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-366ab26c62cd3d6ba15058945ee300fac5b7a5936533f80517f4b07ceac41651 | A cube with side length $s$ contains a convex polyhedron whose projection onto any of the cube's faces covers the entire face. Let $V$ be the volume of the polyhedron. Find the minimum value of $\frac{V}{s^3}$. | [
"\\dfrac{1}{3}"
] | [
"A cube with side length $s$ contains a convex polyhedron whose projection onto any of the cube's faces covers the entire face. Let $V$ be the volume of the polyhedron. Find the minimum value of $\\frac{V}{s^3}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d7dd0e7893d263d60ae7e8cf0e78bacd01044a3e60c0b2795fdb1eb49041a81a | A cube with side length 1 cm is sectioned by a plane passing through vertices A, B, and C. Find the volume of the pyramid formed by this sectioning. | [
"\\dfrac{1}{6}"
] | [
"A cube with side length 1 cm is sectioned by a plane passing through vertices A, B, and C. Find the volume of the pyramid formed by this sectioning."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
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omr-ws-bea48044bac4f6da65278bfd4bf0078abb4c7ab305cf80d5caab8c19ac18a067 | A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labeled $A$. The three vertices adjacent to vertex $A$ are at heights 10, 11, and 12 above the plane. The distance from vertex $A$ to the plane can be expressed as $ \frac{r-\sqrt{s}}{t}$, where $r$, $s$, and $t$ are positive inte... | [
"330"
] | [
"A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labeled $A$. The three vertices adjacent to vertex $A$ are at heights 10, 11, and 12 above the plane. The distance from vertex $A$ to the plane can be expressed as $ \\frac{r-\\sqrt{s}}{t}$, where $r$, $s$, and $t$ are positi... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e83c0cf980806a7dcb2a1f3f71db8406893353664aa5998669af4031adadef48 | A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labeled $A$. The three vertices adjacent to vertex $A$ are at heights 10, 11, and 12 above the plane. The distance from vertex $A$ to the plane can be expressed as $\frac{r-\sqrt{s}}{t}$, where $r$, $s$, and $t$ are positive integ... | [
"330"
] | [
"A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labeled $A$. The three vertices adjacent to vertex $A$ are at heights 10, 11, and 12 above the plane. The distance from vertex $A$ to the plane can be expressed as $\\frac{r-\\sqrt{s}}{t}$, where $r$, $s$, and $t$ are positiv... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 47,
"genselect": 7,
"tir": 7
} | 61 | 0 | 0 |
omr-ws-c11f8283dc07c14652c21c3ca25901e4458faba2ac465727845bc9844d097cab | A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labeled $A$. The three vertices adjacent to vertex $A$ are at heights 10, 11, and 12 above the plane. The distance from vertex $A$ to the plane can be expressed as $\frac{r-\sqrt{s}}{t}$, where $r$, $s$, and $t$ are positive integ... | [
"330"
] | [
"A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labeled $A$. The three vertices adjacent to vertex $A$ are at heights 10, 11, and 12 above the plane. The distance from vertex $A$ to the plane can be expressed as $\\frac{r-\\sqrt{s}}{t}$, where $r$, $s$, and $t$ are positiv... | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 3
} | 26 | 0 | 0 |
omr-ws-f6e8ba6fcb444318537e8d7e6bf0558fdc1b1120045ced744748f37455fd6b47 | A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labeled A. The three vertices adjacent to vertex A are at heights 10, 11, and 12 above the plane. The distance from vertex A to the plane can be expressed as \(\frac{r-\sqrt{s}}{t}\), where \(r\), \(s\), and \(t\) are positive int... | [
"330"
] | [
"A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labeled A. The three vertices adjacent to vertex A are at heights 10, 11, and 12 above the plane. The distance from vertex A to the plane can be expressed as \\(\\frac{r-\\sqrt{s}}{t}\\), where \\(r\\), \\(s\\), and \\(t\\) a... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 45,
"tir": 6
} | 51 | 0 | 0 |
omr-ws-271ac58d2e32c8277b191904c5a129883367bed309e2bdaca0550bd6148b6005 | A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labelled $A$. The three vertices adjacent to vertex $A$ are at heights 10, 11, and 12 above the plane. The distance from vertex $A$ to the plane can be expressed as $\tfrac{r-\sqrt{s}}{t}$, where $r$, $s$, and $t$ are positive int... | [
"330"
] | [
"A cube with side length 10 is suspended above a plane. The vertex closest to the plane is labelled $A$. The three vertices adjacent to vertex $A$ are at heights 10, 11, and 12 above the plane. The distance from vertex $A$ to the plane can be expressed as $\\tfrac{r-\\sqrt{s}}{t}$, where $r$, $s$, and $t$ are posit... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 4
} | 51 | 0 | 0 |
omr-ws-0d6689b684214ecb3a74e01810bd481e22f361cb3b5afc31cea81b649d0b07ea | A cube with side length 12 has faces ABCD and BCEF. Point P is on face BCEF, 5 units away from edges BC and CE. A beam of light leaves vertex A, travels to point P, and is reflected inside the cube each time it strikes a face. How far does the beam travel before it hits a vertex? | [
"12\\sqrt{218}"
] | [
"A cube with side length 12 has faces ABCD and BCEF. Point P is on face BCEF, 5 units away from edges BC and CE. A beam of light leaves vertex A, travels to point P, and is reflected inside the cube each time it strikes a face. How far does the beam travel before it hits a vertex?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-135ff1cf42c2fe8e5d6e715ee5d0f6019e7746ab329d6798d8243f6757255b9f | A cube with side length 2 units is placed on a table such that one of its faces is in contact with the table. A red point is marked at the center of the top face. The cube is rotated along a straight line such that at least two vertices remain in contact with the table until the red point returns to its initial positio... | [
"9"
] | [
"A cube with side length 2 units is placed on a table such that one of its faces is in contact with the table. A red point is marked at the center of the top face. The cube is rotated along a straight line such that at least two vertices remain in contact with the table until the red point returns to its initial po... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 29,
"genselect": 4
} | 33 | 0 | 0 |
omr-ws-e5b00bd387239b1208ce243e388382c7e38b518ae47faf420439f23fe8205ae0 | A cube with side length 3 is divided into 27 unit cubes, each labeled uniquely from 1 to 27. A move consists of swapping the cube labeled 27 with one of its 6 neighbors. Find the minimum number of moves required for cube 27 to return to its original position and for each cube $n$ to move to the position originally occu... | [
"48"
] | [
"A cube with side length 3 is divided into 27 unit cubes, each labeled uniquely from 1 to 27. A move consists of swapping the cube labeled 27 with one of its 6 neighbors. Find the minimum number of moves required for cube 27 to return to its original position and for each cube $n$ to move to the position originally... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5836788346473c66e562525e2875fafc4405bb62c2ae3623a0aeb0d189a46f46 | A cube with side length 5 is divided into 125 smaller cubes, each with side length 1. If two of these smaller cubes are chosen randomly, what is the probability that they share a common side? | [
"\\(\\frac{6}{155}\\)"
] | [
"A cube with side length 5 is divided into 125 smaller cubes, each with side length 1. If two of these smaller cubes are chosen randomly, what is the probability that they share a common side?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-0853461cf8d9ffe1d10e837ab016de4e7607b7395c7ad69cbec0a09574141068 | A cube with side length 8 is divided into smaller unit cubes. An ant is sitting in one of these unit cubes, and its location is unknown. What is the smallest number of partitions between the unit cubes that must be removed so that the ant can reach the border of the large cube, regardless of its initial position? (Reac... | [
"4"
] | [
