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-394992c7f57f30688c604c0b80ef84999481ef5545a47a95de9ec7d4fb13a78a | A circular disc with diameter $D$ is placed on an $8 \times 8$ checkerboard with width $D$ so that the centers coincide. Determine the number of checkerboard squares which are completely covered by the disc. | [
"32"
] | [
"A circular disc with diameter $D$ is placed on an $8 \\times 8$ checkerboard with width $D$ so that the centers coincide. Determine the number of checkerboard squares which are completely covered by the disc."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 6,
"tir": 15
} | 29 | 0 | 0 |
omr-ws-ea7474d0a31403e4850fd5881a79b3ab3f9a00a94754a83804b0e724a419120a | A circular disc with diameter \( D \) is placed on an \( 8 \times 8 \) checkerboard with width \( D \) so that the centers coincide. How many checkerboard squares are completely covered by the disc? | [
"32"
] | [
"A circular disc with diameter \\( D \\) is placed on an \\( 8 \\times 8 \\) checkerboard with width \\( D \\) so that the centers coincide. How many checkerboard squares are completely covered by the disc?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-d44f2cefdda5f313e0e21419982a95095364d10843480f9240b53d7ad7776c91 | A circular disk centered at vertex $A$ of an equilateral triangle $\triangle ABC$ with side length $l$ rests on $BC$. The disk is moved such that its center moves to $A'$, with $AA' \parallel BC$ and $AA' < \frac{l}{2}$. If the disk initially intersects $AB$ and $AC$ at points $M$ and $N$ respectively, and after the mo... | [
"3"
] | [
"A circular disk centered at vertex $A$ of an equilateral triangle $\\triangle ABC$ with side length $l$ rests on $BC$. The disk is moved such that its center moves to $A'$, with $AA' \\parallel BC$ and $AA' < \\frac{l}{2}$. If the disk initially intersects $AB$ and $AC$ at points $M$ and $N$ respectively, and afte... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-03bce52d89649c796067417307c9a3f276dc9ccc8fe8f5f3da603f74fdac3f33 | A circular disk is divided by $2n$ equally spaced radii ($n>0$) and one secant line. What is the maximum number of non-overlapping areas into which the disk can be divided? | [
"3n+1"
] | [
"A circular disk is divided by $2n$ equally spaced radii ($n>0$) and one secant line. What is the maximum number of non-overlapping areas into which the disk can be divided?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 3,
"tir": 4
} | 22 | 0 | 0 |
omr-ws-291ad6a8e7eb8c2b70d61a6dc7d33af9ae55a4acb0210c83cbefe0f630c66eda | A circular disk is partitioned into $2n$ equal sectors by $n$ straight lines through its center. These $2n$ sectors are colored such that exactly $n$ sectors are colored blue and the other $n$ sectors are colored red. The red sectors are numbered from $1$ to $n$ in counter-clockwise direction, and the blue sectors are ... | [
"2n-1"
] | [
"A circular disk is partitioned into $2n$ equal sectors by $n$ straight lines through its center. These $2n$ sectors are colored such that exactly $n$ sectors are colored blue and the other $n$ sectors are colored red. The red sectors are numbered from $1$ to $n$ in counter-clockwise direction, and the blue sectors... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-412e5e8b3cc2e0a4b590266f45ca6880a397f380f987086a046151ba7451e7bf | A circular disk of radius 3 inches is centered at the origin and another disk of the same size is centered at (10, 24). Suppose the disk at the origin is moved along a line through the origin in an effort to make contact with the disk at (10, 24). Give values \(a\) and \(b\) (with \(a\) as small as possible and \(b\) a... | [
"\\frac{15 + 3\\sqrt{10}}{4}"
] | [
"A circular disk of radius 3 inches is centered at the origin and another disk of the same size is centered at (10, 24). Suppose the disk at the origin is moved along a line through the origin in an effort to make contact with the disk at (10, 24). Give values \\(a\\) and \\(b\\) (with \\(a\\) as small as possible ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-2f0001d04851735dd4cf28968c272d6ee124dc3ca7aca3f8243a133cf5110582 | A circular disk slides down a pit defined by the equation $y=x^2$, maintaining two points of contact with the pit. What is the radius of the disk when the two points of contact coalesce into one? | [
"\\(\\frac{1}{2}\\)"
] | [
"A circular disk slides down a pit defined by the equation $y=x^2$, maintaining two points of contact with the pit. What is the radius of the disk when the two points of contact coalesce into one?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 15
} | 30 | 0 | 0 |
