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-51fb3844826cf3abd96facad23b8560bc669887952c44833ea1adf1dcba47dc3 | Through the center \( O \) of a circle circumscribed about a regular pentagon \( ABCDE \), draw straight lines parallel to sides \( AB \) and \( BC \). Their intersection points with \( BC \) and \( CD \) are denoted by \( F \) and \( G \) respectively. What is the ratio of the areas of the quadrilateral \( OFCG \) to ... | [
"\\(\\frac{2}{5}\\)"
] | [
"Through the center \\( O \\) of a circle circumscribed about a regular pentagon \\( ABCDE \\), draw straight lines parallel to sides \\( AB \\) and \\( BC \\). Their intersection points with \\( BC \\) and \\( CD \\) are denoted by \\( F \\) and \\( G \\) respectively. What is the ratio of the areas of the quadril... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e055370098ce54caddb14c84f37987b1873a3767a5b0c3b9ffc25019aaee00f4 | Through the center of the excircle of triangle \( ABC \) tangent to the side \( AC \), a line is drawn perpendicular to the bisector of angle \( B \), intersecting the extensions of sides \( AB \) and \( BC \) at points \( X \) and \( Y \) respectively. Given that \( AX = a \) and \( CY = b \), find the length of \( XY... | [
"\\[ XY = 2\\sqrt{ab} \\]"
] | [
"Through the center of the excircle of triangle \\( ABC \\) tangent to the side \\( AC \\), a line is drawn perpendicular to the bisector of angle \\( B \\), intersecting the extensions of sides \\( AB \\) and \\( BC \\) at points \\( X \\) and \\( Y \\) respectively. Given that \\( AX = a \\) and \\( CY = b \\), f... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 4,
"tir": 1
} | 9 | 0 | 0 |
omr-ws-60f4456065ca9ed3312650d10f067a71afacd9930973d3bb27f0564050c78ac5 | Through the edge of the base of a regular quadrangular pyramid is drawn a plane which cuts from the opposite face a triangle of area $a^2$. Find the lateral surface area of a pyramid that is cut out of the given pyramid by the given plane if the lateral surface area of the original pyramid is equal to $b^2$. | [
"$1/4(2a+b)^2$"
] | [
"Through the edge of the base of a regular quadrangular pyramid is drawn a plane which cuts from the opposite face a triangle of area $a^2$. Find the lateral surface area of a pyramid that is cut out of the given pyramid by the given plane if the lateral surface area of the original pyramid is equal to $b^2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9a9f30b336ca87ec178d96f4b7be64a50e95cfbc26f3c165e37497b420738e6b | Through the endpoints \( A \) and \( B \) of a diameter \( AB \) of a given circle, the tangents \( \ell \) and \( m \) have been drawn. Let \( C \ne A \) be a point on \( \ell \) and let \( q_1, q_2 \) be two rays from \( C \). Ray \( q_i \) cuts the circle in \( D_i \) and \( E_i \) with \( D_i \) between \( C \) and... | [
"no_valid_answer_found"
] | [
"Through the endpoints \\( A \\) and \\( B \\) of a diameter \\( AB \\) of a given circle, the tangents \\( \\ell \\) and \\( m \\) have been drawn. Let \\( C \\ne A \\) be a point on \\( \\ell \\) and let \\( q_1, q_2 \\) be two rays from \\( C \\). Ray \\( q_i \\) cuts the circle in \\( D_i \\) and \\( E_i \\) wi... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f1917c672fb84db9dece7cdd8216518e5386ed91e39e86d522a63a14c2a772bf | Through the ends \( A \) and \( B \) of a diameter \( AB \) of a given circle, the tangents \( \ell \) and \( m \) are drawn, parallel to each other. Let \( C \) be a point on \( \ell \) other than \( A \) and let \( q_1 \), \( q_2 \) be two rays with origin at \( C \). Suppose that \( q_i \) (\( i = 1, 2 \)) intersect... | [
"no_valid_answer_found"
] | [
"Through the ends \\( A \\) and \\( B \\) of a diameter \\( AB \\) of a given circle, the tangents \\( \\ell \\) and \\( m \\) are drawn, parallel to each other. Let \\( C \\) be a point on \\( \\ell \\) other than \\( A \\) and let \\( q_1 \\), \\( q_2 \\) be two rays with origin at \\( C \\). Suppose that \\( q_i... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-38b7b748bb10aac20cb7e8e01f0e33f0c0a8aea2019fd22b96083493c7bcdf24 | Through the hypotenuse of a right-angled isosceles triangle a plane $P$ is drawn at an angle $\alpha $ to the plane of the triangle. Determine the perimeter and area of the figure obtained by projecting the triangle on the plane $P$. The hypotenuse of the triangle is equal to $c$. | [
"By hypothesis, the inclined lines $AC$ and $CB$ (Fig. 96) are equal to each other. Hence, their projections are also equal: $AD = DB$. The angle $DEC$ ($E$ is the midpoint of $A B$) is the plane angle of the dihedral angle $\\alpha $. Since the triangle $ACB$ at the vertex $C$ is a right-angled one, $$CE=AE=\\frac... | [
"Through the hypotenuse of a right-angled isosceles triangle a plane $P$ is drawn at an angle $\\alpha $ to the plane of the triangle. Determine the perimeter and area of the figure obtained by projecting the triangle on the plane $P$. The hypotenuse of the triangle is equal to $c$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-60386cdafd54ca0fc13265ed9f1b7f18708297a2407a5320573b96ff83344383 | Through the midpoint \( C \) of an arbitrary chord \( AB \) on a circle, chords \( KL \) and \( MN \) are drawn such that points \( K \) and \( M \) lie on one side of \( AB \). Segments \( KN \) and \( ML \) intersect \( AB \) at points \( Q \) and \( P \) respectively. If the length of \( AB \) is 10 units, calculate... | [
"no_valid_answer_found"
] | [
"Through the midpoint \\( C \\) of an arbitrary chord \\( AB \\) on a circle, chords \\( KL \\) and \\( MN \\) are drawn such that points \\( K \\) and \\( M \\) lie on one side of \\( AB \\). Segments \\( KN \\) and \\( ML \\) intersect \\( AB \\) at points \\( Q \\) and \\( P \\) respectively. If the length of \\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-db8f123329e7163f2e6f5d86561dcde2350f4d449dd81caec3a3dc4e3760fe1e | Through the midpoint \( M \) of the chord \( AB \) in the circle \( C(0, r) \), a chord \( CD \) is drawn such that \( D \) is on the major arc \( AB \). Let \( E \) be the symmetric point of \( D \) with respect to \( M \). If \( AB \) is tangent to the circumcircle of \( \triangle CEB \), find the length of \( AB \) ... | [
"2\\sqrt{19}"
] | [
"Through the midpoint \\( M \\) of the chord \\( AB \\) in the circle \\( C(0, r) \\), a chord \\( CD \\) is drawn such that \\( D \\) is on the major arc \\( AB \\). Let \\( E \\) be the symmetric point of \\( D \\) with respect to \\( M \\). If \\( AB \\) is tangent to the circumcircle of \\( \\triangle CEB \\), ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0912473adc363463a2a5063269fc224a21ac6089ceb28a2ade994a2555372665 | Through the midpoint \( M \) of the chord \( AB \) in the circle \( C(0, r) \), a chord \( CD \) is drawn such that \( D \) is on the major arc \( AB \). Let \( E \) be the symmetric point of \( D \) with respect to \( M \). If \( MB = 5 \) and \( MC = 12 \), calculate the value of \( ME \cdot MC \). | [
"25"
] | [
"Through the midpoint \\( M \\) of the chord \\( AB \\) in the circle \\( C(0, r) \\), a chord \\( CD \\) is drawn such that \\( D \\) is on the major arc \\( AB \\). Let \\( E \\) be the symmetric point of \\( D \\) with respect to \\( M \\). If \\( MB = 5 \\) and \\( MC = 12 \\), calculate the value of \\( ME \\c... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-044a5c000c4aed71fb68545d2a7dc5e757fb0558fb28c88f3d06a480424c1284 | Through the midpoint \( M \) of the chord \( AB \) in the circle \( C(0, r) \), a chord \( CD \) is drawn such that \( D \) is on the major arc \( AB \). Let \( E \) be the symmetric point of \( D \) with respect to \( M \). If \( \triangle CEB \) is a right isosceles triangle with \( CE = CB \), compute \( AB \) in te... | [
"\\( AB = r\\sqrt{2} \\)"
] | [
"Through the midpoint \\( M \\) of the chord \\( AB \\) in the circle \\( C(0, r) \\), a chord \\( CD \\) is drawn such that \\( D \\) is on the major arc \\( AB \\). Let \\( E \\) be the symmetric point of \\( D \\) with respect to \\( M \\). If \\( \\triangle CEB \\) is a right isosceles triangle with \\( CE = CB... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-bba75480c5b3ab3df98a6de56557b2d9cda8f22d182cf932a36073d652e5250d | Through the midpoint of the chord $AB$ of the circle $\omega$, draw the chord $CD$. The tangents to the circle, drawn at points $A$ and $B$, meet at point $E$. If the measure of angle $AEC$ is $x^\circ$, find the value of $x$ given that the measure of angle $BED$ is $58^\circ$. | [
"58"
] | [
"Through the midpoint of the chord $AB$ of the circle $\\omega$, draw the chord $CD$. The tangents to the circle, drawn at points $A$ and $B$, meet at point $E$. If the measure of angle $AEC$ is $x^\\circ$, find the value of $x$ given that the measure of angle $BED$ is $58^\\circ$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f2f7b7c3fe8b087718c3cf15e3a905950eed789f048479c591711f567b736b46 | Through the point \( B: (0, -b) \), draw a chord of the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) where \(a > b > 0\). Find the maximum length of the chord. | [
"\\(\\frac{a^2}{\\sqrt{a^2-b^2}}\\) if \\(a \\geq b\\sqrt{2}\\), or \\(2b\\) if \\(b \\leq a \\leq b\\sqrt{2}\\)."