"A cube with side length 8 is divided into smaller unit cubes. An ant is sitting in one of these unit cubes, and its location is unknown. What is the smallest number of partitions between the unit cubes that must be removed so that the ant can reach the border of the large cube, regardless of its initial position? ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-25dda5cb8596ae217643bc23d944c62f9f72325f3e905542dd1d37e59f08f0e4 | A cube with side length 8 is divided into smaller unit cubes. An ant is sitting in one of these unit cubes, but its location is unknown. What is the smallest number of partitions between the unit cubes that must be removed so that the ant can reach the border of the large cube, regardless of its initial position? (Reac... | [
"216"
] | [
"A cube with side length 8 is divided into smaller unit cubes. An ant is sitting in one of these unit cubes, but its location is unknown. What is the smallest number of partitions between the unit cubes that must be removed so that the ant can reach the border of the large cube, regardless of its initial position? ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-38aafcc88bcd0b9b21b603d06eac6ee74b08fd6cffea1ab09ae41cd9ecadade5 | A cube with side length of $4$ units is divided into $64$ unit cubes. $16$ unit cubes are selected and colored red such that every $1 \times 1 \times 4$ rectangular prism in the cube contains exactly $1$ red cube. Find the number of ways to select the $16$ red cubes. | [
"576"
] | [
"A cube with side length of $4$ units is divided into $64$ unit cubes. $16$ unit cubes are selected and colored red such that every $1 \\times 1 \\times 4$ rectangular prism in the cube contains exactly $1$ red cube. Find the number of ways to select the $16$ red cubes."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-698b0782bc7b534817c06e1b3317ec9f64eceaf6b26599e7a8348455ccb73b3a | A cube with sides of length 3 cm is painted red and then cut into 27 smaller cubes with sides of length 1 cm. Let \( a \) be the number of small cubes that are not painted at all, \( b \) be the number painted on one side, \( c \) be the number painted on two sides, and \( d \) be the number painted on three sides. Det... | [
"-9"
] | [
"A cube with sides of length 3 cm is painted red and then cut into 27 smaller cubes with sides of length 1 cm. Let \\( a \\) be the number of small cubes that are not painted at all, \\( b \\) be the number painted on one side, \\( c \\) be the number painted on two sides, and \\( d \\) be the number painted on thr... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-21f89d0415566d69f8f4d017a4d150a4ffb9cd6502d2f1a81947e9f29f4225e3 | A cube-shaped basin has a side length of 30 cm. When a body is placed in the basin, the water level rises by 3 cm. Calculate the volume of the body. | [
"2700 cm\\(^3\\)"
] | [
"A cube-shaped basin has a side length of 30 cm. When a body is placed in the basin, the water level rises by 3 cm. Calculate the volume of the body."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-a72afc37178b8d1d5aa0f9bf31d244da606f19b20c45708f4581d8130ca31550 | A cube-shaped box has a side length of 1. A pencil, which consists of a rectangular pyramid with a rectangular prism attached to its base, is placed in the cube such that the tip of the pencil (the apex of the pyramid) is at one of the vertices of the cube. If the ratio of the volume of the pencil to the volume of the ... | [
"2198"
] | [
"A cube-shaped box has a side length of 1. A pencil, which consists of a rectangular pyramid with a rectangular prism attached to its base, is placed in the cube such that the tip of the pencil (the apex of the pyramid) is at one of the vertices of the cube. If the ratio of the volume of the pencil to the volume of... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-99f21268b1b76ff80912c926a1481c8118e5b485082100a9a4434115736e84da | A cube-shaped container has vertices $A,$ $B,$ $C,$ and $D,$ where $\overline{AB}$ and $\overline{CD}$ are parallel edges of the cube, and $\overline{AC}$ and $\overline{BD}$ are diagonals of faces of the cube, as shown. Vertex $A$ of the cube is set on a horizontal plane $\mathcal{P}$ so that the plane of the rectangl... | [
"751"
] | [
"A cube-shaped container has vertices $A,$ $B,$ $C,$ and $D,$ where $\\overline{AB}$ and $\\overline{CD}$ are parallel edges of the cube, and $\\overline{AC}$ and $\\overline{BD}$ are diagonals of faces of the cube, as shown. Vertex $A$ of the cube is set on a horizontal plane $\\mathcal{P}$ so that the plane of ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ec6eea2c3b60879a938294963cf5ea6070df1e6aad79e710183503ade9ae7214 | A cube-shaped container has vertices \(A\), \(B\), \(C\), and \(D\) where \(\overline{AB}\) and \(\overline{CD}\) are parallel edges of the cube, and \(\overline{AC}\) and \(\overline{BD}\) are diagonals of the faces of the cube. Vertex \(A\) of the cube is set on a horizontal plane \(\mathcal{P}\) so that the plane of... | [
"751"
] | [
"A cube-shaped container has vertices \\(A\\), \\(B\\), \\(C\\), and \\(D\\) where \\(\\overline{AB}\\) and \\(\\overline{CD}\\) are parallel edges of the cube, and \\(\\overline{AC}\\) and \\(\\overline{BD}\\) are diagonals of the faces of the cube. Vertex \\(A\\) of the cube is set on a horizontal plane \\(\\math... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ba8fced6717b8c35fa185caf6a896ac34365fd3420ede7a507445bb732583637 | A cubic bottle with side length of 3 centimeters is filled with liquid. When the bottle is placed such that a large diagonal of the cube is perpendicular to the ground, the liquid reaches two-thirds of the length of this diagonal. Determine the fraction of the volume of the bottle that the liquid actually represents. | [
"\\(\\frac{1}{6}\\)"
] | [
"A cubic bottle with side length of 3 centimeters is filled with liquid. When the bottle is placed such that a large diagonal of the cube is perpendicular to the ground, the liquid reaches two-thirds of the length of this diagonal. Determine the fraction of the volume of the bottle that the liquid actually represen... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f4637ed5ef553dc67d32cb31dc70c7e0ef301c83363935c2d5bab2d37935ad8b | A cubic box with side length $1.2$ m is placed on the ground next to a high wall. A ladder of length $3.5$ m leans against the wall and just touches the top edge of the box. Determine the distance from the box to the foot of the ladder and the height above the box where the ladder touches the wall. | [
"$0.9$ m and $1.6$ m"
] | [
"A cubic box with side length $1.2$ m is placed on the ground next to a high wall. A ladder of length $3.5$ m leans against the wall and just touches the top edge of the box. Determine the distance from the box to the foot of the ladder and the height above the box where the ladder touches the wall."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 25
} | 25 | 0 | 0 |
omr-ws-61ff45e3b5f3c1aebe5d98c953b95e2396f95be617823062bf3831581ad9f76e | A cubic equation has three roots which are perfect squares such that $a^2 + b^2 = c^2$, where $a^2$, $b^2$, and $c^2$ are the three roots. If the equation is $x^3 + px^2 + qx + r = 0$, find the relation that holds among $p, q$, and $r$. | [
"\\( q = \\frac{2r}{p} + \\frac{p^2}{4} \\)"
] | [
"A cubic equation has three roots which are perfect squares such that $a^2 + b^2 = c^2$, where $a^2$, $b^2$, and $c^2$ are the three roots. If the equation is $x^3 + px^2 + qx + r = 0$, find the relation that holds among $p, q$, and $r$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0ff243d13b32c04a13ae62e0b8caa6e236e5960bd13833e1e0c3f3b299a7e662 | A cubic function $f(x)$ satisfies the equation $\sin 3t = f(\sin t)$ for all real numbers $t$. Evaluate $\int_0^1 f(x)^2 \sqrt{1-x^2} \, dx$. | [
"\\(\\frac{\\pi}{8}\\)"
] | [
"A cubic function $f(x)$ satisfies the equation $\\sin 3t = f(\\sin t)$ for all real numbers $t$. Evaluate $\\int_0^1 f(x)^2 \\sqrt{1-x^2} \\, dx$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-49bc909d087f679964eebe2729c7490a0a99d81563f087e45c52a39181390586 | A cubic function \( f(x) \) has a minimum and a maximum at points \((g, h)\) and \((j, k)\), in no particular order. The function can be written in the form \( f(x) = ax^3 + bx^2 + cx + d \). Define \( a, b, c, \) and \( d \) (individually) in terms of \( g, h, j, \) and \( k \). | [
"\\dfrac{kg^3 - 3kg^2j + 3hgj^2 - hj^3}{(g - j)^3}"
] | [
