id stringlengths 71 71 | problem stringlengths 8 3.62k | reference_answers listlengths 1 7 | original_problem_variants listlengths 1 3 | problem_sources listlengths 0 4 | source_splits listlengths 1 4 | source_split_row_counts unknown | source_row_count int64 1 364 | missing_reference_answer_rows int64 0 0 | blank_reference_answer_rows int64 0 25 |
|---|---|---|---|---|---|---|---|---|---|
omr-ws-c90c596bda3db164d9117ab1ee1b8b320916de23f7eb6e9ffe9a9b9b15a036ce | A straight line cuts two concentric circles at points \( A, B, C, \) and \( D \) in that order. \( AE \) and \( BF \) are parallel chords, one in each circle. \( GC \) is perpendicular to \( BF \) at \( G \), and \( DH \) is perpendicular to \( AE \) at \( H \). If the radii of the two circles are \( r \) and \( R \) (... | [
"0"
] | [
"A straight line cuts two concentric circles at points \\( A, B, C, \\) and \\( D \\) in that order. \\( AE \\) and \\( BF \\) are parallel chords, one in each circle. \\( GC \\) is perpendicular to \\( BF \\) at \\( G \\), and \\( DH \\) is perpendicular to \\( AE \\) at \\( H \\). If the radii of the two circles ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 8
} | 10 | 0 | 0 |
omr-ws-b306ea917e980ef1d571decffd5850af29c7a94ed58f139b7f15f763f43cc994 | A straight line given by \( lx + my = 1 \) intersects the parabola \( y^2 = 4x \) at points \( P \) and \( Q \). Find the point \( R \) on the parabola such that the normals at \( P \), \( Q \), and \( R \) are concurrent. | [
"\\( R = (t_3^2, 2t_3) \\text{ where } t_3 = -(t_1 + t_2) \\)."
] | [
"A straight line given by \\( lx + my = 1 \\) intersects the parabola \\( y^2 = 4x \\) at points \\( P \\) and \\( Q \\). Find the point \\( R \\) on the parabola such that the normals at \\( P \\), \\( Q \\), and \\( R \\) are concurrent."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9c4b939d5e7a8ea4a8dd22a2394422ca7184f90a8804c185b1b674cd80d5e23d | A straight line intersects a hyperbola \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) at two points \( A \) and \( B \), and intersects the asymptotes of the hyperbola at two points \( C \) and \( D \). If the coordinates of \( A \) are \( (x_1, y_1) \) and the coordinates of \( B \) are \( (x_2, y_2) \), find the value o... | [
"\\dfrac{2ab(1 + m)\\sqrt{b^2 + c^2 - a^2 m^2}}{b^2 - a^2 m^2}"
] | [
"A straight line intersects a hyperbola \\( \\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1 \\) at two points \\( A \\) and \\( B \\), and intersects the asymptotes of the hyperbola at two points \\( C \\) and \\( D \\). If the coordinates of \\( A \\) are \\( (x_1, y_1) \\) and the coordinates of \\( B \\) are \\( (x_2, y... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-8cc8a0277337af248829b67e91b67ebd29cb5df48800b5617742c2d6374a9763 | A straight line intersects the sides $AB$, $BC$, and the extension of the side $AC$ of a triangle $ABC$ at points $D$, $E$, and $F$ respectively. Let $M$, $N$, and $P$ be the midpoints of the line segments $DC$, $AE$, and $BF$ respectively. If $A = (0,0)$, $B = (2,0)$, $C = (1,3)$, $D = (0.5, 0)$, $E = (1.5, 1.5)$, and... | [
"\\text{undefined}"
] | [
"A straight line intersects the sides $AB$, $BC$, and the extension of the side $AC$ of a triangle $ABC$ at points $D$, $E$, and $F$ respectively. Let $M$, $N$, and $P$ be the midpoints of the line segments $DC$, $AE$, and $BF$ respectively. If $A = (0,0)$, $B = (2,0)$, $C = (1,3)$, $D = (0.5, 0)$, $E = (1.5, 1.5)$... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bdb012abd0a60bfdb4116854461eb89aa5dcb7601e8a55a7b6543ce4cd3afc43 | A straight line intersects two concentric circles at points $A$, $B$, $C$, and $D$ in that order. Chords $AE$ and $BF$ are parallel, one in each circle. $GC$ is perpendicular to $BF$ at $G$, and $DH$ is perpendicular to $AE$ at $H$. Determine the value of $\frac{HE}{GF}$. | [
"1"
] | [
"A straight line intersects two concentric circles at points $A$, $B$, $C$, and $D$ in that order. Chords $AE$ and $BF$ are parallel, one in each circle. $GC$ is perpendicular to $BF$ at $G$, and $DH$ is perpendicular to $AE$ at $H$. Determine the value of $\\frac{HE}{GF}$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-329bc171e781366e6c92c0358d2ec6d17d1a2cab817f03bb2e84199e59634b88 | A straight line intersects two concentric circles at points \(A_1, A_2\) and \(B_1, B_2\). Let \(P, Q, R, S\) be the intersection points of the tangents at \(A_1, A_2\) and \(B_1, B_2\). If the radii of the two concentric circles are \(r\) and \(R\) with \(R > r\), and the distance between the centers of the concentric... | [
"\\dfrac{rR}{d}"
] | [
"A straight line intersects two concentric circles at points \\(A_1, A_2\\) and \\(B_1, B_2\\). Let \\(P, Q, R, S\\) be the intersection points of the tangents at \\(A_1, A_2\\) and \\(B_1, B_2\\). If the radii of the two concentric circles are \\(r\\) and \\(R\\) with \\(R > r\\), and the distance between the cent... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-f746eb89430d5fa2f018920fcc97105931cae4012684b9cd07294214ec263244 | A straight line intersects two concentric circles at points \(A_1, A_2\) and \(B_1, B_2\). Let \(P\) be the intersection of the tangents at \(A_1\) and \(B_1\), \(Q\) be the intersection of the tangents at \(A_1\) and \(B_2\), \(R\) be the intersection of the tangents at \(A_2\) and \(B_1\), and \(S\) be the intersecti... | [
"2R"
] | [
"A straight line intersects two concentric circles at points \\(A_1, A_2\\) and \\(B_1, B_2\\). Let \\(P\\) be the intersection of the tangents at \\(A_1\\) and \\(B_1\\), \\(Q\\) be the intersection of the tangents at \\(A_1\\) and \\(B_2\\), \\(R\\) be the intersection of the tangents at \\(A_2\\) and \\(B_1\\), ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-79079f70422fa6a6fff615e7ab43e8ede35e3543409e2586e4781c6dfbe3ae54 | A straight line is drawn from the apex \( A \) of triangle \( ABC \) and intersects the opposite side \( BC \) at a point \( D \) such that \( D \) divides \( BC \) in the ratio 3:2 starting from \( C \). How does this straight line divide \( BC \)? | [
"\\( \\frac{3}{2} \\)"
] | [
"A straight line is drawn from the apex \\( A \\) of triangle \\( ABC \\) and intersects the opposite side \\( BC \\) at a point \\( D \\) such that \\( D \\) divides \\( BC \\) in the ratio 3:2 starting from \\( C \\). How does this straight line divide \\( BC \\)?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-fe352beb86cdccdfd4d107514d651076dc80094c13c67d7d465dcdd7b61f6fb4 | A straight line is drawn from the top left corner to the bottom right corner of a chessboard with A rows and B columns, where A and B have no common divisors other than 1. Find a general formula in terms of A and B that tells you how many squares the diagonal line passes through (excluding squares where the line only t... | [
"\\( A + B - 1 \\)"
] | [
"A straight line is drawn from the top left corner to the bottom right corner of a chessboard with A rows and B columns, where A and B have no common divisors other than 1. Find a general formula in terms of A and B that tells you how many squares the diagonal line passes through (excluding squares where the line o... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-689a53dd031dfe9e8b60c5f2dbff5e5f0ae02123b99c373a997360f755eb5111 | A straight line is drawn parallel to the conjugate axis of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ and intersects the hyperbola at point $P$ and the conjugate hyperbola at point $Q$. Find the locus of the intersection of the tangents and normals at $P$ and $Q$ to the hyperbola and its conjugate hyperbola,... | [
"\\( b^{2}x^{2}(y^{4}+4b^{4})-a^{2}y^{6}=0 \\) for \\(x > a\\) and \\(x < -a\\)."