] | [
"Through the point \\( B: (0, -b) \\), draw a chord of the ellipse \\(\\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1\\) where \\(a > b > 0\\). Find the maximum length of the chord."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 8,
"tir": 10
} | 19 | 0 | 0 |
omr-ws-2336b10f82a2554d1fe0005074c014f98a32b91212917bcbfeea6b7cba0fba0a | Through the point \( F \), located outside the circle \( k \), a tangent \( FA \) and a secant \( FB \) are drawn, intersecting the circle at points \( B \) and \( C \) (with \( C \) between \( F \) and \( B \)). A tangent through point \( C \) intersects the segment \( FA \) at point \( E \). The segment \( FX \) is t... | [
"2"
] | [
"Through the point \\( F \\), located outside the circle \\( k \\), a tangent \\( FA \\) and a secant \\( FB \\) are drawn, intersecting the circle at points \\( B \\) and \\( C \\) (with \\( C \\) between \\( F \\) and \\( B \\)). A tangent through point \\( C \\) intersects the segment \\( FA \\) at point \\( E \... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-7f554d7a55ca1f0bea2b9848ca987a1ea217dd19ad5769c0e77761f47f5aa735 | Through the point \( F(0, \frac{1}{4}) \) of the coordinate plane, two perpendicular lines pass and intersect the parabola \( y = x^2 \) at points \( A, B, C, D \) such that \( A_x < B_x < C_x < D_x \). The difference of the projections of segments \( AD \) and \( BC \) onto the \( Ox \) line is \( m \). Find the area ... | [
"\\(\\frac{1}{2}m^2\\)"
] | [
"Through the point \\( F(0, \\frac{1}{4}) \\) of the coordinate plane, two perpendicular lines pass and intersect the parabola \\( y = x^2 \\) at points \\( A, B, C, D \\) such that \\( A_x < B_x < C_x < D_x \\). The difference of the projections of segments \\( AD \\) and \\( BC \\) onto the \\( Ox \\) line is \\(... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 24,
"genselect": 7,
"tir": 11
} | 42 | 0 | 0 |
omr-ws-a57c83ae27c1720f7352c3eca73ce6ec9dd0a40c6e24caf176377d1aa80b4dfa | Through the point \( K \) lying outside the circle \( \omega \), the tangents are drawn \( KB \) and \( KD \) to this circle (\( B \) and \( D \) are tangency points) and a line intersecting the circle at points \( A \) and \( C \). The bisector of angle \( ABC \) intersects the segment \( AC \) at the point \( E \) an... | [
"90"
] | [
"Through the point \\( K \\) lying outside the circle \\( \\omega \\), the tangents are drawn \\( KB \\) and \\( KD \\) to this circle (\\( B \\) and \\( D \\) are tangency points) and a line intersecting the circle at points \\( A \\) and \\( C \\). The bisector of angle \\( ABC \\) intersects the segment \\( AC \... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-81584ee59af6774fb63a88757f85177a19d26fa5ebd29c3f3ac64157d3f33b2c | Through the point \(M\), which lies inside the circle, 4 different chords \(A_kB_k\) are drawn, where \(k = 1, 2, 3, 4\). Let \(P\) be the intersection point of lines \(A_1A_2\) and \(A_3A_4\), and let \(Q\) be the intersection point of lines \(B_1B_2\) and \(B_3B_4\). If the coordinates of \(M\) are \((2, 3)\), and th... | [
"(0, 1)"
] | [
"Through the point \\(M\\), which lies inside the circle, 4 different chords \\(A_kB_k\\) are drawn, where \\(k = 1, 2, 3, 4\\). Let \\(P\\) be the intersection point of lines \\(A_1A_2\\) and \\(A_3A_4\\), and let \\(Q\\) be the intersection point of lines \\(B_1B_2\\) and \\(B_3B_4\\). If the coordinates of \\(M\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5318738e9d7e1ab0c9c4b0da634a84ff3ff29b947657077dc36a167990a2934c | Through the vertex $A$ of a tetrahedron $ABCD$ is drawn a plane tangent to its circumsphere. The lines formed by the intersection of this plane with the planes $ABC$, $ACD$, and $ADB$ form six equal angles. If $AB = 3$, $AC = 4$, $AD = 5$, $BC = 6$, $BD = 7$, and $CD = 8$, find the value of $AB \cdot CD + AC \cdot BD +... | [
"82"
] | [
"Through the vertex $A$ of a tetrahedron $ABCD$ is drawn a plane tangent to its circumsphere. The lines formed by the intersection of this plane with the planes $ABC$, $ACD$, and $ADB$ form six equal angles. If $AB = 3$, $AC = 4$, $AD = 5$, $BC = 6$, $BD = 7$, and $CD = 8$, find the value of $AB \\cdot CD + AC \\cd... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-674cfcf8fcd1e5f19d26099aa186fef7f5b84f0407a0af006557b7c5560674f8 | Through the vertex $B$ of a triangle $ABC$, draw the line $L$ parallel to the side $AC$. The bisector of angle $A$ intersects $L$ at point $M$ and the exterior bisector of angle $C$ intersects line $L$ at point $N$. If $AB = 24$ and $BC = 36$, calculate $MN$. | [
"12"
] | [
"Through the vertex $B$ of a triangle $ABC$, draw the line $L$ parallel to the side $AC$. The bisector of angle $A$ intersects $L$ at point $M$ and the exterior bisector of angle $C$ intersects line $L$ at point $N$. If $AB = 24$ and $BC = 36$, calculate $MN$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 4
} | 11 | 0 | 0 |
omr-ws-0039f81f3c88647d7f4fb39ce67afa27bd295375880df4400110f8a8919dde2d | Through the vertex $D$ of the rectangular parallelepiped $ABCDA_1B_1C_1D_1$, a plane intersects the edges $AA_1$, $BB_1$, and $CC_1$ at points $K$, $M$, and $N$, respectively. Find the ratio of the volume of the pyramid with the vertex at point $P$ and the base $DKMN$ to the volume of the given parallelepiped, given th... | [
"\\(\\frac{m}{3(m+n)}\\)"
] | [
"Through the vertex $D$ of the rectangular parallelepiped $ABCDA_1B_1C_1D_1$, a plane intersects the edges $AA_1$, $BB_1$, and $CC_1$ at points $K$, $M$, and $N$, respectively. Find the ratio of the volume of the pyramid with the vertex at point $P$ and the base $DKMN$ to the volume of the given parallelepiped, giv... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 7
} | 22 | 0 | 0 |
omr-ws-39bc1a3ddbbcac1ce596d0b5f4a16e6637fb58295402404c16300e9e61acd076 | Through the vertex of a cone a plane is drawn at an angle $\alpha $ to the base of the cone. This plane intersects the base along the chord $AB$ of length $a$ subtending an arc of the base of the cone with central angle $\beta $. Find the volume of the cone. | [
"\\(\\frac{\\pi a^3}{24} \\cdot \\frac{\\tan\\alpha \\cos\\frac{\\beta }{2}}{\\sin^3 \\frac{\\beta }{2}}\\)"
] | [
"Through the vertex of a cone a plane is drawn at an angle $\\alpha $ to the base of the cone. This plane intersects the base along the chord $AB$ of length $a$ subtending an arc of the base of the cone with central angle $\\beta $. Find the volume of the cone."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dc621ab63a07727592972d3d66ee75f973e322b7ca025533dd069164eaa2595a | Through the vertex of a regular quadrangular pyramid a cutting plane is drawn parallel to a side of the base and at an angle $\varphi $ to the base of the pyramid. The side of the base of the pyramid is equal to $a$, and the face angle at the vertex of the pyramid, to $\alpha $. Find the area of the section. | [
"$S=\\frac{a^2\\sqrt {\\cos \\alpha }}{4\\sin\\frac{\\alpha }{2}\\sin\\varphi }$"
] | [
"Through the vertex of a regular quadrangular pyramid a cutting plane is drawn parallel to a side of the base and at an angle $\\varphi $ to the base of the pyramid. The side of the base of the pyramid is equal to $a$, and the face angle at the vertex of the pyramid, to $\\alpha $. Find the area of the section."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9b2933296241ac50481137124d257310b34c9fd4024ea5ec059b618196999f7f | Through vertices $A$ and $B$ of the unit square $ABCD$, a circle intersects lines $AD$ and $AC$ at points $K$ and $M$, other than $A$. Find the length of the projection $KM$ onto $AC$. | [
"\\dfrac{\\sqrt{2}}{2}"
] | [
