diff --git "a/data/math500.jsonl" "b/data/math500.jsonl"
new file mode 100644--- /dev/null
+++ "b/data/math500.jsonl"
@@ -0,0 +1,100 @@
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Distinct prime factorizations correspond to distinct integers, so there are $\\boxed{9}$ divisors of 196."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:9", "label": "9", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "9", "count": 100}]}], "n_distinct": 1, "modal_answer": "9", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 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Therefore, $\\boxed{\\text{Evelyn}}$ is our answer."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:Evelyn", "label": "Evelyn", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "Evelyn", "count": 100}]}], "n_distinct": 1, "modal_answer": "Evelyn", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 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+{"instance_id": "math500-0005", "gold": ["42"], "meta": {"row_pos": 5, "level": 2, "subject": "Prealgebra", "unique_id": "test/prealgebra/1622.json", "solution": "The side length of the hexagon is equal to the side length of one of the equilateral triangles.  Since the hexagon has six sides and the triangle has three sides, the perimeter of the hexagon is twice as large as the perimeter of a triangle.  Therefore, the perimeter of the hexagon is $2(21\\text{ inches})=\\boxed{42}$ inches.\n\n[asy]\nunitsize(10mm);\ndefaultpen(linewidth(.7pt)+fontsize(8pt));\n\nint i;\n\nfor(i=1;i<=6;++i)\n{\n\ndraw(dir(60*i)--dir(60*(i+1)));\n}\n\nfor(i=1;i<=3;++i)\n{\n\ndraw(dir(60*i)--dir(60*(i+3)));\n}[/asy]"}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:42", "label": "42", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "42", "count": 100}]}], "n_distinct": 1, "modal_answer": "42", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 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+{"instance_id": "math500-0006", "gold": ["27"], "meta": {"row_pos": 6, "level": 3, "subject": "Number Theory", "unique_id": "test/number_theory/515.json", "solution": "The sum of three consecutive integers takes the form $(k-1)+(k)+(k+1)=3k$ and hence is a multiple of 3.  Conversely, if a number $n$ is a multiple of 3, then $n/3-1$, $n/3$, and $n/3+1$ are three consecutive integers that sum to give $n$.  Therefore, a number is a sum of three consecutive integers if and only if it is a multiple of 3.  The smallest positive perfect cube that is a multiple of 3 is $3^3=\\boxed{27}$."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:27", "label": "27", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "27", "count": 100}]}], "n_distinct": 1, "modal_answer": "27", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 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+{"instance_id": "math500-0007", "gold": ["90^\\circ"], "meta": {"row_pos": 7, "level": 4, "subject": "Precalculus", "unique_id": "test/precalculus/927.json", "solution": "For the first line, let $t = 2x = 3y = -z.$  Then\n\\[\\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix} = \\begin{pmatrix} t/2 \\\\ t/3 \\\\ -t \\end{pmatrix} = \\frac{t}{6} \\begin{pmatrix} 3 \\\\ 2 \\\\ -6 \\end{pmatrix}.\\]Thus, the direction vector of the first line is $\\begin{pmatrix} 3 \\\\ 2 \\\\ -6 \\end{pmatrix}.$\n\nFor the second line, let $t = 6x = -y = -4z.$  Then\n\\[\\begin{pmatrix} x \\\\ y \\\\ z \\end{pmatrix} = \\begin{pmatrix} t/6 \\\\ -t \\\\ -t/4 \\end{pmatrix} = \\frac{t}{12} \\begin{pmatrix} 2 \\\\ -12 \\\\ -3 \\end{pmatrix}.\\]Thus, the direction vector of the first line is $\\begin{pmatrix} 2 \\\\ -12 \\\\ -3 \\end{pmatrix}.$\n\nNote that\n\\[\\begin{pmatrix} 3 \\\\ 2 \\\\ -6 \\end{pmatrix} \\cdot \\begin{pmatrix} 2 \\\\ -12 \\\\ -3 \\end{pmatrix} = 0.\\]Hence, the angle between the lines is 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We get \\begin{align*}\n2\\cdot 3 \\cdot 4 \\cdot (5 + 1) &= 144, \\\\\n2\\cdot 3 \\cdot (4 \\cdot 5 + 1) &= 126,\\\\\n2\\cdot (3 \\cdot 4 \\cdot 5 + 1) &= 122, \\\\\n(2\\cdot 3 \\cdot 4 \\cdot 5) + 1 \\hphantom{)} &= 121.\n\\end{align*}In total there are $\\boxed{4}$ possible values for the expression."}, "n_samples": 100, "n_success": 86, "n_answered": 86, "n_bare": 0, "n_boxed": 86, "n_bold": 0, "n_correct": 61, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:4", "label": "4", "count": 61, "match": "exact", "judge": null, "members": [{"raw": "4", "count": 61}]}, {"bucket": "math:6", "label": "6", "count": 9, "match": "none", "judge": null, "members": [{"raw": "6", "count": 9}]}, {"bucket": "math:7", "label": "7", "count": 6, "match": "none", "judge": null, "members": [{"raw": "7", "count": 6}]}, {"bucket": "math:8", "label": "8", "count": 4, "match": "none", "judge": null, "members": [{"raw": "8", "count": 4}]}, {"bucket": "math:5", "label": "5", "count": 2, 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Since $M$ is minimal, we take exactly three 2's and do not have any additional 0's: $M=\\boxed{2220}$."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:2220", "label": "2220", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "2220", "count": 100}]}], "n_distinct": 1, "modal_answer": "2220", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 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+{"instance_id": "math500-0011", "gold": ["\\frac{3}{56}"], "meta": {"row_pos": 11, "level": 5, "subject": "Intermediate Algebra", "unique_id": "test/intermediate_algebra/1197.json", "solution": "Let $q(x) = (x^2 - 1) p(x) - x.$  Then $q(x)$ has degree 7, and $q(n) = 0$ for $n = 2$, 3, 4, $\\dots,$ 7, so\n\\[q(x) = (ax + b)(x - 2)(x - 3) \\dotsm (x - 7)\\]for some constants $a$ and $b.$\n\nWe know that $q(1) = (1^2 - 1)p(1) - 1 = -1.$  Setting $x = 1$ in the equation above, we get\n\\[q(1) = 720(a + b),\\]so $a + b = -\\frac{1}{720}.$\n\nWe also know that $q(-1) = ((-1)^2 - 1)p(-1) + 1 = 1.$  Setting $x = -1$ in the equation above, we get\n\\[q(-1) = 20160(-a + b),\\]so $-a + b = \\frac{1}{20160}.$  Solving for $a$ and $b,$ we find $a = -\\frac{29}{40320}$ and $b = -\\frac{3}{4480}.$  Hence,\n\\begin{align*}\nq(x) &= \\left( -\\frac{29}{40320} x - \\frac{3}{4480} \\right) (x - 2)(x - 3) \\dotsm (x - 7) \\\\\n&= -\\frac{(29x + 27)(x - 2)(x - 3) \\dotsm (x - 7)}{40320}.\n\\end{align*}In 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The sum of the proper divisors of $284$ is\n\\begin{align*}\n1+2+2^2+71+2 \\cdot 71 &= (1+2+2^2)(1+71)-284 \\\\\n&= 220 \\\\\n&= 2^2\\cdot5\\cdot11.\n\\end{align*}Here we have used the observation that multiplying out $(1+2+2^2)(1+71)$ by distributing yields an expression which is the sum of all $6$ factors of $284.$ Applying this observation again, we find that the sum of the proper divisors of $220$ is $$(1+2+2^2)(1+5)(1+11)-220=7\\cdot 6\\cdot 12-220=\\boxed{284}.$$"}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:284", "label": "284", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "284", "count": 100}]}], "n_distinct": 1, "modal_answer": "284", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 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The height of the cylinder is $\\boxed{5}$ cm."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:5", "label": "5", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "5", "count": 100}]}], "n_distinct": 1, "modal_answer": "5", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99], 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+{"instance_id": "math500-0018", "gold": ["28"], "meta": {"row_pos": 18, "level": 1, "subject": "Geometry", "unique_id": "test/geometry/434.json", "solution": "Angle $\\angle BCA$ and the angle we're trying to measure are alternate interior angles, so they are congruent. Thus, $\\angle BCA=x^\\circ$:\n\n[asy]\ndraw((0,0)--(10,0));\ndraw((0,3)--(10,3));\ndraw((2,3)--(8,0));\ndraw((2,3)--(4,0));\nlabel(\"$A$\",(2,3),N);\nlabel(\"$B$\",(4,0),S);\nlabel(\"$C$\",(8,0),S);\nlabel(\"$124^{\\circ}$\",(2,3),SW);\nlabel(\"$x^{\\circ}$\",(4.5,3),S);\nlabel(\"$x^{\\circ}$\",(6,0),N);\n[/asy]\n\nSince $AB=BC$, we know that $\\triangle ABC$ is isosceles with equal angles at $C$ and $A$. Therefore, $\\angle BAC = x^\\circ$:\n\n[asy]\ndraw((0,0)--(10,0));\ndraw((0,3)--(10,3));\ndraw((2,3)--(8,0));\ndraw((2,3)--(4,0));\nlabel(\"$A$\",(2,3),N);\nlabel(\"$B$\",(4,0),S);\nlabel(\"$C$\",(8,0),S);\nlabel(\"$124^{\\circ}$\",(2,3),SW);\nlabel(\"$x^{\\circ}$\",(4.5,3),S);\nlabel(\"$x^{\\circ}$\",(6,0),N);\nlabel(\"$x^{\\circ}$\",(3.6,1.7));\n[/asy]\n\nThe sum of the three angles at $A$ is $180^\\circ$, since they form a straight angle. 