {} }, { "index": 5638, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5639, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5639.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2 = 1);H: Line;Expression(H) = (3*x - 4*y - 4 = 0)", "coords": {} }, { "index": 5640, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5640.png", "answer": "12", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;x1:Number;y1:Number;x2:Number;y2:Number;Expression(G) = (x^2 = 8*y);Focus(G)=F;Coordinate(A) = (x1,y1);Coordinate(B)=(x2,y2);PointOnCurve(F,H);Intersection(H,G) = {B, A};y1 + y2 = 8;F:Point", "coords": {} }, { "index": 5641, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5641.png", "answer": "sqrt(2)", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;B: Point;A: Point;P: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (2, 1);Slope(l)=2;Intersection(l,C) = {A, B};MidPoint(LineSegmentOf(A,B))=P", "coords": { "P": [ 2.0, 1.0 ] } }, { "index": 5642, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5642.png", "answer": "2*sqrt(21)/5", "fact_expr": "G: Ellipse;H: Line;Expression(G) = (x^2/4 + y^2/3 = 1);Expression(H) = (3*x + 4*y = 0);P:Point;PointOnCurve(P,G)", "coords": {} }, { "index": 5643, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5643.png", "answer": "2*x+2*y-3=0", "fact_expr": "l: Line;G: Ellipse;Expression(G) = (x^2/2 + y^2 = 1);Intersection(l, G) = {A, B};A: Point;B: Point;Coordinate(MidPoint(LineSegmentOf(A,B))) = (1,1/2)", "coords": {} }, { "index": 5644, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4506710767745972, "b": 2.0511727333068848, "orientation": "horizontal" }, "optimization": { "final_loss": 4.657696081267204e-06, "converged": true, "iterations": 116 }, "output_path": "results/train/hyperbola/problem_5644.png", "answer": "pm*2", "fact_expr": "G: Hyperbola;a: Real;Expression(G) = (x^2/2 - y^2/a^2 = 1);Expression(OneOf(Asymptote(G))) = (y = sqrt(2)*x)", "coords": {} }, { "index": 5645, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5645.png", "answer": "sqrt(2)+1", "fact_expr": "G: Hyperbola;a: Number;b: Number;a>0;b>0;F1: Point;F2: Point;H: Line;A: Point;B: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, H);IsPerpendicular(H, xAxis);Intersection(H, G) = {A, B};DotProduct(VectorOf(F1, A), VectorOf(F1, B)) = 0", "coords": {} }, { "index": 5646, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5646.png", "answer": "(a^2+b^2)/(a^2*b^2)", "fact_expr": "G: Ellipse;b: Number;a: Number;O: Origin;A: Point;B: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(A,G);PointOnCurve(B,G);Center(G)=O;IsPerpendicular(LineSegmentOf(A,O),LineSegmentOf(B,O))", "coords": {} }, { "index": 5647, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5647.png", "answer": "sqrt(2)", "fact_expr": "l: Line;Slope(l) = 2;C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;M: Point;Coordinate(M) = (2, 1);MidPoint(InterceptChord(l, C)) = M", "coords": { "M": [ 2.0, 1.0 ] } }, { "index": 5648, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5649, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5649.png", "answer": "y^2/64 - x^2/36 = 1", "fact_expr": "G: Hyperbola;H: Hyperbola;A: Point;Expression(G) = (x^2/9 - y^2/16 = 1);Coordinate(A) = (6, 8);PointOnCurve(A, H);Asymptote(G) = Asymptote(H)", "coords": { "A": [ 6.0, 8.0 ] } }, { "index": 5650, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5650.png", "answer": "2*sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;p: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 2*(p*x));Distance(LeftVertex(G),Focus(H))=4;Coordinate(Intersection(OneOf(Asymptote(G)),Directrix(H)))=(-2,-1)", "coords": {} }, { "index": 5651, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5651.png", "answer": "[4*sqrt(15)/15, 2*sqrt(3)/3)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;l: Line;OneOf(Asymptote(C)) = l;Slope(l)>0;E: Curve;Expression(E) = (sqrt(-x^2 + 4*x));A: Point;B: Point;D: Point;Negation(A=B);Negation(A=D);Negation(B=D);PointOnCurve(A, E);PointOnCurve(B, E);PointOnCurve(D, E);Distance(A, l) = 1;Distance(B, l) = 1;Distance(D, l) = 1", "coords": {} }, { "index": 5652, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5653, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5654, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5654.png", "answer": "[6-sqrt(2),6+sqrt(2)]", "fact_expr": "G: Ellipse;P: Point;F1: Point;M: Point;Expression(G) = (x^2/9 + y^2/5 = 1);Coordinate(M) = (1, 1);LeftFocus(G) = F1;PointOnCurve(P, G)", "coords": { "M": [ 1.0, 1.0 ] } }, { "index": 5655, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.0625, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5655.png", "answer": "y=-1/16", "fact_expr": "G: Parabola;Expression(G) = (x^2 = y/4)", "coords": {} }, { "index": 5656, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5656.png", "answer": "1/2", "fact_expr": "G: Parabola;H: Ellipse;m: Number;Expression(G) = (x^2 = 4*y);Expression(H) = (m*y^2 + x^2 = 1);OneOf(Focus(H)) = Focus(G)", "coords": {} }, { "index": 5657, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5657.png", "answer": "10", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;x1: Number;y1: Number;x2: Number;y2: Number;Expression(G) = (y^2 = 4*x);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);PointOnCurve(Focus(G), H);Intersection(H, G) = {A, B};x1 + x2 = 8", "coords": {} }, { "index": 5658, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 2.6457513110645907, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5658.png", "answer": "x^2/16+y^2/7=1", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(G) = (x^2/2 - y^2/7 = 1);Focus(G)=Focus(H);Length(MajorAxis(H))=8", "coords": {} }, { "index": 5659, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 8.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5659.png", "answer": "1", "fact_expr": "C: Ellipse;M: Circle;r: Number;P: Point;A: Point;B: Point;Expression(C) = (x^2/8 + y^2/2 = 1);0 < r;r < sqrt(2);Expression(M) = (y^2 + (x + sqrt(2))^2 = r^2);UpperVertex(C) = P;l1: Line;l2: Line;TangentOfPoint(P, M) = {l1, l2};Intersection(l1, C) = A;Intersection(l2, C) = B;Negation(A = P);Negation(B = P)", "coords": {} }, { "index": 5660, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5661, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5662, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5662.png", "answer": "4", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1)", "coords": {} }, { "index": 5663, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5663.png", "answer": "2*sqrt(15)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);L: Line;k: Number;Expression(L) = (y = k*x - 2);A: Point;B: Point;Intersection(L,G) = {A,B};XCoordinate(MidPoint(LineSegmentOf(A, B))) = 2", "coords": {} }, { "index": 5664, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5664.png", "answer": "[sqrt(5)/2,sqrt(5)]", "fact_expr": "l: Line;G: Hyperbola;b: Number;a: Number;c: Number;s: Number;H: Point;I: Point;J: Point;K: Point;e: Number;s>=(4/5)*c;a>1;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(H) = (a, 0);Coordinate(I) = (0, b);Coordinate(J) = (1, 0);Coordinate(K) = (-1, 0);FocalLength(G) = 2*c;PointOnCurve(H,l);PointOnCurve(I,l);Distance(J,l)+Distance(K,l) = s;Eccentricity(G) = e", "coords": { "J": [ 1.0, 0.0 ], "K": [ -1.0, 0.0 ] } }, { "index": 5665, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5665.png", "answer": "sqrt(5)", "fact_expr": "G: Ellipse;H: Line;A: Point;B: Point;Expression(G) = (x^2 + 4*y^2 = 4);Expression(H) = (x - 2*y + 2 = 0);Intersection(H, G) = {A, B}", "coords": {} }, { "index": 5666, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5667, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5667.png", "answer": "sqrt(10)/2", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;l: Line;PointOnCurve(F2, l);Intersection(l, RightPart(C)) = {P, Q};P: Point;Q: Point;Abs(LineSegmentOf(P, F1)) = 3*Abs(LineSegmentOf(P, F2));Abs(LineSegmentOf(P, Q)) = 4*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 5668, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5668.png", "answer": "sqrt(3)/2", "fact_expr": "C1: Ellipse;Expression(C1) = (x^2/a1^2+y^2/b1^2=1);a1: Number;b1: Number;a1>b1;b1>0;C2: Hyperbola;Expression(C2) = (x^2/a2^2-y^2/b2^2=1);a2: Number;b2: Number;a2>0;b2>0;F1: Point;F2: Point;Focus(C1) = {F1, F2};Focus(C2) = {F1, F2};P: Point;OneOf(Intersection(C1, C2)) = P;e1: Number;e2: Number;Eccentricity(C1) = e1;Eccentricity(C2) = e2;AngleOf(F1, P, F2) = pi/3", "coords": {} }, { "index": 5669, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5669.png", "answer": "x^2=-4*y", "fact_expr": "G: Parabola;P: Point;O: Origin;Coordinate(P) = (a,-2);Vertex(G) = O;PointOnCurve(Focus(G), yAxis);PointOnCurve(P, G);Distance(P, Focus(G)) = 3;a:Number", "coords": {} }, { "index": 5670, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5670.png", "answer": "y^2=16*x", "fact_expr": "C: Parabola;Expression(C)=(y^2=2*p*x);p:Number;p>0;M: Point;F: Point;Focus(C)=F;m: Number;Coordinate(M) = (2, m);Abs(LineSegmentOf(M,F))=6;PointOnCurve(M,C)", "coords": {} }, { "index": 5671, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_focus_mismatch", "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5671.png", "answer": "5", "fact_expr": "G: Ellipse;Expression(G) = (y^2/16 + x^2/a^2 = 1);a: Number;a>0;F: Point;OneOf(Focus(G)) = F;Coordinate(F) = (3, 0)", "coords": { "F": [ 3.0, 0.0 ] }, "validation_reasons": [ "ellipse_focus_mismatch" ] }, { "index": 5672, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5672.png", "answer": "7*sqrt(2)/8", "fact_expr": "G: Parabola;H: Line;P: Point;Expression(G) = (y = x^2);Expression(H) = (y = x - 2);PointOnCurve(P, G)", "coords": {} }, { "index": 5673, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5673.png", "answer": "(0,2]+[8,+oo)", "fact_expr": "G: Ellipse;m: Number;e:Number;Expression(G) = (x^2/4 + y^2/m = 1);Eccentricity(G)=e;Range(e)=[sqrt(2)/2,1)", "coords": {} }, { "index": 5674, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5674.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/16 + y^2/9 = 1);LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 5675, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5675.png", "answer": "10", "fact_expr": "l: Line;C: Parabola;F: Point;A: Point;B: Point;Expression(C) = (x^2 = 8*y);Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};XCoordinate(A)=2", "coords": {} }, { "index": 5676, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5677, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5677.png", "answer": "5/4", "fact_expr": "l: Line;G: Parabola;B: Point;A: Point;F: Point;Expression(G) = (y = x^2);Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B};Abs(LineSegmentOf(A, B)) = 3", "coords": {} }, { "index": 5678, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.6457513110645907, "major_axis": "x", "x_coef": 16.0, "y_coef": 7.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5678.png", "answer": "2*sqrt(14)", "fact_expr": "C: Ellipse;G: Line;P: Point;F: Point;Q: Point;O: Origin;Expression(C) = (x^2/16 + y^2/7 = 1);LeftFocus(C) = F;PointOnCurve(O, G);Intersection(G, C) = {P, Q};Abs(LineSegmentOf(P, F))*Abs(LineSegmentOf(Q, F)) = 9", "coords": {} }, { "index": 5679, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5679.png", "answer": "2*sqrt(5)/5", "fact_expr": "P: Point;Coordinate(P) = (x0, y0);x0: Number;y0: Number;Negation(x0=pm*a);E: Ellipse;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>b;b>0;PointOnCurve(P,E) = True;M: Point;N: Point;LeftVertex(E) = M;RightVertex(E) = N;Slope(LineOf(P,M)) * Slope(LineOf(P,N)) = -1/5", "coords": {} }, { "index": 5680, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5681, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5682, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5682.png", "answer": "2/3", "fact_expr": "G: Ellipse;E: Point;P: Point;Q: Point;Expression(G) = (x^2 + y^2/4 = 1);PointOnCurve(P,G);PointOnCurve(Q,G);Coordinate(E)=(0, 1);IsPerpendicular(LineSegmentOf(E, P), LineSegmentOf(E, Q))", "coords": { "E": [ 0.0, 1.0 ] } }, { "index": 5683, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5683.png", "answer": "[1, +\\infty)", "fact_expr": "H: Ellipse;Expression(H) = (y^2 + x^2/a^2 = 1);a: Number;a>0;F1: Point;F2: Point;G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;F: Point;Focus(G) = F;VectorOf(F1, F) = 3*VectorOf(F, F2);z: Number;z>=a^2-p^2;Focus(H)={F1,F2}", "coords": {} }, { "index": 5684, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5684.png", "answer": "sqrt(2)", "fact_expr": "l: Line;G: Hyperbola;b: Number;a: Number;P: Point;Q: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Slope(l)=sqrt(2)/2;Intersection(l, G) = {P, Q};F1:Point;F2:Point;Focus(G)={F1,F2};Projection(P,xAxis)=F1;Projection(Q,xAxis)=F2;Negation(P=Q)", "coords": {} }, { "index": 5685, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5685.png", "answer": "2", "fact_expr": "G: Parabola;p: Number;A: Point;p>0;Expression(G) = (y^2 = 2*p*x);Coordinate(A) = (x0, 2);x0:Number;PointOnCurve(A,G);Distance(A,Focus(G))=2*Distance(A,yAxis)", "coords": {} }, { "index": 5686, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5686.png", "answer": "(-2, 0)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2 = 1)", "coords": {} }, { "index": 5687, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5688, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5688.png", "answer": "sqrt(5)/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x + 2*y - 1 = 0);IsParallel(OneOf(Asymptote(G)), H)", "coords": {} }, { "index": 5689, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5690, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5690.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(G) = F;M: Point;IsPerpendicular(LineSegmentOf(M, F), xAxis);P: Point;MidPoint(LineSegmentOf(M, F)) = P;A: Point;B: Point;Vertex(G) = {A, B};WhenMax(AngleOf(A, P, B));PointOnCurve(M, G)", "coords": {} }, { "index": 5691, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5691.png", "answer": "2", "fact_expr": "C: Hyperbola;a: Number;b: Number;G: Circle;A: Point;O: Origin;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2 + y^2 = a^2);l1:Line;l2:Line;PointOnCurve(OneOf(Focus(C)),l1);PointOnCurve(OneOf(Focus(C)),l2);TangentOfPoint(OneOf(Focus(C)), G) = {l1, l2} ;TangentPoint(l1,G)=A;TangentPoint(l2,G)=B;AngleOf(A, O, B) = ApplyUnit(120,degree)", "coords": {} }, { "index": 5692, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "major_axis": "x", "x_coef": 3.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5692.png", "answer": "4*sqrt(3)", "fact_expr": "G: Ellipse;H: Line;A: Point;B: Point;F2: Point;F1: Point;Expression(G) = (x^2/3 + y^2 = 1);Focus(G) = {F1, F2};PointOnCurve(F1, H);Intersection(H, G) = {A, B}", "coords": {} }, { "index": 5693, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5693.png", "answer": "{5,3}", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (y^2/4 + x^2/m = 1);FocalLength(G) = 2", "coords": {} }, { "index": 5694, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -2.0, 0.0 ], "radius": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_5694.png", "answer": "{1, 4}", "fact_expr": "C1: Circle;C2: Circle;C3: Circle;Omega: Curve;r: Number;r > 0;Expression(C1) = (y^2 + (x - 2)^2 = 4);Expression(C2) = ((x + r)^2 + y^2 = r^2);IsOutTangent(C3, C1);IsOutTangent(C3, C2);Locus(Center(C3))=Omega;Eccentricity(Omega) = 3", "coords": {} }, { "index": 5695, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5695.png", "answer": "pm*sqrt(3)/2", "fact_expr": "l: Line;G: Hyperbola;b: Number;a: Number;H: Circle;M: Point;F2: Point;F1: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2 + y^2 = b^2);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, l);TangentPoint(l, H) = M;Abs(LineSegmentOf(M, F2)) = 2*Abs(LineSegmentOf(M, F1))", "coords": {} }, { "index": 5696, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5697, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5697.png", "answer": "[1,2]", "fact_expr": "G: Ellipse;O: Origin;P: Point;Expression(G) = (4*x^2 + y^2 = 4);PointOnCurve(P, G)", "coords": {} }, { "index": 5698, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5699, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "y", "x_coef": 4.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5699.png", "answer": "x^2+y^2=4", "fact_expr": "M: Point;PointOnCurve(M, G);G: Ellipse;Expression(G) = (x^2/4 + y^2/9 = 1);IsPerpendicular(LineSegmentOf(M, D), xAxis);D: Point;FootPoint(LineSegmentOf(M, D), xAxis) = D;P: Point;PointOnCurve(P, LineSegmentOf(M, D));Abs(LineSegmentOf(D, M))/Abs(LineSegmentOf(D, P)) = 3/2", "coords": {} }, { "index": 5700, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5700.png", "answer": "5/6", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 2*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 25/12;Abs(LineSegmentOf(A, F)) < Abs(LineSegmentOf(B, F))", "coords": {} }, { "index": 5701, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "y", "x_coef": 3.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5701.png", "answer": "6", "fact_expr": "G: Ellipse;M: Circle;N: Circle;l1: Line;l2: Line;P: Point;A: Point;B: Point;C: Point;D: Point;Expression(G) = (x^2/3 + y^2/4 = 1);Expression(M) = (x^2 + (y - 1)^2 = 1);Expression(N) = (x^2 + (y + 1)^2 = 1);PointOnCurve(M1, l1);PointOnCurve(N1, l2);Intersection(l1, M) = {A, B};Intersection(l2, N) = {C, D};PointOnCurve(P, G);M1: Point;N1: Point;Center(M) = M1;Center(N) = N1", "coords": {} }, { "index": 5702, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5702.png", "answer": "9/2", "fact_expr": "G: Parabola;p: Number;l: Line;O: Origin;A: Point;B: Point;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G)=F;PointOnCurve(F,l);Intersection(l,G) = {A, B};DotProduct(VectorOf(O, A), VectorOf(O, B)) = -3/4", "coords": {} }, { "index": 5703, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5703.png", "answer": "15", "fact_expr": "G: Ellipse;P: Point;M: Point;F1: Point;F2: Point;Expression(G) = (x^2/25 + y^2/16 = 1);Coordinate(M) = (6, 4);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G)", "coords": { "M": [ 6.0, 4.0 ] } }, { "index": 5704, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5704.png", "answer": "-8*sqrt(2)/5", "fact_expr": "G: Ellipse;b: Number;a: Number;C: Point;F: Point;A: Point;B: Point;D: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (-a, 0);Coordinate(B) = (0, b);Coordinate(C) = (0, -b);LeftFocus(G) = F;Intersection(LineOf(C,F),LineOf(A, B))=D;Eccentricity(G) = e;e=1/3;e:Number;Vertex(G)={A,B,C}", "coords": {} }, { "index": 5705, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5705.png", "answer": "x^2/18 - y^2/8 = 1", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/4 = 1);C:Hyperbola;Asymptote(G)=Asymptote(C)", "coords": {} }, { "index": 5706, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.23606797749979, "b": 2.0, "major_axis": "y", "x_coef": 4.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5706.png", "answer": "4*sqrt(3)/3", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 + y^2/5 = 1);PointOnCurve(P, G);Focus(G)={F1,F2};AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 5707, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.8284271247461903, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5707.png", "answer": "y=pm*(1/2)*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/8 - y^2/2 = 1)", "coords": {} }, { "index": 5708, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5708.png", "answer": "sqrt(3) - 1", "fact_expr": "G: Hyperbola;C: Ellipse;a: Number;b: Number;A: Point;F: Point;B: Point;Expression(G) = (x^2 - y^2/3 = 1);a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;Intersection(OneOf(Asymptote(G)), C) = {A, B};IsPerpendicular(LineSegmentOf(A, F), LineSegmentOf(B, F))", "coords": {} }, { "index": 5709, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5710, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5710.png", "answer": "7", "fact_expr": "G: Parabola;P: Point;A: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (6, 3);Focus(G) = F;PointOnCurve(P, G)", "coords": { "A": [ 6.0, 3.0 ] } }, { "index": 5711, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_5711.png", "answer": "sqrt(3)*x+pm*y=0", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;Eccentricity(G) = 2;OneOf(Focus(G)) = Focus(H);H: Parabola;Expression(H) = (y^2 = 16*x)", "coords": {} }, { "index": 5712, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5712.png", "answer": "(\\sqrt{5}/2,+\\infty)", "fact_expr": "P: Point;C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;PointOnCurve(P, C) = True;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;Diameter(InscribedCircle(TriangleOf(P, F1, F2))) = a", "coords": {} }, { "index": 5713, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.0, "b": 0.5, "major_axis": "y", "x_coef": 0.25, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5713.png", "answer": "[-sqrt(5)/2,sqrt(5)/2]", "fact_expr": "G: Ellipse;H: Line;m: Real;Expression(G) = (4*x^2 + y^2 = 1);Expression(H) = (y = m + x);IsIntersect(G, H)", "coords": {} }, { "index": 5714, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5714.png", "answer": "8", "fact_expr": "C: Parabola;p: Number;G: Circle;P: Point;O: Origin;A: Point;B: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(P) = (-2, 0);Center(G)=P;Radius(G)=5;Intersection(G, C) = {A, B};Abs(LineSegmentOf(O, A)) = sqrt(17)", "coords": { "P": [ -2.0, 0.0 ] } }, { "index": 5715, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5715.png", "answer": "4", "fact_expr": "G: Hyperbola;H: Parabola;p: Number;Expression(G) = (x^2 - y^2/3 = 1);Expression(H) = (y^2 = 2*(p*x));Focus(H) = RightFocus(G)", "coords": {} }, { "index": 5716, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5716.png", "answer": "3", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 - y^2/5 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, LeftPart(G));Abs(LineSegmentOf(P, F1)) + Abs(LineSegmentOf(P, F2)) = 8", "coords": {} }, { "index": 5717, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5717.png", "answer": "4*sqrt(2)", "fact_expr": "l: Line;C: Parabola;M: Point;O: Origin;A: Point;F: Point;B: Point;S1: Number;S2: Number;Expression(C) = (y^2 = 4*x);Coordinate(M) = (4, 0);Focus(C) = F;PointOnCurve(M, l);Intersection(l, C) = {A, B};Area(TriangleOf(O, A, F)) = S1;Area(TriangleOf(O, B, F)) = S2", "coords": { "M": [ 4.0, 0.0 ] } }, { "index": 5718, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5718.png", "answer": "5", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);G: Circle;A: Point;F: Point;M: Point;Coordinate(M) = (0, 2);PointOnCurve(A, C);Focus(C) = F;IsDiameter(LineSegmentOf(A, F), G) ;NumIntersection(G, yAxis) = 1;Intersection(G, yAxis) = M", "coords": { "M": [ 0.0, 2.0 ] } }, { "index": 5719, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5719.png", "answer": "sqrt(2)+1", "fact_expr": "C: Hyperbola;b: Number;a: Number;A: Point;B: Point;F1: Point;F2: Point;a>0;b>0;H: Line;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1,H);IsPerpendicular(H,xAxis);Intersection(C,H) = {A,B};Abs(LineSegmentOf(A,B)) = 2*Abs(LineSegmentOf(F1,F2))", "coords": {} }, { "index": 5720, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.449489742783178, "b": 1.0, "major_axis": "x", "x_coef": 6.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5720.png", "answer": "4*sqrt(6)", "fact_expr": "A: Point;B: Point;G: Ellipse;F2: Point;F1: Point;Expression(G) = (x^2/6 + y^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1,LineOf(A, B));Intersection(LineOf(A, B), G) = {A,B}", "coords": {} }, { "index": 5721, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5721.png", "answer": "pm*4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;M: Point;m: Number;Coordinate(M) = (1, m);PointOnCurve(M, G);Distance(M, Focus(G)) = 5", "coords": {} }, { "index": 5722, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.4641016151377544, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 12.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5722.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Ellipse;Expression(G) = (x^2/12 + y^2/3 = 1);F1: Point;LeftFocus(G) = F1;P: Point;PointOnCurve(P, G);M: Point;MidPoint(LineSegmentOf(P, F1)) = M;PointOnCurve(M, yAxis)", "coords": {} }, { "index": 5723, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5724, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5725, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5725.png", "answer": "2", "fact_expr": "C: Hyperbola;a: Number;b: Number;G: Circle;A: Point;O: Origin;B: Point;l1: Line;l2: Line;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2 + y^2 = a^2);TangentOfPoint(OneOf(Focus(C)), G) = {l1, l2};TangentPoint(l1, G)= A ;TangentPoint(l2, G)= B ;AngleOf(A, O, B) = ApplyUnit(120, degree)", "coords": {} }, { "index": 5726, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5727, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5727.png", "answer": "{-1, 1, -sqrt(5)/2, sqrt(5)/2}", "fact_expr": "G: Hyperbola;H: Line;k: Number;Expression(G) = (x^2 - y^2 = 4);Expression(H) = (y = k*x - 1);NumIntersection(H, G) = 1", "coords": {} }, { "index": 5728, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5729, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5729.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;b: Number;a: Number;F: Point;M: Point;E: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);OneOf(Focus(G)) = F;L: Line;PointOnCurve(F, L);IsPerpendicular(L, Asymptote(G));FootPoint(L, Asymptote(G)) = M;Intersection(OverlappingLine(LineSegmentOf(F, M)), yAxis) = E;LineSegmentOf(F, M) = LineSegmentOf(M, E)", "coords": {} }, { "index": 5730, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5731, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5731.png", "answer": "4+sqrt(5)", "fact_expr": "G: Ellipse;A: Point;B: Point;P: Point;Expression(G) = (x^2/4 + y^2/3 = 1);Coordinate(A) = (1, 0);Coordinate(B) = (1, 1);PointOnCurve(P, G)", "coords": { "A": [ 1.0, 0.0 ], "B": [ 1.0, 1.0 ] } }, { "index": 5732, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5732.png", "answer": "sqrt(3)", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;e: Number;Eccentricity(G) = e;In(e, [1/2, sqrt(2)/2]);H: Line;Expression(H) = (y = 1 - x);Intersection(H, G) = {M, N};M: Point;N: Point;O: Origin;DotProduct(VectorOf(O, M), VectorOf(O, N)) = 0", "coords": {} }, { "index": 5733, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5733.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;Q: Point;F1: Point;F2: Point;P: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(F1,H);L1:Line;L2:Line;Asymptote(G)={L1,L2};Intersection(H,L1)=P;Intersection(H,L2)=Q;MidPoint(LineSegmentOf(F1,Q))=P;IsPerpendicular(LineSegmentOf(Q, F1), LineSegmentOf(Q, F2))", "coords": {} }, { "index": 5734, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5734.png", "answer": "(1,2]", "fact_expr": "C: Hyperbola;G: Circle;x0: Number;y0: Number;H: Line;b: Number;a: Number;a>0;b>0;P: Point;Expression(G) = ((x - x0)^2 + (y - y0)^2 = 1);Expression(H) = (2*a - a*y + b*x = 0);Coordinate(P) = (x0, y0);Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, H);NumIntersection(G, RightPart(C))=0", "coords": {} }, { "index": 5735, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2631799883090584, "b": 2.5275303393962245, "orientation": "horizontal" }, "optimization": { "final_loss": 4.481942266905428e-06, "converged": true, "iterations": 95 }, "output_path": "results/train/hyperbola/problem_5735.png", "answer": "3", "fact_expr": "G: Hyperbola;a: Number;a>0;Expression(G) = (-y^2/(a + 3)^2 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G))) = (y = 2*x)", "coords": {} }, { "index": 5736, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5736.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/6 - y^2/m = 1);m: Number;Length(ImageinaryAxis(G)) = 6*sqrt(2)", "coords": {} }, { "index": 5737, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5738, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5738.png", "answer": "sqrt(6)/3", "fact_expr": "C1:Ellipse;C2:Circle;A: Point;O: Origin;B: Point;Expression(C1) = (x^2/36 + y^2/b^2 = 1);Expression(C2)=(x^2+y^2=9);PointOnCurve(P,C2);Intersection(TangentOfPoint(P,C2),C1)={A,B};AngleOf(A, O, B) = pi/2;P:Point;b: Number;0 < b;b < 6", "coords": {} }, { "index": 5739, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 5.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5739.png", "answer": "5*x + pm*4*y = 0", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/25 = 1)", "coords": {} }, { "index": 5740, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5740.png", "answer": "-8", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 2*(a*y));a: Number;Expression(Directrix(G)) = (y = 4)", "coords": {} }, { "index": 5741, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.046616818012331, "b": 3.9644919697671575, "orientation": "horizontal" }, "optimization": { "final_loss": 4.0501961192983754e-07, "converged": true, "iterations": 93 }, "output_path": "results/train/hyperbola/problem_5741.png", "answer": "13", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);P: Point;PointOnCurve(P, G);F1: Point;Coordinate(F1) = (-5, 0);Distance(P, F1) = 7;F2: Point;Coordinate(F2) = (5, 0)", "coords": { "F1": [ -5.0, 0.0 ], "F2": [ 5.0, 0.0 ] } }, { "index": 5742, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5742.png", "answer": "6", "fact_expr": "G: Parabola;F: Point;A: Point;B: Point;C: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(A, G);PointOnCurve(B, G);PointOnCurve(C, G);VectorOf(F, A) + VectorOf(F, B) + VectorOf(F, C) = 0", "coords": {} }, { "index": 5743, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5744, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5744.png", "answer": "sqrt(3)/3", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;LeftFocus(C) = F;O: Origin;PointOnCurve(O, G) = True;G: Line;Intersection(G, C) = {P, Q};P: Point;Q: Point;AngleOf(P, F, Q) = ApplyUnit(120, degree);Abs(LineSegmentOf(O, F)) = sqrt(3);Abs(LineSegmentOf(O, P)) = sqrt(7)", "coords": {} }, { "index": 5745, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 3.0, 2.0 ], "radius": 2.23606797749979 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_5745.png", "answer": "x^2/4 - y^2/12 = 1", "fact_expr": "C: Circle;Expression(C) = (-4*y - 6*x + x^2 + y^2 + 8 = 0);A: Point;B: Point;Intersection(C, axis) = {A, B};G: Hyperbola;OneOf(Focus(G)) = A;OneOf(Vertex(G)) = B", "coords": {} }, { "index": 5746, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5746.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F2: Point;F1: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(P,Directrix(G));IsPerpendicular(LineSegmentOf(P,F1),LineSegmentOf(P,F2));DotProduct(VectorOf(P,F1),VectorOf(P,F2))=4*a*b", "coords": {} }, { "index": 5747, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5747.png", "answer": "(x-5)^2+y^2=16", "fact_expr": "G: Hyperbola;H: Ellipse;C:Circle;Expression(G) = (x^2/9 - y^2/16 = 1);Expression(H) = (x^2/169 + y^2/144 = 1);Center(C)=RightFocus(H);IsTangent(Asymptote(G),C)", "coords": {} }, { "index": 5748, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5749, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5749.png", "answer": "[\\sqrt{17}, +\\infty)", "fact_expr": "P: Point;x1: Number;y1: Number;Coordinate(P) = (x1, y1);C: Ellipse;Expression(C) = (x^2/4 + y^2 = 1);PointOnCurve(P, C);a: Number;2*x1 - y1 + a >=0", "coords": {} }, { "index": 5750, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5751, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5752, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5753, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5754, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 3.872983346207417, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5754.png", "answer": "5", "fact_expr": "G: Hyperbola;P: Point;Expression(G) = (x^2 - y^2/15 = 1);PointOnCurve(P, G);F1:Point;F2:Point;OneOf(Focus(G))=F1;OneOf(Focus(G))=F2;Negation(F1=F2);Distance(P, F1) = 3", "coords": {} }, { "index": 5755, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5755.png", "answer": "-4/3", "fact_expr": "G: Ellipse;l: Line;A: Point;B: Point;M: Point;m:Number;Expression(G) = (x^2/4 + y^2/3 = 1);Slope(l)=1;Intersection(l,G)={A,B};Coordinate(M) = (m, 1);MidPoint(LineSegmentOf(A, B)) = M", "coords": {} }, { "index": 5756, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5756.png", "answer": "Slight", "fact_expr": "G: Parabola;Expression(G) = (y = x^2);P:Point;PointOnCurve(P,G);Inclination(TangentOnPoint(P,G))=pi/4", "coords": {} }, { "index": 5757, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5757.png", "answer": "(4,0)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1)", "coords": {} }, { "index": 5758, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5758.png", "answer": "7", "fact_expr": "C: Parabola;p: Number;G: Line;A: Point;F: Point;O: Origin;B: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;PointOnCurve(F, G);Intersection(G, C) = {A, B};Abs(LineSegmentOf(A, F)) = 8*Abs(LineSegmentOf(O, F))", "coords": {} }, { "index": 5759, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5759.png", "answer": "{(0,-sqrt(2)),(0,sqrt(2))}", "fact_expr": "G: Ellipse;P: Point;F1:Point;F2:Point;m:Number;Expression(G) = (x^2/4 + y^2/2 = 1);PointOnCurve(P, G);Distance(P,F1)*Distance(P,F2)=m;WhenMax(m);Focus(G)={F1,F2}", "coords": {} }, { "index": 5760, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5760.png", "answer": "pm*sqrt(2)/2", "fact_expr": "C: Parabola;B: Circle;B1:Point;O: Origin;F: Point;A: Point;Expression(C) = (x^2 = 4*y);Focus(C) = F;PointOnCurve(A, C);CircumCircle(TriangleOf(O,F,A))=B;Center(B)=B1;IsTangent(Directrix(C),B);H:Line;PointOnCurve(B,H);IsTangent(H,C)", "coords": {} }, { "index": 5761, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5761.png", "answer": "6", "fact_expr": "G: Parabola;F: Point;A: Point;B: Point;C: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(A,G);PointOnCurve(B,G);PointOnCurve(C,G);VectorOf(F, A) + VectorOf(F, B) + VectorOf(F, C) = 0", "coords": {} }, { "index": 5762, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5762.png", "answer": "x^2/2 - y^2/4 = 1", "fact_expr": "G:Hyperbola;H: Point;C:Hyperbola;Expression(G)=(x^2-y^2/2=1);Coordinate(H)=(2,2);Asymptote(G)=Asymptote(C);PointOnCurve(H,C)", "coords": { "H": [ 2.0, 2.0 ] } }, { "index": 5763, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5763.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;b: Number;a: Number;O: Origin;F: Point;L1:Line;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);OneOf(Focus(G))=F;PointOnCurve(F,L1);IsPerpendicular(OneOf(Asymptote(G)),L1);PointOnCurve(FootPoint(OneOf(Asymptote(G)),L1),PerpendicularBisector(LineSegmentOf(O,F)))", "coords": {} }, { "index": 5764, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "major_axis": "x", "x_coef": 3.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5764.png", "answer": "{(0, 1), (0, -1)}", "fact_expr": "G: Ellipse;F1: Point;A: Point;F2: Point;B: Point;Expression(G) = (x^2/3 + y^2 = 1);Focus(G) = {F1, F2};PointOnCurve(A, G);PointOnCurve(B, G);VectorOf(F1, A) = 5*VectorOf(F2, B)", "coords": {} }, { "index": 5765, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5765.png", "answer": "3/2", "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2 = 1);F: Point;RightFocus(C) = F;P: Point;PointOnCurve(P, LeftPart(C));A: Point;Coordinate(A) = (0, sqrt(2));WhenMin(Perimeter(TriangleOf(A, P, F)))", "coords": { "A": [ 0.0, 1.4142135623730951 ] } }, { "index": 5766, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5766.png", "answer": "x-y=0", "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;Expression(C) = (y^2 = 4*x);Coordinate(MidPoint(LineSegmentOf(A,B))) = (2, 2);Intersection(l, C) = {A, B}", "coords": {} }, { "index": 5767, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5767.png", "answer": "6", "fact_expr": "G: Hyperbola;Expression(G) = (4*x^2 - 9*y^2 = 36)", "coords": {} }, { "index": 5768, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5768.png", "answer": "[\\sqrt{2}/2,1)", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C)= F2;l: Line;RightDirectrix(C)=l;P: Point;PointOnCurve(P,l);AngleOf(F1,F2,P)=ApplyUnit(120,degree);ApplyUnit(30,degree)<=AngleOf(F1,P,F2);AngleOf(F1,P,F2)0;O: Origin;F: Point;Focus(C) = F;P: Point;PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P, F), xAxis);Q: Point;PointOnCurve(Q, xAxis);IsPerpendicular(LineSegmentOf(P, Q), LineSegmentOf(O, P));Abs(LineSegmentOf(F, Q)) = 6", "coords": {} }, { "index": 5770, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5771, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5771.png", "answer": "4", "fact_expr": "C: Parabola;A: Point;P: Point;E: Point;l:Line;x0:Number;y0:Number;Negation(x0=0);Expression(C) = (x^2 = 4*y);Coordinate(A) = (0, 3);Coordinate(P) = (x0, y0);PointOnCurve(P,C);PointOnCurve(P, l);TangentOfPoint(P,C)=l;IsPerpendicular(l,LineSegmentOf(P, A));Intersection(l, yAxis) = E", "coords": { "A": [ 0.0, 3.0 ] } }, { "index": 5772, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5772.png", "answer": "y = x", "fact_expr": "l: Line;C: Parabola;F: Point;A: Point;B: Point;O: Origin;Coordinate(MidPoint(LineSegmentOf(A, B))) = (2, 2);Coordinate(F) = (1, 0);Vertex(C) = O;Intersection(l, C) = {A, B};Focus(C) = F", "coords": { "F": [ 1.0, 0.0 ] } }, { "index": 5773, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5773.png", "answer": "0", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;M: Point;l: Line;C:Line;Expression(G) = (y^2 = 4*x);Expression(H) = (x - y - 1 = 0);Expression(l) = (x = -1);Intersection(H,G)={A,B};PointOnCurve(MidPoint(LineSegmentOf(A,B)),C);IsPerpendicular(C,l);FootPoint(C,l)=M", "coords": {} }, { "index": 5774, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5775, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5775.png", "answer": "13", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1);P: Point;PointOnCurve(P, RightPart(G));H: Circle;Expression(H) = (y^2 + (x + 5)^2 = 9);Z: Circle;Expression(Z) = (y^2 + (x - 5)^2 = 4);M: Point;PointOnCurve(M, H);N: Point;PointOnCurve(N, Z)", "coords": {} }, { "index": 5776, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 1.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5776.png", "answer": "4*sqrt(3)", "fact_expr": "C: Ellipse;Expression(C) = (x^2/9 + y^2 = 1);P: Point;PointOnCurve(P, C);E: Circle;Expression(E) = (x^2 + (y - 4)^2 = 3);Q: Point;PointOnCurve(Q, E)", "coords": {} }, { "index": 5777, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5777.png", "answer": "3", "fact_expr": "l: Line;C: Parabola;p: Number;A: Point;F: Point;D: Point;B: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;PointOnCurve(F, l);Intersection(l,C)={A,B};Intersection(l,Directrix(C))=D;Abs(LineSegmentOf(A, F)) = 6;VectorOf(D, B) = 2*VectorOf(B, F)", "coords": {} }, { "index": 5778, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5779, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5779.png", "answer": "12", "fact_expr": "l: Line;C: Parabola;p: Number;O: Origin;F: Point;A: Point;B: Point;p>0;Expression(C) = (x^2 = 2*(p*y));Focus(C) = F;Abs(LineSegmentOf(O, F)) = 3/4;PointOnCurve(F, l);Slope(l)=sqrt(3);Intersection(l, C) = {A,B}", "coords": {} }, { "index": 5780, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5780.png", "answer": "sqrt(3)/2", "fact_expr": "H: Parabola;Expression(H) = (y^2 = 4*x);G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1)", "coords": {} }, { "index": 5781, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5781.png", "answer": "sqrt(13)/2", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;G: Circle;F1: Point;P: Point;T: Point;F2: Point;a > b;b > 0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2 + y^2 = a^2);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, l);TangentPoint(l, G) = T;Intersection(l, RightPart(C)) = P;LineSegmentOf(F1, P) = 3*LineSegmentOf(F1, T)", "coords": {} }, { "index": 5782, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5782.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1)", "coords": {} }, { "index": 5783, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5783.png", "answer": "ApplyUnit(45,degree)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);M: Point;PointOnCurve(M, C) = True;F: Point;Focus(C) = F;O: Origin;Abs(LineSegmentOf(M,F)) = 2;K: Point;Intersection(Directrix(C),xAxis) = K", "coords": {} }, { "index": 5784, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5785, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5785.png", "answer": "y=(1/2)*x^2+1", "fact_expr": "G: Parabola;Expression(G) = (y = x^2/4);F: Point;Focus(G) = F;H: Line;PointOnCurve(F, H);A: Point;B: Point;Intersection(H, G) = {A, B}", "coords": {} }, { "index": 5786, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "major_axis": "x", "x_coef": 3.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5786.png", "answer": "1", "fact_expr": "Gamma: Ellipse;F1: Point;A: Point;F2: Point;B: Point;lambda: Real;Expression(Gamma) = (x^2/3 + y^2 = 1);LeftFocus(Gamma) = F1;RightFocus(Gamma) = F2;PointOnCurve(A, Gamma);PointOnCurve(B, Gamma);Negation(A = Vertex(Gamma));Negation(B = Vertex(Gamma));lambda*VectorOf(F2, B) + VectorOf(F1, A) = 0;lambda > 0", "coords": {} }, { "index": 5787, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5787.png", "answer": "sqrt(2)/2", "fact_expr": "G: Parabola;P: Point;F: Point;A: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (-1, 0);PointOnCurve(P, G);Focus(G) = F", "coords": { "A": [ -1.0, 0.0 ] } }, { "index": 5788, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5788.png", "answer": "(1/2,2/3)", "fact_expr": "C: Ellipse;b: Number;a: Number;G: Line;A:Point;B: Point;F: Point;e: Number;k:Number;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftVertex(C)=A;PointOnCurve(A,G);Slope(G) = k;1/30;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (-6*x + x^2 + y^2 + 5 = 0);Length(InterceptChord(Asymptote(G),H))=2", "coords": {} }, { "index": 5790, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 6.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5790.png", "answer": "16", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/36 - y^2/16 = 1);PointOnCurve(P,G);Focus(G)={F1,F2};IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2))", "coords": {} }, { "index": 5791, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5791.png", "answer": "(5, 0)", "fact_expr": "G: Ellipse;M: Point;F1: Point;F2: Point;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(M, G);LeftFocus(G) = F1;RightFocus(G) = F2;Abs(LineSegmentOf(M, F1)) = 4*Abs(LineSegmentOf(M, F2))", "coords": {} }, { "index": 5792, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 1.0, 0.0 ], "radius": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_5792.png", "answer": "4*x^2/25+4*y^2/21=1", "fact_expr": "C: Circle;Expression(C) = (y^2 + (x + 1)^2 = 25);A: Point;Coordinate(A) = (1, 0);Q: Point;PointOnCurve(Q, C) = True;Intersection(PerpendicularBisector(LineSegmentOf(A, Q)), LineSegmentOf(C, Q)) = M;M: Point", "coords": {} }, { "index": 5793, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5793.png", "answer": "1", "fact_expr": "l: Line;G: Parabola;A: Point;F: Point;B: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B}", "coords": {} }, { "index": 5794, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5794.png", "answer": "9/2", "fact_expr": "G: Parabola;A: Point;P: Point;Expression(G) = (y^2 = 2*x);Coordinate(A) = (7/2, 4);PointOnCurve(P, G);Distance(P, yAxis) = d1;Distance(P, A) = d2;d1:Number;d2:Number", "coords": { "A": [ 3.5, 4.0 ] } }, { "index": 5795, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5795.png", "answer": "0", "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2 = 1);F1: Point;F2: Point;Focus(C) = {F1,F2};M: Point;PointOnCurve(M,C);I: Point;Incenter(TriangleOf(M,F1,F2)) = I;YCoordinate(I) = 2-sqrt(3)", "coords": {} }, { "index": 5796, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5796.png", "answer": "[\\pi/4, 3\\pi/4]", "fact_expr": "l: Line;C: Parabola;A: Point;F: Point;B: Point;Expression(C) = (y^2 = 8*x);Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, F))*Abs(LineSegmentOf(B, F)) <= 32", "coords": {} }, { "index": 5797, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5797.png", "answer": "9", "fact_expr": "C: Parabola;G: Line;A: Point;D: Point;B: Point;F: Point;l: Line;Expression(C) = (y^2 = 8*x);Focus(C) = F;PointOnCurve(F, G);Intersection(G, C) = {A, B};Directrix(C) = l;Intersection(G, l) = D;MidPoint(LineSegmentOf(A, D)) = B;IsChordOf(LineSegmentOf(A, B), C)", "coords": {} }, { "index": 5798, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5798.png", "answer": "16", "fact_expr": "C: Ellipse;M: Point;F1: Point;F2: Point;Expression(C) = (x^2/16 + y^2/9 = 1);Focus(C) = {F1, F2};PointOnCurve(M, C)", "coords": {} }, { "index": 5799, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5799.png", "answer": "8", "fact_expr": "C: Parabola;G: Line;H: Point;F: Point;M: Point;N: Point;Expression(C) = (y^2 = 4*x);Coordinate(H) = (-2, 0);Focus(C)=F;PointOnCurve(H, G);Slope(G) = 2/3;Intersection(G, C) = {M, N}", "coords": { "H": [ -2.0, 0.0 ] } }, { "index": 5800, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5801, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5802, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5802.png", "answer": "10", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);l: Line;PointOnCurve(Focus(G), l);A: Point;B: Point;Intersection(l, G) = {A, B};XCoordinate(MidPoint(LineSegmentOf(A, B))) = 4", "coords": {} }, { "index": 5803, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5803.png", "answer": "x-y-5=0", "fact_expr": "G: Hyperbola;H: Line;A: Point;Expression(G) = (x^2/9 - y^2/16 = 1);Coordinate(A) = (4, -1);PointOnCurve(A,H);PointOnCurve(RightFocus(G),H)", "coords": { "A": [ 4.0, -1.0 ] } }, { "index": 5804, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5804.png", "answer": "x^2-4*y+4=0", "fact_expr": "G: Parabola;P: Point;F: Point;Expression(G) = (y = x^2/8);Focus(G) = F;PointOnCurve(P, G)", "coords": {} }, { "index": 5805, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5805.png", "answer": "2*sqrt(2)", "fact_expr": "G: Hyperbola;b: Number;a: Number;F1: Point;A: Point;P: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (0, 1);Coordinate(F1) = (2*sqrt(2), 0);RightFocus(G) = F1;PointOnCurve(P, LeftPart(G));Min(Perimeter(TriangleOf(A, P, F1))) = 8", "coords": { "A": [ 0.0, 1.0 ], "F1": [ 2.8284271247461903, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 5806, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5807, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5807.png", "answer": "20", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);F2: Point;F1: Point;Focus(G) = {F1,F2};L: Line;PointOnCurve(F1,L) = True;Intersection(L,G) = {P,Q};P: Point;Q: Point", "coords": {} }, { "index": 5808, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5809, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5809.png", "answer": "x^2 = 2*y - 1", "fact_expr": "C: Parabola;F: Point;Coordinate(F) = (0, 1);Focus(C) = F;Directrix(C) = xAxis", "coords": { "F": [ 0.0, 1.0 ] } }, { "index": 5810, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5810.png", "answer": "-1", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Eccentricity(C) = sqrt(3)/3;l: Line;Intersection(l, C) = {A, B};A: Point;B: Point;M: Point;Coordinate(M) = (3, 2);MidPoint(LineSegmentOf(A, B)) = M", "coords": { "M": [ 3.0, 2.0 ] } }, { "index": 5811, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5811.png", "answer": "(0, 2]+[8, +\\infty)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/m = 1);m: Number;Eccentricity(G) = e;In(e, [sqrt(2)/2, 1));e: Number", "coords": {} }, { "index": 5812, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5812.png", "answer": "{1,-1}", "fact_expr": "H: Ellipse;Expression(H) = (x^2/4 + y^2/a^2 = 1);G: Hyperbola;Expression(G) = (-y^2/2 + x^2/a^2 = 1);a: Number;Focus(H) = Focus(G)", "coords": {} }, { "index": 5813, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5814, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_5814.png", "answer": "(x-2)^2+y^2=4", "fact_expr": "G: Circle;A: Point;P: Point;M: Point;Expression(G) = (x^2 + y^2 = 9);Coordinate(A) = (6, 0);PointOnCurve(M, G);VectorOf(A, P) = 2*VectorOf(P, M)", "coords": {} }, { "index": 5815, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5815.png", "answer": "(0,\\sqrt{6})", "fact_expr": "l: Line;G: Hyperbola;F1: Point;P: Point;Q: Point;F2: Point;Expression(G) = (x^2/2 - y^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, l);Slope(l) = k;k:Number;k<-1;Intersection(l, RightPart(G)) = {P, Q};Area(TriangleOf(F1, P, Q)) = 4", "coords": {} }, { "index": 5816, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 1.9902384281158447, "b": 1.827926735623088, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 1.2790036214234405e-06, "converged": true, "iterations": 106 }, "output_path": "results/train/ellipse/problem_5816.png", "answer": "(\\sqrt{5}, +\\infty)", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(H) = (2, 1);PointOnCurve(H, G)", "coords": { "H": [ 2.0, 1.0 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 5817, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5817.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);H: Circle;IsDiameter(MinorAxis(G), H);IsTangent(LineSegmentOf(P, F1), H);TangentPoint(LineSegmentOf(P, F1), H) = MidPoint(LineSegmentOf(P, F1))", "coords": {} }, { "index": 5818, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.23606797749979, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5818.png", "answer": "pi/3", "fact_expr": "G: Hyperbola;F1: Point;P: Point;F2: Point;Expression(G) = (3*x^2 - 5*y^2 = 15);PointOnCurve(P, G);Focus(G) = {F1, F2};Area(TriangleOf(F1, P, F2)) = 3*sqrt(2)", "coords": {} }, { "index": 5819, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5819.png", "answer": "9", "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/9 = 1);P: Point;PointOnCurve(P, G);F1: Point;F2: Point;Focus(G) = {F1, F2};AngleOf(F1, P, F2) = ApplyUnit(90, degree)", "coords": {} }, { "index": 5820, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5820.png", "answer": "5", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;H: Point;Coordinate(H) = (p/2, 1);PointOnCurve(H, G);K: Point;PointOnCurve(K, G);YCoordinate(K) = 3", "coords": {} }, { "index": 5821, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5821.png", "answer": "(\\sqrt{6}, 4)", "fact_expr": "C: Hyperbola;b: Number;a: Number;F1: Point;F2: Point;P: Point;c: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);RightFocus(C) = F2;LeftFocus(C) = F1;Expression(Asymptote(C)) = (y = pm*(2*x));FocalLength(C) = 2*sqrt(10);PointOnCurve(P, C);Quadrant(P) = 1;Area(TriangleOf(P, F1, F2)) = 4*sqrt(10)", "coords": {} }, { "index": 5822, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5823, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5823.png", "answer": "8", "fact_expr": "l: Line;G: Parabola;M: Point;O: Origin;A: Point;B: Point;F: Point;t:Number;Expression(G) = (y^2 = 4*x);Coordinate(M) = (t, 2);Quadrant(M) = 1;Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B};VectorOf(O, M) = (VectorOf(O, A) + VectorOf(O, B))/2", "coords": {} }, { "index": 5824, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5824.png", "answer": "(1,2)", "fact_expr": "G: Hyperbola;a: Number;b: Number;H: Circle;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2 + (y + 2)^2 = 1);NumIntersection(Asymptote(G), H) = 0", "coords": {} }, { "index": 5825, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5825.png", "answer": "3", "fact_expr": "G: Curve;B: Point;A: Point;C: Point;a:Number;b:Number;Expression(G) = (x^2 = 4*y);Coordinate(C) = (-1, -1);L1:Line;L2:Line;TangentOfPoint(C,G)={L1,L2};TangentPoint(L1,G)=A;TangentPoint(L2,G)=B;Expression(LineOf(A,B)) = (y/b + x/a = 1)", "coords": { "C": [ -1.0, -1.0 ] } }, { "index": 5826, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5826.png", "answer": "sqrt(3)", "fact_expr": "G: Parabola;A: Point;B: Point;P: Point;Expression(G) = (y^2 = 2*x);Coordinate(A) = (-1, 0);Coordinate(B) = (1, 0);PointOnCurve(P,G);Abs(LineSegmentOf(P,A))=m*Abs(LineSegmentOf(P,B));m:Number", "coords": { "A": [ -1.0, 0.0 ], "B": [ 1.0, 0.0 ] } }, { "index": 5827, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5828, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5829, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5830, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5830.png", "answer": "-2", "fact_expr": "G: Hyperbola;P: Point;A1: Point;F2: Point;Expression(G) = (x^2 - y^2/3 = 1);LeftVertex(G) = A1;RightFocus(G)=F2;PointOnCurve(P,RightPart(G))", "coords": {} }, { "index": 5831, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5832, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.449489742783178, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 6.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5832.png", "answer": "-4", "fact_expr": "G: Parabola;p: Number;H: Ellipse;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (x^2/6 + y^2/2 = 1);Focus(G) = LeftFocus(H)", "coords": {} }, { "index": 5833, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5833.png", "answer": "m-a", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Ellipse;m: Number;n: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b + x^2/a = 1);m > n;n > 0;Expression(H) = (y^2/n + x^2/m = 1);Focus(G)={F1,F2};Focus(H)={F1,F2};Intersection(H,G)=P", "coords": {} }, { "index": 5834, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5834.png", "answer": "{y^2=-8*x,x^2=8*y}", "fact_expr": "G: Parabola;H: Line;O: Origin;Expression(H) = (x - y + 2 = 0);Vertex(G) = O;SymmetryAxis(G)=axis;PointOnCurve(Focus(G), H)", "coords": {} }, { "index": 5835, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5836, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5836.png", "answer": "2*sqrt(2)", "fact_expr": "C: Ellipse;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;G: Line;PointOnCurve(F2, G);A: Point;B: Point;Intersection(G, C) = {A, B};AngleOf(A, F1, B) = ApplyUnit(90, degree);M: Circle;IsTangent(M, OverlappingLine(LineSegmentOf(F1, A)));IsTangent(M, OverlappingLine(LineSegmentOf(F1, B)));IsTangent(M, LineOf(A, B));Expression(C)=(x^2/2+y^2=1)", "coords": {} }, { "index": 5837, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5838, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5838.png", "answer": "9", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Focus(G) = {F1, F2};Expression(G) = (x^2/4 + y^2/3 = 1);PointOnCurve(P, G)", "coords": {} }, { "index": 5839, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5839.png", "answer": "6", "fact_expr": "G: Ellipse;H: Circle;C:Circle;P: Point;Q: Point;R: Point;Expression(G) = (x^2/4 + y^2/3 = 1);Expression(H) = (y^2 + (x + 1)^2 = 1);Expression(C) = ((x-1)^2+y^2=1);PointOnCurve(P,G);PointOnCurve(Q,H);PointOnCurve(R,C)", "coords": {} }, { "index": 5840, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5841, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5841.png", "answer": "sqrt(5)/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;e: Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (2*x + y + 1 = 0);IsPerpendicular(OneOf(Asymptote(G)), H);Eccentricity(G) = e", "coords": {} }, { "index": 5842, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5842.png", "answer": "y=pm*(sqrt(3)/3)*x", "fact_expr": "G: Hyperbola;b: Number;H: Ellipse;F1: Point;F2: Point;Expression(G) = (x^2/3 - y^2/b^2 = 1);Expression(H) = (x^2/6 + y^2/2 = 1);Focus(H) = {F1, F2};Focus(G) = {F1, F2}", "coords": {} }, { "index": 5843, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5843.png", "answer": "3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;F: Point;Focus(G) = F;l: Line;Directrix(G) = l;A: Point;Coordinate(A) = (3, y0);y0: Number;PointOnCurve(A, G);L: Line;PointOnCurve(A, L);IsPerpendicular(L, l);FootPoint(L, l) = B;B: Point;C: Point;Coordinate(C) = ((7/2)*p, 0);Intersection(LineSegmentOf(A, F), LineSegmentOf(B, C)) = E;E: Point;Abs(LineSegmentOf(F, E)) = 2*Abs(LineSegmentOf(A, E))", "coords": {} }, { "index": 5844, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5844.png", "answer": "5/3", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;F1: Point;F2: Point;P: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;IsDiameter(LineSegmentOf(F1, F2), G);OneOf(Intersection(G,Asymptote(C)))=P;Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 5845, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 5.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5845.png", "answer": "8", "fact_expr": "E: Hyperbola;Expression(E) = (25*x^2 - 16*y^2 = 400)", "coords": {} }, { "index": 5846, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5846.png", "answer": "9", "fact_expr": "C: Hyperbola;b: Number;a: Number;E: Circle;M: Point;N: Point;F2: Point;F1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(E) = (x^2 + (y + sqrt(6))^2 = 1);LeftFocus(C) =F1;RightFocus(C)=F2;Length(RealAxis(C)) = 6;Expression(Asymptote(C)) = (y = pm*(x/3));PointOnCurve(M, LeftPart(C));PointOnCurve(N, E)", "coords": {} }, { "index": 5847, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5847.png", "answer": "{(2*x - 3*y = 0), (2*x + 3*y = 0)}", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/4 = 1)", "coords": {} }, { "index": 5848, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5849, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5849.png", "answer": "5*sqrt(3)", "fact_expr": "G: Parabola;M: Point;P: Point;Expression(G) = (y^2 = 10*x);Coordinate(M) = (10, 0);PointOnCurve(P, G)", "coords": { "M": [ 10.0, 0.0 ] } }, { "index": 5850, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2135674953460693, "b": 2.4270472526550293, "orientation": "horizontal" }, "optimization": { "final_loss": 4.453341261978494e-06, "converged": true, "iterations": 125 }, "output_path": "results/train/hyperbola/problem_5850.png", "answer": "y=pm*(1/2)", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = sqrt(5)/2", "coords": {} }, { "index": 5851, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5851.png", "answer": "8", "fact_expr": "G: Ellipse;H: Line;F2: Point;A: Point;B: Point;F1: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(G) ={F1,F2} ;PointOnCurve(F1, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(F2, A)) + Abs(LineSegmentOf(F2, B)) = 12", "coords": {} }, { "index": 5852, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5852.png", "answer": "12+8*sqrt(3)", "fact_expr": "C: Hyperbola;b: Number;A: Point;F1: Point;F2: Point;Expression(C) = (x^2/4 - y^2/b = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(A, C);IsPerpendicular(LineSegmentOf(A, F1), LineSegmentOf(A, F2));AngleOf(A, F1, F2) = ApplyUnit(30, degree)", "coords": {} }, { "index": 5853, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 8.0, "b": 6.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5853.png", "answer": "32/5", "fact_expr": "G: Hyperbola;P: Point;Expression(G) = (x^2/64 - y^2/36 = 1);PointOnCurve(P, G);Distance(P, RightFocus(G)) = 8", "coords": {} }, { "index": 5854, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5854.png", "answer": "1", "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;P: Point;E: Point;M: Point;F: Point;Q: Point;Expression(C) = (x^2 = 8*y);Coordinate(P) = (0, -1);Coordinate(E) = (3, 0);Focus(C) = F;PointOnCurve(P,l);Slope(l) = k;Intersection(l, C) = {A, B};MidPoint(LineSegmentOf(A,B))=Q;Distance(Q, xAxis) = 3;PointOnCurve(M, l);k:Number;k>0", "coords": { "P": [ 0.0, -1.0 ], "E": [ 3.0, 0.0 ] } }, { "index": 5855, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5856, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5856.png", "answer": "y^2=-8*x", "fact_expr": "H: Parabola;Expression(H) = (y^2 = 2*(p*x));p: Number;G: Hyperbola;Expression(G) = (x^2/3 - y^2 = 1);Focus(H) = LeftFocus(G)", "coords": {} }, { "index": 5857, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5858, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5858.png", "answer": "3*sqrt(2)/4", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;A: Point;F: Point;B: Point;p>0;Coordinate(A) = (0, 2);Focus(C) = F;PointOnCurve(B, C);MidPoint(LineSegmentOf(A,F))=B", "coords": { "A": [ 0.0, 2.0 ] } }, { "index": 5859, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5859.png", "answer": "3", "fact_expr": "C: Parabola;A: Point;B: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(A, C);PointOnCurve(B, C);Abs(LineSegmentOf(A, F)) = 2;Abs(LineSegmentOf(B, F)) = 6;IsChordOf(LineSegmentOf(A, B), C)", "coords": {} }, { "index": 5860, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5861, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5861.png", "answer": "y^2/64 - x^2/36 = 1", "fact_expr": "G: Hyperbola;C:Hyperbola;A: Point;Expression(G) = (x^2/9 - y^2/16 = 1);Coordinate(A) = (6, 8*sqrt(2));Asymptote(G) = Asymptote(C);PointOnCurve(A, C)", "coords": { "A": [ 6.0, 11.313708498984761 ] } }, { "index": 5862, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5862.png", "answer": "y = pm*sqrt(3)*x", "fact_expr": "G: Hyperbola;b: Number;H: Point;Expression(G) = (x^2/4 - y^2/b^2 = 1);Coordinate(H) = (4, 0);RightFocus(G) = H", "coords": { "H": [ 4.0, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 5863, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5864, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5864.png", "answer": "sqrt(5)/2", "fact_expr": "C1: Parabola;Expression(C1) = (x^2 = 2*p*y);p: Real;p>0;C2: Hyperbola;Expression(C2) = (y^2/9 - x^2/16 = 1);Distance(Focus(C1), Asymptote(C2)) = sqrt(5)/5", "coords": {} }, { "index": 5865, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5865.png", "answer": "(12 - sqrt(41))/5", "fact_expr": "C: Ellipse;G: Line;P: Point;Expression(C) = (x^2/2 + y^2 = 1);Expression(G) = (x/3 + y/4 = 1);PointOnCurve(P, C)", "coords": {} }, { "index": 5866, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5866.png", "answer": "(1/2, pm*sqrt(2))", "fact_expr": "G: Parabola;O: Origin;Vertex(G) = O;F: Point;Coordinate(F) = (1, 0);Focus(G) = F;H: Line;PointOnCurve(Focus(G), H);A: Point;B: Point;Intersection(H, G) = {A, B};M: Point;MidPoint(LineSegmentOf(A, B)) = M;P: Point;Z: Line;PointOnCurve(M, Z);IsPerpendicular(Z, Directrix(G));Intersection(Z, G) = P;Abs(LineSegmentOf(A, B)) = 6", "coords": { "F": [ 1.0, 0.0 ] } }, { "index": 5867, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5867.png", "answer": "1/2", "fact_expr": "G: Hyperbola;P: Point;F: Point;O: Origin;M: Point;Expression(G) = (x^2 - y^2/4 = 1);RightFocus(G) = F;H:Line;PointOnCurve(F,H);IsParallel(H,OneOf(Asymptote(G)));Intersection(H,G)=P;PointOnCurve(M,LineOf(P,F));DotProduct(VectorOf(O, M), VectorOf(P, F)) = 0", "coords": {} }, { "index": 5868, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5869, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5869.png", "answer": "x-2*y-4=0", "fact_expr": "G: Ellipse;M:Point;A: Point;B: Point;Expression(G) = (x^2/16 + y^2/4 = 1);Coordinate(M) = (2, 1);MidPoint(LineSegmentOf(A,B))=M;IsChordOf(LineSegmentOf(A,B),G)", "coords": { "M": [ 2.0, 1.0 ] } }, { "index": 5870, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5870.png", "answer": "pm*2*sqrt(2)/3", "fact_expr": "G: Parabola;H: Line;F: Point;A: Point;B: Point;k:Number;Expression(G) = (y^2 = 8*x);k>0;Expression(H) = (y = k*(x + 2));Intersection(H, G) = {A, B};Focus(G) = F;LineSegmentOf(F, A) = 2*LineSegmentOf(F, B)", "coords": {} }, { "index": 5871, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "y", "x_coef": 9.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5871.png", "answer": "3", "fact_expr": "l: Line;C: Ellipse;P: Point;M: Point;N: Point;Expression(C) = (x^2/9 + y^2/16 = 1);PointOnCurve(P, C);Expression(l)=(x-y+1=0);Intersection(l,xAxis)=M;Intersection(l,yAxis)=N", "coords": {} }, { "index": 5872, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "y", "x_coef": 4.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5872.png", "answer": "(-3*sqrt(2),3*sqrt(2))", "fact_expr": "l: Line;C: Ellipse;A: Point;B: Point;m: Real;Expression(C) = (x^2/4 + y^2/9 = 1);Expression(l)=(y = m + (3/2)*x);Intersection(C, l) = {A, B}", "coords": {} }, { "index": 5873, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5873.png", "answer": "(\\sqrt{7}-2,1)", "fact_expr": "G: Ellipse;b: Number;a: Number;a > b;b > 0;P:Point;e:Number;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);XCoordinate(P)=a/3;Distance(P,LeftFocus(G))>Distance(P,RightDirectrix(G));Eccentricity(G)=e;PointOnCurve(P,G)", "coords": {} }, { "index": 5874, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 5.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5874.png", "answer": "{2, 22}", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/25 - y^2/9 = 1);Focus(G)={F1,F2};PointOnCurve(P,G);Abs(LineSegmentOf(P,F1))=12", "coords": {} }, { "index": 5875, "success": false, "error": "Circle from diameter: missing point coordinates" }, { "index": 5876, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5877, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5878, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5879, "success": false, "error": "Circle from diameter: missing point coordinates" }, { "index": 5880, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5880.png", "answer": "y=pm*(2/3)*x", "fact_expr": "G: Hyperbola;a: Number;a>0;Expression(G) = (-x^2/9 + y^2/a^2 = 1);Coordinate(OneOf(Focus(G))) = (0, sqrt(13))", "coords": {} }, { "index": 5881, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5882, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5882.png", "answer": "4*sqrt(13)/13", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Ellipse;F1: Point;P: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2/25 + y^2/9 = 1);Focus(G)={F1,F2};Focus(H)={F1,F2};OneOf(Intersection(G,H))=P;AngleOf(F1, P, F2) = pi/3", "coords": {} }, { "index": 5883, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5883.png", "answer": "0", "fact_expr": "G: Parabola;A: Point;B: Point;P: Point;F:Point;Expression(G) = (y^2 = 4*x);PointOnCurve(P, Directrix(G));IsChordOf(LineSegmentOf(A,B),G);PointOnCurve(F,LineSegmentOf(A,B));Focus(G) = F", "coords": {} }, { "index": 5884, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5884.png", "answer": "x-y-1=0", "fact_expr": "l: Line;C: Parabola;M: Point;A: Point;B: Point;F: Point;Expression(C) = (y^2 = 4*x);Coordinate(M) = (-1, 2);Focus(C)=F;PointOnCurve(F, l);Intersection(l, C) = {A, B};AngleOf(A, M, B) = ApplyUnit(90, degree)", "coords": { "M": [ -1.0, 2.0 ] } }, { "index": 5885, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5885.png", "answer": "[\\sqrt{2}/2,1)", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;H: Circle;Expression(H) = (x^2 + y^2 = a^2 + b^2);L:Line;P: Point;A: Point;B: Point;a > b;b > 0;PointOnCurve(P, G);Intersection(L,H)={A,B};Abs(LineSegmentOf(P, A))*Abs(LineSegmentOf(P, B)) = (a^2 - b^2);PointOnCurve(P,L)", "coords": {} }, { "index": 5886, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 3.0, "major_axis": "x", "x_coef": 36.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5886.png", "answer": "24", "fact_expr": "G: Ellipse;H: Line;A: Point;F1: Point;B: Point;F2: Point;Expression(G) = (x^2/36 + y^2/9 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, H);Intersection(H, G) = {A, B}", "coords": {} }, { "index": 5887, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5887.png", "answer": "(\\sqrt{5}/2,+\\infty)", "fact_expr": "H: Line;Expression(H) = (y = x/2);G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;NumIntersection(H, G) = 2", "coords": {} }, { "index": 5888, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5888.png", "answer": "y=pm*((sqrt(3)/2)*x)", "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/4 + y^2/3 = 1)", "coords": {} }, { "index": 5889, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5889.png", "answer": "3/4", "fact_expr": "C: Ellipse;A: Point;F1: Point;F2: Point;O: Origin;B: Point;Expression(C) = (x^2/4 + y^2/3 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(A, C);IsPerpendicular(LineSegmentOf(A, F2), LineSegmentOf(F1, F2));Intersection(LineSegmentOf(A, F1), yAxis) = B", "coords": {} }, { "index": 5890, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5890.png", "answer": "5", "fact_expr": "G: Parabola;H: Line;B: Point;A: Point;M: Point;x1:Number;x2:Number;y1:Number;y2:Number;Expression(G) = (y^2 = 8*x);PointOnCurve(Focus(G),H);Coordinate(A)=(x1,y1);Coordinate(B)=(x2,y2);Intersection(H, G) = {A,B};x1+x2=6;MidPoint(LineSegmentOf(A,B)) = M", "coords": {} }, { "index": 5891, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5891.png", "answer": "1", "fact_expr": "G: Hyperbola;H: Circle;F: Point;P: Point;M: Point;N: Point;O: Origin;Expression(G) = (x^2/16 - y^2/9 = 1);Expression(H) = (x^2 + y^2 = 16);LeftFocus(G) = F;PointOnCurve(P, RightPart(G));TangentPoint(LineSegmentOf(P,F),H)=N;MidPoint(LineSegmentOf(P,F))=M", "coords": {} }, { "index": 5892, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5893, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5894, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5894.png", "answer": "(1,-1)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = x);P: Point;PointOnCurve(P, G);A: Point;Coordinate(A) = (3, -1);WhenMin(Distance(P, A) + Distance(P, Focus(G)))", "coords": { "A": [ 3.0, -1.0 ] } }, { "index": 5895, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5895.png", "answer": "10", "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/9 + y^2/5 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C)", "coords": {} }, { "index": 5896, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5896.png", "answer": "pm*2*sqrt(2)", "fact_expr": "C: Parabola;G: Line;M: Point;A: Point;B: Point;l: Line;k: Number;Expression(C) = (y^2 = 4*x);Coordinate(M) = (1, 0);Directrix(C) = l;PointOnCurve(M,G);Slope(G)=k;Intersection(l,G)=A;OneOf(Intersection(G, C)) = B;VectorOf(A, M) = 2*VectorOf(M, B)", "coords": { "M": [ 1.0, 0.0 ] } }, { "index": 5897, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5897.png", "answer": "12", "fact_expr": "G: Ellipse;H: Line;A: Point;B: Point;F2: Point;F1: Point;Expression(G) = (x^2/9 + y^2/4 = 1);Focus(G) = {F1, F2};PointOnCurve(F1, H);Intersection(H, G) = {A, B}", "coords": {} }, { "index": 5898, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5898.png", "answer": "sqrt(6)/2", "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/3 + y^2/6 = 1)", "coords": {} }, { "index": 5899, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5899.png", "answer": "2*x^2-2*y^2=1", "fact_expr": "G: Hyperbola;H: Ellipse;O: Origin;Expression(H) = (x^2/2 + y^2 = 1);Center(G) = O;Focus(G) = Focus(H);InterReciprocal(Eccentricity(G), Eccentricity(H))", "coords": {} }, { "index": 5900, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5901, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5902, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5902.png", "answer": "6", "fact_expr": "G: Ellipse;H: Line;A: Point;F2: Point;B: Point;F1: Point;Expression(G) = (x^2/25 + y^2/16 = 1);Focus(G)={F1,F2};PointOnCurve(F1, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, F2)) - Abs(LineSegmentOf(B, F1)) = 4", "coords": {} }, { "index": 5903, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 4.898979485566356, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5903.png", "answer": "24", "fact_expr": "F1: Point;F2: Point;Focus(G) = {F1, F2};G: Hyperbola;Expression(G) = (x^2 - y^2/24 = 1);P: Point;PointOnCurve(P, G) = True;Abs(LineSegmentOf(P, F1)) + Abs(LineSegmentOf(P, F2)) = 14", "coords": {} }, { "index": 5904, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5904.png", "answer": "4*sqrt(3)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;l: Line;Directrix(G) = l;P: Point;PointOnCurve(P, G);A: Point;IsPerpendicular(LineSegmentOf(P, A), l);FootPoint(LineSegmentOf(P, A), l) = A;Slope(LineOf(A, F)) = -sqrt(3)", "coords": {} }, { "index": 5905, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5905.