] | [
"A straight line is drawn parallel to the conjugate axis of the hyperbola $\\frac{x^2}{a^2} - \\frac{y^2}{b^2} = 1$ and intersects the hyperbola at point $P$ and the conjugate hyperbola at point $Q$. Find the locus of the intersection of the tangents and normals at $P$ and $Q$ to the hyperbola and its conjugate hyp... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0d0b1053bae383515b1509d22012cc72148a5f52d6c3b7c6b827b7741d35bace | A straight line is drawn through an arbitrary internal point $K$ of the trapezoid $ABCD$, intersecting the bases $BC$ and $AD$ at points $P$ and $Q$, respectively. The circles circumscribed around the triangles $BPK$ and $DQK$ intersect, besides the point $K$, also at the point $L$. If $BD$ intersects $PQ$ at point $M$... | [
"\\dfrac{4}{9}"
] | [
"A straight line is drawn through an arbitrary internal point $K$ of the trapezoid $ABCD$, intersecting the bases $BC$ and $AD$ at points $P$ and $Q$, respectively. The circles circumscribed around the triangles $BPK$ and $DQK$ intersect, besides the point $K$, also at the point $L$. If $BD$ intersects $PQ$ at poin... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-1a9a88b49841430e1f8d953c3201ebe65127916797dc03a826b8c2acd3679413 | A straight line joins the points $(-1, 1)$ and $(3, 9)$. What is the $x$-intercept of this line? | [
"\\(-\\frac{3}{2}\\)"
] | [
"A straight line joins the points $(-1, 1)$ and $(3, 9)$. What is the $x$-intercept of this line?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-c57c9d01493d342c709893e697952b3a7a8605e3d9ae480a25942faefbac1427 | A straight line makes equal angles with the edges of a right trihedral angle. Find the angles. | [
"$\\cos \\alpha =\\cos \\beta =\\cos \\gamma =\\frac{1}{\\sqrt 3}$"
] | [
"A straight line makes equal angles with the edges of a right trihedral angle. Find the angles."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-48715c024e5b60cdfe44708fd22e9923bbb2059f8a52b089850691dda3737dc5 | A straight line makes positive intercepts on the coordinate axes. The sum of the intercepts is 7. If the line passes through the point (-3, 8), find the equation of the line. | [
"4x + 3y = 12"
] | [
"A straight line makes positive intercepts on the coordinate axes. The sum of the intercepts is 7. If the line passes through the point (-3, 8), find the equation of the line."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3d3e4227c2277703b3b048a2ba5ed619eed465ae37066c9a497607ed1a738c2d | A straight line parallel to the base of a triangle, whose area is equal to $S$, cuts off it a triangle with an area equal to $q$. Determine the area of a quadrilateral, whose three vertices coincide with those of the smaller triangle and the fourth one lies on the base of the larger triangle. | [
"$\\sqrt {Sq}$"
] | [
"A straight line parallel to the base of a triangle, whose area is equal to $S$, cuts off it a triangle with an area equal to $q$. Determine the area of a quadrilateral, whose three vertices coincide with those of the smaller triangle and the fourth one lies on the base of the larger triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a6413992224abdac169b02a6070d1a27310e7616965120828c51a965aff35fb5 | A straight line parallel to the bases of a right trapezoid cuts it into two trapezoids, each of which can inscribe a circle. Given that the lateral sides of the original trapezoid are $c$ and $d$ with $c < d$, find the lengths of the bases of the original trapezoid. | [
"\\(c\\) and \\(d\\)"
] | [
"A straight line parallel to the bases of a right trapezoid cuts it into two trapezoids, each of which can inscribe a circle. Given that the lateral sides of the original trapezoid are $c$ and $d$ with $c < d$, find the lengths of the bases of the original trapezoid."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8ede99b90713e5d2cfb8abffb19cc6c2a0c4ee2ddd03e8252194078f28726783 | A straight line passes inside a trihedral angle, all plane angles of which are equal to $\alpha $. The line is inclined identically to all the edges. Find the angle of inclination of this line to each edge of the trihedral angle. | [
"$\\arcsin \\left[\\left(2/\\sqrt 3\\right)\\sin (\\alpha /2)\\right]$"
] | [
"A straight line passes inside a trihedral angle, all plane angles of which are equal to $\\alpha $. The line is inclined identically to all the edges. Find the angle of inclination of this line to each edge of the trihedral angle."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fd7ad6350c26caa64e757c8616487b9e293bb59fa2046c379e134ceb6615b822 | A straight line passes through the center of a square with side \(a\). Find the sum of the squared distances from all the vertices of the square to this line. | [
"a^2"
] | [
"A straight line passes through the center of a square with side \\(a\\). Find the sum of the squared distances from all the vertices of the square to this line."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-db2d8d99ba19ed27f4acb5f66b44980b8bf58da413dc284ef8a3d59877fef5e1 | A straight line passes through the origin and the region defined by $x^2 + (y-1)^2 \leq 1$ and $x \geq \frac{\sqrt{2}}{3}$. Let $L$ be the length of the line that passes through the region. Find the greatest value of $L$. | [
"\\(\\sqrt{\\frac{2}{3}}\\)"
] | [
"A straight line passes through the origin and the region defined by $x^2 + (y-1)^2 \\leq 1$ and $x \\geq \\frac{\\sqrt{2}}{3}$. Let $L$ be the length of the line that passes through the region. Find the greatest value of $L$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 8,
"tir": 25
} | 45 | 0 | 0 |
omr-ws-9b178ee2b8b7ea19772a13235f71d928986a59ad0e309decfca6b29f7936ba9d | A straight line passes through the point \(A(3,4)\) and is bisected at this point. Find the equation of the line. | [
"4x + 3y = 24"
] | [
"A straight line passes through the point \\(A(3,4)\\) and is bisected at this point. Find the equation of the line."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-79287c4049c3edc505497bf0bb398c2c398ebc880bd464d777a868d54c27a6d5 | A straight line passing through the center of the circumscribed circle and the intersection point of the heights of the non-equilateral triangle \(ABC\) divides its perimeter and area in the same ratio. Find this ratio. | [
"1"
] | [
"A straight line passing through the center of the circumscribed circle and the intersection point of the heights of the non-equilateral triangle \\(ABC\\) divides its perimeter and area in the same ratio. Find this ratio."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-91290db772ae8e20661349922ad1d97c86209b42c043bcd63508e7adf92f1ae6 | A straight line passing through the point A(-1, -25) and parallel to the line \(5x - 4y + 95 = 0\) cuts the x-axis and y-axis at points B and C, respectively. Consider the unit grid in the plane. Find the number of squares in the grid containing the line segment BC in their interior. | [
"42"
] | [
"A straight line passing through the point A(-1, -25) and parallel to the line \\(5x - 4y + 95 = 0\\) cuts the x-axis and y-axis at points B and C, respectively. Consider the unit grid in the plane. Find the number of squares in the grid containing the line segment BC in their interior."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fc331c701153a3e10d20b205074e529245f08a3eaafae8b2da5c9ce87f3af25f | A straight line segment $AB$ of length $a$ connects two light sources $A$ (with intensity $p$) and $B$ (with intensity $q$). Find the point $M$ which is illuminated the worst. (The illumination of a point is inversely proportional to the square of the distance from the light source.) | [
"$AM=a\\sqrt [3]{p}\\left(\\sqrt [3]{p}+\\sqrt [3]{q}\\right)^{-1}$"
] | [
"A straight line segment $AB$ of length $a$ connects two light sources $A$ (with intensity $p$) and $B$ (with intensity $q$). Find the point $M$ which is illuminated the worst. (The illumination of a point is inversely proportional to the square of the distance from the light source.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ba27434fa2c4eef1c4b4a03998c2d95669cc944e3134e25f01910b19f8564031 | A straight line segment of length $L$ subtends equal angles at two fixed points $A$ and $B$. If the distance between $A$ and $B$ is $d$, find the radius $r$ of the fixed circle that the line segment always touches, in terms of $L$ and $d$. | [
"no_valid_answer_found"
] | [
"A straight line segment of length $L$ subtends equal angles at two fixed points $A$ and $B$. If the distance between $A$ and $B$ is $d$, find the radius $r$ of the fixed circle that the line segment always touches, in terms of $L$ and $d$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9ebedf1c616fccfa09ef4ce8a9861c8a4984c214a6304b5d0c6fdd3def2fb386 | A straight line tangent to a circle circumscribed about an isosceles triangle $ABC$ ($AB = AC$) at point $B$ intersects line $AC$ at point $P$. Point $E$ is the midpoint of $AB$. What is the projection of $DE$ onto $AB$ if $PA = a$? | [