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+{"instance_id": "math500-0019", "gold": ["3"], "meta": {"row_pos": 19, "level": 3, "subject": "Intermediate Algebra", "unique_id": "test/intermediate_algebra/1000.json", "solution": "Note that $x = -1$ is always a root of $x^3 + ax^2 + ax + 1 = 0,$ so we can factor out $x + 1,$ to get\n\\[(x + 1) (x^2 + (a - 1) x + 1) = 0.\\]The quadratic factor has real roots if and only if its discriminant is nonnegative:\n\\[(a - 1)^2 - 4 \\ge 0.\\]This reduces to $a^2 - 2a - 3 \\ge 0,$ which factors as $(a + 1)(a - 3) \\ge 0.$  The smallest positive value that satisfies this inequality is $\\boxed{3}.$"}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:3", "label": "3", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "3", "count": 100}]}], "n_distinct": 1, "modal_answer": "3", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, 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+{"instance_id": "math500-0021", "gold": ["13535"], "meta": {"row_pos": 21, "level": 3, "subject": "Intermediate Algebra", "unique_id": "test/intermediate_algebra/607.json", "solution": "Let $x = \\sqrt{7} + \\sqrt{5}$ and $y = \\sqrt{7} - \\sqrt{5}.$\n\nFirst, we can square $x = \\sqrt{7} + \\sqrt{5}$ and $y = \\sqrt{7} - \\sqrt{5},$ to get\n\\begin{align*}\nx^2 &= (\\sqrt{7} + \\sqrt{5})^2 = 7 + 2 \\sqrt{35} + 5 = 12 + 2 \\sqrt{35}, \\\\\ny^2 &= (\\sqrt{7} - \\sqrt{5})^2 = 7 - 2 \\sqrt{35} + 5 = 12 - 2 \\sqrt{35}.\n\\end{align*}Note that $x^2$ and $y^2$ are radical conjugates.  Also, $x^2 y^2 = (12 + 2 \\sqrt{35})(12 - 2 \\sqrt{35}) = 12^2 - 2^2 \\cdot 35 = 4,$ so\n\\[y^2 = \\frac{4}{x^2} = \\frac{4}{12 + 2 \\sqrt{35}} < 1.\\]Then\n\\[x^4 = (12 + 2 \\sqrt{35})^2 = 12^2 + 2 \\cdot 12 \\cdot 2 \\sqrt{35} + 2^2 \\cdot 35 = 284 + 48 \\sqrt{35},\\]and\n\\begin{align*}\nx^6 &= x^2 \\cdot x^4 \\\\\n&= (12 + 2 \\sqrt{35})(284 + 48 \\sqrt{35}) \\\\\n&= 12 \\cdot 284 + 12 \\cdot 48 \\sqrt{35} + 2 \\sqrt{35} \\cdot 284 + 2 \\cdot \\sqrt{35} \\cdot 48 \\cdot \\sqrt{35} \\\\\n&= 6768 + 1144 \\sqrt{35}.\n\\end{align*}Then $y^6$ is the radical conjugate of $x^6,$ so $y^6 = 6768 - 1144 \\sqrt{35}.$  Hence,\n\\[x^6 + y^6 = (6768 + 1144 \\sqrt{35}) + (6768 - 1144 \\sqrt{35}) = 13536.\\]Since $0 < y^6 < 1,$ the greatest integer less than $x^6$ is $\\boxed{13535}.$"}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 99, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:13535", "label": "13535", 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+{"instance_id": "math500-0023", "gold": ["x=5"], "meta": {"row_pos": 23, "level": 5, "subject": "Algebra", "unique_id": "test/algebra/2193.json", "solution": "We first isolate the square root, so we can then square both sides to get rid of it.  Subtracting 4 from both sides gives $x-4 = \\!\\sqrt{11-2x}$.  Squaring both sides gives $x^2 - 8x + 16 = 11-2x$, or $x^2 -6x + 5=0$.  Factoring gives $(x-5)(x-1)=0$, so $x=5$ or $x=1$.  Because we squared the equation, we must  check if our solutions are extraneous. For $x=5$, the equation reads $5 = \\!\\sqrt{11-10} + 4$, which is true. If $x=1$, we have $1 = \\!\\sqrt{11-2} + 4$, which is not true, so $x=1$ is extraneous. Therefore, our only solution is $\\boxed{x=5}$."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:5", "label": "5", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "5", "count": 100}]}], "n_distinct": 1, "modal_answer": "5", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99], 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+{"instance_id": "math500-0024", "gold": ["10"], "meta": {"row_pos": 24, "level": 5, "subject": "Algebra", "unique_id": "test/algebra/2427.json", "solution": "If the interest rate is $r$, it follows that $$20000(1+r)^2 + 20000(1+r) + 20000 \\ge 66200.$$ If we set $x = 1+r$ and divide through the inequality by $200$, it follows that $$100x^2 + 100x - 231 \\ge 0.$$ Since $231 = 11 \\cdot 21$, we can factor the quadratic as $(10x - 11)(10x + 21) \\ge 0$, so it follows that $x \\ge \\frac {11}{10}$ or $x \\le \\frac{-21}{10}$. Since we are looking for an interest rate percentage, it follows that $x \\ge \\frac{11}{10} = 1.1$, and $r = x - 1 = \\boxed{10}\\%$."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:10", "label": "10", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "10", "count": 100}]}], "n_distinct": 1, "modal_answer": "10", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 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+{"instance_id": "math500-0025", "gold": ["1,-2"], "meta": {"row_pos": 25, "level": 5, "subject": "Intermediate Algebra", "unique_id": "test/intermediate_algebra/1388.json", "solution": "Setting $x = y = 0,$ we get\n\\[2f(0) = f(0) - 1,\\]so $f(0) = -1.$\n\nSetting $y = 1,$ we get\n\\[f(x) + 1 = f(x + 1) - x - 1,\\]so\n\\[f(x + 1) - f(x) = x + 2.\\]Thus,\n\\begin{align*}\nf(2) - f(1) &= 1 + 2, \\\\\nf(3) - f(2) &= 2 + 2, \\\\\nf(4) - f(3) &= 3 + 2, \\\\\n&\\dots, \\\\\nf(n) - f(n - 1) &= (n - 1) + 2.\n\\end{align*}Adding all the equations, we get\n\\[f(n) - f(1) = 1 + 2 + 3 + \\dots + (n - 1) + 2(n - 1) = \\frac{(n - 1)n}{2} + 2n - 2 = \\frac{n^2 + 3n - 4}{2},\\]so\n\\[f(n) = \\frac{n^2 + 3n - 2}{2}\\]for all positive integers $n.$\n\nSetting $x = -n$ and $y = n,$ where $n$ is a positive integer, we get\n\\[f(-n) + f(n) = f(0) + n^2 - 1.\\]Then\n\\[f(-n) = n^2 - f(n) + f(0) - 1 = n^2 - \\frac{n^2 + 3n - 2}{2} - 2 = \\frac{n^2 - 3n - 2}{2}.\\]Thus, the formula\n\\[f(n) = \\frac{n^2 + 3n - 2}{2}\\]holds for all integers $n.$\n\nWe want to solve $f(n) = n,$ or\n\\[\\frac{n^2 + 3n - 2}{2} = n.\\]Then $n^2 + 3n - 2 = 2n,$ or $n^2 + n - 2 = 0.$  This factors as $(n - 1)(n + 2) = 0,$ so the solutions are $n = \\boxed{1,-2}.$"}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 99, "n_bold": 0, "n_correct": 42, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:1", "label": "1", "count": 58, "match": "none", "judge": null, "members": [{"raw": "1", "count": 58}]}, {"bucket": "math:-2, 1", "label": "-2, 1", "count": 42, "match": "exact", "judge": null, "members": [{"raw": "-2, 1", "count": 42}]}], "n_distinct": 2, "modal_answer": "1", "modal_frac": 0.58, "modal_is_correct": false, "top_wrong_frac": 0.58, "collapsed_wrong": true, "entropy": 0.9814538950336537, "acc": 0.42, "pass_any": true, "majority_correct": 0.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 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+{"instance_id": "math500-0026", "gold": ["144"], "meta": {"row_pos": 26, "level": 5, "subject": "Counting & Probability", "unique_id": "test/counting_and_probability/525.json", "solution": "After Pierre sits, we can place Rosa either two seats from Pierre (that is, with one seat between them) or three seats from Pierre. We tackle these two cases separately:\n\nCase 1: Rosa is two seats from Pierre. There are $2$ such seats. For either of these, there are then four empty seats in a row, and one empty seat between Rosa and Pierre. Thomas can sit in either of the middle two of the four empty seats in a row. So, there are $2\\cdot 2 = 4$ ways to seat Rosa and Thomas in this case. There are then $4$ seats left, which the others can take in $4! = 24$ ways. So, there are $4\\cdot 24 = 96$ seatings in this case.