omr-ws-c9345e495e41c25a8ebcdacf349cc4f9982e5d3ca38d79f8513730f772d47602 | A circular disk with radius $10$ cm is externally tangent to a clock face with radius $20$ cm at the $12$ o'clock position. The disk has an arrow initially pointing upward. If the disk rolls clockwise around the clock face, at what point on the clock face will the disk be tangent when the arrow is next pointing in the ... | [
"\\(\\textbf{(C) } \\mathrm{4\\, o'clock}\\)"
] | [
"A circular disk with radius $10$ cm is externally tangent to a clock face with radius $20$ cm at the $12$ o'clock position. The disk has an arrow initially pointing upward. If the disk rolls clockwise around the clock face, at what point on the clock face will the disk be tangent when the arrow is next pointing in... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-193b0163348bb9a1e5b53e22d9fd4dbeffc9c7d60d37183f081cb2bf76c1dc48 | A circular disk with radius $10$ cm is externally tangent to a clock face with radius $20$ cm at the 12 o'clock position. The disk has an arrow initially pointing upward. If the disk rolls clockwise around the clock face, at what point on the clock face will the disk be tangent when the arrow is next pointing upward? | [
"4 o' clock"
] | [
"A circular disk with radius $10$ cm is externally tangent to a clock face with radius $20$ cm at the 12 o'clock position. The disk has an arrow initially pointing upward. If the disk rolls clockwise around the clock face, at what point on the clock face will the disk be tangent when the arrow is next pointing upwa... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-9e2591b37ecd136f194ac591a96e846108e1d51070c9103d9e05157ed1aa346f | A circular disk with radius 10 cm is externally tangent to a clock face with radius 20 cm at the 12 o'clock position. The disk has an arrow initially pointing upward. If the disk rolls clockwise around the clock face, at what point on the clock face will the disk be tangent when the arrow is next pointing in the upward... | [
"4",
"4 o' clock",
"4 o'clock"
] | [
"A circular disk with radius 10 cm is externally tangent to a clock face with radius 20 cm at the 12 o'clock position. The disk has an arrow initially pointing upward. If the disk rolls clockwise around the clock face, at what point on the clock face will the disk be tangent when the arrow is next pointing in the u... | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 106,
"genselect": 6,
"tir": 35
} | 147 | 0 | 0 |
omr-ws-9afd3794ee6478ca0cd4b50e2c06c295464b36462379de22bf1928ce16f89d00 | A circular disk with radius 10 cm is externally tangent to a clock face with radius 20 cm at the 12 o'clock position. The disk has an arrow initially pointing upward. If the disk rolls clockwise around the clock face, at what point on the clock face will the disk be tangent when the arrow is next pointing upward? | [
"4"
] | [
"A circular disk with radius 10 cm is externally tangent to a clock face with radius 20 cm at the 12 o'clock position. The disk has an arrow initially pointing upward. If the disk rolls clockwise around the clock face, at what point on the clock face will the disk be tangent when the arrow is next pointing upward?"... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 4
} | 19 | 0 | 0 |
omr-ws-61606b8729f6b9efc0101e9b1299f830bd5c2035287d4064ddda4881d319396a | A circular disk with radius 10 cm is externally tangent to a clock face with radius 20 cm at the 12 o'clock position. The disk has an arrow initially pointing upward. The disk rolls clockwise around the clock face. At what point on the clock face will the disk be tangent when the arrow is next pointing in the upward ve... | [
"4"
] | [
"A circular disk with radius 10 cm is externally tangent to a clock face with radius 20 cm at the 12 o'clock position. The disk has an arrow initially pointing upward. The disk rolls clockwise around the clock face. At what point on the clock face will the disk be tangent when the arrow is next pointing in the upwa... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 5
} | 19 | 0 | 0 |
omr-ws-8d80b443f9f3abd6221d92eb130c095d7c74bd549b1a5c4d676fb2a3e8e5ab9c | A circular field has a radius of $80$ m, and a tall, circular column with a radius of $40$ m is centered at the center of the field. Alexandra and Hayley each run to a random spot on the circumference of the field. What is the probability that they can see each other? | [
"\\(\\frac{2}{3}\\)"
] | [