"\\dfrac{a}{4}"
] | [
"A straight line tangent to a circle circumscribed about an isosceles triangle $ABC$ ($AB = AC$) at point $B$ intersects line $AC$ at point $P$. Point $E$ is the midpoint of $AB$. What is the projection of $DE$ onto $AB$ if $PA = a$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-770130d6952f8f4532737b6f951fa017c8780be2afbb93004d00d1e5fbacb181 | A straight line tangent to a cone forms with its generatrix, at the point of tangency, an acute angle $\alpha $ and is inclined to the plane of the base of the cone at an angle $\beta $. Determine the angle between the generatrix and the base plane. | [
"$\\arcsin (\\sin \\beta \\sec \\alpha )$"
] | [
"A straight line tangent to a cone forms with its generatrix, at the point of tangency, an acute angle $\\alpha $ and is inclined to the plane of the base of the cone at an angle $\\beta $. Determine the angle between the generatrix and the base plane."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8cbd841a865456d35fad7a459dffb28fab5de7553902706a72e2356fb5940aa9 | A straight line through the circumcenter of triangle \(ABC\) intersects \(AB\), \(BC\), and \(CA\) at points \(R\), \(Q\), and \(P\) respectively. Let \(X\) and \(Y\) be the two common points of the circles with diameters \(AP\), \(BQ\), and \(CR\), where \(X\) lies on the circumcircle and \(Y\) lies on the nine-point ... | [
"2\\sqrt{R^2 - 4d^2}"
] | [
"A straight line through the circumcenter of triangle \\(ABC\\) intersects \\(AB\\), \\(BC\\), and \\(CA\\) at points \\(R\\), \\(Q\\), and \\(P\\) respectively. Let \\(X\\) and \\(Y\\) be the two common points of the circles with diameters \\(AP\\), \\(BQ\\), and \\(CR\\), where \\(X\\) lies on the circumcircle an... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-ae8f95bc86005b7a0dfb7b293e3937ad06f49220e6c423d62e4dcc4e437cb240 | A straight line with a negative slope passing through the point $(1,4)$ meets the coordinate axes at points $A$ and $B$. What is the minimum value of $OA + OB$? | [
"9"
] | [
"A straight line with a negative slope passing through the point $(1,4)$ meets the coordinate axes at points $A$ and $B$. What is the minimum value of $OA + OB$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-6a7a1ee9eff7e0da28f4dc2baf0893cb54fb9493ef10580213260d3f770ad568 | A straight line with negative slope passes through $A(1, 2)$. If the sum of the two intercepts reaches its minimum, determine the slope. | [
"-\\sqrt{2}"
] | [
"A straight line with negative slope passes through $A(1, 2)$. If the sum of the two intercepts reaches its minimum, determine the slope."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4b34463d881cb4cecac4a04c722eb298897f03c58024197e06e4f9fc2aca3d9a | A straight line, tangent to the curved surface of a cone, forms an angle $\theta $ with the element passing through the point of tangency. What angle $(\varphi )$ does this line form with the plane of the base $P$ of the cone, if its generator is inclined to the plane $P$ at an angle $\alpha $? | [
"$\\varphi =\\arcsin(\\sin\\alpha \\cos\\theta )$"
] | [
"A straight line, tangent to the curved surface of a cone, forms an angle $\\theta $ with the element passing through the point of tangency. What angle $(\\varphi )$ does this line form with the plane of the base $P$ of the cone, if its generator is inclined to the plane $P$ at an angle $\\alpha $?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-459a77fa0d2d03abf9b5b6266bc91f8b6a749a8663db3f5d33b13e7cb15135d6 | A straight needle of length 2015 is placed in a vertical position in a coordinate grid, not passing through any lattice point. It is then rotated 90 degrees around some point in the plane, such that the needle ends up in a horizontal position. What is the smallest possible number of lattice points the needle could have... | [
"0"
] | [
"A straight needle of length 2015 is placed in a vertical position in a coordinate grid, not passing through any lattice point. It is then rotated 90 degrees around some point in the plane, such that the needle ends up in a horizontal position. What is the smallest possible number of lattice points the needle could... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-b32c58d7521617efd7062d2652be93aeed46584a9d140184627827a1ffe62b8f | A straight one-mile stretch of highway, 40 feet wide, is closed. Robert rides his bike on a path composed of semicircles. If he rides at 5 miles per hour, how many hours will it take to cover the one-mile stretch? Note: 1 mile = 5280 feet. | [
"\\(\\frac{\\pi}{10}\\)"
] | [
"A straight one-mile stretch of highway, 40 feet wide, is closed. Robert rides his bike on a path composed of semicircles. If he rides at 5 miles per hour, how many hours will it take to cover the one-mile stretch?\n\nNote: 1 mile = 5280 feet."
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-24e26c0dbb32c978a3a34d7d0444a3503648d7ed84c7407bdf8266ae5793748d | A straight piece of wire is cut at random, and the larger piece is then cut at random. What is the probability that the three pieces of wire can form the sides of a triangle? | [
"\\( 2 \\ln 2 - 1 \\)"
] | [
"A straight piece of wire is cut at random, and the larger piece is then cut at random. What is the probability that the three pieces of wire can form the sides of a triangle?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-dbc94a2034020117326f5f75ca17cd61ba0bab9446fd533811ff02fd20bd9abc | A straight pyramid has a tip at \( S(2, 2, 1) \) and a height of 3 cm. The base area \( ABCD \) is a square with an edge length of 2 cm. Calculate the length of the side edges. | [
"\\sqrt{11}"
] | [
"A straight pyramid has a tip at \\( S(2, 2, 1) \\) and a height of 3 cm. The base area \\( ABCD \\) is a square with an edge length of 2 cm. Calculate the length of the side edges."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1bafd03f77fb128a69b7668b3da3069e693a2ade0527bb5750585421f21af3d9 | A straight pyramid has a tip at \( S(2, 2, 1) \) and a height of 3 cm. The base area \( ABCD \) is a square with an edge length of 2 cm. Determine the coordinates of the vertices \( A, B, C, \) and \( D \). | [
"A(3, 3, -2), B(3, 1, -2), C(1, 1, -2), D(1, 3, -2)"
] | [
"A straight pyramid has a tip at \\( S(2, 2, 1) \\) and a height of 3 cm. The base area \\( ABCD \\) is a square with an edge length of 2 cm. Determine the coordinates of the vertices \\( A, B, C, \\) and \\( D \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-ceaf79b0d8f58317067b17b9ae4d84b8fc5be842bb9ac88f11432a139a402699 | A straight river that is $264$ meters wide flows from west to east at a rate of $14$ meters per minute. Melanie and Sherry sit on the south bank of the river with Melanie a distance of $D$ meters downstream from Sherry. Relative to the water, Melanie swims at $80$ meters per minute, and Sherry swims at $60$ meters per ... | [
"550"
] | [
"A straight river that is $264$ meters wide flows from west to east at a rate of $14$ meters per minute. Melanie and Sherry sit on the south bank of the river with Melanie a distance of $D$ meters downstream from Sherry. Relative to the water, Melanie swims at $80$ meters per minute, and Sherry swims at $60$ meters... | [
"MATH_training_set",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 19
} | 66 | 0 | 0 |
omr-ws-78bd0d0563291628b482867970ccf6209ba84ab96f5e492ee0187a36e7ce5fc3 | A straight road consists of alternating green and red segments, with the first and last segments being green. The lengths of all segments are more than a centimeter and less than a meter, and each subsequent segment is longer than the previous one. A grasshopper jumps forward along the road, stepping on each green segm... | [
"5"
] | [
"A straight road consists of alternating green and red segments, with the first and last segments being green. The lengths of all segments are more than a centimeter and less than a meter, and each subsequent segment is longer than the previous one. A grasshopper jumps forward along the road, stepping on each green... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-580ef73e0e84f9de9942da435f0caaeef7e34d5b79382686fd2095127e081822 | A straight rod is constructed at an angle of $45^\circ$ to the water surface and reaches a depth of $100$ metres. A diver is connected to the rod by a flexible cable that allows the diver to move away from the rod by no more than $10$ metres. Considering the diver's size to be negligible, find the volume of the accessi... | [
"\\( 10000\\pi \\) cubic meters, 31416 cubic meters."