\n\nCase 2: Rosa is three seats from Pierre (that is, there are $2$ seats between them). There are $2$ such seats. Thomas can't sit in either of the $2$ seats directly between them, but after Rosa sits, there are $3$ empty seats in a row still, and Thomas can only sit in the middle seat of these three. Once again, there are $4$ empty seats remaining, and the $4$ remaining people can sit in them in $4! = 24$ ways. So, we have $2\\cdot 24 = 48$ seatings in this case.\n\nPutting our two cases together gives a total of $96+48 = \\boxed{144}$ seatings."}, "n_samples": 100, "n_success": 97, "n_answered": 97, "n_bare": 0, "n_boxed": 97, "n_bold": 0, "n_correct": 94, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:144", "label": "144", "count": 94, "match": "exact", "judge": null, "members": [{"raw": "144", "count": 94}]}, {"bucket": "math:288", "label": "288", "count": 2, "match": "none", "judge": null, "members": [{"raw": "288", "count": 2}]}, {"bucket": "math:48", "label": "48", "count": 1, "match": "none", "judge": null, "members": [{"raw": "48", "count": 1}]}], "n_distinct": 3, "modal_answer": "144", "modal_frac": 0.9690721649484536, "modal_is_correct": true, "top_wrong_frac": 0.020618556701030927, "collapsed_wrong": false, "entropy": 0.22742467664385024, "acc": 0.9690721649484536, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, 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true, true, true], "answer": ["144", "144", "144", "144", "288", "144", "144", "48", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "288", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", null, "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", null, "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", null, "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144", "144"], "answer_source": ["boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", 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+{"instance_id": "math500-0027", "gold": ["78"], "meta": {"row_pos": 27, "level": 2, "subject": "Prealgebra", "unique_id": "test/prealgebra/1840.json", "solution": "To find the profit, we want to find out how much the math club earned from selling the various baked goods and subtract the cost of producing those goods, $\\$15$, from the number we get.\n\nFirst let's calculate how much the math club earned from selling cookies. The cookies were sold at a price of three for $\\$1$, so the math club earned $54\\div 3\\cdot\\$1=18\\cdot\\$1=\\$18$ from selling cookies.\n\nNext, let's calculate how much the club earned from selling cupcakes. At a price of $\\$2$ each, the club earned $20\\cdot \\$2=\\$40$ from selling cupcakes.\n\nFinally, let's calculate how much the club earned from selling brownies. At a price of $\\$1$ each, the club earned $35\\cdot\\$1=\\$35$ from selling brownies.\n\nNow let's add up these numbers to find out how much the club earned in total and subtract $\\$15$ from that number to find the club's profit. We obtain \\begin{align*}\n\\$18+\\$40+\\$35-\\$15&=\\$18+\\$40+\\$35-\\$15\\\\\n&=\\$18+\\$40+\\$35+(-\\$15)\\\\\n&=\\$18+\\$40+(\\$35+(-\\$15))\\\\\n&=\\$18+\\$40+(\\$20)\\\\\n&=\\boxed{78}.\n\\end{align*}Notice how we used the definition of subtraction, $a-b=a+(-b)$ to $\\$35-\\$15$ as $\\$35+(-\\$15)$ and the associative property of addition to group the numbers together."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:78", "label": "78", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "78", "count": 100}]}], "n_distinct": 1, "modal_answer": "78", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 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The number of different battalions, then, is $\\binom{5}{4}\\cdot \\binom{10}{8} = \\boxed{225}$."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 91, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:225", "label": "225", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "225", "count": 100}]}], "n_distinct": 1, "modal_answer": "225", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 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false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false], "correct": [true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true]}
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Then $\\sqrt{242} = \\sqrt{11^2} \\cdot \\sqrt2 = \\boxed{11\\sqrt2}$."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:11\\sqrt{2}", "label": "11\\sqrt{2}", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "11\\sqrt{2}", "count": 100}]}], "n_distinct": 1, "modal_answer": "11\\sqrt{2}", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 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The answer is $3! \\times 5! = \\boxed{720}$."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 87, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:720", "label": "720", "count": 87, "match": "exact", "judge": null, "members": [{"raw": "720", "count": 87}]}, {"bucket": "math:480", "label": "480", "count": 8, "match": "none", "judge": null, "members": [{"raw": "480", "count": 8}]}, {"bucket": "math:240", "label": "240", "count": 5, "match": "none", "judge": null, "members": [{"raw": "240", "count": 5}]}], "n_distinct": 3, "modal_answer": "720", "modal_frac": 0.87, "modal_is_correct": true, "top_wrong_frac": 0.08, "collapsed_wrong": false, "entropy": 0.682398943641963, "acc": 0.87, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 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+{"instance_id": "math500-0039", "gold": ["23"], "meta": {"row_pos": 39, "level": 3, "subject": "Number Theory", "unique_id": "test/number_theory/45.json", "solution": "We first recognize that $132=11\\times 12$, so its prime factorization is $132 = 2^2 \\cdot 3 \\cdot 11$. We only need to see if these three prime factors will divide into $6432$. Indeed, $6432$ will satisfy the divisibility properties for both $3$ and $4$, and we can long divide to see that $11$ does not divide into $6432$. Thus, the greatest common factor is $3 \\times 4 = 12$. The greatest common factor increased by 11 is $12+11 = \\boxed{23}$."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:23", "label": "23", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "23", "count": 100}]}], "n_distinct": 1, "modal_answer": "23", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99], "ok": [true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true], "answer": ["23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", "23", 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"math:23", "math:23", "math:23", "math:23", "math:23", "math:23", "math:23", "math:23", "math:23", "math:23", "math:23", "math:23", "math:23", "math:23", "math:23", "math:23", "math:23", "math:23", "math:23", "math:23", "math:23", "math:23", "math:23", "math:23", "math:23", "math:23"], "truncated": [false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false], "correct": [true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true]}
+{"instance_id": "math500-0040", "gold": ["12"], "meta": {"row_pos": 40, "level": 2, "subject": "Prealgebra", "unique_id": "test/prealgebra/930.json", "solution": "The hexagon has a side length of 16 centimeters, so its perimeter is $16\\times 6 = 96$ centimeters. Since the octagon and the hexagon have the same perimeter, it follows that each side of the octagon has a length of $96/8 = \\boxed{12}$ centimeters."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:12", "label": "12", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "12", "count": 100}]}], "n_distinct": 1, "modal_answer": "12", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99], "ok": [true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true], "answer": ["12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12", "12"], "answer_source": ["boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed"], "bucket": ["math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", "math:12", 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false, false, false, false, false, false, false, false, false, false, false], "correct": [true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true]}