"Through vertices $A$ and $B$ of the unit square $ABCD$, a circle intersects lines $AD$ and $AC$ at points $K$ and $M$, other than $A$. Find the length of the projection $KM$ onto $AC$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d5554c394212e6adb3548ca0fb7f9b09dd9f8482995daa6d43766edb7412ba61 | Through what angle $\theta$ must the point $(2,4)$ be rotated to produce $(x', y') = (-2 + \sqrt{3}, 1 + 2\sqrt{3})$? | [
"\\(\\frac{5\\pi}{6}\\)"
] | [
"Through what angle $\\theta$ must the point $(2,4)$ be rotated to produce $(x', y') = (-2 + \\sqrt{3}, 1 + 2\\sqrt{3})$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-425448f70bebfc798ece3294376136d5b32c6ec78ea436b60615d06ae3dd5b16 | Throughout a basketball season, a player's successful free throws and total attempts are tracked. Let $S(N)$ denote the number of successful free throws out of $N$ total attempts. If the player ends the season with a success rate of exactly 80%, find the smallest possible value of $N$ such that there was a moment durin... | [
"10"
] | [
"Throughout a basketball season, a player's successful free throws and total attempts are tracked. Let $S(N)$ denote the number of successful free throws out of $N$ total attempts. If the player ends the season with a success rate of exactly 80%, find the smallest possible value of $N$ such that there was a moment ... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 3
} | 10 | 0 | 0 |
omr-ws-4537eb42fcff020d023a43f95c818ad15e950d3c64e6ca1bb66927d8dfe18c0d | Throw \( n \) balls into \( 2n \) boxes, where each ball has an equal probability of entering any box. Let \( p_n \) be the probability that no box contains more than one ball. Find the limit \( \lim_{n \to \infty} \frac{\ln p_n}{n} \). | [
"\\(\\ln 2 - 1\\)"
] | [
"Throw \\( n \\) balls into \\( 2n \\) boxes, where each ball has an equal probability of entering any box. Let \\( p_n \\) be the probability that no box contains more than one ball. Find the limit \\( \\lim_{n \\to \\infty} \\frac{\\ln p_n}{n} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-9adfc140d95cd1fc0361618626ef802e821879d5cf6b268dd935cce8bc174d45 | Throw two dice and express the result as a set $\{a, b\}$. Repeat this process to generate two distinct sets. What is the probability that, when you choose one of these two sets, the sum of the elements of the chosen set is larger than or equal to 7? | [
"\\(\\frac{5}{7}\\)"
] | [
"Throw two dice and express the result as a set $\\{a, b\\}$. Repeat this process to generate two distinct sets. What is the probability that, when you choose one of these two sets, the sum of the elements of the chosen set is larger than or equal to 7?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cd730a461cb9f704fc01d137a80c1efaf0e4bd3e179d7448cbd19e4f7e451136 | Throwing the frisbee a distance \(d\) takes \(d^2\) seconds, and after catching the frisbee, it takes a person 1 second before they can throw it. Steven is in a circle with 12 total people and starts by throwing the frisbee. It goes clockwise around the circle back to Steven in \(N\) throws, with him catching it after ... | [
"0"
] | [
"Throwing the frisbee a distance \\(d\\) takes \\(d^2\\) seconds, and after catching the frisbee, it takes a person 1 second before they can throw it. Steven is in a circle with 12 total people and starts by throwing the frisbee. It goes clockwise around the circle back to Steven in \\(N\\) throws, with him catchin... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-1aefdf8c4718d51f630ac472fa8110005f521b998fcd0b31bc53db4ada66f357 | Tick, Trick, and Track have 20, 23, and 25 tickets respectively for the carousel at the fair in Duckburg. They agree that they will only ride all three together, for which they must each give up one of their tickets. Before a ride, if they want, they can redistribute tickets among themselves according to the rule: If o... | [
"\\text{No}"
] | [
"Tick, Trick, and Track have 20, 23, and 25 tickets respectively for the carousel at the fair in Duckburg. They agree that they will only ride all three together, for which they must each give up one of their tickets. Before a ride, if they want, they can redistribute tickets among themselves according to the rule:... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-5c2d72d77e782912d2ba6acad83b2ff92528a7426ccc381cc672aeb917f56369 | Tick, Trick, and Track have 20, 23, and 25 tickets respectively for the carousel at the fair in Duckburg. They agree that they will only ride all three together, for which they must each give up one of their tickets. Before a ride, if they want, they can redistribute tickets among themselves according to the rule: If o... | [
"Yes"
] | [
"Tick, Trick, and Track have 20, 23, and 25 tickets respectively for the carousel at the fair in Duckburg. They agree that they will only ride all three together, for which they must each give up one of their tickets. Before a ride, if they want, they can redistribute tickets among themselves according to the rule:... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-6a6c0c71e7a67bb26676e474a5593fca5ae20bd04696db5f87487353347039dd | Tick, Trick, and Track have 20, 23, and 25 tickets respectively for the carousel at the fair in Duckburg. They agree that they will only ride all three together, for which they must each give up one of their tickets. Before a ride, if they want, they can redistribute tickets among themselves according to the rule: If o... | [
"\\text{No}"
] | [
"Tick, Trick, and Track have 20, 23, and 25 tickets respectively for the carousel at the fair in Duckburg. They agree that they will only ride all three together, for which they must each give up one of their tickets. Before a ride, if they want, they can redistribute tickets among themselves according to the rule:... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 2
} | 14 | 0 | 0 |
omr-ws-19e5a664017ce92055d207c828cdf79855c0397333e46796b2930fbe05136a7e | Ticket prices for the afternoon movie are $5.50 for adults, $4.50 for children, and $4.00 for senior citizens. If 100 tickets were sold, the proceeds were $465, and more senior citizens than children attended, find the maximum number of children that could have been at the afternoon movie. | [
"31"
] | [
"Ticket prices for the afternoon movie are $5.50 for adults, $4.50 for children, and $4.00 for senior citizens. If 100 tickets were sold, the proceeds were $465, and more senior citizens than children attended, find the maximum number of children that could have been at the afternoon movie."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-fc9d94917cdb31abad01dc46a37e7649a47b50eeaa28e6351632134669e35ce8 | Tiffany is constructing a fence around a rectangular tennis court. She must use exactly 300 feet of fencing. The fence must enclose all four sides of the court. Regulation states that the length of the fence enclosure must be at least 80 feet and the width must be at least 40 feet. Tiffany wants the area enclosed by th... | [
"5600"
] | [
"Tiffany is constructing a fence around a rectangular tennis court. She must use exactly 300 feet of fencing. The fence must enclose all four sides of the court. Regulation states that the length of the fence enclosure must be at least 80 feet and the width must be at least 40 feet. Tiffany wants the area enclosed ... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b6e4e9ab120f64b67677a52cba6cdec2293d5e22be33fca9fa045efd93132b59 | Tigger is in Paris! Tigger is a pro jumper, and he can jump double the height of his previous jump, starting with a jump of $2$ feet, and can go $4$ feet in a second. At what time is the sum of all the current heights first greater than the height of the Eiffel Tower of $1063$ feet, if he starts jumping continuously at... | [