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;LeftFocus(G) = F;H: Circle;Expression(H) = (x^2 + y^2 = a^2);Z: Line;TangentOfPoint(F, H) = Z;E: Point;TangentPoint(Z, H) = E;P: Point;Intersection(OverlappingLine(LineSegmentOf(F, E)), G) = P;O: Origin;VectorOf(O, E) = (1/2)*(VectorOf(O, F) + VectorOf(O, P))", "coords": {} }, { "index": 5906, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5907, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5908, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5908.png", "answer": "17", "fact_expr": "G: Hyperbola;P: Point;F1:Point;F2:Point;Expression(G) = (-x^2/16 + y^2/64 = 1);PointOnCurve(P, G);OneOf(Focus(G))=F1;OneOf(Focus(G))=F2;Negation(F1=F2);Distance(P,F1) = 1", "coords": {} }, { "index": 5909, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5909.png", "answer": "3/2", "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2/b^2 = 1);b: Number;0 < b;b < 2;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) = 3;AngleOf(F1, P, F2) = pi/3", "coords": {} }, { "index": 5910, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5910.png", "answer": "sqrt(3)/3", "fact_expr": "C: Ellipse;Expression(C) = (x^2/b^2 + y^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;UpperFocus(C) = F1;LowerFocus(C) = F2;B1: Point;B2: Point;LeftVertex(C) = B1;RightVertex(C) = B2;P: Point;PointOnCurve(P, C) ;VectorOf(F2, P) = VectorOf(F1, B1)/2", "coords": {} }, { "index": 5911, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5912, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5912.png", "answer": "5", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;G: Line;PointOnCurve(F, G);A: Point;B: Point;Intersection(G, C) = {A, B};M: Point;Coordinate(M) = (-1, 1);IsPerpendicular(LineSegmentOf(M, A), LineSegmentOf(M, B));IsChordOf(LineSegmentOf(A, B), C)", "coords": { "M": [ -1.0, 1.0 ] } }, { "index": 5913, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 3.0, "major_axis": "x", "x_coef": 36.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5913.png", "answer": "Answer missing", "fact_expr": "l: Line;G: Ellipse;Expression(G) = (x^2/36 + y^2/9 = 1);A: Point;B: Point;Intersection(l, G) = {A, B};I: Point;Coordinate(I) = (4, 2);MidPoint(LineSegmentOf(A, B)) = I", "coords": { "I": [ 4.0, 2.0 ] } }, { "index": 5914, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5914.png", "answer": "3", "fact_expr": "G: Parabola;l1:Line;l2:Line;P:Point;Expression(G) = (y^2 = 4*x);Expression(l1)=(4*x-3*y+11=0);Expression(l2)=(x=-1);PointOnCurve(P,G)", "coords": {} }, { "index": 5915, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5915.png", "answer": "x^2=8*y", "fact_expr": "C: Parabola;Expression(C) = (x^2 = 2*(p*y));p: Number;p>0;P: Point;PointOnCurve(P, C);Distance(P, Focus(C)) = 8;Distance(P, xAxis) = 6", "coords": {} }, { "index": 5916, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5916.png", "answer": "4", "fact_expr": "G: Parabola;H: Circle;P: Point;Q:Point;Expression(G) = (y^2 = 4*x);Expression(H) = ((x + 3)^2 + (y - 3)^2 = 1);PointOnCurve(P, G);Distance(P,Directrix(G)) = d1;PointOnCurve(Q, H);Distance(P,Q)=d2;d1:Number;d2:Number", "coords": {} }, { "index": 5917, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5918, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.8284271247461903, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5918.png", "answer": "y^2/8 - x^2/4 = 1", "fact_expr": "C: Hyperbola;G: Point;C1:Hyperbola;Expression(C) = (x^2/4 - y^2/8 = 1);Coordinate(G) = (2, 4);PointOnCurve(G, C1);Asymptote(C1)=Asymptote(C)", "coords": { "G": [ 2.0, 4.0 ] } }, { "index": 5919, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5919.png", "answer": "5+sqrt(2)", "fact_expr": "N: Parabola;A: Point;B: Point;P: Point;Expression(N) = (x^2 = 8*y);Coordinate(A) = (-1, 3);Coordinate(B) = (0, 2);PointOnCurve(P, N)", "coords": { "A": [ -1.0, 3.0 ], "B": [ 0.0, 2.0 ] } }, { "index": 5920, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5920.png", "answer": "3", "fact_expr": "C: Parabola;l1: Line;F: Point;l2: Line;A: Point;Expression(C) = (x^2 = 4*y);Expression(l1)=(sqrt(3)*x + y + 5 = 0);Focus(C) = F;Directrix(C) = l2;PointOnCurve(A, C);d1:Number;d2:Number;Distance(A,l1)=d1;Distance(A,l2)=d2", "coords": {} }, { "index": 5921, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5921.png", "answer": "2*sqrt(2)", "fact_expr": "G: Line;Expression(G) = (y = k*x + m);k: Number;k>0;m: Number;C: Parabola;Expression(C) = (y^2 = 4*x);M: Point;Intersection(G, C) = M;N: Point;Intersection(G, Directrix(C)) = N;F: Point;Focus(C) = F;3*VectorOf(F, M) = VectorOf(M, N)", "coords": {} }, { "index": 5922, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5923, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5923.png", "answer": "pm*2*sqrt(2)", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;A: Point;F1: Point;F2: Point;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F2, l);Intersection(l, RightPart(C)) = {A, B};Radius(InscribedCircle(TriangleOf(A,F1,F2)))=r1;Radius(InscribedCircle(TriangleOf(B,F1,F2)))=r2;r1:Number;r2:Number;r1 = 2*r2", "coords": {} }, { "index": 5924, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5925, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5925.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(C) = F;A: Point;B: Point;Quadrant(A) = 1;Quadrant(B) = 1;PointOnCurve(A, C);PointOnCurve(B, C);O: Origin;AngleOf(A, F, O) = AngleOf(A, F, B);AngleOf(A, F, B) = pi/3;Abs(LineSegmentOf(A, B)) = 2*sqrt(7)", "coords": {} }, { "index": 5926, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5927, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5927.png", "answer": "sqrt(2)", "fact_expr": "l: Line;C: Parabola;Expression(C) = (y^2 = x);Intersection(l, C) = {A, B};A: Point;B: Point;l1: Line;Expression(l1) = (y = 1);PointOnCurve(MidPoint(LineSegmentOf(A,B)),l1) = True;DotProduct(VectorOf(O, A), VectorOf(O, B)) = 0;O: Origin", "coords": {} }, { "index": 5928, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5928.png", "answer": "8", "fact_expr": "G: Hyperbola;A: Point;B: Point;Expression(G) = (x^2/4 - y^2/5 = 1);PointOnCurve(A, RightPart(G));PointOnCurve(B, RightPart(G));Distance(MidPoint(LineSegmentOf(A,B)),yAxis)=4", "coords": {} }, { "index": 5929, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5929.png", "answer": "(\\sqrt{2}-1,1)", "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;e: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) =F1;RightFocus(G)=F2;Abs(LineSegmentOf(P,F1))=e*Abs(LineSegmentOf(P,F2));Eccentricity(G) = e;PointOnCurve(P,G)", "coords": {} }, { "index": 5930, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5930.png", "answer": "8*sqrt(3)/5", "fact_expr": "l: Line;C: Ellipse;a: Number;F: Point;A: Point;M: Point;a>sqrt(3);Expression(C) = (y^2/3 + x^2/a^2 = 1);LowerVertex(C) = A;LeftFocus(C) = F;PointOnCurve(A, l);Inclination(l) = ApplyUnit(60, degree);Intersection(l, C) = M;Negation(M=A);Perimeter(TriangleOf(F, A, M)) = 4*a", "coords": {} }, { "index": 5931, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5931.png", "answer": "pm*sqrt(2)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, C) = {A, B};A1: Point;B1: Point;Projection(A, yAxis) = A1;Projection(B, yAxis) = B1;Abs(LineSegmentOf(A, B)) = (3/2)*(Abs(LineSegmentOf(A, A1)) + Abs(LineSegmentOf(B, B1)))", "coords": {} }, { "index": 5932, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5933, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5934, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5935, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5936, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5936.png", "answer": "2*sqrt(8-m^2)", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2/8 + y^2/m^2 = 1);PointOnCurve(Focus(G), xAxis)", "coords": {} }, { "index": 5937, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_5937.png", "answer": "sqrt(3)/2", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Eccentricity(C) = 2;G: Ellipse;Expression(G) = (x^2/5 + y^2 = 1);Focus(C) = Focus(G);P: Point;PointOnCurve(P, C);Z1: Line;Z2: Line;L1: Line;L2: Line;Asymptote(C) = {L1, L2};PointOnCurve(P, Z1);PointOnCurve(P, Z2);IsPerpendicular(Z1, L1);IsPerpendicular(Z2, L2);A: Point;B: Point;FootPoint(Z1, L1) = A;FootPoint(Z2, L2) = B", "coords": {} }, { "index": 5938, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5938.png", "answer": "1", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);H: Line;M: Point;m: Number;Coordinate(M) = (0, m);PointOnCurve(M, H);A: Point;B: Point;Intersection(H, G) = {A,B};x1: Number;x2: Number;y1: Number;y2: Number;Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);x1*x2 = -4", "coords": {} }, { "index": 5939, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5939.png", "answer": "36", "fact_expr": "E: Parabola;Expression(E) = (y^2 = 4*x);F: Point;Focus(E) = F;l1: Line;l2: Line;PointOnCurve(F, l1) = True;PointOnCurve(F, l2) = True;IsPerpendicular(l1, l2);Intersection(l1, E) = {A, B};Intersection(l2, E) = {C, D};A: Point;B: Point;C: Point;D: Point", "coords": {} }, { "index": 5940, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5940.png", "answer": "a - 2*p", "fact_expr": "G: Parabola;p: Number;M: Point;p>0;a: Number;a > p/2;Expression(G) = (x^2 = 2*(p*y));PointOnCurve(M, G);Distance(M, Focus(G)) = a", "coords": {} }, { "index": 5941, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5942, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5942.png", "answer": "(1, \\sqrt{2}]", "fact_expr": "C: Ellipse;Expression(C) = (y^2/4 + x^2/a^2 = 1);a: Number;a>2;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C) = True;IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2)) = True;Gamma: Hyperbola;Expression(Gamma) = (x^2/a^2 - y^2/8 = 1)", "coords": {} }, { "index": 5943, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5943.png", "answer": "sqrt(3)/3", "fact_expr": "P: Point;PointOnCurve(P, G);G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;F1: Point;F2: Point;Focus(G) = {F1, F2};WhenMax(AngleOf(F1, P, F2));Cos(AngleOf(F1, P, F2)) = 1/3", "coords": {} }, { "index": 5944, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5944.png", "answer": "1/4", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2 + y^2/m = 1);PointOnCurve(Focus(G), xAxis);FocalLength(G)=sqrt(3)", "coords": {} }, { "index": 5945, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5945.png", "answer": "sqrt(2)", "fact_expr": "l: Line;C: Parabola;A: Point;F: Point;B: Point;O: Origin;m:Number;Expression(C) = (y^2 = 2*x);Focus(C) = F;Expression(l)=(y=m*(2*x-1));Intersection(l, C) = {A, B};Quadrant(A)=1;Abs(LineSegmentOf(A, F)) = 2*Abs(LineSegmentOf(B, F))", "coords": {} }, { "index": 5946, "success": false, "conic_type": "parabola", "error": "validation: parabola_focus_mismatch", "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5946.png", "answer": "sqrt(5)", "fact_expr": "M: Point;Coordinate(M) = (-2, -3);C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Coordinate(F) = (2, 0);Focus(C) = F;P: Point;Quadrant(P) = 1;PointOnCurve(P, C)", "coords": { "M": [ -2.0, -3.0 ], "F": [ 2.0, 0.0 ] }, "validation_reasons": [ "parabola_focus_mismatch" ] }, { "index": 5947, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5947.png", "answer": "y=(pm*1/2)*x", "fact_expr": "G: Hyperbola;m: Number;H: Parabola;Expression(G) = (-y^2 + x^2/m = 1);Expression(H) = (y^2 = 8*x);PointOnCurve(Focus(H),G)", "coords": {} }, { "index": 5948, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5949, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5949.png", "answer": "2", "fact_expr": "G: Parabola;H: Ellipse;O: Origin;Expression(H) = (x^2/4 + y^2/3 = 1);Vertex(G) = O;Focus(G)=OneOf(Focus(H))", "coords": {} }, { "index": 5950, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5950.png", "answer": "4", "fact_expr": "C: Hyperbola;a: Number;Expression(C) = (-y^2/3 + x^2/a^2 = 1);Expression(OneOf(Asymptote(C))) = (sqrt(3)*x + y = 0)", "coords": {} }, { "index": 5951, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5952, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5953, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5953.png", "answer": "(0,2)", "fact_expr": "G: Ellipse;a: Real;Expression(G) = (x^2/2 + y^2/a = 1);PointOnCurve(Focus(G), xAxis)", "coords": {} }, { "index": 5954, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.449489742783178, "major_axis": "x", "x_coef": 16.0, "y_coef": 6.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5954.png", "answer": "x^2/8 - y^2/2 = 1", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/16 + y^2/6 = 1);Focus(G) = Focus(H);Expression(Asymptote(G)) = (y = pm*(x/2))", "coords": {} }, { "index": 5955, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5955.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Parabola;p: Number;A: Point;B: Point;F: Point;M: Point;N: Point;p>0;Expression(G) = (y^2 = 2*p*x);Focus(G) = F;PointOnCurve(B, G);PointOnCurve(A, G);AngleOf(A, F, B) = pi/2;MidPoint(LineSegmentOf(A, B)) = M;Projection(M, Directrix(G)) = N;IsChordOf(LineSegmentOf(A, B),G)", "coords": {} }, { "index": 5956, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5956.png", "answer": "2*sqrt(5)/5", "fact_expr": "G: Ellipse;b: Number;a: Number;M: Point;Q: Point;A: Point;B: Point;P: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (0, b);Coordinate(Q) = (0, -b);Endpoint(MinorAxis(G)) = {P, Q};OneOf(Endpoint(MajorAxis(G))) = M;IsChordOf(LineSegmentOf(A, B), G);PointOnCurve(Center(G), LineSegmentOf(A, B));Negation(OverlappingLine(LineSegmentOf(A, B), axis));Slope(LineSegmentOf(P, A))*Slope(LineSegmentOf(P, B)) = -1/4", "coords": {} }, { "index": 5957, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5957.png", "answer": "4*p", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;H: Line;PointOnCurve(Focus(G), H);A: Point;B: Point;x1: Number;x2: Number;y1: Number;y2: Number;Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);Intersection(H, G) = {A, B};x1 + x2 = 3*p", "coords": {} }, { "index": 5958, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5958.png", "answer": "sqrt(3)/2", "fact_expr": "C: Ellipse;b: Number;a: Number;G: Line;A: Point;B: Point;M: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(M) = (2, 1);PointOnCurve(M, G);Slope(G) = -1/2;Intersection(G, C) = {A, B};MidPoint(LineSegmentOf(A,B)) = M", "coords": { "M": [ 2.0, 1.0 ] } }, { "index": 5959, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5959.png", "answer": "y=-1", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y)", "coords": {} }, { "index": 5960, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5960.png", "answer": "y^2 - x^2/3 = 1", "fact_expr": "C: Hyperbola;G: Parabola;Expression(G) = (x^2 = 8*y);Eccentricity(C) = 2;OneOf(Focus(C)) = Focus(G)", "coords": {} }, { "index": 5961, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5961.png", "answer": "4", "fact_expr": "C: Parabola;p: Number;G: Circle;H: Line;A: Point;B: Point;M:Point;p>0;Expression(C) = (y^2 = 2*(p*x));PointOnCurve(Focus(C), H);Slope(H) = 2;Intersection(H, C) = {A, B};IsDiameter(LineSegmentOf(A,B),G);Intersection(Directrix(C),G)=M;YCoordinate(M) = 2", "coords": {} }, { "index": 5962, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 9.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5962.png", "answer": "ApplyUnit(120,degree)", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 + y^2/2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 4", "coords": {} }, { "index": 5963, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5963.png", "answer": "5", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;P: Point;y0: Number;Coordinate(P) = (4, y0);PointOnCurve(P, G)", "coords": {} }, { "index": 5964, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5964.png", "answer": "2*sqrt(2)", "fact_expr": "C: Parabola;p: Number;M: Point;O: Origin;F: Point;p>0;y0:Number;Expression(C) = (x^2 = 2*p*y);Coordinate(M) = (2,y0);Focus(C) = F;PointOnCurve(M,C);Abs(LineSegmentOf(M,O))=Abs(LineSegmentOf(M,F))", "coords": {} }, { "index": 5965, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5966, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5966.png", "answer": "5/6", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x);F: Point;Focus(G) = F;Z: Line;PointOnCurve(F, Z);A: Point;B: Point;Intersection(Z, G) = {A, B};Abs(LineSegmentOf(A, B)) = 25/12;Abs(LineSegmentOf(A, F)) b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1,H);Inclination(H)=ApplyUnit(45,degree);OneOf(Intersection(H,G))=M;IsPerpendicular(LineSegmentOf(M,F2),xAxis)", "coords": {} }, { "index": 5971, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5972, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5972.png", "answer": "sqrt(5)", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;H: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y = 2*x);Coordinate(H) = (a, b);PointOnCurve(H, G)", "coords": {} }, { "index": 5973, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_5973.png", "answer": "4", "fact_expr": "C: Parabola;a: Number;Expression(C) = (y^2 = a*x);Expression(Directrix(C)) = (x = -1)", "coords": {} }, { "index": 5974, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5974.png", "answer": "{sqrt(10), sqrt(10)/3}", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;A: Point;RightVertex(G) = A;m: Line;Slope(m) = 1;PointOnCurve(A, m);Z1: Line;Z2: Line;Asymptote(G) = {Z1, Z2};B: Point;Intersection(m, Z1) = B;C: Point;Intersection(m, Z2) = C;Abs(LineSegmentOf(A, B)) = 2*Abs(LineSegmentOf(A, C))", "coords": {} }, { "index": 5975, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5975.png", "answer": "10/3", "fact_expr": "F: Point;G: Parabola;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, l);l: Line;Intersection(l, G) = {A, B};A: Point;B: Point;O: Origin;Abs(LineSegmentOf(A, F)) = 10", "coords": {} }, { "index": 5976, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5976.png", "answer": "(-1, 1)", "fact_expr": "H: Line;Expression(H) = (y = k*x - 1);k: Number;G: Hyperbola;Expression(G) = (x^2 - y^2 = 1);NumIntersection(H, LeftPart(G)) = 1;NumIntersection(H, RightPart(G)) = 1", "coords": {} }, { "index": 5977, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5978, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_5978.png", "answer": "2", "fact_expr": "l: Line;C: Parabola;A: Point;F: Point;O: Origin;B: Point;Expression(C) = (y = x^2/4);Focus(C) = F;PointOnCurve(F,l);Intersection(l,C)={A,B};Abs(LineSegmentOf(A, F)) = 3", "coords": {} }, { "index": 5979, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5980, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.6457513110645907, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 7.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5980.png", "answer": "-1/2", "fact_expr": "G: Parabola;m: Number;H: Ellipse;Expression(G) = (y^2 = (4/m)*x);Expression(H) = (x^2/7 + y^2/3 = 1);Focus(G) = LeftFocus(H)", "coords": {} }, { "index": 5981, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5982, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 2.1278254985809326, "b": 1.8086377171290415, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 2.482879284737182e-05, "converged": true, "iterations": 79 }, "output_path": "results/train/ellipse/problem_5982.png", "answer": "x^2/8+y^2/4=1", "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (sqrt(2), sqrt(3));FocalLength(G) = 4;PointOnCurve(A,G)", "coords": { "A": [ 1.4142135623730951, 1.7320508075688772 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 5983, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5984, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5984.png", "answer": "4", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Eccentricity(G) = (2*sqrt(3))/3;FocalLength(G) = 2*c;c: Number;2*a^2 = 3*c;P: Point;PointOnCurve(P,G) = True;DotProduct(VectorOf(P,F1), VectorOf(P,F2)) = 2;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2", "coords": {} }, { "index": 5985, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5985.png", "answer": "(2/7)*sqrt(21)", "fact_expr": "G: Hyperbola;a: Number;b: Number;F1: Point;P: Point;F2: Point;a>0;b>0;R: Number;r: Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, G);Focus(G) = {F1, F2};AngleOf(F1, P, F2) = pi/3;R = 4*r;Radius(CircumCircle(TriangleOf(F1, P, F2))) = R;Radius(InscribedCircle(TriangleOf(F1, P, F2))) = r", "coords": {} }, { "index": 5986, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 5.0, 0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_5986.png", "answer": "10", "fact_expr": "C1: Curve;C2: Curve;C3: Curve;P: Point;Q: Point;R: Point;Expression(C1) = (x^2/16-y^2/9=1);Expression(C2) = ((x-5)^2+y^2=1);Expression(C3) = ((x+5)^2+y^2=1);PointOnCurve(P,C1);PointOnCurve(Q,C2);PointOnCurve(R,C3)", "coords": {} }, { "index": 5987, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5987.png", "answer": "sqrt(3)/2", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1)", "coords": {} }, { "index": 5988, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "y", "x_coef": 9.0, "y_coef": 25.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5988.png", "answer": "y^2/4 - x^2/12 = 1", "fact_expr": "C: Hyperbola;G: Ellipse;Expression(G) = (x^2/9 + y^2/25 = 1);Focus(C) = Focus(G);Eccentricity(C)+Eccentricity(G)=14/5", "coords": {} }, { "index": 5989, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5989.png", "answer": "4", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/2 = 1)", "coords": {} }, { "index": 5990, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5991, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "y", "x_coef": 9.0, "y_coef": 25.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5991.png", "answer": "10", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/25 = 1)", "coords": {} }, { "index": 5992, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5993, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/train/hyperbola/problem_5993.png", "answer": "2", "fact_expr": "C: Hyperbola;a: Number;b: Number;G: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(G) = (2, 3);PointOnCurve(G, C);FocalLength(C) = 4", "coords": { "G": [ 2.0, 3.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 5994, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5995, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5996, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_5996.png", "answer": "-32", "fact_expr": "G: Hyperbola;k: Number;Expression(G) = (x^2/4 + y^2/k = 1);Eccentricity(G) = e;e:Number;e = 3", "coords": {} }, { "index": 5997, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.463192105293274, "b": 2.070901870727539, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4661259228887502e-06, "converged": true, "iterations": 106 }, "output_path": "results/train/hyperbola/problem_5997.png", "answer": "sqrt(5)/2", "fact_expr": "G: Hyperbola;a: Number;b: Number;H: Ellipse;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a > b;Expression(H) = (y^2/b^2 + x^2/a^2 = 1);Eccentricity(H) = sqrt(3)/2", "coords": {} }, { "index": 5998, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 5999, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_5999.png", "answer": "(\\sqrt{3}+1)/2", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);AngleOf(P,F1,F2)=AngleOf(F1,P,F2);AngleOf(F1,P,F2)=ApplyUnit(30,degree)", "coords": {} }, { "index": 6000, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6001, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6002, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6002.png", "answer": "5", "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (y^2 = 8*x);Coordinate(A) = (3, 2);Focus(G) = F;PointOnCurve(P, G)", "coords": { "A": [ 3.0, 2.0 ] } }, { "index": 6003, "success": false, "conic_type": "parabola", "error": "validation: parabola_focus_mismatch", "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6003.png", "answer": "y^2=8*x", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Coordinate(F) = (2, 0);Focus(C) = F", "coords": { "F": [ 2.0, 0.0 ] }, "validation_reasons": [ "parabola_focus_mismatch" ] }, { "index": 6004, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.125, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6004.png", "answer": "(2+sqrt(3))/2", "fact_expr": "G: Line;C: Curve;A: Point;B:Point;F: Point;Expression(C) = (y = 2*x^2);Inclination(G)=ApplyUnit(60,degree);PointOnCurve(F,G);Intersection(G,C)={A,B};Negation(A=B);Quadrant(A)=1;Focus(C)=F", "coords": {} }, { "index": 6005, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6006, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6006.png", "answer": "6", "fact_expr": "G: Hyperbola;H: Parabola;p: Real;Expression(G) = (x^2/4 - y^2/5 = 1);Expression(H) = (y^2 = 2*(p*x));Focus(H) = RightFocus(G)", "coords": {} }, { "index": 6007, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6007.png", "answer": "2", "fact_expr": "l: Line;C: Parabola;M: Point;N: Point;P: Point;Expression(C) = (y^2 = 4*x);Coordinate(P) = (2, 1);PointOnCurve(P, l);Intersection(l, C) = {M, N};MidPoint(LineSegmentOf(M, N)) = P", "coords": { "P": [ 2.0, 1.0 ] } }, { "index": 6008, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6008.png", "answer": "{pm*(sqrt(5)/2),pm*1}", "fact_expr": "G: Hyperbola;H: Line;k: Number;Expression(G) = (x^2-y^2=4);Expression(H) = (y=k*x - 1);NumIntersection(H, G) = 1", "coords": {} }, { "index": 6009, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6009.png", "answer": "1/2", "fact_expr": "G: Parabola;Expression(G) = (y = x^2)", "coords": {} }, { "index": 6010, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6010.png", "answer": "x + pm*y = 0", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(C) = F;H: Line;PointOnCurve(F,H) = True;IsPerpendicular(H,xAxis) = True;Intersection(H,C) = {A,B};A: Point;B: Point;IsDiameter(LineSegmentOf(A,B),G) = True;G: Circle;IsTangent(G,Asymptote(C)) = True", "coords": {} }, { "index": 6011, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6012, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6012.png", "answer": "pm*2*sqrt(2)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;p>0;F: Point;Focus(C) = F;P: Point;Coordinate(P) = (2, m);m: Number;PointOnCurve(P, C);Abs(LineSegmentOf(P, F)) = 3", "coords": {} }, { "index": 6013, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6013.png", "answer": "6", "fact_expr": "G: Parabola;p: Number;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(Focus(G)) = (3, 0)", "coords": {} }, { "index": 6014, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6014.png", "answer": "2", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;Coordinate(F) = (2, 0);RightFocus(C) = F;PointOnCurve(F, H);IsPerpendicular(H, Asymptote(C));H: Line;G: Circle;Expression(G) = (4*x + x^2 + y^2 = 0);IsTangent(H, G)", "coords": { "F": [ 2.0, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 6015, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6015.png", "answer": "y^2/16 - x^2/9 = 1", "fact_expr": "G1: Hyperbola;Expression(G1) = (x^2/9 - y^2/16 = 1);G2: Hyperbola;Asymptote(G1) = Asymptote(G2);A: Point;Coordinate(A) = (-3, 4*sqrt(2));PointOnCurve(A, G2) = True", "coords": { "A": [ -3.0, 5.656854249492381 ] } }, { "index": 6016, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6016.png", "answer": "8", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;O: Origin;D: Point;E: Point;a>0;b>0;L1:Line;L2:Line;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x = a);Asymptote(C)={L1,L2};Intersection(G,L1)=D;Intersection(G,L2)=E;Area(TriangleOf(O, D, E)) = 8", "coords": {} }, { "index": 6017, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6017.png", "answer": "1", "fact_expr": "A: Point;B: Point;G: Hyperbola;H: Parabola;Expression(G) = (x^2/4 - y^2 = 1);Expression(H) = (y^2 = 4*x);l:Line;Directrix(H)=l;L1:Line;L2:Line;Asymptote(G)={L1,L2};Intersection(L1,l)=A;Intersection(L2,l)=B", "coords": {} }, { "index": 6018, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6018.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x)", "coords": {} }, { "index": 6019, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6019.png", "answer": "4*x-y-7=0", "fact_expr": "l: Line;G: Hyperbola;A: Point;B: Point;M: Point;Expression(G) = (x^2 - y^2/2 = 1);Coordinate(M) = (2, 1);PointOnCurve(M, l);Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A, B)) = M", "coords": { "M": [ 2.0, 1.0 ] } }, { "index": 6020, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6021, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.872983346207417, "major_axis": "x", "x_coef": 25.0, "y_coef": 15.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6021.png", "answer": "x^2/10 - y^2/15 = 1", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/25 + y^2/15 = 1);Vertex(H)=Focus(G);Focus(H)=Vertex(G)", "coords": {} }, { "index": 6022, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -2.0, 1.0 ], "radius": 2.23606797749979 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_6022.png", "answer": "x^2/12+y^2/3=1", "fact_expr": "Gamma:Ellipse;C: Circle;A: Point;B: Point;b:Number;Expression(Gamma) = (x^2+4*y^2=4*b^2);Expression(C) = ((x + 2)^2 + (y - 1)^2 = 5/2);Intersection(Gamma, C)={A,B};IsDiameter(LineSegmentOf(A, B), C)", "coords": {} }, { "index": 6023, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6024, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6024.png", "answer": "3/4", "fact_expr": "C1: Ellipse;Expression(C1) = (x^2/a1^2+y^2/b1^2=1);a1: Number;b1: Number;a1>b1;b1>0;F1: Point;F2: Point;C2: Hyperbola;Expression(C2) = (x^2/a2^2-y^2/b2^2=1);a2: Number;b2: Number;a2>0;b2>0;Focus(C1) = {F1, F2};Focus(C2) = {F1, F2};M: Point;Quadrant(M) = 1;Intersection(C1, C2) = M;AngleOf(F1, M, F2) = ApplyUnit(90, degree);e1: Number;e2: Number;Eccentricity(C1) = e1;Eccentricity(C2) = e2;e2 = (3*sqrt(2))/2", "coords": {} }, { "index": 6025, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6025.png", "answer": "sqrt(37)/5", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;l: Line;PointOnCurve(F2, l);A: Point;B: Point;Intersection(l, RightPart(C)) = {A, B};DotProduct(VectorOf(A, B), VectorOf(A, F1)) = 0;12*Abs(VectorOf(A, B)) = 5*Abs(VectorOf(A, F1))", "coords": {} }, { "index": 6026, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6026.png", "answer": "4/5", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;H: Line;PointOnCurve(F, H);Inclination(H) = pi/4;A: Point;B: Point;Intersection(H, G) = {A, B};IsChordOf(LineSegmentOf(A, B), G);I: Point;Coordinate(I) = (0, 2);PointOnCurve(I, PerpendicularBisector(LineSegmentOf(A, B)))", "coords": { "I": [ 0.0, 2.0 ] } }, { "index": 6027, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6027.png", "answer": "2", "fact_expr": "l: Line;C:Parabola;p: Number;A: Point;B: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Expression(l) = (y = b + 2*x);PointOnCurve(Focus(C), l);Intersection(l,C)={A,B};Abs(LineSegmentOf(A, B)) = 5;b:Number", "coords": {} }, { "index": 6028, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6029, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6029.png", "answer": "1", "fact_expr": "G: Ellipse;Expression(G) = (y^2 + x^2/m = 1);H: Hyperbola;Expression(H) = (-y^2 + x^2/n = 1);m: Number;n: Number;m>1;n>0;F1: Point;F2: Point;Focus(G) = {F1, F2};Focus(H) = {F1, F2};P: Point;OneOf(Intersection(G, H)) = P", "coords": {} }, { "index": 6030, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 9.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6030.png", "answer": "2\n4*sqrt(3)", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 + y^2/2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 4", "coords": {} }, { "index": 6031, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.125, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6031.png", "answer": "(1, 2)", "fact_expr": "P: Point;PointOnCurve(P, G);G: Parabola;Expression(G) = (y = 2*x^2);l: Line;Expression(l) = (4*x - y - 6 = 0);WhenMin(Distance(P, l))", "coords": {} }, { "index": 6032, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6033, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6033.png", "answer": "6", "fact_expr": "G: Ellipse;P: Point;F1: Point;O: Origin;M: Point;F2: Point;Expression(G) = (x^2/25 + y^2/16 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);MidPoint(LineSegmentOf(P,F1)) = M;Abs(LineSegmentOf(O, M)) = 2", "coords": {} }, { "index": 6034, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6035, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6036, "success": false, "error": "Circle from diameter: missing point coordinates" }, { "index": 6037, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6037.png", "answer": "12", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;l: Line;PointOnCurve(Focus(C), l);IsPerpendicular(l, SymmetryAxis(C));A: Point;B: Point;Intersection(l, C) = {A, B};P: Point;PointOnCurve(P, Directrix(C));Area(TriangleOf(A, B, P)) = 36;H: LineSegment;PointOnCurve(Focus(C), H);IsChordOf(H, C)", "coords": {} }, { "index": 6038, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6038.png", "answer": "(x-2)^2+(y-1)^2=4", "fact_expr": "l: Line;C: Parabola;G: Circle;A: Point;Expression(C) = (x^2 = 4*y);Expression(l)=(y = b + x) ;TangentPoint(l,C)=A;Center(G)=A;b:Number;IsTangent(Directrix(C),G)", "coords": {} }, { "index": 6039, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6039.png", "answer": "y^2=2*x", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(C) = F;l: Line;Directrix(C) = l;G: Line;Inclination(G) = ApplyUnit(120, degree);PointOnCurve(F, G);A: Point;Intersection(G, l) = A;B: Point;Intersection(LineSegmentOf(A, F), C) = B;Abs(LineSegmentOf(A, B)) = 4/3", "coords": {} }, { "index": 6040, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6040.png", "answer": "6", "fact_expr": "G: Parabola;C: Circle;M: Point;A: Point;F: Point;Expression(G) = (y^2 = 4*x);Expression(C) = ((x - 6)^2 + (y + 1)^2 = 1);PointOnCurve(M, G);PointOnCurve(A, C);Focus(G)=F", "coords": {} }, { "index": 6041, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6041.png", "answer": "4", "fact_expr": "G: Parabola;M: Point;A: Point;F: Point;C: Curve;Expression(G)=(y = x^2/4);PointOnCurve(M, G);Focus(G) = F;Expression(C)=((x - 1)^2 + (y - 4)^2 = 1);PointOnCurve(A, C)", "coords": {} }, { "index": 6042, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6043, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6043.png", "answer": "16", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 - y^2/16 = 1);Focus(G) = {F1,F2};PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2))", "coords": {} }, { "index": 6044, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2135674953460693, "b": 2.4270472526550293, "orientation": "horizontal" }, "optimization": { "final_loss": 4.453341261978494e-06, "converged": true, "iterations": 125 }, "output_path": "results/train/hyperbola/problem_6044.png", "answer": "4", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/4 + x^2/a^2 = 1);a: Number;a>0;Eccentricity(G) = sqrt(5)/2", "coords": {} }, { "index": 6045, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6045.png", "answer": "(-3/2, 3/2)", "fact_expr": "C: Ellipse;Expression(C) = (x^2 + y^2/4 = 1);P: Point;m: Number;Coordinate(P) = (0, m);PointOnCurve(P, yAxis);P1: Circle;Center(P1) = P;NumIntersection(P1, C) = 4", "coords": {} }, { "index": 6046, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6046.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "C: Ellipse;b: Number;a: Number;O: Circle;A: Point;P: Point;B: Point;L1: Line;L2: Line;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Expression(O) = (x^2 + y^2 = b^2);PointOnCurve(P, C);TangentOfPoint(P, O) = {L1,L2};TangentPoint(L1,O)=A;TangentPoint(L2,O)=B;AngleOf(A, P, B) = ApplyUnit(60, degree)", "coords": {} }, { "index": 6047, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6047.png", "answer": "2", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;LeftFocus(C) = F;L1: Line;L2: Line;OneOf(Asymptote(C))=L1;OneOf(Asymptote(C))=L2;Negation(L1=L2);l: Line;PointOnCurve(F, l);IsPerpendicular(l, L1);A: Point;FootPoint(l, L1)=A;B: Point;Intersection(l, L2)=B;MidPoint(LineSegmentOf(F, B))=A", "coords": {} }, { "index": 6048, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6048.png", "answer": "x=-2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x)", "coords": {} }, { "index": 6049, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6049.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;F2: Point;P: Point;Q: Point;F1: Point;e: Number;l:Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = -x*b/a);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, l);FootPoint(l, H)=Q;Intersection(l,LeftPart(G)) = P;Abs(LineSegmentOf(F2, P)) = 2*Abs(LineSegmentOf(F2, Q));Eccentricity(G) = e;IsPerpendicular(H, l)", "coords": {} }, { "index": 6050, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6051, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6051.png", "answer": "11", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/b^2 = 1);b: Number;b>0;F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P,G) = True;Abs(LineSegmentOf(P, F1)) = 5", "coords": {} }, { "index": 6052, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.3300262689590454, "b": 2.30381441116333, "orientation": "horizontal" }, "optimization": { "final_loss": 6.658357051492203e-06, "converged": true, "iterations": 122 }, "output_path": "results/train/hyperbola/problem_6052.png", "answer": "x^2 - y^2 / 3 = 1", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x);H: Parabola;Expression(H) = (y^2 = 8*x);PointOnCurve(OneOf(Focus(G)), Directrix(H))", "coords": {} }, { "index": 6053, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6053.png", "answer": "sqrt(2)/2", "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;F: Point;P: Point;B: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);UpperVertex(G) = A;LowerVertex(G) = B;RightFocus(G) = F;Intersection(LineOf(A,F), G) = {A,P};WhenMax(Slope(LineOf(B,P)))", "coords": {} }, { "index": 6054, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6055, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6055.png", "answer": "sqrt(6)/3", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y/b + x/a = 1);Focus(G)={F1,F2};Distance(F1,H)+Distance(F2,H)=sqrt(3)*b;Eccentricity(G)=e;e:Number;F1: Point;F2: Point", "coords": {} }, { "index": 6056, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6056.png", "answer": "8*sqrt(3)/3", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2/4 = 1);L: LineSegment;IsChordOf(L,G) = True;PointOnCurve(Focus(G),L) = True;IsPerpendicular(L,xAxis) = True", "coords": {} }, { "index": 6057, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6057.png", "answer": "15", "fact_expr": "A: Point;G: Ellipse;Expression(G) = (x^2/9 + y^2/5 = 1);PointOnCurve(A, G);M: Point;N: Point;H: Circle;Expression(H) = (y^2 + (x - 1)^2 = 1);IsDiameter(LineSegmentOf(M, N), H) = True", "coords": {} }, { "index": 6058, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6058.png", "answer": "2*sqrt(2)", "fact_expr": "l: Line;C: Parabola;p: Number;A: Point;O: Origin;B: Point;p>0;Expression(C) = (x^2 = 2*p*y);Inclination(l)=pi/4;PointOnCurve(F,l);Intersection(l,C)={A,B};Abs(LineSegmentOf(A, B)) = 8;F:Point;Focus(C)=F", "coords": {} }, { "index": 6059, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6059.png", "answer": "2*sqrt(2)", "fact_expr": "G: Curve;A: Point;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (-1, 2);PointOnCurve(P, G);l:Line;Expression(l)=(x=-1)", "coords": { "A": [ -1.0, 2.0 ] } }, { "index": 6060, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6061, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6061.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(G) = F;FocalLength(G) = 8;A: Point;LeftVertex(G) = A;B: Point;Coordinate(B) = (0, b);PointOnCurve(B, yAxis) = True;DotProduct(VectorOf(B, A), VectorOf(B, F)) = 2*a", "coords": {} }, { "index": 6062, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6062.png", "answer": "32", "fact_expr": "G: Parabola;H: Line;P: Point;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Coordinate(P) = (4, 0);PointOnCurve(P, H);x1: Number;y1: Number;x2: Number;y2: Number ;Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);Intersection(H,G)={A,B}", "coords": { "P": [ 4.0, 0.0 ] } }, { "index": 6063, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6063.png", "answer": "11", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 - y^2/16 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 5", "coords": {} }, { "index": 6064, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 7.0, "b": 4.898979485566356, "major_axis": "x", "x_coef": 49.0, "y_coef": 24.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6064.png", "answer": "24", "fact_expr": "G: Ellipse;Expression(G) = (x^2/49 + y^2/24 = 1);P: Point;PointOnCurve(P,G) = True;F1: Point;F2: Point;Focus(G) = {F1,F2}\t;IsPerpendicular(LineSegmentOf(P,F1),LineSegmentOf(P,F2)) = True", "coords": {} }, { "index": 6065, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6065.png", "answer": "2", "fact_expr": "C: Parabola;A: Point;M: Point;F: Point;l:Line;Expression(C) = (y = x^2/4);Focus(C) = F;Directrix(C)=l;Intersection(l, yAxis) = A;PointOnCurve(M, C);Abs(LineSegmentOf(M,A))/Abs(LineSegmentOf(M,F))=sqrt(2)", "coords": {} }, { "index": 6066, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6067, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6067.png", "answer": "3", "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;C: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B};Intersection(l,Directrix(G))=C;VectorOf(A, B) = 2*VectorOf(B, C)", "coords": {} }, { "index": 6068, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6068.png", "answer": "2*sqrt(6)/3", "fact_expr": "G: Ellipse;P: Point;Q: Point;Expression(G) = (x^2/16 + y^2/4 = 1);Coordinate(Q) = (2, 0);PointOnCurve(P, G)", "coords": { "Q": [ 2.0, 0.0 ] } }, { "index": 6069, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6069.png", "answer": "{(x+y-2=0),(x-y-2=0)}", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 8*x);F: Point;Focus(G) = F;P: Point;PointOnCurve(P, Directrix(G)) = True;OneOf(Intersection(LineOf(P, F), G)) = Q;Q: Point;VectorOf(P, Q) = sqrt(2)*VectorOf(Q, F)", "coords": {} }, { "index": 6070, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6070.png", "answer": "5*sqrt(2)", "fact_expr": "G: Parabola;H: Line;P: Point;A: Point;B: Point;Expression(G) = (y^2 = x);Coordinate(P) = (2, 2);PointOnCurve(P, H);Slope(H) = -1;Intersection(H, G) = {A, B}", "coords": { "P": [ 2.0, 2.0 ] } }, { "index": 6071, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6072, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6072.png", "answer": "16", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(P, G);Focus(G)={F1,F2};AngleOf(F1, P, F2) = ApplyUnit(90, degree)", "coords": {} }, { "index": 6073, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6073.png", "answer": "1/2", "fact_expr": "O: Origin;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;G: Ellipse;Expression(G) = (x^2/a^2 + y^2/b^2 = 1);a: Number;b: Number;a>b;b>0;P: Point;PointOnCurve(P,G) = True;AngleOf(F1,P,F2) = pi/3;Abs(LineSegmentOf(O,P)) = a*sqrt(3)/2", "coords": {} }, { "index": 6074, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6075, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6075.png", "answer": "9", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/9 = 1);P: Point;PointOnCurve(P, G);F1: Point;F2: Point;OneOf(Focus(G))=F1;OneOf(Focus(G))=F2;Negation(F1=F2);Distance(P, F1) = 5", "coords": {} }, { "index": 6076, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 0.8176298651812968, "b": 2.452642014841466, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4992967561765154e-06, "converged": true, "iterations": 283 }, "output_path": "results/train/hyperbola/problem_6076.png", "answer": "x^2 - y^2/9 = \\pm 1", "fact_expr": "G: Hyperbola;FocalLength(G) = 2*sqrt(10);Expression(Asymptote(G)) = (y = pm*3*x)", "coords": {} }, { "index": 6077, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6078, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6079, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6080, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6080.png", "answer": "x^2/16+y^2/12=1", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/12 = 1);H: Ellipse;Focus(G) = Vertex(H);Vertex(G) = Focus(H)", "coords": {} }, { "index": 6081, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6081.png", "answer": "3", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;M: Point;A: Point;N: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightVertex(C) = A;Center(G)=A;Radius(G)=b;Intersection(OneOf(Asymptote(C)),G)={M,N};AngleOf(M, A, N) = ApplyUnit(60, degree)", "coords": {} }, { "index": 6082, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142164151010692, "b": 1.4142164151010692, "orientation": "horizontal" }, "optimization": { "final_loss": 2.2160979279094996e-06, "converged": true, "iterations": 95 }, "output_path": "results/train/hyperbola/problem_6082.png", "answer": "x^2/2 - y^2/2 = 1", "fact_expr": "C: Hyperbola;F1: Point;LeftFocus(C) = F1;Coordinate(F1) = (-2,0);F2:Point;RightFocus(C) = F2;Coordinate(F2) = (2,0);P: Point;PointOnCurve(P,C) = True;Coordinate(P) = (2,sqrt(2))", "coords": { "F1": [ -2.0, 0.0 ], "F2": [ 2.0, 0.0 ], "P": [ 2.0, 1.4142135623730951 ] } }, { "index": 6083, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6084, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.3612699570633513, "b": 1.5726654040333643, "orientation": "horizontal" }, "optimization": { "final_loss": 6.189494861157194e-07, "converged": true, "iterations": 81 }, "output_path": "results/train/hyperbola/problem_6084.png", "answer": "{x^2/18 - y^2/8 = 1, y^2/8 - x^2/18 = 1}", "fact_expr": "G: Hyperbola;Expression(Asymptote(G)) = (y = pm*(2/3)*x);FocalLength(G)=2*sqrt(26)", "coords": {} }, { "index": 6085, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.6457513110645907, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 7.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6085.png", "answer": "2", "fact_expr": "G: Ellipse;H: Circle;I: Line;m: Number;n: Number;P: Point;Expression(G) = (x^2/7 + y^2/3 = 1);Expression(H) = (x^2 + y^2 = 3);Expression(I) = (m*x + n*y - 3 = 0);Coordinate(P) = (m, n);NumIntersection(I,H)=0;L:Line;PointOnCurve(P,L)", "coords": {} }, { "index": 6086, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6086.png", "answer": "y = pm*sqrt(2)*x", "fact_expr": "G: Hyperbola;a: Number;b: Number;H: Circle;F1: Point;M: Point;F2: Point;l: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2 + y^2 = a^2);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, l);TangentOfPoint(F1,H)=l;Intersection(l,RightPart(G))=M;AngleOf(F1, M, F2) = ApplyUnit(45,degree)", "coords": {} }, { "index": 6087, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6087.png", "answer": "y=pm*sqrt(6)*x", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;O: Origin;F2: Point;F1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P,RightPart(C));IsPerpendicular(LineSegmentOf(P,O),LineSegmentOf(P,F2));AngleOf(F1, P, O) = ApplyUnit(30, degree)", "coords": {} }, { "index": 6088, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6088.png", "answer": "2*x-3*y-2*sqrt(5)=0", "fact_expr": "l: Line;C: Ellipse;b: Number;a: Number;B: Point;A: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftVertex(C) = A;UpperVertex(C)=B;LineSegmentOf(A, B) = sqrt(13);Eccentricity(C) = sqrt(5)/3;F2:Point;RightFocus(C)=F2;PointOnCurve(F2,l);IsParallel(l,LineOf(A,B))", "coords": {} }, { "index": 6089, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6089.png", "answer": "{(sqrt(7),0) , (-sqrt(7),0)}", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/3 = 1)", "coords": {} }, { "index": 6090, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6090.png", "answer": "-3/4", "fact_expr": "C: Ellipse;P: Point;A1: Point;A2: Point;k1: Number;k2: Number;Expression(C) = (x^2/4 + y^2/3 = 1);LeftVertex(C) = A1;RightVertex(C) = A2;PointOnCurve(P, C);Negation(P = A1);Negation(P = A2);Slope(LineOf(P, A1)) = k1;Slope(LineOf(P, A2)) = k2", "coords": {} }, { "index": 6091, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.6457513110645907, "major_axis": "x", "x_coef": 16.0, "y_coef": 7.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6091.png", "answer": "7", "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;O: Origin;Expression(C) = (x^2/16 + y^2/7 = 1);LeftFocus(C) =F1;RightFocus(C)=F2;PointOnCurve(P, C);Abs(LineSegmentOf(O, P)) = 3", "coords": {} }, { "index": 6092, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6092.png", "answer": "3*sqrt(3)", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/25 + y^2/9 = 1);PointOnCurve(P, G);Focus(G) = {F1,F2};AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 6093, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6093.png", "answer": "6", "fact_expr": "C: Parabola;p: Number;P: Point;F: Point;l: Line;K:Point;H:Line;t:Number;p>0;Expression(C) = (y^2 = 2*p*x);Coordinate(P) = (3, t);Focus(C) = F;Directrix(C)=l;Intersection(l,xAxis)=K;PointOnCurve(K,H);Inclination(H)=ApplyUnit(45,degree);Intersection(H,C)=P", "coords": {} }, { "index": 6094, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.872983346207417, "major_axis": "x", "x_coef": 16.0, "y_coef": 15.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6094.png", "answer": "[5,21]", "fact_expr": "P: Point;G: Ellipse;Expression(G) = (x^2/16 + y^2/15 = 1);PointOnCurve(P, G);E: Point;F: Point;IsDiameter(LineSegmentOf(E, F), N) ;N: Circle;Expression(N) = (y^2 + (x - 1)^2 = 4)", "coords": {} }, { "index": 6095, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_asymptote_mismatch", "params": { "a": 1.4506710767745972, "b": 2.0511727333068848, "orientation": "vertical" }, "optimization": { "final_loss": 4.657696081267204e-06, "converged": true, "iterations": 116 }, "output_path": "results/train/hyperbola/problem_6095.png", "answer": "sqrt(6)/2", "fact_expr": "C: Hyperbola;Expression(C) = (-x^2/b^2 + y^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Expression(OneOf(Asymptote(C))) = (y = sqrt(2)*x)", "coords": {}, "validation_reasons": [ "hyperbola_asymptote_mismatch" ] }, { "index": 6096, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6097, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.7562712752322653, "b": 3.892939649129513, "orientation": "horizontal" }, "optimization": { "final_loss": 3.003583788532906e-05, "converged": true, "iterations": 105 }, "output_path": "results/train/hyperbola/problem_6097.png", "answer": "{x^2/75+y^2/50=1,y^2/75+x^2/50=1}", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(G) = (x^2/16 - y^2/9 = 1);Eccentricity(H) = sqrt(3)/3;FocalLength(G) = FocalLength(H)", "coords": {} }, { "index": 6098, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6098.png", "answer": "x^2/5+y^2/4=1", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Circle;A: Point;B: Point;J: Point;L1:Line;L2:Line;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2 + y^2 = 1);Coordinate(J) = (1, 1/2);TangentOfPoint(J, H)={L1,L2};TangentPoint(L1,H)=A;TangentPoint(L2,H)=B;PointOnCurve(RightFocus(G),LineOf(A,B));PointOnCurve(UpperVertex(G),LineOf(A,B))", "coords": { "J": [ 1.0, 0.5 ] } }, { "index": 6099, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.23606797749979, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 5.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6099.png", "answer": "2*sqrt(5)", "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/5 + y^2/3 = 1);PointOnCurve(P, G);J1: Point;J2: Point;Focus(G) = {J1, J2}", "coords": {} }, { "index": 6100, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6100.png", "answer": "6*sqrt(5)/5-1", "fact_expr": "l: Line;G: Parabola;A: Point;Expression(G) = (y^2 = -4*x);Expression(l) = (2*x + y - 4 = 0);PointOnCurve(A,G);Distance(A, yAxis) = m;Distance(A, l) = n;m:Number;n:Number", "coords": {} }, { "index": 6101, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6101.png", "answer": "{x^2/4+y^2/3=1,x^2/4+3*y^2/16=1}", "fact_expr": "G: Parabola;H: Ellipse;O:Origin;Expression(G) = (y^2 = 8*x);Center(H)=O;OneOf(Vertex(H))=Focus(G);Eccentricity(H)=1/2", "coords": {} }, { "index": 6102, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6103, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6103.png", "answer": "4*sqrt(3)", "fact_expr": "C: Hyperbola;b: Number;b>0;e:Number;Expression(C) = (x^2/4 - y^2/b^2 = 1);Eccentricity(C) = e ;e= 2", "coords": {} }, { "index": 6104, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 2.0, "major_axis": "x", "x_coef": 8.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6104.png", "answer": "sqrt(2)/2", "fact_expr": "G: Ellipse;Expression(G) = (x^2/8 + y^2/4 = 1)", "coords": {} }, { "index": 6105, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6105.png", "answer": "(x-2)^2+(y-5/3)^2=25/9", "fact_expr": "G: Ellipse;H: Circle;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 + y^2/5 = 1);PointOnCurve(P,G);Quadrant(P)=1;LeftFocus(G) = F1;RightFocus(G) = F2;Abs(LineSegmentOf(P, F1))-Abs(LineSegmentOf(P, F2)) = 8/3;Center(H)=P;Radius(H)=Abs(LineSegmentOf(P,F2))", "coords": {} }, { "index": 6106, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "major_axis": "x", "x_coef": 3.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6106.png", "answer": "(-1,0)+(0,1)", "fact_expr": "l: Line;G: Ellipse;A: Point;M: Point;k:Number;Negation(k=0);Expression(G) = (x^2/3 + y^2 = 1);Coordinate(A) = (0, -1);Slope(l)=k;Intersection(l,G)={M,N};Negation(M=N);Distance(M,A)=Distance(N,A);N:Point", "coords": { "A": [ 0.0, -1.0 ] } }, { "index": 6107, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6107.png", "answer": "3", "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (2, 2);Focus(G) = F;PointOnCurve(P, G)", "coords": { "A": [ 2.0, 2.0 ] } }, { "index": 6108, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6108.png", "answer": "1-(\\sqrt{3}/3)", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;A: Point;B: Point;Endpoint(MajorAxis(C)) = {A, B};M: Point;PointOnCurve(M, C);AngleOf(M, A, B) = ApplyUnit(30, degree);AngleOf(M, B, A) = ApplyUnit(45, degree);e: Number;Eccentricity(C) = e", "coords": {} }, { "index": 6109, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6109.png", "answer": "sqrt(6)/2", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/4 - y^2/2 = 1)", "coords": {} }, { "index": 6110, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6110.png", "answer": "2*sqrt(5)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*x);M: Point;Coordinate(M) = (2, -2);PointOnCurve(M, C);A: Point;B: Point;PointOnCurve(A, C);PointOnCurve(B, C);Negation(M=A);Negation(M=B);DotProduct(VectorOf(M, A), VectorOf(M, B)) = 0", "coords": { "M": [ 2.0, -2.0 ] } }, { "index": 6111, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6112, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 8.0, "b": 6.0, "major_axis": "x", "x_coef": 64.0, "y_coef": 36.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6112.png", "answer": "12", "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/64 + y^2/36 = 1);PointOnCurve(P, G);F1:Point;F2:Point;OneOf(Focus(G))=F1;OneOf(Focus(G))=F2;Negation(F1=F2);Distance(P, F1) = 4", "coords": {} }, { "index": 6113, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6114, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6114.png", "answer": "3/2", "fact_expr": "G: Hyperbola;a: Number;H: Point;Expression(G) = (-y^2/5 + x^2/a^2 = 1);Coordinate(H) = (3, 0);RightFocus(G) = H", "coords": { "H": [ 3.0, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 6115, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.125, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6115.png", "answer": "(0,-1/8)", "fact_expr": "G: Parabola;Expression(G) = (x^2 = y/2)", "coords": {} }, { "index": 6116, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6116.png", "answer": "5", "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/16 + y^2/9 = 1);Coordinate(P) = (x1, y1);x1:Number;y1:Number;PointOnCurve(P, G)", "coords": {} }, { "index": 6117, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "major_axis": "x", "x_coef": 3.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6117.png", "answer": "y = x + pm*1", "fact_expr": "l: Line;C: Ellipse;A: Point;B: Point;Expression(C) = (x^2/3 + y^2 = 1);Slope(l) = 1;Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, B)) = 3*sqrt(2)/2", "coords": {} }, { "index": 6118, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6118.png", "answer": "sqrt(2)/2", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;A1: Point;A2: Point;LeftVertex(C) = A1;RightVertex(C) = A2;M: Point;PointOnCurve(M, C);Negation(M = A1);Negation(M = A2);Slope(LineOf(M, A1)) * Slope(LineOf(M, A2)) = -1/2", "coords": {} }, { "index": 6119, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6119.png", "answer": "3/2", "fact_expr": "G: Parabola;Q: Point;M: Point;P: Point;O:Origin;Expression(G) = (y^2 = x);PointOnCurve(P,G);Negation(P=O);IsPerpendicular(LineSegmentOf(P,Q),xAxis);FootPoint(LineSegmentOf(P,Q),xAxis)=Q;L:Line;PointOnCurve(MidPoint(LineSegmentOf(P,Q)),L);IsParallel(L,xAxis);Intersection(L,G)=M;Intersection(LineOf(Q,M),yAxis)=N;N:Point", "coords": {} }, { "index": 6120, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6120.png", "answer": "sqrt(5)/2", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;N: Point;M: Point;a > b;b > 0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(l)=(x-3*y+c=0);l1:Line;l2:Line;Asymptote(C)={l1,l2};Intersection(l,l1)=M;Intersection(l,l2)=N;PointOnCurve(RightFocus(C),PerpendicularBisector(LineSegmentOf(M,N)));Coordinate(RightFocus(C)) = (c, 0);c:Number", "coords": {} }, { "index": 6121, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6121.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;b: Number;a: Number;A: Point;F1: Point;O: Origin;B: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(A, Asymptote(G));IsPerpendicular(LineSegmentOf(A, F1), LineSegmentOf(A, O));Intersection(LineSegmentOf(A, F1), G) = B;Abs(LineSegmentOf(A, B)) = Abs(LineSegmentOf(B, F1))", "coords": {} }, { "index": 6122, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 5.0, 0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_6122.png", "answer": "10", "fact_expr": "P: Point;C3: Curve;Expression(C3) = (x^2/16 - y^2/9 = 1);PointOnCurve(P, C3);Q: Point;C1: Curve;Expression(C1) = (y^2 + (x + 5)^2 = 1);PointOnCurve(Q, C1);R: Point;C2: Curve;Expression(C2) = (y^2 + (x - 5)^2 = 1);PointOnCurve(R, C2)", "coords": {} }, { "index": 6123, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6123.png", "answer": "(1,2)", "fact_expr": "Gamma: Hyperbola;Expression(Gamma) = (x^2/a^2 - y^2/b^2 = 1);a: Number;b: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(Gamma) = F1;RightFocus(Gamma) = F2;P: Point;PointOnCurve(P, RightPart(Gamma));Q: Point;PointOnCurve(Q, OverlappingLine(LineSegmentOf(P, F2)));IsPerpendicular(LineSegmentOf(Q, F1), LineSegmentOf(Q, F2));Sin(AngleOf(P, F1, Q)) = 3/5", "coords": {} }, { "index": 6124, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6124.png", "answer": "[sqrt(5)/5, 1/2]", "fact_expr": "G: Ellipse;b: Number;a: Number;c: Number;F1: Point;F2: Point;M: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);Focus(G) = {F1, F2};PointOnCurve(M, G);DotProduct(VectorOf(F1, M), VectorOf(F2, M)) = 3*c^2", "coords": {} }, { "index": 6125, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6125.png", "answer": "6", "fact_expr": "G: Parabola;p: Number;H: Ellipse;p>0;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (x^2/25 + y^2/16 = 1);PointOnCurve(LeftFocus(H), Directrix(G))", "coords": {} }, { "index": 6126, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6126.png", "answer": "sqrt(2)/2", "fact_expr": "F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;P: Point;PointOnCurve(P, G) = True;AngleOf(F1, P, F2) = ApplyUnit(90, degree)", "coords": {} }, { "index": 6127, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6128, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6128.png", "answer": "y**2=16*x", "fact_expr": "G: Parabola;H: Line;O: Origin;Expression(H) = (3*x - 4*y - 12 = 0);Vertex(G) = O;SymmetryAxis(G)=xAxis;PointOnCurve(Focus(G),H)", "coords": {} }, { "index": 6129, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6130, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6130.png", "answer": "[sqrt(6)/2, 2*sqrt(10)/5]", "fact_expr": "C: Hyperbola;b: Number;a: Number;F: Point;B: Point;A: Point;l1: Line;l2: Line;l: Line;lambda: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;Asymptote(C) = {l1, l2};PointOnCurve(F, l);IsPerpendicular(l, l1);FootPoint(l, l1) = A;Intersection(l, l2) = B;VectorOf(F, B) = lambda*VectorOf(A, F);3 <= lambda;lambda <= 4", "coords": {} }, { "index": 6131, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 0.929953396320343, "b": 2.629857301712036, "orientation": "horizontal" }, "optimization": { "final_loss": 4.594453912432073e-06, "converged": true, "iterations": 175 }, "output_path": "results/train/hyperbola/problem_6131.png", "answer": "1/3", "fact_expr": "G: Hyperbola;a: Number;b: Number;P: Point;F1: Point;F2: Point;M: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, RightPart(G));Incenter(TriangleOf(P, F1, F2)) = M;lambda:Number;Area(TriangleOf(M,P,F1))=Area(TriangleOf(M,P,F2))+lambda*Area(TriangleOf(M,F1,F2));Eccentricity(G) = 3", "coords": {} }, { "index": 6132, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6132.png", "answer": "pm*(2/5)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;H: Line;PointOnCurve(F, H);x1: Number;x2: Number;y1: Number;y2: Number;M: Point;N: Point;Coordinate(M) = (x1, y1);Coordinate(N) = (x2, y2);Intersection(H, C) = {M, N};P: Point;Coordinate(P) = (x2, -y2);Area(TriangleOf(M, P, F)) = 10", "coords": {} }, { "index": 6133, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6133.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;b: Number;a: Number;F2: Point;Q: Point;P: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G)=F2;l:Line;PointOnCurve(F2,l);L1:Line;L2:Line;OneOf(Asymptote(G))=L1;OneOf(Asymptote(G))=L2;Negation(L1=L2);IsPerpendicular(l,L1);FootPoint(l,L1)=P;Intersection(l,L2)=Q;VectorOf(F2,Q)=3*VectorOf(F2,P)", "coords": {} }, { "index": 6134, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_6134.png", "answer": "x^2 - y^2 / 3 = 1", "fact_expr": "C: Hyperbola;b: Number;a: Number;c:Number;Q: Point;P: Point;F1: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(Q) = (0, sqrt(3)*c);Eccentricity(C) = 2;LeftFocus(C) = F1;HalfFocalLength(C)=c;PointOnCurve(P, RightPart(C));Min(Abs(LineSegmentOf(P, F1)) + Abs(LineSegmentOf(P, Q))) = 6", "coords": {} }, { "index": 6135, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_focus_mismatch", "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6135.png", "answer": "x^2/18+y^2/9=1", "fact_expr": "E: Ellipse;H: Line;A: Point;B: Point;F: Point;a:Number;b:Number;a>b;b>0;Expression(E)=(x^2/a^2+y^2/b^2=1);Coordinate(F) = (3, 0);RightFocus(E) = F;PointOnCurve(F, H);Intersection(H, E) = {A,B};Coordinate(MidPoint(LineSegmentOf(A, B))) = (1,-1)", "coords": { "F": [ 3.0, 0.0 ] }, "validation_reasons": [ "ellipse_focus_mismatch" ] }, { "index": 6136, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6136.png", "answer": "y=-2", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 8*y)", "coords": {} }, { "index": 6137, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6138, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.581065485469099, "b": 1.581065485469099, "orientation": "horizontal" }, "optimization": { "final_loss": 1.6812139285133018e-06, "converged": true, "iterations": 95 }, "output_path": "results/train/hyperbola/problem_6138.png", "answer": "sqrt(5)/2", "fact_expr": "G: Hyperbola;F1: Point;Coordinate(F1) = (-sqrt(5), 0);F2: Point;Coordinate(F2) = (sqrt(5), 0);Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2));Abs(LineSegmentOf(P, F1))*Abs(LineSegmentOf(P, F2)) = 2", "coords": { "F1": [ -2.23606797749979, 0.0 ], "F2": [ 2.23606797749979, 0.0 ] } }, { "index": 6139, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6139.png", "answer": "1/2", "fact_expr": "E: Ellipse;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(E) = F1;RightFocus(E) = F2;P: Point;PointOnCurve(P, E) = True;Abs(LineSegmentOf(P, F2)) = Abs(LineSegmentOf(F1, F2));Distance(F2, LineOf(P,F1)) = b", "coords": {} }, { "index": 6140, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6140.png", "answer": "5/2", "fact_expr": "C: Parabola;Expression(C) = (x^2 = 2*y);F: Point;Focus(C) = F;Slope(l) = 1/2;PointOnCurve(F, l) = True;l: Line;Intersection(l, C) = {A, B};A: Point;B: Point", "coords": {} }, { "index": 6141, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6142, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6143, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6144, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6144.png", "answer": "4", "fact_expr": "G: Parabola;p: Number;A: Point;F: Point;m: Number;p>0;Expression(G) = (x^2 = 2*(p*y));Coordinate(A) = (m, 1);PointOnCurve(A, G);Focus(G) = F;Abs(LineSegmentOf(A, F)) = 3", "coords": {} }, { "index": 6145, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6145.png", "answer": "2", "fact_expr": "C: Hyperbola;Expression(C)=(x^2/a^2 - y^2/b^2 = 1);a: Number;b: Number;a>0;b>0;RightFocus(C) = F;O: Origin;D: Circle;F:Point;F = Center(D);M: Point;N: Point;Quadrant(M) = 1;Quadrant(N) = 1;Radius(D)=Abs(LineSegmentOf(O,F));Intersection(D,C)=M;Intersection(D,Asymptote(C))=N;P: Point;P=MidPoint(LineSegmentOf(M,F));Abs(LineSegmentOf(O, N)) = Abs(LineSegmentOf(O, P))", "coords": {} }, { "index": 6146, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6147, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6147.png", "answer": "(3,pm*2*sqrt(3))", "fact_expr": "C: Parabola;P: Point;N: Point;F: Point;Q: Point;Expression(C) = (y^2 = 4*x);PointOnCurve(P, C);Focus(C) = F;Intersection(Directrix(C), xAxis) = N;Intersection(LineOf(N,P),C)=Q;Abs(LineSegmentOf(P, F)) = 3*Abs(LineSegmentOf(Q, F))", "coords": {} }, { "index": 6148, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 8.0, "b": 4.0, "major_axis": "x", "x_coef": 64.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6148.png", "answer": "x^2/36 - y^2/12 = 1", "fact_expr": "C: Hyperbola;G: Ellipse;H: Line;Expression(G) = (x^2 + 4*y^2 = 64);Expression(H) = (x + sqrt(3)*y = 0);Focus(C) = Focus(G);OneOf(Asymptote(C))=H", "coords": {} }, { "index": 6149, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6149.png", "answer": "4/5", "fact_expr": "G: Parabola;p: Number;P: Point;Q: Point;M: Point;l: Line;p>0;Expression(G) = (y^2 = 2*(p*x));Directrix(G) = l;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, Q),l);FootPoint(LineSegmentOf(P, Q),l)=Q;Coordinate(M) = (2, 0);Negation(M=Focus(G));Abs(LineSegmentOf(P, Q)) = Abs(LineSegmentOf(P, M));AngleOf(M, P, Q) = ApplyUnit(120, degree)", "coords": { "M": [ 2.0, 0.0 ] } }, { "index": 6150, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6150.png", "answer": "(1,4)", "fact_expr": "H: Line;Expression(H) = (y = k*x + k);k: Number;G: Ellipse;Expression(G) = (y^2/4 + x^2/m = 1);m: Real;PointOnCurve(Focus(G),yAxis) = True;NumIntersection(H,G) = 2", "coords": {} }, { "index": 6151, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6151.png", "answer": "(x-2)^2 + (y - 2\\sqrt{2})^2 = 9", "fact_expr": "C: Parabola;G: Circle;A: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(A,C);Quadrant(A)=1;Abs(LineSegmentOf(A, F)) = 3;Center(G)=A;Radius(G)=LineSegmentOf(A,F)", "coords": {} }, { "index": 6152, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6152.png", "answer": "(a^2/c,0)", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;F: Point;P: Point;A: Point;c:Number;a > b;b > 0;l:Line;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (c, 0);RightFocus(G) = F;PointOnCurve(F, H);Intersection(H,G) = P;TangentOnPoint(P,G)=l;Intersection(l,xAxis)=A;IsPerpendicular(H,xAxis)", "coords": {} }, { "index": 6153, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6153.png", "answer": "-1/3", "fact_expr": "C: Ellipse;G: Line;O: Origin;A: Point;B: Point;Expression(C) = (x^2/2 + y^2 = 1);Slope(G)=1;PointOnCurve(LeftFocus(C),G);Intersection(G,C)={A,B}", "coords": {} }, { "index": 6154, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6155, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.23606797749979, "b": 2.0, "major_axis": "x", "x_coef": 5.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6155.png", "answer": "5/3", "fact_expr": "G: Ellipse;H: Line;O: Origin;A: Point;B: Point;Expression(G) = (x^2/5 + y^2/4 = 1);PointOnCurve(RightFocus(G), H);Slope(H)=2;Intersection(H,G) = {B, A}", "coords": {} }, { "index": 6156, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "major_axis": "x", "x_coef": 3.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6156.png", "answer": "(0, pm*1)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/3 + y^2 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;A: Point;B: Point;PointOnCurve(A, G);PointOnCurve(B, G);VectorOf(F1, A) = 5*VectorOf(F2, B)", "coords": {} }, { "index": 6157, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6157.png", "answer": "3", "fact_expr": "l: Line;G: Parabola;O: Origin;P: Point;A: Point;B: Point;m:Number;n:Number;m<=n;Expression(G) = (y^2 = 4*x);Expression(l) = (sqrt(3)*x - y - sqrt(3) = 0);Intersection(l, G) = {A, B};Intersection(l,xAxis)=P;VectorOf(O, P) = m*VectorOf(O, A) + n*VectorOf(O, B)", "coords": {} }, { "index": 6158, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6158.png", "answer": "0", "fact_expr": "G: Hyperbola;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 - y^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);Area(TriangleOf(F1, P, F2)) = 1", "coords": {} }, { "index": 6159, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6160, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6160.png", "answer": "2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G);IsPerpendicular(LineSegmentOf(A, B), xAxis);Abs(LineSegmentOf(A, B)) = 4*sqrt(3);F: Point;Focus(G) = F", "coords": {} }, { "index": 6161, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6162, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.4641016151377544, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6162.png", "answer": "(\\sqrt{15}, \\pm 1)", "fact_expr": "G: Hyperbola;P: Point;F1:Point;F2:Point;Expression(G) = (x^2/12 - y^2/4 = 1);PointOnCurve(P,RightPart(G));Area(TriangleOf(P,F1,F2))=4;Focus(G)={F1,F2}", "coords": {} }, { "index": 6163, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6163.png", "answer": "2*x-8*y-9=0", "fact_expr": "l: Line;E: Ellipse;B: Point;A: Point;Expression(E) = (x^2/4 + y^2/2 = 1);Intersection(l, E) = {A, B};Coordinate(MidPoint(LineSegmentOf(A,B)))=(1/2,-1)", "coords": {} }, { "index": 6164, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.4641016151377544, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6164.png", "answer": "1", "fact_expr": "G: Hyperbola;H: Parabola;Expression(G) = (x^2/12 - y^2/4 = 1);Expression(H) = (y^2 = 8*x)", "coords": {} }, { "index": 6165, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6166, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "major_axis": "x", "x_coef": 3.