"A cubic function \\( f(x) \\) has a minimum and a maximum at points \\((g, h)\\) and \\((j, k)\\), in no particular order. The function can be written in the form \\( f(x) = ax^3 + bx^2 + cx + d \\). Define \\( a, b, c, \\) and \\( d \\) (individually) in terms of \\( g, h, j, \\) and \\( k \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-979350f342ee2ca545be038f2eaebd52531a2ad836826ebc53740f9ceb2746e9 | A cubic function \( y = ax^3 + bx^2 + cx + d \) (where \( a \neq 0 \)) touches a line \( y = px + q \) at \( x = \alpha \) and intersects it at \( x = \beta \) (\( \alpha \neq \beta \)). Find the area of the region bounded by these graphs in terms of \( a, \alpha, \beta \). | [
"\\(\\frac{|a|}{12}(\\beta - \\alpha)^4\\)"
] | [
"A cubic function \\( y = ax^3 + bx^2 + cx + d \\) (where \\( a \\neq 0 \\)) touches a line \\( y = px + q \\) at \\( x = \\alpha \\) and intersects it at \\( x = \\beta \\) (\\( \\alpha \\neq \\beta \\)). Find the area of the region bounded by these graphs in terms of \\( a, \\alpha, \\beta \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-a1c1a2fe9276bba07e1c80047f52d33b4a138b2f64030d1aac1e713779049059 | A cubic lattice is formed by all possible planes parallel to the coordinate planes and intersecting the coordinate axes at integer points. The lattice is dissected by the plane \( ax + by + cz = 0 \), where \( a \), \( b \), and \( c \) are natural numbers that are coprime in the aggregate. The lines of intersection wi... | [
"5"
] | [
"A cubic lattice is formed by all possible planes parallel to the coordinate planes and intersecting the coordinate axes at integer points. The lattice is dissected by the plane \\( ax + by + cz = 0 \\), where \\( a \\), \\( b \\), and \\( c \\) are natural numbers that are coprime in the aggregate. The lines of in... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-74036c1ec62f0d80ae707322e12234230adb9f5a995cc856164a27e08f3de1c5 | A cubic polynomial $p$ has the property that its roots are all rational and its coefficients are all positive prime integers. If $p(1) = 144$, compute the least possible value of $p(2)$. | [
"369"
] | [
"A cubic polynomial $p$ has the property that its roots are all rational and its coefficients are all positive prime integers. If $p(1) = 144$, compute the least possible value of $p(2)$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 8,
"tir": 2
} | 15 | 0 | 0 |
omr-ws-8018209b610048247a1aec6893c23059f046a67e6007cb158f3e0abce7b087d8 | A cubic polynomial $p(x)$ satisfies \[p(n) = \frac{1}{n^2}\]for $n = 1, 2, 3,$ and $4.$ Find $p(5).$ | [
"-\\frac{5}{12}"
] | [
"A cubic polynomial $p(x)$ satisfies\n\\[p(n) = \\frac{1}{n^2}\\]for $n = 1, 2, 3,$ and $4.$ Find $p(5).$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-da5bed5ff300ff6fb1d6d9e384719a164f38c088b53be81b271a0631b1533b1b | A cubic polynomial has the equation \( f(x) = a(x + b)^2(x + c) \) for integers \( b \) and \( c \), and real number \( a \). The graph is tangent to the x-axis at \( x = -2 \) and \( b - c = 6 \). If \( f(6) = -c^2 \), then give the y-intercept of the graph of \( f \). | [
"2"
] | [
"A cubic polynomial has the equation \\( f(x) = a(x + b)^2(x + c) \\) for integers \\( b \\) and \\( c \\), and real number \\( a \\). The graph is tangent to the x-axis at \\( x = -2 \\) and \\( b - c = 6 \\). If \\( f(6) = -c^2 \\), then give the y-intercept of the graph of \\( f \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-cb49bb1502348aefc244b22afd5aa197637576ef851b9037aa43d7764602b936 | A cubic polynomial is defined as *elegant* if its three roots form the vertices of an equilateral triangle in the complex plane. Let \(a\), \(b\), and \(c\) be complex numbers. Consider the following three polynomials: $$ p(z) = z^3 + az^2 + bz + c, $$ $$ q(z) = z^3 + bz^2 + cz + a, $$ $$ r(z) = z^3 + cz^2 + az + b. $$... | [
"37."
] | [
"A cubic polynomial is defined as *elegant* if its three roots form the vertices of an equilateral triangle in the complex plane. Let \\(a\\), \\(b\\), and \\(c\\) be complex numbers. Consider the following three polynomials:\n$$\np(z) = z^3 + az^2 + bz + c,\n$$\n$$\nq(z) = z^3 + bz^2 + cz + a,\n$$\n$$\nr(z) = z^3 ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2f1b32babf400c78fce5348d22df04dfc97dfb40d0a35601a39cd66e85b998f3 | A cubic polynomial of the form $ax^3+bx^2+cx+d$ has real roots $r_1$, $r_2$, and $r_3$, with $r_1 \geq r_2 \geq r_3$. The centroid of the triangle with vertices at $(-1, r_1)$, $(0, r_2)$, and $(1, r_3)$ is the inflection point of the polynomial. What is the maximum number of terms of the function? | [
"2"
] | [
"A cubic polynomial of the form $ax^3+bx^2+cx+d$ has real roots $r_1$, $r_2$, and $r_3$, with $r_1 \\geq r_2 \\geq r_3$. The centroid of the triangle with vertices at $(-1, r_1)$, $(0, r_2)$, and $(1, r_3)$ is the inflection point of the polynomial. What is the maximum number of terms of the function?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-9d216ba08b13a30a54880c6879627a554faaa3c776620a67697f75aca8e4ba8c | A cubical box with sides of length $7$ has vertices at $(0,0,0)$, $(7,0,0)$, $(0,7,0)$, $(7,7,0)$, $(0,0,7)$, $(7,0,7)$, $(0,7,7)$, $(7,7,7)$. The inside of the box is lined with mirrors and from the point $(0,1,2)$, a beam of light is directed to the point $(1,3,4)$. The light then reflects repeatedly off the mirrors ... | [
"\\(\\sqrt{362}\\)"
] | [
"A cubical box with sides of length $7$ has vertices at $(0,0,0)$, $(7,0,0)$, $(0,7,0)$, $(7,7,0)$, $(0,0,7)$, $(7,0,7)$, $(0,7,7)$, $(7,7,7)$. The inside of the box is lined with mirrors and from the point $(0,1,2)$, a beam of light is directed to the point $(1,3,4)$. The light then reflects repeatedly off the mir... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c5545c754004286aa265532193b992d5b682cc9c94946a28e779fd10aaba749a | A cubical box with sides of length 7 has vertices at (0,0,0), (7,0,0), (0,7,0), (7,7,0), (0,0,7), (7,0,7), (0,7,7), (7,7,7). The inside of the box is lined with mirrors. A beam of light is directed from the point (0,1,2) to the point (1,3,4) and reflects repeatedly off the mirrors. Determine how far the beam of light t... | [
"42"
] | [
"A cubical box with sides of length 7 has vertices at (0,0,0), (7,0,0), (0,7,0), (7,7,0), (0,0,7), (7,0,7), (0,7,7), (7,7,7). The inside of the box is lined with mirrors. A beam of light is directed from the point (0,1,2) to the point (1,3,4) and reflects repeatedly off the mirrors. Determine how far the beam of li... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 28,
"genselect": 7,
"tir": 1
} | 36 | 0 | 0 |
omr-ws-0164affd6a12e562187907faf4ab4ce9519236b6a0e742e0b7dfb166c5be8504 | A cuboctahedron has 6 square faces and 8 equilateral triangle faces, with each edge adjacent to both a square and a triangle. Suppose the ratio of the volume of an octahedron to a cuboctahedron with the same side length is \( r \). Find \( 100r^2 \). | [
"4"
] | [
"A cuboctahedron has 6 square faces and 8 equilateral triangle faces, with each edge adjacent to both a square and a triangle. Suppose the ratio of the volume of an octahedron to a cuboctahedron with the same side length is \\( r \\). Find \\( 100r^2 \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-46b72b9826e6b61ac22bb95337976a784cf2da496b1e180dcaa52291eb6890f5 | A cuboid and a cube have the same volume and height. The length of the cuboid is three times the side length of the cube, and the width of the cuboid is half the side length of the cube. Calculate the longest diagonal of both the cube and the cuboid. | [
"Diagonal of cube: \\( x \\sqrt{3} \\), Diagonal of cuboid: \\( x \\sqrt{10.25} \\)"
] | [
"A cuboid and a cube have the same volume and height. The length of the cuboid is three times the side length of the cube, and the width of the cuboid is half the side length of the cube. Calculate the longest diagonal of both the cube and the cuboid."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 6
} | 7 | 0 | 0 |
omr-ws-17fef2dfbf8adf6e88e0bbb4e1febb442fbd1b4b2656c39fb31e2699a134538a | A cuboid has a surface area of $6$ and the sum of its twelve edge lengths is $20$. Determine the greatest length that one of its edges can have. | [
"\\dfrac{13}{3}"
] | [
"A cuboid has a surface area of $6$ and the sum of its twelve edge lengths is $20$. Determine the greatest length that one of its edges can have."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-86120ff755434ad2d0e9053d612910fe7e4b2566fe9c0a9ece33d663008f3cd4 | A cuboid has a volume of 42, a surface area of 80, and the sum of its edge lengths is 46. What are the dimensions of the cuboid? | [