"10:08:31.5"
] | [
"Tigger is in Paris! Tigger is a pro jumper, and he can jump double the height of his previous jump, starting with a jump of $2$ feet, and can go $4$ feet in a second. At what time is the sum of all the current heights first greater than the height of the Eiffel Tower of $1063$ feet, if he starts jumping continuous... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 4
} | 16 | 0 | 0 |
omr-ws-6a9621083b769520f397e1e20731fd7623c52310bcb8763a9dd403a929f356fc | Tiham and Shakur are playing a game. Tiham chooses a positive integer $a$ with $1 < a \leq 58$. Shakur then chooses a positive integer $b$ with $b < a$. Let $f(a,b)$ be the smallest positive integer such that $\frac{bf(a,b)}{a}$ is an integer. Tiham aims to maximize $f(a,b)$, while Shakur aims to minimize it. What valu... | [
"53"
] | [
"Tiham and Shakur are playing a game. Tiham chooses a positive integer $a$ with $1 < a \\leq 58$. Shakur then chooses a positive integer $b$ with $b < a$. Let $f(a,b)$ be the smallest positive integer such that $\\frac{bf(a,b)}{a}$ is an integer. Tiham aims to maximize $f(a,b)$, while Shakur aims to minimize it. Wh... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-a41c009e8c0b1f5c69bbb23896ff048e5d2b3740e1ad91d5b13168523d5f1c0d | Tile the plane with regular hexagons. Assign a real number between 0 and 1 to each vertex such that the value of each vertex is equal to the average of its three neighbors. Let \( V \) be the set of all vertices in a finite section of the plane containing exactly 7 hexagons. Calculate the sum of all possible values ass... | [
"18"
] | [
"Tile the plane with regular hexagons. Assign a real number between 0 and 1 to each vertex such that the value of each vertex is equal to the average of its three neighbors. Let \\( V \\) be the set of all vertices in a finite section of the plane containing exactly 7 hexagons. Calculate the sum of all possible val... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6801670c06ecb191820e4e3675dd3ba725c45374232e9cd09deb1e6b2f4ae89d | Tim and Allen play a game starting with an empty set \( S \). On each turn, the player (starting with Tim) picks a positive integer greater than 1 which cannot be written as a sum of elements in \( S \) (possibly with repetition), and adds it to \( S \). The player who cannot make a move loses. Let \( W \) be the numbe... | [
"0"
] | [
"Tim and Allen play a game starting with an empty set \\( S \\). On each turn, the player (starting with Tim) picks a positive integer greater than 1 which cannot be written as a sum of elements in \\( S \\) (possibly with repetition), and adds it to \\( S \\). The player who cannot make a move loses. Let \\( W \\)... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-f5b1127bf03d9cdf1d733f117fc56b976f04bc500df336ae1b52ebdf36cbd217 | Tim and Allen play a game starting with an empty set \( S \). On each turn, the player (starting with Tim) picks a positive integer greater than 1 which cannot be written as a sum of elements in \( S \) (possibly with repetition), and adds it to \( S \). The player who cannot make a move loses. Let \( W(n) \) be the nu... | [
"no_valid_answer_found"
] | [
"Tim and Allen play a game starting with an empty set \\( S \\). On each turn, the player (starting with Tim) picks a positive integer greater than 1 which cannot be written as a sum of elements in \\( S \\) (possibly with repetition), and adds it to \\( S \\). The player who cannot make a move loses. Let \\( W(n) ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-de861866e62a09ae7820388fdf8f94a77b8858092fa8c269c331a35f0b667b26 | Tim draws the diagonals of an isosceles trapezoid, which creates 4 separate triangular regions. Two of these regions have areas 4 and 8. Find the sum of all possible areas of the trapezoid. Write the integer closest to your answer. | [
"77"
] | [
"Tim draws the diagonals of an isosceles trapezoid, which creates 4 separate triangular regions. Two of these regions have areas 4 and 8. Find the sum of all possible areas of the trapezoid. Write the integer closest to your answer."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 6,
"tir": 3
} | 14 | 0 | 0 |
omr-ws-d49ec0d641f0319288717736c273a0ea2894ac61785e7804480a66aac6ad152e | Tim had 5 red marbles and 2 green marbles. If he adds \( n \) more green marbles to his collection, the probability of drawing a green marble triples. What is the value of \( n \)? | [
"28"
] | [
"Tim had 5 red marbles and 2 green marbles. If he adds \\( n \\) more green marbles to his collection, the probability of drawing a green marble triples. What is the value of \\( n \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f376b33eb2928ec64bb7ba6e33c9a5cbdca88ed527079f8f96153d7b66887b4f | Tim has 3 buttons that each have a counter which records the number of times that the button was pressed. Currently, all three counters are set to 0. Every second, Tim presses a button at random. After 27 seconds have elapsed, he records the numbers on the three counters. What is the expected value of the product of th... | [
"650"
] | [
"Tim has 3 buttons that each have a counter which records the number of times that the button was pressed. Currently, all three counters are set to 0. Every second, Tim presses a button at random. After 27 seconds have elapsed, he records the numbers on the three counters. What is the expected value of the product ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-392053a7fea6cca6c4dc0792303653bc16c7ae6552bc07eb4c9332770720a71b | Tim has a test consisting of 3 questions. The probability that he gets a question correct is initially $\frac{2}{3}$. If he gets a question incorrect, the probability of getting the next question correct is halved. The expected value of the number of questions he gets correct can be written in the form $\frac{a}{b}$, w... | [
"5"
] | [
"Tim has a test consisting of 3 questions. The probability that he gets a question correct is initially $\\frac{2}{3}$. If he gets a question incorrect, the probability of getting the next question correct is halved. The expected value of the number of questions he gets correct can be written in the form $\\frac{a}... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fd8d1724ed4474786e363a5640e957e2bf38e3758d68de447a4dfa983d5a82d3 | Tim has bought a lot of candies for Halloween, but he only remembers that the number of candies is an even number less than 2020. He finds that there will be 1 candy left if he puts seven in each box, 6 left if he puts eleven in each box, and 3 left if he puts thirteen in each box. Find the total number of candies Tim ... | [
"666"
] | [
"Tim has bought a lot of candies for Halloween, but he only remembers that the number of candies is an even number less than 2020. He finds that there will be 1 candy left if he puts seven in each box, 6 left if he puts eleven in each box, and 3 left if he puts thirteen in each box. Find the total number of candies... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6941868b13f4c59c212508329217287af8c74b3a3fd1d8d7862c82c1ec6971f7 | Tim wants to invest some money in a bank which compounds quarterly with an annual interest rate of $7\%$. To the nearest dollar, how much money should he invest if he wants a total of $\$60,\!000$ at the end of $5$ years? | [
"\\$42409"
] | [
"Tim wants to invest some money in a bank which compounds quarterly with an annual interest rate of $7\\%$. To the nearest dollar, how much money should he invest if he wants a total of $\\$60,\\!000$ at the end of $5$ years?"
] | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 5
} | 20 | 0 | 0 |
omr-ws-d226e111c906e9cc085f9e83f87951dc0c9421a6b4dc337fbcef72611b0424d8 | Timmy has a large bag of stones consisting of 1 stone of mass 1 gram, 2 stones of mass 2 grams, 3 stones of mass 3 grams, ..., 10 stones of mass 10 grams, and 11 stones of mass 11 grams. Timmy randomly draws two stones without replacement. What is the probability that the sum of the weights of the two stones is exactly... | [
"\\dfrac{28}{429}"
] | [
"Timmy has a large bag of stones consisting of 1 stone of mass 1 gram, 2 stones of mass 2 grams, 3 stones of mass 3 grams, ..., 10 stones of mass 10 grams, and 11 stones of mass 11 grams. Timmy randomly draws two stones without replacement. What is the probability that the sum of the weights of the two stones is ex... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1f7d38d913270d5a3d83850230456721fa7361ee94f2c26aacfe73f847ac9c56 | Tina and Paul are playing a game on a square \( S \). First, Tina selects a point \( T \) inside \( S \). Next, Paul selects a point \( P \) inside \( S \) at random. Paul then colors blue all the points inside \( S \) that are closer to \( P \) than \( T \). Tina wins if the blue region is the interior of a triangle. ... | [
"\\(\\frac{1}{4}\\)"
] | [
"Tina and Paul are playing a game on a square \\( S \\). First, Tina selects a point \\( T \\) inside \\( S \\). Next, Paul selects a point \\( P \\) inside \\( S \\) at random. Paul then colors blue all the points inside \\( S \\) that are closer to \\( P \\) than \\( T \\). Tina wins if the blue region is the int... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8f94fa1cb8711acc9d4251675f151fa07d8b927b5a48dd7d030662d64e3bb694 | Tina left her office at some time after 2pm. When she returned later, the minute and hour hands on her office clock had swapped positions. At what time, to the nearest minute, did Tina leave her office? | [
"2:16"
] | [
"Tina left her office at some time after 2pm. When she returned later, the minute and hour hands on her office clock had swapped positions. At what time, to the nearest minute, did Tina leave her office?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-27b4f9fd498655a2e77c473be5fdac9bceae3fb58e797a934191f0d0d94085f5 | Tina randomly selects 2 distinct numbers from the set \(\{1, 2, 3, 4, 5\}\) and Sergio randomly selects a number from the set \(\{1, 2, \ldots, 10\}\). What is the probability that Sergio's number is larger than the sum of Tina's numbers? | [
"\\(\\frac{2}{5}\\)"
] | [
"Tina randomly selects 2 distinct numbers from the set \\(\\{1, 2, 3, 4, 5\\}\\) and Sergio randomly selects a number from the set \\(\\{1, 2, \\ldots, 10\\}\\). What is the probability that Sergio's number is larger than the sum of Tina's numbers?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-70393b88ac6f130a86ac1d36f1099f9ba21fe17bad4410ba06cc0e89abdeec11 | Tina randomly selects two distinct numbers from the set $\{ 1, 2, 3, 4, 5 \}$, and Sergio randomly selects a number from the set $\{ 1, 2, \ldots, 10 \}$. What is the probability that Sergio's number is larger than the sum of the two numbers chosen by Tina? | [
"$\\frac{2}{5}$",
"\\dfrac{2}{5}"
] | [
"Tina randomly selects two distinct numbers from the set $\\{ 1, 2, 3, 4, 5 \\}$, and Sergio randomly selects a number from the set $\\{ 1, 2, \\ldots, 10 \\}$. What is the probability that Sergio's number is larger than the sum of the two numbers chosen by Tina?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 26
} | 74 | 0 | 0 |
omr-ws-6c4cc811e33906377aea38e658c586515f4a1e748e4f50a79c0fefaa38ef759c | Tina randomly selects two distinct numbers from the set $\{1, 2, 3, 4, 5\}$, and Sergio selects a number from the set $\{1, 2, \ldots, 10\}$. What is the probability that Sergio's number is larger than the sum of the two numbers chosen by Tina? | [
"\\(\\frac{21}{55}\\)"
] | [
"Tina randomly selects two distinct numbers from the set $\\{1, 2, 3, 4, 5\\}$, and Sergio selects a number from the set $\\{1, 2, \\ldots, 10\\}$. What is the probability that Sergio's number is larger than the sum of the two numbers chosen by Tina?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-534e3754052f1517b4a53198c25ff548772c11348c59ddd23c2c8d23d444386a | Tina randomly selects two distinct numbers from the set $\{1,2,3,4,5\}$ and Sergio randomly selects a number from the set $\{1,2,\ldots,10\}$. What is the probability that Sergio's number is larger than the sum of the two numbers chosen by Tina? | [
"\\(\\frac{2}{5}\\)"
] | [
"Tina randomly selects two distinct numbers from the set $\\{1,2,3,4,5\\}$ and Sergio randomly selects a number from the set $\\{1,2,\\ldots,10\\}$. What is the probability that Sergio's number is larger than the sum of the two numbers chosen by Tina?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-b4ed5034b2a2d51ceec21e847504ea8fa739e18c3a46963d032923ca9a864764 | Tina randomly selects two distinct numbers from the set \(\{1, 2, 3, 4, 5\}\), and Sergio randomly selects a number from the set \(\{1, 2, \ldots, 10\}\). What is the probability that Sergio's number is larger than the sum of the two numbers chosen by Tina? | [
"\\(\\frac{2}{5}\\)"
] | [
"Tina randomly selects two distinct numbers from the set \\(\\{1, 2, 3, 4, 5\\}\\), and Sergio randomly selects a number from the set \\(\\{1, 2, \\ldots, 10\\}\\). What is the probability that Sergio's number is larger than the sum of the two numbers chosen by Tina?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-fb7ec24bd2167855066805f8f107ee2ea2aa125835c70f3edd392e9a403f15e9 | Tina randomly selects two distinct numbers from the set \(\{1, 2, 3, 4, 5\}\), and Sergio randomly selects a number from the set \(\{1, 2, \ldots, 10\}\). What is the probability that Sergio’s number is larger than the sum of the two numbers chosen by Tina? | [
"\\(\\frac{2}{5}\\)"
] | [
"Tina randomly selects two distinct numbers from the set \\(\\{1, 2, 3, 4, 5\\}\\), and Sergio randomly selects a number from the set \\(\\{1, 2, \\ldots, 10\\}\\). What is the probability that Sergio’s number is larger than the sum of the two numbers chosen by Tina?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-4e98e3e6a900cec0213c817000a7e26036b739e9c741f757677c0d71dc402383 | Tina randomly selects two distinct numbers from the set \(\{1,2,3,4,5\}\), and Sergio randomly selects a number from the set \(\{1,2,3,4,5,6,7,8,9,10\}\). What is the probability that Sergio's number is larger than the sum of the two numbers chosen by Tina? | [
"\\dfrac{2}{5}"
] | [
"Tina randomly selects two distinct numbers from the set \\(\\{1,2,3,4,5\\}\\), and Sergio randomly selects a number from the set \\(\\{1,2,3,4,5,6,7,8,9,10\\}\\). What is the probability that Sergio's number is larger than the sum of the two numbers chosen by Tina?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-161dd3829a2f736ae39ce063c202549774b133999193b9f48e17debc19d4ae1a | Tina randomly selects two distinct numbers from the set {1, 2, 3, 4, 5}, and Sergio randomly selects a number from the set {1, 2, 3, ..., 10}. What is the probability that Sergio's number is greater than the sum of the two numbers chosen by Tina? | [
"2/5"
] | [
"Tina randomly selects two distinct numbers from the set {1, 2, 3, 4, 5}, and Sergio randomly selects a number from the set {1, 2, 3, ..., 10}. What is the probability that Sergio's number is greater than the sum of the two numbers chosen by Tina?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-f6d22af03c9e5034cac0516db1647c9d6d775650d61c61a4aa0c46328cf753ca | Tina wrote a positive number on each of five pieces of paper. She revealed their pairwise sums as $17, 20, 28, 14, 42, 36, 28, 39, 25, 31$. Determine the five numbers she wrote. | [
"3, 11, 14, 17, 25"
] | [
"Tina wrote a positive number on each of five pieces of paper. She revealed their pairwise sums as $17, 20, 28, 14, 42, 36, 28, 39, 25, 31$. Determine the five numbers she wrote."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 24
} | 25 | 0 | 0 |
omr-ws-43abc7dccd47cb23a3dd3a529ec9a3ef376348c72b63040bafa9efc7701e7bca | Tita the Frog sits on the number line at an integer $k > 1$. If she is sitting on the number $n$, she hops to the number $f(n) + g(n)$, where $f(n)$ and $g(n)$ are, respectively, the biggest and smallest positive prime numbers that divide $n$. Find all values of $k$ such that Tita can hop to infinitely many distinct in... | [
"\\emptyset"
] | [
"Tita the Frog sits on the number line at an integer $k > 1$. If she is sitting on the number $n$, she hops to the number $f(n) + g(n)$, where $f(n)$ and $g(n)$ are, respectively, the biggest and smallest positive prime numbers that divide $n$. Find all values of $k$ such that Tita can hop to infinitely many distin... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-46fb1f18181612d7a86b9577bb49708fd85900644f857f77af5aa54d0c8e6fd9 | Titu has a set of five weights: one 1-pound weight, one 2-pound weight, one 3-pound weight, one 4-pound weight, and one 5-pound weight. The weights are unmarked, and he can only determine their relative weights by using a balance. In five weighings, Titu identifies each of the weights. If $a$, $b$, $c$, $d$, and $e$ re... | [
"55"
] | [