] | [
"A straight rod is constructed at an angle of $45^\\circ$ to the water surface and reaches a depth of $100$ metres. A diver is connected to the rod by a flexible cable that allows the diver to move away from the rod by no more than $10$ metres. Considering the diver's size to be negligible, find the volume of the a... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-84d31a5fcfd0f7b13785ab02be35c2542a3946474c0dd3f66793fe09c2a8aad1 | A straight rod of 2 meters in length is cut into \( N \) sticks, where the length of each stick is an integer number of centimeters. For which smallest \( N \) can one guarantee that it is possible to form the contour of some rectangle using all the sticks without breaking them further? | [
"102"
] | [
"A straight rod of 2 meters in length is cut into \\( N \\) sticks, where the length of each stick is an integer number of centimeters. For which smallest \\( N \\) can one guarantee that it is possible to form the contour of some rectangle using all the sticks without breaking them further?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-215263410bb3ed989b2e62419ac581ca1206caa27f0e331c37cfc36ec314937f | A straight rod of length 4 inches has its ends sliding along the perimeter of a square with side length 12 inches. A paint brush is attached to the rod so that it can slide between the two ends of the rod. Determine the total possible area of the square that can be painted by the brush. | [
"80"
] | [
"A straight rod of length 4 inches has its ends sliding along the perimeter of a square with side length 12 inches. A paint brush is attached to the rod so that it can slide between the two ends of the rod. Determine the total possible area of the square that can be painted by the brush."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-b45d70df32ba7906178217b76324ccfb60584bde1bc421a0dfea0f87e12934b4 | A straight stick of finite length is broken into 3 parts at random. What is the probability that these 3 parts can be used to form a triangle? | [
"\\(\\frac{1}{4}\\)"
] | [
"A straight stick of finite length is broken into 3 parts at random. What is the probability that these 3 parts can be used to form a triangle?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 33
} | 49 | 0 | 0 |
omr-ws-660f9076c6ef216fc04c7c36d49d0f09eabd8fafd1fbb78d89af7b136d6b57f2 | A street has 19 streetlamps. Some of them work and some are broken. The working streetlamps are never next to each other, and there are never three broken ones in a row. How many such arrangements are possible? | [
"351"
] | [
"A street has 19 streetlamps. Some of them work and some are broken. The working streetlamps are never next to each other, and there are never three broken ones in a row. How many such arrangements are possible?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dd0fc9cb4bbfc332586ceed03a37f87e89d7d50a82db01a15013a3241a2da33e | A street has 20 houses on each side, for a total of 40 houses. The addresses on the south side of the street form an arithmetic sequence, as do the addresses on the north side of the street. On the south side, the addresses are 4, 10, 16, etc., and on the north side they are 3, 9, 15, etc. A sign painter paints house n... | [
"84"
] | [
"A street has 20 houses on each side, for a total of 40 houses. The addresses on the south side of the street form an arithmetic sequence, as do the addresses on the north side of the street. On the south side, the addresses are 4, 10, 16, etc., and on the north side they are 3, 9, 15, etc. A sign painter paints ho... | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-66275c8d368dd1de29843c198354e3615b054f019e239c51828cd18becbd0c34 | A street has parallel curbs 10 meters apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is 6 meters and each stripe is 15 meters long. Find the distance, in meters, between the stripes. | [
"4 meters."
] | [
"A street has parallel curbs 10 meters apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is 6 meters and each stripe is 15 meters long. Find the distance, in meters, between the stripes."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-a57e473047587defa7a71d3f8ed6c9c5f27b15d871f3b291806509c53053b3e6 | A street has parallel curbs 40 feet apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is 15 feet and each stripe is 50 feet long. Find the distance, in feet, between the stripes. | [
"12"
] | [
"A street has parallel curbs 40 feet apart. A crosswalk bounded by two parallel stripes crosses the street at an angle. The length of the curb between the stripes is 15 feet and each stripe is 50 feet long. Find the distance, in feet, between the stripes."
] | [
"MATH_training_set",
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 94,
"tir": 83
} | 177 | 0 | 0 |
omr-ws-3e267b072848fe000a36a7747e56fcac56c9663315f428f4a61374c5f466cc5b | A street has parallel curbs 40 feet apart. A crosswalk bounded by two parallel strips crosses the street at an angle. The length of the curb between the strips is 15 feet, and each strip is 50 feet long. Find the distance between the strips. | [
"12"
] | [
"A street has parallel curbs 40 feet apart. A crosswalk bounded by two parallel strips crosses the street at an angle. The length of the curb between the strips is 15 feet, and each strip is 50 feet long. Find the distance between the strips."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 6,
"tir": 17
} | 38 | 0 | 0 |
omr-ws-5fa845579d98fb66e7eebdad45cd53a03571453adb24b32ce5f3c382838146da | A street light is at the top of a 20 ft tall pole. A woman 6 ft tall walks away from the pole with a speed of 8 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 30 ft from the base of the pole? | [
"\\(\\frac{80}{7}\\) ft/sec"
] | [
"A street light is at the top of a 20 ft tall pole. A woman 6 ft tall walks away from the pole with a speed of 8 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 30 ft from the base of the pole?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-71de10a3f596faf7b0bebf835b04a8e8f81ac325b98eb4fa42c1dd9b57e12f3b | A street light is mounted at the top of a 15 ft tall pole. A man 6 ft tall walks away from the pole with a speed of 5 ft/sec along a straight path. How fast is the tip of his shadow moving when he is 40 ft from the pole? | [
"\\(\\frac{25}{3}\\)"
] | [
"A street light is mounted at the top of a 15 ft tall pole. A man 6 ft tall walks away from the pole with a speed of 5 ft/sec along a straight path. How fast is the tip of his shadow moving when he is 40 ft from the pole?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-bd97151d6da462466eb50f5be5009d21079528e5437a02509821528d3f7c607c | A street light is mounted on a 15ft pole. A six-foot tall person walks away from the street light at a rate of 5ft/sec. How fast is the tip of his shadow moving from his perspective? | [
"\\dfrac{10}{3}"
] | [
"A street light is mounted on a 15ft pole. A six-foot tall person walks away from the street light at a rate of 5ft/sec. How fast is the tip of his shadow moving from his perspective?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9fbe213b9586b6d5dbc5d2ef345f6bbd361d7465d3028f8dd0b8757746819f06 | A strictly increasing geometric sequence of positive integers \(a_1, a_2, a_3, \cdots\) satisfies the following properties: - Each term leaves a common remainder when divided by 7. - The first term \(a_1\) is an integer from 1 to 6. - The common ratio is a perfect square. Let \(N\) be the smallest possible value of \(\... | [
"76"
] | [
"A strictly increasing geometric sequence of positive integers \\(a_1, a_2, a_3, \\cdots\\) satisfies the following properties:\n- Each term leaves a common remainder when divided by 7.\n- The first term \\(a_1\\) is an integer from 1 to 6.\n- The common ratio is a perfect square.\n\nLet \\(N\\) be the smallest pos... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-d6c8e40114d3bfac8fc8f7bbbfe5756f64e2db9ca2507cba4f856e244358d762 | A string is wound symmetrically around a right circular cylinder which has height 12 cm and circumference 6 cm, starting at one end of the cylinder and finishing at the other end, having gone exactly 6 times around the cylinder. How long is the string? | [
"\\( 12\\sqrt{10} \\) cm"
] | [