+{"instance_id": "math500-0041", "gold": ["17"], "meta": {"row_pos": 41, "level": 4, "subject": "Geometry", "unique_id": "test/geometry/627.json", "solution": "Name the points $A(5,3)$, $B(6,8)$, $C(7,4)$, and $D(x,y)$ and sketch the first three.  We find that there are three possible locations for $D$ (see figure).  Only the one to the right has an $x$-coordinate greater than 7.  Since $AC$ is parallel to $BD$ and equal in length to it, $D$ is two units to the right and one unit up from $B$, just as $C$ is two units to the right and one unit up from $A$.  Therefore, the coordinates of $D$ are $(8,9)$, and $x+y=8+9=\\boxed{17}$.\n\n[asy]\nsize(5cm);\nimport graph;\ndefaultpen(linewidth(0.7)+fontsize(10));\ndotfactor=5;\nreal x = 7;\npair A=(5,3), B=(6,8), C=(7,4), D1=(8,9), D2=(4,7),D3=(6,-1);\npair[] dots = {A,B,C};\ndot(dots);\nxaxis(-2,10,Ticks(\" \",1.0,begin=false,end=false,NoZero),Arrows(4));\nyaxis(-2,10,Ticks(\" \",1.0,begin=false,end=false,NoZero),Arrows(4));\ndraw(A--C--D1--B--cycle);//linetype(\"8 8\"));\ndraw(A--D3--C);\ndraw(A--C--B--D2--cycle);//,linetype(\"1 2 3 1\"));\nlabel(\"$A(5,3)$\",A,SW);\nlabel(\"$B(6,8)$\",B,NW);\nlabel(\"$C(7,4)$\",C,E);\ndot(D1,UnFill);\ndot(D2,UnFill);\ndot(D3,UnFill);[/asy]"}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:17", "label": "17", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "17", "count": 100}]}], "n_distinct": 1, "modal_answer": "17", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99], "ok": [true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, 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+{"instance_id": "math500-0042", "gold": ["4"], "meta": {"row_pos": 42, "level": 2, "subject": "Algebra", "unique_id": "test/algebra/2214.json", "solution": "Since everything in sight is even, we should begin by dividing by 2.  That gives \\[-2<x-1<4.\\] To isolate $x$, we add 1, so \\[-1<x<5.\\] Since $a=-1$ and $b=5$, we get $a+b=-1+5=\\boxed{4}$."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:4", "label": "4", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "4", "count": 100}]}], "n_distinct": 1, "modal_answer": "4", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 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+{"instance_id": "math500-0044", "gold": ["1.25"], "meta": {"row_pos": 44, "level": 5, "subject": "Prealgebra", "unique_id": "test/prealgebra/1558.json", "solution": "The diagram on the left shows the path of Bill's walk. As the diagram on the right illustrates, he could also have walked from $A$ to $B$ by first walking 1 mile south then $\\frac{3}{4}$ mile east. [asy]\npair a=(0,1), b=(.75, 0), c=(0,.5), d=(.75,.5), o=(0,0);\ndraw(a--b--d--c--cycle);\nlabel(\"$A$\", a, NW);\nlabel(\"$B$\", b, SE);\nlabel(\"$\\frac{1}{2}$\", (0,0.75), W);\nlabel(\"$\\frac{3}{4}$\", (.7, 0.66),W);\nlabel(\"$\\frac{1}{2}$\", (.75, .25), E);\n\npicture pic;\ndraw(pic, a--b--o--cycle);\nlabel(pic, \"$A$\", a, NW);\nlabel(pic, \"$B$\", b, SE);\nlabel(pic, \"$\\frac{3}{4}$\", (.375,0), S);\nlabel(pic, \"1\", (0, .5), W);\nadd(shift(1.5,0)*pic);\n[/asy] By the Pythagorean Theorem, \\[(AB)^2=1^2+\\left(\\frac{3}{4}\\right)^2=1+\\frac{9}{16}=\\frac{25}{16},\\]so $AB=\\frac{5}{4}=1\\frac{1}{4}$, or $\\boxed{1.25}$."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:1.25", "label": "1.25", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "1.25", "count": 100}]}], "n_distinct": 1, "modal_answer": "1.25", "modal_frac": 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+{"instance_id": "math500-0045", "gold": ["2"], "meta": {"row_pos": 45, "level": 3, "subject": "Geometry", "unique_id": "test/geometry/178.json", "solution": "The triangle is shown below:\n\n[asy]\npair A,B,C;\nA = (0,0);\nB = (5,0);\nC = (5,10);\ndraw(A--B--C--A);\ndraw(rightanglemark(C,B,A,16));\nlabel(\"$A$\",A,SW);\nlabel(\"$B$\",B,SE);\nlabel(\"$C$\",C,N);\n[/asy]\n\nWe have $\\sin A = \\frac{BC}{AC}$ and $\\cos A = \\frac{AB}{AC}$, so $\\sin A = 2\\cos A$ gives us $\\frac{BC}{AC} = 2\\cdot\\frac{AB}{AC}$.  Multiplying both sides by $AC$ gives $BC = 2AB$, so $\\frac{BC}{AB} = 2$.  Finally, we have $\\tan A = \\frac{BC}{AB} = \\boxed{2}$.\n\nWe also could have noted that $\\tan A = \\frac{\\sin A}{\\cos A} = \\frac{2\\cos A}{\\cos A } =\\boxed{2}$."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:2", "label": "2", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "2", "count": 100}]}], "n_distinct": 1, "modal_answer": "2", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 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Therefore, the smallest possible value of $n$ is $\\boxed{6}$."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:6", "label": "6", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "6", "count": 100}]}], "n_distinct": 1, "modal_answer": "6", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 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We can also see that when we divide $x$ out of the numerator and denominator, we get \\[\\frac{2x}{x^2 - 5x - 14} = \\frac{\\frac{2x}{x}}{\\frac{x^2-5x-14}{x}}=\\frac{2}{x-5-\\frac{14}{x}}.\\]As $x$ approaches infinity or negative infinity, the expression approaches 0.\n\nSo, our answer is $7 + (-2) + 0 = \\boxed{5}$."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:5", "label": "5", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "5", "count": 100}]}], "n_distinct": 1, "modal_answer": "5", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 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The left-hand side simplifies to $2^{2x}$, so we may set exponents equal to find $2x=3$, which implies $x=\\boxed{\\frac{3}{2}}$."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:\\dfrac{3}{2}", "label": "\\dfrac{3}{2}", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "\\dfrac{3}{2}", "count": 100}]}], "n_distinct": 1, "modal_answer": "\\dfrac{3}{2}", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 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Adding $9$ to both sides yields $ x = \\boxed{83}$."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:83", "label": "83", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "83", "count": 100}]}], "n_distinct": 1, "modal_answer": "83", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 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The least such integer is 29, so you had $29 \\cdot 7 = \\boxed{203}$ coins before finding the bag of 53 coins."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:203", "label": "203", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "203", "count": 100}]}], "n_distinct": 1, "modal_answer": "203", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 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+{"instance_id": "math500-0051", "gold": ["x^5 - x^4 + x^3 - x^2 + x - 1"], "meta": {"row_pos": 51, "level": 2, "subject": "Intermediate Algebra", "unique_id": "test/intermediate_algebra/1217.json", "solution": "We can perform long division.  Alternatively, by the Remainder Theorem, the remainder upon division is $(-1)^6 - 3 = -2.$  Thus, we can write\n\\begin{align*}\n\\frac{x^6 - 3}{x + 1} &= \\frac{(x^6 - 1) - 2}{x + 1} \\\\\n&= \\frac{x^6 - 1}{x + 1} - \\frac{2}{x + 1} \\\\\n&= \\frac{(x^3 - 1)(x^3 + 1)}{x + 1} - \\frac{2}{x + 1} \\\\\n&= \\frac{(x^3 - 1)(x + 1)(x^2 - x + 1)}{x + 1} - \\frac{2}{x + 1} \\\\\n&= (x^3 - 1)(x^2 - x + 1) - \\frac{2}{x + 1} \\\\\n&= x^5 - x^4 + x^3 - x^2 + x - 1 - \\frac{2}{x + 1}.\n\\end{align*}Thus, the quotient is $\\boxed{x^5 - x^4 + x^3 - x^2 + x - 1}.$"}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:x^5 - x^4 + x^3 - x^2 + x - 1", "label": "x^5 - x^4 + x^3 - x^2 + x - 1", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "x^5 - x^4 + x^3 - x^2 + x - 1", "count": 100}]}], "n_distinct": 1, "modal_answer": "x^5 - x^4 + x^3 - x^2 + x - 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+{"instance_id": "math500-0056", "gold": ["11"], "meta": {"row_pos": 56, "level": 3, "subject": "Algebra", "unique_id": "test/algebra/769.json", "solution": "The first $N$ positive odd integers are 1, 3, $\\dots$, $2N - 1$.  The sum of an arithmetic series is equal to the average of the first and last term, multiplied by the number of terms, so the sum of the first $N$ positive odd integers is \\[\\frac{1 + (2N - 1)}{2} \\cdot N = N^2.\\]If $N^2 = 121$, then $N = \\boxed{11}$."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:11", "label": "11", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "11", "count": 100}]}], "n_distinct": 1, "modal_answer": "11", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 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false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false, false], "correct": [true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true]}