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6166.png", "answer": "y=pm*x", "fact_expr": "C1: Ellipse;Expression(C1) = (x^2/3 + y^2 = 1);C2: Hyperbola;F1: Point;F2: Point;Focus(C1) = {F1, F2};Focus(C2) = {F1, F2};P: Point;Intersection(C1, C2) = P;O: Origin;LineSegmentOf(O, P) = LineSegmentOf(O, F)", "coords": {} }, { "index": 6167, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6167.png", "answer": "12", "fact_expr": "G: Ellipse;m: Number;H: Line;A: Point;B: Point;Expression(G) = (y^2/4 + x^2/m = 1);Expression(H) = (y = x + 2);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 3*sqrt(2)", "coords": {} }, { "index": 6168, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6168.png", "answer": "[9/4,13/3]", "fact_expr": "l: Line;C: Ellipse;M: Point;F: Point;N: Point;O:Origin;Expression(C) = (x^2/4 + y^2/3 = 1);RightFocus(C)=F;PointOnCurve(O,l);Intersection(l, C) = {M, N}", "coords": {} }, { "index": 6169, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6169.png", "answer": "(\\sqrt{10}, +\\infty)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;PointOnCurve(RightVertex(G), L) = True;Slope(L) = 3;L: Line;IsIntersect(L, LeftPart(G)) = True;IsIntersect(L, RightPart(G)) = True", "coords": {} }, { "index": 6170, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6170.png", "answer": "16/5", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);F1: Point;F2: Point;Focus(G) = {F1,F2};P: Point;PointOnCurve(P, G) = True;IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2)) = True", "coords": {} }, { "index": 6171, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6171.png", "answer": "{sqrt(5),sqrt(5)/2}", "fact_expr": "C: Hyperbola;G: Parabola;O:Origin;P: Point;F: Point;Expression(G) = (y^2 = 8*x);Focus(G)=F;Center(C)=O;SymmetryAxis(C)=axis;Intersection(OneOf(Asymptote(C)),G)=P;Abs(LineSegmentOf(P, F)) = 4", "coords": {} }, { "index": 6172, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6172.png", "answer": "y^2/12 - x^2/3 = 1", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/4 = 1);G1: Hyperbola;Asymptote(G) = Asymptote(G1);H: Point;Coordinate(H) = (-1, 4);PointOnCurve(H, G1)", "coords": { "H": [ -1.0, 4.0 ] } }, { "index": 6173, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 5.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6173.png", "answer": "25", "fact_expr": "G: Hyperbola;P: Point;Q: Point;R: Point;Expression(G) = (x^2/25 - y^2/16 = 1);PointOnCurve(P, G);Z: Line;L1: Line;L2: Line;Asymptote(G) = {L1, L2};PointOnCurve(P, Z);IsParallel(Z, xAxis);Intersection(Z, L1) = Q;Intersection(Z, L2) = R", "coords": {} }, { "index": 6174, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6174.png", "answer": "sqrt(5)", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F2: Point;F1: Point;Q: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, RightPart(C));IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2));Intersection(LineOf(P, F2), yAxis) = Q;VectorOf(F2, P) = (2/3)*VectorOf(P, Q)", "coords": {} }, { "index": 6175, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6175.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/k = 1);k: Number;e:Number;Eccentricity(G) = e;In(e, (1, 2)) = True", "coords": {} }, { "index": 6176, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6176.png", "answer": "6", "fact_expr": "G: Parabola;P: Point;F: Point;Expression(G) = (y^2 = 8*x);Focus(G) = F;PointOnCurve(P, G);XCoordinate(P)=4", "coords": {} }, { "index": 6177, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.0, "b": 0.5, "major_axis": "y", "x_coef": 0.25, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6177.png", "answer": "1-sqrt(3)/2", "fact_expr": "G: Ellipse;P: Point;F: Point;Expression(G) = (4*x^2 + y^2 = 1);PointOnCurve(P, G);OneOf(Focus(G))=F", "coords": {} }, { "index": 6178, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6178.png", "answer": "{1, 2}", "fact_expr": "G: Parabola;p: Number;A: Point;B: Point;M: Point;l1: Line;l2: Line;p>0;Expression(G) = (x^2 = 2*(p*y));Coordinate(M) = (2, -2*p);TangentOfPoint(M, G) = {l1, l2};TangentPoint(l1, G) = A;TangentPoint(l2, G) = B;YCoordinate(MidPoint(LineSegmentOf(A, B))) = 6", "coords": {} }, { "index": 6179, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6179.png", "answer": "sqrt(3)/3", "fact_expr": "H: Line;Expression(H) = (x - sqrt(3)*y + sqrt(3) = 0);G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;LeftFocus(G) = F;PointOnCurve(F, H);A: Point;B: Point;Intersection(H, G) = {A, B};C: Point;Intersection(H, yAxis) = C;VectorOf(F, A) = 2*VectorOf(F, C)", "coords": {} }, { "index": 6180, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6180.png", "answer": "4", "fact_expr": "C: Parabola;p: Number;P: Point;p>0;Expression(C) = (y^2 = 2*(p*x));PointOnCurve(P, C);Distance(P, Focus(C)) = 7;Distance(P,yAxis) = 5", "coords": {} }, { "index": 6181, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6181.png", "answer": "17", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2:Point;Expression(G) = (-x^2/20 + y^2/16 = 1);PointOnCurve(P, G);OneOf(Focus(G)) = F1;Distance(P, F1) = 9", "coords": {} }, { "index": 6182, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6183, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6183.png", "answer": "5/2", "fact_expr": "G: Parabola;Expression(G) = (y = x^2);A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G);Distance(MidPoint(LineSegmentOf(A, B)), xAxis) = 1", "coords": {} }, { "index": 6184, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4970015082757973, "b": 1.1157990585381212, "major_axis": "x", "x_coef": 2.25, "y_coef": 1.2500000000000002 }, "optimization": { "final_loss": 1.232595164407831e-31, "converged": true, "iterations": 2 }, "output_path": "results/train/ellipse/problem_6184.png", "answer": "6", "fact_expr": "G: Ellipse;H: Line;F1: Point;F2: Point;A: Point;B: Point;Coordinate(F1) = (-1, 0);Coordinate(F2) = (1, 0);Focus(G) = {F1, F2};Eccentricity(G) = 2/3;PointOnCurve(F1, H);Intersection(H, G) = {A, B}", "coords": { "F1": [ -1.0, 0.0 ], "F2": [ 1.0, 0.0 ] } }, { "index": 6185, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6185.png", "answer": "(3/2, sqrt(3))", "fact_expr": "C: Parabola;P: Point;N: Point;F: Point;L:Line;Focus(C)=F;Expression(C) = (y^2 = 2*x);PointOnCurve(P,C);PointOnCurve(P,L);Quadrant(P)=1;TangentOfPoint(P,C)=L;Intersection(L,Directrix(C))=N;AngleOf(P, N, F) = ApplyUnit(60, degree)", "coords": {} }, { "index": 6186, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6186.png", "answer": "32/3", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;Expression(G) = (y^2 = 8*x);PointOnCurve(Focus(G), H);Inclination(H)=pi/3;Intersection(H, G) = {A, B}", "coords": {} }, { "index": 6187, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6187.png", "answer": "6", "fact_expr": "G: Hyperbola;H: Line;B: Point;A: Point;P: Point;Expression(G) = (x^2 - y^2/3 = 1);Coordinate(P) = (2, 1);PointOnCurve(P, H);Intersection(H, G) = {A, B};MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ 2.0, 1.0 ] } }, { "index": 6188, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6188.png", "answer": "y^2=4*x", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;G: Ellipse;Expression(G) = (x^2/2 + y^2 = 1);Focus(C) = RightFocus(G)", "coords": {} }, { "index": 6189, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6189.png", "answer": "2*sqrt(3)", "fact_expr": "G: Line;Inclination(G) = pi/4;C: Hyperbola;Expression(C) = (x^2/3 - y^2 = 1);PointOnCurve(Focus(C), G);A: Point;B: Point;Intersection(G, C) = {A, B}", "coords": {} }, { "index": 6190, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6190.png", "answer": "(2, 2)", "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (y^2 = 2*x);Coordinate(A) = (3, 2);Focus(G) = F;PointOnCurve(P, G);WhenMin(Abs(LineSegmentOf(P, A)) + Abs(LineSegmentOf(P, F)))", "coords": { "A": [ 3.0, 2.0 ] } }, { "index": 6191, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6192, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6192.png", "answer": "(2,4)", "fact_expr": "M: Point;Coordinate(M) = (3, 4);F: Point;Focus(G) = F;G: Parabola;Expression(G) = (y^2 = 8*x);P: Point;PointOnCurve(P, G) = True;WhenMin(Abs(LineSegmentOf(P,M)) + Abs(LineSegmentOf(P,F)))", "coords": { "M": [ 3.0, 4.0 ] } }, { "index": 6193, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6193.png", "answer": "2*sqrt(3)", "fact_expr": "G: Parabola;H: Circle;Expression(G) = (y^2 = 4*x);Expression(H) = (-4*y + x^2 + y^2 = 0)", "coords": {} }, { "index": 6194, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6194.png", "answer": "y^2 = 8*x", "fact_expr": "H: Parabola;Vertex(H) = O;O: Origin;G: Hyperbola;Expression(G) = (x^2/3 - y^2 = 1);Focus(H) = RightFocus(G)", "coords": {} }, { "index": 6195, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6195.png", "answer": "1", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;O: Origin;M: Point;A: Point;B: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = x/2 + 1);Eccentricity(G) = sqrt(3)/2;Intersection(H, G) = {A, B};PointOnCurve(M, G);LineSegmentOf(O, M) = LineSegmentOf(O, A)/2 + (sqrt(3)/2)*LineSegmentOf(O, B)", "coords": {} }, { "index": 6196, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6196.png", "answer": "2", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;F1: Point;F2: Point;A: Point;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;IsDiameter(LineSegmentOf(F1, F2), G);L1: Line;L2: Line;Asymptote(C) = {L1, L2};Intersection(G, L1) = A;MidPoint(LineSegmentOf(A, F1)) = B;PointOnCurve(B, L2)", "coords": {} }, { "index": 6197, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6197.png", "answer": "2", "fact_expr": "H: Ellipse;Expression(H) = (y^2/3 + x^2/a^2 = 1);a: Number;a>0;P: Point;PointOnCurve(P, H);G: Parabola;Expression(G) = (y^2 = (9/4)*x);PointOnCurve(P, G);F: Point;Focus(G) = F;Abs(LineSegmentOf(P, F)) = 25/16", "coords": {} }, { "index": 6198, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6199, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6199.png", "answer": "sqrt(2)/2", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;A: Point;B: Point;RightVertex(C)=A;UpperVertex(C)=B;Abs(LineSegmentOf(A, B)) = (sqrt(3)/2)*Abs(LineSegmentOf(F1, F2))", "coords": {} }, { "index": 6200, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6200.png", "answer": "1", "fact_expr": "G: Hyperbola;b: Number;b>0;Expression(G) = (x^2/4 - y^2/b^2 = 1);Expression(OneOf(Asymptote(G))) = (y = x/2)", "coords": {} }, { "index": 6201, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6201.png", "answer": "3", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/9 = 1)", "coords": {} }, { "index": 6202, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6202.png", "answer": "(-sqrt(3)/3, sqrt(3)/3)", "fact_expr": "M: Point;Coordinate(M) = (x0, y0);x0: Number;y0: Number;C: Hyperbola;Expression(C) = (x^2/2 - y^2 = 1);PointOnCurve(M, C) = True;F1: Point;F2: Point;Focus(C) = {F1, F2};DotProduct(VectorOf(M, F1), VectorOf(M, F2)) < 0", "coords": {} }, { "index": 6203, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6203.png", "answer": "3*sqrt(26)/13", "fact_expr": "H: Parabola;Expression(H) = (y^2 = 2*(a*x));a: Number;a>0;F: Point;Focus(H) = F;G: Hyperbola;Expression(G) = (-x^2/9 + y^2/4 = 1);M: Point;N: Point;Intersection(Directrix(H), G) = {M, N};AngleOf(M, F, N) = ApplyUnit(120, degree)", "coords": {} }, { "index": 6204, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.9560877084732056, "b": 1.622148733645621, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 2.7588219936233933e-05, "converged": true, "iterations": 62 }, "output_path": "results/train/ellipse/problem_6204.png", "answer": "[sqrt(3)/3,sqrt(3)/2]", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(H) = (1, sqrt(2));PointOnCurve(H, G);Range(Length(MajorAxis(G)))=[4,6]", "coords": { "H": [ 1.0, 1.4142135623730951 ] } }, { "index": 6205, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6206, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6206.png", "answer": "1", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;M: Point;B: Point;A: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(M) = (0, 1);PointOnCurve(M, l);Slope(l) = 1;VectorOf(B, M) = 2*VectorOf(A, M);FocalLength(C) = 2*sqrt(10);L1:Line;L2:Line;Asymptote(C)={L1,L2};Intersection(l,L1)=A;Intersection(l,L2)=B", "coords": { "M": [ 0.0, 1.0 ] } }, { "index": 6207, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6207.png", "answer": "2", "fact_expr": "C: Hyperbola;b: Number;a: Number;F: Point;D: Point;O: Origin;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C)=F;PointOnCurve(F,LineSegmentOf(F,D));IsPerpendicular(LineSegmentOf(F,D),Asymptote(C));FootPoint(LineSegmentOf(F,D),Asymptote(C))=D;Abs(LineSegmentOf(F, D)) = (sqrt(3)/2)*Abs(LineSegmentOf(O, F))", "coords": {} }, { "index": 6208, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6209, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6210, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6210.png", "answer": "7/4", "fact_expr": "A: Point;B: Point;LineSegmentOf(A,B) = 4;Endpoint(LineSegmentOf(A,B)) = {A,B};G: Parabola;Expression(G) = (y^2=x);MidPoint(LineSegmentOf(A,B)) = M;M: Point;PointOnCurve(Endpoint(LineSegmentOf(A,B)),G)", "coords": {} }, { "index": 6211, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6211.png", "answer": "y^2=4*x", "fact_expr": "C: Parabola;p: Number;F: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;Distance(F,Directrix(C)) = 2", "coords": {} }, { "index": 6212, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6212.png", "answer": "2", "fact_expr": "G: Parabola;M: Point;N: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(M, G);PointOnCurve(N, G);Abs(LineSegmentOf(M, F)) + Abs(LineSegmentOf(N, F)) = 6", "coords": {} }, { "index": 6213, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6213.png", "answer": "20", "fact_expr": "G: Ellipse;B: Point;A: Point;F1: Point;F2: Point;Expression(G) = (x^2/25 + y^2/9 = 1);RightFocus(G)=F2;PointOnCurve(F2,LineOf(A,B));Intersection(LineOf(A,B),G) = {A, B};LeftFocus(G) = F1", "coords": {} }, { "index": 6214, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 5.196152422706632, "major_axis": "y", "x_coef": 27.0, "y_coef": 36.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6214.png", "answer": "y^2/4 - x^2/5 = 1", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/27 + y^2/36 = 1);IsIntersect(G, H);Focus(G) = Focus(H);Coordinate(OneOf(Intersection(G, H))) = (sqrt(15), 4)", "coords": {} }, { "index": 6215, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6215.png", "answer": "9", "fact_expr": "G: Hyperbola;A: Point;P: Point;F: Point;Expression(G) = (x^2/4 - y^2/12 = 1);Coordinate(A) = (1, 4);LeftFocus(G) = F;PointOnCurve(P, RightPart(G))", "coords": { "A": [ 1.0, 4.0 ] } }, { "index": 6216, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6217, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6217.png", "answer": "sqrt(10)/3", "fact_expr": "G: Hyperbola;b: Number;a: Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G)))=(x+3*y=0)", "coords": {} }, { "index": 6218, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6219, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6220, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 1.0, -0.0 ], "radius": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_6220.png", "answer": "x^2/4 + y^2/3 = 1", "fact_expr": "G: Ellipse;PointOnCurve(Focus(G), xAxis);Eccentricity(G) = 0.5;C: Circle;Expression(C) = (-2*x + x^2 + y^2 - 15 = 0);Length(MajorAxis(G)) = Radius(C)", "coords": {} }, { "index": 6221, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6221.png", "answer": "13", "fact_expr": "H: Parabola;Expression(H) = (y = x^2/16);G: Hyperbola;Expression(G) = (x^2/3 - y^2/m = -1);m: Number;Focus(H) = UpperFocus(G)", "coords": {} }, { "index": 6222, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6222.png", "answer": "sqrt(2) + 1", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;H: Line;PointOnCurve(F2, H);IsPerpendicular(H, xAxis) ;Intersection(H, G) = {A, B};A: Point;B: Point;DotProduct(VectorOf(F1, A), VectorOf(F1, B)) = 0", "coords": {} }, { "index": 6223, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6223.png", "answer": "[1,5)", "fact_expr": "G: Ellipse;m: Real;H: Line;k: Number;Expression(G) = (x^2/5 + y^2/m = 1);Expression(H) = (y = k*x + 1);PointOnCurve(Focus(G),xAxis);IsIntersect(H,G)", "coords": {} }, { "index": 6224, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6225, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6226, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6227, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.4641016151377544, "major_axis": "x", "x_coef": 16.0, "y_coef": 12.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6227.png", "answer": "4*sqrt(5)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/12 = 1);P: Point;PointOnCurve(P, G);H: Line;Expression(H) = (x - 2*y - 12 = 0)", "coords": {} }, { "index": 6228, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6228.png", "answer": "2*sqrt(2)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/2 + y^2 = 1);F: Point;LeftFocus(G) = F;PointOnCurve(F, LineOf(M, N)) = True;Intersection(LineOf(M, N), G) = {M, N};N: Point;M: Point;O: Origin;PointOnCurve(O, LineOf(P, Q)) = True;IsParallel(LineOf(P, Q), LineSegmentOf(M, N)) = True;Intersection(LineSegmentOf(P, Q), G) = {P, Q};Q: Point;P: Point", "coords": {} }, { "index": 6229, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6229.png", "answer": "6", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);l: Line;PointOnCurve(Focus(G), l);P: Point;x1: Number;y1: Number;Coordinate(P) = (x1, y1);Q: Point;x2: Number;y2: Number;Coordinate(Q) = (x2, y2);Intersection(l, G) = {P, Q};x1+x2=4", "coords": {} }, { "index": 6230, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6230.png", "answer": "3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;P: Point;PointOnCurve(P, G);A: Point;Coordinate(A) = (1, 1)", "coords": { "A": [ 1.0, 1.0 ] } }, { "index": 6231, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.3300262689590454, "b": 2.30381441116333, "orientation": "horizontal" }, "optimization": { "final_loss": 6.658357051492203e-06, "converged": true, "iterations": 122 }, "output_path": "results/train/hyperbola/problem_6231.png", "answer": "3", "fact_expr": "G: Hyperbola;a: Number;H: Parabola;M: Point;y0:Number;Expression(G) = (-y^2/3 + x^2/a = 1);Expression(H) = (y^2 = 4*(a*x));Coordinate(M) = (2, y0);Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x);PointOnCurve(M,H);Focus(H)=F;F:Point", "coords": {} }, { "index": 6232, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6232.png", "answer": "pm*sqrt(5)", "fact_expr": "G: Hyperbola;a: Number;H: Parabola;Expression(G) = (-y^2/4 + x^2/a^2 = 1);Expression(H) = (y^2 = -12*x);Focus(H) = OneOf(Focus(G))", "coords": {} }, { "index": 6233, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6233.png", "answer": "(0, \\sqrt{2}/2]", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;B: Point;UpperVertex(C) = B;P: Point;PointOnCurve(P,C);Abs(LineSegmentOf(P,B)) <= 2*b", "coords": {} }, { "index": 6234, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 8.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6234.png", "answer": "2", "fact_expr": "P: Point;Coordinate(P) = (x1, y1);x1: Number;y1: Number;G: Ellipse;Expression(G) = (x^2/8 + y^2/2 = 1);PointOnCurve(P, G) = True;Q: Point;Coordinate(Q) = (x2, y2);x2: Number;y2: Number;H: Line;Expression(H) = (x + 2*y - 8 = 0);PointOnCurve(Q, H) = True", "coords": {} }, { "index": 6235, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6236, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6236.png", "answer": "y=pm*x", "fact_expr": "C: Hyperbola;b: Number;a: Number;H: Circle;O: Origin;F: Point;P: Point;Q: Point;D:Circle;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(D) = (x^2 + y^2 = a^2);RightFocus(C)=F;IsDiameter(LineSegmentOf(O,F),H);Intersection(H,D)={P,Q};Abs(LineSegmentOf(P, Q)) = Abs(LineSegmentOf(O, F))", "coords": {} }, { "index": 6237, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6238, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6238.png", "answer": "y=-1", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y)", "coords": {} }, { "index": 6239, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6239.png", "answer": "x^2/20 - y^2/5 = 1", "fact_expr": "H: Line;Expression(H) = (2*x + y - 10 = 0);G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;PointOnCurve(OneOf(Focus(G)),H) = True;IsPerpendicular(H,OneOf(Asymptote(G))) = True", "coords": {} }, { "index": 6240, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6240.png", "answer": "(x-1)^2+y^2=4", "fact_expr": "G: Parabola;H: Circle;F: Point;l: Line;Expression(G) = (y^2 = 4*x);Focus(G) = F;Directrix(G) = l;Center(H)=F;IsTangent(l,H)", "coords": {} }, { "index": 6241, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6241.png", "answer": "{y=0,x-y+2=0,x+y+2=0}", "fact_expr": "l: Line;G: Parabola;P: Point;Expression(G) = (y^2 = 8*x);Coordinate(P) = (-2, 0);PointOnCurve(P,l);NumIntersection(l,G)=1", "coords": { "P": [ -2.0, 0.0 ] } }, { "index": 6242, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6242.png", "answer": "sqrt(3)/3", "fact_expr": "G: Ellipse;b: Number;a: Number;F2: Point;A: Point;P: Point;F1: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) =F2;LowerVertex(G)=A;Intersection(LineOf(A,F2),G) ={A,P};Abs(LineSegmentOf(P, F1)) = Abs(LineSegmentOf(P, A))", "coords": {} }, { "index": 6243, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6243.png", "answer": "x^2/16 - y^2/9 = 1", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/25 + y^2/9 = 1);Focus(G)=Endpoint(MajorAxis(H));Focus(H)=Vertex(G)", "coords": {} }, { "index": 6244, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6244.png", "answer": "2", "fact_expr": "G: Parabola;p: Number;P: Point;Expression(G) = (y^2 = 2*(p*x));PointOnCurve(P, G);XCoordinate(P)=2;Distance(P, Focus(G)) = 3", "coords": {} }, { "index": 6245, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6245.png", "answer": "x^2+2*y^2-8*x-4*y+17=0", "fact_expr": "G: Curve;M: Point;A: Point;P: Point;x1:Number;y1:Number;Expression(G) = (x^2 + 2*y^2 = 4);Coordinate(Z) = (8, 2);Coordinate(M) = (x1, y1);PointOnCurve(M, G);MidPoint(LineSegmentOf(M, A)) = P;Z: Point;A = Z", "coords": { "Z": [ 8.0, 2.0 ] } }, { "index": 6246, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6246.png", "answer": "4/5", "fact_expr": "C: Ellipse;A: Point;P: Point;B: Point;Q: Point;M: Point;N: Point;Expression(C) = (x^2/4 + y^2 = 1);LeftVertex(C) = A;RightVertex(C) = B;PointOnCurve(M, xAxis);L2:Line;PointOnCurve(M,L2);IsPerpendicular(L2,xAxis);Intersection(C, L2) = {P, Q};L1:Line;PointOnCurve(M,L1);IsPerpendicular(L1,LineSegmentOf(A, P));Intersection(L1, LineSegmentOf(B, Q)) = N", "coords": {} }, { "index": 6247, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6248, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6248.png", "answer": "3*sqrt(6)", "fact_expr": "G: Parabola;p: Number;H: Line;A: Point;F: Point;B: Point;O: Origin;p>0;Expression(G) = (y^2 = 2*p*x);Focus(G)=F;PointOnCurve(F,H);Intersection(H,G)={A,B};VectorOf(A, F) = 2*VectorOf(F, B);Area(TriangleOf(O,A,B))=sqrt(3)*Abs(LineSegmentOf(A,B))", "coords": {} }, { "index": 6249, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6249.png", "answer": "1", "fact_expr": "G: Hyperbola;H: Parabola;Expression(G) = (x^2/3 - y^2 = 1);Expression(H) = (y^2 = 8*x)", "coords": {} }, { "index": 6250, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6250.png", "answer": "6", "fact_expr": "G: Parabola;p: Number;H: Line;P0: Point;p>0;Expression(G) = (x^2 = 2*(p*y));Expression(H) = (y = x - 5);PointOnCurve(P0, G);Min(Distance(P0, H)) = sqrt(2)", "coords": {} }, { "index": 6251, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6251.png", "answer": "(1, 3]", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, RightPart(C));Max(Abs(LineSegmentOf(P, F1))^2/(4*a^2 + Abs(LineSegmentOf(P, F2))^2)) = 2", "coords": {} }, { "index": 6252, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6252.png", "answer": "(2,+oo)", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;M: Point;N: Point;P: Point;F1:Point;l:Line;t:Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (1, 0);RightVertex(C)=F1;Coordinate(P) = (0, -1);PointOnCurve(P,G);Intersection(G,RightPart(C))={M,N};Negation(M=N);Expression(Asymptote(C)) = (y = pm*x);PerpendicularBisector(LineSegmentOf(M,N))=l;Intercept(l,yAxis)=t", "coords": { "F1": [ 1.0, 0.0 ], "P": [ 0.0, -1.0 ] } }, { "index": 6253, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6253.png", "answer": "10", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/16 - y^2/9 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(P, G);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0", "coords": {} }, { "index": 6254, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6254.png", "answer": "(-3/2, 3/2)", "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2 = 1);P: Point;a: Number;Coordinate(P) = (a, 0);PointOnCurve(P, xAxis);O: Circle;Center(O) = P;NumIntersection(C, O) = 4", "coords": {} }, { "index": 6255, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6255.png", "answer": "x^2-4*y^2=1", "fact_expr": "P: Point;PointOnCurve(P, G) = True;G: Curve;Expression(G) = (x^2/4 - y^2 = 1);O: Origin;M: Point;MidPoint(LineSegmentOf(O, P)) = M", "coords": {} }, { "index": 6256, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6256.png", "answer": "2", "fact_expr": "C: Parabola;A: Point;B: Point;F: Point;Expression(C) = (y^2 = 4*x);Coordinate(A) = (5, 4);Focus(C) = F;PointOnCurve(B, C);WhenMin(Perimeter(TriangleOf(A, B, F)))", "coords": { "A": [ 5.0, 4.0 ] } }, { "index": 6257, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6258, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6258.png", "answer": "(3+2*sqrt(2))/4", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G);F: Point;RightFocus(G) = F;PointOnCurve(F, LineSegmentOf(A, B))", "coords": {} }, { "index": 6259, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6259.png", "answer": "-1", "fact_expr": "H: Line;Expression(H) = (a*x - y + 1 = 0);a: Real;G: Parabola;Expression(G) = (y^2 = 4*x);PointOnCurve(Focus(G), H)", "coords": {} }, { "index": 6260, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6261, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6261.png", "answer": "(pm*5, 0)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);WhenMin(Distance(P, F1)*Distance(P, F2))", "coords": {} }, { "index": 6262, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6262.png", "answer": "24", "fact_expr": "M: Parabola;l1:Line;l2:Line;A: Point;C: Point;B: Point;D: Point;F:Point;Expression(M) = (y^2 = 8*x);Focus(M)=F;PointOnCurve(F,l1);PointOnCurve(F,l2);Slope(l1)*Slope(l2)=-2;Intersection(l1,M)={A,C};Intersection(l2,M)={B,D}", "coords": {} }, { "index": 6263, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6264, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6264.png", "answer": "y=-1/4", "fact_expr": "C: Parabola;Expression(C) = (y = x^2)", "coords": {} }, { "index": 6265, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.3300262689590454, "b": 2.30381441116333, "orientation": "horizontal" }, "optimization": { "final_loss": 6.658357051492203e-06, "converged": true, "iterations": 122 }, "output_path": "results/train/hyperbola/problem_6265.png", "answer": "(0, 1/2]", "fact_expr": "D: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(D) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(D) = F1;RightFocus(D) = F2;PointOnCurve(P, RightPart(D));Expression(Asymptote(D)) = (y = pm*sqrt(3)*x)", "coords": {} }, { "index": 6266, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6267, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6267.png", "answer": "9", "fact_expr": "G: Parabola;H: Line;A: Point;F: Point;B: Point;Expression(G) = (y^2 = 8*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, F)) = 6", "coords": {} }, { "index": 6268, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 3.0, "major_axis": "x", "x_coef": 36.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6268.png", "answer": "x + 2*y - 8 = 0", "fact_expr": "G: Ellipse;P: Point;Expression(G) = (x^2/36 + y^2/9 = 1);Coordinate(P) = (4, 2);Z: LineSegment;MidPoint(Z) = P;IsChordOf(Z, G)", "coords": { "P": [ 4.0, 2.0 ] } }, { "index": 6269, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6269.png", "answer": "(-2, -1)", "fact_expr": "H: Line;Expression(H) = (y = x + 1);G: Parabola;Expression(G) = (y^2 = -2*x)", "coords": {} }, { "index": 6270, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6270.png", "answer": "5/3", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1)", "coords": {} }, { "index": 6271, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.1622776601683795, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6271.png", "answer": "4*sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/10 - y^2/2 = 1)", "coords": {} }, { "index": 6272, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6272.png", "answer": "x=-2", "fact_expr": "G: Hyperbola;H: Parabola;p: Number;Expression(G) = (x^2 - y^2/3 = 1);Expression(H) = (y^2 = 2*p*x);Focus(H) = RightFocus(G)", "coords": {} }, { "index": 6273, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6273.png", "answer": "1/3", "fact_expr": "C: Ellipse;a: Number;b: Number;G: Line;A: Point;F: Point;B: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;PointOnCurve(F, G);Intersection(G, C) = {A, B};LineSegmentOf(A, F)/LineSegmentOf(B, F) = 2", "coords": {} }, { "index": 6274, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6274.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;F2: Point;P: Point;F1: Point;e: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(P,LeftPart(G));Abs(LineSegmentOf(P, F1)) = Abs(LineSegmentOf(F1, F2));Distance(F1,LineOf(P,F2)) = sqrt(7)*a;Eccentricity(G) = e", "coords": {} }, { "index": 6275, "success": false, "error": "Circle from diameter: cannot parse diameter pattern" }, { "index": 6276, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6277, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6278, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6278.png", "answer": "sqrt(2)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;A: Point;Coordinate(A) = (0, 2);l: Line;Directrix(G) = l;B: Point;Intersection(LineSegmentOf(F, A), G) = B;L: Line;PointOnCurve(B, L);IsPerpendicular(L, l);M: Point;FootPoint(L, l) = M;IsPerpendicular(LineSegmentOf(A, M), LineSegmentOf(M, F))", "coords": { "A": [ 0.0, 2.0 ] } }, { "index": 6279, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_6279.png", "answer": "x^2 + y^2/(3/4) = 1", "fact_expr": "G: Circle;P: Point;P0: Point;M: Point;Expression(G) = (x^2 + y^2 = 1);PointOnCurve(P, G);PointOnCurve(P,LineSegmentOf(P,P0));IsPerpendicular(xAxis,LineSegmentOf(P,P0));FootPoint(xAxis,LineSegmentOf(P,P0))=P0;VectorOf(M, P0) = (sqrt(3)/2)*VectorOf(P, P0);Locus(M)=C;C:Curve", "coords": {} }, { "index": 6280, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6281, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6281.png", "answer": "b", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1)", "coords": {} }, { "index": 6282, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6282.png", "answer": "1/2", "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;B: Point;F: Point;P:Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F;RightVertex(G)=A;PointOnCurve(B,G);IsPerpendicular(LineSegmentOf(B,F),xAxis);Intersection(LineOf(A,B),yAxis)=P;VectorOf(A,P)=2*VectorOf(P,B)", "coords": {} }, { "index": 6283, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6283.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;B: Point;F: Point;A: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);OneOf(Focus(G))=F;OneOf(Endpoint(ImageinaryAxis(G)))=B;Intersection(LineSegmentOf(B,F),OneOf(Asymptote(G)))= A;VectorOf(F, A) = 2*VectorOf(A, B)", "coords": {} }, { "index": 6284, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6284.png", "answer": "pm*2*sqrt(2)", "fact_expr": "G: Parabola;H: Line;B: Point;A: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G)=F;PointOnCurve(F, H);Intersection(H, G) = {A, B};VectorOf(F, A) = 2*VectorOf(B, F)", "coords": {} }, { "index": 6285, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6285.png", "answer": "sqrt(13)/2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Expression(Asymptote(G)) = (y = pm*x*3/2);e: Number;Eccentricity(G) = e", "coords": {} }, { "index": 6286, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6287, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6287.png", "answer": "{(x+2*y-1=0),(x-2*y+1=0)}", "fact_expr": "A: Point;B: Point;G: Parabola;Expression(G) = (y^2 = 4*x);PointOnCurve(Focus(G), LineSegmentOf(A, B)) = True;IsChordOf(LineSegmentOf(A, B), G) = True;LineSegmentOf(A, B) = 20", "coords": {} }, { "index": 6288, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6288.png", "answer": "2", "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (x^2/4 - y^2/m = 1);Expression(Asymptote(G)) = (y = pm*x*(sqrt(2)/2))", "coords": {} }, { "index": 6289, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6290, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6290.png", "answer": "y^2 = -12x", "fact_expr": "G: Hyperbola;H: Parabola;Expression(G) = (16*x^2 - 9*y^2 = 144);Vertex(H) = Center(G);Focus(H)=LeftVertex(G)", "coords": {} }, { "index": 6291, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6291.png", "answer": "2*sqrt(10)/5", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;G: Line;Expression(G) = (2*x + y + 1 = 0);Q: Point;Intersection(G, xAxis) = Q;P: Point;PointOnCurve(P, C)", "coords": {} }, { "index": 6292, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6293, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6293.png", "answer": "sqrt(17)/3", "fact_expr": "C: Hyperbola;a: Number;b: Number;G: Circle;O: Origin;A: Point;B: Point;F: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;Center(G)=F;Radius(G)=a;Intersection(G, OneOf(Asymptote(C))) = {A, B};VectorOf(O, A) = 2*VectorOf(O, B)", "coords": {} }, { "index": 6294, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6294.png", "answer": "pi/3, 2*pi/3", "fact_expr": "C: Parabola;F: Point;A: Point;Expression(C) = (y^2 = 8*x);Focus(C) = F;PointOnCurve(A, C);Abs(LineSegmentOf(F, A)) = 8", "coords": {} }, { "index": 6295, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6295.png", "answer": "{y^2=4*x,y^2=16*x}", "fact_expr": "C: Parabola;p: Number;G: Circle;H: Point;M: Point;F: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(H) = (0, 2);Focus(C) = F;PointOnCurve(M, C);Abs(LineSegmentOf(M, F)) = 5;IsDiameter(LineSegmentOf(M, F),G);PointOnCurve(H, G)", "coords": { "H": [ 0.0, 2.0 ] } }, { "index": 6296, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6296.png", "answer": "1/4", "fact_expr": "A: Point;Coordinate(A) = (x1, y1);x1: Number;y1: Number;B: Point;Coordinate(B) = (x2, y2);x2: Number;y2: Number;C: Parabola;Expression(C) = (x^2 = 2*(p*y));p: Number;p>0;PointOnCurve(A, C) = True;PointOnCurve(B, C) = True;Negation(A=B);Expression(PerpendicularBisector(LineSegmentOf(A,B))) = (y=x+b);b: Number;x1+x2=-1/2", "coords": {} }, { "index": 6297, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6297.png", "answer": "sqrt(5)/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Point;a>0;b>0;Expression(G) = (-x^2/b^2 + y^2/a^2 = 1);Coordinate(H) = (1, 2);PointOnCurve(H, Asymptote(G))", "coords": { "H": [ 1.0, 2.0 ] } }, { "index": 6298, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6298.png", "answer": "3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;A: Point;PointOnCurve(A, yAxis);B: Point;MidPoint(LineSegmentOf(A, F)) = B;PointOnCurve(B, G)", "coords": {} }, { "index": 6299, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6299.png", "answer": "x^2/8+y^2/2=1", "fact_expr": "G: Hyperbola;C: Ellipse;e: Number;Expression(G) = (x^2 - 2*y^2 = 4);Eccentricity(C) = e;e = sqrt(3)/2;Focus(C) = Focus(G)", "coords": {} }, { "index": 6300, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6300.png", "answer": "4", "fact_expr": "G: Parabola;H: Point;I: Point;M:Point;Expression(G) = (y = -x^2/4);Coordinate(H) = (0, -1);Coordinate(I) = (1, -3);PointOnCurve(M,G)", "coords": { "H": [ 0.0, -1.0 ], "I": [ 1.0, -3.0 ] } }, { "index": 6301, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.3300262689590454, "b": 2.30381441116333, "orientation": "horizontal" }, "optimization": { "final_loss": 6.658357051492203e-06, "converged": true, "iterations": 122 }, "output_path": "results/train/hyperbola/problem_6301.png", "answer": "2", "fact_expr": "H: Parabola;Expression(H) = (y^2 = 4*x);G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Focus(H) = RightVertex(G);Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x)", "coords": {} }, { "index": 6302, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6303, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6303.png", "answer": "5", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P: Point;PointOnCurve(P, G);A: Point;B: Point;Coordinate(A) = (1, 0);Coordinate(B) = (4, 2)", "coords": { "A": [ 1.0, 0.0 ], "B": [ 4.0, 2.0 ] } }, { "index": 6304, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6305, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6305.png", "answer": "{pm*1, pm*sqrt(2)}", "fact_expr": "l: Line;C: Hyperbola;k: Number;Expression(C) = (x^2 - y^2 = 1);Expression(l) = (y = k*x + 1);NumIntersection(l, C) = 1", "coords": {} }, { "index": 6306, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.463192105293274, "b": 2.070901870727539, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4661259228887502e-06, "converged": true, "iterations": 106 }, "output_path": "results/train/hyperbola/problem_6306.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;F: Point;M: Point;P: Point;l: Line;lambda: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;PointOnCurve(F, l);IsPerpendicular(Asymptote(G), l);FootPoint(Asymptote(G), l) = M;Intersection(l, yAxis) = P;VectorOf(F, M) = lambda*VectorOf(M, P);Eccentricity(G) = sqrt(3)", "coords": {} }, { "index": 6307, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6307.png", "answer": "sqrt(3)", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;RightFocus(C) = F;G: Circle;Expression(G) = (x^2 + y^2 = b^2);Z1: Line;Z2: Line;TangentOfPoint(F, G) = {Z1, Z2};IsPerpendicular(Z1, Z2)", "coords": {} }, { "index": 6308, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6308.png", "answer": "{x+y+1=0, x-y+1=0}", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;P: Point;PointOnCurve(P, C);A: Point;Coordinate(A) = (-1, 0);WhenMax(Abs(LineSegmentOf(P, A))/Abs(LineSegmentOf(P, F)))", "coords": { "A": [ -1.0, 0.0 ] } }, { "index": 6309, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6309.png", "answer": "x^2/3 - y^2/12 = 1", "fact_expr": "C: Hyperbola;D: Hyperbola;Expression(C) = (x^2 - y^2/4 = 1);Asymptote(C) = Asymptote(D);P: Point;Coordinate(P) = (2, 2);PointOnCurve(P, D)", "coords": { "P": [ 2.0, 2.0 ] } }, { "index": 6310, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6311, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6311.png", "answer": "{pi/3,2*pi/3}", "fact_expr": "l: Line;C: Parabola;A: Point;F: Point;B: Point;p:Number;p>0;Expression(C) = (y^2 = 2*p*x);Focus(C) = F;PointOnCurve(F,l);Intersection(l,C)={A,B};Abs(LineSegmentOf(A, F)) = 3*Abs(LineSegmentOf(B, F))", "coords": {} }, { "index": 6312, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6312.png", "answer": "0", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;M: Point;Coordinate(M) = (2*p, 0);H: Line;PointOnCurve(M, H);A: Point;B: Point;Intersection(H, G) = {A, B};O: Origin", "coords": {} }, { "index": 6313, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 5.0, "b": 3.3166247903554, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6313.png", "answer": "6/5", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/25 - y^2/11 = 1)", "coords": {} }, { "index": 6314, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6314.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;PointOnCurve(H, OneOf(Asymptote(G)));H: Point;Coordinate(H) = (1, 2)", "coords": { "H": [ 1.0, 2.0 ] } }, { "index": 6315, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6315.png", "answer": "-8", "fact_expr": "H: Line;Expression(H) = (2*x + y + 4 = 0);G: Parabola;Expression(G) = (y^2 = a*x);a: Real;PointOnCurve(Focus(G),H) = True", "coords": {} }, { "index": 6316, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6316.png", "answer": "(\\sqrt{5}-1)/2", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;M: Point;LeftVertex(G) = M;N: Point;UpperVertex(G) = N;F: Point;RightFocus(G) = F;DotProduct(VectorOf(N, M), VectorOf(N, F)) = 0", "coords": {} }, { "index": 6317, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6318, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6318.png", "answer": "[4*sqrt(2),+oo)", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;PointOnCurve(P, C);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;Area(TriangleOf(P, F1, F2)) = 4", "coords": {} }, { "index": 6319, "success": false, "error": "\n= 1 $ $ (a > b > 0) $, the left and right foci are $ F_{1} $ and $\n ^\nParseException: Expected end of text, found '$' (at char 4), (line:1, col:5)" }, { "index": 6320, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6321, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6322, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6323, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6323.png", "answer": "sqrt(5)", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(l)=(2*x+y-3=0);IsParallel(OneOf(Asymptote(C)),l)", "coords": {} }, { "index": 6324, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6324.png", "answer": "-3", "fact_expr": "G: Parabola;H: Line;O: Origin;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G)=F;PointOnCurve(F, H);Intersection(H, G) = {A, B}", "coords": {} }, { "index": 6325, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6325.png", "answer": "3*sqrt(2)", "fact_expr": "G: Parabola;H: Line;A: Point;F: Point;B: Point;C: Point;D: Point;l: Line;Expression(G) = (y^2 = 4*x);Focus(G) = F;Directrix(G) = l;PointOnCurve(F,H);Intersection(H, G) = {A, B};L1:Line;L2:Line;PointOnCurve(A,L1);PointOnCurve(B,L2);IsPerpendicular(L1,l);IsPerpendicular(L2,l);FootPoint(L1,l)=C;FootPoint(L2,l)=D;Abs(LineSegmentOf(A, F)) = 2*Abs(LineSegmentOf(B, F))", "coords": {} }, { "index": 6326, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6326.png", "answer": "3*sqrt(2)/4", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;P: Point;F1: Point;e: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = x*(b/(3*a)));Intersection(H, LeftPart(G)) = P;LeftFocus(G) = F1;IsPerpendicular(LineSegmentOf(P, F1), xAxis);Eccentricity(G) = e", "coords": {} }, { "index": 6327, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6327.png", "answer": "sqrt(5)/2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;O: Origin;A: Point;B: Point;F:Point;RightFocus(G)=F;L1:Line;L2:Line;L3:Line;Asymptote(G)={L2,L3};a > b;b > 0;PointOnCurve(F,L1);Intersection(L1,L2)=A;Intersection(L1,L3)=B;AngleOf(O, A, B) = ApplyUnit(90, degree);Radius(InscribedCircle(TriangleOf(O,A,B)))=a/3", "coords": {} }, { "index": 6328, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6329, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6330, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.125, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6330.png", "answer": "4*x-y-1=0", "fact_expr": "l: Line;A: Point;Coordinate(A) = (1, 3);G: Curve;Expression(G) = (y = 2*x^2);PointOnCurve(A, l);M: Point;N: Point;Intersection(l, G) = {M, N};MidPoint(LineSegmentOf(M, N)) = A", "coords": { "A": [ 1.0, 3.0 ] } }, { "index": 6331, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6332, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 5.196152422706632, "major_axis": "y", "x_coef": 27.0, "y_coef": 36.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6332.png", "answer": "y^2/4 - x^2/5 = 1", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/27 + y^2/36 = 1);Focus(G) = Focus(H);IsIntersect(G, H) = True;Coordinate(OneOf(Intersection(G, H))) = (sqrt(15), 4)", "coords": {} }, { "index": 6333, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6333.png", "answer": "y = pm*sqrt(3)*x", "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2/3 = 1)", "coords": {} }, { "index": 6334, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6334.png", "answer": "7", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 - y^2/3 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(P,G);LineSegmentOf(P, F1) = 3", "coords": {} }, { "index": 6335, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6335.png", "answer": "{(x-1)^2+y^2=2,(x+1)^2+y^2=2}", "fact_expr": "G: Parabola;H: Line;F: Point;P: Point;Q: Point;Expression(G) = (x^2 = 4*y);Focus(G) = F;Intersection(Directrix(G),yAxis)=Q;PointOnCurve(Q,H);TangentPoint(H,G)=P", "coords": {} }, { "index": 6336, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.125, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6336.png", "answer": "{(x=1),(y=4*x-2)}", "fact_expr": "l: Line;M: Point;Coordinate(M) = (1, 2);PointOnCurve(M, l);G: Parabola;Expression(G) = (y = 2*x^2);NumIntersection(l, G) = 1", "coords": { "M": [ 1.0, 2.0 ] } }, { "index": 6337, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6337.png", "answer": "3", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/2 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);PointOnCurve(MidPoint(LineSegmentOf(P, F1)), yAxis)", "coords": {} }, { "index": 6338, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6338.png", "answer": "27*sqrt(13)/13", "fact_expr": "G: Hyperbola;Expression(G) = (4*x^2 - 9*y^2 = 36);P: Point;PointOnCurve(P, G);Distance(P, RightFocus(G)) = 3", "coords": {} }, { "index": 6339, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6339.png", "answer": "8", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Ellipse;a1: Number;b1: Number;F1: Point;P: Point;F2: Point;e1:Number;e2:Number;a1>0;b1>0;Expression(G) = (-y^2/b1^2 + x^2/a1^2 = 1);a > b;b > 0;Expression(H) = (y^2/b^2 + x^2/a^2 = 1);Focus(H)=Focus(G);Focus(H)={F1,F2};Focus(G)={F1,F2};OneOf(Intersection(H, G)) = P;AngleOf(F1, P, F2) = pi/2;Eccentricity(H) = e1;Eccentricity(G) = e2;e1*e2 = 2", "coords": {} }, { "index": 6340, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6340.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 + (x - sqrt(2))^2 = 1);IsTangent(Asymptote(G),H)", "coords": {} }, { "index": 6341, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6342, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6343, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6343.png", "answer": "y=pm*sqrt(3)*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/12 = 1)", "coords": {} }, { "index": 6344, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6344.png", "answer": "sqrt(3)+sqrt(2)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;LeftFocus(G) = F1;Slope(H) = sqrt(2);PointOnCurve(F1, H) = True;H: Line;Intersection(H, yAxis) = A;Intersection(H, RightPart(G)) = B;A: Point;B: Point;VectorOf(F1, A) = VectorOf(A, B)", "coords": {} }, { "index": 6345, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6345.png", "answer": "2*sqrt(13)", "fact_expr": "C: Hyperbola;b: Number;b>0;Expression(C) = (x^2/4 - y^2/b^2 = 1);Expression(OneOf(Asymptote(C)))=(3*x+2*y=0)", "coords": {} }, { "index": 6346, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6346.png", "answer": "8", "fact_expr": "G: Parabola;H: Line;Expression(G) = (y^2 = 4*x);Expression(H) = (y = x - 1)", "coords": {} }, { "index": 6347, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.732385540716972, "b": 2.4497476064802375, "orientation": "horizontal" }, "optimization": { "final_loss": 2.601017381791051e-06, "converged": true, "iterations": 91 }, "output_path": "results/train/hyperbola/problem_6347.png", "answer": "x^2/3 - y^2/6 = 1", "fact_expr": "G: Hyperbola;F1: Point;F2: Point;Coordinate(F1) = (-3, 0);Coordinate(F2) = (3, 0);Focus(G) = {F1, F2};Expression(Asymptote(G)) = (y = pm*(sqrt(2)*x))", "coords": { "F1": [ -3.0, 0.0 ], "F2": [ 3.0, 0.0 ] } }, { "index": 6348, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 2.8284271247461903, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6348.png", "answer": "2", "fact_expr": "C: Hyperbola;H: Line;F1: Point;P: Point;Q: Point;F2: Point;Expression(C) = (x^2/9 - y^2/8 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F2, H);Intersection(H, RightPart(C)) = {P, Q};AngleOf(F1, P, Q) = ApplyUnit(90, degree);r:Number;Radius(InscribedCircle(TriangleOf(F1,P,Q)))=r", "coords": {} }, { "index": 6349, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6349.png", "answer": "y^2=8*x", "fact_expr": "G: Hyperbola;H: Parabola;Expression(G) = (x^2/3 - y^2 = 1);RightFocus(G)=Focus(H)", "coords": {} }, { "index": 6350, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6350.png", "answer": "5", "fact_expr": "G: Parabola;Expression(G) = (x^2 = a*y);a: Number;a>0;Distance(Focus(G), Directrix(G)) = 5/2", "coords": {} }, { "index": 6351, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6351.png", "answer": "4", "fact_expr": "G: Hyperbola;p: Number;H: Parabola;Expression(G) = (x^2/3 - 16*y^2/p^2 = 1);Expression(H) = (y^2 = 2*p*x);PointOnCurve(LeftFocus(G),Directrix(H))", "coords": {} }, { "index": 6352, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6352.png", "answer": "20", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};H: Line;PointOnCurve(F1, H);A: Point;B: Point;Intersection(H, G) = {A, B}", "coords": {} }, { "index": 6353, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6354, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6355, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6356, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6356.png", "answer": "x^2/4 - y^2/4 = 1", "fact_expr": "C1: Hyperbola;C2: Hyperbola;Vertex(C1) = Vertex(C2);Expression(C1) = (x^2/4 - y^2 = 1);Slope(OneOf(Asymptote(C2))) = 2*Slope(OneOf(Asymptote(C1)))", "coords": {} }, { "index": 6357, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6357.png", "answer": "5/4", "fact_expr": "C: Parabola;Expression(C) = (x^2 = 2*p*y);p: Number;p>0;F: Point;Focus(C) = F;G: Line;PointOnCurve(F, G) = True;Intersection(G, C) = {A, B};A: Point;B: Point;4*Abs(LineSegmentOf(A, F)) = Abs(LineSegmentOf(B, F));O: Origin", "coords": {} }, { "index": 6358, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6358.png", "answer": "2*sqrt(3)/3", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Expression(OneOf(Asymptote(G))) = (sqrt(3)*x + 3*y = 0)", "coords": {} }, { "index": 6359, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6360, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6360.png", "answer": "x=-1/2", "fact_expr": "C: Parabola;Expression(C) =(y^2 = 2*(p*x));p:Number;p>0;L:Line;Inclination(L)=pi/4;Intersection(L,C)={A,B};A: Point;B: Point;YCoordinate(MidPoint(LineSegmentOf(A,B))) = 1", "coords": {} }, { "index": 6361, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6361.png", "answer": "[\\sqrt{3}/3,1)", "fact_expr": "F1: Point;F2: Point;Focus(C) = {F1, F2};C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;P: Point;PointOnCurve(P, Directrix(C)) = True;LineSegmentOf(P, F1) = 2*LineSegmentOf(P, F2)", "coords": {} }, { "index": 6362, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6362.png", "answer": "sqrt(2)/2", "fact_expr": "l: Line;C: Ellipse;b: Number;a: Number;A: Point;B: Point;O: Origin;E: Point;k1: Number;k2: Number;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Negation(PointOnCurve(O, l));Intersection(l, C) = {A, B};MidPoint(LineSegmentOf(A, B)) = E;Slope(LineOf(A, B)) = k1;Slope(LineOf(O, E)) = k2;k1*k2 = -1/2", "coords": {} }, { "index": 6363, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6364, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6364.png", "answer": "y^2/4 + x^2 = 1", "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;H:LineSegment;a > b;b > 0;Expression(G) = (x^2/b^2 + y^2/a^2 = 1);Coordinate(A) = (1, 0);RightVertex(G)=A;PointOnCurve(Focus(G),H);IsPerpendicular(H,MajorAxis(G));Length(H)=1;IsChordOf(l,G)", "coords": { "A": [ 1.0, 0.0 ] } }, { "index": 6365, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6365.png", "answer": "10", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1)", "coords": {} }, { "index": 6366, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6366.png", "answer": "3", "fact_expr": "G: Parabola;p: Number;H: Line;A: Point;F: Point;B: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;PointOnCurve(F, H);Slope(H) = sqrt(3);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, F)) > Abs(LineSegmentOf(B, F))", "coords": {} }, { "index": 6367, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6368, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.8284271247461903, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6368.png", "answer": "(-11/7, 24/7)", "fact_expr": "C: Hyperbola;A: Point;P: Point;F: Point;Expression(C) = (x^2 - y^2/8 = 1);Coordinate(A) = (0, 36/5);RightFocus(C) = F;PointOnCurve(P, LeftPart(C));WhenMin(Perimeter(TriangleOf(A,P,F)))", "coords": { "A": [ 0.0, 7.2 ] } }, { "index": 6369, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6370, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6371, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.4149149626832767, "b": 2.450642358201144, "orientation": "horizontal" }, "optimization": { "final_loss": 1.0319236343469396e-05, "converged": true, "iterations": 94 }, "output_path": "results/train/hyperbola/problem_6371.png", "answer": "2*sqrt(3)/3", "fact_expr": "C: Hyperbola;G: Point;Coordinate(G) = (sqrt(2), 3);PointOnCurve(G, C) = True;Expression(Asymptote(C)) = (y = pm*sqrt(3)*x)", "coords": { "G": [ 1.4142135623730951, 3.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 6372, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6372.png", "answer": "x^2/5 - y^2/4 = 1", "fact_expr": "O: Origin;C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Eccentricity(C) = (3*sqrt(5))/5;F: Point;RightFocus(C) = F;L: Line;PointOnCurve(F, L);IsPerpendicular(L, Asymptote(C));A: Point;FootPoint(L, Asymptote(C)) = A;Area(TriangleOf(A, F, O)) = sqrt(5)", "coords": {} }, { "index": 6373, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6373.png", "answer": "6", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 8*x);l1: Line;Directrix(C) = l1;P: Point;PointOnCurve(P, C);l2: Line;Expression(l2) = (3*x - 4*y + 24 = 0)", "coords": {} }, { "index": 6374, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6374.png", "answer": "2*sqrt(2)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(C) = F;l: Line;PointOnCurve(F, l);M: Point;N: Point;Quadrant(M) = 1;Intersection(l, C) = {M, N};VectorOf(M, N) = 3*VectorOf(F, N)", "coords": {} }, { "index": 6375, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2631799883090584, "b": 2.5275303393962245, "orientation": "horizontal" }, "optimization": { "final_loss": 4.481942266905428e-06, "converged": true, "iterations": 95 }, "output_path": "results/train/hyperbola/problem_6375.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;b>0;Expression(G) = (x^2 - y^2/b^2 = 1);Expression(OneOf(Asymptote(G))) = (y = 2*x)", "coords": {} }, { "index": 6376, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6376.png", "answer": "sqrt(3)-1", "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;e: Number;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(P,G);AngleOf(P, F1, F2) = pi/6;AngleOf(P, F2, F1) = pi/3;Eccentricity(G) = e", "coords": {} }, { "index": 6377, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6377.png", "answer": "26", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;l: Line;PointOnCurve(F2,l) = True;Intersection(l, RightPart(G)) = {A, B};A: Point;B: Point;Abs(LineSegmentOf(A, B)) = 5", "coords": {} }, { "index": 6378, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/train/hyperbola/problem_6378.png", "answer": "9", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/m = 1);m: Number;Eccentricity(G) = 5/4", "coords": {} }, { "index": 6379, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.8284271247461903, "b": 2.8284271247461903, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6379.png", "answer": "12", "fact_expr": "C: Hyperbola;A: Point;M: Point;F: Point;Expression(C) = (x^2/8 - y^2/8 = 1);Coordinate(A) = (0, 4);LeftFocus(C) = F;PointOnCurve(M, RightPart(C));WhenMin(Perimeter(TriangleOf(M,A,F)))", "coords": { "A": [ 0.0, 4.0 ] } }, { "index": 6380, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2175992727279663, "b": 2.434081792831421, "orientation": "horizontal" }, "optimization": { "final_loss": 3.452532837400213e-06, "converged": true, "iterations": 124 }, "output_path": "results/train/hyperbola/problem_6380.png", "answer": "(pm*sqrt(5)/2, 0)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Expression(OneOf(Asymptote(G))) = (y = 2*x);Length(ImageinaryAxis(G)) = 2", "coords": {} }, { "index": 6381, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6382, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6382.png", "answer": "(x-1)^2+(y-3)^2=8", "fact_expr": "l: Line;Expression(l) = (x-y-2=0);M: Point;Coordinate(M) = (3, 1);H: Circle;TangentPoint(l, H) = M;G: Parabola;Expression(G) = (y^2 = -4*x);PointOnCurve(Center(H), Directrix(G))", "coords": { "M": [ 3.0, 1.0 ] } }, { "index": 6383, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6384, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6384.png", "answer": "pm*2", "fact_expr": "G: Parabola;p: Number;P: Point;p>0;Expression(G)=(x^2 = 2*p*y);Coordinate(P)=(t,1);PointOnCurve(P,G);Distance(P,Directrix(G))=2;t:Number", "coords": {} }, { "index": 6385, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6385.png", "answer": "3", "fact_expr": "G: Parabola;P:Point;F:Point;Expression(G) = (y^2 = 8*x);Focus(G)=F;PointOnCurve(P,G);XCoordinate(P)=1", "coords": {} }, { "index": 6386, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6386.png", "answer": "-32", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/k = 1);k: Number;e: Number;Eccentricity(G) = e;e = 3", "coords": {} }, { "index": 6387, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6388, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6388.png", "answer": "4/3", "fact_expr": "C: Ellipse;F1: Point;A: Point;B: Point;F2: Point;Expression(C) = (x^2/2 + y^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;UpperVertex(C) = A;Intersection(LineOf(A,F1),C)={A,B}", "coords": {} }, { "index": 6389, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6390, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6390.png", "answer": "sqrt(5)", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F1: Point;O: Origin;e: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, C);LeftFocus(C)=F1;IsParallel(LineSegmentOf(P, F1), Asymptote(C));Abs(LineSegmentOf(P, O)) = Abs(LineSegmentOf(F1, O));Eccentricity(C) = e", "coords": {} }, { "index": 6391, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6391.png", "answer": "2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;I: Curve;Expression(I) = And((4*x^2 - 3*y^2/4 = 1), (y > 0));Intersection(Directrix(G), I) = P;P: Point;F: Point;Focus(G) = F;Inclination(LineOf(P,F)) = ApplyUnit(135, degree)", "coords": {} }, { "index": 6392, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6392.png", "answer": "2", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);P: Point;Coordinate(P) = (m, n);m: Number;n: Number;PointOnCurve(P, G) = True;F: Point;Focus(G) = F;Distance(P, F) = 3", "coords": {} }, { "index": 6393, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6393.png", "answer": "sqrt(5)/3", "fact_expr": "E: Ellipse;b: Number;a: Number;A: Point;M: Point;F: Point;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (0, b);RightFocus(E) = F;PointOnCurve(M, E);Max(Perimeter(TriangleOf(A, M, F))) = 6*b", "coords": {} }, { "index": 6394, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/train/hyperbola/problem_6394.png", "answer": "y^2/15 - x^2/12 = 1", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Expression(OneOf(Asymptote(C))) = (y = x*(sqrt(5)/2));G: Point;Coordinate(G) = (2, 2*sqrt(5));PointOnCurve(G,C) = True", "coords": { "G": [ 2.0, 4.47213595499958 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 6395, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6395.png", "answer": "(1,-2)", "fact_expr": "G: Parabola;Q: Point;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(Q) = (2, -2);PointOnCurve(P, G);WhenMin(Distance(P,Q)+Distance(P,Focus(G)))", "coords": { "Q": [ 2.0, -2.0 ] } }, { "index": 6396, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6396.png", "answer": "8", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x);F: Point;Focus(G) = F;l1: Line;l2: Line;A: Point;B: Point;C: Point;D: Point;PointOnCurve(F, l1);PointOnCurve(F, l2);Intersection(l1, G) = {A, B};Intersection(l2, G) = {C, D};k1: Number;k2: Number;Slope(l1) = k1;Slope(l2) = k2;k1^2+k2^2 = 2", "coords": {} }, { "index": 6397, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.125, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6397.png", "answer": "(0,1/8)", "fact_expr": "G: Parabola;Expression(G) = (y = 2*x^2)", "coords": {} }, { "index": 6398, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6398.png", "answer": "sqrt(3)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;A: Point;PointOnCurve(A, RightPart(C)) = True;Intersection(LineSegmentOf(A, F1), C) = B;B: Point;DotProduct(VectorOf(F2, A), VectorOf(F2, B)) = 0;Abs(VectorOf(F2, A)) = Abs(VectorOf(F2, B))", "coords": {} }, { "index": 6399, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6400, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6401, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6402, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6402.png", "answer": "[-9/20,9/20]", "fact_expr": "E: Ellipse;G: Line;k: Number;Q: Point;P: Point;l:Line;Expression(E) = (x^2/4 + y^2 = 1);Expression(G) = (y = k*x - 3/5);Intersection(G, E) = {P, Q};PerpendicularBisector(LineSegmentOf(P,Q))=l", "coords": {} }, { "index": 6403, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.2352789464429823, "b": 2.2352789464429823, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4312591569844774e-06, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_6403.png", "answer": "x^2/8 - y^2/2 = 1", "fact_expr": "G: Hyperbola;F1: Point;F2: Point;Coordinate(F1) = (-sqrt(10), 0);Coordinate(F2) = (sqrt(10), 0);Expression(Asymptote(G)) = (y = pm*(x/2));Focus(G) = {F1, F2}", "coords": { "F1": [ -3.1622776601683795, 0.0 ], "F2": [ 3.1622776601683795, 0.0 ] } }, { "index": 6404, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6404.png", "answer": "4", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/16 = 1)", "coords": {} }, { "index": 6405, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.75, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6405.png", "answer": "sqrt(3)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 3*x);F: Point;Focus(C) = F;G: Line;PointOnCurve(F, G);A: Point;B: Point;Intersection(G, C) = {A, B};O: Origin;D: Point;Intersection(LineOf(O, A), Directrix(C)) = D", "coords": {} }, { "index": 6406, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6406.png", "answer": "8", "fact_expr": "E: Parabola;P: Point;A: Point;B: Point;l1: Line;l2: Line;Expression(E) = (x^2 = 4*y);Coordinate(P) = (2, -1);TangentOfPoint(P, E) = {l1, l2};TangentPoint(l1, E) = A;TangentPoint(l2, E) = B", "coords": { "P": [ 2.0, -1.0 ] } }, { "index": 6407, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6407.png", "answer": "17/2", "fact_expr": "W: Parabola;Expression(W) = (x^2 = 8*y);F: Point;Focus(W) = F;l: Line;PointOnCurve(F,l) = True;Intersection(l, W) = {A, B};A: Point;B: Point;l1: Line;Expression(l1) = (x-2*y-4=0);WhenMin(Distance(A,l1) + Distance(B,l1)) = True", "coords": {} }, { "index": 6408, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6408.png", "answer": "{1, 16}", "fact_expr": "E: Ellipse;m: Number;e: Number;Expression(E) = (x^2/4 + y^2/m = 1);Eccentricity(E) = e ;e = sqrt(3)/2", "coords": {} }, { "index": 6409, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6410, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6410.png", "answer": "2*sqrt(3)", "fact_expr": "C: Parabola;E: Ellipse;A: Point;B: Point;O:Origin;Expression(C) = (y^2 = 4*x);Center(E)=O;RightFocus(E) = Focus(C);Intersection(E, Directrix(C)) = {A, B};LineSegmentOf(A, B) = 3", "coords": {} }, { "index": 6411, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6411.png", "answer": "x^2/4+y^2/3=1", "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;F: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftVertex(G) = A;LeftFocus(G) = F;Distance(UpperVertex(G), Focus(G)) = 2;e: Number;Eccentricity(G) = e;e = 1/2", "coords": {} }, { "index": 6412, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6412.png", "answer": "3\n8\n6/5", "fact_expr": "F1: Point;F2: Point;Focus(G) = {F1, F2};G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);P: Point;PointOnCurve(P, G);Distance(P, F1) = 1;H: Line;PointOnCurve(F1, H);A: Point;B: Point;Intersection(H, G) = {A, B};AngleOf(P, F1, F2) = ApplyUnit(120, degree)", "coords": {} }, { "index": 6413, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6413.png", "answer": "24/7", "fact_expr": "l: Line;G: Ellipse;A: Point;B: Point;M: Point;Expression(G) = (x^2/4 + y^2/3 = 1);PointOnCurve(RightFocus(G),l);Intersection(l, G) = {A, B};Coordinate(M) = (4/7, -3/7);MidPoint(LineSegmentOf(A, B)) = M;IsChordOf(LineSegmentOf(A,B),G)", "coords": { "M": [ 0.5714285714285714, -0.42857142857142855 ] } }, { "index": 6414, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6415, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6415.png", "answer": "20", "fact_expr": "G: Ellipse;H: Line;P: Point;Q: Point;F2: Point;F1: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(G)={F1,F2};PointOnCurve(F1, H);Intersection(H, G) = {P, Q}", "coords": {} }, { "index": 6416, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6417, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6417.png", "answer": "[-1,17/8]", "fact_expr": "G: Parabola;H: Ellipse;a: Real;Expression(G) = (x^2 = 2*y);Expression(H) = (x^2 + 4*(-a + y)^2 = 4);IsIntersect(H, G)", "coords": {} }, { "index": 6418, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6418.png", "answer": "[2*sqrt(2),4)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);P: Point;PointOnCurve(P, C);M: Circle;Expression(M) = (y^2 + (x - 3)^2 = 4);L1: Line;L2: Line;TangentOfPoint(P, M) = {L1, L2};TangentPoint(L1, M) = A;TangentPoint(L2, M) = B;A: Point;B: Point", "coords": {} }, { "index": 6419, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6419.png", "answer": "(1+sqrt(5))/2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;Focus(G) = {F1, F2};Z: Line;PointOnCurve(F2, Z);IsPerpendicular(Z, xAxis);A: Point;B: Point;Intersection(Z, G) = {A, B};Radius(InscribedCircle(TriangleOf(A, B, F1))) = a", "coords": {} }, { "index": 6420, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6421, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6421.png", "answer": "6-sqrt(2)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/5 = 1);F1: Point;LeftFocus(G) = F1;P: Point;PointOnCurve(P, G);A: Point;Coordinate(A) = (1, 1)", "coords": { "A": [ 1.0, 1.0 ] } }, { "index": 6422, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6423, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6423.png", "answer": "y = pm*2*x", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;F2: Point;F1: Point;P: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Focus(G) = {F1, F2};IsDiameter(LineSegmentOf(F1,F2),H);OneOf(Intersection(H,G))=P;LineSegmentOf(P, F1) = 2*LineSegmentOf(P, F2)", "coords": {} }, { "index": 6424, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6424.png", "answer": "7*sqrt(2)/8", "fact_expr": "G: Parabola;H: Line;P0: Point;Expression(G) = (y = x^2);Expression(H) = (x - y - 2 = 0);PointOnCurve(P0, G)", "coords": {} }, { "index": 6425, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6425.png", "answer": "x^2/4-y^2=1", "fact_expr": "l: Line;G: Hyperbola;b: Number;a: Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(l) = (y=2*x);IsPerpendicular(l,OneOf(Asymptote(G)));Distance(RightFocus(G),l)=2", "coords": {} }, { "index": 6426, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6426.png", "answer": "[0, 12]", "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;P: Point;F: Point;e: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftVertex(G) = A;LeftFocus(G) = F;PointOnCurve(P, G);Distance(UpperVertex(G), Focus(G))=2;Eccentricity(G) = e;e = 1/2", "coords": {} }, { "index": 6427, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6427.png", "answer": "sqrt(2)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/2 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) - Abs(LineSegmentOf(P, F2)) = 2", "coords": {} }, { "index": 6428, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.6259966743174588, "b": 2.298015331604694, "orientation": "horizontal" }, "optimization": { "final_loss": 1.93011531933813e-06, "converged": true, "iterations": 91 }, "output_path": "results/train/hyperbola/problem_6428.png", "answer": "sqrt(2)/2", "fact_expr": "G: Hyperbola;a: Number;a>0;Expression(G) = (-y^2 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G))) = (y = sqrt(2)*x)", "coords": {} }, { "index": 6429, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6429.png", "answer": "16 + pm*8*sqrt(3)", "fact_expr": "C: Parabola;p: Number;B: Point;F: Point;A: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Expression(Directrix(C)) = (x = -2);Focus(C) = F;Intersection(Directrix(C), xAxis) = A;PointOnCurve(B, C);sqrt(5)*Abs(LineSegmentOf(B, F)) = 2*Abs(LineSegmentOf(A, B))", "coords": {} }, { "index": 6430, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6430.png", "answer": "2*sqrt(5)", "fact_expr": "G: Parabola;H: Line;A: Point;P: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (x = 1);Coordinate(A) = (3, 4);PointOnCurve(P, G);Distance(P, H) = d;d:Number", "coords": { "A": [ 3.0, 4.0 ] } }, { "index": 6431, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6431.png", "answer": "(-16/5,16/5)", "fact_expr": "G: Ellipse;A: Point;C: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Coordinate(A) = (x1, y1);Coordinate(C) = (x2, y2);x1:Number;y1:Number;x2:Number;y2:Number;PointOnCurve(A, G);PointOnCurve(B, G);x1 + x2 = 8;Expression(PerpendicularBisector(LineSegmentOf(A, C))) = (y = k*x + m);k:Number;m:Number", "coords": {} }, { "index": 6432, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6433, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6434, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6434.png", "answer": "[1/3, 1)", "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(P,G);Abs(LineSegmentOf(P,F1))=2*Abs(LineSegmentOf(P,F2))", "coords": {} }, { "index": 6435, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6436, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6436.png", "answer": "(6, pm*6*sqrt(2))", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 12*x);P:Point;PointOnCurve(P, G);Distance(P, Focus(G)) = 9", "coords": {} }, { "index": 6437, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6437.png", "answer": "2", "fact_expr": "G: Parabola;H: Circle;P: Point;Q: Point;F: Point;Expression(G) = (y^2 = 4*x);Expression(H) = ((x - 2)^2 + (y - 1)^2 = 1);Focus(G) = F;PointOnCurve(P, G);PointOnCurve(Q, H)", "coords": {} }, { "index": 6438, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.6457513110645907, "major_axis": "x", "x_coef": 16.0, "y_coef": 7.