"Titu has a set of five weights: one 1-pound weight, one 2-pound weight, one 3-pound weight, one 4-pound weight, and one 5-pound weight. The weights are unmarked, and he can only determine their relative weights by using a balance. In five weighings, Titu identifies each of the weights. If $a$, $b$, $c$, $d$, and $... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-56eae0337c0457ed0efe02952b68c70ac20458a8125f1c7b85109aed30e08dd7 | To a vessel containing pure water are added 6 liters of a solution containing 64\% (by volume) alcohol. The solution is thoroughly mixed and an equal quantity (6 liters) of the solution is drawn off. How much water was there originally in the vessel if after performing this operation three times the vessel contained a ... | [
"18 liters"
] | [
"To a vessel containing pure water are added 6 liters of a solution containing 64\\% (by volume) alcohol. The solution is thoroughly mixed and an equal quantity (6 liters) of the solution is drawn off. How much water was there originally in the vessel if after performing this operation three times the vessel contai... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3b05589c9e14d365d108b2ddf2c9f79dc84038a6ce8327b3fb290cb9fbb0dddf | To approximate a function \( g \) over the interval \([0, 1]\) by a polynomial \( p \) of degree \( n \) (or less), we minimize the criterion \[ f(a) = \int_{0}^{1} [g(x) - p(x)]^2 \, dx, \] where \( p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_0 \). Find the equations satisfied by the optimal coefficients \( a = (a_0... | [
"\n\\sum_{i=0}^n \\frac{a_i}{i + k + 1} = \\int_{0}^{1} g(x) x^k \\, dx \\quad \\text{for } k = 0, 1, \\ldots, n\n"
] | [
"To approximate a function \\( g \\) over the interval \\([0, 1]\\) by a polynomial \\( p \\) of degree \\( n \\) (or less), we minimize the criterion \n\\[ f(a) = \\int_{0}^{1} [g(x) - p(x)]^2 \\, dx, \\]\nwhere \\( p(x) = a_n x^n + a_{n-1} x^{n-1} + \\cdots + a_0 \\). Find the equations satisfied by the optimal c... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c6a8482dcc813f1493339a27a114885bd9c7751c92a1fb736190198989329c52 | To bake $12$ cookies, I use $2$ quarts of milk. There are $2$ pints in a quart. How many pints of milk do I need to bake $3$ cookies? | [
"1"
] | [
"To bake $12$ cookies, I use $2$ quarts of milk. There are $2$ pints in a quart. How many pints of milk do I need to bake $3$ cookies?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f32157ef851c74389c256fbe4a3a7096f23f516ae3f08dd27ee884746e57dd1b | To bake wheat-bread, a baker took a certain amount of flour equal (in kg) to the percentage of the gain in weight of bread for this amount of flour. To bake rye-bread, he took 10 kg more flour than for wheat-bread so that the mass of the flour (in kg) was equal to the gain for rye-flour. How much wheat- and rye-flour w... | [
"35 kg of wheat-flour and 45 kg of rye-flour"
] | [
"To bake wheat-bread, a baker took a certain amount of flour equal (in kg) to the percentage of the gain in weight of bread for this amount of flour. To bake rye-bread, he took 10 kg more flour than for wheat-bread so that the mass of the flour (in kg) was equal to the gain for rye-flour. How much wheat- and rye-fl... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3cb8e33f42543c56071d577da2270508f5b198564cead3cd4437eb517c0ddc30 | To be able to walk to the center $C$ of a circular fountain, a repair crew places a 16-foot plank from $A$ to $B$ and then a 10-foot plank from $D$ to $C$, where $D$ is the midpoint of $\overline{AB}$ . What is the area of the circular base of the fountain? Express your answer in terms of $\pi$. [asy] size(250); import... | [
"164 \\pi \\mbox{ square feet}"
] | [
"To be able to walk to the center $C$ of a circular fountain, a repair crew places a 16-foot plank from $A$ to $B$ and then a 10-foot plank from $D$ to $C$, where $D$ is the midpoint of $\\overline{AB}$ . What is the area of the circular base of the fountain? Express your answer in terms of $\\pi$. [asy]\n\nsize(25... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ff6294ea52f22d57db5e5ae296b99a15e70755243898e8507695e9a99980a47b | To be eligible for an algebra class a student must have an average of at least $83\%$ over all four quarters in his or her pre-algebra class. If Fisher had an $82\%$, $77\%$, and $75\%$ in the first three quarters, what is the minimum score he must earn in the 4th quarter to move on to algebra? | [
"98\\%"
] | [
"To be eligible for an algebra class a student must have an average of at least $83\\%$ over all four quarters in his or her pre-algebra class. If Fisher had an $82\\%$, $77\\%$, and $75\\%$ in the first three quarters, what is the minimum score he must earn in the 4th quarter to move on to algebra?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f674fa496082c0b0c164365fb135ee9528ed55352f510175eeb0ea2e4311ba30 | To calculate $31^2$, Emily mentally figures the value $30^2$ and adds 61. Emily subtracts a number from $30^2$ to calculate $29^2$. What number does she subtract? | [
"59"
] | [
"To calculate $31^2$, Emily mentally figures the value $30^2$ and adds 61. Emily subtracts a number from $30^2$ to calculate $29^2$. What number does she subtract?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3596fc992e9b3f7341cf2e2e86c5f95733fe174abe8d7ec5cebce109a1f27fe2 | To calculate $41^2$, David mentally figures the value $40^2$ and adds 81. David subtracts a number from $40^2$ to calculate $39^2$. What number does he subtract? | [
"79"
] | [
"To calculate $41^2$, David mentally figures the value $40^2$ and adds 81. David subtracts a number from $40^2$ to calculate $39^2$. What number does he subtract?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-847b6543ec4d72e0515e24079b1d8e3b1ebceac7bdb6dd143c7d05d3eb9452d8 | To celebrate the 20th Thailand Mathematical Olympiad (TMO), Ratchasima Witthayalai School put up flags around the Thao Suranari Monument such that: - Each flag is painted in exactly one color, and at least 2 distinct colors are used. - The number of flags is odd. - Every flag is placed on a regular polygon such that ea... | [
"3"
] | [
"To celebrate the 20th Thailand Mathematical Olympiad (TMO), Ratchasima Witthayalai School put up flags around the Thao Suranari Monument such that:\n- Each flag is painted in exactly one color, and at least 2 distinct colors are used.\n- The number of flags is odd.\n- Every flag is placed on a regular polygon such... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b34781fdee8480f6a4a5f30464beb1bee9e40d3761dcd8d48739548addaf12e7 | To control her blood pressure, Jill's grandmother takes one half of a pill every other day. If one supply of medicine contains 60 pills, then the supply of medicine would last approximately how long? | [
"8\\text{ months}"
] | [
"To control her blood pressure, Jill's grandmother takes one half of a pill every other day. If one supply of medicine contains 60 pills, then the supply of medicine would last approximately how long?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 2,
"tir": 25
} | 38 | 0 | 0 |
omr-ws-81266eb7b0eac778089f3e7b8a26466823c0be09841871e2d160a68566a50aac | To cover all sides of a complete graph of order 21 with a complete subgraph of order 5, which can be covered repeatedly, at least how many subgraphs are needed? | [
"21"
] | [
"To cover all sides of a complete graph of order 21 with a complete subgraph of order 5, which can be covered repeatedly, at least how many subgraphs are needed?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d6052c9666e84266d73398dbf87c9745f1348119cc73ed6e64e310fc8df9255e | To cross a shallow river of width 14 meters without getting my feet wet, I would like stepping stones placed in a river so that they form a straight path perpendicular to the riverbank. Stones can be placed 1, 2, 3, ..., 12, 13 feet away from the shore. Assuming I can only make a leap of at most 4 meters, in how many w... | [
"5525"
] | [
"To cross a shallow river of width 14 meters without getting my feet wet, I would like stepping stones placed in a river so that they form a straight path perpendicular to the riverbank. Stones can be placed 1, 2, 3, ..., 12, 13 feet away from the shore. Assuming I can only make a leap of at most 4 meters, in how m... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 6,
"tir": 2
} | 12 | 0 | 0 |
omr-ws-061118941eff342dd94c1dc247133f9968c2da9e53323cadf601982ff81c0310 | To determine if Michael can walk again, he must score in the top 90% on a range of motion test. The average score for the test is 21, with a standard deviation of 3.3. What score must Michael achieve to be in the top 90%? | [
"25.224"
] | [
"To determine if Michael can walk again, he must score in the top 90% on a range of motion test. The average score for the test is 21, with a standard deviation of 3.3. What score must Michael achieve to be in the top 90%?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 2,
"tir": 27
} | 36 | 0 | 0 |
omr-ws-df24e686ca29bd07cdc20c1e8245efe3248be0e5bf18ca977d3e9a0b528047fc | To each element of the set \( S = \{1, 2, \ldots, 1000\} \) a colour is assigned. Suppose that for any two elements \( a, b \) of \( S \), if \( 15 \) divides \( a + b \) then they are both assigned the same colour. What is the maximum possible number of distinct colours used? | [
"8"
] | [