"A string is wound symmetrically around a right circular cylinder which has height 12 cm and circumference 6 cm, starting at one end of the cylinder and finishing at the other end, having gone exactly 6 times around the cylinder. How long is the string?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-d65c8b639e8de4dcb0feee302857cc954938fa8ec852606af8e72f4f807168b3 | A string is wrapped tightly around two circular cans that are externally tangent with radii of 2 feet and 6 feet, respectively. What is the minimum length of string needed to do this? | [
"\\( 8\\sqrt{3} + \\frac{28\\pi}{3} \\)"
] | [
"A string is wrapped tightly around two circular cans that are externally tangent with radii of 2 feet and 6 feet, respectively. What is the minimum length of string needed to do this?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b72e6ab2c41598a61cd34ce91ab4327c5272c0045268716edb97f9ecd027736e | A string of 15 letters consisting of only the letters \(a, b, c, d\) is written. There are 7 distinct words that can be made by looking at consecutive letters: \(a, ab, add, bad, cab, dab, dad\). The expected value of the total number of such words made from such a string can be represented as \(\frac{m}{n}\) in simple... | [
"425"
] | [
"A string of 15 letters consisting of only the letters \\(a, b, c, d\\) is written. There are 7 distinct words that can be made by looking at consecutive letters: \\(a, ab, add, bad, cab, dab, dad\\). The expected value of the total number of such words made from such a string can be represented as \\(\\frac{m}{n}\... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-84acff546343993f3776b10caceb592361f682d7c1d92be508996c46936e03d1 | A string of 4 letters is called desirable if it consists of one vowel with one consonant on one side of the vowel and two consonants on the other side that are not in the same half of the alphabet as the first consonant and are adjacent to each other in the alphabet. For example, the string “cefg” does not work because... | [
"3140"
] | [
"A string of 4 letters is called desirable if it consists of one vowel with one consonant on one side of the vowel and two consonants on the other side that are not in the same half of the alphabet as the first consonant and are adjacent to each other in the alphabet. For example, the string “cefg” does not work be... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-0f028d7d8a23e8d5205513adc584ad2bcf84511a09d847f45d34169cd44d82c2 | A string of digits is created by randomly choosing the next digit with equal probability. You stop choosing digits when you have all 10 digits (0 through 9) in some order at the end. What is the expected value of the length of the string of digits? | [
"\\(\\frac{3536219}{1134}\\)"
] | [
"A string of digits is created by randomly choosing the next digit with equal probability. You stop choosing digits when you have all 10 digits (0 through 9) in some order at the end. What is the expected value of the length of the string of digits?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-b382d348deb1a5f09c1e3242386879e765fde049c96530ad743e66f25ca7a269 | A string of length $n$ is randomly created such that each index has a probability $p$ of being an 'A' and a probability $1-p$ of being a 'B'. Find the expected length of the longest substring of 'A's in the string. | [
"\\sum_{k=1}^{n} \\left(1 - Q(n, k)\\right)"
] | [
"A string of length $n$ is randomly created such that each index has a probability $p$ of being an 'A' and a probability $1-p$ of being a 'B'. Find the expected length of the longest substring of 'A's in the string."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 4
} | 11 | 0 | 0 |
omr-ws-c95d21c02e7a9314610356ff022c58e4602df0af698dfe47453a8807ac2591e4 | A string of size \( n \ge 42 \) of the numbers \(-1\) and \(1\) is written on the blackboard such that the sum of every 40 consecutive numbers is 0 and the sum of every 42 consecutive numbers is not equal to 0. Let \( S_n \) denote the sum of all the numbers on the blackboard. Find \(\max S_n\). | [
"20"
] | [
"A string of size \\( n \\ge 42 \\) of the numbers \\(-1\\) and \\(1\\) is written on the blackboard such that the sum of every 40 consecutive numbers is 0 and the sum of every 42 consecutive numbers is not equal to 0. Let \\( S_n \\) denote the sum of all the numbers on the blackboard. Find \\(\\max S_n\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 2
} | 10 | 0 | 0 |
omr-ws-198be9d73b987d0cb04bfb545a6d7c48792ab961114c5a598a685c06ac1a7d28 | A string of three decimal digits is chosen at random. The probability that there exists a perfect cube ending in those three digits can be expressed as \( \frac{a}{b} \), where \( a \) and \( b \) are relatively prime positive integers. Find \( a + b \). | [
"301"
] | [
"A string of three decimal digits is chosen at random. The probability that there exists a perfect cube ending in those three digits can be expressed as \\( \\frac{a}{b} \\), where \\( a \\) and \\( b \\) are relatively prime positive integers. Find \\( a + b \\)."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 30
} | 51 | 0 | 0 |
omr-ws-de15dbf760c53565a4b981b7b85d51323bd21d1c715c3c423d003b4d71c9926f | A strip of width $1$ is to be divided by rectangular panels of common width $1$ and lengths $a_1, a_2, a_3, \ldots$ (where $a_1 \ne 1$). Each panel from the second onward is similar but not congruent to the already paved part of the strip. When the first $n$ panels are laid, the length of the paved part of the strip is... | [
"3"
] | [
"A strip of width $1$ is to be divided by rectangular panels of common width $1$ and lengths $a_1, a_2, a_3, \\ldots$ (where $a_1 \\ne 1$). Each panel from the second onward is similar but not congruent to the already paved part of the strip. When the first $n$ panels are laid, the length of the paved part of the s... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-aac5218b2b035a247aed5fa735ce9df8e9578be31a4fb4afe8ba2394b1a1f1c6 | A strobogrammatic number reads the same upside down, e.g., \(619\). A triangular number is of the form \(n(n + 1)/2\) for some \(n \in \mathbb{N}\), e.g., \(15\). Let \(a\) be the third strobogrammatic prime number. Let \(b\) be the smaller number of the smallest pair of triangular numbers whose sum and difference are ... | [
"2715"
] | [
"A strobogrammatic number reads the same upside down, e.g., \\(619\\). A triangular number is of the form \\(n(n + 1)/2\\) for some \\(n \\in \\mathbb{N}\\), e.g., \\(15\\). Let \\(a\\) be the third strobogrammatic prime number. Let \\(b\\) be the smaller number of the smallest pair of triangular numbers whose sum ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7ed10bfad97040857015482ac35c90962b0de729e1f8960152a0c147fd6ec3e6 | A student accidentally added five to both the numerator and denominator of a fraction, changing the fraction's value to $\frac12$. If the original numerator was a 2, what was the original denominator? | [
"9"
] | [
"A student accidentally added five to both the numerator and denominator of a fraction, changing the fraction's value to $\\frac12$. If the original numerator was a 2, what was the original denominator?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-283da05a7ef68735f1c2701156824532f102de9339a0e35e608c2d6ec4e8014b | A student adds up rational fractions incorrectly: (*) \( \frac{a}{b} + \frac{x}{y} = \frac{a+x}{b+y} \) Despite that, he sometimes obtains correct results. For a given fraction \( \frac{a}{b} \) with \( a, b \in \mathbb{Z} \) and \( b > 0 \), find all fractions \( \frac{x}{y} \) with \( x, y \in \mathbb{Z} \) and \( y ... | [
"\\frac{x}{y} = -\\frac{a y}{b^2} \\text{ where } y \\text{ is a positive integer such that } b^2 \\text{ divides } a y^2"
] | [
"A student adds up rational fractions incorrectly:\n\n(*) \\( \\frac{a}{b} + \\frac{x}{y} = \\frac{a+x}{b+y} \\)\n\nDespite that, he sometimes obtains correct results. For a given fraction \\( \\frac{a}{b} \\) with \\( a, b \\in \\mathbb{Z} \\) and \\( b > 0 \\), find all fractions \\( \\frac{x}{y} \\) with \\( x, ... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 4
} | 12 | 0 | 0 |
omr-ws-7864a3d5b8d65d8b3eebef301718353688cfe3eaebbfbf23becbf8e56f08af31 | A student appears for tests T1, T2, and T3. The student is successful if he passes either in tests T1 and T2 or in tests T1 and T3. The probabilities of the student passing in tests T1, T2, and T3 are \(a\), \(b\), and \(\frac{1}{2}\) respectively. If the probability that the student is successful is \(\frac{1}{2}\), f... | [
"\\(a(1 + b) = 1\\)"
] | [