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6438.png", "answer": "[1, 7]", "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/7 = 1);P: Point;PointOnCurve(P, G);F: Point;LeftFocus(G) = F", "coords": {} }, { "index": 6439, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6440, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6440.png", "answer": "pi/6", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);F: Point;Focus(G) = F;M: Point;Intersection(Directrix(G), yAxis) = M;N: Point;PointOnCurve(N, G);Abs(LineSegmentOf(N, F)) = (sqrt(3)/2)*Abs(LineSegmentOf(M, N))", "coords": {} }, { "index": 6441, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6441.png", "answer": "-1", "fact_expr": "F: Point;RightFocus(G) = F;G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);P: Point;PointOnCurve(P, G) = True;A: Point;PointOnCurve(A, H) = True;H: Line;Expression(H) = (3*x - 4*y - 12 = 0)", "coords": {} }, { "index": 6442, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_focus_mismatch", "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6442.png", "answer": "sqrt(6)/3", "fact_expr": "l: Line;G: Ellipse;b: Number;a: Number;B: Point;A: Point;F: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(MidPoint(LineSegmentOf(A,B))) = (-3, 1);Coordinate(F) = (-4, 0);OneOf(Focus(G)) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B}", "coords": { "F": [ -4.0, 0.0 ] }, "validation_reasons": [ "ellipse_focus_mismatch" ] }, { "index": 6443, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/train/hyperbola/problem_6443.png", "answer": "(5,0)", "fact_expr": "G: Hyperbola;a: Number;a>0;Expression(G) = (-y^2/9 + x^2/a^2 = 1);Eccentricity(G) = 5/4", "coords": {} }, { "index": 6444, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6444.png", "answer": "sqrt(3)", "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2/3 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;M: Point;N: Point;PointOnCurve(M, Asymptote(C));PointOnCurve(N, LeftPart(C));WhenMin(Abs(LineSegmentOf(M,N))+Abs(LineSegmentOf(N,F2)))", "coords": {} }, { "index": 6445, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6445.png", "answer": "6", "fact_expr": "G: Parabola;A: Point;B: Point;C: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(Vertex(TriangleOf(A, B, C)), G);VectorOf(F, A) + VectorOf(F, B) + VectorOf(F, C) = 0", "coords": {} }, { "index": 6446, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 0.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6446.png", "answer": "x \\pm 2y = 0", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - 4*y^2 = 1)", "coords": {} }, { "index": 6447, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.4142135623730951, "major_axis": "y", "x_coef": 2.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6447.png", "answer": "4", "fact_expr": "G: Ellipse;Expression(G) = (x^2/2 + y^2/3 = 1);F1: Point;F2: Point;Focus(G) = {F1,F2};A: Point;B: Point;IsChordOf(LineSegmentOf(A,B),G) = True ;PointOnCurve(F1,LineSegmentOf(A,B)) = True", "coords": {} }, { "index": 6448, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6448.png", "answer": "y=pm*(sqrt(2)/2)*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/2 - y^2 = 1)", "coords": {} }, { "index": 6449, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6449.png", "answer": "(1, \\sqrt{5})", "fact_expr": "E: Hyperbola;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;G: Line;PointOnCurve(RightFocus(E), G);Slope(G) = 2;NumIntersection(G, RightPart(E)) = 2", "coords": {} }, { "index": 6450, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6450.png", "answer": "x^2=16*y", "fact_expr": "C1: Hyperbola;Expression(C1) = (x^2/a^2 - y^2/b^2 = 1);a: Number;b: Number;a>0;b>0;Eccentricity(C1) = 2;C2: Parabola;Expression(C2) = (x^2 = 2*(p*y));p: Number;p>0;Distance(Focus(C2), Asymptote(C1)) = 2", "coords": {} }, { "index": 6451, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6451.png", "answer": "y^2=2*x", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;l: Line;Expression(l) = (x = 2);P: Point;Q: Point;Intersection(l, C) = {P, Q};O: Origin;IsPerpendicular(LineSegmentOf(O, P), LineSegmentOf(O, Q))", "coords": {} }, { "index": 6452, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_ecc_mismatch", "params": { "a": 1.9257268905639648, "b": 1.7993911322594118, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 2.5576271215184017e-15, "converged": true, "iterations": 2 }, "output_path": "results/train/ellipse/problem_6452.png", "answer": "{3/2, 8/3}", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2/2 + y^2/m = 1);Eccentricity(G) = 1/2", "coords": {}, "validation_reasons": [ "ellipse_ecc_mismatch" ] }, { "index": 6453, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6453.png", "answer": "7/2", "fact_expr": "G: Parabola;H: Line;A: Point;P: Point;B: Point;Z: Line;Expression(G) = (y^2 = 8*x);Expression(H) = (x = -2);Coordinate(A) = (5, 0);PointOnCurve(P, G);PointOnCurve(P, Z);IsPerpendicular(H, Z);FootPoint(H, Z) = B;Abs(LineSegmentOf(P, B)) = Abs(LineSegmentOf(P, A))", "coords": { "A": [ 5.0, 0.0 ] } }, { "index": 6454, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6454.png", "answer": "16", "fact_expr": "G: Hyperbola;P: Point;Q: Point;F2: Point;F1: Point;Expression(G) = (x^2/16 - y^2/9 = 1);Focus(G) = {F1, F2};PointOnCurve(F1,LineSegmentOf(P, Q));IsChordOf(LineSegmentOf(P, Q),G);Inclination(LineSegmentOf(P,Q))=ApplyUnit(60, degree)", "coords": {} }, { "index": 6455, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6456, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.4149149626832767, "b": 2.450642358201144, "orientation": "horizontal" }, "optimization": { "final_loss": 1.0319236343469396e-05, "converged": true, "iterations": 94 }, "output_path": "results/train/hyperbola/problem_6456.png", "answer": "x^2/(2/3)-y^2/2=1", "fact_expr": "G: Hyperbola;H: Point;Coordinate(H) = (1, 1);Expression(Asymptote(G)) = (y = pm*sqrt(3)*x);PointOnCurve(H, G)", "coords": { "H": [ 1.0, 1.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 6457, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6457.png", "answer": "9", "fact_expr": "G: Hyperbola;A: Point;B: Point;P: Point;Expression(G) = (x^2/4 - y^2/5 = 1);Coordinate(A) = (3, 0);Coordinate(B) = (9, 5);PointOnCurve(P, RightPart(G))", "coords": { "A": [ 3.0, 0.0 ], "B": [ 9.0, 5.0 ] } }, { "index": 6458, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6459, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6459.png", "answer": "(a+b)/2", "fact_expr": "C: Parabola;p: Number;p>0;Expression(C) = (y^2 = 2*p*x);A:Point;B:Point;M:Point;a:Number;b:Number;Negation(a*b=0);PointOnCurve(A,C);PointOnCurve(B,C);YCoordinate(A)=a;YCoordinate(B)=b;Negation(A=B);Intersection(TangentOnPoint(A,C),TangentOnPoint(B,C))=M", "coords": {} }, { "index": 6460, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2631799883090584, "b": 2.5275303393962245, "orientation": "horizontal" }, "optimization": { "final_loss": 4.481942266905428e-06, "converged": true, "iterations": 95 }, "output_path": "results/train/hyperbola/problem_6460.png", "answer": "x^2/5 - y^2/20 = 1", "fact_expr": "G: Hyperbola;PointOnCurve(Focus(G), xAxis);Expression(OneOf(Asymptote(G))) = (y = 2*x);FocalLength(G) = 10", "coords": {} }, { "index": 6461, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6461.png", "answer": "3", "fact_expr": "G: Parabola;p: Number;A: Point;F: Point;M: Point;p>0;Expression(G) = (y^2 = 2*p*x);Focus(G)=F;PointOnCurve(M,G);Coordinate(A) = ((7/2)*p, 0);Abs(LineSegmentOf(A, F)) = 2*Abs(LineSegmentOf(M, F));Area(TriangleOf(A,M,F))=27*sqrt(2)/2", "coords": {} }, { "index": 6462, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6463, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6463.png", "answer": "9/4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = x);H: Line;I: Point;Coordinate(I) = (1/4, 0);PointOnCurve(I, H);A: Point;B: Point;Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 4;IsChordOf(LineSegmentOf(A, B), G);L: Line;Expression(L) = (x + 1/2 = 0)", "coords": { "I": [ 0.25, 0.0 ] } }, { "index": 6464, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6464.png", "answer": "(p/2, p)", "fact_expr": "M: Point;PointOnCurve(M, C);C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;A: Point;Coordinate(A) = (p, p);F: Point;Coordinate(F) = (p/2, 0);WhenMin(Distance(M, A) + Distance(M, F))", "coords": {} }, { "index": 6465, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6465.png", "answer": "1", "fact_expr": "l: Line;C: Parabola;M: Point;A: Point;B: Point;F: Point;k: Number;Expression(C) = (y^2 = 4*x);Coordinate(M) = (-1, 2);Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};DotProduct(VectorOf(M, A), VectorOf(M, B)) = 0;Slope(l) = k", "coords": { "M": [ -1.0, 2.0 ] } }, { "index": 6466, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.23606797749979, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6466.png", "answer": "6", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/5 - y^2/9 = 1)", "coords": {} }, { "index": 6467, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6468, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6468.png", "answer": "1", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(P, G)", "coords": {} }, { "index": 6469, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6470, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6470.png", "answer": "sqrt(5)/2", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;O: Origin;P: Point;Q: Point;A: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(A, xAxis);Negation(A = O);Center(G) = A;Intersection(G, Asymptote(C)) = {P, Q};VectorOf(O, P) = 3*VectorOf(O, Q);DotProduct(VectorOf(A, P), VectorOf(A, Q)) = 0", "coords": {} }, { "index": 6471, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6472, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6473, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6474, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6475, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 2.23606797749979, "major_axis": "x", "x_coef": 8.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6475.png", "answer": "x^2/3 - y^2/5 = 1", "fact_expr": "G: Hyperbola;H: Ellipse;I: Point;Expression(H) = (x^2/8 + y^2/5 = 1);Coordinate(I) = (3, sqrt(10));PointOnCurve(I, G);Endpoint(MajorAxis(H))=Focus(G)", "coords": { "I": [ 3.0, 3.1622776601683795 ] } }, { "index": 6476, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6477, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6477.png", "answer": "sqrt(2)", "fact_expr": "l: Line;Expression(l) = (x - y = 0);G: Ellipse;Expression(G) = (x^2/2 + y^2 = 1);A: Point;B: Point;Intersection(l, G) = {A, B};C: Point;PointOnCurve(C, G)", "coords": {} }, { "index": 6478, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6479, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6479.png", "answer": "4", "fact_expr": "G: Parabola;p: Number;H: Line;E: Point;B: Point;A: Point;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(E) = (-p/2, 0);PointOnCurve(E, H);Intersection(H, G) = {A, B};Focus(G) = F;MidPoint(LineSegmentOf(E,B)) = A;Abs(LineSegmentOf(A, F)) = 3", "coords": {} }, { "index": 6480, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.4641016151377544, "major_axis": "x", "x_coef": 16.0, "y_coef": 12.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6480.png", "answer": "(2*sqrt(3), sqrt(3))", "fact_expr": "G: Ellipse;A: Point;P: Point;F: Point;Expression(G) = (x^2/16 + y^2/12 = 1);Coordinate(A) = (-2, sqrt(3));RightFocus(G) = F;PointOnCurve(P, G);WhenMin(Abs(LineSegmentOf(P, A)) + 2*Abs(LineSegmentOf(P, F)))", "coords": { "A": [ -2.0, 1.7320508075688772 ] } }, { "index": 6481, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6482, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6482.png", "answer": "x=1/2", "fact_expr": "G: Hyperbola;a: Number;H: Parabola;a>0;Expression(G) = (-y^2/3 + x^2/a^2 = 1);Expression(H) = (y^2 = 8*x);Focus(H) = RightFocus(G)", "coords": {} }, { "index": 6483, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6484, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6485, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6485.png", "answer": "5/3", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;M: Point;A: Point;LeftVertex(C)=A;F: Point;RightFocus(C)=F;PointOnCurve(F,l);IsPerpendicular(l, OneOf(Asymptote(C)));FootPoint(l,OneOf(Asymptote(C))) = M;Tan(AngleOf(M, A, F)) = 1/2;l:Line", "coords": {} }, { "index": 6486, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6487, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6488, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6489, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6490, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6490.png", "answer": "[(sqrt(5)-1)/2,1)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/a^2 + y^2/b^2 = 1);a: Number;b: Number;a > b;b > 0;F: Point;c:Number;Coordinate(F) = (c, 0);RightFocus(G) = F;l: Line;Intersection(l,G) = {A,B};A: Point;B: Point;IsPerpendicular(LineSegmentOf(O, A), LineSegmentOf(O, B)) = True;PointOnCurve(F, l) = True;O:Origin", "coords": {} }, { "index": 6491, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6492, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6492.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;Expression(G) = (-x^2 + y^2/4 = 1);L1:Line;L2: Line;Asymptote(G) = {L1,L2};H: Parabola;Expression(H) = (y^2 = 2*p*x);p: Number;p>0;Intersection(L1, Directrix(H)) = A;Intersection(L2, Directrix(H)) = B;A: Point;O: Origin;B: Point;Area(TriangleOf(A, O, B)) = 1", "coords": {} }, { "index": 6493, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6493.png", "answer": "x^2/2 - y^2/2 = 1", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;F: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (2, 0);OneOf(Focus(G)) = F;PointOnCurve(F, H);Slope(H) = 1;NumIntersection(H, G) = 1", "coords": { "F": [ 2.0, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 6494, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6494.png", "answer": "ApplyUnit(30, degree)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;Intersection(Directrix(G),xAxis) = M;M: Point;PointOnCurve(N,G);N: Point;Abs(LineSegmentOf(N, F)) = (sqrt(3)/2)*Abs(LineSegmentOf(M, N))", "coords": {} }, { "index": 6495, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 3.0, "major_axis": "x", "x_coef": 36.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6495.png", "answer": "6+3*sqrt(3)", "fact_expr": "G: Ellipse;P: Point;F: Point;Expression(G) = (x^2 + 4*y^2 = 36);PointOnCurve(P, G);LeftFocus(G) = F", "coords": {} }, { "index": 6496, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6497, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6497.png", "answer": "2*sqrt(2)/3", "fact_expr": "C: Parabola;G: Line;F: Point;A: Point;B: Point;k:Number;Expression(C) = (y^2 = 8*x);k>0;Expression(G) = (y = k*(x + 2));Intersection(G, C) = {A, B};Focus(C) = F;Abs(LineSegmentOf(F, A)) = 2*Abs(LineSegmentOf(F, B))", "coords": {} }, { "index": 6498, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6498.png", "answer": "2", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;Expression(G) = (y^2 = 4*x);IsPerpendicular(H,xAxis);Intersection(H, G) = {A, B};Length(LineSegmentOf(A, B)) = 4*sqrt(3)", "coords": {} }, { "index": 6499, "success": false, "error": "\n$ $ (a > b > 0) $ with eccentricity $ e $, the line $ l $: $ y = e x +\n^\nParseException: Expected end of text, found '$' (at char 0), (line:1, col:1)" }, { "index": 6500, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6500.png", "answer": "4/3", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;PointOnCurve(F, l) = True;l: Line;Intersection(l, C) = {P, Q};P: Point;Q: Point;Intersection(Directrix(C), l) = M;M: Point;VectorOf(F, M) = 3*VectorOf(F, P)", "coords": {} }, { "index": 6501, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6502, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6503, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6503.png", "answer": "16", "fact_expr": "G: Hyperbola;m: Number;F: Point;Expression(G) = (-x^2/9 + y^2/m = 1);Coordinate(F) = (0, 5);OneOf(Focus(G)) = F", "coords": { "F": [ 0.0, 5.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 6504, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6504.png", "answer": "{11,29}", "fact_expr": "G: Ellipse;k: Number;Expression(G) = (x^2/20 + y^2/k = 1);FocalLength(G) = 6", "coords": {} }, { "index": 6505, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6505.png", "answer": "2*sqrt(2)", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x - y - 2 = 0);PointOnCurve(OneOf(Focus(C)), G);IsPerpendicular(G, OneOf(Asymptote(C)))", "coords": {} }, { "index": 6506, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6506.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;Q: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P,Asymptote(G));Quadrant(P)=1;PointOnCurve(Q,G);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;VectorOf(P,F2)=4*VectorOf(P,Q)", "coords": {} }, { "index": 6507, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6507.png", "answer": "{2, 6}", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;l: Line;Directrix(G) = l;C: Circle;Expression(C) = ((x-a)^2 + (y-2*sqrt(3))^2 = 16);PointOnCurve(F, C);IsTangent(C, l)", "coords": {} }, { "index": 6508, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6508.png", "answer": "12", "fact_expr": "l: Line;G: Hyperbola;P: Point;F1:Point;F2: Point;Q: Point;Expression(G) = (x^2 - y^2/4 = 1);LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(F1, l);Intersection(l, LeftPart(G)) = {P, Q};Abs(LineSegmentOf(P, Q)) = 4", "coords": {} }, { "index": 6509, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.0, "b": 0.7071067811865476, "major_axis": "x", "x_coef": 1.0, "y_coef": 0.5 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6509.png", "answer": "y^2=4*x", "fact_expr": "H: Ellipse;Expression(H) = (x^2 + 2*y^2 = 1);Vertex(G) = Center(H);Focus(G) = RightVertex(H);G: Parabola", "coords": {} }, { "index": 6510, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6510.png", "answer": "1\n2", "fact_expr": "C1: Hyperbola;C2: Hyperbola;F: Point;a:Number;b:Number;a>0;b>0;Expression(C1) = (x^2/a^2-y^2/b^2=1);Expression(C2) = (x^2/4-y^2/16=1);Asymptote(C1)=Asymptote(C2);RightFocus(C1)=F;Coordinate(F) = (sqrt(5), 0)", "coords": { "F": [ 2.23606797749979, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 6511, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.449489742783178, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 6.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6511.png", "answer": "sqrt(2)", "fact_expr": "C1: Curve;C2: Curve;P: Point;F1: Point;F2: Point;Expression(C1) = (x^2/6 + y^2/2 = 1);Expression(C2) = (x^2/3 - y^2 = 1);Focus(C1) = {F1, F2};OneOf(Intersection(C1, C2)) = P", "coords": {} }, { "index": 6512, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.6457513110645907, "major_axis": "x", "x_coef": 16.0, "y_coef": 7.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6512.png", "answer": "[1,7]", "fact_expr": "G: Ellipse;P: Point;F: Point;Expression(G) = (x^2/16 + y^2/7 = 1);PointOnCurve(P, G);LeftFocus(G) = F", "coords": {} }, { "index": 6513, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6513.png", "answer": "(26/3, -1)", "fact_expr": "P: Point;Coordinate(P) = (1, -1);G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);F: Point;RightFocus(G) = F;M: Point;PointOnCurve(M, G);WhenMin(Abs(LineSegmentOf(M, P)) + 2*Abs(LineSegmentOf(M, F)))", "coords": { "P": [ 1.0, -1.0 ] } }, { "index": 6514, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "y", "x_coef": 4.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6514.png", "answer": "9*x + 4*y - 13 = 0", "fact_expr": "l: Line;C: Curve;A: Point;B: Point;M: Point;Expression(C) = (x^2/4 + y^2/9 = 1);Coordinate(M) = (1, 1);PointOnCurve(M, l);Intersection(l, C) = {A, B};IsChordOf(LineSegmentOf(A, B), C);MidPoint(LineSegmentOf(A, B)) = M", "coords": { "M": [ 1.0, 1.0 ] } }, { "index": 6515, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6515.png", "answer": "sqrt(2)", "fact_expr": "H: Parabola;Expression(H) = (y^2 = 8*x);G: Hyperbola;Expression(G) = (-y^2/2 + x^2/a^2 = 1);a: Number;a>0;Focus(H) = RightFocus(G)", "coords": {} }, { "index": 6516, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6516.png", "answer": "[1/2,1)", "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;F2:Point;P: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Focus(G)={F1,F2};PointOnCurve(P,G);AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 6517, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6518, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6519, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6519.png", "answer": "3/2", "fact_expr": "C: Hyperbola;b: Number;a: Number;x0:Number;P: Point;F1: Point;F2: Point;G: Point;O: Origin;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;Coordinate(P)=(x0,5/2);PointOnCurve(P, C);Radius(InscribedCircle(TriangleOf(P,F1,F2)))=1;Center(InscribedCircle(TriangleOf(P,F1,F2)))=G;Distance(G, O) = sqrt(5)", "coords": {} }, { "index": 6520, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6520.png", "answer": "y^2=(x-1)/4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x);P: Point;PointOnCurve(P,G) = True", "coords": {} }, { "index": 6521, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6521.png", "answer": "8/5", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);LineSegmentOf(P, F1) = 5*LineSegmentOf(P, F2)", "coords": {} }, { "index": 6522, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6523, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6523.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;P: Point;F: Point;b: Number;a: Number;L: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;PointOnCurve(F,L);IsPerpendicular(L,Asymptote(G));FootPoint(L,Asymptote(G))=P;PointOnCurve(MidPoint(LineSegmentOf(P, F)), G)", "coords": {} }, { "index": 6524, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6524.png", "answer": "x^2 - y^2 / 3 = 1", "fact_expr": "Focus(C1) = Focus(C2);C1: Hyperbola;C2: Hyperbola;Expression(C1) = (x^2/3 - y^2 = 1);Inclination(OneOf(Asymptote(C2))) = Inclination(OneOf(Asymptote(C1)))*2", "coords": {} }, { "index": 6525, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6526, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6526.png", "answer": "6", "fact_expr": "G: Parabola;H: Line;A: Point;F: Point;B: Point;Expression(G) = (y^2 = 8*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};VectorOf(A, F) = 2*VectorOf(F, B)", "coords": {} }, { "index": 6527, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6527.png", "answer": "{4, 9/4}", "fact_expr": "G: Ellipse;Expression(G) = (y^2/3 + x^2/m = 1);m: Number;m>0;F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;Quadrant(P) = 1;PointOnCurve(P, G);(Sin(AngleOf(P, F1, F2)) + Sin(AngleOf(P, F2, F1)))/Sin(AngleOf(F1, P, F2)) = 2", "coords": {} }, { "index": 6528, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6528.png", "answer": "y=pm*x", "fact_expr": "G: Hyperbola;b: Number;H: Parabola;Expression(G) = (x^2/4 - y^2/b^2 = 1);Expression(H) = (y^2 = 8*sqrt(2)*x);RightFocus(G) = Focus(H)", "coords": {} }, { "index": 6529, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6529.png", "answer": "sqrt(13)/3", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;G: Line;Expression(G) = (2*x - 3*y = 0);OneOf(Asymptote(C)) = G", "coords": {} }, { "index": 6530, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6531, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6531.png", "answer": "(0,1)", "fact_expr": "G: Parabola;Expression(G) = (y = x^2/4)", "coords": {} }, { "index": 6532, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6532.png", "answer": "4", "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/4 + y^2/3 = 1);Focus(C)={F1,F2};PointOnCurve(P,C)", "coords": {} }, { "index": 6533, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6533.png", "answer": "y^2/3 - x^2/6 = 1", "fact_expr": "G: Hyperbola;C:Hyperbola;H: Point;Expression(G) = (x^2/2 - y^2 = 1);Coordinate(H) = (sqrt(2), -2);Asymptote(G)=Asymptote(C);PointOnCurve(H,C)", "coords": { "H": [ 1.4142135623730951, -2.0 ] } }, { "index": 6534, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6534.png", "answer": "sqrt(5)", "fact_expr": "G: Parabola;H: Line;A: Point;P: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (x = -1);Coordinate(A) = (-1, 1);PointOnCurve(P, G)", "coords": { "A": [ -1.0, 1.0 ] } }, { "index": 6535, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6535.png", "answer": "x^2/9 - y^2/16 = 1", "fact_expr": "C: Hyperbola;G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);Focus(C) = Endpoint(MajorAxis(G));Endpoint(RealAxis(C)) = Focus(G)", "coords": {} }, { "index": 6536, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6537, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6538, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6538.png", "answer": "sqrt(13)/4", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;P: Point;PointOnCurve(P, G);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;AngleOf(F1, P, F2) = ApplyUnit(120, degree);Abs(LineSegmentOf(P, F1)) = 3*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 6539, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6539.png", "answer": "(1,2]", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Focus(G)={F1,F2};PointOnCurve(P,G);Abs(LineSegmentOf(P, F1)) = 3*Abs(LineSegmentOf(P, F2));Eccentricity(G)=e;e:Number", "coords": {} }, { "index": 6540, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6540.png", "answer": "8", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);l: Line;PointOnCurve(Focus(G), l);A: Point;B: Point;Intersection(l, G) = {A, B};XCoordinate(MidPoint(LineSegmentOf(A, B))) = 3", "coords": {} }, { "index": 6541, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6541.png", "answer": "pm*1", "fact_expr": "G: Hyperbola;b: Number;a: Number;A1: Point;B: Point;A2: Point;C: Point;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F;LeftVertex(G) = A1;RightVertex(G) = A2;L:Line;PointOnCurve(F,L);IsPerpendicular(L,xAxis);Intersection(L,G)={B,C};IsPerpendicular(LineSegmentOf(A1, B), LineSegmentOf(A2, C))", "coords": {} }, { "index": 6542, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_6542.png", "answer": "(pm*2,0)\nsqrt(3)*x+pm*y=0", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 8*x);Eccentricity(G) = 2;OneOf(Focus(G))=Focus(H)", "coords": {} }, { "index": 6543, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6544, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6544.png", "answer": "2*a^2*b^2/a^2+b^2", "fact_expr": "G: Ellipse;b: Number;a: Number;O: Origin;P: Point;Q: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P,G);PointOnCurve(Q,G);IsPerpendicular(LineSegmentOf(O, P), LineSegmentOf(O, Q))", "coords": {} }, { "index": 6545, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6545.png", "answer": "6", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;A: Point;B: Point;C: Point;PointOnCurve(A, G);PointOnCurve(B, G);PointOnCurve(C, G);VectorOf(F, A) + VectorOf(F, B) + VectorOf(F, C) = 0", "coords": {} }, { "index": 6546, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6547, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6548, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "major_axis": "x", "x_coef": 3.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6548.png", "answer": "(pm*sqrt(2),0)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/3 + y^2 = 1)", "coords": {} }, { "index": 6549, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6549.png", "answer": "9", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/12 = 1);F: Point;LeftFocus(G) = F;A: Point;Coordinate(A) = (1, 4);P: Point;PointOnCurve(P, RightPart(G))", "coords": { "A": [ 1.0, 4.0 ] } }, { "index": 6550, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 3.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6550.png", "answer": "4*sqrt(3)", "fact_expr": "C: Ellipse;Expression(C) = (x^2/3 + y^2/2 = 1);F: Point;RightFocus(C) = F;P: Point;PointOnCurve(P, C);A: Point;Coordinate(A) = (1, 2*sqrt(2))", "coords": { "A": [ 1.0, 2.8284271247461903 ] } }, { "index": 6551, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6551.png", "answer": "25/3, 3", "fact_expr": "G: Ellipse;k: Real;Expression(G) = (x^2/5 + y^2/k = 1);Eccentricity(G) = e ;e: Number;e= sqrt(10)/5", "coords": {} }, { "index": 6552, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6552.png", "answer": "(pm*2, 0)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2 = 1)", "coords": {} }, { "index": 6553, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6553.png", "answer": "8", "fact_expr": "C: Parabola;P: Point;F: Point;Q: Point;Expression(C) = (y^2 = 16*x);Focus(C) = F;PointOnCurve(P, C);Intersection(OverlappingLine(LineSegmentOf(P,F)),yAxis)=Q;VectorOf(P,F)=(2/3)*VectorOf(P,Q)", "coords": {} }, { "index": 6554, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 5.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6554.png", "answer": "{2, 22}", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/25 - y^2/9 = 1);P: Point;PointOnCurve(P, G);F1: Point;F2: Point;OneOf(Focus(G)) = F1;OneOf(Focus(G)) = F2;Negation(F1=F2);Distance(P, F1) = 12", "coords": {} }, { "index": 6555, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_ecc_mismatch", "params": { "a": 1.9257268905639648, "b": 1.7993911322594118, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 2.5576271215184017e-15, "converged": true, "iterations": 2 }, "output_path": "results/train/ellipse/problem_6555.png", "answer": "32", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (y^2/2 + x^2/m = 1);PointOnCurve(Focus(G), yAxis);Eccentricity(G) = 1/2", "coords": {}, "validation_reasons": [ "ellipse_ecc_mismatch" ] }, { "index": 6556, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6556.png", "answer": "4", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G)", "coords": {} }, { "index": 6557, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6557.png", "answer": "x^2/9+y^2/6=1", "fact_expr": "G: Ellipse;m: Number;0 < m;m < 3;Expression(G) = (x^2/9 + y^2/m^2 = 1);Eccentricity(G) = sqrt(3)/3", "coords": {} }, { "index": 6558, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6558.png", "answer": "-3/4", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;A: Point;Coordinate(A) = (-2, 3);PointOnCurve(A, Directrix(C));F: Point;Focus(C) = F", "coords": { "A": [ -2.0, 3.0 ] } }, { "index": 6559, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.23606797749979, "b": 4.47213595499958, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6559.png", "answer": "x^2/10 - y^2/40 = 1", "fact_expr": "G: Hyperbola;C:Hyperbola;H: Point;Expression(G) = (x^2/5 - y^2/20 = 1);Coordinate(H) = (-5, 2*sqrt(15));PointOnCurve(H, C);Asymptote(G)=Asymptote(C)", "coords": { "H": [ -5.0, 7.745966692414834 ] } }, { "index": 6560, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 9.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6560.png", "answer": "ApplyUnit(120,degree)", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/9 + y^2/2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 4", "coords": {} }, { "index": 6561, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6561.png", "answer": "y^2=2*x", "fact_expr": "G: Parabola;p: Number;A: Point;B: Point;F: Point;p>0;x0:Number;y0:Number;x0>p/2;Expression(G) = (y^2 = 2*(p*x));Coordinate(A) = (x0,y0);Focus(G)=F;PointOnCurve(A,G);L:Line;PointOnCurve(A,L);IsPerpendicular(L,Directrix(G));FootPoint(L,Directrix(G))=B;Abs(LineSegmentOf(A, F)) = 4 ;Distance(B,LineOf(A, F)) = sqrt(7)", "coords": {} }, { "index": 6562, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6562.png", "answer": "x^2/4+y^2/9=1", "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/9 = 1);P: Point;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P,D), yAxis) = True;FootPoint(LineSegmentOf(P, D), yAxis) = D;D: Point", "coords": {} }, { "index": 6563, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6563.png", "answer": "10", "fact_expr": "C: Curve;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/25 + y^2/9 = 1);Coordinate(P) = (x1, y1);Coordinate(F1) = (-4, 0);Coordinate(F2)= (4, 0);PointOnCurve(P, C);x1:Number;y1:Number", "coords": { "F1": [ -4.0, 0.0 ], "F2": [ 4.0, 0.0 ] } }, { "index": 6564, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6564.png", "answer": "3", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;LeftFocus(G) = F;P: Point;PointOnCurve(P, RightPart(G)) = True;H: Circle;Center(H) = O;O: Origin;Radius(H) = Abs(LineSegmentOf(O,F));PointOnCurve(MidPoint(LineSegmentOf(P,F)),H) = True;Slope(LineOf(P,F)) = sqrt(5)/2", "coords": {} }, { "index": 6565, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6566, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6567, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6567.png", "answer": "y=pm*(sqrt(3)*x)", "fact_expr": "H: Parabola;Expression(H) = (y^2 = 8*x);G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;PointOnCurve(LeftFocus(G), Directrix(H));Length(InterceptChord(Directrix(H), G)) = 6", "coords": {} }, { "index": 6568, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6569, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6569.png", "answer": "{4,1/4}", "fact_expr": "G: Ellipse;a: Number;Expression(G) = (x^2/a^2 + y^2/a = 1);Length(MajorAxis(G)) = 2 * Length(MinorAxis(G))", "coords": {} }, { "index": 6570, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6570.png", "answer": "2", "fact_expr": "C: Ellipse;M: Point;F1: Point;F2: Point;Expression(C) = (x^2/4 + y^2/2 = 1);Focus(C) = {F1, F2};PointOnCurve(M, C)", "coords": {} }, { "index": 6571, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6572, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.4641016151377544, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 12.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6572.png", "answer": "x-2*y+4=0", "fact_expr": "G: Ellipse;H: Circle;A: Point;B: Point;Expression(G) = (x^2/12 + y^2/3 = 1);Expression(H) = ((x + 2)^2 + (y - 1)^2 = 5/2);IsDiameter(LineSegmentOf(A, B), H);PointOnCurve(A,G);PointOnCurve(B,G)", "coords": {} }, { "index": 6573, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6573.png", "answer": "y = pm*sqrt(2)*x", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;FocalLength(C) = 2*c;c: Number;P: Point;PointOnCurve(P, C);Perimeter(TriangleOf(F1, P, F2)) = 6*a + 2*c;Area(TriangleOf(F1, P, F2)) = 2*a*c", "coords": {} }, { "index": 6574, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6575, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6576, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6576.png", "answer": "-1/4", "fact_expr": "G: Parabola;A: Point;P: Point;B: Point;Expression(G) = (x^2 = y);PointOnCurve(A, G);PointOnCurve(B, G);l1: Line;l2: Line;TangentOnPoint(A, G) = l1;TangentOnPoint(B, G) = l2;Intersection(l1, l2) = P;AngleOf(A, P, B) = ApplyUnit(90, degree)", "coords": {} }, { "index": 6577, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6578, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6579, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6579.png", "answer": "1", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*x);H: Line;P: Point;Coordinate(P) = (2, 1);PointOnCurve(P, H);A: Point;B: Point;Intersection(H, G) = {A, B};IsChordOf(LineSegmentOf(A, B), G);MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ 2.0, 1.0 ] } }, { "index": 6580, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6581, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6581.png", "answer": "-b^2*x0/a^2*y0", "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;x0:Number;y0:Number;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(P) = (x0, y0);H:Line;PointOnCurve(P,G);Negation(y0=0);TangetOnPoint(P,G) = H", "coords": {} }, { "index": 6582, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6582.png", "answer": "x^2/9 - y^2/16 = 1", "fact_expr": "G: Hyperbola;b: Number;a: Number;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (5, 0);RightFocus(G) = F;Expression(OneOf(Asymptote(G)))=(y=(4/3)*x)", "coords": { "F": [ 5.0, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 6583, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6583.png", "answer": "{2*sqrt(2)-2,2*sqrt(2)+2}", "fact_expr": "G: Hyperbola;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 - y^2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 6584, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6584.png", "answer": "5", "fact_expr": "l: Line;E: Ellipse;P: Point;M: Point;N: Point;Expression(E) = (x^2/4 + y^2 = 1);PointOnCurve(P, MajorAxis(E));PointOnCurve(P, l);Slope(l) = 1/2;Intersection(l, E) = {M, N}", "coords": {} }, { "index": 6585, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6585.png", "answer": "-8", "fact_expr": "G: Parabola;a: Number;Expression(G) = (x^2 = a*y);Expression(Directrix(G)) = (y = 2)", "coords": {} }, { "index": 6586, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -1.0, 2.0 ], "radius": 2.23606797749979 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_6586.png", "answer": "x-y+1=0", "fact_expr": "l: Line;G: Circle;Expression(G) = (a - 4*y + 2*x + x^2 + y^2 = 0);a: Number;a<3;Intersection(l,G) = {A,B};IsChordOf(LineSegmentOf(A,B),G) = True;Coordinate(MidPoint(LineSegmentOf(A,B))) = (0,1);A: Point;B: Point", "coords": {} }, { "index": 6587, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.4641016151377544, "b": 2.0, "major_axis": "x", "x_coef": 12.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6587.png", "answer": "[-4,4]", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/12 + y^2/4 = 1);PointOnCurve(P, G);Focus(G) = {F1, F2}", "coords": {} }, { "index": 6588, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6588.png", "answer": "5/4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;H: Line;PointOnCurve(F, H);A: Point;B: Point;Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, F)) = 5", "coords": {} }, { "index": 6589, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6589.png", "answer": "21", "fact_expr": "C1: Ellipse;Expression(C1) = (x^2/25 + y^2/16 = 1);C2: Hyperbola;Expression(C2) = (x^2/4 - y^2/5 = 1);A: Point;OneOf(Intersection(C1, C2)) = A;Focus(C1) = {F1, F2};F1: Point;F2: Point", "coords": {} }, { "index": 6590, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6590.png", "answer": "[7,13]", "fact_expr": "G: Ellipse;H1: Circle;H2: Circle;P: Point;M: Point;N: Point;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(P,G) = True;Expression(H1) = (y^2 + (x + 3)^2 = 4);PointOnCurve(M,H1) = True;Expression(H2) = (y^2 + (x - 3)^2 = 1);PointOnCurve(N,H2) = True", "coords": {} }, { "index": 6591, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2135674953460693, "b": 2.4270472526550293, "orientation": "horizontal" }, "optimization": { "final_loss": 4.453341261978494e-06, "converged": true, "iterations": 125 }, "output_path": "results/train/hyperbola/problem_6591.png", "answer": "4", "fact_expr": "l: Line;C1: Hyperbola;C2: Parabola;P: Point;F: Point;O:Origin;Expression(C1)=(x^2/a^2-y^2/b^2=1);Expression(C2)=(y^2=8*x);Eccentricity(C1)=sqrt(5);OneOf(Asymptote(C1))=l;Focus(C2)=F;Intersection(l,C2)=P;Negation(P=O);a: Number;b: Number", "coords": {} }, { "index": 6592, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6592.png", "answer": "4", "fact_expr": "G: Ellipse;b: Number;M: Point;P: Point;l: Line;0 < b;b < 2;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2/b^2 = 1);Coordinate(M) = (2, 2);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(M, l);TangentOfPoint(M, G) = l;TangentPoint(G, l) = P;Negation(PointOnCurve(P, xAxis))", "coords": { "M": [ 2.0, 2.0 ] } }, { "index": 6593, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6593.png", "answer": "(-oo, -sqrt(5)/4) + (sqrt(5)/4, +oo)", "fact_expr": "G: Ellipse;H: Line;k: Real;Expression(G) = (x^2/16 + y^2/4 = 1);Expression(H) = (k*x - y + 3 = 0);NumIntersection(H, G)= 2", "coords": {} }, { "index": 6594, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6594.png", "answer": "sqrt(3)+1", "fact_expr": "C: Hyperbola;b: Number;a: Number;O: Origin;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, RightPart(C));Abs(LineSegmentOf(O, P)) = Abs(LineSegmentOf(O, F1));Abs(LineSegmentOf(P, F1)) = sqrt(3)*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 6595, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6595.png", "answer": "5/3", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;G: Circle;F1: Point;F2:Point;P: Point;T: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2 + y^2 = a^2);LeftFocus(C) =F1;RightFocus(C)=F2;PointOnCurve(F1, l);TangentPoint(l,G)=T;Intersection(l, RightPart(C)) = P;LineSegmentOf(F1, P) = 4*LineSegmentOf(F1, T)", "coords": {} }, { "index": 6596, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6596.png", "answer": "sqrt(3)", "fact_expr": "C: Parabola;P: Point;O: Origin;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(P, C);Abs(LineSegmentOf(P, F)) = 4", "coords": {} }, { "index": 6597, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6597.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;a:Number;b:Number;a>0;b>0;P: Point;F1: Point;F2: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(P, RightPart(G));Tan(AngleOf(P, F1, F2)) = 1/2;Tan(AngleOf(P, F2, F1)) = 2", "coords": {} }, { "index": 6598, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6599, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6599.png", "answer": "0", "fact_expr": "G: Parabola;A: Point;B: Point;P: Point;Expression(G) = (y^2 = 8*x);Coordinate(A) = (0, -4);Coordinate(B) = (3, 2);PointOnCurve(P, G)", "coords": { "A": [ 0.0, -4.0 ], "B": [ 3.0, 2.0 ] } }, { "index": 6600, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6600.png", "answer": "sqrt(33)/3", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;l: Line;PointOnCurve(F2, l) = True;Intersection(l, RightPart(C)) = {A, B};A: Point;B: Point;M: Point;VectorOf(A, B) + VectorOf(A, F1) = 2*VectorOf(A, M);DotProduct(VectorOf(A, M), VectorOf(B, F1)) = 0;Cos(AngleOf(A, F1, B)) = 1/3", "coords": {} }, { "index": 6601, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6602, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6602.png", "answer": "-4", "fact_expr": "G: Parabola;H: Line;c: Number;Expression(G) = (x^2 = 4*y);Expression(H) = (c + 2*x - y = 0);IsTangent(H, G)", "coords": {} }, { "index": 6603, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6603.png", "answer": "sqrt(3)", "fact_expr": "C: Parabola;P: Point;O: Origin;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(P, C);Abs(LineSegmentOf(P, F)) = 4", "coords": {} }, { "index": 6604, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6604.png", "answer": "1", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);H: Line;Expression(H) = (y = k*x + m);k: Number;m: Number;P: Point;Q: Point;Intersection(H, G) = {P, Q};Coordinate(MidPoint(LineSegmentOf(P,Q))) = (x0, 2);x0: Number", "coords": {} }, { "index": 6605, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6605.png", "answer": "16", "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;Expression(G) = (y^2 = 8*x);PointOnCurve(Focus(G),l);Inclination(l) = pi/4;Intersection(l, G) = {A, B};IsChordOf(LineSegmentOf(A,B),G)", "coords": {} }, { "index": 6606, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6606.png", "answer": "sqrt(2)-1", "fact_expr": "A: Point;G: Parabola;Expression(G) = (x^2 = 8*y);Intersection(SymmetryAxis(G), Directrix(G)) = A;B: Point;Focus(G) = B;P: Point;PointOnCurve(P, G);Sin(TriangleOf(P, A, B)) = m * Sin(TriangleOf(P, B, A));m: Real;WhenMin(m);PointOnCurve(P, H) = True;Focus(H) = {A, B};H: Ellipse", "coords": {} }, { "index": 6607, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6607.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;b: Number;a: Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Distance(RightFocus(G),RightDirectrix(G))=(1/3)*FocalLength(G)", "coords": {} }, { "index": 6608, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6608.png", "answer": "sqrt(2)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;G: Circle;Center(G) = Vertex(C);A: Point;B: Point;Intersection(G, C) = {A, B};D: Point;E: Point;Intersection(G, Directrix(C)) = {D, E};Abs(LineSegmentOf(A, B)) = 2*sqrt(6);Abs(LineSegmentOf(D, E)) = 2*sqrt(10)", "coords": {} }, { "index": 6609, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6610, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.3300262689590454, "b": 2.30381441116333, "orientation": "horizontal" }, "optimization": { "final_loss": 6.658357051492203e-06, "converged": true, "iterations": 122 }, "output_path": "results/train/hyperbola/problem_6610.png", "answer": "x^2/4 - y^2/12 = 1", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 16*x);Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x);OneOf(Focus(G))=Focus(H)", "coords": {} }, { "index": 6611, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6612, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6612.png", "answer": "(-1/4, 1)", "fact_expr": "A: Point;Coordinate(A) = (-2, 1);G: Curve;Expression(G) = (y^2 = -4*x);F: Point;Focus(G) = F;P: Point;PointOnCurve(P, G);WhenMin(Abs(LineSegmentOf(P, A))+Abs(LineSegmentOf(P, F)))", "coords": { "A": [ -2.0, 1.0 ] } }, { "index": 6613, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6613.png", "answer": "12/5", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/9 - y^2/16 = 1);P: Point;PointOnCurve(P, C);M: Point;O: Origin;Abs(VectorOf(O, M)) = 1;DotProduct(VectorOf(O, M), VectorOf(P, M)) = 0;WhenMin(Abs(VectorOf(P, M)))", "coords": {} }, { "index": 6614, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6615, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6615.png", "answer": "12", "fact_expr": "H: Parabola;Expression(H) = (y^2 = 16*x);D: Point;PointOnCurve(D, H) ;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;Distance(D, RightFocus(G)) = Distance(D, LeftDirectrix(G)) + 1", "coords": {} }, { "index": 6616, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6617, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6618, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6618.png", "answer": "2*sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;p: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);p>0;Expression(H) = (y^2 = 2*(p*x));Coordinate(Intersection(OneOf(Asymptote(G)),Directrix(H))) = (-2, -1);Distance(LeftVertex(G),Focus(H)) = 4", "coords": {} }, { "index": 6619, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6619.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G))) = (y = x)", "coords": {} }, { "index": 6620, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6620.png", "answer": "5", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P : Point;PointOnCurve(P, G);Distance(P, Focus(G)) = 6", "coords": {} }, { "index": 6621, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6622, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.4641016151377544, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6622.png", "answer": "2", "fact_expr": "H: Parabola;Expression(H) = (y^2 = 16*x);G: Hyperbola;Expression(G) = (x^2/12 - y^2/4 = 1)", "coords": {} }, { "index": 6623, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6623.png", "answer": "y^2 = 8*x", "fact_expr": "G: Parabola;m: Number;H: Line;m>0;Expression(G) = (y^2 = m*x);Expression(H) = (x = 1);Distance(Directrix(G) , H) = 3", "coords": {} }, { "index": 6624, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6625, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6626, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6627, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6628, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6629, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6630, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6630.png", "answer": "1", "fact_expr": "l: Line;Expression(l) = (4*x - 3*y + 6 = 0);C: Parabola;Expression(C) = (y^2 = 4*x);P: Point;PointOnCurve(P, C)", "coords": {} }, { "index": 6631, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6631.png", "answer": "9", "fact_expr": "P: Point;G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);PointOnCurve(P, G);H1: Circle;Expression(H1) = (y^2 + (x - 5)^2 = 4);H2: Circle;Expression(H2) = (y^2 + (x + 5)^2 = 1);M: Point;N: Point;PointOnCurve(M, H1) ;PointOnCurve(N,H2)", "coords": {} }, { "index": 6632, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6633, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6633.png", "answer": "(\\sqrt{2}/2, 1)", "fact_expr": "G: Ellipse;b: Number;a: Number;O: Origin;M: Point;A: Point;e: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftVertex(G) = A;PointOnCurve(M, G);IsPerpendicular(LineSegmentOf(O, M), LineSegmentOf(M, A));Eccentricity(G) = e", "coords": {} }, { "index": 6634, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6634.png", "answer": "2*p", "fact_expr": "E: Parabola;Expression(E) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(E) = F;l: Line;Inclination(l) = ApplyUnit(30, degree);PointOnCurve(F, l);A: Point;B: Point;Intersection(l, E) = {A, B};L1: Line;L2: Line;C: Point;D: Point;PointOnCurve(A, L1);IsPerpendicular(L1, Directrix(E));FootPoint(L1, Directrix(E)) = C;PointOnCurve(B, L2);IsPerpendicular(L2, Directrix(E));FootPoint(L2, Directrix(E)) = D;M: Point;MidPoint(LineSegmentOf(C, D)) = M", "coords": {} }, { "index": 6635, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6635.png", "answer": "sqrt(3)/3", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;p>0;F: Point;Focus(C) = F;l: Line;Directrix(C) = l;K: Point;Intersection(l, xAxis) = K;G: Line;PointOnCurve(F, G);A: Point;B: Point;Intersection(G, C) = {A, B};alpha: Number;Slope(G) = alpha;AngleOf(A, K, B) = ApplyUnit(60, degree)", "coords": {} }, { "index": 6636, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6636.png", "answer": "4", "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (3, 1);Focus(G) = F;PointOnCurve(P, G)", "coords": { "A": [ 3.0, 1.0 ] } }, { "index": 6637, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6638, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6638.png", "answer": "5/4", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;E: Circle;Expression(E) = ((x-5)^2 + y^2 = 9);IsTangent(Asymptote(C), E) = True", "coords": {} }, { "index": 6639, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6639.png", "answer": "(\\sqrt{6}+\\sqrt{2})/2", "fact_expr": "G: Parabola;p: Number;P: Point;Q: Point;F: Point;M: Point;M1: Point;p>0;Expression(G) = (x^2 = 2*(p*y));Focus(G) = F;PointOnCurve(P, G);PointOnCurve(Q, G);AngleOf(P, F, Q) = 5*pi/6;MidPoint(LineSegmentOf(P, Q)) = M;L:Line;PointOnCurve(M,L);Directrix(G)=l;IsPerpendicular(L,l);FootPoint(L,l)=M1;IsChordOf(LineSegmentOf(P,Q),G);l:Line", "coords": {} }, { "index": 6640, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.125, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6640.png", "answer": "11/8", "fact_expr": "G: Parabola;Expression(G) = (y = 2*x^2);A: Point;B: Point;IsChordOf(LineSegmentOf(A,B),G) = True;MidPoint(LineSegmentOf(A,B)) = M;M: Point;Length(LineSegmentOf(A,B)) = 3", "coords": {} }, { "index": 6641, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6642, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6642.png", "answer": "sqrt(3)+2", "fact_expr": "E: Hyperbola;b: Number;a: Number;F2: Point;P: Point;F1: Point;a>0;b>0;Expression(E) = (-y^2/b + x^2/a^2 = 1);LeftFocus(E) = F1;RightFocus(E) = F2;PointOnCurve(P, E);PointOnCurve(MidPoint(LineSegmentOf(P,F2)), yAxis);AngleOf(P, F2, F1) = ApplyUnit(60, degree)", "coords": {} }, { "index": 6643, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6643.png", "answer": "7", "fact_expr": "G: Ellipse;H: Circle;C:Circle;P: Point;M: Point;N: Point;Expression(G) = (x^2/25 + y^2/16 = 1);Expression(H) = (y^2 + (x + 3)^2 = 1);Expression(C) = (y^2 + (x - 3)^2 = 4);PointOnCurve(P, G);PointOnCurve(M, H);PointOnCurve(N, C)", "coords": {} }, { "index": 6644, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6644.png", "answer": "8*sqrt(2)/3", "fact_expr": "G: Hyperbola;H: Line;Expression(G) = (x^2 - y^2/4 = 1);Expression(H) = (y = x + 1)", "coords": {} }, { "index": 6645, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6645.png", "answer": "(1, 5/4]", "fact_expr": "E: Hyperbola;b: Number;a: Number;C: Parabola;P: Point;A: Point;F: Point;a>0;b>0;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);Expression(C) = (y^2 = 16*(a*x));RightVertex(E) = A;Focus(C) = F;PointOnCurve(P, Asymptote(E));DotProduct(VectorOf(P, A), VectorOf(P, F)) = 0", "coords": {} }, { "index": 6646, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6646.png", "answer": "sqrt(10)/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;F1: Point;A: Point;F2: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(A, G);AngleOf(F1, A, F2) = ApplyUnit(90, degree);Abs(LineSegmentOf(A, F1)) = 3*Abs(LineSegmentOf(A, F2))", "coords": {} }, { "index": 6647, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_6647.png", "answer": "2", "fact_expr": "G: Hyperbola;a: Number;b: Number;H: Parabola;p: Number;A: Point;O: Origin;B: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);p>0;Expression(H) = (y^2 = 2*(p*x));L1:Line;L2:Line;Asymptote(G)={L1,L2};Intersection(L1,Directrix(H))=A;Intersection(L2,Directrix(H))=B;Eccentricity(G) = 2;Area(TriangleOf(A, O, B)) = sqrt(3)", "coords": {} }, { "index": 6648, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 3.753767905018077, "b": 2.2935134786480793, "major_axis": "x", "x_coef": 14.999999999999998, "y_coef": 5.000000000000001 }, "optimization": { "final_loss": 0.00019830568017831433, "converged": true, "iterations": 62 }, "output_path": "results/train/ellipse/problem_6648.png", "answer": "x^2/25 + y^2/5 = 1", "fact_expr": "G: Ellipse;P: Point;Q: Point;Coordinate(P) = (sqrt(3), -2);Coordinate(Q) = (-2*sqrt(3), 1);PointOnCurve(Q, G);PointOnCurve(P, G)", "coords": { "P": [ 1.7320508075688772, -2.0 ], "Q": [ -3.4641016151377544, 1.0 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 6649, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6649.png", "answer": "8", "fact_expr": "G: Hyperbola;P: Point;A: Point;O: Origin;B: Point;t: Real;Expression(G) = (x^2/9 - y^2/16 = 1);PointOnCurve(P, G);VectorOf(P, A) = VectorOf(O, P)*(t - 1);DotProduct(VectorOf(O, A), VectorOf(O, P)) = 60;Coordinate(VectorOf(O, B)) = (0, 1)", "coords": {} }, { "index": 6650, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 9.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6650.png", "answer": "-4", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/2 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G) = True;Abs(LineSegmentOf(P, F1)) = 4", "coords": {} }, { "index": 6651, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6651.png", "answer": "3*sqrt(2)/16", "fact_expr": "G: Parabola;H: Line;P: Point;Q: Point;O: Origin;M: Point;Expression(G) = (y = x^2);Expression(H) = (y = x + 1);PointOnCurve(P, H);PointOnCurve(Q, G);MidPoint(LineSegmentOf(P, Q)) = M", "coords": {} }, { "index": 6652, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6652.png", "answer": "sqrt(3)*2/3\npm*4", "fact_expr": "C: Hyperbola;G: Parabola;m: Number;Expression(C) = (x^2/3 - y^2 = 1);Expression(G) = (y^2 = 2*(m*x));Focus(G) = Focus(C)", "coords": {} }, { "index": 6653, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6653.png", "answer": "4*sqrt(38)/5", "fact_expr": "G: Ellipse;H: Line;Expression(G) = (x^2 + 4*y^2 = 16);Expression(H) = (y = x + 1)", "coords": {} }, { "index": 6654, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6654.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Parabola;H: Line;P: Point;Expression(G) = (y^2 = 2*x);Coordinate(P) = (0, 1);PointOnCurve(P,H);NumIntersection(G, H) = 1", "coords": { "P": [ 0.0, 1.0 ] } }, { "index": 6655, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_focus_mismatch", "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6655.png", "answer": "sqrt(2)", "fact_expr": "C: Hyperbola;a: Number;b: Number;F: Point;O: Origin;E: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (2, 0);RightFocus(C) = F;l:Line;PointOnCurve(F,l);IsPerpendicular(l,OneOf(Asymptote(C)));FootPoint(l,OneOf(Asymptote(C)))=E;WhenMax(Area(TriangleOf(O,E,F)))", "coords": { "F": [ 2.0, 0.0 ] }, "validation_reasons": [ "hyperbola_focus_mismatch" ] }, { "index": 6656, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6656.png", "answer": "(6, 4*sqrt(3))", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 8*x);F: Point;Focus(C) = F;P: Point;PointOnCurve(P, C);Quadrant(P) = 1;Abs(LineSegmentOf(P, F)) = 8", "coords": {} }, { "index": 6657, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6657.png", "answer": "4", "fact_expr": "M: Point;G: Parabola;Expression(G) = (y^2 = 4*x);PointOnCurve(M, G) = True;F: Point;Focus(G) = F;A: Point;C: Circle;Expression(C) = ((x - 4)^2 + (y - 1)^2 = 1);PointOnCurve(A, C) = True", "coords": {} }, { "index": 6658, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 2.0, "major_axis": "x", "x_coef": 8.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6658.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Ellipse;H: Line;P0: Point;Expression(G) = (x^2/8 + y^2/4 = 1);Expression(H) = (x - y + 6 = 0);PointOnCurve(P0, G)", "coords": {} }, { "index": 6659, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6659.png", "answer": "pm*y+x=0", "fact_expr": "C: Hyperbola;b: Number;a: Number;F1: Point;P: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);AngleOf(F1, P, F2) = pi/2;Area(TriangleOf(F1, P, F2)) = a^2", "coords": {} }, { "index": 6660, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6661, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6661.png", "answer": "sqrt(37)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);L: Line;PointOnCurve(Focus(G), L);Inclination(L) = ApplyUnit(60, degree);H: Circle;Expression(H) = (4*(sqrt(3)*y) - 4*x + x^2 + y^2 = 0)", "coords": {} }, { "index": 6662, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6662.png", "answer": "-3/8", "fact_expr": "l: Line;C: Ellipse;P: Point;A2: Point;A1: Point;Expression(C) = (x^2/4 + y^2/3 = 1);LeftVertex(C)=A1;RightVertex(C)=A2;PointOnCurve(A1,l);Intersection(l,C)={A1, P};Slope(l)=2", "coords": {} }, { "index": 6663, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6664, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.23606797749979, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6664.png", "answer": "2*sqrt(5)\n4\n(pm*3,0)\n3*sqrt(5)/5\ny=pm*(2*sqrt(5)/5)*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/5 - y^2/4 = 1)", "coords": {} }, { "index": 6665, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6665.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-x^2/b^2 + y^2/a^2 = 1);Expression(OneOf(Asymptote(G))) = (x - 2*y = 0);Eccentricity(G)=e;e:Number", "coords": {} }, { "index": 6666, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6666.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(Focus(G), H);Inclination(H) = pi/3;Intersection(H, G) = {A, B}", "coords": {} }, { "index": 6667, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6667.png", "answer": "7/4", "fact_expr": "l: Line;G: Parabola;Expression(G) = (y^2 = x);PointOnCurve(Focus(G), l);A: Point;B: Point;Intersection(l, G) = {A, B};Abs(LineSegmentOf(A, B)) = 4;IsChordOf(LineSegmentOf(A, B), G)", "coords": {} }, { "index": 6668, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6668.png", "answer": "(2, 0)", "fact_expr": "G: Ellipse;M: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2/3 = 1);PointOnCurve(M, G);LeftFocus(G) = F1;RightFocus(G) = F2;Abs(LineSegmentOf(M, F1)) = 3*Abs(LineSegmentOf(M, F2))", "coords": {} }, { "index": 6669, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6669.png", "answer": "(\\sqrt{5}, +\\infty)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;H: Line;Expression(H) = (y = 2*x);IsIntersect(G, H)", "coords": {} }, { "index": 6670, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6670.png", "answer": "x^2 - y^2/4 = 1", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Expression(OneOf(Asymptote(G))) = (2*x + y = 0);H: Point;Coordinate(H) = (sqrt(5), 0);OneOf(Focus(G)) = H", "coords": { "H": [ 2.23606797749979, 0.0 ] } }, { "index": 6671, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6671.png", "answer": "1", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x)", "coords": {} }, { "index": 6672, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6672.png", "answer": "(4, pm*4*sqrt(2))", "fact_expr": "G: Parabola;A: Point;F: Point;Expression(G) = (y^2 = 8*x);Focus(G) = F;PointOnCurve(A, G);Abs(LineSegmentOf(A, F))=6", "coords": {} }, { "index": 6673, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6673.png", "answer": "4", "fact_expr": "G: Parabola;H: Line;Q: Point;A: Point;B: Point;F: Point;D: Point;Expression(G) = (y^2 = 4*x);Coordinate(Q) = (0, 3);Focus(G) = F;PointOnCurve(Q, H);Intersection(H, G) = {A, B};Intersection(PerpendicularBisector(LineSegmentOf(A, B)), xAxis) = D;Abs(LineSegmentOf(A, F)) + Abs(LineSegmentOf(B, F)) = 6", "coords": { "Q": [ 0.0, 3.0 ] } }, { "index": 6674, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6674.png", "answer": "1/2", "fact_expr": "l: Line;C: Ellipse;b: Number;a: Number;O: Circle;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Expression(O) = (x^2 + y^2 = b^2);IsTangent(l, O);e:Number;Eccentricity(C) = e;Inclination(l) = pi/3;PointOnCurve(RightVertex(C),l)", "coords": {} }, { "index": 6675, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6675.png", "answer": "x-sqrt(3)*y+1=0", "fact_expr": "l: Line;C: Parabola;G: Circle;t: Number;A: Point;B: Point;M: Point;F: Point;N: Point;Expression(C) = (y^2 = 4*x);t>0;Expression(l) = (-t*y + x + 1 = 0);Intersection(l, C) = {A, B};Negation(A=B);MidPoint(LineSegmentOf(A, B)) = M;Focus(C) = F;IsDiameter(LineSegmentOf(M,F),G);Intersection(l,G)={M,N};Abs(LineSegmentOf(M,N))=3*sqrt(3)*Abs(LineSegmentOf(N,F))", "coords": {} }, { "index": 6676, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6677, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.8284271247461903, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6677.png", "answer": "sqrt(6)/2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/8 - y^2/4 = 1)", "coords": {} }, { "index": 6678, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 10.0, "b": 6.0, "major_axis": "x", "x_coef": 100.0, "y_coef": 36.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6678.png", "answer": "36", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/100 + y^2/36 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(90, degree)", "coords": {} }, { "index": 6679, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6679.png", "answer": "(pm*4, 0)\nsqrt(3)*x + pm*y = 0", "fact_expr": "G: Hyperbola;a: Number;b: Number;H: Ellipse;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2/25 + y^2/9 = 1);Eccentricity(G) = 2;Focus(G) = Focus(H)", "coords": {} }, { "index": 6680, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6680.png", "answer": "1", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;H: Point;Intersection(Directrix(G), xAxis) = H;P: Point;PointOnCurve(P, G);Abs(LineSegmentOf(P, H)) = sqrt(2)*Abs(LineSegmentOf(P, F))", "coords": {} }, { "index": 6681, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6681.png", "answer": "sqrt(3)+1", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, RightPart(G));O: Origin;DotProduct((VectorOf(O, F2) + VectorOf(O, P)), VectorOf(F2, P)) = 0;Abs(LineSegmentOf(P, F1)) = sqrt(3)*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 6682, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 3.0, "major_axis": "x", "x_coef": 36.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6682.png", "answer": "23/3", "fact_expr": "G: Ellipse;K: Point;M: Point;N: Point;Expression(G) = (x^2/36 + y^2/9 = 1);Coordinate(K) = (2, 0);DotProduct(VectorOf(K, M), VectorOf(K, N)) = 0;PointOnCurve(M, G);PointOnCurve(N, G)", "coords": { "K": [ 2.0, 0.0 ] } }, { "index": 6683, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.23606797749979, "b": 2.0, "major_axis": "x", "x_coef": 5.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6683.png", "answer": "y^2=4*x", "fact_expr": "G: Parabola;H: Ellipse;Expression(H) = (x^2/5 + y^2/4 = 1);Focus(G) = RightFocus(H);Vertex(G)=Center(H)", "coords": {} }, { "index": 6684, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6684.png", "answer": "x^2/3-y^2=1", "fact_expr": "E: Hyperbola;a: Number;b: Number;a>0;b>0;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(OneOf(Focus(E))) = (2, 0);Distance(OneOf(Focus(E)),Asymptote(E))=1", "coords": {} }, { "index": 6685, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6685.png", "answer": "2*p", "fact_expr": "C: Parabola;p: Number;G: Line;Q: Point;B: Point;F: Point;A: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;PointOnCurve(F, G);Intersection(Directrix(C), xAxis) = Q;Intersection(G, C) = {A, B};AngleOf(Q, B, F) = ApplyUnit(90, degree)", "coords": {} }, { "index": 6686, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6686.png", "answer": "pm*(3/2)", "fact_expr": "G: Hyperbola;H: Line;k: Real;A: Point;B: Point;l1:Line;l2:Line;Expression(G) = (x^2 - y^2/3 = 1);Expression(H) = (y = k*x);Intersection(H, G) = {A, B};IsPerpendicular(l1,xAxis);IsPerpendicular(l2,xAxis);PointOnCurve(A,l1);PointOnCurve(B,l2);FootPoint(l1,xAxis)=F1;FootPoint(l2,xAxis)=F2;Focus(G)={F1,F2};F1:Point;F2:Point", "coords": {} }, { "index": 6687, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6687.png", "answer": "7/4", "fact_expr": "P: Point;G: Parabola;Expression(G) = (y^2 = x);PointOnCurve(P, G);M: Circle;Expression(M) = (y^2 + (x - 3)^2 = 1);l1: Line;A: Point;TangentOfPoint(P, M) = l1;TangentPoint(l1, M) = A", "coords": {} }, { "index": 6688, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.125, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6688.png", "answer": "(5/4,+oo)", "fact_expr": "l: Line;G: Parabola;B: Point;A: Point;Expression(G) = (y = 2*x^2);PointOnCurve(A, G);PointOnCurve(B, G);PerpendicularBisector(LineSegmentOf(A,B))=l;Slope(l) = 1/2", "coords": {} }, { "index": 6689, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6689.png", "answer": "8", "fact_expr": "G: Ellipse;H: Line;F2: Point;A: Point;B: Point;F1: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(G) = {F1, F2};PointOnCurve(F1, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(F2, A)) + Abs(LineSegmentOf(F2, B)) = 12", "coords": {} }, { "index": 6690, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6691, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6691.png", "answer": "9", "fact_expr": "G: Hyperbola;A: Point;B: Point;F2: Point;F1: Point;Expression(G) = (x^2/9 - y^2/16 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;IsChordOf(LineSegmentOf(A,B),G);PointOnCurve(F1,LineSegmentOf(A,B));PointOnCurve(A,LeftPart(G));PointOnCurve(B,LeftPart(G));Perimeter(TriangleOf(A, B, F2)) = 30", "coords": {} }, { "index": 6692, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6692.png", "answer": "[(sqrt(5)-1)/2, 1)", "fact_expr": "E: Ellipse;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(E) = F1;RightFocus(E) = F2;A: Point;B: Point;RightVertex(E) = A;UpperVertex(E) = B;P: Point;PointOnCurve(P, LineOf(A, B));IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2));e: Number;Eccentricity(E) = e", "coords": {} }, { "index": 6693, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6693.png", "answer": "16/3", "fact_expr": "P: Point;Coordinate(P) = (1, 0);l: Line;PointOnCurve(P, l);Inclination(l) = ApplyUnit(60, degree);G: Parabola;Expression(G) = (y^2 = 4*x);A: Point;B: Point;Intersection(l, G) = {A, B};IsChordOf(LineSegmentOf(A, B), G)", "coords": { "P": [ 1.0, 0.0 ] } }, { "index": 6694, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6694.png", "answer": "-sqrt(13)", "fact_expr": "P: Point;G:Curve;x1:Number;y1:Number;Coordinate(P) = (x1, y1);Expression(G)=(x^2/25 + y^2/16 = 1);PointOnCurve(P,G)", "coords": {} }, { "index": 6695, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6695.