"A circular field has a radius of $80$ m, and a tall, circular column with a radius of $40$ m is centered at the center of the field. Alexandra and Hayley each run to a random spot on the circumference of the field. What is the probability that they can see each other?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-daccaf9fcd6b3555984b312491334fe611d85a92d6b89776bcac51605c025b85 | A circular frame with radius $3$ is fixed onto a table centered at a nail. Disk A with radius $1$ is attached to the nail, and Disk B with radius $1$ is placed tangent to Disk A and the frame. Disk A rotates $120^{\circ}$ clockwise about its center, causing Disk B to roll without slipping. Relative to the frame, how ma... | [
"80 degrees"
] | [
"A circular frame with radius $3$ is fixed onto a table centered at a nail. Disk A with radius $1$ is attached to the nail, and Disk B with radius $1$ is placed tangent to Disk A and the frame. Disk A rotates $120^{\\circ}$ clockwise about its center, causing Disk B to roll without slipping. Relative to the frame, ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2b8ae74e407689c05722bc108f90af7361eecce4e213f44d7a8ebe9621fcf89d | A circular game board is divided into $n \ge 3$ sectors. Each sector is either empty or occupied by a marker. In each step, one chooses an occupied sector, removes its marker, and then switches each of the two adjacent sectors from occupied to empty or vice-versa. Starting with a single occupied sector, for which value... | [
"\\( n \\) must be odd."
] | [
"A circular game board is divided into $n \\ge 3$ sectors. Each sector is either empty or occupied by a marker. In each step, one chooses an occupied sector, removes its marker, and then switches each of the two adjacent sectors from occupied to empty or vice-versa. Starting with a single occupied sector, for which... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-059397326f7fe7088c874e18878a67aee52e4661fcd8a9016f05a219b6421a35 | A circular garbage can with a diameter of 48 cm is wedged into a rectangular corner. Find the distance from the corner point to the center of the can (PA). | [
"\\( 24\\sqrt{2} \\)"
] | [
"A circular garbage can with a diameter of 48 cm is wedged into a rectangular corner. Find the distance from the corner point to the center of the can (PA)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-54e40f7e73d267c48278f97f4e04f6046b281368ef27febe882e4f0f08406c7f | A circular garden is enlarged so that the new diameter is twice the old diameter. What is the ratio of the original area to the enlarged area? Express your answer as a common fraction. | [
"\\frac{1}{4}"
] | [
"A circular garden is enlarged so that the new diameter is twice the old diameter. What is the ratio of the original area to the enlarged area? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-db07b79b5d37b925de3ee50c71f3c406e08c660ec4cb8a1aecbefe06c420c659 | A circular grass plot with a diameter of 12 feet is cut by a straight gravel path 3 feet wide, one edge of which passes through the center of the plot. What is the area of the remaining grass area in square feet? | [
"\\(30\\pi - 9\\sqrt{3}\\)"
] | [
"A circular grass plot with a diameter of 12 feet is cut by a straight gravel path 3 feet wide, one edge of which passes through the center of the plot. What is the area of the remaining grass area in square feet?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-7014861571d95f6025542f2a6317a6750c18f509b5110fd0328a2eda3f682686 | A circular hole with radius 5 is drilled all the way through the center of a sphere of clay with radius 13. The remaining clay is rolled to form a new perfectly spherical "bead". What is the radius of this bead? | [
"11.91"
] | [
"A circular hole with radius 5 is drilled all the way through the center of a sphere of clay with radius 13. The remaining clay is rolled to form a new perfectly spherical \"bead\". What is the radius of this bead?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2b2b35bc81e68d5e68cdd0c99f44957a95412c3bd7dd0af94fbb716548675eed | A circular lighthouse floor has a circular rug in the center. If a 10-foot pole placed on the floor touches the wall at both ends and is tangent to the rug, find the area of the floor that is not covered by the rug. | [
"\\( 25\\pi \\)"
] | [
"A circular lighthouse floor has a circular rug in the center. If a 10-foot pole placed on the floor touches the wall at both ends and is tangent to the rug, find the area of the floor that is not covered by the rug."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-f221884cf37b95e27aefce0b93fc3e7f3fa2958adf9ecd8c4b380acc342470b4 | A circular lighthouse floor has a circular rug in the center. The lighthouse keeper observes that if he places a 10-foot pole on the floor so that each end touches the wall, then it is tangent to the rug. Find the area of the floor that is not covered by the rug. | [