"To each element of the set \\( S = \\{1, 2, \\ldots, 1000\\} \\) a colour is assigned. Suppose that for any two elements \\( a, b \\) of \\( S \\), if \\( 15 \\) divides \\( a + b \\) then they are both assigned the same colour. What is the maximum possible number of distinct colours used?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a7b72b67d1ce3e3af6193f0eb23bcdeb3f8aaaac5250305efb2456e3d1f501fe | To each of the first two of the 4 numbers 1, 19, 203, 2003 is added \( x \) and to each of the last two \( y \). The numbers form a geometric progression (G.P.). Find all such ordered pairs \((x, y)\) of real numbers. | [
"\\((1, -3)\\) and \\((-2.63, -366)\\)"
] | [
"To each of the first two of the 4 numbers 1, 19, 203, 2003 is added \\( x \\) and to each of the last two \\( y \\). The numbers form a geometric progression (G.P.). Find all such ordered pairs \\((x, y)\\) of real numbers."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 11
} | 12 | 0 | 0 |
omr-ws-97384d53fda9c7bc37b98940366dfddf4d382fc23028aaf004d28830f0153354 | To each positive integer \( n \) it is assigned a non-negative integer \( f(n) \) such that the following conditions are satisfied: (1) \( f(rs) = f(r) + f(s) \) (2) \( f(n) = 0 \), if the first digit (from right to left) of \( n \) is 3. (3) \( f(10) = 0 \). Find \( f(1985) \). Justify your answer. | [
"0"
] | [
"To each positive integer \\( n \\) it is assigned a non-negative integer \\( f(n) \\) such that the following conditions are satisfied:\n(1) \\( f(rs) = f(r) + f(s) \\)\n(2) \\( f(n) = 0 \\), if the first digit (from right to left) of \\( n \\) is 3.\n(3) \\( f(10) = 0 \\).\n\nFind \\( f(1985) \\). Justify your an... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-156693fe2dfad26417afe5fb4f5f943a1fc91fbe389212668e665e3f8f534aab | To each vertex of a regular $n$-gon, a real number is assigned such that the sum of all $n$ numbers is positive. If three consecutive vertices are assigned the numbers $x, y, z$ respectively and $y < 0$, the following operation is allowed: replace $x, y, z$ with $x + y, -y, z + y$ respectively. This operation is perfor... | [
"15"
] | [
"To each vertex of a regular $n$-gon, a real number is assigned such that the sum of all $n$ numbers is positive. If three consecutive vertices are assigned the numbers $x, y, z$ respectively and $y < 0$, the following operation is allowed: replace $x, y, z$ with $x + y, -y, z + y$ respectively. This operation is p... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-6c6160da2ad5134bd6fb9f2f4632e9be72b8423971e780f5a5f88f3c6bdb9b43 | To each vertex of a regular $n$-gon, a real number is assigned such that the sum of all $n$ numbers is positive. If three consecutive vertices are assigned the numbers $x, y, z$ respectively, and $y < 0$, then the operation $x, y, z \rightarrow x+y, -y, z+y$ is allowed. Let $S$ be the sum of the absolute values of all ... | [
"6"
] | [
"To each vertex of a regular $n$-gon, a real number is assigned such that the sum of all $n$ numbers is positive. If three consecutive vertices are assigned the numbers $x, y, z$ respectively, and $y < 0$, then the operation $x, y, z \\rightarrow x+y, -y, z+y$ is allowed. Let $S$ be the sum of the absolute values o... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-a31c984bbcfbd2eb4cf10286ca98ddf7a42c35216d188f48176068358edacf96 | To each vertex of a regular n-gon, a real number is assigned such that the sum of all n numbers is positive. If three consecutive vertices are assigned the numbers \( x \), \( y \), \( z \) respectively and \( y < 0 \), then the operation \( (x, y, z) \to (x+y, -y, z+y) \) is allowed. This operation is performed repeat... | [
"15"
] | [
"To each vertex of a regular n-gon, a real number is assigned such that the sum of all n numbers is positive. If three consecutive vertices are assigned the numbers \\( x \\), \\( y \\), \\( z \\) respectively and \\( y < 0 \\), then the operation \\( (x, y, z) \\to (x+y, -y, z+y) \\) is allowed. This operation is ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-54b5204d543b99c427d5a55cdffdee84389207d4c55c5afe604b2bf1f27c1b96 | To each vertex of a regular pentagon, an integer is assigned such that the sum of all five numbers is positive. If three consecutive vertices are assigned the numbers $x, y, z$ respectively, and $y < 0$, then the operation $x, y, z$ are replaced by $x+y, -y, z+y$ respectively is allowed. After performing the operation ... | [
"1"
] | [
"To each vertex of a regular pentagon, an integer is assigned such that the sum of all five numbers is positive. If three consecutive vertices are assigned the numbers $x, y, z$ respectively, and $y < 0$, then the operation $x, y, z$ are replaced by $x+y, -y, z+y$ respectively is allowed. After performing the opera... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1e1b726b6e158becb8539c76424df72f2c94013fab8bb0b3073b91a41c2915fc | To eliminate the effects of the wind and obtain the car's speed \( V \) in a windless situation, should we find the average of \( \frac{d}{t} \) and \( \frac{d}{t'} \) (method 1) or should we divide \( d \) by the average of \( t \) and \( t' \)? | [
"divide \\( d \\) by the average of \\( t \\) and \\( t' \\)"
] | [
"To eliminate the effects of the wind and obtain the car's speed \\( V \\) in a windless situation, should we find the average of \\( \\frac{d}{t} \\) and \\( \\frac{d}{t'} \\) (method 1) or should we divide \\( d \\) by the average of \\( t \\) and \\( t' \\)?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-462a211b7b59ef2229138b02a96adb387729b37eb6e9ebe05036b00e0465fb18 | To estimate the height of the Space Needle in Seattle, Washington, James held a book up to his eyes so that the top and bottom of the building were in line with the bottom edge and binding of the cover. If James' eye level is 6 feet from the ground and he is standing 60 feet from the tower, how tall is the tower? | [
"66"
] | [
"To estimate the height of the Space Needle in Seattle, Washington, James held a book up to his eyes so that the top and bottom of the building were in line with the bottom edge and binding of the cover. If James' eye level is 6 feet from the ground and he is standing 60 feet from the tower, how tall is the tower?"... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4f10d4a654464090197f3228ebd51340bb0c156d2cf6fecb31610d401f254c05 | To every pair of positive real numbers \( x \) and \( y \), we assign a positive real number \( x * y \) satisfying the two properties that \( x * x = 1 \) and \( x * (y * z) = (x * y) \cdot z \) where \( \cdot \) represents the standard multiplication of real numbers. Determine \( 61 * 2013 \). | [
"\\(\\frac{1}{33}\\)"
] | [
"To every pair of positive real numbers \\( x \\) and \\( y \\), we assign a positive real number \\( x * y \\) satisfying the two properties that \\( x * x = 1 \\) and \\( x * (y * z) = (x * y) \\cdot z \\) where \\( \\cdot \\) represents the standard multiplication of real numbers. Determine \\( 61 * 2013 \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-d16fc0199bb8a3840e750f93009d74f53dec26ca1fbfbe105dc25a5faffa27a7 | To every vertex of a regular pentagon, a real number is assigned such that the sum of all five numbers is initially equal to zero. We may repeatedly choose two adjacent vertices and replace each of the two numbers assigned to these vertices by their arithmetic mean. Let $S$ be the set of all possible configurations of ... | [
""
] | [
"To every vertex of a regular pentagon, a real number is assigned such that the sum of all five numbers is initially equal to zero. We may repeatedly choose two adjacent vertices and replace each of the two numbers assigned to these vertices by their arithmetic mean. Let $S$ be the set of all possible configuration... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 1 |
omr-ws-c68a5e88ef8b156bbf69a21ee6bd44cde695bc72e1dafd5965eae7353d23589f | To express 20 as a sum of different powers of 2, we would write $20 = 2^4 + 2^2$. The sum of the exponents of these powers is $4 + 2 = 6$. If 400 were expressed as a sum of at least two distinct powers of 2, what would be the least possible sum of the exponents of these powers? | [
"19"
] | [
"To express 20 as a sum of different powers of 2, we would write $20 = 2^4 + 2^2$. The sum of the exponents of these powers is $4 + 2 = 6$. If 400 were expressed as a sum of at least two distinct powers of 2, what would be the least possible sum of the exponents of these powers?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f804fc87aa50daed79caa6041f980ba90af1cff141436d8f2aff070eea9f34a9 | To express 20 as a sum of distinct powers of 2, we would write $20 = 2^4 + 2^2$. The sum of the exponents of these powers is $4 + 2 = 6$. If 1562 were expressed as a sum of distinct powers of 2, what would be the least possible sum of the exponents of these powers? | [
"27"
] | [
"To express 20 as a sum of distinct powers of 2, we would write $20 = 2^4 + 2^2$. The sum of the exponents of these powers is $4 + 2 = 6$. If 1562 were expressed as a sum of distinct powers of 2, what would be the least possible sum of the exponents of these powers?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9585beb37cb7a1ebaeb5f1973020f8d53f66ef7d377adeab0802fb0fa50f693a | To fill 12 vacancies, there are 25 candidates, of which 5 are from a reserved category. If 3 vacancies are reserved for the reserved category candidates and the rest are open to all candidates, find the number of ways in which the selection can be made. | [
"4974200"
] | [