+{"instance_id": "math500-0057", "gold": ["16"], "meta": {"row_pos": 57, "level": 2, "subject": "Intermediate Algebra", "unique_id": "test/intermediate_algebra/1168.json", "solution": "We know that magnitudes of complex numbers are multiplicative: the magnitude of $|ab|$ is the product $|a|\\cdot |b|$. Thus, \\[\\left|\\left(1-i\\right)^8\\right|=\\left|1-i\\right|^8\\]The magnitude of $1-i$ is $\\sqrt{1^2+(-1)^2}=\\sqrt{2}$; thus our answer is $\\left(\\sqrt{2}\\right) ^8=\\boxed{16}$. Ringo didn't give much of a reprieve."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:16", "label": "16", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "16", "count": 100}]}], "n_distinct": 1, "modal_answer": "16", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99], "ok": 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"boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed", "boxed"], "bucket": ["math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", "math:16", 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+{"instance_id": "math500-0058", "gold": ["9901"], "meta": {"row_pos": 58, "level": 1, "subject": "Algebra", "unique_id": "test/algebra/722.json", "solution": "Factoring the first two terms, we have:\n\n$99^2+99+1=99(99+1)+1=99\\cdot 100+1=9900+1=\\boxed{9901}$."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:9901", "label": "9901", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "9901", "count": 100}]}], "n_distinct": 1, "modal_answer": "9901", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 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+{"instance_id": "math500-0059", "gold": ["5"], "meta": {"row_pos": 59, "level": 2, "subject": "Prealgebra", "unique_id": "test/prealgebra/1247.json", "solution": "Among the $50-6=44$ students that participate in either MATHCOUNTS or science club, $44-28=16$ students do not participate in MATHCOUNTS.  All 16 of these students only participate in science club.  The other $21-16=\\boxed{5}$ science club participants participate in MATHCOUNTS as well."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:5", "label": "5", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "5", "count": 100}]}], "n_distinct": 1, "modal_answer": "5", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 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+{"instance_id": "math500-0060", "gold": ["(6,31,-1)"], "meta": {"row_pos": 60, "level": 5, "subject": "Intermediate Algebra", "unique_id": "test/intermediate_algebra/956.json", "solution": "Let $\\alpha$ be a root of $x^3 - 3x^2 + 4x - 1 = 0,$ so $\\alpha^3 = 3 \\alpha^2 - 4 \\alpha + 1.$  Then\n\\[\\alpha^4 = 3 \\alpha^3 - 4 \\alpha^2 + \\alpha = 3 (3 \\alpha^2 - 4 \\alpha + 1) - 4 \\alpha^2 + \\alpha = 5 \\alpha^2 - 11 \\alpha + 3.\\]Hence,\n\\begin{align*}\n\\alpha^6 &= (3 \\alpha^2 - 4 \\alpha + 1)^2 \\\\\n&= 9 \\alpha^4 - 24 \\alpha^3 + 22 \\alpha^2 - 8 \\alpha + 1 \\\\\n&= 9 (5 \\alpha^2 - 11 \\alpha + 3) - 24 (3 \\alpha^2 - 4 \\alpha + 1) + 22 \\alpha^2 - 8 \\alpha + 1 \\\\\n&= -5 \\alpha^2 - 11 \\alpha + 4,\n\\end{align*}and\n\\begin{align*}\n\\alpha^9 &= \\alpha^3 \\cdot \\alpha^6 \\\\\n&= (3 \\alpha^2 - 4 \\alpha + 1)(-5 \\alpha^2 - 11 \\alpha + 4) \\\\\n&= -15 \\alpha^4 - 13 \\alpha^3 + 51 \\alpha^2 - 27 \\alpha + 4 \\\\\n&= -15 (5 \\alpha^2 - 11 \\alpha + 3) - 13 (3 \\alpha^2 - 4 \\alpha + 1) + 51 \\alpha^2 - 27 \\alpha + 4 \\\\\n&= -63 \\alpha^2 + 190 \\alpha - 54.\n\\end{align*}Then\n\\begin{align*}\n\\alpha^9 + p \\alpha^6 + q \\alpha^3 + r &= (-63 \\alpha^2 + 190 \\alpha - 54) + p (-5 \\alpha^2 - 11 \\alpha + 4) + q (3 \\alpha^2 - 4 \\alpha + 1) + r \\\\\n&= (-5p + 3q - 63) \\alpha^2 + (-11p - 4q + 190) \\alpha + (4p + q + r - 54).\n\\end{align*}We want this to reduce to 0, so we set\n\\begin{align*}\n-5p + 3q &= 63, \\\\\n11p + 4q &= 190, \\\\\n4p + q + r &= 54.\n\\end{align*}Solving, we find $(p,q,r) = \\boxed{(6,31,-1)}.$   For these values, $\\alpha^9 + p \\alpha^6 + q \\alpha^3 + r$ reduces to 0 for any root $\\alpha$ of $x^3 - 3x^2 + 4x - 1,$ so $x^9 + px^6 + qx^3 + r$ will be divisible by $x^3 - 3x^2 + 4x - 1.$"}, "n_samples": 100, "n_success": 98, "n_answered": 98, "n_bare": 0, "n_boxed": 98, "n_bold": 0, "n_correct": 97, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:(6, 31, -1)", "label": "(6, 31, -1)", 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Since \\[\n8x^{3}+4ax^{2}+2bx+a=8(x-r_1)(x-r_2)(x-r_3),\n\\]it follows that $a = -8r_1r_2r_3= \\boxed{-256}$."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:-256", "label": "-256", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "-256", "count": 100}]}], "n_distinct": 1, "modal_answer": "-256", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 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+{"instance_id": "math500-0062", "gold": ["4"], "meta": {"row_pos": 62, "level": 5, "subject": "Precalculus", "unique_id": "test/precalculus/675.json", "solution": "Let $\\bold{v} = \\begin{pmatrix} x \\\\ y \\end{pmatrix}$.  Then\n\\[\\|\\bold{v}\\| = \\left\\| \\begin{pmatrix} x \\\\ y \\end{pmatrix} \\right\\| = \\sqrt{x^2 + y^2},\\]and\n\\begin{align*}\n\\left\\| \\begin{pmatrix} 2 & 3 \\\\ 0 & -2 \\end{pmatrix} \\bold{v} \\right\\| &= \\left\\| \\begin{pmatrix} 2 & 3 \\\\ 0 & -2 \\end{pmatrix} \\begin{pmatrix} x \\\\ y \\end{pmatrix} \\right\\| \\\\\n&= \\left\\| \\begin{pmatrix} 2x + 3y \\\\ -2y \\end{pmatrix} \\right\\| \\\\\n&= \\sqrt{(2x + 3y)^2 + (-2y)^2} \\\\\n&= \\sqrt{4x^2 + 12xy + 13y^2},\n\\end{align*}so the given inequality becomes\n\\[\\sqrt{4x^2 + 12xy + 13y^2} \\le C \\sqrt{x^2 + y^2},\\]or\n\\[\\sqrt{\\frac{4x^2 + 12xy + 13y^2}{x^2 + y^2}} \\le C.\\]Thus, we can think of $C$ as the maximum value of the expression in the left-hand side.\n\nMaximizing the expression in the left-hand side is equivalent to maximizing its square, namely\n\\[\\frac{4x^2 + 12xy + 13y^2}{x^2 + y^2}.\\]Let $k$ be a possible value of this expression, which means the equation\n\\[\\frac{4x^2 + 12xy + 13y^2}{x^2 + y^2} = k\\]has a solution in $x$ and $y$.  We can re-write this equation as\n\\[(4 - k) x^2 + 12xy + (13 - k) y^2 = 0.\\]For this quadratic expression to have a solution in $x$ and $y$, its discriminant must be nonnegative.  In other words,\n\\[12^2 - 4 (4 - k)(13 - k) \\ge 0,\\]or $4k^2 - 68k + 64 \\le 0$.  This inequality factors as $4(k - 1)(k - 16) \\le 0$.  The largest value of $k$ that satisfies this inequality is 16, so the value of $C$ we seek is $\\sqrt{16} = \\boxed{4}$.  Note that equality occurs for\n\\[\\bold{v} = \\begin{pmatrix} 1 \\\\ 2 \\end{pmatrix}.\\]"}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:4", "label": "4", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "4", "count": 100}]}], "n_distinct": 1, "modal_answer": "4", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 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Thus, the area decreased by $\\boxed{10}$ percent."