"6"
] | [
"A cuboid has a volume of 42, a surface area of 80, and the sum of its edge lengths is 46. What are the dimensions of the cuboid?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-6e271be05ece30cbeda175196edbd57908b8cdaaa43661e082cfcfafae7d4ff8 | A cuboid has vertices $Z$ at $(0, 0, 0)$, $A$ at $(a, 0, 0)$, $Y$ at $(a, 0, c)$, and $X$ at $(0, b, c)$. Given that $XY = 9$, $XZ = 8$, and $YZ = \sqrt{55}$, what is the length of the diagonal $XA$? | [
"10"
] | [
"A cuboid has vertices $Z$ at $(0, 0, 0)$, $A$ at $(a, 0, 0)$, $Y$ at $(a, 0, c)$, and $X$ at $(0, b, c)$. Given that $XY = 9$, $XZ = 8$, and $YZ = \\sqrt{55}$, what is the length of the diagonal $XA$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-c7d5800062e5b18fd49d42c1bb0f131736545fab8ed65d6b1f55e009a4ed7779 | A cuboid with a square base has a diagonal of length $d$ and a surface area of $b$. Calculate the sum of the lengths of all edges of the cuboid. | [
"\\(6a + \\frac{b}{a}\\)"
] | [
"A cuboid with a square base has a diagonal of length $d$ and a surface area of $b$. Calculate the sum of the lengths of all edges of the cuboid."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f20a7b4502942b30ff5828ef1c15113f02f94512f6b4c6cd7c90c4c0f496f6d3 | A cuboidal container has dimensions 30 cm (length), 25 cm (breadth), and 15 cm (height). It is filled with water up to 5 cm from the bottom. A cubical piece of metal with a side length of 10 cm is placed in the container such that one of its faces touches the bottom. How much water must be added to the container so tha... | [
"2750"
] | [
"A cuboidal container has dimensions 30 cm (length), 25 cm (breadth), and 15 cm (height). It is filled with water up to 5 cm from the bottom. A cubical piece of metal with a side length of 10 cm is placed in the container such that one of its faces touches the bottom. How much water must be added to the container s... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 7,
"tir": 15
} | 33 | 0 | 0 |
omr-ws-16aee79bc83b9c937959f11cfc89a012d2c5370a2692284573a1d1d75f70601b | A cuboidal wall of dimensions $12 \times 18 \times 16$ is made up of $1 \times 1 \times 1$ bricks. A straight line fault appears in the wall which goes exactly through the right diagonal of the wall. How many bricks are damaged? | [
"36"
] | [
"A cuboidal wall of dimensions $12 \\times 18 \\times 16$ is made up of $1 \\times 1 \\times 1$ bricks. A straight line fault appears in the wall which goes exactly through the right diagonal of the wall. How many bricks are damaged?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-55d5da00c912cd0c96c933b6ce6940c5e51b88f2f8910be233474fbb649bbc00 | A cue ball is shot out of the corner of a $3 \times 1001$ pool table at a $45$ degree angle. Which corner of the pool table will the ball reach first, and how many times will it hit the sides until it does so? | [
"1002"
] | [
"A cue ball is shot out of the corner of a $3 \\times 1001$ pool table at a $45$ degree angle. Which corner of the pool table will the ball reach first, and how many times will it hit the sides until it does so?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 2,
"tir": 2
} | 12 | 0 | 0 |
omr-ws-81f5d215248655f45e1646bdb2bac6cb227db66b54a10a1a3313915e18b005da | A cue ball is shot out of the corner of a $3 \times 5$ pool table at a $45$ degree angle. Which corner of the pool table will the ball reach first, and how many times will it hit the sides until it does so? | [
"7"
] | [
"A cue ball is shot out of the corner of a $3 \\times 5$ pool table at a $45$ degree angle. Which corner of the pool table will the ball reach first, and how many times will it hit the sides until it does so?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-45f0948c9d451a1784816b62d1a9882b219360a949c1eed96875fbe4ce9e1782 | A cult leader converts a disciple in January. Each subsequent month, each disciple converts either 1 or 2 more disciples. What is the expected number of adherents at the end of December? | [
"\\(\\left(\\frac{5}{2}\\right)^{11}\\)"
] | [
"A cult leader converts a disciple in January. Each subsequent month, each disciple converts either 1 or 2 more disciples. What is the expected number of adherents at the end of December?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 21
} | 36 | 0 | 0 |
omr-ws-e4c4993b104e57c76cd7c739deb13dc80aae847097f7fdb601f2e85cc89d071c | A cultivator has $100 and wants to buy exactly 100 animals. The prices are as follows: goats cost $0.50 each, pigs cost $2.00 each, and turkeys cost $0.10 each. How many combinations of these animals can the cultivator buy to get exactly 100 animals for exactly $100? | [
"1\n\nThe forum discussion indicates that there is only one solution to the problem: 44 pigs, 16 goats, and 40 turkeys. Therefore, there is only one combination of animals that the cultivator can buy to get exactly 100 animals for exactly $100."
] | [
"A cultivator has $100 and wants to buy exactly 100 animals. The prices are as follows: goats cost $0.50 each, pigs cost $2.00 each, and turkeys cost $0.10 each. How many combinations of these animals can the cultivator buy to get exactly 100 animals for exactly $100?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-22908aebac50118fc31c287d10b22b7974fe0cbe26b395d09b8b6c3ec992520f | A cultural event will be held from June 24 to August 26. Concerts will take place on different dates, with no two consecutive days and no two days having the same total number of days and months (e.g., $25/6$ and $24/7$ both sum to 31). What is the maximum number of concerts that can take place during this event? | [
"14"
] | [
"A cultural event will be held from June 24 to August 26. Concerts will take place on different dates, with no two consecutive days and no two days having the same total number of days and months (e.g., $25/6$ and $24/7$ both sum to 31). What is the maximum number of concerts that can take place during this event?"... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1e8d6266524583a0612ca3faa4f10f0f32fc47a57398d37c08dcd13fab14f381 | A culture of bacteria doubles in size every 12 hours. If a dish will be full with 1,000,000 bacteria after 13 days, how long will it take to fill a dish whose capacity is 2,000,000 bacteria (in days)? | [
"13.5 days"
] | [
"A culture of bacteria doubles in size every 12 hours. If a dish will be full with 1,000,000 bacteria after 13 days, how long will it take to fill a dish whose capacity is 2,000,000 bacteria (in days)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 8
} | 23 | 0 | 0 |
omr-ws-a27be58b08b1a75a7e50e61b89dea2516bf6927a8d6acbd0fe0d717e0cffe739 | A cup of boiling water ($212^{\circ}\text{F}$) is placed to cool in a room whose temperature remains constant at $68^{\circ}\text{F}$. The difference between the water temperature and the room temperature is halved every $5$ minutes. What is the water temperature, in degrees Fahrenheit, after $15$ minutes? | [
"86"
] | [
"A cup of boiling water ($212^{\\circ}\\text{F}$) is placed to cool in a room whose temperature remains constant at $68^{\\circ}\\text{F}$. The difference between the water temperature and the room temperature is halved every $5$ minutes. What is the water temperature, in degrees Fahrenheit, after $15$ minutes?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-cba0176c03d870d0e6a76f96784f62243b8e3f774112870fbb2bf22caa8b3dac | A cup of coffee has a temperature of 200°F and is in a room that has a temperature of 70°F. After 10 minutes, the temperature of the coffee is 150°F. a) What is the temperature of the coffee after 15 minutes? b) When will the coffee have cooled to 100°F? | [
"30.2"
] | [
"A cup of coffee has a temperature of 200°F and is in a room that has a temperature of 70°F. After 10 minutes, the temperature of the coffee is 150°F.\na) What is the temperature of the coffee after 15 minutes?\nb) When will the coffee have cooled to 100°F?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-641a0f8caffbb9dc7c958b61099e23157e4a9d73028f618c07bfa2d9489a4041 | A curve $C$ is determined by the parametric equations $x = at^2$ and $y = 2at$, where $a > 0$. Let $P$ be a point on the curve with parameter $t = p$. Calculate the value of $p^2$ that minimizes $|PN|^2$, where $N$ is the point $(4a, 0)$. | [