"A student appears for tests T1, T2, and T3. The student is successful if he passes either in tests T1 and T2 or in tests T1 and T3. The probabilities of the student passing in tests T1, T2, and T3 are \\(a\\), \\(b\\), and \\(\\frac{1}{2}\\) respectively. If the probability that the student is successful is \\(\\f... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-801156fb115275a275cd57829ead7c8cadd3a66f5ad84a1f53c291c40ab684b5 | A student attempts $N$ problems on a piece of paper, with each problem being solved independently. Let $X$ be the number of problems the student solves. If $N=100$, calculate the maximum possible value of the variance of $X$, denoted $D(X)$. | [
"25"
] | [
"A student attempts $N$ problems on a piece of paper, with each problem being solved independently. Let $X$ be the number of problems the student solves. If $N=100$, calculate the maximum possible value of the variance of $X$, denoted $D(X)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1098f529d931798a4d08efee2d17267de262e1cc82128380055505c4c54f029c | A student attempts to copy a convex polygon \( M \) and draw it inside a circle \( C(O,1) \). The student draws each side with an angle identical to the corresponding angle in \( M \), but the relative error in the length of each side is no more than \( k \). After completing the drawing, the first and last vertices ar... | [
"0.2"
] | [
"A student attempts to copy a convex polygon \\( M \\) and draw it inside a circle \\( C(O,1) \\). The student draws each side with an angle identical to the corresponding angle in \\( M \\), but the relative error in the length of each side is no more than \\( k \\). After completing the drawing, the first and las... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fafb2c7da825774957a209adaef3f6ff51e4a98bdd1c7ccc87878b31e61e0d23 | A student bought \( x \) pencils for \( y \) one dollar bills. The seller advised him to buy 10 more pencils so that he would get all for $2 and save 80 cents per dozen. Find \( x \). | [
"5"
] | [
"A student bought \\( x \\) pencils for \\( y \\) one dollar bills. The seller advised him to buy 10 more pencils so that he would get all for $2 and save 80 cents per dozen. Find \\( x \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 3,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-ab6089e22047488adacac5f0d2da164fa2bab6710a9711d9cc011ba7c9f763d7 | A student can register for a Mathematics class at 8:00, 10:00, or 11:00; an English class at 9:00, 10:00, or 11:00; and a Science class at 8:00, 10:00, or 11:00. In how many different ways can the student schedule the three courses? | [
"10"
] | [
"A student can register for a Mathematics class at 8:00, 10:00, or 11:00; an English class at 9:00, 10:00, or 11:00; and a Science class at 8:00, 10:00, or 11:00. In how many different ways can the student schedule the three courses?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b97824a28344573a8330b652d0098e96b75c8564ba31b6bca23caf6a094b2948 | A student can return home either through the train station or through the airport. He chooses the road through the station in 1/3 of all cases and through the airport in 2/3 of the cases. If he goes through the station, he returns home by 5 pm in 75% of cases, and if he goes through the airport, he returns home by 5 pm... | [
"\\(\\frac{17}{60}\\)"
] | [
"A student can return home either through the train station or through the airport. He chooses the road through the station in 1/3 of all cases and through the airport in 2/3 of the cases. If he goes through the station, he returns home by 5 pm in 75% of cases, and if he goes through the airport, he returns home by... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-e495b1097d814458b26fd9056cb9901f45de002e35b1dbaf5c1cea1d935fcbcf | A student counts only numbers that are relatively prime to 2009. If he counts 2009 such numbers, what is the last number counted? | [
"2402"
] | [
"A student counts only numbers that are relatively prime to 2009. If he counts 2009 such numbers, what is the last number counted?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-b03e5534e3d917ad2a01ef6e85d5c0d41d024dc9249852c37e12ffaf97fc4426 | A student did not notice the multiplication sign between two 7-digit numbers and wrote one 14-digit number which turned out to be 3 times the intended product. What are the initial 7-digit numbers? | [
"\\( 1666667 \\) and \\( 3333334 \\)"
] | [
"A student did not notice the multiplication sign between two 7-digit numbers and wrote one 14-digit number which turned out to be 3 times the intended product. What are the initial 7-digit numbers?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 28
} | 28 | 0 | 0 |
omr-ws-aa24f94d0e819a4e957adcb81d95c61ce0502bdf6cfb061ba27494c3b8c08148 | A student did not notice the multiplication sign between two 7-digit numbers and wrote one 14-digit number which turns out to be 3 times the product of the two numbers. What are the initial 7-digit numbers? | [
"16666667 and 3333334"
] | [
"A student did not notice the multiplication sign between two 7-digit numbers and wrote one 14-digit number which turns out to be 3 times the product of the two numbers. What are the initial 7-digit numbers?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-0781b622fb2fe4956b31bac97aabc6c74d073cd274914975f8887f2729b18d98 | A student did not notice the multiplication sign between two three-digit numbers and wrote it as a six-digit number. The result was 7 times more than it should be. Find these numbers. | [
"(143, 143)",
"143 and 143"
] | [
"A student did not notice the multiplication sign between two three-digit numbers and wrote it as a six-digit number. The result was 7 times more than it should be. Find these numbers."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 4,
"tir": 60
} | 74 | 0 | 0 |
omr-ws-c0aeb65f4fd84127f3eaafc10de5e21ff9a4c010905f00704ebd7179874b96dd | A student draws a flower by drawing a regular hexagon and then constructing semicircular petals on each side of the hexagon. If the hexagon has side length 2, what is the area of the flower? | [
"\\(6\\sqrt{3} + 3\\pi\\)"
] | [
"A student draws a flower by drawing a regular hexagon and then constructing semicircular petals on each side of the hexagon. If the hexagon has side length 2, what is the area of the flower?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 2,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-b4be40ef50af8a8b24fb93b05d0432baf001f69e78d0b2d39e2a09356f9684d5 | A student draws some circles on a plane and all possible tangents to those circles. It is known that these tangents form a 2011-regular polygon with 2011 sides and 2011 angles. Find the minimum number of circles the student must draw. | [
"1006"
] | [
"A student draws some circles on a plane and all possible tangents to those circles. It is known that these tangents form a 2011-regular polygon with 2011 sides and 2011 angles. Find the minimum number of circles the student must draw."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 26,
"genselect": 6
} | 32 | 0 | 0 |
omr-ws-d75ffbbb14f33fda1be22001eb6e4d79b9f7c5c6cb6685ca0002c80feedf762c | A student enters an integer \( n > 20 \) into her calculator and multiplies it by 123. She then takes the square root of the result twice (\(\sqrt{\sqrt{123n}}\)). She repeats this process for about 8 to 15 times. Eventually, the number stabilizes. To the nearest whole integer, what is that stabilized number? | [
"5"
] | [
"A student enters an integer \\( n > 20 \\) into her calculator and multiplies it by 123. She then takes the square root of the result twice (\\(\\sqrt{\\sqrt{123n}}\\)). She repeats this process for about 8 to 15 times. Eventually, the number stabilizes. To the nearest whole integer, what is that stabilized number... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-01fe36c4e3502b7804f16aa8c33b22bdfb0dce1b4a0f9177bc0d8b63264fafc7 | A student had 18 papers. He selected some of these papers, cut each of them into 18 pieces, then selected some of these pieces and cut each of them into 18 pieces, repeating this process until he stopped. If the student ended up with exactly 2012 pieces, find the remainder when the total number of cuts he made is divid... | [
"5"
] | [