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:10", "label": "10", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "10", "count": 100}]}], "n_distinct": 1, "modal_answer": "10", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 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+{"instance_id": "math500-0069", "gold": ["-1"], "meta": {"row_pos": 69, "level": 4, "subject": "Intermediate Algebra", "unique_id": "test/intermediate_algebra/207.json", "solution": "Since $z^5 = 1,$ $z^5 - 1 = 0,$ which factors as\n\\[(z - 1)(z^4 + z^3 + z^2 + z + 1) = 0.\\]Since $z \\neq 1,$ $z^4 + z^3 + z^2 + z + 1 = 0.$\n\nThen\n\\[z + \\frac{1}{z} + z^2 + \\frac{1}{z^2} = \\frac{z^3 + z + z^4 + 1}{z^2} = \\frac{-z^2}{z^2} = \\boxed{-1}.\\]"}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:-1", "label": "-1", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "-1", "count": 100}]}], "n_distinct": 1, "modal_answer": "-1", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, 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+{"instance_id": "math500-0070", "gold": ["40_9"], "meta": {"row_pos": 70, "level": 2, "subject": "Number Theory", "unique_id": "test/number_theory/46.json", "solution": "This subtraction is fairly straightforward: we just subtract the respective digits. No borrowing occurs: $$ \\begin{array}{c@{}c@{\\;}c@{}c} & & 5 & 8_9 \\\\ &- & 1 & 8_9 \\\\ \\cline{2-4} & & 4 & 0_9 \\\\ \\end{array} $$ Thus, the answer is $\\boxed{40_9}.$"}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:40", "label": "40", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "40", "count": 51}, {"raw": "40_9", "count": 49}]}], "n_distinct": 1, "modal_answer": "40", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 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+{"instance_id": "math500-0071", "gold": ["2516_8"], "meta": {"row_pos": 71, "level": 4, "subject": "Number Theory", "unique_id": "test/number_theory/516.json", "solution": "Since $2^3=8$, we may convert between base 2 and base 8 representations by replacing each block of three digits in base 2 with its equivalent in base 8. In this case, we begin by noticing that the last three digits are worth $110_2=6_8$. The next block of three digits is $001_2=1_8$. Continuing, we find that the next two digits (moving right-to-left) are $101_2=5_8$ and $010_2=2_8$. Altogether, we find that $10101001110_{2}=\\boxed{2516_8}$."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:2516", "label": "2516", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "2516", "count": 100}]}], "n_distinct": 1, "modal_answer": "2516", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 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+{"instance_id": "math500-0072", "gold": ["3"], "meta": {"row_pos": 72, "level": 2, "subject": "Geometry", "unique_id": "test/geometry/456.json", "solution": "The volume of a sphere is $\\frac{4}{3}\\pi r^3$ and the surface area is $4\\pi r^2$, so\n\\[\\frac{4}{3} \\pi r^3 = 4 \\pi r^2.\\]We can divide both sides by $4 \\pi r^2$, to get\n\\[\\frac{1}{3} r = 1.\\]Therefore, $r = \\boxed{3}.$"}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:3", "label": "3", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "3", "count": 100}]}], "n_distinct": 1, "modal_answer": "3", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 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+{"instance_id": "math500-0073", "gold": ["\\frac{3\\sqrt{3}}{4}"], "meta": {"row_pos": 73, "level": 4, "subject": "Algebra", "unique_id": "test/algebra/2253.json", "solution": "We have $9\\&2 = \\frac{\\sqrt{(9)(2)+9}}{\\sqrt{(9)(2)-2}} = \\frac{\\sqrt{27}}{\\sqrt{16}} = \\boxed{\\frac{3\\sqrt{3}}{4}}.$"}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:\\dfrac{3\\sqrt{3}}{4}", "label": "\\dfrac{3\\sqrt{3}}{4}", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "\\dfrac{3\\sqrt{3}}{4}", "count": 99}, {"raw": "\\frac{3\\sqrt{3}}{4}", "count": 1}]}], "n_distinct": 1, "modal_answer": "\\dfrac{3\\sqrt{3}}{4}", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 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"answer": ["\\dfrac{3\\sqrt{3}}{4}", "\\dfrac{3\\sqrt{3}}{4}", "\\dfrac{3\\sqrt{3}}{4}", "\\dfrac{3\\sqrt{3}}{4}", "\\dfrac{3\\sqrt{3}}{4}", "\\dfrac{3\\sqrt{3}}{4}", "\\dfrac{3\\sqrt{3}}{4}", "\\dfrac{3\\sqrt{3}}{4}", "\\dfrac{3\\sqrt{3}}{4}", "\\dfrac{3\\sqrt{3}}{4}", "\\dfrac{3\\sqrt{3}}{4}", "\\dfrac{3\\sqrt{3}}{4}", "\\dfrac{3\\sqrt{3}}{4}", "\\dfrac{3\\sqrt{3}}{4}", "\\dfrac{3\\sqrt{3}}{4}", "\\dfrac{3\\sqrt{3}}{4}", "\\dfrac{3\\sqrt{3}}{4}", "\\dfrac{3\\sqrt{3}}{4}", "\\dfrac{3\\sqrt{3}}{4}", "\\dfrac{3\\sqrt{3}}{4}", "\\dfrac{3\\sqrt{3}}{4}", "\\dfrac{3\\sqrt{3}}{4}", "\\dfrac{3\\sqrt{3}}{4}", "\\dfrac{3\\sqrt{3}}{4}", "\\dfrac{3\\sqrt{3}}{4}", "\\dfrac{3\\sqrt{3}}{4}", "\\dfrac{3\\sqrt{3}}{4}", "\\dfrac{3\\sqrt{3}}{4}", "\\dfrac{3\\sqrt{3}}{4}", "\\dfrac{3\\sqrt{3}}{4}", "\\dfrac{3\\sqrt{3}}{4}", "\\dfrac{3\\sqrt{3}}{4}", "\\dfrac{3\\sqrt{3}}{4}", "\\dfrac{3\\sqrt{3}}{4}", "\\frac{3\\sqrt{3}}{4}", "\\dfrac{3\\sqrt{3}}{4}", "\\dfrac{3\\sqrt{3}}{4}", "\\dfrac{3\\sqrt{3}}{4}", 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+{"instance_id": "math500-0078", "gold": ["(-2,1)"], "meta": {"row_pos": 78, "level": 4, "subject": "Intermediate Algebra", "unique_id": "test/intermediate_algebra/47.json", "solution": "By the remainder theorem, $f(x) - f(a)$ is divisible by $x - a,$ so we can take out a factor of $x - a$ accordingly:\n\\begin{align*}\nf(x) - f(a) &= (x^3 + 3x^2 + 1) - (a^3 + 3a^2 + 1) \\\\\n&= (x^3 - a^3) + 3(x^2 - a^2) \\\\\n&= (x - a)(x^2 + ax + a^2) + 3(x - a)(x + a) \\\\\n&= (x - a)(x^2 + ax + a^2 + 3x + 3a) \\\\\n&= (x - a)(x^2 + (a + 3) x + a^2 + 3a).\n\\end{align*}Thus, we want\n\\[x^2 + (a + 3) x + a^2 + 3a = (x - a)(x - b) = x^2 - (a + b) x + ab.\\]Matching coefficients, we get\n\\begin{align*}\na + 3 &= -a - b, \\\\\na^2 + 3a &= ab.\n\\end{align*}Since $a \\neq 0,$ we can divide both sides of the second equation by $a,$ to get $a + 3 = b.$  Then $-a - b = b,$ so $a = -2b.$  Then\n\\[-2b + 3 = 2b - b,\\]which gives us $b = 1$.  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+{"instance_id": "math500-0079", "gold": ["2"], "meta": {"row_pos": 79, "level": 1, "subject": "Prealgebra", "unique_id": "test/prealgebra/192.json", "solution": "Since the prime factorization of 72 is $72=2^3\\cdot 3^2$, we have $x=\\boxed{2}$."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:2", "label": "2", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "2", "count": 100}]}], "n_distinct": 1, "modal_answer": "2", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 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+{"instance_id": "math500-0080", "gold": ["501"], "meta": {"row_pos": 80, "level": 5, "subject": "Intermediate Algebra", "unique_id": "test/intermediate_algebra/1849.json", "solution": "First, note that if $k < 0,$ then $\\log(kx)$ is defined for $x \\in (-\\infty, 0),$ and is strictly decreasing on that interval. Since $2\\log(x+2)$ is defined for $x \\in (-2, \\infty)$ and is strictly increasing on that interval, it follows that $\\log(kx) = 2\\log(x+2)$ has exactly one real solution, which must lie in the interval $(-2, 0).$ Therefore, all the values $k = -500, -499, \\ldots, -2, -1$ satisfy the condition.\n\nIf $k = 0,$ then the left-hand side is never defined, so we may assume now that $k > 0.$ In this case, converting to exponential form, we have \\[ kx = (x+2)^2\\]or \\[x^2 + (4-k)x + 4 = 0.\\]Any solution of this equation satisfies $\\log(kx) = 2\\log(x+2)$ as well, as long as the two logarithms are defined; since $k > 0,$ the logarithms are defined exactly when $x > 0.$ Therefore, this quadratic must have exactly one positive root.\n\nBut by Vieta's formulas, the product of the roots of this quadratic is $4,$ which is positive, so the only way for it to have exactly one positive root is if it has $\\sqrt{4} = 2$ as a double root. That is, \\[x^2 + (4-k)x + 4 = (x-2)^2 = x^2 - 4x + 4\\]for all $x,$ so $4-k=-4,$ and $k=8,$ which is the only positive value of $k$ satisfying the condition.