"2"
] | [
"A curve $C$ is determined by the parametric equations $x = at^2$ and $y = 2at$, where $a > 0$. Let $P$ be a point on the curve with parameter $t = p$. Calculate the value of $p^2$ that minimizes $|PN|^2$, where $N$ is the point $(4a, 0)$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4b6bd22572f126eaa0835a95930cef6dc115b9d45caa1fc21cf94b5276b6b071 | A curve $C$ is determined by the parametric equations $x = at^2$ and $y = 2at$, where $a > 0$. The normal to $C$ at a point $P$ with non-zero parameter $p$ meets $C$ again at a point $N$ with parameter $n$. If $p = 3$, find the value of $n$. | [
"-\\dfrac{11}{3}"
] | [
"A curve $C$ is determined by the parametric equations $x = at^2$ and $y = 2at$, where $a > 0$. The normal to $C$ at a point $P$ with non-zero parameter $p$ meets $C$ again at a point $N$ with parameter $n$. If $p = 3$, find the value of $n$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ef8e461356fcfb322376da02ab63ea8cfd1e9954f9334f5c5e3a592b1c767a8b | A curve $C$ is determined by the parametric equations $x = at^2$ and $y = 2at$, where $a > 0$. The point $Q$ with parameter $q$ is the point at which the circle with diameter $PN$ cuts $C$ again. Given that $P$ has parameter $p$, calculate the value of $p^2 - q^2 + \frac{2q}{p}$ and find the length $|PN|$ when $Q$ is a... | [
"no_valid_answer_found"
] | [
"A curve $C$ is determined by the parametric equations $x = at^2$ and $y = 2at$, where $a > 0$. The point $Q$ with parameter $q$ is the point at which the circle with diameter $PN$ cuts $C$ again. Given that $P$ has parameter $p$, calculate the value of $p^2 - q^2 + \\frac{2q}{p}$ and find the length $|PN|$ when $Q... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5d5951e863850d3ceab3b6c8182cb9122faa5a8b49b71be884e98f9e1315a857 | A curve $C$ lies in the first quadrant. The tangent at any point on $C$ meets the positive directions of the coordinate axes at points $A$ and $B$. Let $d$ be the minimum distance of the curve $C$ from the origin $O$. If $AB = 1$, find the value of $d$. | [
"\\(\\frac{\\sqrt{2}}{2}\\)"
] | [
"A curve $C$ lies in the first quadrant. The tangent at any point on $C$ meets the positive directions of the coordinate axes at points $A$ and $B$. Let $d$ be the minimum distance of the curve $C$ from the origin $O$. If $AB = 1$, find the value of $d$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6fcb5807a6982b928eb5b8a4cf297a2d0117d69bf168cdffcc5116ab562a5ea3 | A curve $C$ lies in the first quadrant. The tangent at any point on $C$ meets the positive directions of the coordinate axes at points $A$ and $B$. Let $d$ be the minimum distance of the curve $C$ from the origin $O$. If $OA + OB = 1$, find the numerical value of $d$. | [
"\\(\\frac{\\sqrt{2}}{2}\\)"
] | [
"A curve $C$ lies in the first quadrant. The tangent at any point on $C$ meets the positive directions of the coordinate axes at points $A$ and $B$. Let $d$ be the minimum distance of the curve $C$ from the origin $O$. If $OA + OB = 1$, find the numerical value of $d$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5991e0c89bf328f9379bb9dbcd0965b8244b6b9c79e51320e67ca8f1e9998745 | A curve $C$ lies in the first quadrant. The tangent at any point on $C$ meets the positive directions of the coordinate axes at points $A$ and $B$. Let $d$ be the minimum distance of the curve $C$ from the origin $O$. If $OA \cdot OB = 4$, find the value of $d$. | [
"\\( 2\\sqrt{2} \\)"
] | [
"A curve $C$ lies in the first quadrant. The tangent at any point on $C$ meets the positive directions of the coordinate axes at points $A$ and $B$. Let $d$ be the minimum distance of the curve $C$ from the origin $O$. If $OA \\cdot OB = 4$, find the value of $d$."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-3e90ffb7518adbbd518182fa71b4c2a931331a0de13e745845bb75dc3c41d654 | A curve $y = f(x)$ passes through the origin and the slope of the tangent at any point $(x, y)$ of the curve is given by $\frac{x^4 + 2xy - 1}{1 + x^2}$. Determine the value of the greatest integer less than or equal to $f(-1)$. | [
"1"
] | [
"A curve $y = f(x)$ passes through the origin and the slope of the tangent at any point $(x, y)$ of the curve is given by $\\frac{x^4 + 2xy - 1}{1 + x^2}$. Determine the value of the greatest integer less than or equal to $f(-1)$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-40d249d0c75760e6cae664810c67177732eb19f008dae4994f2e3fd61b3c6e39 | A curve \( C \) has the property that the slope of the tangent at any given point \((x, y)\) on \( C \) is \(\frac{x^2 + y^2}{2xy}\). Find the general equation for such a curve. | [
"\\( y^2 = x^2 + ax \\)"
] | [
"A curve \\( C \\) has the property that the slope of the tangent at any given point \\((x, y)\\) on \\( C \\) is \\(\\frac{x^2 + y^2}{2xy}\\). Find the general equation for such a curve."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-28efe85232e566be52f4f819ba587be3945c9e233478e5862d75c9f1fed6ac35 | A curve \( C \) which is not a straight line lies in the first quadrant. The tangent at any point on \( C \) meets the positive directions of the coordinate axes at points \( A \) and \( B \). Let \( d \) be the minimum distance of the curve \( C \) from the origin \( O \). If \( OA + OB = 1 \), then find \( d \). | [
"\\(\\frac{1}{2\\sqrt{2}}\\)"
] | [
"A curve \\( C \\) which is not a straight line lies in the first quadrant. The tangent at any point on \\( C \\) meets the positive directions of the coordinate axes at points \\( A \\) and \\( B \\). Let \\( d \\) be the minimum distance of the curve \\( C \\) from the origin \\( O \\). If \\( OA + OB = 1 \\), th... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-3ee13b4d2f4e9717463016ebdfd9f386dff2dca8cad7fd3f79a4281f2d87a62e | A curve in the $XY$ plane is given by the parametric equations $x=t^2+t+1$, $y=t^2-t+1$, where the parameter $t$ varies over all non-negative real numbers. Find the number of straight lines passing through the point $(1,1)$ which are tangent to the curve. | [
"1"
] | [
"A curve in the $XY$ plane is given by the parametric equations $x=t^2+t+1$, $y=t^2-t+1$, where the parameter $t$ varies over all non-negative real numbers. Find the number of straight lines passing through the point $(1,1)$ which are tangent to the curve."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-eaa59f5d00273ce8c6587478550f1868b6d3920a8d119ce46bebd2fd7b2ddbb1 | A curve in the plane is defined parametrically by the equations \( x = t^3 + t \) and \( y = t^4 + 2t^2 \). Find the equation of the line tangent to the curve at \( t = 1 \). | [
"\\( y = 2x - 1 \\)"
] | [
"A curve in the plane is defined parametrically by the equations \\( x = t^3 + t \\) and \\( y = t^4 + 2t^2 \\). Find the equation of the line tangent to the curve at \\( t = 1 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-5e174b1bab8cade1072a7e5fce9d743c693737e4944104ce11aa51bc092b200c | A curve is defined by $x(t)=t^2-2t$ and $y(t)=-t^3+t^2$ for all $t \in \mathbb{R}$. Let $R$ be the region enclosed by the loop of the curve. Find the centroid of $R$. | [
"\\left( -\\dfrac{5}{14}, -\\dfrac{9}{7} \\right)"
] | [
"A curve is defined by $x(t)=t^2-2t$ and $y(t)=-t^3+t^2$ for all $t \\in \\mathbb{R}$. Let $R$ be the region enclosed by the loop of the curve. Find the centroid of $R$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-46a2c737269e7d2e1da6653de54ac54dd41e9844379a82903aae48320acb22c6 | A curve is defined by $x(t)=t^2-2t$ and $y(t)=-t^3+t^2$ for all $t \in \mathbb{R}$. Let $R$ be the region enclosed by the loop of the curve. If $R$ is rotated about the line $x = -1$, find the volume of the resulting solid. | [
"\\(\\frac{8\\pi}{5}\\)"
] | [
"A curve is defined by $x(t)=t^2-2t$ and $y(t)=-t^3+t^2$ for all $t \\in \\mathbb{R}$. Let $R$ be the region enclosed by the loop of the curve. If $R$ is rotated about the line $x = -1$, find the volume of the resulting solid."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6524344804b4e0945ffbb14b62404dd7f28ea6081d7b985746db87cdba33bb7f | A curve is defined parametrically by $x=t^{2}$ and $y=t(1+t^{2})$. The tangent at the point with parameter $t$, where $t \neq 0$, meets the curve again at the point with parameter $T$, where $T \neq t$. Find the value of $T$ in terms of $t$ and calculate the sum of all possible values of $t$ for which $3t^{2} = 1$. | [