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;b > a;a > 0;c: Number;FocalLength(G) = 2*c;l: Line;Q: Point;W: Point;Coordinate(Q) = (a, 0);Coordinate(W) = (0, b);PointOnCurve(Q, l);PointOnCurve(W, l);E: Point;Coordinate(E) = (-a, 0);Distance(E, l) = c*(2*sqrt(2)/3)", "coords": {} }, { "index": 6696, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6696.png", "answer": "12/5", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P: Point;PointOnCurve(P, G);d1: Number;Distance(P, Directrix(G)) = d1;H: Line;Expression(H) = (3*x - 4*y + 9 = 0);d2: Number;Distance(P, H) = d2", "coords": {} }, { "index": 6697, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6698, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6698.png", "answer": "4", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2 = 1);PointOnCurve(P, G);Focus(G) = {F1, F2}", "coords": {} }, { "index": 6699, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6699.png", "answer": "sqrt(2)/2", "fact_expr": "G: Parabola;P: Point;F: Point;A: Point;Expression(G) = (y = x^2/4);Focus(G) = F;Intersection(SymmetryAxis(G),Directrix(G))=A;PointOnCurve(P,G);Abs(LineSegmentOf(P,F))=m*Abs(LineSegmentOf(P,A));m:Number", "coords": {} }, { "index": 6700, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6701, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6701.png", "answer": "1/2", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;LeftFocus(C) = F;G: Line;Expression(G) = (x = m);m: Number;A: Point;B: Point;Intersection(G, C) = {A, B};WhenMax(Perimeter(TriangleOf(A, B, F)));Area(TriangleOf(A, B, F)) = b^2;e: Number;Eccentricity(C) = e", "coords": {} }, { "index": 6702, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6703, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6703.png", "answer": "(0,2]", "fact_expr": "C: Parabola;p: Number;M: Point;P: Point;Q: Point;p>0;Expression(C) = (x^2 = 2*p*y);Coordinate(M) = (0, -1);PointOnCurve(P,C);PointOnCurve(Q, C);DotProduct(VectorOf(M, P), VectorOf(M, Q)) >= 0", "coords": { "M": [ 0.0, -1.0 ] } }, { "index": 6704, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6704.png", "answer": "(3, 6)", "fact_expr": "H: Line;Expression(H) = (y = 2*x);P: Point;Quadrant(P) = 1;PointOnCurve(P, H);G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1);L1: Line;L2: Line;Asymptote(G) = {L1, L2};Distance(P, L1)*Distance(P, L2) = 27", "coords": {} }, { "index": 6705, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6705.png", "answer": "sqrt(26)-2*sqrt(3)", "fact_expr": "G: Hyperbola;A: Point;M: Point;F: Point;Expression(G) = (x^2/3 - y^2 = 1);Coordinate(A) = (3, 1);RightFocus(G) = F;PointOnCurve(M,RightPart(G))", "coords": { "A": [ 3.0, 1.0 ] } }, { "index": 6706, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6707, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6707.png", "answer": "1-sqrt(5)/5", "fact_expr": "C: Ellipse;Expression(C)=(x^2/a^2 + y^2/b^2 = 1);b:Number;a>b;F: Point;LeftFocus(C)=F;A:Point;RightVertex(C)=A;PointOnCurve(P,C);IsPerpendicular(LineSegmentOf(P,F),xAxis);Slope(LineOf(P,A))=sqrt(5)/5;b>0;a:Number;P:Point", "coords": {} }, { "index": 6708, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6708.png", "answer": "1+3*sqrt(2)/4", "fact_expr": "l: Line;E: Ellipse;A: Point;F: Point;B: Point;Expression(E) = (x^2/2 + y^2 = 1);LeftFocus(E) = F;PointOnCurve(F, l);Intersection(l, E) = {A, B}", "coords": {} }, { "index": 6709, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6709.png", "answer": "(1, 1)", "fact_expr": "G: Parabola;H: Line;Expression(G) = (y = x^2);Expression(H) = (2*x - y - 4 = 0);P: Point;PointOnCurve(P, G);WhenMin(Distance(P, H))", "coords": {} }, { "index": 6710, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6711, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6711.png", "answer": "(3, 2*sqrt(3))", "fact_expr": "C: Parabola;p: Number;M: Point;F: Point;p>0;y0: Number;y0 > 0;Expression(C) = (y^2 = 2*(p*x));Coordinate(M) = (3, y0);Focus(C) = F;PointOnCurve(M, C);Distance(M, Focus(C)) = Abs(LineSegmentOf(M, F));Abs(LineSegmentOf(M, F)) = 4", "coords": {} }, { "index": 6712, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6712.png", "answer": "(-1, 1/4)", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);P: Point;PointOnCurve(P, G);F: Point;Focus(G) = F;A: Point;Coordinate(A) = (-1, 3);WhenMin(Abs(LineSegmentOf(P, F))+Abs(LineSegmentOf(P, A)))", "coords": { "A": [ -1.0, 3.0 ] } }, { "index": 6713, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/train/hyperbola/problem_6713.png", "answer": "7*sqrt(2)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;P: Point;PointOnCurve(P, G);Eccentricity(G) = 5/4;F1: Point;F2: Point;Focus(G) = {F1, F2};DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;Area(TriangleOf(F1, P, F2)) = 18", "coords": {} }, { "index": 6714, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6715, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6715.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Ellipse;a: Number;b: Number;H: Line;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(Focus(G), H);IsPerpendicular(H, xAxis);Length(InterceptChord(H, G)) = a", "coords": {} }, { "index": 6716, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6716.png", "answer": "5", "fact_expr": "l: Line;C: Parabola;G: Line;B: Point;A: Point;F: Point;M: Point;Expression(C) = (y^2 = 4*x);Expression(G) = (x + 1 = 0);Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, B)) = 10;MidPoint(LineSegmentOf(A,B)) = M", "coords": {} }, { "index": 6717, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6717.png", "answer": "2*sqrt(5)", "fact_expr": "C: Parabola;G: Circle;A: Point;F: Point;Expression(C) = (x^2 = -4*y);Focus(C) = F;PointOnCurve(A,C);Abs(LineSegmentOf(A,F))=3;Center(G)=A;Radius(G)=LineSegmentOf(A,F)", "coords": {} }, { "index": 6718, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6718.png", "answer": "(1,2)", "fact_expr": "G: Parabola;H: Line;P: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (y = x + 4);PointOnCurve(P, G);WhenMin(Distance(P,H))", "coords": {} }, { "index": 6719, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6719.png", "answer": "1", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;M: Point;PointOnCurve(M, G) = True;AngleOf(F1, M, F2) = 2*pi/3;DotProduct(VectorOf(M, F1), VectorOf(M, F2)) = -2", "coords": {} }, { "index": 6720, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6720.png", "answer": "1", "fact_expr": "F1: Point;F2: Point;G: Hyperbola;Expression(G) = (x^2 - 4*y^2 = 4);Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G) = True;DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0", "coords": {} }, { "index": 6721, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6722, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 2.0, "major_axis": "x", "x_coef": 8.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6722.png", "answer": "[0,sqrt(2)]", "fact_expr": "C: Ellipse;P: Point;F1: Point;F2: Point;O: Origin;Expression(C) = (x^2/8 + y^2/4 = 1);PointOnCurve(P, C);LeftFocus(C)=F1;RightFocus(C)=F2", "coords": {} }, { "index": 6723, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6723.png", "answer": "2", "fact_expr": "G: Parabola;p: Number;Q: Point;p>0;Expression(G) = (y^2 = 2*(p*x));PointOnCurve(Q, G);Min(Distance(Q, Focus(G))) = 1", "coords": {} }, { "index": 6724, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6724.png", "answer": "x=pm*3", "fact_expr": "G: Parabola;H: Ellipse;a: Number;Expression(G) = (y^2 = 4*x);Expression(H) = (y^2/2 + x^2/a^2 = 1);PointOnCurve(Focus(H), Directrix(G))", "coords": {} }, { "index": 6725, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6725.png", "answer": "{2*sqrt(5), 2*sqrt(13)}", "fact_expr": "G: Ellipse;k: Real;k > 0;Expression(G) = (x^2/36 + y^2/k^2 = 1);FocalLength(G) = 8", "coords": {} }, { "index": 6726, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6727, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -3.0, -0.0 ], "radius": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_6727.png", "answer": "x^2/36+y^2/27=1", "fact_expr": "G: Circle;C:Circle;H:Circle;Expression(G) = (6*x + x^2 + y^2 + 5 = 0);Expression(C) = (-6*x + x^2 + y^2 - 91 = 0);IsOutTangent(H,G);IsInTangent(H,C)", "coords": {} }, { "index": 6728, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6728.png", "answer": "2*sqrt(2)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/m^2 = 1);m: Number;m>0;Z: Line;Expression(Z) = (y = (sqrt(2)/2)*x)\t;Projection(OneOf(Intersection(Z, G)), xAxis) = RightFocus(G)", "coords": {} }, { "index": 6729, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6729.png", "answer": "9", "fact_expr": "G: Hyperbola;H: Circle;Z: Circle;P: Point;M: Point;N: Point;Expression(G) = (x^2/9 - y^2/16 = 1);Expression(H) = (y^2 + (x + 5)^2 = 4);PointOnCurve(P, RightPart(G));Expression(Z) = (y^2 + (x - 5)^2 = 1);PointOnCurve(M, H);PointOnCurve(N, Z)", "coords": {} }, { "index": 6730, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6730.png", "answer": "2/3", "fact_expr": "G: Ellipse;H: Line;A: Point;O: Origin;B: Point;Expression(G) = (x^2/2 + y^2 = 1);PointOnCurve(RightFocus(G), H);Slope(H) = 1;Intersection(H, G) = {A, B}", "coords": {} }, { "index": 6731, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6731.png", "answer": "sqrt(65)/15", "fact_expr": "M: Ellipse;Expression(M) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;P: Point;Coordinate(P) = (a/3, b/2);F1: Point;F2: Point;LeftFocus(M) = F1;RightFocus(M) = F2;IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2))", "coords": {} }, { "index": 6732, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 4.47213595499958, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6732.png", "answer": "17", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/20 = 1);F1: Point;F2: Point;Focus(G) = {F1,F2};P: Point;PointOnCurve(P, G) = True;Distance(P, F1) = 9", "coords": {} }, { "index": 6733, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6733.png", "answer": "8", "fact_expr": "M: Ellipse;N: Circle;E: Point;F: Point;P: Point;Expression(M) = (x^2/2 + y^2 = 1);Expression(N) = (x^2 + (y - 2)^2 = 1);PointOnCurve(P, M);IsDiameter(LineSegmentOf(E, F), N)", "coords": {} }, { "index": 6734, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6735, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6735.png", "answer": "sqrt(17)/3", "fact_expr": "l: Line;E: Hyperbola;b: Number;a: Number;F2: Point;Q: Point;P: Point;F1: Point;a>0;b>0;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(E) = F1;RightFocus(E) = F2;PointOnCurve(F1,l);Intersection(l,LeftPart(E))={P, Q};Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(F1, Q));IsPerpendicular(LineSegmentOf(F2, Q), LineSegmentOf(P, Q))", "coords": {} }, { "index": 6736, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6736.png", "answer": "-4", "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (x^2 + y^2/m = 1);Length(ImageinaryAxis(G)) = 2*Length(RealAxis(G))", "coords": {} }, { "index": 6737, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6737.png", "answer": "sqrt(2)", "fact_expr": "G: Parabola;H: Ellipse;a: Number;F1: Point;F: Point;F2: Point;Expression(G) = (y^2 = 2*x);a>0;Expression(H) = (y^2 + x^2/a^2 = 1);Focus(H) = {F1, F2};Focus(G) = F;VectorOf(F1, F) = 3*VectorOf(F, F2)", "coords": {} }, { "index": 6738, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_focus_mismatch", "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6738.png", "answer": "8 + sqrt(2)", "fact_expr": "G: Ellipse;a: Number;b: Number;F: Point;A: Point;M: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (2, 0);Coordinate(A) = (-2, sqrt(2));RightFocus(G) = F;l: LineSegment;IsChordOf(l, G);IsPerpendicular(l, xAxis);PointOnCurve(F, l);Length(l) = 6;PointOnCurve(M, G)", "coords": { "F": [ 2.0, 0.0 ], "A": [ -2.0, 1.4142135623730951 ] }, "validation_reasons": [ "ellipse_focus_mismatch" ] }, { "index": 6739, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6739.png", "answer": "sqrt(5)/3", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/4 = 1)", "coords": {} }, { "index": 6740, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6740.png", "answer": "pi/2", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/4 = 1);F1: Point;F2: Point;P: Point;PointOnCurve(P,G) = True;LeftFocus(G) = F1;RightFocus(G) = F2;Abs((LineSegmentOf(P,F1))) = Abs((LineSegmentOf(P,F2)))*2", "coords": {} }, { "index": 6741, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -4.0, 0.0 ], "radius": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_6741.png", "answer": "(x^2/4 - y^2/12 = 1)&(x <= -2)", "fact_expr": "B: Circle;A: Point;C: Circle;Expression(B) = (y^2 + (x - 4)^2 = 16);Coordinate(A) = (-4, 0);PointOnCurve(A, C);IsOutTangent(C, B)", "coords": {} }, { "index": 6742, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6742.png", "answer": "3/2", "fact_expr": "G: Parabola;H: Line;F: Point;P: Point;Q: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(F,H);Focus(G) = F;Intersection(H, G) = {P, Q};Length(LineSegmentOf(P,F)) = 3", "coords": {} }, { "index": 6743, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6743.png", "answer": "9*sqrt(2)/4", "fact_expr": "G: Parabola;P: Point;O: Origin;F: Point;Expression(G) = (y^2 = 6*x);PointOnCurve(P, G);Focus(G) = F;Distance(P, F) = 9/2", "coords": {} }, { "index": 6744, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6745, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6745.png", "answer": "9", "fact_expr": "G: Ellipse;Expression(G) = (x^2/a^2 + y^2/b^2 = 1);a: Number;In(a, [1, 5]);In(b, [2, 4]);b: Number;PointOnCurve(Focus(G), xAxis);Eccentricity(G) < sqrt(3)/2;z: Number;z = a + b", "coords": {} }, { "index": 6746, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4149149626832767, "b": 2.450642358201144, "orientation": "horizontal" }, "optimization": { "final_loss": 1.0319236343469396e-05, "converged": true, "iterations": 94 }, "output_path": "results/train/hyperbola/problem_6746.png", "answer": "x^2/9 - y^2/27 = 1", "fact_expr": "G: Hyperbola;H: Parabola;Expression(H) = (y^2 = 24*x);Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x);PointOnCurve(OneOf(Focus(G)), Directrix(H))", "coords": {} }, { "index": 6747, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.1622776601683795, "b": 1.0, "major_axis": "x", "x_coef": 10.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6747.png", "answer": "sqrt(3)", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/10 + y^2 = 1);PointOnCurve(P, G);Focus(G) = {F1, F2};AngleOf(F1,P,F2)=2*pi/3", "coords": {} }, { "index": 6748, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6749, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.581065485469099, "b": 1.581065485469099, "orientation": "horizontal" }, "optimization": { "final_loss": 1.6812139285133018e-06, "converged": true, "iterations": 95 }, "output_path": "results/train/hyperbola/problem_6749.png", "answer": "x^2/4 - y^2 = 1", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;O: Origin;Coordinate(F1) = (sqrt(5), 0);Coordinate(F2) = (-sqrt(5), 0);Focus(G) = {F1, F2};Center(G) = O;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2));Area(TriangleOf(P, F1, F2)) = 1", "coords": { "F1": [ 2.23606797749979, 0.0 ], "F2": [ -2.23606797749979, 0.0 ] } }, { "index": 6750, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6750.png", "answer": "(0, \\sqrt{2}/2)", "fact_expr": "G: Ellipse;Expression(G) = (x^2 + y^2/b^2 = 1);b: Number;0 < b;b < 1;F: Point;LeftFocus(G) = F;A: Point;C: Point;LeftVertex(G) = A;RightVertex(G) = C;B: Point;UpperVertex(G) = B;PointOnCurve(F, P1) = True;PointOnCurve(B, P1) = True;PointOnCurve(C, P1) = True;P1: Circle;Center(P1) = P;P: Point;Coordinate(P) = (m, n);m: Number;n: Number;m + n > 0", "coords": {} }, { "index": 6751, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6751.png", "answer": "2*x^2-2*y^2=1", "fact_expr": "G: Hyperbola;H: Parabola;Expression(H) = (y^2 = 4*x);Expression(Asymptote(G)) = (y = pm*x);RightFocus(G) = Focus(H)", "coords": {} }, { "index": 6752, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6752.png", "answer": "sqrt(5)/5", "fact_expr": "E: Ellipse;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(E) = F1;RightFocus(E) = F2;A: Point;B: Point;C: Point;PointOnCurve(A, E);PointOnCurve(B, E);PointOnCurve(C, E);VectorOf(A, F1) = VectorOf(F1, B);VectorOf(A, F2) = 2*VectorOf(F2, C)", "coords": {} }, { "index": 6753, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6753.png", "answer": "[1/2, 2]", "fact_expr": "G: Parabola;A: Point;B: Point;M: Point;P: Point;Expression(G) = (x^2 = 4*y);Coordinate(M) = (t, -2);t:Number;In(t, (-1, 1));TangentOfPoint(M,G)={LineOf(M,A),LineOf(M,B)};TangentPoint(LineOf(M,A),G)=A;TangentPoint(LineOf(M,B),G)=B;Intersection(LineOf(A,B), yAxis) = P", "coords": {} }, { "index": 6754, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6755, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6755.png", "answer": "4*sqrt(10)/5", "fact_expr": "Slope(l) = 1;l: Line;G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);Intersection(l, G) = {A, B};A: Point;B: Point", "coords": {} }, { "index": 6756, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6756.png", "answer": "9/2", "fact_expr": "G: Parabola;H: Line;P: Point;Q: Point;F: Point;B: Point;A: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};Intersection(H,Directrix(G))=P;Intersection(H,yAxis)=Q;VectorOf(P, Q) = VectorOf(F, B);IsChordOf(LineSegmentOf(A,B),G)", "coords": {} }, { "index": 6757, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6757.png", "answer": "(-2, 1/2)", "fact_expr": "C: Parabola;A: Point;P: Point;F: Point;Expression(C) = (x^2 = 8*y);Coordinate(A) = (-2, 4);Focus(C) = F;PointOnCurve(P, C);WhenMin(Abs(LineSegmentOf(P, F)) + Abs(LineSegmentOf(P, A)))", "coords": { "A": [ -2.0, 4.0 ] } }, { "index": 6758, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6758.png", "answer": "20", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);A: Point;B: Point;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2,LineOf(A,B));Intersection(LineOf(A,B),G) = {A, B}", "coords": {} }, { "index": 6759, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6759.png", "answer": "16", "fact_expr": "G: Ellipse;m: Real;Expression(G) = (x^2/36 + y^2/m = 1);Length(MinorAxis(G))=8", "coords": {} }, { "index": 6760, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6761, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6762, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6762.png", "answer": "2", "fact_expr": "G: Parabola;H: Line;A: Point;F: Point;B: Point;Expression(G) = (y^2 = 4*x);Focus(G)=F;PointOnCurve(F,H);Intersection(H,G) = {B, A};Abs(LineSegmentOf(A, F)) = 2", "coords": {} }, { "index": 6763, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6763.png", "answer": "2", "fact_expr": "G: Parabola;B: Point;A: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(A, G);PointOnCurve(B,G);Distance(A,Focus(G))+Distance(B,Focus(G))=6", "coords": {} }, { "index": 6764, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6765, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2175992727279663, "b": 2.434081792831421, "orientation": "horizontal" }, "optimization": { "final_loss": 3.452532837400213e-06, "converged": true, "iterations": 124 }, "output_path": "results/train/hyperbola/problem_6765.png", "answer": "10", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/20 + x^2/a^2 = 1);a: Number;a>0;Expression(OneOf(Asymptote(G))) = (y = 2*x)", "coords": {} }, { "index": 6766, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6766.png", "answer": "sqrt(2)", "fact_expr": "G: Parabola;p: Number;O: Origin;F: Point;M: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;Expression(Directrix(G)) = (x = -1);PointOnCurve(M, G);Abs(LineSegmentOf(M, F)) = 3", "coords": {} }, { "index": 6767, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6767.png", "answer": "2*sqrt(3)/3", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2 = 1)", "coords": {} }, { "index": 6768, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6769, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6769.png", "answer": "4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);L: LineSegment;IsChordOf(L, G) = True;PointOnCurve(Focus(G), L) = True;IsPerpendicular(L, SymmetryAxis(G)) = True", "coords": {} }, { "index": 6770, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6771, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6771.png", "answer": "y = pm*sqrt(3)*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1)", "coords": {} }, { "index": 6772, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6772.png", "answer": "8", "fact_expr": "C: Parabola;G: Line;B: Point;A: Point;F: Point;l: Line;Expression(C) = (y^2 = 4*x);Focus(C) = F;Directrix(C) = l;PointOnCurve(F,G);Intersection(l, G) = A;OneOf(Intersection(G,C))=B;MidPoint(LineSegmentOf(A,B)) = F", "coords": {} }, { "index": 6773, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6773.png", "answer": "9/32", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 2*y);P: Point;PointOnCurve(P, G);F: Point;Focus(G) = F;Distance(P, F) = 2;O: Origin", "coords": {} }, { "index": 6774, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6775, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7195594310760498, "b": 1.7224669456481934, "orientation": "horizontal" }, "optimization": { "final_loss": 2.391175712546101e-06, "converged": true, "iterations": 83 }, "output_path": "results/train/hyperbola/problem_6775.png", "answer": "sqrt(2)", "fact_expr": "C: Hyperbola;a: Number;b: Number;G: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(G) = (0, 2);Eccentricity(C) = sqrt(2)", "coords": { "G": [ 0.0, 2.0 ] } }, { "index": 6776, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6776.png", "answer": "(1, 0)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = a*x);a: Number;Expression(Directrix(G)) = (x = -1);H: Line", "coords": {} }, { "index": 6777, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6778, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6778.png", "answer": "1", "fact_expr": "G: Parabola;p: Number;H: Line;A: Point;B: Point;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Slope(H) = sqrt(3);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};Quadrant(A) = 1;Abs(LineSegmentOf(A, F)) = 2", "coords": {} }, { "index": 6779, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6779.png", "answer": "7", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);P: Point;PointOnCurve(P, G);OneOf(Focus(G)) = F1;OneOf(Focus(G)) = F2;Negation(F1 = F2);F1: Point;F2: Point;Distance(P, F1) = 3", "coords": {} }, { "index": 6780, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6781, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6781.png", "answer": "8*sqrt(3)/3", "fact_expr": "C: Parabola;G: Line;M: Point;N: Point;P: Point;Q: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F, G);Inclination(G) = ApplyUnit(60, degree);Intersection(G, C) = {P, Q};Projection(P, Directrix(C)) = M;Projection(Q, Directrix(C)) = N", "coords": {} }, { "index": 6782, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6783, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6783.png", "answer": "x^2/9 - y^2/7 = 1", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;l: Line;Expression(l) = (7*x - 24*y + 28 = 0);PointOnCurve(F1, l);A: Point;B: Point;Intersection(l, C) = {A, B};IsPerpendicular(LineSegmentOf(A, F2), xAxis)", "coords": {} }, { "index": 6784, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6784.png", "answer": "6", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;M:Point;Expression(G) = (y^2 = 4*x);PointOnCurve(Focus(G), H);Intersection(H, G) = {A, B};MidPoint(LineSegmentOf(A,B))=M;XCoordinate(M)=2", "coords": {} }, { "index": 6785, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6786, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6786.png", "answer": "pm*sqrt(3)", "fact_expr": "G: Ellipse;Z: Ellipse;a: Real;Expression(G) = (x^2/4 + y^2/a^2 = 1);Expression(Z) = (y^2/2 + x^2/a^2 = 1);Focus(G) = Focus(Z)", "coords": {} }, { "index": 6787, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6787.png", "answer": "y^2=8*x", "fact_expr": "G: Parabola;H: Ellipse;Expression(H) = (x^2/9 + y^2/5 = 1);Center(H)=Vertex(G);RightFocus(H)=Focus(G)", "coords": {} }, { "index": 6788, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6788.png", "answer": "x^2=12*y", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 2*p*y);p: Number;p>0;D: Point;PointOnCurve(D, G) = True;YCoordinate(D) = 2;Distance(D, Focus(G)) = 5", "coords": {} }, { "index": 6789, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6790, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6790.png", "answer": "5/2", "fact_expr": "G: Ellipse;P: Point;F1: Point;M: Point;Expression(G) = (x^2/16 + y^2/9 = 1);PointOnCurve(P,G);LeftFocus(G)=F1;Distance(P,F1)=3;MidPoint(LineSegmentOf(P, F1)) = M;O:Origin", "coords": {} }, { "index": 6791, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6791.png", "answer": "(1, \\pm 2)", "fact_expr": "G: Parabola;O: Origin;A: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(A, G);DotProduct(VectorOf(O, A), VectorOf(A, F)) = -4", "coords": {} }, { "index": 6792, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 9.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6792.png", "answer": "sqrt(10)/4", "fact_expr": "P: Point;PointOnCurve(P, G) = True;G: Ellipse;Expression(G) = (x^2/9 + y^2/3 = 1);Q: Point;PointOnCurve(Q, H) = True;H: Circle;Expression(H) = (y^2 + (x - 1)^2 = 5/8)", "coords": {} }, { "index": 6793, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6793.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/5 + y^2/a^2 = 1);a: Number;a>0;H: Point;Coordinate(H) = (0, 3);OneOf(Focus(G))=H", "coords": { "H": [ 0.0, 3.0 ] } }, { "index": 6794, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6794.png", "answer": "(\\sqrt{5}+1)/2", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/a^2-y^2/b^2=1);a: Number;b: Number;a>0;b>0;A: Point;LeftVertex(C) = A;F: Point;RightFocus(C) = F;B: Point;Coordinate(B) = (0, b);DotProduct(VectorOf(B, A), VectorOf(B, F)) = 0", "coords": {} }, { "index": 6795, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6796, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6796.png", "answer": "8", "fact_expr": "C: Parabola;G: Line;A: Point;B: Point;M: Point;E: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;Intersection(Directrix(C), xAxis) = E;PointOnCurve(F, G);Intersection(G, C) = {A, B};MidPoint(LineSegmentOf(A, B)) = M;Abs(LineSegmentOf(M, E)) = sqrt(20)", "coords": {} }, { "index": 6797, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6797.png", "answer": "2*sqrt(2)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/4 + x^2/m = 1);m: Number;Expression(OneOf(Asymptote(G))) = (y = x);L1: Line;L2: Line;Directrix(G) = {L1,L2}", "coords": {} }, { "index": 6798, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6799, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6800, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6800.png", "answer": "(0, \\sqrt{6}/4)", "fact_expr": "C1: Ellipse;Expression(C1) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>b;b>0;C2: Circle;Expression(C2) = (x^2 + y^2 = (4*b^2)/5);P: Point;Negation(PointOnCurve(P, C1));Z1: Line;Z2: Line;TangentOfPoint(P, C2) = {Z1, Z2};IsPerpendicular(Z1, Z2)", "coords": {} }, { "index": 6801, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6801.png", "answer": "sqrt(3)+1", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) =F2;PointOnCurve(P, RightPart(G));DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0;Abs(LineSegmentOf(P, F1)) = sqrt(3)*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 6802, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6802.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;A: Point;LeftVertex(G) = A;B: Point;OneOf(Vertex(MinorAxis(G))) = B;F: Point;RightFocus(G) = F;IsPerpendicular(LineSegmentOf(A, B), LineSegmentOf(B, F))", "coords": {} }, { "index": 6803, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6803.png", "answer": "y^2 = 16*x", "fact_expr": "G: Parabola;p: Number;M: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(M) = (4, m);PointOnCurve(M, G);Distance(M, F) = 8;Focus(G) = F;F: Point;m: Number", "coords": {} }, { "index": 6804, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "y", "x_coef": 9.0, "y_coef": 25.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6804.png", "answer": "pm*2", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/9 + y^2/25 = 1);Endpoint(MajorAxis(H))=Focus(G);PointOnCurve(Focus(H),Directrix(G))", "coords": {} }, { "index": 6805, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6805.png", "answer": "(-oo,2]", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);Q: Point;PointOnCurve(Q, G);P: Point;a: Number;Coordinate(P) = (a, 0);Abs(LineSegmentOf(P, Q)) >= Abs(a)", "coords": {} }, { "index": 6806, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6806.png", "answer": "sqrt(5)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;Z: Line;PointOnCurve(F2, Z);IsPerpendicular(Z, OneOf(Asymptote(C)));H: Point;FootPoint(Z, OneOf(Asymptote(C))) = H;M: Point;Intersection(Z, C) = M;VectorOf(F2, M) = 2*VectorOf(M, H)", "coords": {} }, { "index": 6807, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.4641016151377544, "major_axis": "x", "x_coef": 16.0, "y_coef": 12.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6807.png", "answer": "2*sqrt(10)", "fact_expr": "l: Line;G: Ellipse;C:Curve;Expression(l)=(x+y-4=0);Expression(C)=(x^2/16+y^2/12=1);Focus(C)=Focus(G);PointOnCurve(M,l);PointOnCurve(M,G);M:Point", "coords": {} }, { "index": 6808, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6808.png", "answer": "4*x-y-7=0", "fact_expr": "G: Hyperbola;H: Line;P1: Point;P2: Point;P: Point;Coordinate(P) = (2, 1);Expression(G) = (x^2 - y^2/2 = 1);PointOnCurve(P, H);Intersection(H, G) = {P1, P2};MidPoint(LineSegmentOf(P1, P2)) = P", "coords": { "P": [ 2.0, 1.0 ] } }, { "index": 6809, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6809.png", "answer": "2*sqrt(7)-2", "fact_expr": "P: Point;PointOnCurve(P, RightPart(G)) = True;RightFocus(G) = F;F: Point;G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1);A: Point;Coordinate(A) = (3, sqrt(3))", "coords": { "A": [ 3.0, 1.7320508075688772 ] } }, { "index": 6810, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6810.png", "answer": "{0,1}", "fact_expr": "G: Parabola;H: Line;k: Number;Expression(G) = (y^2 = 8*x);Expression(H) = (y = k*x + 2);NumIntersection(H, G) = 1", "coords": {} }, { "index": 6811, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6811.png", "answer": "sqrt(3)", "fact_expr": "P: Point;G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);PointOnCurve(P, G) = True;F1: Point;F2: Point;Focus(G) = {F1, F2};AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 6812, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6813, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6813.png", "answer": "5", "fact_expr": "G: Hyperbola;m: Real;H: Line;Expression(G) = (x^2/4 - y^2/m = 1);Expression(H) = (x - 3*y - 3 = 0);PointOnCurve(Focus(G), H)", "coords": {} }, { "index": 6814, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6815, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6815.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;p: Number;H: Parabola;p>0;Expression(G) = (x^2 - 8*y^2/p^2 = 1);Expression(H) = (y^2 = 2*p*x);PointOnCurve(LeftFocus(G), Directrix(H))", "coords": {} }, { "index": 6816, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6816.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;M: Point;F: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 4*x);Intersection(H,OneOf(Asymptote(G)))=M;Focus(H)=F;Abs(LineSegmentOf(M, F)) = 3", "coords": {} }, { "index": 6817, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 3.872983346207417, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6817.png", "answer": "5", "fact_expr": "G: Hyperbola;H: Circle;D: Circle;P: Point;M: Point;N: Point;Expression(G) = (x^2 - y^2/15 = 1);Expression(H) = (y^2 + (x + 4)^2 = 4);Expression(D) = (y^2 + (x - 4)^2 = 1);PointOnCurve(P, RightPart(G));PointOnCurve(M, H);PointOnCurve(N, D)", "coords": {} }, { "index": 6818, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6819, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6819.png", "answer": "1/5", "fact_expr": "O: Origin;G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;l: Line;PointOnCurve(F, l);M: Point;N: Point;Intersection(l, G) = {M, N};T: Point;IsChordOf(LineSegmentOf(M, N), G);MidPoint(LineSegmentOf(M, N)) = T;Abs(LineSegmentOf(M, F)) = 4*Abs(LineSegmentOf(O, F));Area(TriangleOf(M, F, O)) = 4*sqrt(3);S: Point;Intersection(OverlappingLine(LineSegmentOf(O, T)), G) = S", "coords": {} }, { "index": 6820, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6820.png", "answer": "x = -4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 16*x)", "coords": {} }, { "index": 6821, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6821.png", "answer": "1", "fact_expr": "H: Curve;Expression(H) = (y = x^2);G: Line;Expression(G) = (-a + 2*x - y = 0);a: Number;IsTangent(H,G)", "coords": {} }, { "index": 6822, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6822.png", "answer": "(0, pm*sqrt(m - 4))", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/m = 1);PointOnCurve(Focus(G), yAxis);m: Number", "coords": {} }, { "index": 6823, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6823.png", "answer": "4/5", "fact_expr": "alpha: Number;l: Line;Inclination(l) = alpha;G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1);Slope(l) = Eccentricity(G)", "coords": {} }, { "index": 6824, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6824.png", "answer": "[0,2)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = x);PointOnCurve(P,C) = True;P: Point;A: Point;Coordinate(A) = (-1, 0);B: Point;Coordinate(B) = (1, 0)", "coords": { "A": [ -1.0, 0.0 ], "B": [ 1.0, 0.0 ] } }, { "index": 6825, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6825.png", "answer": "3/2", "fact_expr": "G: Parabola;P: Point;A: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(P,G);Coordinate(A)=(1/2,1)", "coords": { "A": [ 0.5, 1.0 ] } }, { "index": 6826, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6826.png", "answer": "2", "fact_expr": "G: Parabola;p: Number;A: Point;F: Point;m:Number;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(A) = (m, 2*sqrt(2));Focus(G)=F;PointOnCurve(A,G);C:Circle;Center(C)=A;Radius(C)=Abs(LineSegmentOf(A,F));Length(InterceptChord(yAxis,C))=2*sqrt(5)", "coords": {} }, { "index": 6827, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6827.png", "answer": "{y^2 = 4*x, y^2 = -36*x}", "fact_expr": "G: Parabola;H: Line;O: Origin;Expression(H) = (2*x - y - 4 = 0);Vertex(G) = O;PointOnCurve(Focus(G), xAxis);Length(InterceptChord(H, G)) = 3*sqrt(5)", "coords": {} }, { "index": 6828, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6828.png", "answer": "8*sqrt(5)/5", "fact_expr": "G: Hyperbola;L1: Line;L2: Line;Expression(G) = (x^2 - 4*y^2 = 4);Directrix(G) = {L1, L2}", "coords": {} }, { "index": 6829, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 2.449489742783178, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6829.png", "answer": "sqrt(6)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2/6 = 1)", "coords": {} }, { "index": 6830, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2175992727279663, "b": 2.434081792831421, "orientation": "horizontal" }, "optimization": { "final_loss": 3.452532837400213e-06, "converged": true, "iterations": 124 }, "output_path": "results/train/hyperbola/problem_6830.png", "answer": "5*x^2-5*y^2/4=1", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 4*x);Expression(OneOf(Asymptote(G)))=(y=2*x);OneOf(Focus(G))=Focus(H)", "coords": {} }, { "index": 6831, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6832, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.449489742783178, "major_axis": "x", "x_coef": 9.0, "y_coef": 6.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6832.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "P: Point;G: Ellipse;Expression(G) = (x^2/9 + y^2/6 = 1);PointOnCurve(P, G) = True;F1: Point;F2: Point;Focus(G) = {F1, F2};AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 6833, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6833.png", "answer": "x^2/16 - y^2/9 = 1", "fact_expr": "C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Distance(RightFocus(C),Asymptote(C))=3;P: Point;F1: Point;F2: Point;Focus(C) = {F1, F2};PointOnCurve(P, RightPart(C));Distance(P, F1) - Distance(P, F2) = (4/3)*Length(ImageinaryAxis(C))", "coords": {} }, { "index": 6834, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6835, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6836, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6836.png", "answer": "4", "fact_expr": "G: Parabola;H: Line;N: Point;F: Point;M: Point;Expression(G) = (y^2 = 12*x);Focus(G)=F;PointOnCurve(M,G);PointOnCurve(M,H);IsPerpendicular(H,Directrix(G));FootPoint(H,Directrix(G))=N;Slope(LineOf(N,F)) = -sqrt(3)/3", "coords": {} }, { "index": 6837, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 0.7071067811865476, "b": 0.7071067811865476, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6837.png", "answer": "x^2/2+y^2=1", "fact_expr": "G: Hyperbola;H: Ellipse;O: Origin;Expression(G) = (2*x^2 - 2*y^2 = 1);Center(H) = O;Focus(H) = Focus(G);InterReciprocal(Eccentricity(G), Eccentricity(H))", "coords": {} }, { "index": 6838, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 4.898979485566356, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6838.png", "answer": "16", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/24 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G) = True;Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 6839, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6840, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6840.png", "answer": "sqrt(6)", "fact_expr": "C: Hyperbola;b: Number;a: Number;H: Line;A: Point;B: Point;O: Origin;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);F1:Point;F2:Point;LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1,H);L1:Line;L2:Line;Asymptote(C)={L1,L2};Intersection(H,L1)=A;Intersection(H,L2)=B;VectorOf(F1,A) = 2*VectorOf(A,B);DotProduct(VectorOf(F1,A),VectorOf(A,O)) = 0", "coords": {} }, { "index": 6841, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6842, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6842.png", "answer": "-3", "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (x^2 + y^2/m = 1);Inclination(OneOf(Asymptote(G)))=ApplyUnit(60,degree)", "coords": {} }, { "index": 6843, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6843.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Ellipse;b: Number;a: Number;c: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);HalfFocalLength(G)=c", "coords": {} }, { "index": 6844, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6844.png", "answer": "3/5", "fact_expr": "G:Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;e:Number;Eccentricity(G)=e;e=sqrt(3)/2;A:Point;B:Point;LeftVertex(G)=A;RightVertex(G)=B;PointOnCurve(P,G);Negation(P=A);Negation(P=B);Inclination(LineOf(P, A)) = alpha;Inclination(LineOf(P,B)) = beta;alpha:Number;beta:Number;P:Point", "coords": {} }, { "index": 6845, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 8.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6845.png", "answer": "10", "fact_expr": "G: Ellipse;H: Line;P: Point;Q: Point;x1:Number;y1:Number;x2:Number;y2:Number;Expression(G) = (x^2/8 + y^2/2 = 1);Expression(H) = (x + 2*y - 8 = 0);Coordinate(P) = (x1, y1);Coordinate(Q) = (x2, y2);PointOnCurve(P, G);PointOnCurve(Q, H)", "coords": {} }, { "index": 6846, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.449489742783178, "b": 3.1622776601683795, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6846.png", "answer": "8", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/6 - y^2/10 = 1);H: Parabola;Expression(H) = (y^2 = 2*(p*x));p: Real;p>0;Focus(H) = RightFocus(G)", "coords": {} }, { "index": 6847, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6847.png", "answer": "-2", "fact_expr": "G: Ellipse;H: Line;P: Point;F: Point;Q: Point;Expression(G) = (x^2/4 + y^2/3 = 1);RightFocus(G) = F;PointOnCurve(Center(G), H);Intersection(H, G) = {P, Q};WhenMax(Area(TriangleOf(P,F,Q)))", "coords": {} }, { "index": 6848, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6848.png", "answer": "2*sqrt(5)", "fact_expr": "G: Ellipse;H: Line;Expression(G) = (x^2 + 4*y^2 = 16);Expression(H) = (x - 2*y + 4 = 0)", "coords": {} }, { "index": 6849, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6849.png", "answer": "8", "fact_expr": "G: Hyperbola;H: Parabola;p: Number;Expression(G) = (-x^2 + y^2/3 = 1);Expression(H) = (x^2 = p*y);UpperFocus(G)=Focus(H)", "coords": {} }, { "index": 6850, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6850.png", "answer": "x^2/16+y^2/7=1", "fact_expr": "G: Ellipse;C:Ellipse;H: Point;Expression(G) = (x^2/25 + y^2/16 = 1);Coordinate(H) = (-3, 7/4);Focus(G)=Focus(C);PointOnCurve(H,C)", "coords": { "H": [ -3.0, 1.75 ] } }, { "index": 6851, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6851.png", "answer": "2", "fact_expr": "C: Hyperbola;M: Point;F1: Point;F2: Point;I: Point;a: Number;b: Number;a > 0;b > 0;Expression(C) = (x^2/a^2 - y^2/b^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(M, C);Incenter(TriangleOf(M, F1, F2)) = I;Area(TriangleOf(I, M, F1)) + Area(TriangleOf(I, M, F2)) = (3/2)*Area(TriangleOf(I, F1, F2));Abs(LineSegmentOf(M, F1)) = 2*Abs(LineSegmentOf(M, F2))", "coords": {} }, { "index": 6852, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6853, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6853.png", "answer": "sqrt(2)+1", "fact_expr": "M: Hyperbola;b: Number;a: Number;G: Line;F2: Point;A: Point;B: Point;F1: Point;e: Number;a>0;b>0;Expression(M) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(M) = F1;RightFocus(M) = F2;PointOnCurve(F1,G);IsPerpendicular(G,xAxis);Intersection(G, M) = {A, B};DotProduct(VectorOf(F2, A), VectorOf(F2, B)) = 0;Eccentricity(M) = e", "coords": {} }, { "index": 6854, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6854.png", "answer": "2", "fact_expr": "A: Point;B: Point;PointOnCurve(A, G);PointOnCurve(B, G);G: Hyperbola;Expression(G) = (x^2 - y^2/2 = 1);O: Origin;DotProduct(VectorOf(O, A), VectorOf(O, B)) = 0", "coords": {} }, { "index": 6855, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6855.png", "answer": "4", "fact_expr": "G: Parabola;M: Point;F: Point;O: Origin;Expression(G) = (y^2 = 4*x);PointOnCurve(M, G);Focus(G) = F;AngleOf(M, F, O) = ApplyUnit(120, degree)", "coords": {} }, { "index": 6856, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.4142135623730951, "major_axis": "y", "x_coef": 2.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6856.png", "answer": "sqrt(3)/3", "fact_expr": "C: Ellipse;Expression(C) = (x^2/2 + y^2/3 = 1)", "coords": {} }, { "index": 6857, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6857.png", "answer": "3", "fact_expr": "G: Parabola;P: Point;F: Point;x0:Number;y0:Number;Expression(G) = (y^2 = 4*x);Coordinate(P) = (x0,y0);PointOnCurve(P, G);Focus(G) = F;Distance(P, F) = 4", "coords": {} }, { "index": 6858, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6859, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6859.png", "answer": "2", "fact_expr": "G: Ellipse;Expression(G) = (y^2 + x^2/m^2 = 1);m: Real;H: Line;Expression(H) = (m*x + y - 1 = 0)", "coords": {} }, { "index": 6860, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6860.png", "answer": "a/2 - p/2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G);a: Number;a>=2*p;Length(LineSegmentOf(A, B)) = a;M: Point;MidPoint(LineSegmentOf(A, B)) = M", "coords": {} }, { "index": 6861, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6862, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6862.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;L: Line;Q: Point;P: Point;F: Point;O:Origin;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(O,L);Inclination(L)=pi/6;Intersection(L,LeftPart(G))=P;Intersection(L,RightPart(G))=Q;IsDiameter(LineSegmentOf(P,Q),H);PointOnCurve(F,G);RightFocus(G)=F", "coords": {} }, { "index": 6863, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.6457513110645907, "major_axis": "x", "x_coef": 16.0, "y_coef": 7.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6863.png", "answer": "5/2", "fact_expr": "G: Ellipse;Expression(G) = (x^2/16 + y^2/7 = 1);P: Point;PointOnCurve(P,G) = True;XCoordinate(P) = 2", "coords": {} }, { "index": 6864, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6864.png", "answer": "4*x+pm*3*y=0", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1)", "coords": {} }, { "index": 6865, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6865.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;F2: Point;l:Line;Q: Point;P: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G)=F2;PointOnCurve(F2,l);L1:Line;L2:Line;IsPerpendicular(L1,l);FootPoint(L1,l)=P;Intersection(L2,l)=Q;VectorOf(F2, Q) = 2*VectorOf(F2, P);OneOf(Asymptote(G))=L1;OneOf(Asymptote(G))=L2;Negation(L1=L2)", "coords": {} }, { "index": 6866, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 8.0, "b": 6.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6866.png", "answer": "20", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/64 - y^2/36 = 1)", "coords": {} }, { "index": 6867, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6867.png", "answer": "1", "fact_expr": "G: Line;H: Curve;P: Point;Expression(G) = (2*x + y - 1 = 0);Expression(H) = (x^2 = 4*y);Coordinate(P) = (m, n);m:Number;n:Number;IsPerpendicular(TangentOnPoint(P,H),G)", "coords": {} }, { "index": 6868, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6868.png", "answer": "(\\sqrt{5}-1)/2", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F2), xAxis);Abs(LineSegmentOf(F1, F2)) = 2*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 6869, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.560132719491411, "b": 1.280571558186289, "orientation": "horizontal" }, "optimization": { "final_loss": 5.001754676787936e-06, "converged": true, "iterations": 73 }, "output_path": "results/train/hyperbola/problem_6869.png", "answer": "x^2/4-y^2=1", "fact_expr": "G: Hyperbola;PointOnCurve(Focus(G),xAxis);Expression(Asymptote(G))=(y=pm*(1/2)*x);FocalLength(G) = 2*sqrt(5)", "coords": {} }, { "index": 6870, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6871, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6871.png", "answer": "sqrt(5)/2", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;O: Origin;A: Point;B: Point;F: Point;a > b;b > 0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1) ;OneOf(Focus(C)) = F;PointOnCurve(F, l);L1:Line;L2:Line;Asymptote(C)={L1,L2};Intersection(l,L1) = A;Intersection(l,L2 ) = B;DotProduct(VectorOf(O, A), VectorOf(F, A)) = 0;Radius(InscribedCircle(TriangleOf(O,A,B)))=a/3", "coords": {} }, { "index": 6872, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6872.png", "answer": "1/2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = x)", "coords": {} }, { "index": 6873, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6873.png", "answer": "4", "fact_expr": "l: Line;G: Hyperbola;b: Number;a: Number;B: Point;C: Point;O: Origin;A: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);L1:Line;L2:Line;Asymptote(G)={L1,L2};Intersection(L1,l)=A;Intersection(L2,l) = B;Intersection(xAxis,l) = C;MidPoint(LineSegmentOf(B,C))=A;LineSegmentOf(O, A) = LineSegmentOf(O, C)", "coords": {} }, { "index": 6874, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6874.png", "answer": "y^2=2*x-2", "fact_expr": "G: Parabola;H: Line;P: Point;Q: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(Focus(G), H);Intersection(H, G) = {P,Q}", "coords": {} }, { "index": 6875, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6875.png", "answer": "1", "fact_expr": "G: Parabola;H: Line;P: Point;Expression(G) = (y^2 = 2*x);Expression(H) = (3*x - 4*y + 7/2 = 0);PointOnCurve(P, G)", "coords": {} }, { "index": 6876, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6876.png", "answer": "16/3", "fact_expr": "A: Point;B: Point;G: Parabola;p: Number;H: Line;C: Point;F: Point;l:Line;p>0;Expression(G) = (y^2 = 2*p*x);Focus(G)=F;PointOnCurve(F,H);Intersection(H,G) = {A,B};Directrix(G)=l;Intersection(H,l)=C;MidPoint(LineSegmentOf(A, C)) = F;LineSegmentOf(A, F) = 4", "coords": {} }, { "index": 6877, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6878, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6878.png", "answer": "2", "fact_expr": "F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;C: Ellipse;Expression(C) = (x^2/4 + y^2/3 = 1);P: Point;PointOnCurve(P, C)", "coords": {} }, { "index": 6879, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6879.png", "answer": "sqrt(6)/3", "fact_expr": "G: Ellipse;b: Number;a: Number;Q: Point;F: Point;O: Origin;A: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F;PointOnCurve(A, G);Intersection(TangentOfPoint(A,G), yAxis) = Q;AngleOf(Q, F, O) = pi/4;AngleOf(Q, F, A) = pi/6", "coords": {} }, { "index": 6880, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6881, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6882, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6883, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6883.png", "answer": "40", "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (x^2 = 4*y);Coordinate(A) = (12, 39);PointOnCurve(P, G);Focus(G) = F", "coords": { "A": [ 12.0, 39.0 ] } }, { "index": 6884, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -1.0, 0.0 ], "radius": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_6884.png", "answer": "x^2/4 + y^2/3 = 1", "fact_expr": "G: Circle;A: Point;B: Point;C: Point;M: Point;Expression(G) = (y^2 + (x - 1)^2 = 16);Coordinate(A) = (-1, 0);Center(G)=C;PointOnCurve(B,G);Intersection(PerpendicularBisector(LineSegmentOf(A,B)), LineSegmentOf(B, C)) = M", "coords": {} }, { "index": 6885, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.1209959984138265, "b": 2.1209959984138265, "orientation": "horizontal" }, "optimization": { "final_loss": 8.099017116797696e-06, "converged": true, "iterations": 98 }, "output_path": "results/train/hyperbola/problem_6885.png", "answer": "6", "fact_expr": "G: Hyperbola;F1: Point;F2: Point;Coordinate(F1) = (-3, 0);Coordinate(F2) = (3, 0);Focus(G) = {F1, F2}", "coords": { "F1": [ -3.0, 0.0 ], "F2": [ 3.0, 0.0 ] } }, { "index": 6886, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6886.png", "answer": "4/3", "fact_expr": "G: Ellipse;Expression(G) = (y^2 + x^2/a^2 = 1);a: Number;a>1;F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 6887, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6887.png", "answer": "1", "fact_expr": "G: Ellipse;C: Point;D: Point;M: Point;Expression(G)=(x^2/4 + y^2 = 1);Coordinate(C)=(-sqrt(3), 0);Coordinate(D)=(sqrt(3), 0);PointOnCurve(M, G)", "coords": { "C": [ -1.7320508075688772, 0.0 ], "D": [ 1.7320508075688772, 0.0 ] } }, { "index": 6888, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.998991012573242, "direction": "right" }, "optimization": { "final_loss": 1.9470178358460544e-06, "converged": true, "iterations": 122 }, "output_path": "results/train/parabola/problem_6888.png", "answer": "10+(20*sqrt(2)/3)", "fact_expr": "l: Line;C: Parabola;p: Number;B: Point;A: Point;H: Point;M: Point;F: Point;p>0;Expression(C) = (y^2 = 2*p*x);Coordinate(H) = (3, 6);Coordinate(M) = (2, 2);PointOnCurve(H, C);PointOnCurve(M, l);Intersection(l, C)={A, B};MidPoint(LineSegmentOf(A,B))=M;Focus(C)=F", "coords": { "H": [ 3.0, 6.0 ], "M": [ 2.0, 2.0 ] } }, { "index": 6889, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.463192105293274, "b": 2.070901870727539, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4661259228887502e-06, "converged": true, "iterations": 106 }, "output_path": "results/train/hyperbola/problem_6889.png", "answer": "4", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;Eccentricity(G) = sqrt(3);H: Parabola;Expression(H) = (y^2 = 4*x);OneOf(Directrix(G)) = Directrix(H);OneOf(Intersection(G, H)) = P;P: Point;Focus(H) = F;F: Point", "coords": {} }, { "index": 6890, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6890.png", "answer": "(-12,0)", "fact_expr": "G: Hyperbola;k: Number;e:Number;Expression(G) = (y^2/4 + x^2/k = 1);Eccentricity(G) = e;e<2", "coords": {} }, { "index": 6891, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6891.png", "answer": "2", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(C) = F;A: Point;RightVertex(C) = A;L: Line;PointOnCurve(F, L) = True;IsPerpendicular(L, xAxis) = True;Intersection(L, C) = {M, N};M: Point;N: Point;Tan(AngleOf(M, A, N)) = -3/4", "coords": {} }, { "index": 6892, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6892.png", "answer": "{y^2=4*x,y^2=8*x}", "fact_expr": "l: Line;C: Parabola;p:Number;p>0;A: Point;B: Point;M: Point;F:Point;Expression(C)=(y^2=2*p*x);Coordinate(M) = (3, 2);Focus(C) = F;PointOnCurve(F,l);Intersection(l, C) = {A, B};MidPoint(LineSegmentOf(A,B))=M", "coords": { "M": [ 3.0, 2.0 ] } }, { "index": 6893, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6894, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6895, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6895.png", "answer": "x^2=4*y", "fact_expr": "C: Parabola;p: Number;P: Point;F: Point;E: Point;l: Line;p>0;Expression(C) = (x^2 = 2*(p*y));Focus(C) = F;Directrix(C) = l;PointOnCurve(P, C);L:Line;PointOnCurve(P, L);IsPerpendicular(l,L);Intersection(l, L) = E;AngleOf(P, F, E) = ApplyUnit(60, degree);Abs(LineSegmentOf(P, F)) = 4", "coords": {} }, { "index": 6896, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 5.0, "b": 3.0, "major_axis": "y", "x_coef": 9.0, "y_coef": 25.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6896.png", "answer": "y^2/20 + x^2/4 = 1", "fact_expr": "G: Ellipse;PointOnCurve(H,G) = True;H: Point;Coordinate(H) = (sqrt(3), -sqrt(5));G1: Ellipse;Expression(G1) = (x^2/9 + y^2/25 = 1);Focus(G) = Focus(G1)", "coords": { "H": [ 1.7320508075688772, -2.23606797749979 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 6897, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6897.png", "answer": "4", "fact_expr": "G: Parabola;H: Line;A: Point;F: Point;B: Point;O: Origin;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B}", "coords": {} }, { "index": 6898, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.8284271247461903, "b": 2.8284271247461903, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6898.png", "answer": "8", "fact_expr": "G: Hyperbola;H: Parabola;p: Number;Expression(G) = (x^2 - y^2 = 8);p>0;Expression(H) = (y^2 = 2*(p*x));Focus(H) = OneOf(Focus(G))", "coords": {} }, { "index": 6899, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6899.png", "answer": "sqrt(2)", "fact_expr": "C: Hyperbola;G: Point;Expression(C) = (x^2 - y^2 = 1);Coordinate(G) = (2, 0)", "coords": { "G": [ 2.0, 0.0 ] } }, { "index": 6900, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6900.png", "answer": "10", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);H: Line;PointOnCurve(Focus(G), H);A: Point;B: Point;x1: Number;x2: Number;y1: Number;y2: Number;Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 12", "coords": {} }, { "index": 6901, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6901.png", "answer": "16", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 16*x);P: Point;PointOnCurve(P, G);Distance(P, xAxis) = 12;F: Point;Focus(G) = F;Distance(P, F) = Abs(LineSegmentOf(P, F))", "coords": {} }, { "index": 6902, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6902.png", "answer": "3/2", "fact_expr": "G: Parabola;A: Point;B: Point;F: Point;Expression(G)=(y^2=2*x);Focus(G)=F;PointOnCurve(A, G);PointOnCurve(B, G);Abs(LineSegmentOf(A, F)) + Abs(LineSegmentOf(B, F)) = 3", "coords": {} }, { "index": 6903, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6904, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4506710767745972, "b": 2.0511727333068848, "orientation": "horizontal" }, "optimization": { "final_loss": 4.657696081267204e-06, "converged": true, "iterations": 116 }, "output_path": "results/train/hyperbola/problem_6904.png", "answer": "1", "fact_expr": "G: Hyperbola;a: Real;a>0;Expression(G) = (-y^2/2 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G))) = (y = sqrt(2)*x)", "coords": {} }, { "index": 6905, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6905.png", "answer": "2*sqrt(5)/5", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x + 2*y - 2 = 0);PointOnCurve(OneOf(Focus(G)),H);PointOnCurve(OneOf(Vertex(G)),H)", "coords": {} }, { "index": 6906, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6907, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6908, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/train/hyperbola/problem_6908.png", "answer": "x^2/16 - y^2/9 = 1", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Eccentricity(G) = 5/4;A: Point;Coordinate(A) = (-4*sqrt(2), 3);PointOnCurve(A, G)", "coords": { "A": [ -5.656854249492381, 3.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 6909, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_ecc_mismatch;ellipse_focus_mismatch", "params": { "a": 1.9257268905639648, "b": 1.7993911322594118, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 2.5576271215184017e-15, "converged": true, "iterations": 2 }, "output_path": "results/train/ellipse/problem_6909.png", "answer": "-4/3", "fact_expr": "r: Ellipse;a: Number;b: Number;O: Origin;D: Point;A: Point;B: Point;C: Point;E: Point;M: Point;F: Point;a > b;b > 0;Expression(r) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F) = (0, 1);RightFocus(r) = F;Eccentricity(r) = 1/2;PointOnCurve(Vertex(TriangleOf(A, B, C)), r);MidPoint(LineSegmentOf(A, B)) = D;MidPoint(LineSegmentOf(B, C)) = E;MidPoint(LineSegmentOf(A, C)) = M;k1: Number;k2: Number;k3: Number;Slope(OverlappingLine(LineSegmentOf(A, B))) = k1;Slope(OverlappingLine(LineSegmentOf(B, C))) = k2;Slope(OverlappingLine(LineSegmentOf(A, C))) = k3;Negation(k1 = 0);Negation(k2 = 0);Negation(k3 = 0);Slope(LineOf(O, D)) + Slope(LineOf(O, E))+ Slope(LineOf(O, M)) = 1", "coords": { "F": [ 0.0, 1.0 ] }, "validation_reasons": [ "ellipse_ecc_mismatch", "ellipse_focus_mismatch" ] }, { "index": 6910, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6911, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6911.png", "answer": "sqrt(6)", "fact_expr": "G: Ellipse;A: Point;P: Point;Expression(G) = (x^2/4 + y^2/2 = 1);Coordinate(A) = (0, 1);PointOnCurve(P, G)", "coords": { "A": [ 0.0, 1.0 ] } }, { "index": 6912, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6913, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6913.png", "answer": "y^2 = -4 \\sqrt{5} x", "fact_expr": "G: Parabola;H: Ellipse;Expression(H) = (x^2/9 + y^2/4 = 1);Focus(G) = LeftFocus(H);Vertex(G) = Center(H)", "coords": {} }, { "index": 6914, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6915, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6915.png", "answer": "(-3, 0)", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;C: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 4;O: Origin;Orthocenter(TriangleOf(A, B, C)) = O", "coords": {} }, { "index": 6916, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6916.png", "answer": "(1, 1)", "fact_expr": "G: Parabola;H: Line;P:Point;Expression(G) = (y^2 = x);Expression(H) = (x - 2*y + 4 = 0);PointOnCurve(P, G);WhenMin(Distance(P,H))", "coords": {} }, { "index": 6917, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6917.png", "answer": "2", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Circle;c: Number;H: Line;F: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (2*(c*x) + x^2 + y^2 = 0);Coordinate(F) = (c, 0);RightFocus(C) = F;PointOnCurve(F,H);IsPerpendicular(H,Asymptote(C));IsTangent(H,G)", "coords": {} }, { "index": 6918, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6918.png", "answer": "1", "fact_expr": "C: Ellipse;a: Number;Q: Circle;P: Point;F1: Point;F2: Point;a>1;O: Origin;Expression(C) = (y^2 + x^2/a^2 = 1);Focus(C) = {F1, F2};Center(Q) = O ;Diameter(Q) = FocalLength(C);Intersection(Q, C) = P", "coords": {} }, { "index": 6919, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6920, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6921, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.75, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6921.png", "answer": "-5/2", "fact_expr": "G: Parabola;A: Point;B: Point;x1: Number;x2: Number;y1: Number;y2: Number;Expression(G) = (y^2 = 3*x);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);PointOnCurve(A, G);PointOnCurve(B, G);Slope(LineOf(A, B)) = 2;y1 = 4", "coords": {} }, { "index": 6922, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6922.png", "answer": "2", "fact_expr": "H: Parabola;Expression(H) = (y^2 = 2*(p*x));p: Number;p>0;G: Hyperbola;Expression(G) = (x^2 - y^2 = 1);PointOnCurve(LeftVertex(G), Directrix(H))", "coords": {} }, { "index": 6923, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6924, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6924.png", "answer": "3", "fact_expr": "H: Line;Expression(H) = (y = k*x);k: Number;Negation(k=0);C: Hyperbola;Expression(C) = (x^2/a^2 - y^2/b^2 = 1);a: Number;b: Number;a>0;b>0;A: Point;B: Point;Intersection(H, C) = {A, B};E: Circle;IsDiameter(LineSegmentOf(A, B), E);F: Point;RightFocus(C) = F;PointOnCurve(F, E);Area(TriangleOf(O, A, F)) = 4*a^2;O: Origin", "coords": {} }, { "index": 6925, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6925.png", "answer": "4/3", "fact_expr": "l: Line;G: Parabola;A: Point;F: Point;B: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B};Abs(LineSegmentOf(A, F)) = 4", "coords": {} }, { "index": 6926, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.463192105293274, "b": 2.070901870727539, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4661259228887502e-06, "converged": true, "iterations": 106 }, "output_path": "results/train/hyperbola/problem_6926.png", "answer": "y=pm*sqrt(2)*x", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Eccentricity(C) = sqrt(3)", "coords": {} }, { "index": 6927, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6927.png", "answer": "5/3", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G)))=(4*x-3*y=0)", "coords": {} }, { "index": 6928, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6928.png", "answer": "(2,2)", "fact_expr": "C: Hyperbola;O: Origin;P: Point;Expression(C) = (-x^2/4 + y^2/4 = 1);Quadrant(P)=1;PointOnCurve(P,Asymptote(C));Abs(LineSegmentOf(O, P)) = 2*sqrt(2)", "coords": {} }, { "index": 6929, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6929.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "C: Parabola;Expression(C) = (y^2=2*p*x);p: Number;p>0;F: Point;Focus(C) = F;A: Point;B: Point;Quadrant(A) = 1;Quadrant(B) = 1;PointOnCurve(A,C) = True;PointOnCurve(B,C) = True;O: Origin;AngleOf(A,F,O)=AngleOf(A,F,B);AngleOf(A,F,B) = pi/3;Area(TriangleOf(A,F,B)) = 3*sqrt(3)", "coords": {} }, { "index": 6930, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6931, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6932, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6932.png", "answer": "[1, sqrt(2)]", "fact_expr": "G: Parabola;P: Point;A: Point;F: Point;x0: Number;y0: Number;Expression(G) = (y^2 = 8*x);Coordinate(P) = (x0, y0);Coordinate(A) = (-2, 0);Focus(G) = F;PointOnCurve(P, G)", "coords": { "A": [ -2.0, 0.0 ] } }, { "index": 6933, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_6933.png", "answer": "1", "fact_expr": "O: Circle;A: Point;B: Point;M: Point;b:Number;lambda:Number;Negation(b=-2);Expression(O) = (x^2 + y^2 = 1);Coordinate(A) = (-2, 0);Coordinate(B) = (b, 0);PointOnCurve(M,O);Abs(LineSegmentOf(M,B))=lambda*Abs(LineSegmentOf(M,A));Vertex(O)=B", "coords": {} }, { "index": 6934, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 3.0, "major_axis": "x", "x_coef": 36.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6934.png", "answer": "x+2*y-8=0", "fact_expr": "G: Ellipse;Expression(G) = (x^2/36 + y^2/9 = 1);Z: LineSegment;IsChordOf(Z, G);A: Point;Coordinate(A) = (4, 2);MidPoint(Z) = A", "coords": { "A": [ 4.0, 2.0 ] } }, { "index": 6935, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6935.png", "answer": "3/2 - sqrt(2)", "fact_expr": "G: Parabola;M: Point;O: Origin;A: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(M) = (0, 1);Focus(G) =F;Intersection(LineSegmentOf(F,M),G)=A", "coords": { "M": [ 0.0, 1.0 ] } }, { "index": 6936, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6936.png", "answer": "pm*2*sqrt(5)", "fact_expr": "G: Hyperbola;H: Circle;a: Number;Expression(G) = (x^2/4 - y^2 = 1);Expression(H) = (y^2 + (-a + x)^2 = 4);IsTangent(G, Asymptote(H))", "coords": {} }, { "index": 6937, "success": false, "conic_type": "parabola", "error": "validation: parabola_point_off_curve", "params": { "p": 0.11525513976812363, "direction": "right" }, "optimization": { "final_loss": 0.05289620906114578, "converged": true, "iterations": 67 }, "output_path": "results/train/parabola/problem_6937.png", "answer": "1/4", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;H: Point;Coordinate(H) = (2, 1);PointOnCurve(H, G)", "coords": { "H": [ 2.0, 1.0 ] }, "validation_reasons": [ "parabola_point_off_curve" ] }, { "index": 6938, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6938.png", "answer": "(1/2, 1)", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;LeftFocus(G) = F1;A: Point;RightVertex(G) = A;H: Circle;IsDiameter(LineSegmentOf(F1, A), H);NumIntersection(H, G) = 3", "coords": {} }, { "index": 6939, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6939.png", "answer": "(-1, 1)", "fact_expr": "H: Line;Expression(H) = (y = k*x - 1);k: Number;G: Hyperbola;Expression(G) = (x^2 - y^2 = 4);NumIntersection(H, LeftPart(G)) = 1;NumIntersection(H, RightPart(G)) = 1", "coords": {} }, { "index": 6940, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6940.png", "answer": "(2, 2)", "fact_expr": "A: Point;Coordinate(A) = (3, 2);G: Parabola;Expression(G) = (y^2 = 2*x);F: Point;Focus(G) = F;P: Point;PointOnCurve(P, G);WhenMin(Abs(LineSegmentOf(P, A))+Abs(LineSegmentOf(P, F)))", "coords": { "A": [ 3.0, 2.0 ] } }, { "index": 6941, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6941.png", "answer": "y=pm*2*x/3", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/a = 1);a: Number;Coordinate(RightFocus(G)) = (sqrt(13), 0)", "coords": {} }, { "index": 6942, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 1.0, 0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_6942.png", "answer": "(x^2/4+y^2/3=1)&Negation(x=-2)", "fact_expr": "M: Circle;Expression(M) = (y^2 + (x + 1)^2 = 1);N: Circle;Expression(N) = ((x-1)^2 + y^2 = 9);P: Circle;IsOutTangent(P, M) = True;IsInTangent(P, N) = True;C: Curve;P1: Point;Center(P) = P1;Locus(P1) = C", "coords": {} }, { "index": 6943, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6943.png", "answer": "(2,2*sqrt(2))", "fact_expr": "C: Parabola;A: Point;P: Point;Expression(C) = (y^2 = 4*x);Coordinate(A) = (3, 4*sqrt(2));PointOnCurve(P, C);WhenMin(Distance(P, A)+Distance(P, Directrix(C)))", "coords": { "A": [ 3.0, 5.656854249492381 ] } }, { "index": 6944, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6944.png", "answer": "(1/4,1)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);Q: Point;PointOnCurve(Q, C);B: Point;Coordinate(B) = (4, 1);F:Point;Focus(C)=F;WhenMin(Distance(Q,B)+Distance(Q,F))", "coords": { "B": [ 4.0, 1.0 ] } }, { "index": 6945, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6945.png", "answer": "8", "fact_expr": "M: Point;Coordinate(M) = (2, 0);G: Line;PointOnCurve(M, G);Slope(G) = sqrt(3);C: Parabola;Expression(C) = (y^2 = a*x);a: Number;a>0;l: Line;Directrix(C) = l;A: Point;B: Point;Intersection(G, l) = B;OneOf(Intersection(G, C)) = A;VectorOf(B, M) = VectorOf(M, A)", "coords": { "M": [ 2.0, 0.0 ] } }, { "index": 6946, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6946.png", "answer": "3+2*sqrt(2)", "fact_expr": "l: Line;G: Hyperbola;M: Point;F1: Point;F2: Point;N: Point;Expression(G) = (x^2/4 - y^2/2 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(F2,l);Inclination(l)=pi/4;Intersection(l,G)={M,N};Quadrant(M)=1;Radius(InscribedCircle(TriangleOf(M,F1,F2)))=R1;Radius(InscribedCircle(TriangleOf(N,F1,F2)))=R2;R1:Number;R2:Number", "coords": {} }, { "index": 6947, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6948, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6949, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6949.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;a: Number;b: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Negation(a = b);Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(G) = F2;LeftFocus(G) = F1;l: Line;PointOnCurve(F2, l);IsPerpendicular(l, Asymptote(G));FootPoint(l, Asymptote(G)) = P;Tan(AngleOf(P, F1, F2)) = 1/3", "coords": {} }, { "index": 6950, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6951, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6951.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;P: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Focus(G)={F1,F2};PointOnCurve(P,G);AngleOf(F1,P,F2)=ApplyUnit(60,degree)", "coords": {} }, { "index": 6952, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6953, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6953.png", "answer": "2", "fact_expr": "G: Parabola;a: Number;P: Point;F: Point;a>0;y0:Number;Expression(G) = (y^2 = a*x);Coordinate(P) = (3/2, y0);PointOnCurve(P, G);Focus(G) = F;Distance(P, F) = 2", "coords": {} }, { "index": 6954, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6954.png", "answer": "3/2", "fact_expr": "G: Ellipse;M: Point;I: Point;F1: Point;F2: Point;N: Point;Expression(G) = (x^2/9 + y^2/5 = 1);PointOnCurve(M, G);Focus(G) = {F1, F2};Incenter(TriangleOf(M, F1, F2)) = I;Intersection(OverlappingLine(LineSegmentOf(M, I)), LineSegmentOf(F1, F2)) = N", "coords": {} }, { "index": 6955, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6956, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6956.png", "answer": "2", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);O:Origin;Vertex(C) = O;G: Line;PointOnCurve(Focus(C), G) = True;IsPerpendicular(G, xAxis) = True;Intersection(G, C) = {A, B};A: Point;B: Point", "coords": {} }, { "index": 6957, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6958, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6958.png", "answer": "sqrt(105)/6", "fact_expr": "l: Line;C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);FocalLength(C)=2;b: Number;a: Number;A: Point;B: Point;Q: Point;O: Origin;P: Point;a > b;b > 0;Coordinate(Q) = (1, 1);PointOnCurve(P, C);Max(Distance(P,OneOf(Focus(C))) = 3);PointOnCurve(Q, l);Intersection(l, C) = {A, B};MidPoint(LineSegmentOf(A, B)) = Q", "coords": { "Q": [ 1.0, 1.0 ] } }, { "index": 6959, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7994307279586792, "b": 1.695713604329748, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 1.1180587689958606e-05, "converged": true, "iterations": 86 }, "output_path": "results/train/ellipse/problem_6959.png", "answer": "{8,81/8}", "fact_expr": "G: Ellipse;m: Real;Expression(G) = (y^2/9 + x^2/m = 1);Eccentricity(G) = 1/3", "coords": {} }, { "index": 6960, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6960.png", "answer": "(sqrt(2)+2-sqrt(6))/2", "fact_expr": "F1: Point;F2: Point;G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;LeftFocus(G) = F1;RightFocus(G) = F2;M: Point;PointOnCurve(M, G) = True;AngleOf(M, F1, F2) = ApplyUnit(30, degree);AngleOf(M, F2, F1) = ApplyUnit(105, degree)", "coords": {} }, { "index": 6961, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6961.png", "answer": "NO ANSWER", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(P) = (1, 0);PointOnCurve(P, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, P)) = 2", "coords": { "P": [ 1.0, 0.0 ] } }, { "index": 6962, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_6962.png", "answer": "3", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;F: Point;O: Origin;M: Point;L: Line;Expression(L) = (x = -2);Coordinate(F) = (1, 0);Vertex(G) = O;Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 4;MidPoint(LineSegmentOf(A, B)) = M", "coords": { "F": [ 1.0, 0.0 ] } }, { "index": 6963, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6963.png", "answer": "3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);M: Point;PointOnCurve(M, G);F: Point;Focus(G) = F;Distance(M, F) = 4", "coords": {} }, { "index": 6964, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6964.