\n\nIn total, there are $500 + 1 = \\boxed{501}$ values of $k$ satisfying the condition."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:501", "label": "501", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "501", "count": 100}]}], "n_distinct": 1, "modal_answer": "501", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 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+{"instance_id": "math500-0081", "gold": ["3"], "meta": {"row_pos": 81, "level": 3, "subject": "Algebra", "unique_id": "test/algebra/1035.json", "solution": "Let $t,s,g$ be the weight of one treek, the weight of one squig, and the weight of one goolee, respectively. Then the given information tells us \\begin{align*}\n10t &=3s+g\\\\\n2t +g &= s.\n\\end{align*} Since we would like to solve for $s$ in terms of $t$, we want to eliminate $g$. Add the two equations to obtain \\begin{align*}\n10t+2t+g &= 3s+g+s\\\\\n\\Rightarrow 10t+2t &= 3s+s\\\\\n\\Rightarrow 4s &= 12t\\\\\n\\Rightarrow s &=3t.\n\\end{align*} So one squig weighs $\\boxed{3}$ treeks."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:3", "label": "3", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "3", "count": 100}]}], "n_distinct": 1, "modal_answer": "3", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 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The maximum slope is $\\boxed{\\frac{3}{2}}$."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 99, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:\\dfrac{3}{2}", "label": "\\dfrac{3}{2}", "count": 99, "match": "exact", "judge": null, "members": [{"raw": "\\dfrac{3}{2}", "count": 98}, {"raw": "\\frac{3}{2}", "count": 1}]}, {"bucket": "math:3", "label": "3", "count": 1, "match": "none", "judge": null, "members": [{"raw": "3", "count": 1}]}], "n_distinct": 2, "modal_answer": "\\dfrac{3}{2}", "modal_frac": 0.99, "modal_is_correct": true, "top_wrong_frac": 0.01, "collapsed_wrong": false, "entropy": 0.08079313589591118, "acc": 0.99, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 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Trying $x=1$, $x=2$, and $x=3$, we see that $10^x-9^x$ is increasing for $x>0$, and it equals 19 when $x=\\boxed{2}$.\n\nNote: Using calculus, we could prove that $10^x-9^x$ is monotonically increasing for $x>0$, which would prove that the solution we found is unique."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:2", "label": "2", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "2", "count": 100}]}], "n_distinct": 1, "modal_answer": "2", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 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Therefore our answer is $a+b=\\boxed{-1}$."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:-1", "label": "-1", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "-1", "count": 100}]}], "n_distinct": 1, "modal_answer": "-1", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99], 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+{"instance_id": "math500-0085", "gold": ["\\sqrt{5}"], "meta": {"row_pos": 85, "level": 4, "subject": "Geometry", "unique_id": "test/geometry/353.json", "solution": "The volume of the water increases by $20\\pi$ cubic meters each hour, while the height of the water in the tank rises by 4 meters each hour. The volume of a right cylinder is $\\pi r^2h$. If we look at the changes in volume and height in just one hour, we can solve for the radius.  \\begin{align*}\n\\pi r^2h_f-\\pi r^2h_0&=V_f-V_0\\quad\\Rightarrow\\\\\n\\pi r^2(\\Delta h)&=\\Delta V\\quad\\Rightarrow\\\\\n\\pi r^2(4)&=20\\pi\\quad\\Rightarrow\\\\\n4r^2&=20\\quad\\Rightarrow\\\\\nr^2&=5\n\\end{align*} Since the radius must be positive, $r=\\boxed{\\sqrt{5}}$ meters."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:\\sqrt{5}", "label": "\\sqrt{5}", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "\\sqrt{5}", "count": 100}]}], "n_distinct": 1, "modal_answer": "\\sqrt{5}", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 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+{"instance_id": "math500-0086", "gold": ["240"], "meta": {"row_pos": 86, "level": 1, "subject": "Prealgebra", "unique_id": "test/prealgebra/307.json", "solution": "There are 5 options for shirts, 6 options for pants, and 8 options for hats, for a total of $5 \\times 6 \\times 8 = \\boxed{240}$ outfits."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 91, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:240", "label": "240", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "240", "count": 100}]}], "n_distinct": 1, "modal_answer": "240", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 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+{"instance_id": "math500-0087", "gold": ["1"], "meta": {"row_pos": 87, "level": 1, "subject": "Algebra", "unique_id": "test/algebra/567.json", "solution": "We can split the expression $|5x-1|=x+3$ into two separate cases. In the first case, \\begin{align*} 5x-1&=x+3\n\\\\\\Rightarrow \\qquad 4x&=4\n\\\\\\Rightarrow \\qquad x&=1\n\\end{align*}If we plug this value of $x$ back into the original equation in order to check our answer, we see that $|5(1)-1|=1+3$ or $4=4$. Since this is true, we can accept $x=1$ as a valid solution.\n\nIn the second case, \\begin{align*} 5x-1&=-(x+3)\n\\\\ 5x-1&=-x-3\n\\\\\\Rightarrow \\qquad 6x&=-2\n\\\\\\Rightarrow \\qquad x&=-\\frac13.\n\\end{align*}If we plug $-\\frac13$ back into the initial equation, we get that $\\left|5\\left(-\\frac13\\right)-1\\right|=-\\frac13+3$, which simplifies to $\\left|-\\frac{8}{3}\\right|=\\frac{8}{3}$, or $\\frac{8}{3}=\\frac{8}{3}$. Since this is also a true statement, we can accept $x=-\\frac13$ as a valid solution as well. Since both $x=1$ and $x=-\\frac13$ are possible values of $x$, the final answer is the larger of the two: $x=\\boxed{1}$."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:1", "label": "1", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "1", "count": 100}]}], "n_distinct": 1, "modal_answer": "1", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99], "ok": [true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true, true], "answer": ["1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", "1", 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+{"instance_id": "math500-0088", "gold": ["2"], "meta": {"row_pos": 88, "level": 5, "subject": "Algebra", "unique_id": "test/algebra/892.json", "solution": "For $x < a,$ the graph of $y = f(x)$ is the same as the graph of $y = ax+2a,$ which is a line with slope $a$ and which passes through the point $(a, a^2+2a).$ For $x \\ge a,$ the graph of $y = f(x)$ is the same as the graph of $y = ax^2,$ which is a parabola passing through the point $(a, a^3).$\n\nNotice that the parabola only ever takes nonnegative values. Therefore, the line portion of the graph must have positive slope, because it must intersect horizontal lines which lie below the $x-$axis. Thus, $a > 0.$\n\nFor $a > 0,$ the line portion of the graph passes through all horizontal lines with height less than or equal to $a^2+2a,$ and the parabola portion of the graph passes through all horizontal lines with height greater than or equal to $a^3.$ Therefore, all horizontal lines are covered if and only if \\[a^2 + 2a \\ge a^3.\\]Since $ a > 0,$ we can divide by $a$ to get \\[a + 2 \\ge a^2,\\]so $0 \\ge a^2 - a - 2 = (a-2) ( a+1).$ This means that $-1 \\le a \\le 2,$ so the greatest possible value of $a$ is $\\boxed{2}.