"0"
] | [
"A curve is defined parametrically by $x=t^{2}$ and $y=t(1+t^{2})$. The tangent at the point with parameter $t$, where $t \\neq 0$, meets the curve again at the point with parameter $T$, where $T \\neq t$. Find the value of $T$ in terms of $t$ and calculate the sum of all possible values of $t$ for which $3t^{2} = ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9218e964fcc9472d1ea7c59dda841e07a89701098e80a5205da92f3141639baa | A curve is described parametrically by \[(x,y) = (2 \cos t - \sin t, 4 \sin t).\]The graph of the curve can be expressed in the form \[ax^2 + bxy + cy^2 = 1.\]Enter the ordered triple $(a,b,c).$ | [
"\\left( \\frac{1}{4}, \\frac{1}{8}, \\frac{5}{64} \\right)"
] | [
"A curve is described parametrically by\n\\[(x,y) = (2 \\cos t - \\sin t, 4 \\sin t).\\]The graph of the curve can be expressed in the form\n\\[ax^2 + bxy + cy^2 = 1.\\]Enter the ordered triple $(a,b,c).$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0ad404ac2d5c21e6789863b20c687e101407c5bfbd8ce5b50a2daee3119a29b6 | A curve is given by \( y = x^2 \). Point \( R \) is at \((-1, 0)\). Point \( P(s, 0) \) is a variable point on the x-axis, and \( s \) is increasing at a rate of 5 units per second. Point \( Q \) is on the curve such that \( PQ \) is perpendicular to the x-axis. Calculate the rate of increase of the length of \( QR \) ... | [
"19"
] | [
"A curve is given by \\( y = x^2 \\). Point \\( R \\) is at \\((-1, 0)\\). Point \\( P(s, 0) \\) is a variable point on the x-axis, and \\( s \\) is increasing at a rate of 5 units per second. Point \\( Q \\) is on the curve such that \\( PQ \\) is perpendicular to the x-axis. Calculate the rate of increase of the ... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0591dc4d2d2ae96a4d14b117614bbf869ccbedd4e570c4727406db45bada75f6 | A curve is traced by a point \( P(x, y) \) which moves such that its distance from the point \( A(-1, 1) \) is three times its distance from the point \( B(2, -1) \). Determine the equation of the curve. | [
"\\(8y^2 + 20y = -8x^2 + 38x - 43\\)"
] | [
"A curve is traced by a point \\( P(x, y) \\) which moves such that its distance from the point \\( A(-1, 1) \\) is three times its distance from the point \\( B(2, -1) \\). Determine the equation of the curve."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 3
} | 9 | 0 | 0 |
omr-ws-a6b8491b595a20aa5d9931554cba48cd13b35125aaf3c4d7c11b552b43819974 | A curve passes through $(0, 1)$ and satisfies the differential equation $\frac{dy}{dx} = \sqrt{4-y^{2}}$. Find the equation of the curve and sketch its graph. | [
"\\( y(x) = 2\\sin\\left(x + \\frac{\\pi}{6}\\right) \\) for \\( -\\frac{\\pi}{2} \\leq x + \\frac{\\pi}{6} \\leq \\frac{\\pi}{2} \\) which simplifies to \\( -\\frac{2\\pi}{3} \\leq x \\leq \\frac{\\pi}{3} \\). Beyond these intervals, the solution is \\( y = 2 \\) for \\( x > \\frac{\\pi}{3} \\) and \\( y = -2 \\) ... | [
"A curve passes through $(0, 1)$ and satisfies the differential equation $\\frac{dy}{dx} = \\sqrt{4-y^{2}}$. Find the equation of the curve and sketch its graph."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 11
} | 11 | 0 | 0 |
omr-ws-0fc9ca2abc8b94d3163e3fb307e4a6cf2bf194f21d4292a3c11944767c118de9 | A curve with Cartesian equation \( y = f(x) \) passes through the origin. Lines drawn parallel to the coordinate axes through an arbitrary point of the curve form a rectangle with two sides on the axes. The curve divides every such rectangle into two regions \( A \) and \( B \), one of which has an area equal to \( n \... | [
"y = C x^{\\frac{1}{n}}"
] | [
"A curve with Cartesian equation \\( y = f(x) \\) passes through the origin. Lines drawn parallel to the coordinate axes through an arbitrary point of the curve form a rectangle with two sides on the axes. The curve divides every such rectangle into two regions \\( A \\) and \\( B \\), one of which has an area equa... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-c1c56f4f6aa713a7ed2a1e9c00b11d997e0a539ca0200cfde058da2e5cdc8d6a | A customer bought 4 items at a 7-11 store, and the total price was $7.11. The cashier mentioned that the prices were both multiplied and added together to get $7.11. What are the 4 prices, given that they are in perfect dollars and cents? | [
"3.16, 1.20, 1.25, 1.50"
] | [
"A customer bought 4 items at a 7-11 store, and the total price was $7.11. The cashier mentioned that the prices were both multiplied and added together to get $7.11. What are the 4 prices, given that they are in perfect dollars and cents?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 18
} | 18 | 0 | 0 |
omr-ws-a2b0704d4c463d033d3fe93b596b2368a9d73b75f1111b4dbf8517a52a025251 | A customer brings $2 to a bank for change into 20 cent, 10 cent, and 5 cent coins. The number of 10 cent coins is two-thirds the number of 20 cent coins, and the number of 5 cent coins is two more than the number of 20 cent coins. How many coins of each type should the teller give the customer? | [
"20 cent coins: 6, 10 cent coins: 4, 5 cent coins: 8"
] | [
"A customer brings $2 to a bank for change into 20 cent, 10 cent, and 5 cent coins. The number of 10 cent coins is two-thirds the number of 20 cent coins, and the number of 5 cent coins is two more than the number of 20 cent coins. How many coins of each type should the teller give the customer?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 26
} | 33 | 0 | 0 |
omr-ws-591dcbb9412d44bd26bde0fc2aed871a8c029899a8526b20db53246febf797e3 | A customer deposited in a savings bank 1640 roubles at the beginning of a year and withdrew 882 roubles at the end of the year. A year later he had 882 roubles on his account. How many per cent a year does the savings bank add? | [
"5\\%"
] | [
"A customer deposited in a savings bank 1640 roubles at the beginning of a year and withdrew 882 roubles at the end of the year. A year later he had 882 roubles on his account. How many per cent a year does the savings bank add?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-15f3fe14c9b0f9e062d33b0cf8bde4e33421f674f049eb289e789569561a6e6b | A customer paid 2 roubles for a kg of one product and ten kg of another product. If after the seasonal change in prices the first product goes up in price by 15\% and the other product becomes cheaper by 25\%, then the customer will pay 1 rouble and 82 kopecks for the same amount of products. What is the cost of a kg o... | [
"12 kopecks, 80 kopecks"
] | [
"A customer paid 2 roubles for a kg of one product and ten kg of another product. If after the seasonal change in prices the first product goes up in price by 15\\% and the other product becomes cheaper by 25\\%, then the customer will pay 1 rouble and 82 kopecks for the same amount of products. What is the cost of... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bc9f1c0a9591c2574fba7d6d9c641feb2ef563b51669667d53052b8ed9509c10 | A cutter sets out down the river at the same time as a raft in the same direction and travels down the river for $13\frac{1}{3}$ km, and then, without stopping, it travels $9\frac{1}{3}$ km in the reverse direction and meets the raft. How many times is the inherent speed of the cutter higher than the speed of the river... | [
"$\\frac{17}{3}$"
] | [
"A cutter sets out down the river at the same time as a raft in the same direction and travels down the river for $13\\frac{1}{3}$ km, and then, without stopping, it travels $9\\frac{1}{3}$ km in the reverse direction and meets the raft. How many times is the inherent speed of the cutter higher than the speed of th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2881711b848a548f003b77453540711f7b7572b7bf64872095d6edba8a473c44 | A cutting plane divides the lateral edges of a triangular pyramid in the ratios (as measured from the vertex): $\frac{m_1}{n_1}$, $\frac{m_2}{n_2}$, $\frac{m_3}{n_3}$. In what ratio is the volume of the pyramid divided by this plane? | [
"$\\displaystyle\\frac{V_1}{V-V_1}=\\displaystyle\\frac{m_1m_2m_3}{(m_1+n_1)(m_2+n_2)(m_3+n_3)-m_1m_2m_3}$"
] | [
"A cutting plane divides the lateral edges of a triangular pyramid in the ratios (as measured from the vertex): $\\frac{m_1}{n_1}$, $\\frac{m_2}{n_2}$, $\\frac{m_3}{n_3}$. In what ratio is the volume of the pyramid divided by this plane?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bf149bc65c805587a6b29b9ac7484a6aaada0d4298e09e2a1b2cd8b2a98602d5 | A cyclic $n$-gon is divided by non-intersecting (inside the $n$-gon) diagonals into $n-2$ triangles. Each of these triangles is similar to at least one of the remaining ones. For what $n$ is this possible? | [