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;H: Line;Expression(H) = (2*x + y + 3 = 0);IsParallel(OneOf(Asymptote(G)), H) = True", "coords": {} }, { "index": 6965, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6965.png", "answer": "4", "fact_expr": "G: Hyperbola;Expression(G) = (y^2/9 - x^2/a^2 = 1);a: Number;a>0;F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);Abs(LineSegmentOf(P, F2)) = 2;lambda: Number;Sin(AngleOf(P, F2, F1)) = lambda*Sin(AngleOf(P, F1, F2))", "coords": {} }, { "index": 6966, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6966.png", "answer": "sqrt(3)/2", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;A: Point;B: Point;M: Point;e: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(M) = (-2, 1);PointOnCurve(M, H);Slope(H) = 1/2;Intersection(H, G) = {A, B};Negation(A = B);MidPoint(LineSegmentOf(A, B)) = M;Eccentricity(G) = e", "coords": { "M": [ -2.0, 1.0 ] } }, { "index": 6967, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6968, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6968.png", "answer": "3/2", "fact_expr": "G: Parabola;M: Point;O: Origin;Expression(G) = (y^2 = 2*x);PointOnCurve(M, G);Distance(M, O) = sqrt(3)", "coords": {} }, { "index": 6969, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6969.png", "answer": "2*sqrt(10)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/9 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G) = True;DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 0", "coords": {} }, { "index": 6970, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6970.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;b: Number;a: Number;F1: Point;P: Point;F2: Point;H:Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;IsPerpendicular(H,OneOf(Asymptote(G)));PointOnCurve(F1,H);Intersection(H,RightPart(G))=P;AngleOf(F1, P, F2) = pi/4", "coords": {} }, { "index": 6971, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6971.png", "answer": "1-(sqrt(3)/3)", "fact_expr": "C: Ellipse;b: Number;a: Number;A: Point;P: Point;F: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);RightFocus(C)=F;LeftVertex(C)=A;PointOnCurve(P, C);IsPerpendicular(LineSegmentOf(P,F),xAxis);Inclination(LineOf(P,A))=ApplyUnit(30, degree)", "coords": {} }, { "index": 6972, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6972.png", "answer": "2", "fact_expr": "G: Hyperbola;A: Point;C:Hyperbola;Expression(G) = (-x^2/9 + y^2/16 = 1);Coordinate(A) = (-3,2*sqrt(3));Asymptote(G) = Asymptote(C);PointOnCurve(A,C)", "coords": { "A": [ -3.0, 3.4641016151377544 ] } }, { "index": 6973, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6973.png", "answer": "x^2/4+y^2=1", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Eccentricity(C) = sqrt(3)/2;FocalLength(C) = 2*sqrt(3)", "coords": {} }, { "index": 6974, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6974.png", "answer": "3*x+4*y-7=0", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);A: Point;B: Point;IsChordOf(LineSegmentOf(A,B),G) = True;MidPoint(LineSegmentOf(A,B)) = P;Coordinate(P) = (1,1);P: Point", "coords": { "P": [ 1.0, 1.0 ] } }, { "index": 6975, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6976, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6976.png", "answer": "x+y-1=0", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;Slope(l) = -1;l: Line;Intersection(l, C) = {M, N};M: Point;N: Point;Abs(LineSegmentOf(M, F)) + Abs(LineSegmentOf(N, F)) = 8", "coords": {} }, { "index": 6977, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6977.png", "answer": "y=-2", "fact_expr": "l: Line;C: Parabola;F: Point;A: Point;B: Point;P:Point;Expression(C) = (x^2 = 8*y);Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};TangentOnPoint(A,C)=l1;TangentOnPoint(B,C)=l2;Intersection(l1,l2)=P", "coords": {} }, { "index": 6978, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6979, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6979.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = x^2 + 1);NumIntersection(Asymptote(G),H)=2", "coords": {} }, { "index": 6980, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6980.png", "answer": "[sqrt(3)/3, sqrt(2)/2]", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;F1: Point;F2: Point;c: Number;Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = c^2", "coords": {} }, { "index": 6981, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6981.png", "answer": "2", "fact_expr": "l: Line;G: Parabola;A: Point;B: Point;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(P) = (1, 1);PointOnCurve(P, l);Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A,B)) = P", "coords": { "P": [ 1.0, 1.0 ] } }, { "index": 6982, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6982.png", "answer": "7*sqrt(2)/2", "fact_expr": "A: Point;B: Point;G: Ellipse;P: Point;Expression(G) = (x^2/16 + y^2/9 = 1);Coordinate(A) = (-2, 0);Coordinate(B) = (0, 2);PointOnCurve(P, G)", "coords": { "A": [ -2.0, 0.0 ], "B": [ 0.0, 2.0 ] } }, { "index": 6983, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6984, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6985, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 2.0, "major_axis": "x", "x_coef": 8.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6985.png", "answer": "8*sqrt(2)", "fact_expr": "C: Ellipse;l: Line;P: Point;Q: Point;F: Point;Expression(C) = (x^2/8 + y^2/4 = 1);Expression(l) = (x - sqrt(3)*y + 2 = 0);RightFocus(C) = F;Intersection(C,l) = {P,Q}", "coords": {} }, { "index": 6986, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6986.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;PointOnCurve(LeftFocus(G), L) = True;IsPerpendicular(L, xAxis) = True;L: Line;Intersection(L, G) = {M, N};M: Point;N: Point;IsDiameter(LineSegmentOf(M, N), V) = True;PointOnCurve(RightVertex(G), V) = True;V: Circle", "coords": {} }, { "index": 6987, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6987.png", "answer": "[0, \\sqrt{5}-2)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);x0: Number;y0: Number;M: Point;Coordinate(M) = (x0, y0);PointOnCurve(M, G);F: Point;Focus(G) = F;P: Point;Coordinate(P) = (-1, 0);DotProduct(VectorOf(M, F), VectorOf(M, P)) < 0", "coords": { "P": [ -1.0, 0.0 ] } }, { "index": 6988, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "y", "x_coef": 4.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6988.png", "answer": "4*x+9*y-13=0", "fact_expr": "l: Line;M: Point;Coordinate(M) = (1, 1);PointOnCurve(M, l);G: Ellipse;Expression(G) = (x^2/4 + y^2/9 = 1);A: Point;B: Point;Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A, B)) = M", "coords": { "M": [ 1.0, 1.0 ] } }, { "index": 6989, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6989.png", "answer": "7", "fact_expr": "G: Hyperbola;a: Number;P: Point;F1: Point;F2: Point;a>0;Expression(G) = (-y^2/9 + x^2/a^2 = 1);PointOnCurve(P, G);Expression(OneOf(Asymptote(G))) = (3*x - 2*y = 0);LeftFocus(G) = F1;RightFocus(G) = F2;Abs(LineSegmentOf(P, F1)) = 3", "coords": {} }, { "index": 6990, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2175992727279663, "b": 2.434081792831421, "orientation": "horizontal" }, "optimization": { "final_loss": 3.452532837400213e-06, "converged": true, "iterations": 124 }, "output_path": "results/train/hyperbola/problem_6990.png", "answer": "sqrt(5)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Expression(OneOf(Asymptote(C))) = (y = 2*x)", "coords": {} }, { "index": 6991, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_6991.png", "answer": "8", "fact_expr": "G: Hyperbola;H: Parabola;m: Real;Expression(G) = (x^2/4 - y^2/3 = 1);Expression(H) = (y^2 = m*x);Focus(H) = RightVertex(G)", "coords": {} }, { "index": 6992, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6993, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6993.png", "answer": "2*sqrt(2)", "fact_expr": "l: Line;E: Parabola;A: Point;B: Point;O: Origin;C:Point;Expression(l) = (y = k*(x - 1));Expression(E) = (y^2 = 4*x);Intersection(l, E) = {A, B};Intersection(OverlappingLine(LineSegmentOf(A,B)),Directrix(E))=C;Area(TriangleOf(O,A,B))=Area(TriangleOf(O,B,C));k:Number;k>0", "coords": {} }, { "index": 6994, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6994.png", "answer": "3/4", "fact_expr": "C: Parabola;H: Line;B: Point;A: Point;F: Point;G: Point;O: Origin;Expression(C) = (y^2 = 2*x);Focus(C)=F;PointOnCurve(F, H);Intersection(H, C) = {A, B};L:Line;PointOnCurve(F,L);IsPerpendicular(LineSegmentOf(A,B),L);Intersection(L,Directrix(C))=G;Abs(LineSegmentOf(F, G)) = 3/2", "coords": {} }, { "index": 6995, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6995.png", "answer": "{0,-1/2}", "fact_expr": "l: Line;C: Parabola;m: Real;Expression(C) = (y^2 = 8*x);Expression(l)=(y=m*x-4);NumIntersection(l, C) = 1", "coords": {} }, { "index": 6996, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "y", "x_coef": 4.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6996.png", "answer": "9/2", "fact_expr": "G: Ellipse;P: Point;F1:Point;F2:Point;OneOf(Focus(G))=F1;OneOf(Focus(G))=F2;Negation(F1=F2);Expression(G) = (x^2/4 + y^2/9 = 1);PointOnCurve(P, G);Distance(P, F1) = 3/2", "coords": {} }, { "index": 6997, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_6997.png", "answer": "(4, +oo)", "fact_expr": "G: Ellipse;m: Number;Expression(G) = (x^2/4 + y^2/m = 1);PointOnCurve(Focus(G), yAxis)", "coords": {} }, { "index": 6998, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 6999, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_6999.png", "answer": "sqrt(3)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;l: Line;PointOnCurve(F,l) = True;Slope(l) = k;k: Number;k>0;Intersection(l,G) = {A,B};A: Point;B: Point;Directrix(G) = q;q: Line;Intersection(q,l) = M;M: Point;Abs(LineSegmentOf(A, F)) = (3/2)*Abs(LineSegmentOf(B, M))", "coords": {} }, { "index": 7000, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7001, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7001.png", "answer": "5 + sqrt(2)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/5 = 1);H: Circle;Expression(H) = (y^2 + (x + 3)^2 = 2);C: Circle;Expression(C) = (y^2 + (x - 3)^2 = 1);P: Point;M: Point;N: Point;PointOnCurve(P, RightPart(G));PointOnCurve(M, H);PointOnCurve(N, C)", "coords": {} }, { "index": 7002, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7002.png", "answer": "9", "fact_expr": "G: Parabola;P: Point;F: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(P, G);Focus(G) = F;Distance(P, F) = 10", "coords": {} }, { "index": 7003, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7003.png", "answer": "2*b^2/a", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;H: LineSegment;IsChordOf(H, G);PointOnCurve(Focus(G), H);IsPerpendicular(H, xAxis)", "coords": {} }, { "index": 7004, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_7004.png", "answer": "x^2/9 + y^2/4 = 1", "fact_expr": "G: Circle;P: Point;D: Point;M: Point;Expression(G) = (x^2 + y^2 = 9);PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, D), xAxis);FootPoint(LineSegmentOf(P, D), xAxis) = D;PointOnCurve(M, LineSegmentOf(P, D));VectorOf(P, M) = (1/3)*VectorOf(P, D)", "coords": {} }, { "index": 7005, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7006, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7007, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.196152422706632, "b": 5.0990195135927845, "major_axis": "x", "x_coef": 27.0, "y_coef": 26.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7007.png", "answer": "40", "fact_expr": "G: Ellipse;H: Circle;A: Point;B: Point;M: Point;Expression(G) = (x^2/27 + y^2/26 = 1);Expression(H) = (y^2 + (x - 1)^2 = 8);PointOnCurve(M, G);IsDiameter(LineSegmentOf(A, B), H)", "coords": {} }, { "index": 7008, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7008.png", "answer": "2*sqrt(3)/3", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);l: Line;Directrix(G) = l;P: Point;PointOnCurve(P, l) = True;TangentOfPoint(P, G) = {LineOf(P, A), LineOf(P, B)};TangentPoint(LineOf(P, A), G) = A;TangentPoint(LineOf(P, B), G) = B;B: Point;A: Point;Inclination(LineOf(A, B)) = pi/6", "coords": {} }, { "index": 7009, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7010, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7010.png", "answer": "5/7", "fact_expr": "C: Ellipse;a: Number;b: Number;G: Line;A: Point;F: Point;B: Point;O: Origin;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F;PointOnCurve(O, G);Intersection(C, G) = {A, B};Abs(LineSegmentOf(A, B)) = 10;Abs(LineSegmentOf(B, F)) = 8;Cos(AngleOf(A, B, F)) = 4/5", "coords": {} }, { "index": 7011, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7011.png", "answer": "6", "fact_expr": "G: Parabola;l: Line;A: Point;B: Point;M: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(M) = (-1, 0);PointOnCurve(M,l);Intersection(l, G) = {A,B};Focus(G) = F;Abs(LineSegmentOf(B, F)) = (sqrt(3)/2)*Abs(LineSegmentOf(B, M))", "coords": { "M": [ -1.0, 0.0 ] } }, { "index": 7012, "success": false, "error": "\n= 1 $ $ (a > b > 0) $, the left vertex is $ A $, the right focus is $\n ^\nParseException: Expected end of text, found '$' (at char 4), (line:1, col:5)" }, { "index": 7013, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7013.png", "answer": "10", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 8*x);F: Point;Focus(C) = F;M: Point;PointOnCurve(M, C) = True;N: Point;Coordinate(N) = (0, 4);AngleOf(M, N, F) = ApplyUnit(90, degree)", "coords": { "N": [ 0.0, 4.0 ] } }, { "index": 7014, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7014.png", "answer": "8", "fact_expr": "G: Parabola;O: Origin;F: Point;A: Point;B: Point;Expression(G) = (y^2 = 8*x);Focus(G) = F;PointOnCurve(A, G);PointOnCurve(B, G);VectorOf(O, F) = (VectorOf(O, A) + VectorOf(O, B))/2", "coords": {} }, { "index": 7015, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7015.png", "answer": "sqrt(22)/2", "fact_expr": "G: Hyperbola;a: Number;b: Number;P: Point;F2: Point;F1: Point;O:Origin;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);Distance(P, O) = b;Sin(AngleOf(P, F2, F1)) = 2*Sin(AngleOf(P, F1, F2))", "coords": {} }, { "index": 7016, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_focus_mismatch", "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7016.png", "answer": "3", "fact_expr": "G: Ellipse;m: Number;F1: Point;m>0;Expression(G) = (x^2/25 + y^2/m^2 = 1);Coordinate(F1) = (-4, 0);LeftFocus(G) = F1", "coords": { "F1": [ -4.0, 0.0 ] }, "validation_reasons": [ "ellipse_focus_mismatch" ] }, { "index": 7017, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7017.png", "answer": "4/5", "fact_expr": "G: Hyperbola;P: Point;F2: Point;I: Point;lambda:Number;F1: Point;Expression(G) = (x^2/16 - y^2/9 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, RightPart(G));Incenter(TriangleOf(P,F1,F2))=I;Area(TriangleOf(I,P,F2))=Area(TriangleOf(I,P,F1))-lambda*Area(TriangleOf(I,F1,F2))", "coords": {} }, { "index": 7018, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7018.png", "answer": "x = -2", "fact_expr": "O: Origin;C: Parabola;F: Point;P: Point;p: Number;Q: Point;p>0;Expression(C) = (y^2 = 2*p*x);Focus(C) = F;PointOnCurve(P, C) = True;IsPerpendicular(LineSegmentOf(P, F), xAxis) = True;PointOnCurve(Q, yAxis);IsPerpendicular(LineSegmentOf(P, Q), LineSegmentOf(O, P)) = True;Abs(LineSegmentOf(P, Q)) = sqrt(5)", "coords": {} }, { "index": 7019, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7019.png", "answer": "1", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);P: Point;PointOnCurve(P, G)", "coords": {} }, { "index": 7020, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7020.png", "answer": "-9", "fact_expr": "Q: Point;Coordinate(Q) = (2, 0);G: Line;PointOnCurve(Q, G) = True;C: Parabola;Expression(C) = (y^2 = 2*x);OneOf(Intersection(G, C)) = A;A: Point;Intersection(G, yAxis) = D;D: Point;VectorOf(D, A) = 2*VectorOf(D, Q);P: Point;PointOnCurve(P, C) = True", "coords": { "Q": [ 2.0, 0.0 ] } }, { "index": 7021, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7021.png", "answer": "y=pm*(sqrt(2)/2)*x", "fact_expr": "C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);FocalLength(C) = 4*sqrt(3);Length(RealAxis(C)) = 4*sqrt(2)", "coords": {} }, { "index": 7022, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7022.png", "answer": "2*sqrt(3)/3 - 1", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 7023, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7023.png", "answer": "pm*4/3", "fact_expr": "l: Line;C: Parabola;A: Point;M: Point;B: Point;k: Real;Expression(C) = (y^2 = 16*x);Expression(l) = (y = k*x - 4*k);Intersection(SymmetryAxis(C),Directrix(C))=M;Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, M)) = 4*Abs(LineSegmentOf(B, M))", "coords": {} }, { "index": 7024, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.0625, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7024.png", "answer": "2", "fact_expr": "G: Parabola;A: Point;Expression(G) = (y = 4*x^2);PointOnCurve(A, G);Distance(A, Focus(G)) = 33/16", "coords": {} }, { "index": 7025, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_7025.png", "answer": "(x-2)^2+y^2=1/4", "fact_expr": "G: Curve;B: Point;A: Point;P: Point;Expression(G) = (x^2 + y^2 = 1);Coordinate(A) = (4, 0);PointOnCurve(B, G);MidPoint(LineSegmentOf(A,B)) = P", "coords": {} }, { "index": 7026, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7026.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;O: Origin;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);DotProduct(VectorOf(P, F1),(VectorOf(O,F1)+VectorOf(O,P)))=0;Abs(VectorOf(P, F1)) = 2*Abs(VectorOf(P, F2))", "coords": {} }, { "index": 7027, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7027.png", "answer": "4", "fact_expr": "G: Parabola;P: Point;F: Point;m:Number;Expression(G) = (y^2 = 4*x);Coordinate(P) = (m, 2*sqrt(3));PointOnCurve(P, G);Focus(G) = F", "coords": {} }, { "index": 7028, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7028.png", "answer": "(1+sqrt(5))/2", "fact_expr": "C: Hyperbola;b: Number;a: Number;B: Point;F: Point;A: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(B) = (0, b);LeftFocus(C) = F;RightVertex(C) = A;DotProduct(VectorOf(F, B), VectorOf(A, B)) = 0", "coords": {} }, { "index": 7029, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7029.png", "answer": "28", "fact_expr": "G: Hyperbola;A: Point;B: Point;F1:Point;F2: Point;Expression(G) = (x^2/16 - y^2/9 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;IsChordOf(LineSegmentOf(A,B),G);PointOnCurve(F1,LineSegmentOf(A,B));Length(LineSegmentOf(A,B))=6", "coords": {} }, { "index": 7030, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7030.png", "answer": "12", "fact_expr": "C: Parabola;p: Number;M: Point;F: Point;N: Point;p>0;l:Line;Expression(C) = (y^2 = 2*p*x);Focus(C) = F;PointOnCurve(M, C);Directrix(C)=l;PointOnCurve(N,l);VectorOf(F, N) + 3*VectorOf(M, F) = 0;-p + 2*Abs(LineSegmentOf(M, F)) = 4", "coords": {} }, { "index": 7031, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7032, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7032.png", "answer": "y = -2", "fact_expr": "C: Parabola;Expression(C) = (x^2 = 2*p*y);p: Number;p>0;Distance(Focus(C), Directrix(C)) = 4", "coords": {} }, { "index": 7033, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7034, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "y", "x_coef": 4.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7034.png", "answer": "x^2 = -4\\sqrt{5}y", "fact_expr": "G: Parabola;H: Ellipse;Expression(H) = (x^2/4 + y^2/9 = 1);Focus(G) = LowerFocus(H);Vertex(G) = Center(H)", "coords": {} }, { "index": 7035, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7035.png", "answer": "-1/4", "fact_expr": "G: Parabola;Expression(G) = (y = a*x^2);a: Real;F: Point;Coordinate(F) = (0, -1);Focus(G) = F", "coords": { "F": [ 0.0, -1.0 ] } }, { "index": 7036, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 5.196152422706632, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7036.png", "answer": "14", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/27 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};A: Point;PointOnCurve(A, G);Abs(LineSegmentOf(A, F1)) = 8", "coords": {} }, { "index": 7037, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.0416666679084301, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7037.png", "answer": "(0,1/24)", "fact_expr": "G: Parabola;Expression(G) = (y = 6*x^2)", "coords": {} }, { "index": 7038, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7038.png", "answer": "{1, 3}", "fact_expr": "l: Line;C: Parabola;B: Point;D: Point;F: Point;A: Point;m: Line;p: Number;p > 0;Expression(C) = (y^2 = 2*p*x);Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};Directrix(C) = m;Intersection(l, m) = D;Abs(LineSegmentOf(B, D)) = 2*Abs(LineSegmentOf(F, B));Abs(LineSegmentOf(F, A)) = 2", "coords": {} }, { "index": 7039, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7040, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7040.png", "answer": "sqrt(3)", "fact_expr": "C: Parabola;p: Number;G: Line;A: Point;B: Point;F: Point;l: Line;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;Directrix(C) = l;PointOnCurve(F, G);Intersection(G, C) = {A, B};A1: Point;B1: Point;IsPerpendicular(LineSegmentOf(A, A1), l);IsPerpendicular(LineSegmentOf(B, B1), l);FootPoint(LineSegmentOf(A, A1), l) = A1;FootPoint(LineSegmentOf(B, B1), l) = B1;Abs(LineSegmentOf(A1, F)) = 2*sqrt(3);Abs(LineSegmentOf(B1, F)) = 2", "coords": {} }, { "index": 7041, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7041.png", "answer": "y = pm*(3/4)*x", "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/16 + y^2/9 = 1)", "coords": {} }, { "index": 7042, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7042.png", "answer": "x^2/5+y^2=1", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x + 2*y - 2 = 0);PointOnCurve(OneOf(Focus(G)),H);PointOnCurve(OneOf(Vertex(G)),H)", "coords": {} }, { "index": 7043, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 0.7085586085307979, "b": 0.7085586085307979, "orientation": "horizontal" }, "optimization": { "final_loss": 1.0145607782384324e-05, "converged": true, "iterations": 73 }, "output_path": "results/train/hyperbola/problem_7043.png", "answer": "x^2/(2/3)-y^2/(1/3)=1", "fact_expr": "Gamma: Hyperbola;P: Point;PointOnCurve(P, Gamma) = True;F1: Point;Coordinate(F1) = (-1, 0);F2: Point;Coordinate(F2) = (1, 0);d1: Number;d2: Number;Distance(P, F1) = d1;Distance(P, F2) = d2;theta: Number;AngleOf(F1, P, F2) = 2*theta;d1*d2*Sin(theta)^2 = 1/3", "coords": { "F1": [ -1.0, 0.0 ], "F2": [ 1.0, 0.0 ] } }, { "index": 7044, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7044.png", "answer": "x=3", "fact_expr": "G: Parabola;Expression(G) = (y^2 = -12*x)", "coords": {} }, { "index": 7045, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2175992727279663, "b": 2.434081792831421, "orientation": "horizontal" }, "optimization": { "final_loss": 3.452532837400213e-06, "converged": true, "iterations": 124 }, "output_path": "results/train/hyperbola/problem_7045.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G))) = (y = 2*x);Distance(Focus(G), Asymptote(G)) = 2;a>0", "coords": {} }, { "index": 7046, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7046.png", "answer": "x^2 - y^2/3 = 1", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Coordinate(OneOf(Focus(C))) = (2, 0);l: Line;Expression(l) = (x + sqrt(3)*y = 0);IsPerpendicular(OneOf(Asymptote(C)),l)", "coords": {} }, { "index": 7047, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7047.png", "answer": "2", "fact_expr": "C: Ellipse;m: Number;Expression(C) = (x^2/3 + y^2/m = 1);Length(MajorAxis(C)) = 4", "coords": {} }, { "index": 7048, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7049, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7049.png", "answer": "{1, 3}", "fact_expr": "G: Parabola;H: Line;I: Point;O: Origin;A: Point;B: Point;m: Number;Expression(G) = (y^2 = 4*x);Coordinate(I) = (m, 0);PointOnCurve(I, H);Intersection(G, H) = {A, B};LineSegmentOf(O, A)*LineSegmentOf(O, B) = -3", "coords": {} }, { "index": 7050, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7050.png", "answer": "1/10", "fact_expr": "G: Curve;Expression(G) = (y = x^2);Coordinate(A) = (a, a^2);A: Point;PointOnCurve(A, G) = True;a: Number;In(a, (0, 5]);M: Point;Coordinate(M) = (0, 1/(a^2 + a + 4));PointOnCurve(M, yAxis) = True;O: Origin", "coords": {} }, { "index": 7051, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7052, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7052.png", "answer": "4*a", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;H: Line;PointOnCurve(F1, H);A: Point;B: Point;Intersection(H, LeftPart(G)) = {A, B};Abs(LineSegmentOf(A, F2)) + Abs(LineSegmentOf(B, F2)) = 2*Abs(LineSegmentOf(A, B))", "coords": {} }, { "index": 7053, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 5.656854249492381, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7053.png", "answer": "2", "fact_expr": "A: Point;Coordinate(A) = (x0, y0);x0: Number;y0: Number;G: Hyperbola;Expression(G) = (x^2/4 - y^2/32 = 1);PointOnCurve(A, RightPart(G)) = True;Distance(A, RightFocus(G)) = 2*x0", "coords": {} }, { "index": 7054, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7054.png", "answer": "3", "fact_expr": "P: Point;G: Ellipse;Expression(G) = (x^2 + y^2/m = 1);m: Number;m>0;PointOnCurve(P, G);H: Line;Expression(H) = (y = x + 4);Q: Point;PointOnCurve(Q, H);Min(LineSegmentOf(P, Q)) = sqrt(2)", "coords": {} }, { "index": 7055, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7055.png", "answer": "16*sqrt(3)", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (16*x^2 - 9*y^2 = 144);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1))*Abs(LineSegmentOf(P, F2)) = 64", "coords": {} }, { "index": 7056, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7056.png", "answer": "10/3", "fact_expr": "C: Parabola;P: Point;F: Point;Q: Point;l: Line;Expression(C) = (y^2 = 4*x);Focus(C) = F;Directrix(C) = l;PointOnCurve(P, l);Intersection(OverlappingLine(LineSegmentOf(P, F)), C) = Q;2*VectorOf(F, P) + 3*VectorOf(F, Q) =0", "coords": {} }, { "index": 7057, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7058, "success": false, "conic_type": "parabola", "error": "validation: parabola_point_off_curve", "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7058.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "H: Point;Coordinate(H) = (1, 0);G: Line;Inclination(G) = (2*pi)/3;PointOnCurve(H, G);A: Point;B: Point;Z: Curve;Expression(Z) = (y^2 = 4*x);Intersection(G, Z) = {A, B};IsChordOf(LineSegmentOf(A, B), Z)", "coords": { "H": [ 1.0, 0.0 ] }, "validation_reasons": [ "parabola_point_off_curve" ] }, { "index": 7059, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7059.png", "answer": "{x^2 = -20*y, x^2 = 4*y}", "fact_expr": "G: Parabola;Expression(G) = (x^2 = m*y);m: Number;Z: Line;Expression(Z) = (y = 2);Distance(Directrix(G), Z) = 3", "coords": {} }, { "index": 7060, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7060.png", "answer": "sqrt(2)", "fact_expr": "C: Parabola;l1:Line;l2:Line;P: Point;Expression(C) = (y^2 = 4*x);Expression(l2) = (y=x+1);Expression(l1) = (x = -1);PointOnCurve(P, C)", "coords": {} }, { "index": 7061, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7062, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7062.png", "answer": "2", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;G: Circle;Expression(G) = (x^2 + y^2 = a^2);L1: Line;L2: Line;TangentOfPoint(OneOf(Focus(C)), G) = {L1, L2};TangentPoint(L1, G) = A;TangentPoint(L2, G) = B;A: Point;B: Point;AngleOf(A, O, B) = ApplyUnit(120, degree);O: Origin", "coords": {} }, { "index": 7063, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7063.png", "answer": "4/3", "fact_expr": "l: Line;G: Parabola;A: Point;F: Point;B: Point;Expression(G) = (y^2 = 4*x);Focus(G)= F;PointOnCurve(F,l);Intersection(l,G) = {A, B};Abs(LineSegmentOf(A, F)) = 4", "coords": {} }, { "index": 7064, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7064.png", "answer": "(10, 0)", "fact_expr": "O: Origin;C: Parabola;Expression(C) = (y^2 = 8*x);F: Point;Focus(C) = F;P: Point;PointOnCurve(P, C) ;IsPerpendicular(LineSegmentOf(P, F), xAxis) ;Q: Point;PointOnCurve(Q, xAxis) ;PointOnCurve(P, G) ;IsDiameter(LineSegmentOf(O, Q), G);G: Circle", "coords": {} }, { "index": 7065, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7065.png", "answer": "(0, 1]", "fact_expr": "G: Parabola;p: Number;A: Point;B: Point;H: Point;N: Point;F: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;PointOnCurve(A,G);PointOnCurve(B,G);AngleOf(A, F, B) = ApplyUnit(60, degree);IsChordOf(LineSegmentOf(A,B),G);MidPoint(LineSegmentOf(A,B))=H;PointOnCurve(H,LineOf(H,N));IsPerpendicular(Directrix(G),LineOf(H,N));FootPoint(Directrix(G),LineOf(H,N))=N", "coords": {} }, { "index": 7066, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7067, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7067.png", "answer": "y=pm*(1/2)*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1)", "coords": {} }, { "index": 7068, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7068.png", "answer": "2*sqrt(3)/3", "fact_expr": "C: Hyperbola;b: Number;a: Number;E: Curve;l: Line;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(E) = (y^2 + (x - 2)^2 = 4);OneOf(Asymptote(C)) = l;Slope(l) > 0;F1: Point;F2: Point;F3: Point;PointOnCurve(F1, E);PointOnCurve(F2, E);PointOnCurve(F3, E);Distance(F1, l) = 1;Distance(F2, l) = 1;Distance(F3, l) = 1;Negation(F1 = F2);Negation(F2 = F3);Negation(F1 = F3)", "coords": {} }, { "index": 7069, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7070, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7070.png", "answer": "[41,49]", "fact_expr": "l: Line;G: Ellipse;a:Number;b:Number;C: Point;A: Point;B: Point;Expression(G) = (x^2 + y^2/9 = 1);Expression(l) = (a*x + b*y = 0);Coordinate(C) = (5, 5);Intersection(l, G) = {A, B}", "coords": { "C": [ 5.0, 5.0 ] } }, { "index": 7071, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7072, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7073, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7073.png", "answer": "[4*sqrt(3)/3,+oo)", "fact_expr": "C: Curve;Expression(C) = (x^2/4 + y^2/b^2 = 1);b: Number;b>0;l: Line;Expression(l) = (k + k*x - y + 2 = 0);k: Number;IsIntersect(C, l) = True", "coords": {} }, { "index": 7074, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7074.png", "answer": "(x-1)^2+y^2=4", "fact_expr": "G: Ellipse;H: Circle;Expression(G)=(x^2/4+y^2/3=1);RightFocus(G)=F;Center(H)=F;F:Point;PointOnCurve(Endpoint(MinorAxis(G)),H)", "coords": {} }, { "index": 7075, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7075.png", "answer": "7", "fact_expr": "G: Hyperbola;a: Number;H: Circle;M: Point;N: Point;F: Point;a>0;Expression(G) = (-y^2/12 + x^2/a^2 = 1);Expression(H) = (x^2 + (y - 3)^2 = 4);Expression(OneOf(Asymptote(G))) = (sqrt(3)*x - y = 0);LeftFocus(G) = F;PointOnCurve(M, RightPart(G));PointOnCurve(N, H)", "coords": {} }, { "index": 7076, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_7076.png", "answer": "sqrt(7)/4", "fact_expr": "C: Hyperbola;b: Number;a: Number;O: Circle;c: Number;P: Point;F1: Point;F2: Point;N: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(O) = (x^2 + y^2 = c^2);LeftFocus(C) = F1;RightFocus(C) = F2;Eccentricity(C) = 2;PointOnCurve(P, RightPart(C));MidPoint(LineSegmentOf(P, F1)) = N;PointOnCurve(N, O);HalfFocalLength(C) = c", "coords": {} }, { "index": 7077, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7077.png", "answer": "(1, 2*sqrt(3)/3)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(G) = F;PointOnCurve(F, H) = True;Inclination(H) = ApplyUnit(30, degree);H: Line;NumIntersection(H, RightPart(G)) = 2", "coords": {} }, { "index": 7078, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7079, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7079.png", "answer": "5/3", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1)", "coords": {} }, { "index": 7080, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7080.png", "answer": "2", "fact_expr": "G: Ellipse;M: Point;N: Point;F: Point;Expression(G) = (x^2/4 + y^2/2 = 1);RightFocus(G) = F;PointOnCurve(M, yAxis);PointOnCurve(N, yAxis);DotProduct(VectorOf(M, F), VectorOf(N, F)) = 0", "coords": {} }, { "index": 7081, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7081.png", "answer": "4", "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2 = 1);F1: Point;P: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 7082, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7082.png", "answer": "[\\sqrt{2}/2,1)", "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F2: Point;F1: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P,G);DotProduct(VectorOf(P,F1),VectorOf(P,F2))=0", "coords": {} }, { "index": 7083, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7084, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 5.196152422706632, "major_axis": "x", "x_coef": 36.0, "y_coef": 27.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7084.png", "answer": "6", "fact_expr": "G: Ellipse;O: Origin;B: Point;A: Point;F1: Point;C: Point;F2: Point;Expression(G) = (x^2/36 + y^2/27 = 1);LeftFocus(G)= F1;RightFocus(G) = F2;PointOnCurve(A, G);VectorOf(O, B) = (VectorOf(O, A) + VectorOf(O, F1))/2;VectorOf(O, C) = 1/2*(VectorOf(O,A)+VectorOf(O,F2))", "coords": {} }, { "index": 7085, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7085.png", "answer": "sqrt(10)/2", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Focus(C)={F1,F2};PointOnCurve(P,C);IsPerpendicular(LineSegmentOf(P,F1),LineSegmentOf(P,F2));Abs(LineSegmentOf(P,F1))=3*Abs(LineSegmentOf(P,F2))", "coords": {} }, { "index": 7086, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7086.png", "answer": "sqrt(3)/2", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);F1: Point;F2: Point;Focus(G) = {F1, F2};P: Point;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F2), xAxis)", "coords": {} }, { "index": 7087, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7087.png", "answer": "(2*sqrt(21))/3", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);P: Point;PointOnCurve(P, G);A: Point;Coordinate(A) = (0, 2)", "coords": { "A": [ 0.0, 2.0 ] } }, { "index": 7088, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7088.png", "answer": "(2, 0)\nsqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1)", "coords": {} }, { "index": 7089, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7089.png", "answer": "2", "fact_expr": "G: Hyperbola;C: Circle;r: Number;Expression(G) = (x^2 - y^2 = 1);r>0;Expression(C) = (y^2 + (x - 2)^2 = r^2);Length(InterceptChord(OneOf(Asymptote(G)),C))=2*sqrt(2)", "coords": {} }, { "index": 7090, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7090.png", "answer": "2*sqrt(5) - 1", "fact_expr": "G: Parabola;P: Point;F: Point;Q: Point;Coordinate(Q) = (3, 4);Focus(G) = F;Expression(G) = (y^2 = 4*x);PointOnCurve(P, G)", "coords": { "Q": [ 3.0, 4.0 ] } }, { "index": 7091, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7091.png", "answer": "[\\sqrt{2}/2,1)", "fact_expr": "H: Circle;O: Origin;Center(H) = O;Intersection(H, xAxis) = {A, B};A: Point;B: Point;Focus(G) = {A, B};G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;IsIntersect(H, G) = True", "coords": {} }, { "index": 7092, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7092.png", "answer": "[-2,2]", "fact_expr": "G: Ellipse;Expression(G) = (x^2 + 4*y^2 = 16);P:Point;PointOnCurve(P,G)", "coords": {} }, { "index": 7093, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7094, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7094.png", "answer": "sqrt(2)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;p>0;F: Point;Focus(C) = F;l: Line;Expression(l) = (y = m + x);m: Number;Intersection(l, C) = {M, N};M: Point;N: Point;Abs(LineSegmentOf(M, F)) = 2;Abs(LineSegmentOf(N, F)) = 3", "coords": {} }, { "index": 7095, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7095.png", "answer": "a/(sqrt(a^2+b^2))", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;I: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, RightPart(G));LeftFocus(G) = F1;RightFocus(G) = F2;Incenter(TriangleOf(P, F1, F2))=I;Area(TriangleOf(I,P, F1))=Area(TriangleOf(I,P,F2)) +lambda*Area(TriangleOf(I,F1,F2));lambda:Number", "coords": {} }, { "index": 7096, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7096.png", "answer": "y = -1", "fact_expr": "C: Parabola;p: Number;G: Line;A: Point;B: Point;p>0;Expression(C) = (x^2 = 2*(p*y));Expression(G) = (y = p/2 + x);Intersection(G, C) = {A, B};Abs(LineSegmentOf(A, B)) = 8", "coords": {} }, { "index": 7097, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7097.png", "answer": "x^2=pm*16*y", "fact_expr": "G: Parabola;O: Origin;Vertex(G) = O;SymmetryAxis(G) = yAxis;L: LineSegment;PointOnCurve(Focus(G), L) = True;IsPerpendicular(L, yAxis) = True;IsChordOf(L, G) = True;Length(L) = 16", "coords": {} }, { "index": 7098, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7098.png", "answer": "(1, 1+sqrt(3)]", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, RightPart(G));O: Origin;DotProduct((VectorOf(O, F2) + VectorOf(O, P)), VectorOf(F2, P)) = 0;Abs(LineSegmentOf(P, F1)) >= sqrt(3)*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 7099, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7099.png", "answer": "sqrt(2)/2", "fact_expr": "E: Ellipse;b: Number;a: Number;G: Line;F: Point;c:Number;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);Expression(G) = (-a*y + b*x = 0);Coordinate(F) = (c, 0);RightFocus(E) = F;Distance(F, G) = (sqrt(3)/3)*c", "coords": {} }, { "index": 7100, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7100.png", "answer": "y=pm*(sqrt(2)/2)*x", "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/8 + y^2/4 = 1)", "coords": {} }, { "index": 7101, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 4.898979485566356, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7101.png", "answer": "48*pi/7", "fact_expr": "F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;C: Hyperbola;Expression(C) = (x^2 - y^2/24 = 1);P: Point;PointOnCurve(P, C) = True;Quadrant(P) = 1;Abs(LineSegmentOf(P, F1))/Abs(LineSegmentOf(P, F2)) = 5/4", "coords": {} }, { "index": 7102, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7102.png", "answer": "18", "fact_expr": "G: Ellipse;H: Line;P: Point;F: Point;Q: Point;Expression(G) = (x^2/25 + y^2/16 = 1);PointOnCurve(Center(G), H);Intersection(H, G) = {P, Q};LeftFocus(G) = F", "coords": {} }, { "index": 7103, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7103.png", "answer": "1", "fact_expr": "G: Parabola;P: Point;F: Point;Expression(G) = (y^2 = 8*x);Focus(G) = F;PointOnCurve(P, G);Abs(LineSegmentOf(P, F)) = 3", "coords": {} }, { "index": 7104, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7104.png", "answer": "[sqrt(5)/5, 1/2]", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;c: Number;F1: Point;F2: Point;Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C);DotProduct(VectorOf(P, F1), VectorOf(P, F2)) = 3*c^2;e: Number;Eccentricity(C) = e", "coords": {} }, { "index": 7105, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7105.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;Eccentricity(G) = sqrt(2)/2;l: Line;A: Point;B: Point;Intersection(l, G) = {A, B};M: Point;Coordinate(M) = (1, 1);MidPoint(LineSegmentOf(A, B)) = M", "coords": { "M": [ 1.0, 1.0 ] } }, { "index": 7106, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7106.png", "answer": "y=(pm*sqrt(3)/3)*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - 3*y^2 = 3)", "coords": {} }, { "index": 7107, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7107.png", "answer": "2*a+2*b", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;PointOnCurve(Center(G), H) = True;H: Line;Intersection(H, G) = {P, Q};P: Point;Q: Point;F: Point;OneOf(Focus(G)) = F", "coords": {} }, { "index": 7108, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7108.png", "answer": "x^2/12 - y^2/8 = 1", "fact_expr": "G: Hyperbola;Z: Hyperbola;H: Point;Expression(G) = (x^2/16 - y^2/4 = 1);Coordinate(H) = (3*sqrt(2), 2);PointOnCurve(H, Z);Focus(G) = Focus(Z)", "coords": { "H": [ 4.242640687119286, 2.0 ] } }, { "index": 7109, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7109.png", "answer": "pm*2*sqrt(2)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;F: Point;Focus(G) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, G) = {A, B};C: Point;Intersection(l, Directrix(G)) = C;VectorOf(C, B) = 3*VectorOf(B, F)", "coords": {} }, { "index": 7110, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7110.png", "answer": "4", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Focus(C) = {F1, F2};PointOnCurve(P, C);IsPerpendicular(VectorOf(P, F2),VectorOf(P, F1));Area(TriangleOf(P, F1, F2)) = 16", "coords": {} }, { "index": 7111, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7111.png", "answer": "y^2=4*x", "fact_expr": "F: Point;Focus(G) = F;G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;A: Point;B: Point;C: Point;D: Point;IsChordOf(LineSegmentOf(A, B), G);IsChordOf(LineSegmentOf(C, D), G);PointOnCurve(F, LineSegmentOf(A, B));PointOnCurve(F, LineSegmentOf(C, D));IsPerpendicular(LineSegmentOf(A, B), LineSegmentOf(C, D));Min(Abs(LineSegmentOf(A, B)) + Abs(LineSegmentOf(C, D))) = 16", "coords": {} }, { "index": 7112, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7112.png", "answer": "y=x", "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;F: Point;O: Origin;Coordinate(F) = (1, 0);Focus(C)=F;Vertex(C) = O;Intersection(l,C)={A,B};Coordinate(MidPoint(LineSegmentOf(A,B)))=(2,2)", "coords": { "F": [ 1.0, 0.0 ] } }, { "index": 7113, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7195594310760498, "b": 1.7224669456481934, "orientation": "horizontal" }, "optimization": { "final_loss": 2.391175712546101e-06, "converged": true, "iterations": 83 }, "output_path": "results/train/hyperbola/problem_7113.png", "answer": "y=pm*x", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = sqrt(2)", "coords": {} }, { "index": 7114, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7115, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7115.png", "answer": "8", "fact_expr": "C: Parabola;p: Number;G: Line;B: Point;A: Point;M: Point;N: Point;p>0;Expression(C) = (y^2 = 2*p*x);Expression(G) = (x = 1);Intersection(G, C) = M;Intersection(G,xAxis)=N;PointOnCurve(A, C);PointOnCurve(B, C);Slope(LineOf(A,B)) = 2;Distance(N,LineOf(M,A))=Distance(N,LineOf(M,B))", "coords": {} }, { "index": 7116, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7117, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7118, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7118.png", "answer": "1/2", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;O:Origin;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(OneOf(Focus(G)),H);PointOnCurve(Endpoint(MinorAxis(G)),H);Distance(OneOf(Focus(G)),H)=b/2", "coords": {} }, { "index": 7119, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7119.png", "answer": "4", "fact_expr": "C: Hyperbola;F1: Point;P: Point;F2: Point;Expression(C) = (x^2 - y^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 7120, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7120.png", "answer": "(1, 2+sqrt(5))", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;c: Number;HalfFocalLength(G) = c;-4*a*c + b^2 < 0", "coords": {} }, { "index": 7121, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7121.png", "answer": "1/4", "fact_expr": "l: Line;G: Curve;A: Point;Expression(G) = (y = x^2);Coordinate(A) = (t, 0);Inclination(l) = ApplyUnit(45, degree);PointOnCurve(A,l);IsTangent(l, G);t:Number", "coords": {} }, { "index": 7122, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7122.png", "answer": "2*sqrt(7)-2", "fact_expr": "G: Hyperbola;P: Point;F: Point;A: Point;Expression(G) = (x^2 - y^2/3 = 1);Coordinate(A) = (3, sqrt(3));PointOnCurve(P, RightPart(G));RightFocus(G) = F", "coords": { "A": [ 3.0, 1.7320508075688772 ] } }, { "index": 7123, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.125, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7123.png", "answer": "(0,1/8)", "fact_expr": "G: Parabola;Expression(G) = (y = 2*x^2)", "coords": {} }, { "index": 7124, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7124.png", "answer": "5", "fact_expr": "Slope(G) = 2;G: Line;C: Parabola;Expression(C) = (y^2 = 4*x);PointOnCurve(Focus(C), G) = True;Intersection(G, C) = {A, B};A: Point;B: Point", "coords": {} }, { "index": 7125, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7125.png", "answer": "81", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/9 + x^2/m = 1);m: Number;m>0;Expression(OneOf(Asymptote(C))) = (y = x/3)", "coords": {} }, { "index": 7126, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7127, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.449489742783178, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 6.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7127.png", "answer": "1/3", "fact_expr": "l: Line;C: Ellipse;P: Point;A: Point;B: Point;b:Number;Expression(C) = (x^2/6 + y^2/3 = 1);PointOnCurve(P,C);Coordinate(P) = (2, 1);Expression(l)=(y=-x+b);Intersection(l, C) = {A,B};IsPerpendicular(LineSegmentOf(P, A), LineSegmentOf(P, B));Negation(A=B)", "coords": { "P": [ 2.0, 1.0 ] } }, { "index": 7128, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7128.png", "answer": "4*sqrt(6)/3", "fact_expr": "G: Hyperbola;P: Point;Expression(G) = (x^2/4 - y^2/2 = 1);PointOnCurve(P, G);Distance(P, RightFocus(G)) = 2", "coords": {} }, { "index": 7129, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.1622776601683795, "b": 1.0, "major_axis": "x", "x_coef": 10.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7129.png", "answer": "48", "fact_expr": "l: Line;G: Ellipse;H:Circle;P: Point;A: Point;B: Point;Expression(G) = (x^2/10 + y^2 = 1);Expression(H) = (x^2 + (y - 6)^2 = 2);PointOnCurve(P, G);PointOnCurve(Center(H),l);Intersection(l, H) = {A, B}", "coords": {} }, { "index": 7130, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7130.png", "answer": "x^2/25+y^2/16=1", "fact_expr": "G: Ellipse;b: Number;a: Number;B: Point;A: Point;F: Point;C: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftVertex(G) = A;RightFocus(G) = F;UpperVertex(G) = B;LowerVertex(G) = C;T:Point;Coordinate(T)=(3*a,16);Intersection(LineOf(A,B),LineOf(C,F))=T", "coords": {} }, { "index": 7131, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7132, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7132.png", "answer": "sqrt(3)/2", "fact_expr": "Gamma: Ellipse;M: Point;A: Point;B: Point;C: Point;D: Point;a:Number;b:Number;a>b;b>0;Expression(Gamma) = (x^2/a^2+y^2/b^2=1);Endpoint(MajorAxis(Gamma))={A,B};Endpoint(MinorAxis(Gamma))={C,D};Abs(LineSegmentOf(M, A))/Abs(LineSegmentOf(M, B)) = 2;Max(Area(TriangleOf(M, A, B))) = 8;Min(Area(TriangleOf(M, C, D))) = 1", "coords": {} }, { "index": 7133, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7133.png", "answer": "3", "fact_expr": "G: Ellipse;Expression(G) = (x^2/b^2 + y^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;P: Point;PointOnCurve(P, G);F1: Point;F2: Point;Focus(G) = {F1, F2};Distance(P, F1) + Distance(P, F2) = 6", "coords": {} }, { "index": 7134, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7134.png", "answer": "pm*2*sqrt(2)", "fact_expr": "G: Hyperbola;H: Parabola;p: Number;I: Line;A: Point;F: Point;B: Point;k:Number;e:Number;Expression(G) = (x^2/4 - y^2/12 = 1);Expression(H) = (y^2 = 2*(p*x));Expression(I) = (y = k*(-p/2 + x));Eccentricity(G) = e;Focus(H)=F;Intersection(H, I) = {A, B};Abs(LineSegmentOf(A, F))/Abs(LineSegmentOf(F, B)) = e", "coords": {} }, { "index": 7135, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7136, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7136.png", "answer": "4*x^2/9 - y^2/4 = 1", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);G1: Hyperbola;Asymptote(G) = Asymptote(G1);H: Point;Coordinate(H) = (-3, 2*sqrt(3));PointOnCurve(H, G1) = True", "coords": { "H": [ -3.0, 3.4641016151377544 ] } }, { "index": 7137, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7137.png", "answer": "(1, 2)", "fact_expr": "P: Point;PointOnCurve(P, G);G: Parabola;Expression(G) = (y^2 = 4*x);A: Point;Coordinate(A) = (2, 2);F: Point;Focus(G) = F;WhenMin(Abs(LineSegmentOf(P, A)) + Abs(LineSegmentOf(P, F)))", "coords": { "A": [ 2.0, 2.0 ] } }, { "index": 7138, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7139, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7140, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7140.png", "answer": "sqrt(3)", "fact_expr": "C: Hyperbola;a: Number;F: Point;a>0;Expression(C) = (-y^2/3 + x^2/a^2 = 1);RightFocus(C) = F", "coords": {} }, { "index": 7141, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7141.png", "answer": "{sqrt(21)/3, sqrt(13)}", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;L1: Line;L2: Line;Asymptote(G) = {L1, L2};H: Parabola;Expression(H) = (y^2 = 4*x);O: Origin;A: Point;Intersection(L1, H) = A;B: Point;Intersection(L2, H) = B;Negation(O=A);Negation(O=B);F: Point;Focus(H) = F;AngleOf(A, F, B) = (2*pi)/3", "coords": {} }, { "index": 7142, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_7142.png", "answer": "2*sqrt(3)/3", "fact_expr": "C1: Hyperbola;Expression(C1) = (x^2/a1^2 - y^2/b1^2 = 1);a1: Number;b1: Number;a1>0;b1>0;C2: Hyperbola;Expression(C2) = (y^2/a2^2 - x^2/b2^2 = 1);a2: Number;b2: Number;a2>0;b2>0;Asymptote(C1) = Asymptote(C2);Eccentricity(C1) = 2", "coords": {} }, { "index": 7143, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7144, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.449489742783178, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7144.png", "answer": "6", "fact_expr": "G: Hyperbola;H: Parabola;p: Number;Expression(G) = (x^2/6 - y^2/3 = 1);Expression(H) = (y^2 = 2*(p*x));Focus(H) = RightFocus(G)", "coords": {} }, { "index": 7145, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7145.png", "answer": "3/2", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2/3 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) - Abs(LineSegmentOf(P, F2)) = 1", "coords": {} }, { "index": 7146, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7146.png", "answer": "pm*4/3", "fact_expr": "l: Line;G: Parabola;A: Point;F: Point;B: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B};VectorOf(A, F) = 4*VectorOf(F, B)", "coords": {} }, { "index": 7147, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "y", "x_coef": 3.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7147.png", "answer": "8", "fact_expr": "C: Ellipse;G: Line;F2: Point;M: Point;N: Point;F1: Point;Expression(C) = (x^2/3 + y^2/4 = 1);Focus(C) = {F1, F2};PointOnCurve(F1, G);Intersection(G, C) = {M,N}", "coords": {} }, { "index": 7148, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7148.png", "answer": "y=pm*(sqrt(3)/3)*x", "fact_expr": "G: Hyperbola;a: Number;H: Parabola;a>0;Expression(G) = (-y^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 8*x);RightFocus(G) = Focus(H)", "coords": {} }, { "index": 7149, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7149.png", "answer": "sqrt(3)", "fact_expr": "F1: Point;F2: Point;LeftFocus(E) = F1;RightFocus(E) = F2;E: Hyperbola;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;l: Line;PointOnCurve(F1, l);NumIntersection(l, E) = 1;O1: Circle;Expression(O1) = (x^2 + y^2 = 2*a^2);IsTangent(l,O1) = True", "coords": {} }, { "index": 7150, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7150.png", "answer": "a", "fact_expr": "P: Point;PointOnCurve(P, RightPart(G));G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;FocalLength(G) = 2*c;c: Number", "coords": {} }, { "index": 7151, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7152, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7152.png", "answer": "-1/2", "fact_expr": "G: Hyperbola;H: Point;a: Number;b: Number;Expression(G) = (x^2 - y^2 = 1);Coordinate(H) = (a, b);PointOnCurve(H, LeftPart(G));Expression(L) = (y = x);Distance(H, L) = sqrt(2);OneOf(Asymptote(G)) = L;L:Line", "coords": {} }, { "index": 7153, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7153.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;H: Parabola;Expression(H) = (y^2 = 4*x);OneOf(Focus(G)) = Focus(H);Distance(Focus(G), Asymptote(G)) = sqrt(3)/2", "coords": {} }, { "index": 7154, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7155, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7155.png", "answer": "21/10", "fact_expr": "l: Line;G: Parabola;p: Number;H:Line;A: Point;B: Point;F: Point;p>0;alpha:Number;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (x = -1/2);Coordinate(F) = (-Sin(alpha)*Cos(alpha), 0);Tan(alpha) = -2;Focus(G) = F;PointOnCurve(F, l);Intersection(l, G) = {A, B};Abs(LineSegmentOf(A, B)) = 4", "coords": {} }, { "index": 7156, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7156.png", "answer": "[4, +oo)", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/2 - y^2/lambda = 1);lambda: Number;Negation(Eccentricity(C) > sqrt(2*lambda - 5))", "coords": {} }, { "index": 7157, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7157.png", "answer": "7/9", "fact_expr": "K: Point;Coordinate(K) = (-1, 0);PointOnCurve(K, l) = True;l: Line;C: Parabola;Expression(C) = (y^2 = 4*x);Intersection(l, C) = {A, B};A: Point;B: Point;F: Point;Focus(C) = F;Abs(LineSegmentOf(B, F)) = 2*Abs(LineSegmentOf(A, F))", "coords": { "K": [ -1.0, 0.0 ] } }, { "index": 7158, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7159, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7159.png", "answer": "sqrt(3)-1", "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;F2: Point;P: Point;e: Number;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);AngleOf(P, F1, F2) = ApplyUnit(30, degree);AngleOf(P, F2, F1) = ApplyUnit(60, degree);Eccentricity(G) = e", "coords": {} }, { "index": 7160, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7160.png", "answer": "7", "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (6, 3);PointOnCurve(P, G);Focus(G) = F", "coords": { "A": [ 6.0, 3.0 ] } }, { "index": 7161, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7161.png", "answer": "3", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;Focus(C) = {F1, F2};P: Point;PointOnCurve(P, C);IsPerpendicular(VectorOf(P, F2), VectorOf(P, F1));Area(TriangleOf(P, F1, F2)) = 9", "coords": {} }, { "index": 7162, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7162.png", "answer": "2*sqrt(6)/3", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;F2: Point;F1: Point;LeftFocus(C) = F1;RightFocus(C) = F2;G: Line;PointOnCurve(F1, G);A: Point;B: Point;Intersection(G,LeftPart(C))=A;Intersection(G,RightPart(C))=B;Abs(LineSegmentOf(A, B)) = Abs(LineSegmentOf(A, F2));Cos(AngleOf(B, A, F2)) = 7/8", "coords": {} }, { "index": 7163, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 7.917515754699707, "b": 6.981808353473198, "major_axis": "y", "x_coef": 47.99999999999999, "y_coef": 64.0 }, "optimization": { "final_loss": 9.99869816637308e-15, "converged": true, "iterations": 2 }, "output_path": "results/train/ellipse/problem_7163.png", "answer": "y^2/64 + x^2/48 = 1", "fact_expr": "G: Ellipse;O: Origin;Center(G)=O;PointOnCurve(Focus(G), yAxis);Eccentricity(G)=1/2;FocalLength(G) = 8", "coords": {} }, { "index": 7164, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7164.png", "answer": "3", "fact_expr": "C: Parabola;A: Point;F: Point;B: Point;l: Line;Expression(C) = (x^2 = 4*y);Focus(C) = F;PointOnCurve(A, C);Intersection(OverlappingLine(LineSegmentOf(F, A)), xAxis) = B;Expression(l) = (y = -1);d: Number;Distance(A, l) = d;Distance(A, B) = d", "coords": {} }, { "index": 7165, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7165.png", "answer": "{1, 13}", "fact_expr": "C: Hyperbola;a: Number;P: Point;F1: Point;F2: Point;Expression(C) = (-y^2/4 + x^2/a^2 = 1);Expression(OneOf(Asymptote(C))) = (2*x + 3*y = 0);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) = 7", "coords": {} }, { "index": 7166, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "y", "x_coef": 9.0, "y_coef": 25.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7166.png", "answer": "16", "fact_expr": "G: Ellipse;P: Point;A: Point;B: Point;C:Curve;D:Curve;Expression(G) = (x^2/9 + y^2/25 = 1);PointOnCurve(P, G);PointOnCurve(A, C);PointOnCurve(B, D);Expression(C)=(x^2+(y-4)^2=16);Expression(D)=(x^2+(y+4)^2=4)", "coords": {} }, { "index": 7167, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7167.png", "answer": "sqrt(13)", "fact_expr": "F: Point;OneOf(Focus(C)) = F;C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;P: Point;PointOnCurve(P,C) = True;Coordinate(MidPoint(LineSegmentOf(P,F))) = (0,-sqrt(3)*b)", "coords": {} }, { "index": 7168, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7168.png", "answer": "2", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y)", "coords": {} }, { "index": 7169, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7169.png", "answer": "-9", "fact_expr": "G: Parabola;A: Point;P: Point;B: Point;Expression(G) = (y^2 = 2*x);Coordinate(A) = (-3, 0);Coordinate(B) = (3,0);PointOnCurve(P, G)", "coords": { "A": [ -3.0, 0.0 ], "B": [ 3.0, 0.0 ] } }, { "index": 7170, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7171, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7171.png", "answer": "sqrt(2)/2", "fact_expr": "H: Line;Expression(H) = (y = 1 - x);G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;A: Point;B: Point;Intersection(H, G) = {A, B};M: Point;MidPoint(LineSegmentOf(A, B)) = M;Z: Line;Expression(Z) = (x - 2*y = 0);PointOnCurve(M, Z)", "coords": {} }, { "index": 7172, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7172.png", "answer": "18", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G)", "coords": {} }, { "index": 7173, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7174, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7174.png", "answer": "b^2", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;M: Point;c:Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, G);Negation(P=Vertex(G));LeftFocus(G) = F1;RightFocus(G) = F2;HalfFocalLength(G)=c;TangentPoint(InscribedCircle(TriangleOf(P,F1,F2)),LineSegmentOf(F1,F2))=M", "coords": {} }, { "index": 7175, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7175.png", "answer": "[7, 13]", "fact_expr": "G: Ellipse;H: Circle;D: Circle;P: Point;M: Point;N: Point;Expression(G) = (x^2/25 + y^2/16 = 1);Expression(H) = (y^2 + (x + 3)^2 = 4);Expression(D) = (y^2 + (x - 3)^2 = 1);PointOnCurve(P, G);PointOnCurve(M, H);PointOnCurve(N, D)", "coords": {} }, { "index": 7176, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7176.png", "answer": "(0,-1/2)", "fact_expr": "G: Parabola;Expression(G) = (y = -x^2/2)", "coords": {} }, { "index": 7177, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2631799883090584, "b": 2.5275303393962245, "orientation": "horizontal" }, "optimization": { "final_loss": 4.481942266905428e-06, "converged": true, "iterations": 95 }, "output_path": "results/train/hyperbola/problem_7177.png", "answer": "x^2/1 - y^2/4 = 1", "fact_expr": "G: Hyperbola;Expression(Asymptote(G)) = (y = pm*2*x);H: Parabola;Expression(H) = (y^2 = 4*(sqrt(5)*x));OneOf(Focus(G)) = Focus(H)", "coords": {} }, { "index": 7178, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_7178.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;b: Number;a: Number;k:Number;a>0;b>0;k>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(OneOf(Asymptote(G)))=(y=k*x);Eccentricity(G)=2", "coords": {} }, { "index": 7179, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7180, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.3300262689590454, "b": 2.30381441116333, "orientation": "horizontal" }, "optimization": { "final_loss": 6.658357051492203e-06, "converged": true, "iterations": 122 }, "output_path": "results/train/hyperbola/problem_7180.png", "answer": "x^2 - y^2 / 3 = 1", "fact_expr": "H: Parabola;Expression(H) = (y^2 = 4*x);F: Point;Focus(H) = F;G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;RightVertex(G) = F;Expression(Asymptote(G)) = (y = pm*sqrt(3)*x)", "coords": {} }, { "index": 7181, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7181.png", "answer": "5/3", "fact_expr": "G: Hyperbola;n: Number;m: Number;H: Ellipse;Q: Point;F1: Point;F2: Point;m>0;n>0;Expression(G) = (-y^2/n^2 + x^2/m^2 = 1);Expression(H) = (x^2/25 + y^2/16 = 1);Focus(G)={F1,F2};Focus(H)={F1,F2};OneOf(Intersection(G,H))=Q;AngleOf(Q, F1, F2) = ApplyUnit(90, degree)", "coords": {} }, { "index": 7182, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7182.png", "answer": "10", "fact_expr": "H: Line;Expression(H) = (y = k*(x - 2));k: Number;G: Parabola;Expression(G) = (y^2 = 8*x);Intersection(H, G) = {A, B};B: Point;A: Point;XCoordinate(MidPoint(LineSegmentOf(A, B))) = 3", "coords": {} }, { "index": 7183, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7183.png", "answer": "6", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 8*y);P: Point;PointOnCurve(P, G);A: Point;Coordinate(A) = (-2, 4);F: Point;Focus(G) = F", "coords": { "A": [ -2.0, 4.0 ] } }, { "index": 7184, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7184.png", "answer": "sqrt(6)", "fact_expr": "G: Ellipse;Expression(G) = (y^2/2 + x^2/a^2 = 1);a: Number;a>0;H: Point;Coordinate(H) = (2, 0);OneOf(Focus(G)) = H", "coords": { "H": [ 2.0, 0.0 ] } }, { "index": 7185, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7186, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7187, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7187.png", "answer": "8", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;F2: Point;F1: Point;l: Line;PointOnCurve(F1,l) = True;Intersection(G,l) = {A,B};A: Point;B: Point", "coords": {} }, { "index": 7188, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7188.png", "answer": "7", "fact_expr": "A: Point;C: Parabola;Expression(C) = (y^2 = 12*x);PointOnCurve(A, C);Distance(A, Focus(C)) = 10", "coords": {} }, { "index": 7189, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 2.6968069696030055, "b": 1.9073494445037336, "major_axis": "x", "x_coef": 8.000000000000004, "y_coef": 3.999999999999999 }, "optimization": { "final_loss": 1.553898377657047e-05, "converged": true, "iterations": 84 }, "output_path": "results/train/ellipse/problem_7189.png", "answer": "sqrt(2)/2", "fact_expr": "C: Ellipse;G: Point;Coordinate(G) = (2, sqrt(2));PointOnCurve(G, C);H: Point;Coordinate(H) = (sqrt(2), sqrt(3));PointOnCurve(H, C)", "coords": { "G": [ 2.0, 1.4142135623730951 ], "H": [ 1.4142135623730951, 1.7320508075688772 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 7190, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7191, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7192, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7192.png", "answer": "(0,pm*sqrt(5))", "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/2 + y^2/3 = 1)", "coords": {} }, { "index": 7193, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -0.0, 2.0 ], "radius": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_7193.png", "answer": "27*sqrt(7)/56", "fact_expr": "Gamma:Parabola;C: Circle;O:Origin;Vertex(Gamma)=O;SymmetryAxis(Gamma)=xAxis;P:Point;Q:Point;Expression(C) = (-4*y + x^2 + y^2 = 0);Intersection(Gamma,C)={P,Q};Distance(P,Q)=3", "coords": {} }, { "index": 7194, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7194.png", "answer": "2", "fact_expr": "G: Parabola;p: Number;H: Circle;p>0;Expression(G) = (y^2 = 2*(p*x));Expression(H) = (-6*x + x^2 + y^2 - 7 = 0);IsTangent(Directrix(G),H)", "coords": {} }, { "index": 7195, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7195.png", "answer": "5", "fact_expr": "G: Parabola;H: Line;A: Point;F: Point;B: Point;C: Point;D: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};L1:Line;L2:Line;PointOnCurve(A,L1);PointOnCurve(B,L2);IsPerpendicular(L1,Directrix(G));IsPerpendicular(L2,Directrix(G));Abs(LineSegmentOf(A, F)) = 4*Abs(LineSegmentOf(B, F));FootPoint(L1,Directrix(G))=C;FootPoint(L2,Directrix(G))=D", "coords": {} }, { "index": 7196, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7196.png", "answer": "sqrt(6)/3", "fact_expr": "C: Ellipse;b: Number;a: Number;G: Circle;H: Line;A1: Point;A2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (2*a*b - a*y + b*x = 0);LeftVertex(C) = A1;RightVertex(C) = A2;IsDiameter(LineSegmentOf(A1,A2),G);IsTangent(H,G)", "coords": {} }, { "index": 7197, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7197.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;a>0;b>0;L1:Line;L2:Line;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);OneOf(Asymptote(G))=L1;OneOf(Asymptote(G))=L2;Negation(L1=L2);PointOnCurve(Center(H),L1);IsTangent(H,xAxis);IsTangent(H,L2)", "coords": {} }, { "index": 7198, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7198.png", "answer": "x + pm*sqrt(3)*y = 0", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;F:Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(l)=(x+3*y-2*b=0);RightFocus(C)=F;PointOnCurve(F,l)", "coords": {} }, { "index": 7199, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7199.png", "answer": "16", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);PointOnCurve(Focus(G),H) = True;Inclination(H) = pi/6;H: Line;A: Point;B: Point;Intersection(H, G) = {A, B}", "coords": {} }, { "index": 7200, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7201, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7201.png", "answer": "[80, 120]", "fact_expr": "G: Ellipse;H: Circle;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/4 + y^2/3 = 1);Expression(H) = ((x - 6)^2 + (y - 8)^2 = 1);PointOnCurve(P, H);Focus(G) = {F1, F2}", "coords": {} }, { "index": 7202, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5002657771110535, "direction": "right" }, "optimization": { "final_loss": 8.410525879298802e-06, "converged": true, "iterations": 102 }, "output_path": "results/train/parabola/problem_7202.png", "answer": "3*x + 6*y + 4 = 0", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;H: Circle;Expression(H) = (y^2 + (x - 2)^2 = 1);A: Point;Coordinate(A) = (2, 2);B: Point;C: Point;PointOnCurve(A,G);PointOnCurve(B,G);PointOnCurve(C,G)\t;IsTangent(LineOf(A,B),H);IsTangent(LineOf(A,C),H)", "coords": { "A": [ 2.0, 2.0 ] } }, { "index": 7203, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 2.8284271247461903, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7203.png", "answer": "12", "fact_expr": "G: Hyperbola;M: Point;F1: Point;F2: Point;Expression(G) = (x^2/16 - y^2/8 = 1);PointOnCurve(M, G);LeftFocus(G) = F1;RightFocus(G) = F2;Abs(LineSegmentOf(M, F1)) = 4", "coords": {} }, { "index": 7204, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7204.png", "answer": "-6", "fact_expr": "G: Hyperbola;m: Real;Expression(G) = (x^2/m + y^2/2 = 1);Eccentricity(G) = 2", "coords": {} }, { "index": 7205, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7205.png", "answer": "Ellipse", "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;M: Point;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(M, G);LeftFocus(G) = F1;MidPoint(LineSegmentOf(M,F1))=P;a>b;b>0;P:Point", "coords": {} }, { "index": 7206, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7207, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7207.png", "answer": "sqrt(2)/2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;A: Point;B: Point;PointOnCurve(A, G);PointOnCurve(B, G);AngleOf(A, F, B) = pi/2;IsChordOf(LineSegmentOf(A, B), G);M: Point;MidPoint(LineSegmentOf(A, B)) = M;N: Point;Projection(M, Directrix(G)) = N", "coords": {} }, { "index": 7208, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 2.1144830911547814, "b": 1.8307220340480923, "orientation": "horizontal" }, "optimization": { "final_loss": 2.1389768270834375e-06, "converged": true, "iterations": 96 }, "output_path": "results/train/hyperbola/problem_7208.png", "answer": "8", "fact_expr": "C: Hyperbola;G: Point;A: Point;P: Point;F: Point;PointOnCurve(G,C);Coordinate(G) = (2*sqrt(2), sqrt(3));Coordinate(A) = (0, 3);PointOnCurve(P,RightPart(C));LeftFocus(C) = F;Expression(Asymptote(C))=(y=pm*(sqrt(3)/2)*x)", "coords": { "G": [ 2.8284271247461903, 1.7320508075688772 ], "A": [ 0.0, 3.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 7209, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7209.png", "answer": "2*sqrt(15)", "fact_expr": "G: Parabola;H: Line;k: Number;A: Point;B: Point;Expression(G) = (y^2 = 8*x);Expression(H) = (y = k*x - 2);Intersection(H, G) = {A, B};XCoordinate(MidPoint(LineSegmentOf(A, B))) = 2", "coords": {} }, { "index": 7210, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 0.7071067811865476, "major_axis": "y", "x_coef": 0.5, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7210.png", "answer": "8", "fact_expr": "G: Ellipse;k: Real;A:Point;B:Point;Expression(G) = (4*x^2 + y^2 = 2);PointOnCurve(A,G);PointOnCurve(B,G);Max(Distance(A,B))=8", "coords": {} }, { "index": 7211, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7212, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7213, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7214, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7214.png", "answer": "(0,1/8)", "fact_expr": "P: Point;G: Hyperbola;Expression(G) = (x^2/4 - y^2 = 1);PointOnCurve(P, RightPart(G));Quadrant(P) = 1;A1: Point;A2: Point;LeftVertex(G) = A1;RightVertex(G) = A2;O: Origin;k1: Number;k2: Number;k3: Number;Slope(LineOf(P, A1)) = k1;Slope(LineOf(P, O)) = k2;Slope(LineOf(P, A2)) = k3", "coords": {} }, { "index": 7215, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7215.png", "answer": "(0,-1)", "fact_expr": "C: Ellipse;P: Point;Q: Point;F: Point;Expression(C) = (x^2/2 + y^2 = 1);Coordinate(Q) = (4, 3);LeftFocus(C) = F;PointOnCurve(P, C);WhenMax(Abs(LineSegmentOf(P, F)) + Abs(LineSegmentOf(P, Q)))", "coords": { "Q": [ 4.0, 3.0 ] } }, { "index": 7216, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_7216.png", "answer": "4", "fact_expr": "C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(C) = 2;Distance(Focus(C),Asymptote(C))=sqrt(3)", "coords": {} }, { "index": 7217, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7218, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7218.png", "answer": "2*sqrt(2)", "fact_expr": "G: Parabola;H: Line;I: Curve;P: Point;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 4*x);Expression(I) = And((y^2=-4*x), (y>=0));Focus(G) = F;PointOnCurve(F, H);Slope(H) = 1;Intersection(H, G) = {A, B};PointOnCurve(P, I)", "coords": {} }, { "index": 7219, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7219.png", "answer": "[2*pi, pi*10/3)", "fact_expr": "G: Hyperbola;I: Line;A: Point;F1: Point;B: Point;F2: Point;Expression(G) = (x^2 - y^2/3 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2, I);Intersection(I, RightPart(G)) = {A, B}", "coords": {} }, { "index": 7220, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 8.0, "b": 6.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7220.png", "answer": "16", "fact_expr": "G: Hyperbola;P: Point;Expression(G) = (x^2/64 - y^2/36 = 1);PointOnCurve(P, G);Distance(P, RightFocus(G)) = 4", "coords": {} }, { "index": 7221, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7221.