"A student had 18 papers. He selected some of these papers, cut each of them into 18 pieces, then selected some of these pieces and cut each of them into 18 pieces, repeating this process until he stopped. If the student ended up with exactly 2012 pieces, find the remainder when the total number of cuts he made is ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-787edb5a5f5a392f75557ceab7aa9b7af02479d01f221ddd0a32fb9f53f78f09 | A student had 18 papers. He selected some of these papers, then cut each of them into 18 pieces. He took these pieces and selected some of them, which he again cut into 18 pieces each. The student repeated this procedure until he got tired. After a while, he counted the pieces and got 2012 pieces. Calculate the remaind... | [
"1"
] | [
"A student had 18 papers. He selected some of these papers, then cut each of them into 18 pieces. He took these pieces and selected some of them, which he again cut into 18 pieces each. The student repeated this procedure until he got tired. After a while, he counted the pieces and got 2012 pieces. Calculate the re... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f7713b6ea5771bb5fb9e8fcbe8194278cbd1cd20b02ed01ef1216a58e57c2736 | A student has 3 colleges to choose from: A, B, or C. The student has 3 advisers to ask about which one to go to. Adviser 1 picks the right college 80% of the time. Adviser 2 picks the right one 10% of the time. Adviser 3 selects the same as Advisers 1 and 2 when these two agree. When Advisers 1 and 2 disagree, Adviser ... | [
"0.8"
] | [
"A student has 3 colleges to choose from: A, B, or C. The student has 3 advisers to ask about which one to go to. Adviser 1 picks the right college 80% of the time. Adviser 2 picks the right one 10% of the time. Adviser 3 selects the same as Advisers 1 and 2 when these two agree. When Advisers 1 and 2 disagree, Adv... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-27fae287653120cafc0f883a17e654dbbe7e4f2994517f31789f70c326b0919b | A student has 3 points and needs to accumulate 8 points to pass an exam. The student can bet any number of points \( n > 0 \) (with \( n \leq X \), where \( X \) is the current number of points). If the student solves a problem correctly (with probability 0.4), they gain \( n \) points; otherwise, they lose \( n \) poi... | [
"\\dfrac{32}{125}"
] | [
"A student has 3 points and needs to accumulate 8 points to pass an exam. The student can bet any number of points \\( n > 0 \\) (with \\( n \\leq X \\), where \\( X \\) is the current number of points). If the student solves a problem correctly (with probability 0.4), they gain \\( n \\) points; otherwise, they lo... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-9c150d23929b5dba6fe55f42905a4305ed5835901f51c3aa687b77e30de69943 | A student has 37 days to prepare for an examination and requires no more than 60 hours of study, studying at least 1 hour each day. Let $S_n$ represent the cumulative hours studied by the end of day $n$. Determine the number of distinct values of $k$ such that there exist days $i$ and $j$ (with $1 \leq i \leq j \leq 37... | [
"13"
] | [
"A student has 37 days to prepare for an examination and requires no more than 60 hours of study, studying at least 1 hour each day. Let $S_n$ represent the cumulative hours studied by the end of day $n$. Determine the number of distinct values of $k$ such that there exist days $i$ and $j$ (with $1 \\leq i \\leq j ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-9761ee3e3aa2b4d5df0cff5d3cef91e505af17554ba8bab2a5ef5b7beae18288 | A student has 7 days to solve 7 problems and wants to make a schedule to work on these problems according to the following rules: 1. He will work on exactly 3 different problems each day. 2. He will not work on the same set of 3 different problems on two different days. 3. He will work on each of the 7 problems at leas... | [
"31169376000"
] | [
"A student has 7 days to solve 7 problems and wants to make a schedule to work on these problems according to the following rules:\n1. He will work on exactly 3 different problems each day.\n2. He will not work on the same set of 3 different problems on two different days.\n3. He will work on each of the 7 problems... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 3
} | 15 | 0 | 0 |
omr-ws-8d29f46e9e0edb15fdec2012d517c903c47e2eb31bac8a6a65792dd8c10bb047 | A student has a 40% probability of getting a scholarship. If the student gets the scholarship, the probability of finishing the course is 0.8. If the student does not get the scholarship, the probability of finishing the course is 0.4. (a) What is the probability of the student finishing the course? (b) If the student ... | [
"0.56 and \\(\\frac{4}{7}\\)"
] | [
"A student has a 40% probability of getting a scholarship. If the student gets the scholarship, the probability of finishing the course is 0.8. If the student does not get the scholarship, the probability of finishing the course is 0.4.\n\n (a) What is the probability of the student finishing the course?\n\n (b) ... | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 9
} | 9 | 0 | 0 |
omr-ws-3804d4c1c5dddbbd91db3e04c188a79492066a1b0abf27829f08e3cdaf9cde1c | A student has a savings account earning 9% simple interest. She must pay $1400 for first-semester tuition by September 1 and $1400 for second-semester tuition by January 1. How much must she earn in the summer (by September 1) in order to pay the first-semester bill on time and still have the remainder of her summer ea... | [
"2759.22"
] | [
"A student has a savings account earning 9% simple interest. She must pay $1400 for first-semester tuition by September 1 and $1400 for second-semester tuition by January 1. How much must she earn in the summer (by September 1) in order to pay the first-semester bill on time and still have the remainder of her summ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-18a42ff9233489f3daaa71e7deaac251b3778ffdb02e9a6d1e693a03d1f9b8d7 | A student has an average grade of exactly 1.12 from a series of tests, where the possible grades are 1, 2, 3, 4, and 5. If the student has received the minimum number of grades of 1 to achieve this average, how many grades of 1 did the student receive? | [
"22"
] | [
"A student has an average grade of exactly 1.12 from a series of tests, where the possible grades are 1, 2, 3, 4, and 5. If the student has received the minimum number of grades of 1 to achieve this average, how many grades of 1 did the student receive?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-1a539742a86982147969853c305fe42e9de76987779988f5d130880c20a04d99 | A student has prepared for 7 out of 10 questions on an exam where she must answer 5 questions. What is the probability that she can answer at least 4 questions on the exam? | [
"\\(\\frac{{7 \\choose 4} \\cdot {3 \\choose 1} + {7 \\choose 5}}{10 \\choose 5}\\)"
] | [
"A student has prepared for 7 out of 10 questions on an exam where she must answer 5 questions. What is the probability that she can answer at least 4 questions on the exam?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 26
} | 73 | 0 | 0 |
omr-ws-443f3e1e33346896d8b04159d2f12d999ab4fa48199b17736a3a1ec503dd9326 | A student has seven pieces of paper. She repeatedly chooses some of them and cuts each chosen piece into seven smaller pieces. Let $N$ be the number of times she needs to perform this operation to have exactly 2016 pieces of paper for the first time. Find the smallest possible value of $N$. | [
"335"
] | [
"A student has seven pieces of paper. She repeatedly chooses some of them and cuts each chosen piece into seven smaller pieces. Let $N$ be the number of times she needs to perform this operation to have exactly 2016 pieces of paper for the first time. Find the smallest possible value of $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-1f066970791b4fa383aad4620f19ded2d3d900fe786115aab6d1eef5bf1dd5df | A student has to take eight hours of classes a week and wants to have fewer hours on Friday than on Thursday. In how many ways can he do this, assuming the number of hours per day must be an integer and he can have days with no classes? | [
"200"
] | [
"A student has to take eight hours of classes a week and wants to have fewer hours on Friday than on Thursday. In how many ways can he do this, assuming the number of hours per day must be an integer and he can have days with no classes?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 5,
"tir": 23
} | 37 | 0 | 0 |
omr-ws-0f908a00458cd78de4608ad574b3510c4db7dca7ba3db3185e84461cbabbc2ac | A student is 5 meters away from an airplane. The airplane moves at a constant velocity of 120 meters per minute at a 60-degree angle from the horizontal. Determine the rate of change of the distance between the student and the airplane when the airplane is 10 meters above the ground. | [