"4"
] | [
"A cyclic $n$-gon is divided by non-intersecting (inside the $n$-gon) diagonals into $n-2$ triangles. Each of these triangles is similar to at least one of the remaining ones. For what $n$ is this possible?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-5925b2620bc349f114e8af10748f1a54331dceb0f3f414db279ea17d84f80cb5 | A cyclic group of order 15 has an element \( x \) such that the set \( \{x^3, x^5, x^9\} \) has exactly two elements. How many elements are in the set \( \{x^{13n} : n \text{ is a positive integer} \} \)? | [
"3"
] | [
"A cyclic group of order 15 has an element \\( x \\) such that the set \\( \\{x^3, x^5, x^9\\} \\) has exactly two elements. How many elements are in the set \\( \\{x^{13n} : n \\text{ is a positive integer} \\} \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-838d12bd7421365538f7a5841168399208299f6e773b662b7bd8878a6beceeef | A cyclic hexagon has five sides of length 81 and one side of length 31. Find the sum of the three diagonals from a vertex on the side of length 31. | [
"384"
] | [
"A cyclic hexagon has five sides of length 81 and one side of length 31. Find the sum of the three diagonals from a vertex on the side of length 31."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 5
} | 20 | 0 | 0 |
omr-ws-77f0a072c76313de1d2d3403c355e5a6e31c7024afdd3a93f51888162cd0d10b | A cyclic hexagon has side lengths $3, 3, 3, 5, 5, 5$ in clockwise order. The circle circumscribing it has a chord with length $X$ which splits the hexagon into trapezoids with side lengths $3, 3, 3, X$ and $5, 5, 5, X$. Find $X$. | [
"\\(\\frac{360}{49}\\)"
] | [
"A cyclic hexagon has side lengths $3, 3, 3, 5, 5, 5$ in clockwise order. The circle circumscribing it has a chord with length $X$ which splits the hexagon into trapezoids with side lengths $3, 3, 3, X$ and $5, 5, 5, X$. Find $X$."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 27
} | 41 | 0 | 0 |
omr-ws-508107ad0049f27e5ea635e2610353c03b7ae5c68d3600bd5392f7780eb8a724 | A cyclic hexagon has side lengths 2, 2, 7, 7, 11, and 11, in that order. Find the length of its circumradius. | [
"7"
] | [
"A cyclic hexagon has side lengths 2, 2, 7, 7, 11, and 11, in that order. Find the length of its circumradius."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2ed5aa2867a95fa4c097eeb52c96f9a91b4d040a1b1d009a0707d3a565de2fef | A cyclic hexagon has side lengths of 2, 2, 7, 7, 11, and 11, in that order. Find the length of its circumradius. | [
"7"
] | [
"A cyclic hexagon has side lengths of 2, 2, 7, 7, 11, and 11, in that order. Find the length of its circumradius."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-3aa5a1938ce89754b09247e3e083ed67760a0526cf041bbf18e084f3eb98e173 | A cyclic quadrilateral \( ABCD \) is inscribed in a circle \( O \) with radius 1. Given \( 136AB = 255BC = 120AC \) and \( BD \perp AC \), compute the length of \( BD \). | [
"\\(\\frac{480}{289}\\)"
] | [
"A cyclic quadrilateral \\( ABCD \\) is inscribed in a circle \\( O \\) with radius 1. Given \\( 136AB = 255BC = 120AC \\) and \\( BD \\perp AC \\), compute the length of \\( BD \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 16
} | 31 | 0 | 0 |
omr-ws-5da8d595c18fd8c3f739a7a2069e96bc74f2edbb8e20ef28d7f663ec06e2135f | A cyclic quadrilateral has sides of lengths 1, 4, 8, and 7. Find the radius of the circumscribed circle. | [
"\\dfrac{\\sqrt{65}}{2}"
] | [
"A cyclic quadrilateral has sides of lengths 1, 4, 8, and 7. Find the radius of the circumscribed circle."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-3588da5e635e86ade718010553d2efab683757b6d9e4d3453f347ecc0f70f7e3 | A cyclic quadrilateral is cut along one of its diagonals. The two angles not cut by any diagonal have measures 60° and 120°. What is the maximum ratio of the area of the smaller triangle to the area of the bigger triangle? | [
"\\dfrac{1}{3}"
] | [
"A cyclic quadrilateral is cut along one of its diagonals. The two angles not cut by any diagonal have measures 60° and 120°. What is the maximum ratio of the area of the smaller triangle to the area of the bigger triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 4
} | 11 | 0 | 0 |
omr-ws-c623881dca107d26486083a205fb5247ef813d3973a61000e2e0a5bc36037eeb | A cyclic quadrilateral with vertices on the Cartesian plane is called $good$ if at least two of its sides are parallel to each other and have slopes of $1$, $-1$, $0$, or $\infty$. Find an expression for the number of $good$ quadrilaterals formed by joining the points in the set $S$, where $S=\big\{(x,y):x,y \in \mathb... | [
"\\left( \\dfrac{n(n-1)}{2} \\right)^2 + \\dfrac{n(n-1)(n-2)}{6}"
] | [
"A cyclic quadrilateral with vertices on the Cartesian plane is called $good$ if at least two of its sides are parallel to each other and have slopes of $1$, $-1$, $0$, or $\\infty$. Find an expression for the number of $good$ quadrilaterals formed by joining the points in the set $S$, where $S=\\big\\{(x,y):x,y \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 5
} | 7 | 0 | 0 |
omr-ws-9e166e367d68463412fec781bc6430102f79ce03bc237902115a958ecc4ba151 | A cyclic trapezoid $ABCD$ has an area of $7$ $cm^2$ and a perimeter of $12$ $cm$. Given that $AB=CD$ and $AD$ is the diameter of the circumscribed circle, calculate the dimensions of the trapezoid. | [
"\\( AD \\approx 5.5604353 \\), \\( BC \\approx 5.4771447 \\), \\( AB \\approx 0.4813 \\), \\( CD \\approx 0.4813 \\)"
] | [
"A cyclic trapezoid $ABCD$ has an area of $7$ $cm^2$ and a perimeter of $12$ $cm$. Given that $AB=CD$ and $AD$ is the diameter of the circumscribed circle, calculate the dimensions of the trapezoid."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3798446cf1fe8820cf09549abf19029eafbb024ef1c47f926b6cbffb6b05485a | A cycling group that has $4n$ members will have several cycling events, such that: a) Two cycling events are done every week; once on Saturday and once on Sunday. b) Exactly $2n$ members participate in any cycling event. c) No member may participate in both cycling events of a week. d) After all cycling events are comp... | [
"0"
] | [
"A cycling group that has $4n$ members will have several cycling events, such that:\na) Two cycling events are done every week; once on Saturday and once on Sunday.\nb) Exactly $2n$ members participate in any cycling event.\nc) No member may participate in both cycling events of a week.\nd) After all cycling events... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e7451df2ba59fe6f1cf292f03d01d71b39b1b2d7d4fe47300b6658c3ea5c3b56 | A cyclist and a bus simultaneously left points $A$ and $B$ to meet each other. It takes the cyclist 2 hours and 40 minutes more to go from $A$ to $B$ than the bus to go from $B$ to $A$, and the sum of the times they take is $\frac{16}{3}$ times the time it takes for them to meet after starting. How much time does it ta... | [
"4 hours and $\\frac{4}{3}$ hours"
] | [
"A cyclist and a bus simultaneously left points $A$ and $B$ to meet each other. It takes the cyclist 2 hours and 40 minutes more to go from $A$ to $B$ than the bus to go from $B$ to $A$, and the sum of the times they take is $\\frac{16}{3}$ times the time it takes for them to meet after starting. How much time does... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d6aa5c60b778f5346bf83b4b916ab3c6661b4b44a437ac070975e181210e3fc3 | A cyclist and a pedestrian left simultaneously point $A$ for point $B$ which is 100 km from $A$. At the same time a car-driver started towards them from point $B$. An hour after the start, the car met the bicycle and then, having covered another 240/17 km, it met the pedestrian. The car-driver reversed the car, took th... | [
"The speed of the cyclist is 20 km/h and that of the car is 80 km/h"
] | [
"A cyclist and a pedestrian left simultaneously point $A$ for point $B$ which is 100 km from $A$. At the same time a car-driver started towards them from point $B$. An hour after the start, the car met the bicycle and then, having covered another 240/17 km, it met the pedestrian. The car-driver reversed the car, t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-11a68f521d96d9f152fde629df01f7824af6281fa219964787832d22624d5f9a | A cyclist covers half the distance between points $A$ and $B$ two hours quicker than a pedestrian covers a third of that distance. During the time needed by the cyclist to cover the whole way from $A$ to $B$, the pedestrian walks 24 km. If the cyclist increased his speed by 7 km/h, then during the time the pedestrian w... | [