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;Distance(RightVertex(G), OneOf(Asymptote(G))) = b/2", "coords": {} }, { "index": 7222, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7223, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7223.png", "answer": "sqrt(3)", "fact_expr": "F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;Endpoint(MinorAxis(C)) = M;M: Point;PointOnCurve(N,C) = True;N: Point;VectorOf(M, F2) = 3*VectorOf(F2, N);Area(TriangleOf(M, N, F1)) = 2", "coords": {} }, { "index": 7224, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 3.0, "major_axis": "x", "x_coef": 36.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7224.png", "answer": "x+2*y-8=0", "fact_expr": "l: Line;G: Ellipse;A: Point;B: Point;M: Point;Expression(G) = (x^2 + 4*y^2 = 36);InterceptChord(l,G)=LineSegmentOf(A,B);MidPoint(LineSegmentOf(A,B))=M;Coordinate(M) = (4, 2)", "coords": { "M": [ 4.0, 2.0 ] } }, { "index": 7225, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7226, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7226.png", "answer": "3*sqrt(3)", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);AngleOf(F1, P, F2)=ApplyUnit(60,degree)", "coords": {} }, { "index": 7227, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7228, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.560132719491411, "b": 1.280571558186289, "orientation": "horizontal" }, "optimization": { "final_loss": 5.001754676787936e-06, "converged": true, "iterations": 73 }, "output_path": "results/train/hyperbola/problem_7228.png", "answer": "x^2/10 - y^2/(5/2) = 1", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (3*x^2 + 13*y^2 = 39);Focus(H)=Vertex(G);Expression(Asymptote(G)) = (y=pm*(1/2)*x)", "coords": {} }, { "index": 7229, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7229.png", "answer": "y^2 = 4*x", "fact_expr": "C: Parabola;p: Number;Z: Circle;H: Line;M: Point;E: Point;G: Point;F: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Expression(H) = (x = p/2);Coordinate(M) = (x0, 2*sqrt(2));x0:Number;x0>p/2;Focus(C) = F;PointOnCurve(M, C);Center(Z)=M;Radius(Z)=sqrt(3);Intersection(Z,H)={E,G};Abs(LineSegmentOf(E, G)) = 2*sqrt(2)", "coords": {} }, { "index": 7230, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7230.png", "answer": "(-3,0);(3,0)", "fact_expr": "G: Ellipse;F1: Point;F2: Point;Expression(G) = (x^2/25 + y^2/16 = 1);LeftFocus(G) = F1;RightFocus(G) = F2", "coords": {} }, { "index": 7231, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7231.png", "answer": "2*sqrt(3)/3", "fact_expr": "G: Hyperbola;M: Point;F1: Point;F2: Point;Expression(G) = (x^2 - y^2/2 = 1);Focus(G) = {F1, F2};PointOnCurve(M, G);DotProduct(VectorOf(M, F1), VectorOf(M, F2)) = 0", "coords": {} }, { "index": 7232, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7232.png", "answer": "4/3", "fact_expr": "l: Line;E: Ellipse;b: Number;A: Point;B: Point;F2: Point;F1: Point;0 < b;b < 1;Expression(E) = (x^2 + y^2/b^2 = 1);LeftFocus(E) = F1;RightFocus(E) = F2;PointOnCurve(F1, l);Intersection(l, E) = {A, B};2*Abs(LineSegmentOf(A, B)) = Abs(LineSegmentOf(A, F2)) + Abs(LineSegmentOf(B, F2))", "coords": {} }, { "index": 7233, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7233.png", "answer": "44", "fact_expr": "P: Point;Q: Point;C: Hyperbola;A: Point;F: Point;Expression(C) = (x^2/9 - y^2/16 = 1);Coordinate(A) = (5, 0);LeftFocus(C) = F;PointOnCurve(P, C);PointOnCurve(Q, C);2*Length(ImageinaryAxis(C)) = Length(LineSegmentOf(P, Q));PointOnCurve(A, LineSegmentOf(P, Q))", "coords": { "A": [ 5.0, 0.0 ] } }, { "index": 7234, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7234.png", "answer": "y=pm*(sqrt(5)/2)*x", "fact_expr": "C: Hyperbola;a: Number;a>0;Expression(C) = (-y^2/5 + x^2/a^2 = 1);Coordinate(OneOf(Focus(C))) = (3, 0)", "coords": {} }, { "index": 7235, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7235.png", "answer": "2*x+4*y-3=0", "fact_expr": "G: Ellipse;H: LineSegment;P: Point;Coordinate(P) = (1/2, 1/2);Expression(G) = (x^2/2 + y^2 = 1);IsChordOf(H,G);MidPoint(H)=P;PointOnCurve(P,H)", "coords": { "P": [ 0.5, 0.5 ] } }, { "index": 7236, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7236.png", "answer": "8", "fact_expr": "C: Ellipse;Expression(C) = (x^2/25 + y^2/9 = 1)", "coords": {} }, { "index": 7237, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7237.png", "answer": "6", "fact_expr": "G: Parabola;H: Ellipse;a: Number;P: Point;F1: Point;F2: Point;F3: Point;a>0;Expression(H) = (y^2/3 + x^2/a^2 = 1);Focus(H)={F1,F2};PointOnCurve(P,H);PointOnCurve(P,G);Focus(G) = F3;Abs(LineSegmentOf(P, F3)) = 25/16;Expression(G)=(y^2=(9/4)*x)", "coords": {} }, { "index": 7238, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7239, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7239.png", "answer": "4*sqrt(5)/5", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/4 - y^2 = 1);E: Circle;Expression(E) = (-2*x + x^2 + y^2 = 0);A: Point;B: Point;Intersection(OneOf(Asymptote(C)), E) = {A, B}", "coords": {} }, { "index": 7240, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7240.png", "answer": "6", "fact_expr": "C: Hyperbola;A: Point;P: Point;F: Point;Expression(C) = (x^2 - y^2 = 1);Coordinate(A) = (0, sqrt(2));RightFocus(C) = F;PointOnCurve(P, LeftPart(C))", "coords": { "A": [ 0.0, 1.4142135623730951 ] } }, { "index": 7241, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7242, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7242.png", "answer": "2", "fact_expr": "G: Hyperbola;m: Number;n: Number;H: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;m>0;n>0;Expression(G) = (-y^2/n^2 + x^2/m^2 = 1);a > b;b > 0;Expression(H) = (y^2/b^2 + x^2/a^2 = 1);Focus(H) = {F1, F2};Focus(G) = {F1, F2};OneOf(Intersection(H,G))=P;Abs(LineSegmentOf(P, F1))*Abs(LineSegmentOf(P, F2)) = 2", "coords": {} }, { "index": 7243, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7243.png", "answer": "sqrt(5)/3", "fact_expr": "l: Line;C: Ellipse;b: Number;a: Number;A: Point;F: Point;B: Point;a > b;b > 0;Expression(C) = (x^2/b^2 + y^2/a^2 = 1);Focus(C) = F;PointOnCurve(F, l);Slope(l) = 1/2;Intersection(l, C) = {A, B};VectorOf(A, F) = (2/3)*VectorOf(A, B)", "coords": {} }, { "index": 7244, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7244.png", "answer": "x+y+3=0", "fact_expr": "G: Parabola;H: Line;C:Line;Expression(G) = (x^2 = 12*y);Expression(H) = (x - y = 0);IsTangent(C,G);IsPerpendicular(C,H)", "coords": {} }, { "index": 7245, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7245.png", "answer": "sqrt(3)/2\nx^2/9-y^2/3=1", "fact_expr": "H: Ellipse;Expression(H) = (x^2 + 4*y^2 = 16);G: Hyperbola;Focus(G) = Focus(H);Expression(OneOf(Asymptote(G))) = (x + sqrt(3)*y = 0)", "coords": {} }, { "index": 7246, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.6495774984359741, "b": 1.117531657218933, "orientation": "horizontal" }, "optimization": { "final_loss": 9.364227298647165e-05, "converged": true, "iterations": 59 }, "output_path": "results/train/hyperbola/problem_7246.png", "answer": "sqrt(2)", "fact_expr": "C: Hyperbola;b: Number;a: Number;P: Point;F2: Point;F1: Point;M: Point;Q: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-2, 0);Coordinate(F2) = (2, 0);LeftFocus(C) = F1;RightFocus(C)=F2;PointOnCurve(M, RightPart(C));Intersection(LineSegmentOf(M, F1), yAxis) = P;TangentPoint(InscribedCircle(TriangleOf(M,P,F2)),LineSegmentOf(P,F2))=Q;Abs(LineSegmentOf(P, Q)) = sqrt(2)", "coords": { "F1": [ -2.0, 0.0 ], "F2": [ 2.0, 0.0 ] } }, { "index": 7247, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7248, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7249, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7249.png", "answer": "x \\pm y + 1 = 0", "fact_expr": "C: Parabola;P: Point;L: Line;Expression(C) = (y^2 = 4*x);Intersection(Directrix(C),SymmetryAxis(C))=P;TangentOfPoint(P, C) = L", "coords": {} }, { "index": 7250, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7251, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7251.png", "answer": "-1", "fact_expr": "G: Parabola;a: Real;l:Line;Expression(G) = (y^2 = 4*x);Expression(l) = (a*x - y + 1 = 0);PointOnCurve(Focus(G), l)", "coords": {} }, { "index": 7252, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7252.png", "answer": "sqrt(6)/3", "fact_expr": "H: Ellipse;Expression(H) = (y^2/2 + x^2/a^2 = 1);a: Number;G: Parabola;Expression(G) = (y^2 = 8*x);OneOf(Focus(H)) = Focus(G)", "coords": {} }, { "index": 7253, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7253.png", "answer": "y=pm*(sqrt(2)/2)*x", "fact_expr": "G: Hyperbola;H: Ellipse;a1: Number;b1: Number;b1 > 0;a1 > b1;a2: Number;b2: Number;a2 > 0;b2 > 0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2/b^2 + x^2/a^2 = 3);Focus(G) = Focus(H)", "coords": {} }, { "index": 7254, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7254.png", "answer": "sqrt(34)", "fact_expr": "G: Parabola;A: Point;F: Point;P: Point;Coordinate(A) = (5, 4);Expression(G) = (x^2 = 4*y);Focus(G) = F;PointOnCurve(P, G)", "coords": { "A": [ 5.0, 4.0 ] } }, { "index": 7255, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7255.png", "answer": "1", "fact_expr": "C: Ellipse;M: Point;P: Point;F1: Point;F2: Point;Expression(C) = (x^2/4 + y^2/3 = 1);Coordinate(M) = (4, 4);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C)", "coords": { "M": [ 4.0, 4.0 ] } }, { "index": 7256, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7256.png", "answer": "{y^2 = 4*x, y^2 = 16*x}", "fact_expr": "G: Parabola;p: Number;H: Line;Q: Point;F: Point;M: Point;P: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;PointOnCurve(P, G);PointOnCurve(P, H);IsPerpendicular(Directrix(G), H);FootPoint(Directrix(G), H) = Q;Coordinate(M) = (0, 2);Intersection(LineOf(F, Q), yAxis) = M;Area(TriangleOf(P, Q, F)) = 10", "coords": { "M": [ 0.0, 2.0 ] } }, { "index": 7257, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7257.png", "answer": "1/2", "fact_expr": "G: Ellipse;b: Number;a: Number;F1: Point;F2: Point;P: Point;e: Number;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-1, 0);Coordinate(F2) = (1, 0);Focus(G) = {F1, F2};PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) + Abs(LineSegmentOf(P, F2)) = 4;Eccentricity(G) = e", "coords": { "F1": [ -1.0, 0.0 ], "F2": [ 1.0, 0.0 ] } }, { "index": 7258, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7259, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7259.png", "answer": "(2*sqrt(5))/5", "fact_expr": "H: Line;Expression(H) = (a*x + y - 4 = 0);a: Number;G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;Coordinate(OneOf(Intersection(H, G))) = (1, 2)", "coords": {} }, { "index": 7260, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7260.png", "answer": "7/4", "fact_expr": "G: Parabola;M: Point;N: Point;P1: Point;P2: Point;P: Point;Expression(G) = (y^2 = x);Length(LineSegmentOf(M, N)) = 4;Endpoint(LineSegmentOf(M, N)) = {P1, P2};PointOnCurve(P1, G);PointOnCurve(P2, G);MidPoint(LineSegmentOf(M, N)) = P", "coords": {} }, { "index": 7261, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.8284271247461903, "major_axis": "x", "x_coef": 9.0, "y_coef": 8.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7261.png", "answer": "[1/2, 2]", "fact_expr": "l: Line;C: Ellipse;A: Point;F: Point;B: Point;Expression(C) = (x^2/9 + y^2/8 = 1);RightFocus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B}", "coords": {} }, { "index": 7262, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7263, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7264, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7265, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7266, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7266.png", "answer": "pm*2*sqrt(2)", "fact_expr": "G: Parabola;O: Origin;Vertex(G) = O;PointOnCurve(Focus(G), xAxis);P: Point;a: Number;Coordinate(P) = (-2, a);PointOnCurve(P, G);Distance(P, Focus(G)) = 3", "coords": {} }, { "index": 7267, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7268, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7268.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/5 = 1);F: Point;OneOf(Focus(G)) = F", "coords": {} }, { "index": 7269, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7269.png", "answer": "1/4", "fact_expr": "C: Hyperbola;P: Point;F1: Point;F2: Point;Expression(C) = (x^2 - y^2/3 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P, C);Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 7270, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7270.png", "answer": "x^2/2-y^2=1", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/4 + y^2 = 1);Focus(G) = Focus(H);P: Point;Coordinate(P) = (2, 1);PointOnCurve(P, G) = True", "coords": { "P": [ 2.0, 1.0 ] } }, { "index": 7271, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7272, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7272.png", "answer": "5+sqrt(13)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);F: Point;Focus(G) = F;A: Point;Coordinate(A) = (4,2);P: Point;PointOnCurve(P,G) =True;Negation(PointOnCurve(P,LineOf(A,F)))", "coords": { "A": [ 4.0, 2.0 ] } }, { "index": 7273, "success": false, "error": "Circle from diameter: missing point coordinates" }, { "index": 7274, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 7.0, "b": 4.898979485566356, "major_axis": "x", "x_coef": 49.0, "y_coef": 24.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7274.png", "answer": "x^2/9 - y^2/16 = 1", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/49 + y^2/24 = 1);Focus(G)=Focus(H);Expression(Asymptote(G))=(y=(pm*4/3)*x)", "coords": {} }, { "index": 7275, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7276, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 8.0, "b": 6.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7276.png", "answer": "16", "fact_expr": "G: Hyperbola;P: Point;Expression(G) = (x^2/64 - y^2/36 = 1);PointOnCurve(P, G);Distance(P, RightFocus(G)) = 4", "coords": {} }, { "index": 7277, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7277.png", "answer": "2*sqrt(2)", "fact_expr": "G: Ellipse;H: Line;P: Point;Expression(G) = (x^2 + y^2/3 = 1);Expression(H) = (x + y - 4 = 0);PointOnCurve(P, G)", "coords": {} }, { "index": 7278, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7279, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7279.png", "answer": "sqrt(3)/3", "fact_expr": "F1: Point;F2: Point;C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C);PointOnCurve(MidPoint(LineSegmentOf(P, F1)), yAxis);AngleOf(P, F1, F2) = ApplyUnit(30, degree)", "coords": {} }, { "index": 7280, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7281, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7281.png", "answer": "2*sqrt(7)-2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/3 = 1);P: Point;PointOnCurve(P, RightPart(G));F: Point;RightFocus(G) = F;A: Point;Coordinate(A) = (3, sqrt(3))", "coords": { "A": [ 3.0, 1.7320508075688772 ] } }, { "index": 7282, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7283, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7283.png", "answer": "sqrt(6)/3", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Circle;F1: Point;F2: Point;P: Point;c:Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);Focus(G)={F1,F2};IsDiameter(Abs(LineSegmentOf(F1,F2)),H);OneOf(Intersection(H,G))=P;AngleOf(P, F1, F2) = 5*AngleOf(P, F2, F1)", "coords": {} }, { "index": 7284, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7284.png", "answer": "[\\sqrt{2},2]", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;a>0;b>0;L1:Line;L2:Line;d1:Number;d2:Number;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, G);Asymptote(G)={L1,L2};Distance(P,L1)=d1;Distance(P,L2)=d2;3*a^2+b^2<=8*d1*d2", "coords": {} }, { "index": 7285, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7285.png", "answer": "2*sqrt(17) - 2", "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2/3 = 1);A: Point;RightVertex(C) = A;M: Point;Coordinate(M) = (2, 4);P: Point;PointOnCurve(P, C) ;L: Line;PointOnCurve(P, L) ;IsPerpendicular(L, LineOf(M, A)) ;FootPoint(L, LineOf(M, A)) = H;H: Point", "coords": { "M": [ 2.0, 4.0 ] } }, { "index": 7286, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7286.png", "answer": "y=pm*(5/4)*x", "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/16 + y^2/25 = 1)", "coords": {} }, { "index": 7287, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7287.png", "answer": "(0, 1/2]", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, C) = True;Coordinate(P) = (x0, y0);x0: Number;y0: Number;x0 >= 0;AngleOf(P, F1, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 7288, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7289, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7289.png", "answer": "sqrt(5)/10", "fact_expr": "m: Number;m > 0;M: Hyperbola;Expression(M) = (x^2/4 - y^2 = 1);N: Circle;Expression(N) = (x^2 + (-m + y)^2 = 5);IsTangent(M, N);A: Point;Coordinate(A) = (-sqrt(5), 0);B: Point;Coordinate(B) = (sqrt(5), 0);P: Point;PointOnCurve(P, N);Abs(LineSegmentOf(P, A)) - Abs(LineSegmentOf(P, B)) = 4", "coords": { "A": [ -2.23606797749979, 0.0 ], "B": [ 2.23606797749979, 0.0 ] } }, { "index": 7290, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7290.png", "answer": "44", "fact_expr": "C: Hyperbola;P: Point;Q: Point;A: Point;F: Point;Expression(C) = (x^2/9 - y^2/16 = 1);Coordinate(A) = (5, 0);LeftFocus(C) = F;PointOnCurve(P, C);PointOnCurve(Q, C);Length(LineSegmentOf(P,Q))=2*Length(ImageinaryAxis(C));PointOnCurve(A,LineSegmentOf(P,Q))", "coords": { "A": [ 5.0, 0.0 ] } }, { "index": 7291, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7291.png", "answer": "(1,3]", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;Focus(G) = {F1,F2};P: Point;PointOnCurve(P,RightPart(G)) = True;Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 7292, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7293, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7293.png", "answer": "sqrt(6)/3", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;A: Point;M: Point;B: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftVertex(G)=A;Slope(H) = 1;Intersection(H,G) = {A, M};Intersection(H,yAxis)=B;Abs(LineSegmentOf(A, M)) = Abs(LineSegmentOf(M, B))", "coords": {} }, { "index": 7294, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7295, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7295.png", "answer": "y^2=8*x", "fact_expr": "l: Line;Slope(l) = 2;G: Parabola;Expression(G) = (y^2 = a*x);a: Number;a>0;F: Point;Focus(G) = F;PointOnCurve(F, l);A: Point;Intersection(l, yAxis) = A;O: Origin;Area(TriangleOf(O, A, F)) = 4", "coords": {} }, { "index": 7296, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7297, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7298, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.8284271247461903, "major_axis": "x", "x_coef": 9.0, "y_coef": 8.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7298.png", "answer": "[3,15]", "fact_expr": "G: Ellipse;T: Circle;A: Point;B: Point;P: Point;Expression(G) = (x^2/9 + y^2/8 = 1);Expression(T) = (y^2 + (x + 1)^2 = 1);PointOnCurve(P, G);IsDiameter(LineSegmentOf(A, B), T)", "coords": {} }, { "index": 7299, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7299.png", "answer": "4*x+3*y-7=0", "fact_expr": "l: Line;G: Ellipse;A: Point;B: Point;M: Point;Expression(G) = (x^2/4 + y^2/3 = 1);Coordinate(M) = (1, 1);PointOnCurve(M, l);MidPoint(LineSegmentOf(A, B)) = M;Intersection(l,G)={A,B}", "coords": { "M": [ 1.0, 1.0 ] } }, { "index": 7300, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7300.png", "answer": "sqrt(3)", "fact_expr": "l: Line;C: Ellipse;P: Point;Q: Point;F: Point;O: Origin;Expression(C) = (x^2/4 + y^2/3 = 1);RightFocus(C)=F;PointOnCurve(O, l);Negation(Slope(l) = 0);Intersection(l, C) = {P, Q};Negation(P=Q)", "coords": {} }, { "index": 7301, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7301.png", "answer": "y=pm*sqrt(2)*x", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;O: Origin;B: Point;C: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = b);Intersection(H, LeftPart(G))=B;Intersection(H,RightPart(G))=C;IsPerpendicular(LineSegmentOf(O,B),LineSegmentOf(O,C))", "coords": {} }, { "index": 7302, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7303, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7304, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.6457513110645907, "major_axis": "x", "x_coef": 16.0, "y_coef": 7.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7304.png", "answer": "7", "fact_expr": "G: Ellipse;F1: Point;P: Point;F2: Point;O: Origin;Expression(G) = (x^2/16 + y^2/7 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);Abs(LineSegmentOf(O, P)) = 3", "coords": {} }, { "index": 7305, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 7.0, "b": 5.744562646538029, "major_axis": "x", "x_coef": 49.0, "y_coef": 33.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7305.png", "answer": "14+sqrt(41)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/49 + y^2/33 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);M: Point;Coordinate(M) = (8, 5)", "coords": { "M": [ 8.0, 5.0 ] } }, { "index": 7306, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7306.png", "answer": "16", "fact_expr": "l: Line;G: Parabola;A: Point;M: Point;B: Point;Expression(G) = (y^2 = 4*x);Focus(G)=F;PointOnCurve(F, l);Intersection(l, G) = {A, B};Intersection(Directrix(G), xAxis) = M;Tan(AngleOf(A, M, B)) = 4/3;F:Point", "coords": {} }, { "index": 7307, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7307.png", "answer": "3", "fact_expr": "P: Point;PointOnCurve(P, G);G: Parabola;Expression(G) = (y^2 = 4*x);Q: Point;Coordinate(Q) = (2, -1)", "coords": { "Q": [ 2.0, -1.0 ] } }, { "index": 7308, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7308.png", "answer": "sqrt(3)/2", "fact_expr": "G: Ellipse;Expression(G) = (x^2 + 4*y^2 = 4)", "coords": {} }, { "index": 7309, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7309.png", "answer": "sqrt(3)+1", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;F: Point;LeftFocus(C) = F;G: Line;PointOnCurve(F, G) = True;Inclination(G) = ApplyUnit(30, degree);A: Point;Intersection(G,C) = A;Quadrant(A) = 1;O: Origin;Abs(LineSegmentOf(O,F)) = Abs(LineSegmentOf(O,A))", "coords": {} }, { "index": 7310, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7310.png", "answer": "(3,pm*2*sqrt(3))", "fact_expr": "G: Parabola;P: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(P, G);Distance(P, Focus(G)) = 4", "coords": {} }, { "index": 7311, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0010501146316528, "direction": "right" }, "optimization": { "final_loss": 3.455598402979376e-07, "converged": true, "iterations": 81 }, "output_path": "results/train/parabola/problem_7311.png", "answer": "sqrt(3)/2", "fact_expr": "l: Line;E: Parabola;p: Number;P: Point;M: Point;A: Point;B: Point;p>0;Expression(E) = (y^2 = 2*(p*x));Coordinate(P) = (1, 2);Coordinate(M) = (1, 0);PointOnCurve(P, E);PointOnCurve(M, l);Intersection(l, E) = {A, B};VectorOf(A, M) = 3*VectorOf(M, B)", "coords": { "P": [ 1.0, 2.0 ], "M": [ 1.0, 0.0 ] } }, { "index": 7312, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7313, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7313.png", "answer": "pi/6", "fact_expr": "G: Parabola;N: Point;F: Point;M: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;Intersection(Directrix(G),xAxis)=M;PointOnCurve(N,G);Abs(LineSegmentOf(N, F)) = (sqrt(3)/2)*Abs(LineSegmentOf(M, N))", "coords": {} }, { "index": 7314, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7315, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7316, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 3.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7316.png", "answer": "2", "fact_expr": "C: Curve;Expression(C) = (x^2/3 + y^2/2 = 1)", "coords": {} }, { "index": 7317, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7317.png", "answer": "x^2 - y^2 / 3 = 1", "fact_expr": "l: Line;C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(l) = (x+sqrt(3)*y=0);IsPerpendicular(OneOf(Asymptote(C)),l);Distance(OneOf(Focus(C)), l) = 1", "coords": {} }, { "index": 7318, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7318.png", "answer": "4/3", "fact_expr": "C: Parabola;p: Number;G: Line;A: Point;F: Point;B: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Focus(C) = F;PointOnCurve(F, G);Intersection(G, C) = {A, B};Abs(LineSegmentOf(A, F)) = 2*Abs(LineSegmentOf(B, F));2*Abs(LineSegmentOf(B, F)) = 2", "coords": {} }, { "index": 7319, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7319.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, G);Focus(G) ={F1,F2};IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2));Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 7320, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7320.png", "answer": "5*sqrt(2)/4", "fact_expr": "G: Parabola;H: Line;P: Point;Expression(G) = (y^2 = 10*x);Expression(H) = (x + y + 5 = 0);PointOnCurve(P, G)", "coords": {} }, { "index": 7321, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7321.png", "answer": "(1, 2)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2 + (y + 2)^2 = 1);NumIntersection(Asymptote(G),H)=0", "coords": {} }, { "index": 7322, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7322.png", "answer": "7/2", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;M: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (x1,y1);Coordinate(B) = (x2,y2);PointOnCurve(Focus(G), H);Intersection(H, G) = {A,B};Abs(LineSegmentOf(A, B)) = 7;MidPoint(LineSegmentOf(A, B)) = M;x1:Number;x2:Number;y1:Number;y2:Number", "coords": {} }, { "index": 7323, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7323.png", "answer": "(6,+oo)", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 8*y);A: Point;Intersection(Directrix(G), yAxis) = A;B: Point;PointOnCurve(B, SymmetryAxis(G));H: Line;PointOnCurve(A, H);M: Point;N: Point;Intersection(H, G) = {M, N};DotProduct(VectorOf(B, M), VectorOf(M, N)) = (-1/2)*VectorOf(M, N)^2;O: Origin", "coords": {} }, { "index": 7324, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7325, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7326, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7326.png", "answer": "x = -4", "fact_expr": "G: Parabola;p: Number;M: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(M) = (1, m);m:Number;PointOnCurve(M, G);Distance(M, Focus(G)) = 5", "coords": {} }, { "index": 7327, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7327.png", "answer": "(0,-2)", "fact_expr": "C: Parabola;Expression(C) = (y = -x^2/8)", "coords": {} }, { "index": 7328, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7328.png", "answer": "sqrt(6)/6", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 1);P: Point;a: Number;b: Number;Coordinate(P) = (a, b);PointOnCurve(P, RightPart(G));H: Line;Expression(H) = (y = x);Distance(P, H) = sqrt(3)", "coords": {} }, { "index": 7329, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7329.png", "answer": "3", "fact_expr": "l: Line;C: Parabola;P: Point;Q: Point;F: Point;Expression(C) = (y^2 = 8*x);Focus(C) = F;PointOnCurve(F,l);Intersection(l, C) = {P,Q};Abs(LineSegmentOf(P, Q)) = 32/3;Abs(LineSegmentOf(P, F)) > Abs(LineSegmentOf(Q, F))", "coords": {} }, { "index": 7330, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7331, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7331.png", "answer": "y^2=12*x", "fact_expr": "G: Parabola;H: Ellipse;Expression(H) = (x^2/25 + y^2/16 = 1);Vertex(G) = Center(H);Focus(G) = RightFocus(H)", "coords": {} }, { "index": 7332, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7332.png", "answer": "(1,2]", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;e: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(P, RightPart(G));Abs(LineSegmentOf(P, F1)) = 3*Abs(LineSegmentOf(P, F2));Eccentricity(G) = e", "coords": {} }, { "index": 7333, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7333.png", "answer": "x^2/4+y^2=1", "fact_expr": "l: Line;C: Ellipse;b: Number;a: Number;P: Point;M: Point;N: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Length(MajorAxis(C)) = 4;PointOnCurve(P, C);O:Origin;PointOnCurve(O,l);Intersection(l, C) = {M, N};Slope(LineSegmentOf(P, M)) = K1;Slope(LineSegmentOf(P, N)) = K2;K1*K2 = -1/4;K1:Number;K2:Number", "coords": {} }, { "index": 7334, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7334.png", "answer": "sqrt(21)/3", "fact_expr": "C: Hyperbola;b: Number;a: Number;M: Point;F1: Point;F2: Point;N: Point;O: Origin;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Focus(C) = {F1, F2};PointOnCurve(M, Asymptote(C));Abs(LineSegmentOf(M, F1)) = 2*Abs(LineSegmentOf(M, F2));VectorOf(M, N) = 2*VectorOf(O, N);AngleOf(M, F2, N) = ApplyUnit(120, degree)", "coords": {} }, { "index": 7335, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7335.png", "answer": "sqrt(10)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;H: Line;Expression(H) = (x + 3*y + 1 = 0);IsPerpendicular(OneOf(Asymptote(G)), H)", "coords": {} }, { "index": 7336, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7336.png", "answer": "y=pm*2*x", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Ellipse;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2/20 + y^2/25 = 1);Distance(Focus(C),Asymptote(C))=2;FocalLength(C)=FocalLength(G)", "coords": {} }, { "index": 7337, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7337.png", "answer": "[1, 5)", "fact_expr": "G: Ellipse;m: Number;H: Line;a: Real;Expression(G) = (x^2/5 + y^2/m = 1);Expression(H) = (y = -1 + a*x);IsIntersect(G, H) ;PointOnCurve(Focus(G),xAxis)", "coords": {} }, { "index": 7338, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7338.png", "answer": "sqrt(7)", "fact_expr": "Gamma: Hyperbola;a: Number;b: Number;lambda:Number;A: Point;B: Point;F1: Point;C: Point;F2: Point;a>0;b>0;Expression(Gamma) = (x^2/a^2 - y^2/b^2 = 1);LeftFocus(Gamma) = F1;RightFocus(Gamma) = F2;PointOnCurve(A,LeftPart(Gamma));PointOnCurve(B,RightPart(Gamma));VectorOf(A,B)=lambda*VectorOf(F1,A);PointOnCurve(C, xAxis);VectorOf(C, B) = 3*VectorOf(F2, A);DotProduct(VectorOf(B, F2), VectorOf(B, F1))/Abs(VectorOf(B, F1)) = DotProduct(VectorOf(B, F2), VectorOf(B, C))/Abs(VectorOf(B, C))", "coords": {} }, { "index": 7339, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7340, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7340.png", "answer": "y=pm*2*x", "fact_expr": "E: Hyperbola;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;A: Point;RightVertex(E) = A;G: Line;PointOnCurve(A, G) = True;Slope(G) = -1;Intersection(G, Asymptote(E)) = {B, C};B: Point;C: Point;2*VectorOf(B, A) + VectorOf(B, C) = 0", "coords": {} }, { "index": 7341, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7342, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7342.png", "answer": "3*sqrt(5)/4", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);P: Point;PointOnCurve(P, G);Q: Point;Coordinate(Q) = (1/4, 0)", "coords": { "Q": [ 0.25, 0.0 ] } }, { "index": 7343, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7343.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "C: Hyperbola;b: Number;a: Number;O: Origin;F1: Point;P: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(P,RightPart(C));Abs(LineSegmentOf(O,P))= Abs(LineSegmentOf(O, F1));Abs(LineSegmentOf(P, F1)) = sqrt(3)*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 7344, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7344.png", "answer": "y = pm*sqrt(3)*x", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;RightFocus(C) = F;F: Point;Coordinate(F) = (c, 0);c: Number;Distance(F, Asymptote(C)) = c*sqrt(3)/2", "coords": {} }, { "index": 7345, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7345.png", "answer": "2*pi/3", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 8*y);F: Point;Focus(G) = F;l: Line;Directrix(G) = l;M: Point;Intersection(l, yAxis) = M;P: Point;PointOnCurve(P, G) = True;IsPerpendicular(LineSegmentOf(P, Q), l) = True;Intersection(LineSegmentOf(P, Q), l) = Q;Q: Point;Abs(LineSegmentOf(P, F)) = 8/3;PointOnCurve(P,LineSegmentOf(P,Q))", "coords": {} }, { "index": 7346, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7346.png", "answer": "sqrt(3) + 1", "fact_expr": "F1: Point;F2: Point;C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;LeftFocus(C) = F1;RightFocus(C) = F2;P: Point;PointOnCurve(P, RightPart(C)) = True;Abs(LineSegmentOf(O, P)) = Abs(LineSegmentOf(O, F1));O: Origin;Abs(LineSegmentOf(P, F1)) = sqrt(3)*Abs(LineSegmentOf(P, F2))", "coords": {} }, { "index": 7347, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7347.png", "answer": "{x=0,3*x+4*y+8=0}", "fact_expr": "l: Line;C: Parabola;A: Point;Expression(C) = (y^2 = 6*x);Coordinate(A) = (0, 2);IsTangent(l,C);PointOnCurve(A,l)=True", "coords": { "A": [ 0.0, 2.0 ] } }, { "index": 7348, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7349, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7350, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7351, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7351.png", "answer": "2", "fact_expr": "l: Line;G: Parabola;p: Number;O: Origin;A: Point;B: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G)=F;PointOnCurve(F,l);Slope(l)=1/2;Intersection(l,G)={A,B};Area(TriangleOf(O,A,B))=2*sqrt(5);F:Point", "coords": {} }, { "index": 7352, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7352.png", "answer": "32", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 2*(p*y));p: Number;p>0;F: Point;Focus(G) = F;O: Origin;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, G) = {A, B};C: Point;Coordinate(C) = (2*p, 0);Intersection(l, xAxis) = C;Abs(LineSegmentOf(A, B)) = 17", "coords": {} }, { "index": 7353, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7353.png", "answer": "sqrt(10)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x - 3*y + 1 = 0);IsPerpendicular(OneOf(Asymptote(G)), H)", "coords": {} }, { "index": 7354, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7355, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 1.820299506187439, "b": 1.6536897822836507, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 1.2588495006847936e-06, "converged": true, "iterations": 97 }, "output_path": "results/train/ellipse/problem_7355.png", "answer": "2", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;P: Point;Coordinate(P) = (0, sqrt(3));a > b;b > 0;PointOnCurve(P,G);Length(MajorAxis(G))=2*FocalLength(G)", "coords": { "P": [ 0.0, 1.7320508075688772 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 7356, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7357, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7357.png", "answer": "x^2/9 - y^2/3 = 1", "fact_expr": "H: Ellipse;Expression(H) = (x^2 + 4*y^2 = 16);G: Hyperbola;Focus(G) = Focus(H);Expression(OneOf(Asymptote(G))) = (x + sqrt(3)*y = 0)", "coords": {} }, { "index": 7358, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7359, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7359.png", "answer": "sqrt(10)", "fact_expr": "G: Parabola;Q: Point;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(Q) = (2, 3);PointOnCurve(P, G)", "coords": { "Q": [ 2.0, 3.0 ] } }, { "index": 7360, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7360.png", "answer": "sqrt(6)/2", "fact_expr": "F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;C: Hyperbola;Expression(C) = (x^2 - y^2 = 1);P: Point;PointOnCurve(P, C);AngleOf(F1, P, F2) = ApplyUnit(60, degree)", "coords": {} }, { "index": 7361, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7362, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7362.png", "answer": "8", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;Expression(G) = (y^2 = -4*x);Slope(H)=-1;PointOnCurve(Focus(G),H);Intersection(H,G)={A,B}", "coords": {} }, { "index": 7363, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7363.png", "answer": "1", "fact_expr": "G: Parabola;M: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(M, G);Distance(M, Focus(G)) = 2", "coords": {} }, { "index": 7364, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7364.png", "answer": "3/2", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 12*x);M: Point;Coordinate(M) = (-3, 4);PointOnCurve(Focus(C), G) = True;Slope(G) = k;k: Number;G: Line;Intersection(G, C) = {A, B};A: Point;B: Point;DotProduct(VectorOf(M, A), VectorOf(M, B)) = 0", "coords": { "M": [ -3.0, 4.0 ] } }, { "index": 7365, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7366, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7366.png", "answer": "(2*sqrt(3),7/2)", "fact_expr": "C: Ellipse;Q: Point;R: Point;O: Origin;P: Point;Expression(C) = (x^2/4 + y^2/3 = 1);PointOnCurve(P,C);Quadrant(P)=2;L:Line;TangentOnPoint(P,C)=L;Intersection(L,yAxis)=Q;PointOnCurve(Q,LineSegmentOf(Q,R));IsPerpendicular(LineSegmentOf(Q,R),LineSegmentOf(O,P));FootPoint(LineSegmentOf(Q,R),LineSegmentOf(O,P))=R", "coords": {} }, { "index": 7367, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7367.png", "answer": "[4, +oo)", "fact_expr": "H: Line;Expression(H) = (x = t);t: Number;G: Parabola;Expression(G) = (y^2 = 4*x);A: Point;B: Point;Intersection(H, G) = {A, B};C: Point;PointOnCurve(C, G);IsPerpendicular(LineSegmentOf(A, C), LineSegmentOf(B, C))", "coords": {} }, { "index": 7368, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7368.png", "answer": "15", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(C) = F;Distance(F, Directrix(C)) = 4;l: Line;PointOnCurve(F, l);P: Point;Q: Point;Intersection(l, C) = {P, Q};Abs(LineSegmentOf(P, F)) = 10", "coords": {} }, { "index": 7369, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7369.png", "answer": "3", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1)", "coords": {} }, { "index": 7370, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7371, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7371.png", "answer": "48", "fact_expr": "G: Hyperbola;P: Point;F2: Point;F1: Point;Expression(G) = (x^2/9 - y^2/16 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(P,RightPart(G));Abs(LineSegmentOf(P, F2)) = Abs(LineSegmentOf(F1, F2))", "coords": {} }, { "index": 7372, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7372.png", "answer": "(7+pm*sqrt(17))/16", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;Focus(G) = F2;P: Point;OneOf(Intersection(C, G)) = P;Cos(AngleOf(P, F1, F2)) = 7/9", "coords": {} }, { "index": 7373, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7373.png", "answer": "(0,1/2)", "fact_expr": "G: Parabola;Expression(G) = (y = x^2/2)", "coords": {} }, { "index": 7374, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7374.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Hyperbola;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 - y^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(90, degree)", "coords": {} }, { "index": 7375, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7376, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7377, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7377.png", "answer": "x^2/4 - y^2/16 = 1\nsqrt(5)", "fact_expr": "C: Hyperbola;a:Number;b:Number;a>0;b>0;G: Line;P: Point;F1: Point;F2: Point;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Expression(G) = (y = 2*x);LeftFocus(C) = F1;RightFocus(C) = F2;OneOf(Asymptote(C))=G;PointOnCurve(P, C);Abs(LineSegmentOf(P,F1))-Abs(LineSegmentOf(P,F2))=4", "coords": {} }, { "index": 7378, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7378.png", "answer": "3/4", "fact_expr": "C: Ellipse;H: Line;A: Point;B: Point;F1: Point;F2: Point;Expression(C) = (x^2/4 + y^2/3 = 1);LeftFocus(C) =F1;RightFocus(C)=F2;IsPerpendicular(xAxis,H);PointOnCurve(F1, H);Intersection(H, C) = {A, B}", "coords": {} }, { "index": 7379, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7379.png", "answer": "sqrt(2)/2", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;RightFocus(G) = F;H: Line;O: Origin;PointOnCurve(O, H);A: Point;P: Point;Intersection(H, G) = {A, P};IsPerpendicular(LineSegmentOf(P, F), xAxis);B: Point;Intersection(LineOf(A, F), G) = B;IsPerpendicular(LineSegmentOf(P, B), LineSegmentOf(P, A));e: Number;Eccentricity(G) = e", "coords": {} }, { "index": 7380, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7381, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7381.png", "answer": "20", "fact_expr": "C: Hyperbola;m: Number;G: Parabola;A: Point;B: Point;m>0;Expression(C) = (2*x^2 - y^2 = m);Expression(G) = (y^2 = 16*x);Intersection(C, Directrix(G)) = {A,B};Abs(LineSegmentOf(A,B))=4*sqrt(3)", "coords": {} }, { "index": 7382, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7382.png", "answer": "(2/3)*sqrt(21)", "fact_expr": "G: Ellipse;P: Point;A: Point;Expression(G) = (x^2/4 + y^2 = 1);Coordinate(A) = (0, 2);PointOnCurve(P, G)", "coords": { "A": [ 0.0, 2.0 ] } }, { "index": 7383, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7383.png", "answer": "-4*sqrt(5)", "fact_expr": "G: Hyperbola;H: Parabola;a: Number;Expression(G) = (x^2/4 - y^2 = 1);Expression(H) = (y^2 = a*x);Focus(H) = LeftFocus(G)", "coords": {} }, { "index": 7384, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "major_axis": "x", "x_coef": 3.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7384.png", "answer": "y^2=4*sqrt(2)*x", "fact_expr": "G: Parabola;H: Ellipse;Expression(H) = (x^2/3 + y^2 = 1);Focus(G)=RightFocus(H)", "coords": {} }, { "index": 7385, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7385.png", "answer": "(1, 2]", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = sqrt(3)*x);NumIntersection(G, H)=0;e: Number;Eccentricity(G) = e", "coords": {} }, { "index": 7386, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7386.png", "answer": "sqrt(6)/3", "fact_expr": "T: Ellipse;Expression(T) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;A: Point;B: Point;In(A,Vertex(T));Coordinate(A) = (a, 0);Coordinate(B) = (0, b);In(B,Vertex(T));C: Point;D: Point;L1: Line;L2: Line;PointOnCurve(A,L1);IsPerpendicular(L1,LineSegmentOf(A,B));PointOnCurve(B,L2);IsPerpendicular(L2,LineSegmentOf(A,B));Intersection(L1,T) = D;Intersection(L2,T) = C;LineSegmentOf(B, C) = 3*LineSegmentOf(A, D);Negation(C = Vertex(T))", "coords": {} }, { "index": 7387, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7388, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7388.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/b^2 = 1);b: Number;H: Parabola;Expression(H) = (y^2 = 12*x);RightFocus(G) = Focus(H)", "coords": {} }, { "index": 7389, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7389.png", "answer": "0", "fact_expr": "C: Ellipse;P: Point;F: Point;A: Point;Expression(C) = (x^2/4 + y^2/3 = 1);LeftFocus(C) = F;RightVertex(C)=A;PointOnCurve(P, C)", "coords": {} }, { "index": 7390, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7391, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7391.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;P: Point;F: Point;Q: Point;Expression(C) = (y^2 = 4*x);Coordinate(P) = (-1, 0);Focus(C) = F;PointOnCurve(P, l);Intersection(l, C) = {A, B};MidPoint(LineSegmentOf(A,B)) = Q;Abs(LineSegmentOf(F, Q)) = 2", "coords": { "P": [ -1.0, 0.0 ] } }, { "index": 7392, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7393, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7393.png", "answer": "{11, 29}", "fact_expr": "G: Ellipse;k: Number;Expression(G) = (x^2/20 + y^2/k = 1);FocalLength(G) = 6", "coords": {} }, { "index": 7394, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.23606797749979, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7394.png", "answer": "5*y^2/4-x^2=1", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/5 - y^2/4 = 1);C: Hyperbola;Asymptote(C) = Asymptote(G);H: Point;Coordinate(H) = (2, 2);PointOnCurve(H, C)", "coords": { "H": [ 2.0, 2.0 ] } }, { "index": 7395, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7395.png", "answer": "pm*9/4", "fact_expr": "P: Point;G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);PointOnCurve(P, G);F1: Point;F2: Point;Focus(G) = {F1, F2};AngleOf(F1, P, F2) = pi/2", "coords": {} }, { "index": 7396, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_7396.png", "answer": "3/16", "fact_expr": "G: Hyperbola;n: Number;m: Number;H: Parabola;Expression(G) = (-y^2/n + x^2/m = 1);Expression(H) = (y^2 = 4*x);Eccentricity(G) = 2;OneOf(Focus(G)) = Focus(H)", "coords": {} }, { "index": 7397, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7397.png", "answer": "1", "fact_expr": "G: Parabola;p: Number;H: Circle;M: Point;C: Point;N: Point;p>0;Expression(G) = (x^2 = 2*p*y);Center(H)=C;PointOnCurve(C,G);Coordinate(A) = (0, p);PointOnCurve(A,H);Intersection(xAxis,H) ={M,N};A:Point", "coords": {} }, { "index": 7398, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7398.png", "answer": "2*sqrt(2)", "fact_expr": "l: Line;E: Parabola;A: Point;B: Point;C:Point;Expression(E) = (y^2 = 4*x);PointOnCurve(Focus(E), l);Intersection(l, E) = {A, B};Intersection(l, Directrix(E)) = C;MidPoint(LineSegmentOf(A, C)) = B;Quadrant(A)=1", "coords": {} }, { "index": 7399, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7400, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7400.png", "answer": "2-sqrt(3)", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;H: Line;PointOnCurve(F2, H);Inclination(H) = ApplyUnit(120, degree);M: Point;OneOf(Intersection(H, G)) = M;IsPerpendicular(LineSegmentOf(M, F1), xAxis)", "coords": {} }, { "index": 7401, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7402, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7402.png", "answer": "x^2/4 + y^2/2 = 1", "fact_expr": "G: Hyperbola;H: Parabola;I: Ellipse;b: Number;a: Number;Expression(G) = (x^2 - y^2 = 1);Expression(H) = (y^2 = 8*x);Expression(I) = (y^2/b^2 + x^2/a^2 = 1);PointOnCurve(Focus(H), I);Focus(I) = Focus(G)", "coords": {} }, { "index": 7403, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7403.png", "answer": "x^2 - y^2 / 3 = 1", "fact_expr": "H: Parabola;Expression(H) = (y^2 = 4*x);G: Hyperbola;Focus(H) = Vertex(G);Vertex(H) = Center(G);Eccentricity(G) = 2", "coords": {} }, { "index": 7404, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7405, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7405.png", "answer": "2", "fact_expr": "G: Parabola;p: Number;H: Line;A: Point;B: Point;F: Point;p>0;Expression(G) = (y^2 = 2*p*x);Focus(G) = F;PointOnCurve(F, H);Inclination(H) = ApplyUnit(45, degree);Intersection(H, G) = {A, B};Length(LineSegmentOf(A,B))=8", "coords": {} }, { "index": 7406, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7406.png", "answer": "y^2/20 - x^2/36 = 1", "fact_expr": "C1:Hyperbola;C2:Hyperbola;M: Point;Expression(C1)=(x^2/9-y^2/5=1);Asymptote(C1)=Asymptote(C2);Coordinate(M) = (3, 5);PointOnCurve(M, C2)", "coords": { "M": [ 3.0, 5.0 ] } }, { "index": 7407, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.125, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7407.png", "answer": "17/8", "fact_expr": "C: Parabola;Expression(C) = (y = 2*x^2);F: Point;Focus(C) = F;P: Point;x0: Number;y0: Number;Coordinate(P) = (x0, y0);PointOnCurve(P, C);x0 = 1", "coords": {} }, { "index": 7408, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7409, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7409.png", "answer": "2*sqrt(3)/3", "fact_expr": "G: Hyperbola;b: Number;a: Number;e: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Inclination(OneOf(Asymptote(G))) = pi/3;Eccentricity(G) = e", "coords": {} }, { "index": 7410, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7410.png", "answer": "pm*sqrt(3)", "fact_expr": "Gamma: Parabola;Expression(Gamma) = (y = x^2);M: Point;PointOnCurve(M, Gamma) = True;Distance(M, Focus(Gamma)) = 1", "coords": {} }, { "index": 7411, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7412, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7413, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7414, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 3.0, 0.0 ], "radius": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_7414.png", "answer": "x^2/36+y^2/27=1", "fact_expr": "C1: Circle;C2: Circle;Expression(C2) = (y^2 + (x + 3)^2 = 4);IsOutTangent(C1,C2) = True;C3: Circle;Expression(C3) = (y^2 + (x - 3)^2 = 100);IsInTangent(C1,C3) = True", "coords": {} }, { "index": 7415, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7415.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Line;m: Number;k: Number;A: Point;B: Point;N: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = k*x + m);C:Curve;Expression(C)=(y=x/k);Intersection(H,Asymptote(G))={A,B};Intersection(H,C)=N;MidPoint(LineSegmentOf(A, B)) = N", "coords": {} }, { "index": 7416, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7417, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7418, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7419, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7420, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7420.png", "answer": "(pm*4*sqrt(2),4)", "fact_expr": "G: Parabola;P: Point;Expression(G) = (x^2 = 8*y);PointOnCurve(P, G);Distance(P, Focus(G)) = 6", "coords": {} }, { "index": 7421, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7421.png", "answer": "(-4/3,2/3)", "fact_expr": "G: Ellipse;H: Line;Expression(G) = (x^2 + 2*y^2 = 4);Expression(H) = (y = x + 2)", "coords": {} }, { "index": 7422, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 5.196152422706632, "major_axis": "y", "x_coef": 27.0, "y_coef": 36.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7422.png", "answer": "-x^2/5 + y^2/4 = 4", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(H) = (x^2/27 + y^2/36 = 1);Focus(G) = Focus(H);YCoordinate(OneOf(Intersection(G, H))) = 4", "coords": {} }, { "index": 7423, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7423.png", "answer": "6", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 6*y);H: LineSegment;IsChordOf(H, G);PointOnCurve(Focus(G), H)", "coords": {} }, { "index": 7424, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7424.png", "answer": "2", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;F: Point;LeftFocus(C) = F;Coordinate(F) = (-c, 0);P: Point;F1: Circle;Expression(F1) = (-2*c*x + x^2 + y^2 = 0);PointOnCurve(P, F1) = True;PerpendicularBisector(LineSegmentOf(F,P)) = OneOf(Asymptote(C));c:Number", "coords": {} }, { "index": 7425, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7425.png", "answer": "[1/3,1)", "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Focus(G) = {F1,F2};PointOnCurve(P,G);Abs(LineSegmentOf(P,F1))=2*Abs(LineSegmentOf(P,F2))", "coords": {} }, { "index": 7426, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7426.png", "answer": "sqrt(10)/5", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;LeftFocus(C) = F;G: Line;PointOnCurve(F, G);Inclination(G) = ApplyUnit(60, degree);A: Point;B: Point;Intersection(G, C) = {A, B};M: Point;Coordinate(M) = (-5, sqrt(3));MidPoint(LineSegmentOf(A, B)) = M", "coords": { "M": [ -5.0, 1.7320508075688772 ] } }, { "index": 7427, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7427.png", "answer": "12", "fact_expr": "G: Parabola;Expression(G) = (x^2 = 4*y);F: Point;Focus(G) = F;P: Point;Coordinate(P) = (1, 5);l: Line;PointOnCurve(P, l);A: Point;B: Point;Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ 1.0, 5.0 ] } }, { "index": 7428, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7428.png", "answer": "2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;L1: Line;L2: Line;Asymptote(G) = {L1, L2};k1: Number;k2: Number;Slope(L1) = k1;Slope(L2) = k2;k1*k2 = -3", "coords": {} }, { "index": 7429, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.8284271247461903, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7429.png", "answer": "8*sqrt(3)", "fact_expr": "G: Hyperbola;F1: Point;A: Point;F2: Point;B: Point;L:Line;Expression(G) = (x^2/4 - y^2/8 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F2,L);Intersection(F2,RightPart(G))={A,B};AngleOf(F1, A, B) = pi/3", "coords": {} }, { "index": 7430, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7430.png", "answer": "x^2-2*y^2=1", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/2 = 1);P: Point;PointOnCurve(P, G);Z: Line;PointOnCurve(P, Z);IsPerpendicular(Z, xAxis);Q: Point;FootPoint(Z, xAxis) = Q;M: Point;MidPoint(LineSegmentOf(P, Q)) = M", "coords": {} }, { "index": 7431, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7431.png", "answer": "3*sqrt(6)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/4 = 1);A: Point;LeftVertex(G) = A;B: Point;C: Point;PointOnCurve(B, LeftPart(G));PointOnCurve(C, RightPart(G));IsParallel(LineSegmentOf(B, C), xAxis);L1: Line;L2: Line;PointOnCurve(B, L1);PointOnCurve(C, L2);IsPerpendicular(L1, LineOf(A, B));IsPerpendicular(L2, LineOf(A, C));D: Point;Intersection(L1, L2) = D;Area(TriangleOf(B, C, D)) = (27*sqrt(3))/4", "coords": {} }, { "index": 7432, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7432.png", "answer": "4/3", "fact_expr": "C: Parabola;p: Number;G: Line;F: Point;B: Point;A: Point;Expression(C) = (y^2 = 2*(p*x));Coordinate(A) = (-2, 3);PointOnCurve(A, Directrix(C));PointOnCurve(A, G);TangentPoint(G, C) = B;Quadrant(B) = 1;Focus(C) = F", "coords": { "A": [ -2.0, 3.0 ] } }, { "index": 7433, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 2.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7433.png", "answer": "2 - 2*sqrt(2)", "fact_expr": "G: Ellipse;H: Line;t: Number;A: Point;M: Point;N: Point;Expression(G) = (x^2/16 + y^2/4 = 1);Expression(H) = (x = t*y + 4);LowerVertex(G) = A;Intersection(H, G) = {M, N};Negation(M = N);WhenMax(XCoordinate(Circumcenter(TriangleOf(A, M, N))))", "coords": {} }, { "index": 7434, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7434.png", "answer": "(4/3, 0)", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;M: Point;PointOnCurve(M, G);Distance(M, Directrix(G)) = 2;Distance(M, Vertex(G)) = Distance(M, Focus(G))", "coords": {} }, { "index": 7435, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7436, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7436.png", "answer": "12", "fact_expr": "C: Parabola;G: Point;P: Point;Expression(C) = (y^2 = 4*x);Coordinate(G) = (6, 12);PointOnCurve(P, C)", "coords": { "G": [ 6.0, 12.0 ] } }, { "index": 7437, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7437.png", "answer": "(0,15)", "fact_expr": "G: Hyperbola;m: Number;Expression(G) = (y^2/5 - x^2/m = 1);Eccentricity(G) = e;e:Number;In(e,(1,2))", "coords": {} }, { "index": 7438, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7438.png", "answer": "x+2*y-3=0", "fact_expr": "G: Ellipse;H: LineSegment;Expression(G) = (x^2/4 + y^2/2 = 1);Coordinate(MidPoint(H)) = (1,1);IsChordOf(H,G)", "coords": {} }, { "index": 7439, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7439.png", "answer": "sqrt(3)", "fact_expr": "G: Parabola;H: Ellipse;b: Number;Expression(G) = (y^2 = -4*x);b>0;Expression(H) = (x^2/4 + y^2/b^2 = 1);PointOnCurve(Focus(H), Directrix(G))", "coords": {} }, { "index": 7440, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7440.png", "answer": "(0, 0)", "fact_expr": "G: Parabola;A: Point;B: Point;P: Point;Expression(G) = (y^2 = -4*x);Coordinate(A) = (2, 0);Coordinate(B) = (4, 0);PointOnCurve(P, G);WhenMin(DotProduct(VectorOf(A, P), VectorOf(B, P)))", "coords": { "A": [ 2.0, 0.0 ], "B": [ 4.0, 0.0 ] } }, { "index": 7441, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7441.png", "answer": "[1, 5)+(5, +\\infty)", "fact_expr": "k: Real;H: Line;Expression(H) = (-k*x + y - 1 = 0);G: Ellipse;Expression(G) = (x^2/5 + y^2/m = 1);m: Real;IsIntersect(H, G)", "coords": {} }, { "index": 7442, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7443, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7443.png", "answer": "sqrt(7)/14", "fact_expr": "G: Ellipse;B: Point;A: Point;F: Point;C: Point;D: Point;Expression(G) = (x^2/4 + y^2/3 = 1);UpperVertex(G)=A;LowerVertex(G)=C;LeftVertex(G)=B;LeftFocus(G) = F;Intersection(LineOf(A,B), LineOf(F, C)) = D", "coords": {} }, { "index": 7444, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7444.png", "answer": "25", "fact_expr": "C: Ellipse;Expression(C)=(x^2/25+y^2/16=1);P: Point;F1: Point;F2: Point;LeftFocus(C)=F1;RightFocus(C)=F2;PointOnCurve(P,C);Abs(LineSegmentOf(P,F1))*Abs(LineSegmentOf(P,F2))=m;m: Number", "coords": {} }, { "index": 7445, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7445.png", "answer": "{0,1/2,-1}", "fact_expr": "H: Line;Expression(H) = (y = k*(x + 2) + 1);G: Parabola;Expression(G) = (y^2 = 4*x);NumIntersection(H, G) = 1;k: Number", "coords": {} }, { "index": 7446, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 2.23606797749979, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7446.png", "answer": "8/3", "fact_expr": "G: Hyperbola;P: Point;Expression(G) = (x^2/4 - y^2/5 = 1);PointOnCurve(P, LeftPart(G));Distance(P, RightFocus(G)) = 8", "coords": {} }, { "index": 7447, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7447.png", "answer": "{2,8}", "fact_expr": "C: Parabola;p: Number;G: Circle;H: Point;M: Point;F: Point;p>0;Expression(C) = (y^2 = 2*p*x);Coordinate(H) = (0, 2);Focus(C) = F;PointOnCurve(M, C);Abs(LineSegmentOf(M, F)) = 5;IsDiameter(LineSegmentOf(M,F),G);PointOnCurve(H,G)", "coords": { "H": [ 0.0, 2.0 ] } }, { "index": 7448, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7448.png", "answer": "(3*sqrt(2)-sqrt(10))/2", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;LeftFocus(C) = F;l: Line;O: Origin;PointOnCurve(O, l);Inclination(l) = pi/3;M: Point;N: Point;Intersection(l, C) = {M, N};AngleOf(M, F, N) = (2*pi)/3", "coords": {} }, { "index": 7449, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7449.png", "answer": "sqrt(3)/2", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;H: Line;PointOnCurve(Focus(G), H);IsPerpendicular(H, xAxis);Length(InterceptChord(H, G)) = a/2", "coords": {} }, { "index": 7450, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7451, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7451.png", "answer": "2*sqrt(2)+2*sqrt(5)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/4 = 1);P:Point;PointOnCurve(P,G);H: Line;Expression(H) = (2*x - y + 10 = 0)", "coords": {} }, { "index": 7452, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7452.png", "answer": "2*sqrt(10)/5", "fact_expr": "l: Line;G: Ellipse;P: Point;Expression(G) = (x^2/4 + y^2 = 1);Expression(l) = (x + 2*y = 0);PointOnCurve(P, G)", "coords": {} }, { "index": 7453, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7453.png", "answer": "8", "fact_expr": "G: Hyperbola;H: Parabola;a: Number;Expression(G) = (-x^2 + y^2 = 2);Expression(H) = (x^2 = a*y);Focus(H) = UpperFocus(G)", "coords": {} }, { "index": 7454, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7454.png", "answer": "16/3", "fact_expr": "Slope(G) = sqrt(3);G: Line;C: Parabola;Expression(C) = (y^2 = 4*x);PointOnCurve(Focus(C), G) = True;Intersection(G, C) = {A, B};A: Point;B: Point", "coords": {} }, { "index": 7455, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7455.png", "answer": "y=x-1", "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F, l);Slope(l) = k;k:Number;k>0;Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, B)) = 8", "coords": {} }, { "index": 7456, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7457, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7458, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7459, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7460, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "down" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7460.png", "answer": "7/2", "fact_expr": "G: Parabola;H: Line;N: Point;F: Point;Expression(G) = (y = -x^2/2);Expression(H) = (y = 1);PointOnCurve(N, G);Focus(G) = F;Distance(N, F) = 3", "coords": {} }, { "index": 7461, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7462, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 8.94427190999916, "b": 4.47213595499958, "major_axis": "x", "x_coef": 80.0, "y_coef": 20.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7462.png", "answer": "1/2", "fact_expr": "G: Ellipse;H: Line;k: Number;P: Point;Q: Point;Expression(G) = (x^2 + 4*y^2 = 80);Expression(H) = (y = k*x - 2);Intersection(H, G) = {P, Q};Negation(P=Q);XCoordinate(MidPoint(LineSegmentOf(P, Q))) = 2", "coords": {} }, { "index": 7463, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.366584062576294, "b": 2.364173412322998, "orientation": "horizontal" }, "optimization": { "final_loss": 0.00017838670464698225, "converged": true, "iterations": 89 }, "output_path": "results/train/hyperbola/problem_7463.png", "answer": "(pm*2,0)", "fact_expr": "C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Length(RealAxis(C))=2;Eccentricity(C)=2", "coords": {} }, { "index": 7464, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7464.png", "answer": "5", "fact_expr": "G: Parabola;A: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(A, G);XCoordinate(A) = 4", "coords": {} }, { "index": 7465, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7466, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7467, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7467.png", "answer": "2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;p>0;Coordinate(Focus(G)) = (1, 0)", "coords": {} }, { "index": 7468, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7468.png", "answer": "10", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/9 - y^2/16 = 1);P: Point;PointOnCurve(P, G);Distance(P, LeftFocus(G)) = 4", "coords": {} }, { "index": 7469, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7469.png", "answer": "x^2 = -12 \\cdot y", "fact_expr": "G: Parabola;O: Origin;Vertex(G) = O;PointOnCurve(Focus(G), yAxis);Expression(Directrix(G)) = (y=3)", "coords": {} }, { "index": 7470, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7470.png", "answer": "{5,3}", "fact_expr": "G: Ellipse;m: Real;Expression(G) = (y^2/4 + x^2/m = 1);FocalLength(G) = 2", "coords": {} }, { "index": 7471, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "x", "x_coef": 9.0, "y_coef": 5.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7471.png", "answer": "pm*8", "fact_expr": "G: Parabola;Expression(G) = (y^2 = m*x);m: Number;H: Ellipse;Expression(H) = (x^2/9 + y^2/5 = 1);Focus(G) = OneOf(Focus(H))", "coords": {} }, { "index": 7472, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.463192105293274, "b": 2.070901870727539, "orientation": "horizontal" }, "optimization": { "final_loss": 1.4661259228887502e-06, "converged": true, "iterations": 106 }, "output_path": "results/train/hyperbola/problem_7472.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;a: Real;a>0;Expression(G) = (-y^2/4 + x^2/a^2 = 1);Eccentricity(G) = sqrt(3)", "coords": {} }, { "index": 7473, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7473.png", "answer": "-2", "fact_expr": "M: Point;Coordinate(M) = (2, 2);C: Parabola;Expression(C) = (y^2 = -8*x);G: Line;k: Number;Slope(G)=k;PointOnCurve(Focus(C),G);A: Point;B: Point;Intersection(G,C)={A,B};IsPerpendicular(VectorOf(M,A),VectorOf(M,B))", "coords": { "M": [ 2.0, 2.0 ] } }, { "index": 7474, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7475, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7476, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7476.png", "answer": "12", "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (y^2 = 16*x);Coordinate(A) = (8, 2);Focus(G) = F;PointOnCurve(P, G)", "coords": { "A": [ 8.0, 2.0 ] } }, { "index": 7477, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7477.png", "answer": "1", "fact_expr": "G: Hyperbola;F1: Point;P: Point;F2: Point;Expression(G) = (x^2/4 - y^2 = 1);Focus(G) = {F1, F2};PointOnCurve(P, G);AngleOf(F1, P, F2) = ApplyUnit(90, degree)", "coords": {} }, { "index": 7478, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7478.png", "answer": "(18, \\pm 12)", "fact_expr": "G: Parabola;P: Point;Expression(G) = (y^2 = 8*x);PointOnCurve(P, G);Distance(P, Focus(G)) = 20", "coords": {} }, { "index": 7479, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7479.png", "answer": "8", "fact_expr": "l: Line;G: Parabola;p: Number;A: Point;B: Point;p>0;Expression(G) = (y^2 = 2*p*x);Expression(Directrix(G)) = (x = -1);Expression(l)=(y=x-1);Intersection(l,G)={A,B};IsChordOf(LineSegmentOf(A,B),G)", "coords": {} }, { "index": 7480, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7480.png", "answer": "1+sqrt(2)", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;l: Line;Slope(l) = 1;PointOnCurve(F1, l);P: Point;Intersection(l, RightPart(C)) = P;AngleOf(F1, F2, P) = ApplyUnit(90, degree)", "coords": {} }, { "index": 7481, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7481.png", "answer": "13", "fact_expr": "G: Parabola;P: Point;F: Point;Expression(G) = (y^2 = 16*x);PointOnCurve(P, G);Distance(P,SymmetryAxis(G))=12;Focus(G) = F", "coords": {} }, { "index": 7482, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7482.png", "answer": "sqrt(5)/3", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;Abs(LineSegmentOf(F1, F2)) = 4*sqrt(5);P: Point;PointOnCurve(P, C);Perimeter(TriangleOf(P, F1, F2)) = 4*sqrt(5) + 12", "coords": {} }, { "index": 7483, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7483.png", "answer": "4", "fact_expr": "G: Hyperbola;A: Circle;M: Point;P: Point;F1: Point;F2: Point;m:Number;Expression(G) = (x^2/16 - y^2/9 = 1);Coordinate(M) = (m, 0);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, RightPart(G));IsInscribedCircle(A, TriangleOf(P,F1,F2));TangentPoint(xAxis,A)=M", "coords": {} }, { "index": 7484, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7484.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;OneOf(Focus(G)) = F;F: Point;L: Line;PointOnCurve(F, L) ;IsPerpendicular(L, Asymptote(G)) ;FootPoint(L, Asymptote(G)) = M;M: Point;Intersection(OverlappingLine(LineSegmentOf(F, M)), yAxis) = E;E: Point;Abs(LineSegmentOf(F, M)) = Abs(LineSegmentOf(M, E))", "coords": {} }, { "index": 7485, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7485.png", "answer": "2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);M: Point;PointOnCurve(M, G);H: Line;Expression(H) = (y = x);Distance(M, H) = 4*sqrt(2);m: Number;n: Number;Coordinate(M) = (m, n);m>0;n>0", "coords": {} }, { "index": 7486, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 4.47213595499958, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7486.png", "answer": "17", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/20 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 9", "coords": {} }, { "index": 7487, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "y", "x_coef": 2.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7487.png", "answer": "sqrt(2)/2", "fact_expr": "G: Ellipse;Expression(G)=(y^2/4+x^2/2=1)", "coords": {} }, { "index": 7488, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7488.png", "answer": "y=pm*sqrt(2)*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2/2 = 1)", "coords": {} }, { "index": 7489, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7490, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7490.png", "answer": "(0,2]+[8,+oo)", "fact_expr": "F1: Point;F2: Point;C: Ellipse;Expression(C) = (y^2/4 + x^2/m = 1);m: Real;Focus(C) = {F1, F2};M: Point;PointOnCurve(M, C) = True;DotProduct(VectorOf(M, F1), VectorOf(M, F2)) = 0", "coords": {} }, { "index": 7491, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7491.png", "answer": "ApplyUnit(60,degree)", "fact_expr": "G: Parabola;P: Point;Q: Point;F: Point;x1:Number;y1:Number;x2:Number;y2:Number;Expression(G) = (y^2 = 4*x);Coordinate(P) = (x1, y1);Coordinate(Q) = (x2, y2);Focus(G)=F;PointOnCurve(P,G);PointOnCurve(Q, G);x1 + x2 + 2 = 2*Abs(LineSegmentOf(P, Q))", "coords": {} }, { "index": 7492, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7492.png", "answer": "y=pm*(2/3)*sqrt(3)*x", "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/3 + y^2/4 = 1)", "coords": {} }, { "index": 7493, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7494, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7494.png", "answer": "(4+sqrt(6))/2", "fact_expr": "C: Hyperbola;b: Number;a: Number;M: Point;P: Point;F2: Point;F1: Point;a > b;b > 0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, LeftPart(C));LeftFocus(C) = F1;RightFocus(C) = F2;OneOf(Endpoint(ImageinaryAxis(C)))=M;Min(Abs(LineSegmentOf(M, P)) + Abs(LineSegmentOf(P, F2))) = Abs(LineSegmentOf(F1, F2))", "coords": {} }, { "index": 7495, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7495.png", "answer": "5", "fact_expr": "G: Parabola;p: Number;A: Point;B: Point;F: Point;P: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Focus(G) = F;IsChordOf(LineSegmentOf(A, B), G);Intersection(PerpendicularBisector(LineSegmentOf(A, B)), xAxis) = P;Abs(LineSegmentOf(A, B)) = 10;PointOnCurve(F, LineSegmentOf(A, B))", "coords": {} }, { "index": 7496, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7496.png", "answer": "8", "fact_expr": "G: Ellipse;H: Line;F2: Point;A: Point;B: Point;F1: Point;Expression(G) = (x^2/25 + y^2/9 = 1);Focus(G) = {F1, F2};PointOnCurve(F1, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(F2, A)) + Abs(LineSegmentOf(F2, B)) = 12", "coords": {} }, { "index": 7497, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7498, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7498.png", "answer": "[2*sqrt(2), 2*sqrt(6)]", "fact_expr": "l: Line;C: Parabola;M: Point;A: Point;F: Point;B: Point;Expression(C) = (y^2 = 4*x);Coordinate(M) = (-2, 0);Focus(C) = F;PointOnCurve(F, l);Intersection(l, C) = {A, B};Quadrant(A) = 1;3/2 <= Area(TriangleOf(M, A, F))/Area(TriangleOf(M, B, F));Area(TriangleOf(M, A, F))/Area(TriangleOf(M, B, F)) <= 2", "coords": { "M": [ -2.0, 0.0 ] } }, { "index": 7499, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7499.png", "answer": "1/2", "fact_expr": "l: Line;G: Parabola;H: Point;A: Point;B: Point;Expression(G) = (y^2 = 4*x);Coordinate(H) = (0, 2);Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);PointOnCurve(H, l);x1:Number;x2:Number;y1:Number;y2:Number;Intersection(l,G)={A,B}", "coords": { "H": [ 0.0, 2.0 ] } }, { "index": 7500, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7500.png", "answer": "2", "fact_expr": "P: Point;Coordinate(P) = (-1, -1);F: Point;Focus(C) = F;C: Parabola;Expression(C) = (y^2 = 2*(p*x));p: Number;p>0;l: Line;PointOnCurve(F,l) = True;Slope(l) = -2;A: Point;B: Point;Intersection(C,l) = {A,B};DotProduct(VectorOf(A, P), VectorOf(B, P)) = 0", "coords": { "P": [ -1.0, -1.0 ] } }, { "index": 7501, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7502, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7502.png", "answer": "sqrt(2)", "fact_expr": "C: Parabola;p: Number;F: Point;P: Point;Q: Point;M: Point;l: Line;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(P) = (0, 2);Focus(C) = F;Intersection(LineSegmentOf(P, F), C) = M;PointOnCurve(M, l);IsPerpendicular(Directrix(C), l);FootPoint(Directrix(C), l) = Q;AngleOf(P, Q, F) = ApplyUnit(90, degree)", "coords": { "P": [ 0.0, 2.0 ] } }, { "index": 7503, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7503.png", "answer": "2/3", "fact_expr": "C: Hyperbola;b: Number;a: Number;F1: Point;M: Point;F2: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(M, RightPart(C));Theta: Number;AngleOf(F1, M, F2) = Theta;DotProduct(VectorOf(M, F1), VectorOf(M, F2)) = 4*b^2", "coords": {} }, { "index": 7504, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7504.png", "answer": "y^2=pm*8*x", "fact_expr": "l: Line;G: Parabola;a: Number;O: Origin;A: Point;F: Point;Expression(G) = (y^2 = a*x);Slope(l)=2;Focus(G)=F;PointOnCurve(F,l);Intersection(l,yAxis)=A;Area(TriangleOf(O,A,F))=4", "coords": {} }, { "index": 7505, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7505.png", "answer": "4*sqrt(5)/7", "fact_expr": "H: Line;Expression(H) = (y = 2*x - 3);G: Hyperbola;Expression(G) = (x^2/2 - y^2 = 1);A: Point;B: Point;Intersection(H, G) = {A, B}", "coords": {} }, { "index": 7506, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7507, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7508, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7508.png", "answer": "(\\sqrt{5}-1)/2", "fact_expr": "M: Ellipse;a: Number;b: Number;A: Point;B: Point;F: Point;a > b;b > 0;Expression(M) = (y^2/b^2 + x^2/a^2 = 1);OneOf(Vertex(MajorAxis(M)))=A;OneOf(Vertex(MinorAxis(M)))=B;RightFocus(M) = F;IsPerpendicular(LineSegmentOf(A, B), LineSegmentOf(B, F))", "coords": {} }, { "index": 7509, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7510, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.23606797749979, "b": 1.0, "major_axis": "x", "x_coef": 5.