"\\(\\frac{1600\\sqrt{3} + 600}{2 \\sqrt{158.33 + 33.33\\sqrt{3}}}\\)"
] | [
"A student is 5 meters away from an airplane. The airplane moves at a constant velocity of 120 meters per minute at a 60-degree angle from the horizontal. Determine the rate of change of the distance between the student and the airplane when the airplane is 10 meters above the ground."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-061b45c8bb3be41b26808fc53016f994f841d1e0f8f9942546f19da0dd6b2cf6 | A student is allowed to select at most $n$ books from a collection of $(2n+1)$ books. If the total number of ways in which he can select a book is 63, find the value of $n$. | [
"3"
] | [
"A student is allowed to select at most $n$ books from a collection of $(2n+1)$ books. If the total number of ways in which he can select a book is 63, find the value of $n$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cf305db97abafd2bb061a0c6897c9364f426d6f469a6972d2d102fef9895d2d3 | A student is lost in a forest that covers a half-plane and knows she is no more than 2 miles from the border. Propose a route for her such that she would come out of the forest having walked no more than 13 miles. The student cannot see the border until she reaches it. | [
"\\(2\\sqrt{2} + 2 + \\frac{7\\pi}{4}\\)"
] | [
"A student is lost in a forest that covers a half-plane and knows she is no more than 2 miles from the border. Propose a route for her such that she would come out of the forest having walked no more than 13 miles. The student cannot see the border until she reaches it."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-e1696480ac6d60a8b8072854b8407a751341a6518d40516b85fce4142cb429ab | A student is offered two different after-school jobs. One pays $10 per hour, and the other pays $100 for the first hour, but the hourly rate decreases by half for each additional hour worked. What is the maximum number of hours the student can work at the second job so that his total earnings from the second job are mo... | [
"19"
] | [
"A student is offered two different after-school jobs. One pays $10 per hour, and the other pays $100 for the first hour, but the hourly rate decreases by half for each additional hour worked. What is the maximum number of hours the student can work at the second job so that his total earnings from the second job a... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7ab4350c34ed8672181525ac8674666cbc9f067500954b0800517a28c3145c49 | A student is playing a game with 2002 positive numbers. The game allows the following operations: - Take two numbers, double the first one, add the second one, and save the sum. - Take another two numbers, double the first one, add the second one, multiply this sum with the previous sum, and save the result. - Repeat t... | [
"\\prod_{k=1}^{1001} (2a_{2k-1} + a_{2k})"
] | [
"A student is playing a game with 2002 positive numbers. The game allows the following operations:\n- Take two numbers, double the first one, add the second one, and save the sum.\n- Take another two numbers, double the first one, add the second one, multiply this sum with the previous sum, and save the result.\n- ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-b7e4c3e27592d31b0de49ea15ee69b612498beb470eaa4b72f33d76160985fda | A student is taking a 20-item quiz where each question has 4 choices. The probability that the student knows the answer given that he is correct is 6/7. How many questions was the student prepared for? | [
"12"
] | [
"A student is taking a 20-item quiz where each question has 4 choices. The probability that the student knows the answer given that he is correct is 6/7. How many questions was the student prepared for?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-75ec6293057b48308685d2d9eac9714beafa6a582f8cdaa2e564d6975f5fb795 | A student is trying to pass a competency exam. Each time she takes the exam, she has a 20% chance of passing, and she is allowed a maximum of three attempts. What is the expected number of times she will fail the exam? | [
"2.2"
] | [
"A student is trying to pass a competency exam. Each time she takes the exam, she has a 20% chance of passing, and she is allowed a maximum of three attempts. What is the expected number of times she will fail the exam?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f2c9570d582f38e08b1691d64c5e2c9df67b3d761b19d04c41adf5d470709986 | A student knows how to solve 7 out of 10 problems. If the final exam consists of a random selection of 5 problems from these 10, what is the probability that the student will answer correctly: (a) all 5 problems; (b) at least 4 of the problems. | [
"(a) \\(\\frac{1}{12}\\) or approximately \\(0.0833\\); (b) \\(\\frac{1}{2}\\) or \\(0.5\\)"
] | [
"A student knows how to solve 7 out of 10 problems. If the final exam consists of a random selection of 5 problems from these 10, what is the probability that the student will answer correctly:\n(a) all 5 problems;\n(b) at least 4 of the problems."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 12
} | 12 | 0 | 0 |
omr-ws-4bfd00216576bcc0abb658cdc15faf9aee9726ccea33fa02af3c35b3614ad7d8 | A student must answer $k 2^{k-1}$ yes-or-no questions on an exam. The student can take mock exams with the same questions and is told the number of correct answers after each mock exam. Determine the minimum number of mock exams the student must take to ensure all answers are correct on the real exam. | [
"k"
] | [
"A student must answer $k 2^{k-1}$ yes-or-no questions on an exam. The student can take mock exams with the same questions and is told the number of correct answers after each mock exam. Determine the minimum number of mock exams the student must take to ensure all answers are correct on the real exam."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-08766d1019b73dfe2c107f354b6b8687d3d65d7bd5bff675430b702d197099a9 | A student must choose a program of four courses from a list of courses consisting of English, Algebra, Geometry, History, Art, and Latin. This program must contain English and at least one mathematics course. In how many ways can this program be chosen? | [
"9"
] | [
"A student must choose a program of four courses from a list of courses consisting of English, Algebra, Geometry, History, Art, and Latin. This program must contain English and at least one mathematics course. In how many ways can this program be chosen?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-34f4227ef7dd559f4a9bb8acae4e9f977b7b286d46ca92c67c56716036cad317 | A student must choose three classes among Physics, Literature, and Mathematics to build a 7-day calendar. Each day, the student must choose only one class. The only restriction is that on four consecutive days, all three branches must be chosen. Determine the number of possible calendars the student can make. | [
"324"
] | [
"A student must choose three classes among Physics, Literature, and Mathematics to build a 7-day calendar. Each day, the student must choose only one class. The only restriction is that on four consecutive days, all three branches must be chosen. Determine the number of possible calendars the student can make."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-29fe7491c8b4e0e5e9c7c6aa2eb7fbb1b8445f461e1ff4c4b4298e736ba30104 | A student needs to pass 3 more courses to graduate and each time he takes a test, he has a 1/4 chance of passing that particular course. What is the expected number of attempts the student will need in order to graduate? | [
"12"
] | [
"A student needs to pass 3 more courses to graduate and each time he takes a test, he has a 1/4 chance of passing that particular course. What is the expected number of attempts the student will need in order to graduate?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8ffd33b22ade48d4e41d4c0f34c05189671e57bfb2365196f6abef455fcf74eb | A student needs to work five days in January and does not want to work on more than one Sunday. In how many ways can he select his five working days, given that January has five Sundays? | [
"\\(_{26}C_5 + 5 \\cdot _{26}C_4\\)"
] | [
"A student needs to work five days in January and does not want to work on more than one Sunday. In how many ways can he select his five working days, given that January has five Sundays?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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