$\n\nThe graph of $y = f(x)$ for $a = 2$ is shown below (not to scale); note how the parabola and line meet at one point: [asy]\nsize(8cm);\nimport graph;\n\nreal a =2;\ndraw((-5,0)--(6,0),EndArrow());\ndraw((0,-6)--(0,14),EndArrow());\n\nreal g(real x) {return 0.5*a*(x-a)^2+a^3;}\nreal f(real x) {return a*x+2*a;}\n\ndraw(graph(f,-4.6,a),BeginArrow());\ndraw(graph(g,a,4.5),EndArrow());\n\nlabel(\"$f(x)$\",(0,15.5));\nlabel(\"$x$\",(6,0),E);\ndot((2,8));\n[/asy]"}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:2", "label": "2", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "2", "count": 100}]}], "n_distinct": 1, "modal_answer": "2", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 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+{"instance_id": "math500-0089", "gold": ["21"], "meta": {"row_pos": 89, "level": 4, "subject": "Number Theory", "unique_id": "test/number_theory/357.json", "solution": "We can perform the Euclidean Algorithm two times.\n\nFirst, we use it for $3339$ and $2961$. \\begin{align*}\n\\text{gcd}\\,(3339,2961) &=\\text{gcd}\\,(3339-2961,2961)\\\\\n&=\\text{gcd}\\,(378,2961)\\\\\n&=\\text{gcd}\\,(378,2961-378 \\cdot 7)\\\\\n&=\\text{gcd}\\,(378,315)\\\\\n&=\\text{gcd}\\,(378-315,315)\\\\\n&=\\text{gcd}\\,(63,315)\\\\\n\\end{align*}Since $63$ is a divisor of $315$, the greatest common divisor of $3339$ and $2961$ is $63$.\n\nNext, we can find the greatest common divisor of $63$ and $1491$, also using the Euclidean Algorithm. \\begin{align*}\n\\text{gcd}\\,(63,1491) &=\\text{gcd}\\,(63,1491-63 \\cdot 23)\\\\\n&=\\text{gcd}\\,(63,42)\\\\\n\\end{align*}Since $63=3 \\cdot 21$ and $42=2 \\cdot 21$, the greatest common divisor is $\\boxed{21}$."}, "n_samples": 100, "n_success": 100, "n_answered": 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+{"instance_id": "math500-0090", "gold": ["\\frac{3}{2}"], "meta": {"row_pos": 90, "level": 3, "subject": "Prealgebra", "unique_id": "test/prealgebra/1761.json", "solution": "Remember that dividing by a fraction is the same thing as multiplying by the reciprocal of the fraction.  The reciprocal of $\\frac{2}{3}$ is $\\boxed{\\frac{3}{2}}$, so that is what Remmy should multiply by."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:\\dfrac{3}{2}", "label": "\\dfrac{3}{2}", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "\\dfrac{3}{2}", "count": 93}, {"raw": "\\frac{3}{2}", "count": 7}]}], "n_distinct": 1, "modal_answer": "\\dfrac{3}{2}", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 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+{"instance_id": "math500-0091", "gold": ["1"], "meta": {"row_pos": 91, "level": 3, "subject": "Algebra", "unique_id": "test/algebra/2023.json", "solution": "In order for $f(x)$ to have a real number value, the expression inside the square root in the numerator must be non-negative and the denominator must not be 0. So we have the two conditions $x-1\\ge0 \\Rightarrow x \\ge 1$ and $x-2 \\ne 0 \\Rightarrow x \\ne 2$. We see that $x=\\boxed{1}$ is the smallest integer value that satisfies both conditions."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:1", "label": "1", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "1", "count": 100}]}], "n_distinct": 1, "modal_answer": "1", "modal_frac": 1.0, "modal_is_correct": true, "top_wrong_frac": 0.0, "collapsed_wrong": false, "entropy": 0.0, "acc": 1.0, "pass_any": true, "majority_correct": 1.0, "frac_truncated": 0.0, "frac_no_answer": 0.0, "sample_idx": [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 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+{"instance_id": "math500-0094", "gold": ["80"], "meta": {"row_pos": 94, "level": 5, "subject": "Prealgebra", "unique_id": "test/prealgebra/1646.json", "solution": "Triangle $AFD$ must have a total angle measure of $180^\\circ$. We know that the other two angles have measures of $2x$ and $2y$, so angle $AFD$ must have a measure of $180-2x-2y=180-(2x+2y)$ degrees. We now look at quadrilateral $ABCD$, whose interior angle measures must sum to $360^\\circ$. Therefore, we have that $110^\\circ +100^\\circ +3y+3x=360^\\circ$, so $3x+3y=150^\\circ$. We want to find $2x+2y$, so we multiply both sides of the equation by $2/3$ to get that $2x+2y=100^\\circ$. We can now substitute in $100^\\circ$ for $2x+2y$ to find that the measure of angle $AFD$ is $180-(2x+2y)=180-100=\\boxed{80}$ degrees."}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 54, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:80", "label": "80", "count": 54, "match": "exact", "judge": null, "members": [{"raw": "80", "count": 54}]}, {"bucket": "math:130", "label": "130", "count": 40, "match": "none", "judge": null, "members": [{"raw": "130", "count": 40}]}, {"bucket": "math:40", "label": "40", "count": 2, "match": "none", "judge": null, "members": [{"raw": "40", "count": 2}]}, {"bucket": "math:100", "label": "100", "count": 2, "match": "none", "judge": null, "members": [{"raw": "100", "count": 2}]}, {"bucket": "math:81", "label": "81", "count": 1, "match": "none", "judge": null, "members": [{"raw": "81", "count": 1}]}, {"bucket": "math:115", "label": "115", "count": 1, "match": "none", 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+{"instance_id": "math500-0095", "gold": ["-4"], "meta": {"row_pos": 95, "level": 2, "subject": "Precalculus", "unique_id": "test/precalculus/34.json", "solution": "The projection of $\\begin{pmatrix} 2 \\\\ y \\\\ -5 \\end{pmatrix}$ onto $\\begin{pmatrix} 1 \\\\ -2 \\\\ 1 \\end{pmatrix}$ is\n\\[\\frac{\\begin{pmatrix} 2 \\\\ y \\\\ -5 \\end{pmatrix} \\cdot \\begin{pmatrix} 1 \\\\ -2 \\\\ 1 \\end{pmatrix}}{\\begin{pmatrix} 1 \\\\ -2 \\\\ 1 \\end{pmatrix} \\cdot \\begin{pmatrix} 1 \\\\ -2 \\\\ 1 \\end{pmatrix}} \\begin{pmatrix} 1 \\\\ -2 \\\\ 1 \\end{pmatrix} = \\frac{-2y - 3}{6} \\begin{pmatrix} 1 \\\\ -2 \\\\ 1 \\end{pmatrix}.\\]Then $-2y - 3 = 5,$ so $y = \\boxed{-4}.$"}, "n_samples": 100, "n_success": 100, "n_answered": 100, "n_bare": 0, "n_boxed": 100, "n_bold": 0, "n_correct": 100, "n_truncated": 0, "n_no_answer": 0, "answer_counts": [{"bucket": "math:-4", "label": "-4", "count": 100, "match": "exact", "judge": null, "members": [{"raw": "-4", "count": 100}]}], "n_distinct": 1, 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+{"instance_id": "math500-0096", "gold": ["1 \\pm \\sqrt{19}"], "meta": {"row_pos": 96, "level": 5, "subject": "Intermediate Algebra", "unique_id": "test/intermediate_algebra/662.json", "solution": "Multiplying out each numerator and denominator, we get\n\\[\\frac{x^2 - 2x - 3}{5(x^2 - 2x - 8)} + \\frac{x^2 - 2x - 15}{9(x^2 - 2x - 24)} - \\frac{2(x^2 - 2x - 35)}{13(x^2 - 2x - 48)} = \\frac{92}{585}.\\]We can write this as\n\\[\\frac{(x^2 - 2x - 8) + 5}{5(x^2 - 2x - 8)} + \\frac{(x^2 - 2x - 24) + 9}{9(x^2 - 2x - 24)} - \\frac{2((x^2 - 2x - 48) + 13)}{13(x^2 - 2x - 48)} = \\frac{92}{585}.\\]Hence,\n\\[\\frac{1}{5} + \\frac{1}{x^2 - 2x - 8} + \\frac{1}{9} + \\frac{1}{x^2 - 2x - 24} - \\frac{2}{13} - \\frac{2}{x^2 - 2x - 48} = \\frac{92}{585}.\\]This simplifies to\n\\[\\frac{1}{x^2 - 2x - 8} + \\frac{1}{x^2 - 2x - 24} - \\frac{2}{x^2 - 2x - 48} = 0.\\]Let $y = x^2 - 2x - 48.$  Then\n\\[\\frac{1}{y + 40} + \\frac{1}{y + 24} - \\frac{2}{y} = 0.\\]Multiplying everything by $y(y + 24)(y + 40),$ we get\n\\[y(y + 24) + y(y + 40) - 2(y + 24)(y + 40) = 0.\\]This simplifies to $64y + 1920 = 0,$ so $y = -30.$  Then $x^2 - 2x - 48 = -30,$ or $x^2 - 2x - 18 = 0.$  By the quadratic formula, $x = \\boxed{1 \\pm \\sqrt{19}}.$  (Since the denominators are nonzero for these values, we know that they are not extraneous.)"}, "n_samples": 100, "n_success": 82, "n_answered": 80, "n_bare": 0, "n_boxed": 80, "n_bold": 0, "n_correct": 13, "n_truncated": 2, "n_no_answer": 2, "answer_counts": [{"bucket": "math:1 - \\sqrt{19}", "label": "1 - \\sqrt{19}", "count": 36, "match": "none", "judge": null, "members": [{"raw": "1 - \\sqrt{19}", "count": 36}]}, {"bucket": "math:1 + \\sqrt{19}", "label": "1 + \\sqrt{19}", "count": 29, "match": "none", "judge": null, "members": [{"raw": "1 + \\sqrt{19}", "count": 29}]}, {"bucket": "math:1 - \\sqrt{19}, 1 + \\sqrt{19}", "label": "1 - \\sqrt{19}, 1 + \\sqrt{19}", "count": 13, "match": "exact", "judge": null, "members": [{"raw": "1 - \\sqrt{19}, 1 + \\sqrt{19}", "count": 7}, {"raw": "1 + \\sqrt{19}, 1 - 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