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7510.png", "answer": "-10", "fact_expr": "l: Line;G: Ellipse;O: Origin;A: Point;E: Point;M: Point;F: Point;N: Point;lambda1:Number;lambda2:Number;Expression(G) = (x^2/5 + y^2 = 1);Center(G)=O;RightFocus(G) = F;OneOf(Endpoint(MinorAxis(G)))=A;PointOnCurve(F, l);Intersection(l, G) = {M, N};Intersection(OverlappingLine(LineSegmentOf(O, A)),l) = E;VectorOf(E, M) = lambda1*VectorOf(M, F);VectorOf(E, N) = lambda2*VectorOf(N, F)", "coords": {} }, { "index": 7511, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7512, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7512.png", "answer": "2", "fact_expr": "C:Ellipse;Expression(C)=(x^2/2+y^2=1)", "coords": {} }, { "index": 7513, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.23606797749979, "b": 2.0, "major_axis": "x", "x_coef": 5.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7513.png", "answer": "sqrt(5)-1", "fact_expr": "G: Ellipse;Expression(G) = (x^2/5 + y^2/4 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;M: Point;PointOnCurve(M,G);Negation(M=Endpoint(MajorAxis(G)));theta: Number;AngleOf(F1, M, F2) = 2*theta;I: Point;Incenter(TriangleOf(M, F1, F2)) = I", "coords": {} }, { "index": 7514, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7514.png", "answer": "sqrt(2)/2", "fact_expr": "G: Ellipse;b: Number;a: Number;H: Line;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y = x + 2);PointOnCurve(OneOf(Focus(G)),H);PointOnCurve(OneOf(Vertex(G)),H)", "coords": {} }, { "index": 7515, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7515.png", "answer": "9", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;G: Line;PointOnCurve(F, G);A: Point;B: Point;Intersection(G, C) = {A, B};N: Point;MidPoint(LineSegmentOf(A, B)) = N;P: Point;Coordinate(P) = (12, 4);WhenMin(Abs(LineSegmentOf(N, A))+Abs(LineSegmentOf(N, P)))", "coords": { "P": [ 12.0, 4.0 ] } }, { "index": 7516, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7516.png", "answer": "2", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);P: Point;PointOnCurve(P, G);O: Origin;F: Point;LeftFocus(G) = F;M: Point;LineSegmentOf(O, M) = (1/2)*(LineSegmentOf(O, F) + LineSegmentOf(O, P))", "coords": {} }, { "index": 7517, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7517.png", "answer": "y=-1", "fact_expr": "l: Line;G: Parabola;H: Circle;A: Point;Expression(G) = (x^2 = 4*y);Coordinate(A) = (0, 1);PointOnCurve(A, H);PointOnCurve(Center(H),G);IsTangent(l,H)", "coords": { "A": [ 0.0, 1.0 ] } }, { "index": 7518, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7519, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7519.png", "answer": "(-1, -\\sqrt{3}/2)", "fact_expr": "G: Hyperbola;H: Line;k: Real;Expression(G) = (x^2/4 - y^2/3 = 1);Expression(H) = (y = k*x + 1);NumIntersection(H, RightPart(G)) = 2", "coords": {} }, { "index": 7520, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.242640687119285, "major_axis": "x", "x_coef": 25.0, "y_coef": 18.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7520.png", "answer": "pi/3", "fact_expr": "G: Ellipse;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/25 + y^2/18 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 6", "coords": {} }, { "index": 7521, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.4641016151377544, "major_axis": "x", "x_coef": 16.0, "y_coef": 12.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7521.png", "answer": "(-2,3)", "fact_expr": "G: Ellipse;A: Point;M: Point;F: Point;Expression(G) = (x^2/16 + y^2/12 = 1);Coordinate(A) = (-2, sqrt(3));RightFocus(G) = F;PointOnCurve(M, G);WhenMin(Abs(LineSegmentOf(A,M))+Abs(LineSegmentOf(M,F)))", "coords": { "A": [ -2.0, 1.7320508075688772 ] } }, { "index": 7522, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7522.png", "answer": "sqrt(2)", "fact_expr": "G: Parabola;H: Line;A: Point;P: Point;Expression(G) = (y^2 = 4*x);Expression(H) = (x = -1);Coordinate(A) = (0, -1);PointOnCurve(P, G)", "coords": { "A": [ 0.0, -1.0 ] } }, { "index": 7523, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 4.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7523.png", "answer": "2*sqrt(41)-6", "fact_expr": "A: Point;Coordinate(A) = (6, 0);P: Point;PointOnCurve(P, G);G: Parabola;Expression(G) = (y^2 = 16*x);B: Point;PointOnCurve(B, H);H: Curve;Expression(H) = (y^2 + (x - 4)^2 = 1)", "coords": { "A": [ 6.0, 0.0 ] } }, { "index": 7524, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 0.9550795555114746, "b": 2.8647773265838623, "orientation": "horizontal" }, "optimization": { "final_loss": 1.6550993677810766e-05, "converged": true, "iterations": 144 }, "output_path": "results/train/hyperbola/problem_7524.png", "answer": "3", "fact_expr": "C: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(C) = sqrt(10)", "coords": {} }, { "index": 7525, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7526, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7526.png", "answer": "3", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F: Point;RightFocus(C) = F;G: Line;Expression(G) = (y = k*x);k: Number;Negation(k=0);Intersection(G, C) = {A, B};A: Point;B: Point;AngleOf(A, F, B) = ApplyUnit(90, degree);O: Origin;Area(TriangleOf(O, A, F)) = 4*a^2", "coords": {} }, { "index": 7527, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.242640687119285, "major_axis": "x", "x_coef": 25.0, "y_coef": 18.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7527.png", "answer": "pi/3", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/18 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, G);Abs(LineSegmentOf(P, F1)) = 6", "coords": {} }, { "index": 7528, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ -0.0, 4.0 ], "radius": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_7528.png", "answer": "(x-1)^2+(y-3)^2=2", "fact_expr": "l: Line;C: Circle;B: Point;A: Point;P: Point;M: Point;O: Origin;Expression(C) = (-8*y + x^2 + y^2 = 0);Coordinate(P) = (2, 2);PointOnCurve(P,l);Intersection(l,C)={A,B};MidPoint(LineSegmentOf(A,B))=M", "coords": {} }, { "index": 7529, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7529.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Circle;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (x^2 + y^2 = 2*a^2);OneOf(Intersection(H,G))=P;LeftFocus(G) = F1;RightFocus(G) = F2;LineSegmentOf(P, F1) = 3*LineSegmentOf(P, F2)", "coords": {} }, { "index": 7530, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7530.png", "answer": "[3, +\\infty)", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;M: Point;PointOnCurve(M, C);N: Point;Coordinate(N) = (2, 2)", "coords": { "N": [ 2.0, 2.0 ] } }, { "index": 7531, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7531.png", "answer": "sqrt(10)", "fact_expr": "P: Point;x0: Number;y0: Number;Coordinate(P) = (x0, y0);G: Parabola;Expression(G) = (y^2 = 4*x);PointOnCurve(P, G);H: Point;Coordinate(H) = (0, 3)", "coords": { "H": [ 0.0, 3.0 ] } }, { "index": 7532, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7532.png", "answer": "4*sqrt(13)/3", "fact_expr": "l: Line;Expression(l) = (y = m + 2*sqrt(3)*x);m: Number;C: Hyperbola;Expression(C) = (x^2 - y^2/3 = 1);PointOnCurve(RightVertex(C),l) = True;L1: Line;L2: Line;Asymptote(C) = {L1, L2};Intersection(l, L1) = A;Intersection(l, L2) = B;A: Point;B: Point", "coords": {} }, { "index": 7533, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7533.png", "answer": "5/3", "fact_expr": "G: Ellipse;Expression(G) = (x^2/25 + y^2/16 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G);PointOnCurve(F1, LineSegmentOf(A, B));Perimeter(InscribedCircle(TriangleOf(A, B, F2))) = pi;x1: Number;y1: Number;x2: Number;y2: Number;Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2)", "coords": {} }, { "index": 7534, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7534.png", "answer": "x^2/6+y^2/3=1", "fact_expr": "G: Hyperbola;H: Ellipse;M: Point;Expression(G) = (x^2 - 2*y^2 = 2);Coordinate(M) = (-sqrt(6), 0);PointOnCurve(M, H);Focus(G)=Focus(H)", "coords": { "M": [ -2.449489742783178, 0.0 ] } }, { "index": 7535, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 3.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7535.png", "answer": "[-6,6]", "fact_expr": "H: Circle;Expression(H) = (x^2 + (-a + y)^2 = 9);G: Ellipse;Expression(G) = (x^2/25 + y^2/9 = 1);IsIntersect(H, G) = True;a: Real", "coords": {} }, { "index": 7536, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7536.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;e: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P,F1),LineSegmentOf(P,F2));Area(TriangleOf(P, F1, F2)) = 2*a*b;Eccentricity(G) = e", "coords": {} }, { "index": 7537, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7537.png", "answer": "8", "fact_expr": "G: Parabola;A: Point;F: Point;P: Point;l: Line;Expression(G) = (y^2 = 8*x);Focus(G) = F;Directrix(G)=l;PointOnCurve(P,G);IsPerpendicular(LineSegmentOf(P,A),l);FootPoint(LineSegmentOf(P,A),l)=A;Slope(LineOf(A,F)) = -sqrt(3)", "coords": {} }, { "index": 7538, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 2.8284271247461903, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7538.png", "answer": "49*pi", "fact_expr": "G: Hyperbola;H: Circle;O: Origin;N: Point;F1: Point;P: Point;M: Point;F2: Point;Expression(G) = (x^2/16 - y^2/8 = 1);PointOnCurve(P, RightPart(G));LeftFocus(G) = F1;RightFocus(G) = F2;lambda: Number;mu: Number;lambda > 0;VectorOf(F1, P) = lambda*VectorOf(P, M);VectorOf(P, N) = mu*((VectorOf(P, F2)/Abs(VectorOf(P, F2))) + (VectorOf(P, M)/Abs(VectorOf(P, M))));DotProduct(VectorOf(P, N), VectorOf(F2, N)) = 0;VectorOf(P, F2) = 3;Center(H) = O;Radius(H) = LineSegmentOf(O, N)", "coords": {} }, { "index": 7539, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7540, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7541, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7541.png", "answer": "4/5", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;I: Point;Expression(G) = (x^2/16 - y^2/9 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, RightPart(G));Incenter(TriangleOf(P, F1, F2)) = I;Area(TriangleOf(I,P,F2))=Area(TriangleOf(I,P,F1))-lambda*Area(TriangleOf(I,F1,F2));lambda:Number", "coords": {} }, { "index": 7542, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7542.png", "answer": "x=-1", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x)", "coords": {} }, { "index": 7543, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7543.png", "answer": "sqrt(3)*x + pm*y = 0", "fact_expr": "G: Hyperbola;Expression(G) = (-x^2/4 + y^2/12 = 1)", "coords": {} }, { "index": 7544, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7545, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7545.png", "answer": "sqrt(7)/4", "fact_expr": "C: Ellipse;a: Number;M: Point;F: Point;O: Origin;N: Point;a>0;Expression(C) = (y^2/9 + x^2/a = 1);RightFocus(C) = F;PointOnCurve(M, C);MidPoint(LineSegmentOf(M,F)) = N;Abs(LineSegmentOf(M,F)) = 2*Abs(LineSegmentOf(O,N));2*Abs(LineSegmentOf(O,N)) = 4", "coords": {} }, { "index": 7546, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7546.png", "answer": "sqrt(5)", "fact_expr": "C: Hyperbola;b: Number;a: Number;B: Point;F: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);OneOf(Focus(C))=F;OneOf(Endpoint(ImageinaryAxis(C)))=B;PointOnCurve(MidPoint(LineSegmentOf(F,B)),C)", "coords": {} }, { "index": 7547, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7547.png", "answer": "6/5", "fact_expr": "C: Hyperbola;b: Number;a: Number;G: Line;A: Point;F: Point;B: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(C) = F;PointOnCurve(F, G);Slope(G)=sqrt(3);Intersection(G, C) = {A,B};VectorOf(A, F) = 4*VectorOf(F, B)", "coords": {} }, { "index": 7548, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.8284271247461903, "major_axis": "x", "x_coef": 9.0, "y_coef": 8.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7548.png", "answer": "1/3", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/8 = 1)", "coords": {} }, { "index": 7549, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7549.png", "answer": "3/2", "fact_expr": "l: Line;G: Parabola;p: Number;F: Point;A: Point;B: Point;p>0;Expression(G) = (y^2 = 2*p*x);Coordinate(F) = (1, 0);Focus(G) = F;PointOnCurve(F,l);Intersection(l, G) = {A, B};Abs(LineSegmentOf(A, F)) = 3", "coords": { "F": [ 1.0, 0.0 ] } }, { "index": 7550, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.449489742783178, "b": 2.449489742783178, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7550.png", "answer": "(-sqrt(15)/3,-1)", "fact_expr": "G: Hyperbola;H: Line;k: Real;Expression(G) = (x^2 - y^2 = 6);Expression(H) = (y = k*x + 2);NumIntersection(H,RightPart(G))=2", "coords": {} }, { "index": 7551, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7551.png", "answer": "4*sqrt(3)/3", "fact_expr": "C: Ellipse;P: Point;B: Point;Expression(C) = (x^2/4 + y^2 = 1);UpperVertex(C)=B;PointOnCurve(P, C)", "coords": {} }, { "index": 7552, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7552.png", "answer": "sqrt(3)", "fact_expr": "C: Ellipse;b: Number;a: Number;G: Circle;F: Point;L1:Line;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Expression(G) = (x^2 + y^2 = b^2);RightFocus(C) = F;TangentOfPoint(F,G)={L1,L2};IsPerpendicular(L1,L2);L2:Line", "coords": {} }, { "index": 7553, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7553.png", "answer": "8*sqrt(2)/3", "fact_expr": "C: Ellipse;Expression(C) = (x^2/2 + y^2 = 1);F1: Point;LeftFocus(C) = F1;l: Line;Expression(l) = (y = x - 1);A: Point;B: Point;Intersection(l, C) = {A, B}", "coords": {} }, { "index": 7554, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7554.png", "answer": "(\\sqrt{5}+1)/2", "fact_expr": "C: Hyperbola;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;A: Point;RightVertex(C) = A;B1: Point;B2: Point;Endpoint(ImageinaryAxis(C)) = {B1, B2};F: Point;RightFocus(C) = F;IsPerpendicular(LineSegmentOf(B2, F), LineSegmentOf(A, B1))", "coords": {} }, { "index": 7555, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7555.png", "answer": "4*sqrt(2)", "fact_expr": "G: Hyperbola;H: Line;P: Point;Q: Point;F:Point;Expression(G) = (x^2 - y^2 = 2);RightFocus(G)=F;PointOnCurve(F,H);Inclination(H)=ApplyUnit(30,degree);Intersection(H,G)={P,Q}", "coords": {} }, { "index": 7556, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7556.png", "answer": "143/12", "fact_expr": "G: Hyperbola;H: Point;Expression(G) = (x^2/4 - y^2 = 1);Coordinate(H) = (x1, y1);PointOnCurve(H, G);x1:Number;y1:Number", "coords": {} }, { "index": 7557, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7558, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7558.png", "answer": "3", "fact_expr": "C: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);Focus(C)={F1,F2};PointOnCurve(P, C);IsPerpendicular(VectorOf(P,F1),VectorOf(P,F2));Area(TriangleOf(P,F1,F2))=9", "coords": {} }, { "index": 7559, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.8284271247461903, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 8.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7559.png", "answer": "1", "fact_expr": "C: Ellipse;Expression(C) = (x^2/8 + y^2/2 = 1);M: Circle;Expression(M) = (-r^2 + 2*(sqrt(2)*x) + x^2 + y^2 + 2 = 0);r: Number;0 < r;r < sqrt(2);P: Point;UpperVertex(C) = P;Z1: Line;Z2: Line;TangentOfPoint(P, M) = {Z1, Z2};A: Point;Intersection(Z1, C) = A;B: Point;Intersection(Z2, C) = B;Negation(P=A);Negation(P=B)", "coords": {} }, { "index": 7560, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7560.png", "answer": "10", "fact_expr": "G: Parabola;M: Point;Expression(G) = (y^2 = 4*x);PointOnCurve(M, G);Distance(M, yAxis) = 9", "coords": {} }, { "index": 7561, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7561.png", "answer": "sqrt(3)-1", "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(P,G);AngleOf(P, F1, F2) = pi/6;AngleOf(P,F2,F1)=pi/3;Eccentricity(G)=e;e:Number", "coords": {} }, { "index": 7562, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 3.0, "b": 2.0, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7562.png", "answer": "x^2/15+y^2/10=1", "fact_expr": "M: Point;Coordinate(M) = (-3, 2);Z: Curve;Expression(Z) = (x^2/9 + y^2/4 = 1);G: Ellipse;PointOnCurve(M, G);Focus(Z) = Focus(G)", "coords": { "M": [ -3.0, 2.0 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 7563, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7563.png", "answer": "8", "fact_expr": "M: Point;Coordinate(M) = (sqrt(3), 0);G: Ellipse;Expression(G) = (x^2/4 + y^2 = 1);H: Line;Expression(H) = (y = k*(x + sqrt(3)));Intersection(G, H) = {A, B};A: Point;B: Point;k:Number", "coords": { "M": [ 1.7320508075688772, 0.0 ] } }, { "index": 7564, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7564.png", "answer": "(\\sqrt{5}-1)/2", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;A: Point;LeftVertex(C) = A;B: Point;UpperVertex(C) = B;F: Point;RightFocus(C) = F;AngleOf(A, B, F) = ApplyUnit(90, degree)", "coords": {} }, { "index": 7565, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.8284271247461903, "b": 4.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7565.png", "answer": "y=pm*sqrt(2)*x", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/8 - y^2/16 = 1)", "coords": {} }, { "index": 7566, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7566.png", "answer": "12", "fact_expr": "N: Parabola;Expression(N) = (y^2 = 2*p*x);p: Number;p>0;G: Line;Expression(G) = (x = m);m: Number;m>0;Intersection(G, N) = {A, B};A: Point;B: Point;O: Origin;IsPerpendicular(LineSegmentOf(O, A), LineSegmentOf(O, B)) = True;Coordinate(Focus(N))=(3,0)", "coords": {} }, { "index": 7567, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7568, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7568.png", "answer": "2*sqrt(3)/3", "fact_expr": "C1:Parabola;C2:Hyperbola;a: Number;a>0;Expression(C2) = (x^2/3 - y^2 = 1);Expression(C1) = (y = x^2/(4*a));Intersection(LineSegmentOf(Focus(C1),RightFocus(C2)),C1)=M;Quadrant(M)=1;IsParallel(TangentOnPoint(M,C1),OneOf(Asymptote(C2)));M:Point", "coords": {} }, { "index": 7569, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7569.png", "answer": "(3*sqrt(2))/7", "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2/3 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;l: Line;A: Point;B: Point;PointOnCurve(F2,l);Inclination(l) = pi/4;Intersection(l, C) = {A, B}", "coords": {} }, { "index": 7570, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7570.png", "answer": "(4,-4*sqrt(3))", "fact_expr": "G: Parabola;M: Point;F: Point;Expression(G) = (y^2 = 12*x);Focus(G) = F;PointOnCurve(M,G);Quadrant(M)=4;Abs(LineSegmentOf(M, F)) = 7", "coords": {} }, { "index": 7571, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7571.png", "answer": "11", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/4 - y^2/3 = 1);F1: Point;F2: Point;LeftFocus(C) = F1;RightFocus(C) = F2;l: Line;PointOnCurve(F1, l);A: Point;B: Point;Intersection(l, LeftPart(C)) = {A, B}", "coords": {} }, { "index": 7572, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7573, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7574, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7574.png", "answer": "sqrt(14)/2", "fact_expr": "C: Parabola;Expression(C) = (y^2 = 2*x);F: Point;Focus(C) = F;l: Line;PointOnCurve(F, l);Intersection(l, C) = {A, B};B: Point;A: Point;Abs(LineSegmentOf(A, F)) = 2*Abs(LineSegmentOf(B, F));IsDiameter(LineSegmentOf(A, B), G) = True;G: Circle", "coords": {} }, { "index": 7575, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7575.png", "answer": "1/5", "fact_expr": "Gamma:Ellipse;A: Point;F: Point;B: Point;C: Point;D: Point;Expression(Gamma) = (x^2/a^2+y^2/b^2=1);LeftVertex(Gamma)=C;LeftFocus(Gamma)=F;LowerVertex(Gamma)=A;UpperVertex(Gamma)=B;Intersection(LineSegmentOf(A,F),LineSegmentOf(B,C))=D;VectorOf(C, D) = 2*VectorOf(D, B)", "coords": {} }, { "index": 7576, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7576.png", "answer": "4+2*sqrt(3)", "fact_expr": "G: Ellipse;P: Point;F1: Point;O: Origin;Expression(G) = (x^2/4 + y^2 = 1);LeftFocus(G) = F1;PointOnCurve(P, G)", "coords": {} }, { "index": 7577, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7577.png", "answer": "(\\sqrt{3} - 1)/2", "fact_expr": "E: Ellipse;b: Number;a: Number;G: Circle;F1: Point;F2: Point;P: Point;Q: Point;c: Number;a > b;b > 0;Expression(E) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);LeftFocus(E) = F1;RightFocus(E) = F2;IsDiameter(LineSegmentOf(F1, F2), G);Intersection(G, E) = {P, Q};Quadrant(P) = 1;Quadrant(Q) = 3;Area(TriangleOf(P, Q, F2)) = c^2*(sqrt(3)/2)", "coords": {} }, { "index": 7578, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7578.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Ellipse;H: Line;A: Point;B: Point;Expression(G) = (x^2/2 + y^2 = 1);Expression(H) = (y = x + 1);Intersection(H, G) = {A, B}", "coords": {} }, { "index": 7579, "success": true, "conic_type": "circle", "error": null, "params": { "center": [ 0.0, 0.0 ], "radius": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/circle/problem_7579.png", "answer": "4*x^2+y^2=1", "fact_expr": "G: Circle;H: LineSegment;Expression(G) = (x^2 + y^2 = 1);PointOnCurve(P,G);IsPerpendicular(H,yAxis);M:Point;MidPoint(H)=M;P:Point", "coords": {} }, { "index": 7580, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.4142135623730951, "major_axis": "x", "x_coef": 4.0, "y_coef": 2.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7580.png", "answer": "sqrt(2)/2", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/2 = 1)", "coords": {} }, { "index": 7581, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7581.png", "answer": "(2*sqrt(2),3*sqrt(2))", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/2 - y^2/b^2 = 1);a: Number;b: Number;a>0;b>0;e: Number;Eccentricity(C) = e;In(e, (sqrt(5), sqrt(10)))", "coords": {} }, { "index": 7582, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7583, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7583.png", "answer": "12/5", "fact_expr": "C: Parabola;p: Number;F: Point;l: Line;P: Point;p>0;Expression(C) = (y^2 = 2*p*x);Focus(C) = F;Directrix(C) = l;PointOnCurve(P, l);L1: Line;L2: Line;TangentOfPoint(P, C) = {L1, L2};A: Point;B: Point;TangentPoint(L1, C) = A;TangentPoint(L2, C) = B;Abs(LineSegmentOf(P, A)) = 3;Abs(LineSegmentOf(P, B)) = 4", "coords": {} }, { "index": 7584, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7584.png", "answer": "8", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);H: Line;PointOnCurve(Focus(G), H) = True;Intersection(H, G) = {A, B};A: Point;B: Point;Coordinate(A) = (x1, y1);x1: Number;y1: Number;Coordinate(B) = (x2, y2);x2: Number;y2: Number;x1 + x2 = 6", "coords": {} }, { "index": 7585, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7585.png", "answer": "sqrt(2)", "fact_expr": "G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);PointOnCurve(P, G);Focus(G) = {F1,F2};AngleOf(P,F1,F2)=ApplyUnit(75,degree);AngleOf(P, F2, F1) = ApplyUnit(15, degree)", "coords": {} }, { "index": 7586, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7587, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7587.png", "answer": "y^2/3 - x^2/3 = 1", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = 1);Z: Hyperbola;Asymptote(G) = Asymptote(Z);H: Point;Coordinate(H) = (1, 2);PointOnCurve(H, Z)", "coords": { "H": [ 1.0, 2.0 ] } }, { "index": 7588, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7588.png", "answer": "12", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);H: Line;PointOnCurve(Focus(G), H);A: Point;B: Point;x1: Number;x2: Number;y1: Number;y2: Number;Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2);Intersection(H, G) = {A, B};x1 + x2 = 10", "coords": {} }, { "index": 7589, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7590, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7591, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.23606797749979, "b": 2.0, "major_axis": "x", "x_coef": 5.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7591.png", "answer": "(\\sqrt{15}/2, 1),(\\sqrt{15}/2, -1),(-\\sqrt{15}/2, 1),(-\\sqrt{15}/2, -1)", "fact_expr": "G: Ellipse;P: Point;F1:Point;F2:Point;Expression(G) = (x^2/5 + y^2/4 = 1);PointOnCurve(P,G);Focus(G)={F1,F2};Area(TriangleOf(P,F1,F2))=1", "coords": {} }, { "index": 7592, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7592.png", "answer": "[\\sqrt{2}/2,1)", "fact_expr": "G: Ellipse;b: Number;a: Number;O: Circle;A: Point;P: Point;B: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Expression(O) = (x^2 + y^2 = b^2);L1:Line;L2:Line;TangentOfPoint(P,O)={L1,L2};TangentPoint(L1,O)=A;TangentPoint(L2,O)=B;AngleOf(A, P, B) = ApplyUnit(90, degree);PointOnCurve(P,G)", "coords": {} }, { "index": 7593, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.23606797749979, "b": 2.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7593.png", "answer": "2*x + pm*sqrt(5)*y = 0", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/5 - y^2/4 = 1)", "coords": {} }, { "index": 7594, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7594.png", "answer": "9/2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = -4*x);F: Point;Focus(G) = F;l: Line;PointOnCurve(F, l);A: Point;B: Point;Intersection(l, G) = {A, B};C: Point;Intersection(l, Directrix(G)) = C;VectorOf(F, C) = 4*VectorOf(F, B)", "coords": {} }, { "index": 7595, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7596, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7597, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 3.4641016151377544, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7597.png", "answer": "x^2/16+y^2/12=1", "fact_expr": "G: Hyperbola;H: Ellipse;Expression(G) = (x^2/4 - y^2/12 = 1);Focus(G) = Vertex(H);Vertex(G) = Focus(H)", "coords": {} }, { "index": 7598, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7599, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7600, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7600.png", "answer": "sqrt(2)/2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*(p*x));p: Number;p>0;F: Point;Focus(G) = F;A: Point;B: Point;PointOnCurve(A, G) = True;PointOnCurve(B, G) = True;IsDiameter(LineSegmentOf(A, B), H) = True;H: Circle;PointOnCurve(F, H) = True;MidPoint(LineSegmentOf(A, B)) = M;M: Point;PointOnCurve(M, LineOf(M, N)) = True;IsPerpendicular(LineOf(M, N), Directrix(G)) = True;FootPoint(LineOf(M, N), Directrix(G)) = N;N: Point", "coords": {} }, { "index": 7601, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7601.png", "answer": "5", "fact_expr": "G: Parabola;C: Circle;M: Point;A: Point;F: Point;Expression(G) = (y^2 = 8*x);Expression(C) = (-2*y - 8*x + x^2 + y^2 + 16 = 0);PointOnCurve(M, G);Focus(G) = F;PointOnCurve(A, C)", "coords": {} }, { "index": 7602, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.23606797749979, "major_axis": "y", "x_coef": 5.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7602.png", "answer": "12", "fact_expr": "l: Line;C: Ellipse;A: Point;F1: Point;B: Point;F2: Point;Expression(C) = (x^2/5 + y^2/9 = 1);Focus(C) = {F1,F2};PointOnCurve(F2, l);Intersection(l, C) = {A, B}", "coords": {} }, { "index": 7603, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7603.png", "answer": "8", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;l:Number;Expression(G) = (y^2 = 4*x);Slope(H)=l;Intersection(H,G)={A,B};PointOnCurve(Focus(G),H)", "coords": {} }, { "index": 7604, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7604.png", "answer": "2", "fact_expr": "E: Hyperbola;b: Number;a: Number;G: Line;P: Point;A: Point;F: Point;B: Point;a>0;b>0;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);RightFocus(E) = F;PointOnCurve(F, G);Intersection(G, E) = {A, B};Intersection(G, yAxis) = P;VectorOf(P, A) = 3*VectorOf(A, F);VectorOf(P, B) = (-7/3)*VectorOf(B, F)", "coords": {} }, { "index": 7605, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7605.png", "answer": "sqrt(3)/3", "fact_expr": "C: Ellipse;b: Number;a: Number;G: Line;A: Point;F1: Point;B: Point;F2: Point;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(F1, G);Intersection(G, C) = {A, B};Abs(LineSegmentOf(A, F1)) = 2*Abs(LineSegmentOf(B, F1));Abs(LineSegmentOf(A, B)) = Abs(LineSegmentOf(B, F2))", "coords": {} }, { "index": 7606, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7607, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7607.png", "answer": "11", "fact_expr": "G: Hyperbola;m: Real;Expression(G) = (-y^2/5 + x^2/m = 1);FocalLength(G) = 8", "coords": {} }, { "index": 7608, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7609, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "y", "x_coef": 9.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7609.png", "answer": "11", "fact_expr": "G: Ellipse;Expression(G) = (x^2/9 + y^2/16 = 1);H: Line;A: Point;B: Point;F1: Point;F2: Point;Focus(G) = {F1, F2};PointOnCurve(F2, H);Intersection(H, G) = {A, B};Abs(LineSegmentOf(A, B)) = 5", "coords": {} }, { "index": 7610, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "left" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7610.png", "answer": "4*x+y+3=0", "fact_expr": "M: Point;Coordinate(M) = (-1, 1);G: Parabola;Expression(G) = (y^2 = -8*x);A: Point;B: Point;IsChordOf(LineSegmentOf(A, B), G);MidPoint(LineSegmentOf(A, B)) = M", "coords": { "M": [ -1.0, 1.0 ] } }, { "index": 7611, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7611.png", "answer": "[1/3,1)", "fact_expr": "G: Ellipse;b: Number;a: Number;P: Point;F1: Point;F2: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Abs(LineSegmentOf(P, F1)) = 2*Abs(LineSegmentOf(P, F2));LeftFocus(G) =F1;RightFocus(G)=F2;PointOnCurve(P,G)", "coords": {} }, { "index": 7612, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "major_axis": "x", "x_coef": 3.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7612.png", "answer": "4*sqrt(3)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/3 + y^2 = 1);A: Point;B: Point;C: Point;PointOnCurve(B, G);PointOnCurve(C, G);K: Point;OneOf(Focus(G)) = A;OneOf(Focus(G)) = K;Negation(A=K);PointOnCurve(K, LineSegmentOf(B, C))", "coords": {} }, { "index": 7613, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7614, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.6277244091033936, "b": 1.818325400352478, "orientation": "horizontal" }, "optimization": { "final_loss": 6.648646149187698e-07, "converged": true, "iterations": 99 }, "output_path": "results/train/hyperbola/problem_7614.png", "answer": "sqrt(5)", "fact_expr": "G: Hyperbola;b: Number;b>0;Expression(G) = (x^2/4 - y^2/b^2 = 1);Eccentricity(G) = 3/2", "coords": {} }, { "index": 7615, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7616, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7616.png", "answer": "6", "fact_expr": "G: Parabola;A: Point;B: Point;M: Point;Expression(G) = (y^2 = 8*x);IsChordOf(LineSegmentOf(A,B),G);Length(LineSegmentOf(A, B)) = 16;MidPoint(LineSegmentOf(A, B)) = M", "coords": {} }, { "index": 7617, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7617.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Hyperbola;F1: Point;A: Point;B: Point;F2: Point;Expression(G) = (x^2/4 - y^2/3 = 1);Focus(G) = {F1, F2};L:Line;PointOnCurve(F2, L);IsPerpendicular(L,xAxis);Intersection(L, G) = {A, B}", "coords": {} }, { "index": 7618, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7618.png", "answer": "-1/3", "fact_expr": "F: Point;C: Parabola;Expression(C) = (y^2 = 4*x);Focus(C) = F;PointOnCurve(F, G) = True;Slope(G) = sqrt(3);G: Line;Intersection(G, C) = {A, B};A: Point;B: Point;Abs(VectorOf(F, A)) < Abs(VectorOf(F, B));VectorOf(F, A) = lambda*VectorOf(F, B);lambda: Number", "coords": {} }, { "index": 7619, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7620, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7620.png", "answer": "-1", "fact_expr": "G: Parabola;H: Line;c: Number;Expression(G) = (x^2 = y);Expression(H) = (-c*y + 2*x + 1 = 0);IsTangent(H, G)", "coords": {} }, { "index": 7621, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7622, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7623, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7623.png", "answer": "[\\sqrt{3}/3, 1)", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;H: Line;c: Number;Expression(H) = (x = a^2/c);P: Point;PointOnCurve(P, H);PointOnCurve(F2, PerpendicularBisector(LineSegmentOf(P, F1)))", "coords": {} }, { "index": 7624, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7624.png", "answer": "2/3", "fact_expr": "G: Ellipse;b: Number;a: Number;A: Point;F1: Point;F2: Point;M: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(A, G);Center(InscribedCircle(TriangleOf(A,F1,F2)))=M;VectorOf(M, F1) + 2*VectorOf(M, F2) + 2*VectorOf(M, A) = 0", "coords": {} }, { "index": 7625, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7625.png", "answer": "10", "fact_expr": "G: Parabola;f: LineSegment;Expression(G) = (y^2 = 4*x);PointOnCurve(Focus(G), f);IsChordOf(f, G);XCoordinate(MidPoint(f)) = 4", "coords": {} }, { "index": 7626, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7626.png", "answer": "(0,1/4)", "fact_expr": "G: Parabola;Expression(G) = (x^2 = y)", "coords": {} }, { "index": 7627, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7627.png", "answer": "8", "fact_expr": "G: Hyperbola;H: Line;F: Point;P: Point;Expression(G) = (x^2/16 - y^2/9 = 1);Expression(H) = (x = 16/5);Coordinate(F)=(5,0);PointOnCurve(P, G);Distance(P,F) = 10", "coords": { "F": [ 5.0, 0.0 ] } }, { "index": 7628, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7629, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.6259966743174588, "b": 2.298015331604694, "orientation": "horizontal" }, "optimization": { "final_loss": 1.93011531933813e-06, "converged": true, "iterations": 91 }, "output_path": "results/train/hyperbola/problem_7629.png", "answer": "y^2/2 - x^2 = 1", "fact_expr": "G: Hyperbola;H: Point;Coordinate(H) = (1, 2);Expression(Asymptote(G)) = (y = pm*(sqrt(2)*x));PointOnCurve(H, G)", "coords": { "H": [ 1.0, 2.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 7630, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7630.png", "answer": "9", "fact_expr": "C: Parabola;P: Point;F: Point;A: Point;B: Point;l: Line;Expression(C) = (y^2 = 8*x);Focus(C) = F;Directrix(C) = l;PointOnCurve(P, l);Intersection(LineOf(P,F), C) = {A, B};VectorOf(P, A) = 3*VectorOf(A, F)", "coords": {} }, { "index": 7631, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7631.png", "answer": "2", "fact_expr": "G: Ellipse;Expression(G) = (y^2 + x^2/a^2 = 1);a: Number;a>0;PointOnCurve(Focus(G), xAxis) = True;Length(MajorAxis(G)) = 2*Length(MinorAxis(G))", "coords": {} }, { "index": 7632, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7632.png", "answer": "{-1/2, 0, 1}", "fact_expr": "l: Line;G: Parabola;P: Point;Expression(G) = (y^2 = 4*x);Coordinate(P) = (-2, -1);PointOnCurve(P,l);NumIntersection(l,G)=1", "coords": { "P": [ -2.0, -1.0 ] } }, { "index": 7633, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7634, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7635, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7636, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7636.png", "answer": "(1, \\sqrt{2}+1]", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);a: Number;b: Number;a>0;b>0;F1: Point;c: Number;LeftFocus(G) = F1;Coordinate(F1) = (-c, 0);H: Line;Expression(H) = (y = k*x);k: Number;A: Point;B: Point;Intersection(H, G) = {A, B};Abs(VectorOf(A, F1)) = (c/a)*Abs(VectorOf(B, F1))", "coords": {} }, { "index": 7637, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7638, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7638.png", "answer": "7", "fact_expr": "F1: Point;LeftFocus(C) = F1;C: Hyperbola;Expression(C) = (x^2/4 - y^2 = 1);l: Line;O: Origin;PointOnCurve(O, l);P: Point;Q: Point;Intersection(l, C) = {P, Q};Abs(LineSegmentOf(P, F1)) = 3", "coords": {} }, { "index": 7639, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7639.png", "answer": "sqrt(2)/2", "fact_expr": "l: Line;G: Parabola;l1:Line;B: Point;A: Point;Q: Point;C: Point;P: Point;O: Origin;Expression(G) = (y^2 = 4*x);Coordinate(A) = (1, 2);PointOnCurve(A,l);Intersection(l,G)={A,B};MidPoint(LineSegmentOf(A,B))=Q;PointOnCurve(Q,l1);IsPerpendicular(l1,yAxis);Intersection(l1,G)=C;VectorOf(Q, C) = VectorOf(C, P)", "coords": { "A": [ 1.0, 2.0 ] } }, { "index": 7640, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7640.png", "answer": "x^2/2 + y^2 = 1", "fact_expr": "C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;OneOf(Focus(C)) = Focus(G);G: Parabola;Expression(G) = (y^2 = 4*x);H: Line;M: Point;Coordinate(M) = (-1/2, 1/2);PointOnCurve(M, H) ;Slope(H) = 1/2;Intersection(H, C) = {A, B};B: Point;A: Point;MidPoint(LineSegmentOf(A, B)) = M", "coords": { "M": [ -0.5, 0.5 ] } }, { "index": 7641, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "major_axis": "x", "x_coef": 2.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7641.png", "answer": "[2,2*sqrt(2))\n0", "fact_expr": "C: Ellipse;G: Line;y0: Number;x0: Number;P: Point;F1: Point;F2: Point;Expression(C)=(x^2/2 + y^2 = 1);Expression(G)=((x0*x)/2+y0*y=1);Focus(C)={F1,F2};Coordinate(P)=(x0,y0);00;Expression(G) = (y^2 = 2*(p*x));IsChordOf(LineSegmentOf(A, B), G);PointOnCurve(Focus(G), LineSegmentOf(A, B));Endpoint(LineSegmentOf(A, B))={A,B};Coordinate(A) = (x1, y1);Coordinate(B) = (x2, y2)", "coords": {} }, { "index": 7645, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7646, "success": false, "conic_type": "ellipse", "error": "validation: ellipse_point_off_curve", "params": { "a": 2.8503371627061314, "b": 1.8676572177554456, "major_axis": "x", "x_coef": 9.0, "y_coef": 4.0 }, "optimization": { "final_loss": 5.51866980271156e-07, "converged": true, "iterations": 103 }, "output_path": "results/train/ellipse/problem_7646.png", "answer": "x^2/9+y^2/4=1", "fact_expr": "G: Ellipse;P: Point;Q: Point;Coordinate(P) = (-3,0);Coordinate(Q) = (0,-2);PointOnCurve(P,G);PointOnCurve(Q,G)", "coords": { "P": [ -3.0, 0.0 ], "Q": [ 0.0, -2.0 ] }, "validation_reasons": [ "ellipse_point_off_curve" ] }, { "index": 7647, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7647.png", "answer": "1/3", "fact_expr": "G: Ellipse;a: Number;b: Number;F1: Point;P: Point;F2: Point;c:Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(F1) = (-c, 0);Coordinate(F2) = (c, 0);Focus(G) = {F1,F2} ;PointOnCurve(P, G);Area(TriangleOf(P, F1, F2)) = b^2*(sqrt(2)/2)", "coords": {} }, { "index": 7648, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7648.png", "answer": "(1/2, 1)", "fact_expr": "G: Parabola;H: Line;P: Point;Expression(G) = (y^2 = 2*x);Expression(H) = (x - y + 3 = 0);PointOnCurve(P, G);WhenMin(Distance(P, H))", "coords": {} }, { "index": 7649, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7649.png", "answer": "y=pm*sqrt(2)*x", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;H: Line;PointOnCurve(F2, H);IsPerpendicular(H, xAxis) ;OneOf(Intersection(H, G)) = P;P: Point;AngleOf(P, F1, F2) = pi/6", "coords": {} }, { "index": 7650, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7651, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7651.png", "answer": "sqrt(2)", "fact_expr": "G: Parabola;p: Number;F: Point;A: Point;M: Point;l: Line;B: Point;p>0;Expression(G) = (y^2 = 2*(p*x));Coordinate(A) = (0, 2);Focus(G) = F;Directrix(G) = l;Intersection(LineSegmentOf(F,A), G) = B;L:Line;PointOnCurve(B, L);IsPerpendicular(l,L);FootPoint(l,L) = M;IsPerpendicular(LineSegmentOf(A, M), LineSegmentOf(M, F))", "coords": { "A": [ 0.0, 2.0 ] } }, { "index": 7652, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7652.png", "answer": "1", "fact_expr": "C: Hyperbola;Expression(C) = (x^2 - y^2 = 1)", "coords": {} }, { "index": 7653, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7653.png", "answer": "2", "fact_expr": "E: Hyperbola;b: Number;a: Number;B: Point;A: Point;F: Point;a>0;b>0;Expression(E) = (-y^2/b^2 + x^2/a^2 = 1);LeftVertex(E) = A;OneOf(Endpoint(ImageinaryAxis(E)))=B;RightFocus(E)=F;Distance(F, LineOf(A,B)) = (3*b)/2", "coords": {} }, { "index": 7654, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 0.9550795555114746, "b": 2.8647773265838623, "orientation": "horizontal" }, "optimization": { "final_loss": 1.6550993677810766e-05, "converged": true, "iterations": 144 }, "output_path": "results/train/hyperbola/problem_7654.png", "answer": "pm*sqrt(6)/2", "fact_expr": "G: Hyperbola;b: Number;a: Number;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Eccentricity(G) = sqrt(10)/2", "coords": {} }, { "index": 7655, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7655.png", "answer": "8/3", "fact_expr": "G: Parabola;A: Point;B: Point;F: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(A, G);PointOnCurve(B, G);VectorOf(A, F) = 3*VectorOf(F, B);IsChordOf(LineSegmentOf(A, B), G)", "coords": {} }, { "index": 7656, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7656.png", "answer": "15", "fact_expr": "G: Ellipse;P: Point;M: Point;F1: Point;F2: Point;Expression(G) = (x^2/25 + y^2/16 = 1);Coordinate(M) = (6, 4);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(P, G)", "coords": { "M": [ 6.0, 4.0 ] } }, { "index": 7657, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7658, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7658.png", "answer": "(1/2, 2/3)", "fact_expr": "C: Ellipse;b: Number;a: Number;G: Line;B: Point;F: Point;e: Number;k: Number;a > b;b > 0;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);LeftVertex(C) = A;A: Point;PointOnCurve(A, G);Slope(G) = k;Intersection(C, G) = {A, B};Projection(B, xAxis) = F;RightFocus(C) = F;k = 1/2;Eccentricity(C) = e", "coords": {} }, { "index": 7659, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7660, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7660.png", "answer": "9", "fact_expr": "G:Parabola;p:Number;p<0;Expression(G) = (y^2 = 2*(p*x));F:Point;Focus(G)=F;l:Line;Directrix(G)=l;H:Point;Coordinate(H)=(-p/2,0);Z:Line;PointOnCurve(H,Z);C:Circle;Center(C)=F;O:Origin;PointOnCurve(O,C);N:Point;TangentPoint(Z,C)=N;M:Point;A:Point;B:Point;Intersection(LineOf(F,N),l)=M;Intersection(LineOf(F,N),G)={A,B};Between(A,M,N)", "coords": {} }, { "index": 7661, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "x", "x_coef": 4.0, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7661.png", "answer": "Preserve all mathematical expressions exactly (including LaTeX format, symbols, and numbers). \nDo not add explanations.", "fact_expr": "G: Hyperbola;P: Point;Coordinate(P) = (2, 1);PointOnCurve(P, G);H: Ellipse;Expression(H) = (x^2/4 + y^2 = 1);Focus(G) = Focus(H)", "coords": { "P": [ 2.0, 1.0 ] } }, { "index": 7662, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7663, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7664, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7664.png", "answer": "{2,8}", "fact_expr": "C: Parabola;p: Number;A: Point;M: Point;F: Point;p>0;Expression(C) = (y^2 = 2*(p*x));Coordinate(A) = (0, 2);Focus(C) = F;PointOnCurve(M, C);Abs(LineSegmentOf(M, F)) = 5;PointOnCurve(A, yAxis);IsPerpendicular(LineSegmentOf(A, M), LineSegmentOf(A, F))", "coords": { "A": [ 0.0, 2.0 ] } }, { "index": 7665, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 0.929953396320343, "b": 2.629857301712036, "orientation": "horizontal" }, "optimization": { "final_loss": 4.594453912432073e-06, "converged": true, "iterations": 175 }, "output_path": "results/train/hyperbola/problem_7665.png", "answer": "1/16", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/4 - y^2/b^2 = 1);b: Number;b>0;F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, RightPart(G));Eccentricity(G) = 3", "coords": {} }, { "index": 7666, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7667, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7667.png", "answer": "sqrt(3)", "fact_expr": "C1: Hyperbola;Expression(C1) = (x^2/a^2 - y^2/b^2 = 1);a: Number;b: Number;a>0;b>0;F1: Point;F2: Point;LeftFocus(C1) = F1;RightFocus(C1) = F2;O: Origin;C2: Parabola;Vertex(C2) = O;Directrix(C2) = LeftDirectrix(C1);P: Point;Intersection(C1, C2) = P;IsPerpendicular(LineSegmentOf(P,F2), LineSegmentOf(F1, F2)) = True", "coords": {} }, { "index": 7668, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7668.png", "answer": "1/4", "fact_expr": "G: Ellipse;b: Number;a: Number;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);P: Point;F1: Point;F2: Point;OneOf(Endpoint(MinorAxis(G))) = P;Focus(G) = {F1, F2};Radius(InscribedCircle(TriangleOf(P, F1, F2))) = b/5", "coords": {} }, { "index": 7669, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7670, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7670.png", "answer": "sqrt(7)/3", "fact_expr": "C: Parabola;M: Circle;G: Line;B: Point;A: Point;P0: Point;M0: Point;Expression(C) = (y^2 = 4*x);Expression(M) = (y^2 + (x + 1)^2 = 7/9);Center(M) = M0;PointOnCurve(M0, G);Intersection(G, C) = {A, B};VectorOf(M, B) = 3*VectorOf(M, A);PointOnCurve(P0, M)", "coords": {} }, { "index": 7671, "success": false, "conic_type": "hyperbola", "error": "validation: hyperbola_point_off_curve", "params": { "a": 1.9960020780563354, "b": 1.4970015287399292, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "iterations": 2 }, "output_path": "results/train/hyperbola/problem_7671.png", "answer": "sqrt(3)", "fact_expr": "G: Hyperbola;m: Number;M: Point;e:Number;Expression(G) = (y^2/2 - x^2/m = 1);Coordinate(M) = (2, 2);PointOnCurve(M, G);Eccentricity(G)=e", "coords": { "M": [ 2.0, 2.0 ] }, "validation_reasons": [ "hyperbola_point_off_curve" ] }, { "index": 7672, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7672.png", "answer": "sqrt(7)/7", "fact_expr": "l: Line;C: Parabola;G: Point;A: Point;B: Point;Expression(C) = (x^2 = 12*y);Coordinate(G) = (0, 3);PointOnCurve(G,l);Intersection(l, C) = {A, B};Abs(LineSegmentOf(A, B)) = 14;Inclination(l)=alpha;alpha:Number", "coords": { "G": [ 0.0, 3.0 ] } }, { "index": 7673, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7674, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7675, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7675.png", "answer": "3", "fact_expr": "G: Ellipse;H: Line;m: Number;F: Point;A: Point;B: Point;Expression(G) = (x^2/4 + y^2/3 = 1);Expression(H) = (x = m);LeftFocus(G) = F;Intersection(H, G) = {A, B};WhenMax(Perimeter(TriangleOf(F,A,B)))", "coords": {} }, { "index": 7676, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7676.png", "answer": "sqrt(3)/3", "fact_expr": "A: Point;B: Point;G: Ellipse;b: Number;a: Number;C: Point;a > b;b > 0;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);Coordinate(A) = (0, b);Intersection(LeftDirectrix(G), xAxis) = B;MidPoint(LineSegmentOf(A, B))=C;PointOnCurve(C, G)", "coords": {} }, { "index": 7677, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 3.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7677.png", "answer": "10", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 12*x);F: Point;Focus(G) = F;P: Point;Coordinate(P) = (2, 1);l: Line;PointOnCurve(P, l);A: Point;B: Point;Intersection(l, G) = {A, B};MidPoint(LineSegmentOf(A, B)) = P", "coords": { "P": [ 2.0, 1.0 ] } }, { "index": 7678, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7678.png", "answer": "[1, 5) + (5, +oo)", "fact_expr": "H: Line;Expression(H) = (-k*x + y - 1 = 0);k: Real;G: Ellipse;Expression(G) = (x^2/5 + y^2/m = 1);m: Number;IsIntersect(H, G)", "coords": {} }, { "index": 7679, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7680, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7681, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7682, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7682.png", "answer": "8", "fact_expr": "C: Hyperbola;A: Point;P: Point;F: Point;Coordinate(A) = (0, 3);LeftFocus(C) = F;Expression(C)=(x^2/4-y^2/3=1);PointOnCurve(P,RightPart(C))", "coords": { "A": [ 0.0, 3.0 ] } }, { "index": 7683, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7683.png", "answer": "y = pm*2*x", "fact_expr": "l: Line;G: Hyperbola;b: Number;a: Number;P: Point;F1: Point;F2: Point;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(G) = F1;RightFocus(G) = F2;PointOnCurve(F1, l);Slope(l) = 2;Intersection(l, LeftPart(G)) = P;IsPerpendicular(LineOf(P, F2), l)", "coords": {} }, { "index": 7684, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7684.png", "answer": "sqrt(2)/2", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - y^2 = a^2);a>0;a: Number;H: Parabola;Expression(H) = (y^2 = 4*x);RightFocus(G) = Focus(H)", "coords": {} }, { "index": 7685, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7685.png", "answer": "-1/2", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);P: Point;Coordinate(P) = (4, 4);PointOnCurve(P, G) = True;PointOnCurve(P, LineOf(P, A)) = True;PointOnCurve(P, LineOf(P, B)) = True;Intersection(LineOf(P, A), G) = A;Intersection(LineOf(P, B), G) = B;A: Point;Coordinate(A) = (x1, y1);x1: Number;y1: Number;B: Point;Coordinate(B) = (x2, y2);x2: Number;y2: Number;Slope(LineOf(P, A))+Slope(LineOf(P, B)) = 0", "coords": { "P": [ 4.0, 4.0 ] } }, { "index": 7686, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7686.png", "answer": "2*sqrt(3)/3", "fact_expr": "G: Hyperbola;b: Number;H: Circle;C: Point;E: Point;D: Point;F: Point;b>0;Expression(G) = (x^2/9 - y^2/b^2 = 1);Expression(H) = (x^2 + y^2 = 9);L1:Line;L2:Line;TangentOfPoint(F, H) = {L1,L2};RightFocus(G)=F;TangentPoint(L1,H)=C;TangentPoint(L2,H)=D;RightVertex(G)=E;AngleOf(C, E, D) = ApplyUnit(150, degree)", "coords": {} }, { "index": 7687, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7687.png", "answer": "(\\sqrt{3}+1)/2", "fact_expr": "G: Hyperbola;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a>0;b>0;P: Point;PointOnCurve(P, RightPart(G));F2: Point;RightFocus(G) = F2;Slope(LineOf(P, F2)) = sqrt(3);M: Point;MidPoint(LineSegmentOf(P, F2)) = M;O: Origin;Abs(LineSegmentOf(O, F2)) = Abs(LineSegmentOf(F2, M))", "coords": {} }, { "index": 7688, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7688.png", "answer": "(\\sqrt{2}/2,1)", "fact_expr": "A2: Point;A1: Point;C: Ellipse;Expression(C) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;LeftVertex(C) = A1;RightVertex(C) = A2;P: Point;PointOnCurve(P, C);Negation(P = A1);Negation(P = A2);IsPerpendicular(LineSegmentOf(P, O), LineSegmentOf(P, A2));O: Origin", "coords": {} }, { "index": 7689, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7690, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2631799883090584, "b": 2.5275303393962245, "orientation": "horizontal" }, "optimization": { "final_loss": 4.481942266905428e-06, "converged": true, "iterations": 95 }, "output_path": "results/train/hyperbola/problem_7690.png", "answer": "x^2 - y^2/4 = 1", "fact_expr": "G: Hyperbola;H: Point;Coordinate(H) = (sqrt(5), 0);Expression(Asymptote(G)) = (y = pm*2*x);OneOf(Focus(G))=H", "coords": { "H": [ 2.23606797749979, 0.0 ] } }, { "index": 7691, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7692, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 7.0, "b": 4.898979485566356, "major_axis": "x", "x_coef": 49.0, "y_coef": 24.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7692.png", "answer": "24", "fact_expr": "G: Ellipse;Expression(G) = (x^2/49 + y^2/24 = 1);P: Point;PointOnCurve(P, G);F1: Point;F2: Point;Focus(G) = {F1, F2};IsPerpendicular(LineSegmentOf(P, F1), LineSegmentOf(P, F2))", "coords": {} }, { "index": 7693, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.2631799883090584, "b": 2.5275303393962245, "orientation": "horizontal" }, "optimization": { "final_loss": 4.481942266905428e-06, "converged": true, "iterations": 95 }, "output_path": "results/train/hyperbola/problem_7693.png", "answer": "x^2/16 - y^2/64 = 1", "fact_expr": "G: Hyperbola;Expression(OneOf(Asymptote(G))) = (y = 2*x);H: Point;Coordinate(H) = (4, 0);PointOnCurve(H, OneOf(Asymptote(G)))", "coords": { "H": [ 4.0, 0.0 ] } }, { "index": 7694, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.25, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7694.png", "answer": "1/2", "fact_expr": "G: Parabola;Expression(G) = (x^2 = y)", "coords": {} }, { "index": 7695, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0599746704101562, "b": 2.576634645462036, "orientation": "horizontal" }, "optimization": { "final_loss": 9.342352313979063e-07, "converged": true, "iterations": 116 }, "output_path": "results/train/hyperbola/problem_7695.png", "answer": "4*sqrt(3)", "fact_expr": "C: Hyperbola;m: Number;F1: Point;F2: Point;M: Point;m>0;Expression(C) = (-y^2/4 + x^2/m = 1);Coordinate(F1) = (-4, 0);Coordinate(F2) = (4, 0);Focus(C) = {F1, F2};PointOnCurve(M, C);AngleOf(F1, M, F2) = ApplyUnit(60, degree)", "coords": { "F1": [ -4.0, 0.0 ], "F2": [ 4.0, 0.0 ] } }, { "index": 7696, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7696.png", "answer": "(2,pm*2*sqrt(2))", "fact_expr": "G: Parabola;H: Line;A: Point;F: Point;B: Point;Expression(G) = (y^2 = 4*x);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};VectorOf(A, F) = 2*VectorOf(F, B)", "coords": {} }, { "index": 7697, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.23606797749979, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 5.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7697.png", "answer": "x^2/10 + y^2/8 = 1 (the answer is not unique)", "fact_expr": "C: Ellipse;Expression(C) = (x^2/5 + y^2/3 = 1);C1: Ellipse;Focus(C) = Focus(C1)", "coords": {} }, { "index": 7698, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 4.0, "b": 3.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7698.png", "answer": "3", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/16 - y^2/9 = 1)", "coords": {} }, { "index": 7699, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7699.png", "answer": "4", "fact_expr": "G: Parabola;H: Line;A: Point;B: Point;C: Point;F: Point;Expression(G) = (x^2 = 4*y);Focus(G) = F;PointOnCurve(F, H);Intersection(H, G) = {A, B};Intersection(TangentOnPoint(A,G),TangentOnPoint(B,G))=C", "coords": {} }, { "index": 7700, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7700.png", "answer": "y^2=4*x", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;H: Circle;p>0;Expression(H) = (-6*x + x^2 + y^2 - 7 = 0);IsTangent(H,Directrix(G))", "coords": {} }, { "index": 7701, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 4.0, "b": 3.0, "major_axis": "x", "x_coef": 16.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7701.png", "answer": "9*x + 8*y - 26 = 0", "fact_expr": "G: Ellipse;H: LineSegment;M: Point;Expression(G) = (x^2/16 + y^2/9 = 1);Coordinate(M) = (2, 1);IsChordOf(H, G);MidPoint(H) = M;PointOnCurve(M, H)", "coords": { "M": [ 2.0, 1.0 ] } }, { "index": 7702, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7702.png", "answer": "2", "fact_expr": "G: Parabola;p: Number;P: Point;Q: Point;F: Point;t: Real;l:Line;p>0;Expression(G) = (x^2 = 2*p*y);Coordinate(P) = (4, t);Focus(G) = F;Directrix(G)=l;Intersection(l, yAxis) = Q;PointOnCurve(P,G);LineSegmentOf(P,Q)=sqrt(2)*LineSegmentOf(P,F)", "coords": {} }, { "index": 7703, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 8.660254037844387, "b": 5.0, "major_axis": "y", "x_coef": 25.0, "y_coef": 75.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7703.png", "answer": "(1/2, -1/2)", "fact_expr": "G: Ellipse;H: Line;l: LineSegment;M: Point;Expression(G) = (x^2/25 + y^2/75 = 1);Expression(H) = (x = 1/2);Slope(l) = 3;IsChordOf(l, G);MidPoint(l) = M;Intersection(G, H) = M", "coords": {} }, { "index": 7704, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7704.png", "answer": "sqrt(3)/4", "fact_expr": "C: Ellipse;G: Line;B: Point;A: Point;F: Point;M: Point;Expression(C) = (x^2/4 + y^2/3 = 1);Coordinate(F) = (1, 0);PointOnCurve(F, G);Intersection(G, C) = {A, B};MidPoint(LineSegmentOf(A,B))=M", "coords": { "F": [ 1.0, 0.0 ] } }, { "index": 7705, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7706, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 6.0, "b": 3.0, "major_axis": "x", "x_coef": 36.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7706.png", "answer": "2", "fact_expr": "C: Ellipse;F1: Point;F2: Point;M: Point;N: Point;O: Origin;Expression(C) = (x^2/36 + y^2/9 = 1);LeftFocus(C) = F1;RightFocus(C) = F2;PointOnCurve(M, C);MidPoint(LineSegmentOf(M, F1)) = N;Abs(LineSegmentOf(M, N)) = 4", "coords": {} }, { "index": 7707, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7708, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.0, "b": 1.7320508075688772, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7708.png", "answer": "9", "fact_expr": "F2: Point;RightFocus(C) = F2;C: Hyperbola;Expression(C) = (x^2/9 - y^2/3 = 1);A: Point;PointOnCurve(A, LeftPart(C));B: Point;PointOnCurve(B, E);E: Circle;Expression(E) = (x^2 + (y + 2)^2 = 1)", "coords": {} }, { "index": 7709, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7709.png", "answer": "sqrt(2)", "fact_expr": "G: Parabola;Expression(G) = (y = x^2/2);F: Point;Focus(G) = F;l: Line;Directrix(G) = l;M: Point;PointOnCurve(M, l);N: Point;Intersection(LineSegmentOf(M, F), G) = N;Abs(LineSegmentOf(M, N)) = sqrt(2)*Abs(LineSegmentOf(N, F))", "coords": {} }, { "index": 7710, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7711, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7711.png", "answer": "x=-1", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 2*p*x);p: Number;Coordinate(Focus(G)) = (1, 0)", "coords": {} }, { "index": 7712, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7712.png", "answer": "sqrt(3)", "fact_expr": "C: Ellipse;M: Point;P: Point;F: Point;x1: Number;y1: Number;Expression(C) = (x^2/25 + y^2/16 = 1);Coordinate(P) = (x1, y1);PointOnCurve(P, C);RightFocus(C) = F;Abs(LineSegmentOf(M, F)) = 1;IsPerpendicular(LineSegmentOf(M, P), LineSegmentOf(M, F))", "coords": {} }, { "index": 7713, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7713.png", "answer": "8", "fact_expr": "l: Line;C: Parabola;A: Point;B: Point;F: Point;Expression(C) = (y^2 = 4*x);Focus(C) = F;Expression(l) = (y = x - 1);Intersection(l, C) = {A, B}", "coords": {} }, { "index": 7714, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 1.4142135623730951, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7714.png", "answer": "y=sqrt(2)*x+1", "fact_expr": "C: Hyperbola;G: Line;Expression(C) = (x^2 - y^2/2 = 1);IsParallel(Asymptote(C),G);Slope(G)>0", "coords": {} }, { "index": 7715, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7715.png", "answer": "sqrt(2)/2", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;F: Point;LeftFocus(G) = F;P: Point;PointOnCurve(P, G);IsPerpendicular(LineSegmentOf(P, F), xAxis);O: Origin;A: Point;B: Point;RightVertex(G) = A;UpperVertex(G) = B;IsParallel(LineSegmentOf(O, P), LineSegmentOf(A, B))", "coords": {} }, { "index": 7716, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 0.5, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7716.png", "answer": "(1/2,1)", "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (y^2 = 2*x);Coordinate(A) = (3, 1);PointOnCurve(P, G);Focus(G)=F;WhenMin(Abs(LineSegmentOf(P,F))+Abs(LineSegmentOf(P,A)))", "coords": { "A": [ 3.0, 1.0 ] } }, { "index": 7717, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7718, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.3300262689590454, "b": 2.30381441116333, "orientation": "horizontal" }, "optimization": { "final_loss": 6.658357051492203e-06, "converged": true, "iterations": 122 }, "output_path": "results/train/hyperbola/problem_7718.png", "answer": "x^2/9 - y^2/27 = 1", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Parabola;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (y^2 = 24*x);Expression(OneOf(Asymptote(G))) = (y = sqrt(3)*x);PointOnCurve(OneOf(Focus(G)), Directrix(H))", "coords": {} }, { "index": 7719, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7719.png", "answer": "-3/4", "fact_expr": "G: Ellipse;Expression(G) = (x^2/4 + y^2/3 = 1);Endpoint(MajorAxis(G)) = {M,N};M: Point;N: Point;P: Point;PointOnCurve(P, G) = True", "coords": {} }, { "index": 7720, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7721, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 8.0, "b": 6.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7721.png", "answer": "33", "fact_expr": "G: Hyperbola;P: Point;F1: Point;F2: Point;Expression(G) = (x^2/64 - y^2/36 = 1);PointOnCurve(P, G);Focus(G) = {F1, F2};Abs(LineSegmentOf(P, F1)) = 17", "coords": {} }, { "index": 7722, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7722.png", "answer": "2", "fact_expr": "G: Hyperbola;b: Number;a: Number;H: Curve;a>0;b>0;Expression(G) = (-y^2/b^2 + x^2/a^2 = 1);Expression(H) = (-4*x + x^2 + y^2 + 1 = 0);IsTangent(Asymptote(G), H)", "coords": {} }, { "index": 7723, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 3.1622776601683795, "b": 2.449489742783178, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7723.png", "answer": "y^0 = -16 * x", "fact_expr": "C: Hyperbola;G: Parabola;O: Origin;Expression(C) = (x^2/10 - y^2/6 = 1);Vertex(G)=O;SymmetryAxis(G)=xAxis;Focus(G)=LeftFocus(C)", "coords": {} }, { "index": 7724, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7724.png", "answer": "y = pm*(1/2)*x", "fact_expr": "E: Hyperbola;Expression(E) = (x^2/4 - y^2 = 1)", "coords": {} }, { "index": 7725, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7726, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.449489742783178, "b": 2.0, "major_axis": "x", "x_coef": 6.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7726.png", "answer": "2*sqrt(2)", "fact_expr": "G: Ellipse;Expression(G) = (x^2/6 + y^2/4 = 1)", "coords": {} }, { "index": 7727, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7728, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7728.png", "answer": "x^2=sqrt(2)*y", "fact_expr": "C1: Parabola;Expression(C1) = (y^2=4*x);C2: Parabola;Expression(C2) = (x^2=2*p*y);p: Number;p>0;O: Origin;Intersection(C1,C2) = A;A: Point;Negation(A=O);Distance(A,Focus(C1)) = 3", "coords": {} }, { "index": 7729, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 3.0, "b": 2.0, "major_axis": "y", "x_coef": 4.0, "y_coef": 9.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7729.png", "answer": "4", "fact_expr": "G: Ellipse;P: Point;F2: Point;F1: Point;Expression(G)=(x^2/4+y^2/9=1);LeftFocus(G)=F1;RightFocus(G)=F2;PointOnCurve(P,G)", "coords": {} }, { "index": 7730, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7731, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7731.png", "answer": "3", "fact_expr": "G: Parabola;A: Point;P: Point;F: Point;Expression(G) = (y^2 = 4*x);Coordinate(A) = (2, 2);Focus(G) = F;PointOnCurve(P, G)", "coords": { "A": [ 2.0, 2.0 ] } }, { "index": 7732, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7733, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.4142135623730951, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7733.png", "answer": "(-1, -\\sqrt{2}/2) + (-\\sqrt{2}/2, \\sqrt{2}/2) + (\\sqrt{2}/2, 1)", "fact_expr": "C: Hyperbola;l: Line;k: Number;A: Point;B: Point;Expression(l) = (y = k*x + 1);Expression(C) = (x^2/2 - y^2 = 1);Intersection(l, C) = {A, B};Negation(A = B)", "coords": {} }, { "index": 7734, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7735, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7735.png", "answer": "3", "fact_expr": "C: Ellipse;Expression(C) = (x^2/4 + y^2/3 = 1);RightFocus(C) = F;F: Point;O: Origin;l: Line;PointOnCurve(O,l) = True;Intersection(l,C) = {A,B};B: Point;A: Point;LineSegmentOf(A, F) = 1", "coords": {} }, { "index": 7736, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7736.png", "answer": "16", "fact_expr": "l: Line;G: Parabola;M: Point;N: Point;A: Point;B: Point;Expression(G) = (y^2 = 8*x);Coordinate(M) = (2, 0);Coordinate(N) = (3, 1);PointOnCurve(M, l);PointOnCurve(N,l);Intersection(l,G)={A,B}", "coords": { "M": [ 2.0, 0.0 ], "N": [ 3.0, 1.0 ] } }, { "index": 7737, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.0, "b": 0.7071067811865476, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7737.png", "answer": "(\\sqrt{6}/2,0)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2 - 2*y^2 = 1)", "coords": {} }, { "index": 7738, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7738.png", "answer": "100", "fact_expr": "T: Point;Coordinate(T) = (-2, 2);TangentOfPoint(T,C) = {l1,l2};l1: Line;l2: Line;C: Parabola;Expression(C) = (y^2 = 2*p*x);p: Number;TangentPoint(l1,C) = A;TangentPoint(l2,C) = B;A: Point;B: Point;Focus(C) = F;F: Point;PointOnCurve(F,LineOf(A,B)) = True", "coords": { "T": [ -2.0, 2.0 ] } }, { "index": 7739, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7739.png", "answer": "6", "fact_expr": "G: Hyperbola;m: Number;H: Parabola;Expression(G) = (-y^2/3 + x^2/m = 1);Expression(H) = (y^2 = 12*x);RightFocus(G) = Focus(H)", "coords": {} }, { "index": 7740, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true }, "output_path": "results/train/parabola/problem_7740.png", "answer": "(2,2*sqrt(2))", "fact_expr": "C: Parabola;p: Number;G: Circle;P: Point;F: Point;Expression(C) = (y^2 = 2*(p*x));Expression(G) = (-2*x + x^2 + y^2 = 0);Quadrant(P) = 1;PointOnCurve(P, C);LineSegmentOf(P, F) = 3;Focus(C)=F;Center(G)=F", "coords": {} }, { "index": 7741, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.7320508075688772, "major_axis": "x", "x_coef": 4.0, "y_coef": 3.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7741.png", "answer": "sqrt(6)/3", "fact_expr": "G: Ellipse;Expression(G) = (y^2/b^2 + x^2/a^2 = 1);b: Number;a: Number;a > b;b > 0;O: Origin;PointOnCurve(O, L) = True;Inclination(L) = pi/6;Intersection(L, G) = {P, Q};Q: Point;P: Point;L: Line;IsDiameter(LineSegmentOf(P, Q), H) = True;H: Circle;F: Point;RightFocus(G) = F;PointOnCurve(F, H) = True", "coords": {} }, { "index": 7742, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7742.png", "answer": "1/5", "fact_expr": "G: Parabola;Expression(G) = (y^2 = 4*x);H: Line;Expression(H) = (y = 2*x - 4);A: Point;B: Point;Intersection(G, H) = {A, B};C: Point;PointOnCurve(C, G);O: Origin;lambda: Real;VectorOf(O, A) + VectorOf(O, B) = lambda*VectorOf(O, C)", "coords": {} }, { "index": 7743, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 5.0, "b": 4.0, "major_axis": "x", "x_coef": 25.0, "y_coef": 16.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7743.png", "answer": "12/5", "fact_expr": "Gamma: Ellipse;Expression(Gamma) = (x^2/25 + y^2/16 = 1);P: Point;l1: Line;l2: Line;PointOnCurve(P,l1) = True;PointOnCurve(P,l2) = True;IsPerpendicular(l1,l2) = True;Intersection(l1,Gamma) = {A,B};A: Point;B: Point;Intersection(l2,Gamma) = {C,D};C: Point;D: Point;O: Origin;1/(Abs(LineSegmentOf(C, P))*Abs(LineSegmentOf(D, P))) + 1/(Abs(LineSegmentOf(A, P))*Abs(LineSegmentOf(B, P))) = 41/256", "coords": {} }, { "index": 7744, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7745, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7745.png", "answer": "4/3", "fact_expr": "G: Hyperbola;a: Number;H: Ellipse;Expression(G) = (-y^2/7 + x^2/a = 1);Expression(H) = (x^2/25 + y^2/9 = 1);Focus(G) = Focus(H)", "coords": {} }, { "index": 7746, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7747, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 1.0, "direction": "right" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7747.png", "answer": "3", "fact_expr": "C: Curve;Expression(C) = (y^2 = 4*x);F: Point;Focus(C) = F;G: Parabola;P: Point;PointOnCurve(P, G) = True;A: Point;Coordinate(A) = (2, 1)", "coords": { "A": [ 2.0, 1.0 ] } }, { "index": 7748, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7749, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 1.0, "b": 0.7071067811865476, "major_axis": "y", "x_coef": 0.5, "y_coef": 1.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7749.png", "answer": "sqrt(2)", "fact_expr": "G: Ellipse;Expression(G) = (2*x^2 + y^2 = 1)", "coords": {} }, { "index": 7750, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.8284271247461903, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7750.png", "answer": "16/7", "fact_expr": "C: Hyperbola;Expression(C) = (x^2/8 - y^2 = 1);RightFocus(C) = F;F: Point;Asymptote(C) = {l1,l2};l1: Line;l2: Line;PointOnCurve(F,l) = True;l: Line;Intersection(l,l1) = A;Intersection(l,l2) = B;A: Point;B: Point;IsPerpendicular(LineSegmentOf(A,B),l2) = True", "coords": {} }, { "index": 7751, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "vertical" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7751.png", "answer": "3", "fact_expr": "H: Ellipse;Expression(H) = (x^2/4 + y^2/a^2 = 1);a: Number;G: Hyperbola;Expression(G) = (y^2/2 - x^2/a = 1);Focus(H) = Focus(G)", "coords": {} }, { "index": 7752, "success": false, "error": "Unsupported or unparseable expression" }, { "index": 7753, "success": true, "conic_type": "ellipse", "error": null, "params": { "a": 2.0, "b": 1.0, "major_axis": "y", "x_coef": 1.0, "y_coef": 4.0 }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/ellipse/problem_7753.png", "answer": "sqrt(6)/3", "fact_expr": "G: Ellipse;Expression(G) = (x^2 + y^2/3 = 1)", "coords": {} }, { "index": 7754, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 2.0, "b": 1.5, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7754.png", "answer": "(1,\\sqrt{5}]", "fact_expr": "C: Hyperbola;b: Number;a: Number;F1: Point;F2: Point;O: Origin;M: Point;a>0;b>0;Expression(C) = (-y^2/b^2 + x^2/a^2 = 1);LeftFocus(C) =F1;RightFocus(C)=F2;PointOnCurve(M, RightPart(C));Abs(LineSegmentOf(F1, F2)) = 2*Abs(LineSegmentOf(O, M));Tan(AngleOf(M, F2, F1)) >= 2", "coords": {} }, { "index": 7755, "success": true, "conic_type": "parabola", "error": null, "params": { "p": 2.0, "direction": "up" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/parabola/problem_7755.png", "answer": "10", "fact_expr": "G: Parabola;P: Point;Expression(G) = (x^2 = 8*y);PointOnCurve(P, G);Distance(P, Focus(G)) = 12", "coords": {} }, { "index": 7756, "success": true, "conic_type": "hyperbola", "error": null, "params": { "a": 1.7320508075688772, "b": 1.0, "orientation": "horizontal" }, "optimization": { "final_loss": 0.0, "converged": true, "note": "using explicit params" }, "output_path": "results/train/hyperbola/problem_7756.png", "answer": "5+2*sqrt(3)", "fact_expr": "G: Hyperbola;Expression(G) = (x^2/3 - y^2 = 1);F1: Point;F2: Point;LeftFocus(G) = F1;RightFocus(G) = F2;P: Point;PointOnCurve(P, RightPart(G));Q: Point;Coordinate(Q) = (-2, 3)", "coords": { "Q": [ -2.0, 3.0 ] } } ] }