"To fill 12 vacancies, there are 25 candidates, of which 5 are from a reserved category. If 3 vacancies are reserved for the reserved category candidates and the rest are open to all candidates, find the number of ways in which the selection can be made."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-d674799e4d869d2a57f5ef1740fe280d4ffb09aaa758dedfb469a0d2c946be05 | To guarantee the capture of a thief with $2m$ accomplices, a strategy involves placing each policeman under surveillance of one accomplice each morning. Every evening, the thief stops trusting one accomplice. The thief is captured if by the $m$-th evening, some policeman shadows exactly those $m$ accomplices who are st... | [
"70"
] | [
"To guarantee the capture of a thief with $2m$ accomplices, a strategy involves placing each policeman under surveillance of one accomplice each morning. Every evening, the thief stops trusting one accomplice. The thief is captured if by the $m$-th evening, some policeman shadows exactly those $m$ accomplices who a... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-2cf6804086ef647042238e44804508bffb3cc4d3f695e80fcbfa6397aa5c8499 | To inscribe in a given acute-angled triangle the triangle of minimum perimeter. | [
"\\text{The orthic triangle}"
] | [
"To inscribe in a given acute-angled triangle the triangle of minimum perimeter."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-23e42166feff7c25f90330403e9a3fde2fb61ad7a3c016d48c2a7fa92b9be20b | To make a rectangular trough, you bend the sides of a 32-inch wide sheet of metal to obtain a cross section. Find the dimensions of the cross section with the maximum area. Given that the width of the metal sheet is 32 inches, and once you bend it, the width of the cross section is \(32 - 2x\), where \(x\) is the heigh... | [
"\\( x = 8 \\) inches, width \\( = 16 \\) inches"
] | [
"To make a rectangular trough, you bend the sides of a 32-inch wide sheet of metal to obtain a cross section. Find the dimensions of the cross section with the maximum area. \n\nGiven that the width of the metal sheet is 32 inches, and once you bend it, the width of the cross section is \\(32 - 2x\\), where \\(x\\)... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-111463a75125444c773ea166cb4e75bc0b86070ab583b5c2ea95f26aca18f021 | To make lemonade, I use a ratio of $7$ parts water to $1$ part lemon juice. If I want to make a gallon of lemonade, and there are four quarts in a gallon, how many quarts of water do I need? Write your answer as a proper or improper fraction. | [
"\\frac{7}{2}"
] | [
"To make lemonade, I use a ratio of $7$ parts water to $1$ part lemon juice. If I want to make a gallon of lemonade, and there are four quarts in a gallon, how many quarts of water do I need? Write your answer as a proper or improper fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b2c31a56bc2e0a16dc48053361c761b8872cf5d59822e5ca2a407ab59269300c | To make pizza dough, Luca mixes 50 mL of milk for every 250 mL of flour. How many mL of milk does he mix with 750 mL of flour? | [
"150"
] | [
"To make pizza dough, Luca mixes 50 mL of milk for every 250 mL of flour. How many mL of milk does he mix with 750 mL of flour?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-62abcc2c6ab45b0b61e7e7d92457c30338c21f978935e1788a10fc0bd7511763 | To make ‘short’ pastry, one uses flour to fat in the ratio 2 : 1. To make ‘flaky’ pastry, the ratio is 4 : 3. Given 29 kg of flour and 20 kg of fat, how much flaky pastry can be made if all ingredients are used? | [
"38.5kg"
] | [
"To make ‘short’ pastry, one uses flour to fat in the ratio 2 : 1. To make ‘flaky’ pastry, the ratio is 4 : 3. Given 29 kg of flour and 20 kg of fat, how much flaky pastry can be made if all ingredients are used?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 7
} | 29 | 0 | 0 |
omr-ws-125a0badbadcc49758f07b6aeeb974d583e22698a066da1f1d790ae7b3a62785 | To measure the height of a mountain, Peter takes two sightings of the peak at a distance of 900 meters apart on a direct line to the mountain. The first observation results in an angle of elevation of 47 degrees and the second results in an angle of elevation of 35 degrees. If the transit is 2 meters high, what is the ... | [
"1818"
] | [
"To measure the height of a mountain, Peter takes two sightings of the peak at a distance of 900 meters apart on a direct line to the mountain. The first observation results in an angle of elevation of 47 degrees and the second results in an angle of elevation of 35 degrees. If the transit is 2 meters high, what is... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-517d7343ada91dcaf32b1535afe4d0e9b6cf75d4a98d430afdaa13d4012b04ab | To number the pages of an encyclopedia, the number 1 was printed 1988 times. How many pages does this encyclopedia have? | [
"4159"
] | [
"To number the pages of an encyclopedia, the number 1 was printed 1988 times. How many pages does this encyclopedia have?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c8b6c16a8cb2b9eae60254e7468faf265f790bb1d1b8b8232b665321afd9ac1d | To pass a certain test, $65$ out of $100$ is needed. The class average is $66$. The average score of the students who pass the test is $71$ and the average score of the students who fail the test is $56$. It is decided to add $5$ to every score, so that a few more students pass the test. Now the average score of the st... | [
"24"
] | [
"To pass a certain test, $65$ out of $100$ is needed. The class average is $66$. The average score of the students who pass the test is $71$ and the average score of the students who fail the test is $56$. It is decided to add $5$ to every score, so that a few more students pass the test. Now the average score of t... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ae1e2e19d446fb04969c658ab0fae3a8ffacbefa5917edf39c3df809d71c44f1 | To pay off a loan of R7000 due now and a loan of R2000 due in 14 months, Mrs. Slovo agrees to make three payments in two, five, and ten months, respectively. The second payment is double the first, and the third payment is triple the first. What is the approximate size of the payment at month five if interest is calcul... | [
"3161.98"
] | [
"To pay off a loan of R7000 due now and a loan of R2000 due in 14 months, Mrs. Slovo agrees to make three payments in two, five, and ten months, respectively. The second payment is double the first, and the third payment is triple the first. What is the approximate size of the payment at month five if interest is c... | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 28
} | 42 | 0 | 0 |
omr-ws-6d6ecf37c32b64115b01bc9834521d4bb67eb3ffe1a25848769e1cb90036de3b | To prepare for her half-marathon, Mary ran a total of $2847_9$ miles. How many miles (in base ten) did Mary run? | [
"2149"
] | [
"To prepare for her half-marathon, Mary ran a total of $2847_9$ miles. How many miles (in base ten) did Mary run?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3ee506c8fe14a70fbb9548fb23f8a81b7b8b0846b069bc2ee9f2c1426c7b4626 | To reach a point like (2,1), find the sum of the number of ways to reach the points (0,0), (1,0), (2,0), (0,1), (1,1), and (2,1). | [
"9"
] | [
"To reach a point like (2,1), find the sum of the number of ways to reach the points (0,0), (1,0), (2,0), (0,1), (1,1), and (2,1)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2ce41bb703171dfce1f45010c189f434aca075e5e07f1f755f874e4b7fae559c | To solve the differential equation, use the substitution \( u = \sqrt{x^2 + y^2} \), and find \( u' \). | [
"\\( u' = \\frac{x + yy'}{u} \\)"
] | [
"To solve the differential equation, use the substitution \\( u = \\sqrt{x^2 + y^2} \\), and find \\( u' \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-89a251f8951444945678e096cb4d6fb7b50a2075292e9e4bb48591c11e7dae45 | To take quizzes, each of 30 students in a class is paired with another student. If the pairing is done randomly, what is the probability that Margo is paired with her best friend, Irma? Express your answer as a common fraction. | [
"\\frac{1}{29}"
] | [
"To take quizzes, each of 30 students in a class is paired with another student. If the pairing is done randomly, what is the probability that Margo is paired with her best friend, Irma? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4d0da62935a9ecc84a578478d7d46b28de5365bcfd01351a166ae9f2ad22995e | To test a number for divisibility by a prime number, remove the final digit, then subtract three times this digit from whatever remains. The new number is divisible by this prime number if and only if the original one is. What is this prime number? | [
"31"
] | [
"To test a number for divisibility by a prime number, remove the final digit, then subtract three times this digit from whatever remains. The new number is divisible by this prime number if and only if the original one is. What is this prime number?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-22c814e31af7d194084b07a4b2ab5989603d9dcdc2d374df125bf3d4003d543c | To test whether an integer, $n$, is prime, it is enough to be sure that none of the primes less than or equal to the square root of $n$ divide $n$. If you want to check that a number between 900 and 950 is prime with this rule, what is the largest prime divisor you need to test? | [
"29"
] | [
"To test whether an integer, $n$, is prime, it is enough to be sure that none of the primes less than or equal to the square root of $n$ divide $n$. If you want to check that a number between 900 and 950 is prime with this rule, what is the largest prime divisor you need to test?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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