"6 km/h"
] | [
"A cyclist covers half the distance between points $A$ and $B$ two hours quicker than a pedestrian covers a third of that distance. During the time needed by the cyclist to cover the whole way from $A$ to $B$, the pedestrian walks 24 km. If the cyclist increased his speed by 7 km/h, then during the time the pedest... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3f347b0032947e37c49160f81d50a2eef2a479e9c69df696072e2d95aeb3676a | A cyclist has to make a trip of 30 km. He leaves 3 minutes late, but travels 1 km/h faster and arrives in time. Determine the speed of the cyclist. | [
"25 km/h"
] | [
"A cyclist has to make a trip of 30 km. He leaves 3 minutes late, but travels 1 km/h faster and arrives in time. Determine the speed of the cyclist."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9ec6e5d93ed39e4ea6a45f74701cde37acf2511b64356f28d6afd9e848fe5969 | A cyclist left $A$ for $B$. Simultaneously, a motor-scooter started from $B$ and met the cyclist after 45 minutes. How much time does it take the cyclist to ride from $A$ to $B$ if the motor-scooter can travel the same distance 2 hours faster? | [
"3 hours"
] | [
"A cyclist left $A$ for $B$. Simultaneously, a motor-scooter started from $B$ and met the cyclist after 45 minutes. How much time does it take the cyclist to ride from $A$ to $B$ if the motor-scooter can travel the same distance 2 hours faster?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2a586882719ef6db3d4fbfb7eaef8a0f0ded2e18a2bb106a6cd37dfd797ba705 | A cyclist left point $A$ for point $B$, which are 70 km apart; some time later a motor-cyclist followed him having also started from point $A$ and traveled at 50 km/h. The motor-cyclist caught up with the cyclist 20 km away from point $A$. He reached point $B$ and 48 minutes later turned back toward $A$. He again came ... | [
"25 km/h"
] | [
"A cyclist left point $A$ for point $B$, which are 70 km apart; some time later a motor-cyclist followed him having also started from point $A$ and traveled at 50 km/h. The motor-cyclist caught up with the cyclist 20 km away from point $A$. He reached point $B$ and 48 minutes later turned back toward $A$. He again ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-345de1094d027337c2eb2da5b7530382283ec2442880b81abe3f94ff05e47097 | A cyclist left point $A$ for point $B$. At the time he covered 1/4 of the distance from $A$ to $B$ a motor-cyclist left $B$ for $A$ and having reached point $A$, started back without delay and arrived at $B$ at the same time as the cyclist. The time it took the motor-cyclist till his first meeting with the cyclist is e... | [
"The speed of the motorcyclist is 4 times as high as that of the cyclist"
] | [
"A cyclist left point $A$ for point $B$. At the time he covered 1/4 of the distance from $A$ to $B$ a motor-cyclist left $B$ for $A$ and having reached point $A$, started back without delay and arrived at $B$ at the same time as the cyclist. The time it took the motor-cyclist till his first meeting with the cycl... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-41582b540157c93561c3390723f6ef020a4bbc4861a6e04fec545cd259d2a167 | A cyclist makes one complete counter-clockwise revolution around the surface given by \( x^2 + y^2 + z = 2\pi \), starting at \( \left(\sqrt{2\pi}, 0, 0\right) \) and ending at \( \left(0, 0, 2\pi\right) \). The vertical rate of climb is constant. Find the parametrization of this path. | [
"\\(\\mathbf{r} = \\left< \\sqrt{2\\pi - t} \\cos t, \\sqrt{2\\pi - t} \\sin t, t \\right> \\quad \\text{for} \\quad t \\in [0, 2\\pi]\\)"
] | [
"A cyclist makes one complete counter-clockwise revolution around the surface given by \\( x^2 + y^2 + z = 2\\pi \\), starting at \\( \\left(\\sqrt{2\\pi}, 0, 0\\right) \\) and ending at \\( \\left(0, 0, 2\\pi\\right) \\). The vertical rate of climb is constant. Find the parametrization of this path."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 4,
"tir": 12
} | 25 | 0 | 0 |
omr-ws-9d0dcd0ce8323c209d0d687ffc7249e0b8af3bb627af1172e2c6517e7e422e43 | A cyclist moving at 4 m/s stops pedaling and lets the wheels come to rest. The retardation of the cycle has two components: a constant $0.08 \, \text{m/s}^2$ due to friction and a resistance of $0.02v^2 \, \text{m/s}^2$, where $v$ is the speed in meters per second. What distance is traversed by the cycle before it come... | [
"\\( 25 \\times \\ln 5 \\) meters"
] | [
"A cyclist moving at 4 m/s stops pedaling and lets the wheels come to rest. The retardation of the cycle has two components: a constant $0.08 \\, \\text{m/s}^2$ due to friction and a resistance of $0.02v^2 \\, \\text{m/s}^2$, where $v$ is the speed in meters per second. What distance is traversed by the cycle befor... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-8832baecb30919da88a19ac7a917526338eef21c75a0c7b46068aba1ddcdea55 | A cyclist takes 36 seconds and 44 seconds to cover the inner and outer edges of a circular path, respectively, while moving at the same speed. If the path is 7 meters wide, find the area of the circular path. | [
"\\( 490\\pi \\) square meters"
] | [
"A cyclist takes 36 seconds and 44 seconds to cover the inner and outer edges of a circular path, respectively, while moving at the same speed. If the path is 7 meters wide, find the area of the circular path."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-0175248e1dce47391952844df787b8f00b3a9a86907a6fab5a49b8f717c97e1f | A cyclist travels from A to B, and a motorcyclist travels from B to A. They meet at 2 pm. If the cyclist doubles their speed, they meet at 1:30 pm. If the motorcyclist doubles their speed, they meet at 1:12 pm. What time did they depart? | [
"12:00"
] | [
"A cyclist travels from A to B, and a motorcyclist travels from B to A. They meet at 2 pm. If the cyclist doubles their speed, they meet at 1:30 pm. If the motorcyclist doubles their speed, they meet at 1:12 pm. What time did they depart?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 44,
"genselect": 7,
"tir": 15
} | 66 | 0 | 0 |
omr-ws-89d70511562294e51d0a4140b9410cef6c437dd62b2033c2f919106c2db86e80 | A cyclist travels from Town A to Town B, where the road is level for half the distance and the other half is part uphill and part downhill. The cyclist averages 12 mph on level roads, 6 mph uphill, and 18 mph downhill. The trip from A to B takes 2 hours and 40 minutes, and the return trip from B to A takes 2 hours. How... | [
"9"
] | [
"A cyclist travels from Town A to Town B, where the road is level for half the distance and the other half is part uphill and part downhill. The cyclist averages 12 mph on level roads, 6 mph uphill, and 18 mph downhill. The trip from A to B takes 2 hours and 40 minutes, and the return trip from B to A takes 2 hours... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-a7aca5e9cde37cc34cc9a7cc97f7aa28f4da8c9d8207241516967a0834bfb6d7 | A cylinder drum with an open top and height of 90 cm is filled to the top with 210 L of water. What is the remaining volume of water (in L) when the drum is tilted 10 degrees with the vertical? (Give your answer to the nearest integer.) | [
"208"
] | [
"A cylinder drum with an open top and height of 90 cm is filled to the top with 210 L of water. What is the remaining volume of water (in L) when the drum is tilted 10 degrees with the vertical? (Give your answer to the nearest integer.)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 7,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-eab069431e46f0686f38cd62bdc7dacdc606a02052fd6ba2ae4b0043631437e2 | A cylinder has a circumference of 4 and a height of 12. A spiral is drawn on the cylinder such that it makes 4 complete revolutions. What is the length of the spiral? | [
"20"
] | [
"A cylinder has a circumference of 4 and a height of 12. A spiral is drawn on the cylinder such that it makes 4 complete revolutions. What is the length of the spiral?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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