"\\( 25\\pi \\)"
] | [
"A circular lighthouse floor has a circular rug in the center. The lighthouse keeper observes that if he places a 10-foot pole on the floor so that each end touches the wall, then it is tangent to the rug. Find the area of the floor that is not covered by the rug."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 28
} | 45 | 0 | 0 |
omr-ws-a8861919a99963290382c245e3d455146408265750e528f72a52da4f288461f3 | A circular mirror in the plane is defined by the equation $x^2 + y^2 = 1$. A light beam starts at the point $(0, 1)$. When the light beam hits the mirror, it follows the law of reflection. How many different directions can the light beam be shone (assuming it must be shone into the unit circle) so that it first comes b... | [
"1632"
] | [
"A circular mirror in the plane is defined by the equation $x^2 + y^2 = 1$. A light beam starts at the point $(0, 1)$. When the light beam hits the mirror, it follows the law of reflection. How many different directions can the light beam be shone (assuming it must be shone into the unit circle) so that it first co... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 2
} | 12 | 0 | 0 |
omr-ws-0f74400e35681ebd6c8363b0ec0067139e5039ec5cc0cd369e0f83740bc995fc | A circular pavement with diameter $2r$ has two vertical columns of height $h$ at its endpoints $A$ and $B$. These columns support a beam $A'B'$ of length $2r$. Numerous taut cables, which are perpendicular to the beam $A'B'$, connect points on the beam to points on the circumference of the pavement. Determine the volum... | [
"\\(\\pi r^2 h\\)"
] | [
"A circular pavement with diameter $2r$ has two vertical columns of height $h$ at its endpoints $A$ and $B$. These columns support a beam $A'B'$ of length $2r$. Numerous taut cables, which are perpendicular to the beam $A'B'$, connect points on the beam to points on the circumference of the pavement. Determine the ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-97f2e0c454a20d74c2a892c4de3afda0ca62769d221eb2e5d2ef5557aa058b09 | A circular piece of metal of maximum size is cut out of a square piece, and then a square piece of maximum size is cut out of the circular piece. What fraction of the area of the circular piece is wasted? | [
"\\dfrac{\\pi - 2}{\\pi}"
] | [
"A circular piece of metal of maximum size is cut out of a square piece, and then a square piece of maximum size is cut out of the circular piece. What fraction of the area of the circular piece is wasted?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3cbc82ec55033e14b6f60f22f0b6a3b75a6cfbf561dc688861af45716e570cf8 | A circular piece of paper with radius 5 cm has a sector containing an angle of 144 degrees removed. The remaining sector is folded to form a right circular cone. What is the volume of the cone in cubic centimeters? | [
"\\( 12\\pi \\) cubic centimeters."
] | [
"A circular piece of paper with radius 5 cm has a sector containing an angle of 144 degrees removed. The remaining sector is folded to form a right circular cone. What is the volume of the cone in cubic centimeters?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-38032b4590c09bb96808902bb11072061f39df922eb2e70f10a3ada0e6042d32 | A circular plate with radius $9$ cm has a square piece of watermelon $3$ cm wide randomly placed in it. Suika stabs the plate at a random location and angle with a fork, where the prongs are $2$ cm wide in one direction and have negligible width in the other direction. Calculate the expected number of times Suika has t... | [
"\\dfrac{81\\pi^2}{9\\pi - 20}"
] | [
"A circular plate with radius $9$ cm has a square piece of watermelon $3$ cm wide randomly placed in it. Suika stabs the plate at a random location and angle with a fork, where the prongs are $2$ cm wide in one direction and have negligible width in the other direction. Calculate the expected number of times Suika ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-6f98d11f940a944274967646391f62c67590b781403fbfa4b544c78a085fa46f | A circular pool has a diameter of 10 feet. One cubic yard of concrete is to be used to create a circular border of uniform width around the pool, with a depth of 4 inches. How wide will the border be? (Note: 1 cubic yard = 27 cubic feet.) | [
"2.1 feet"
] | [
"A circular pool has a diameter of 10 feet. One cubic yard of concrete is to be used to create a circular border of uniform width around the pool, with a depth of 4 inches. How wide will the border be? (Note: 1 cubic yard = 27 cubic feet.)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-44695cfe4f65ef66cbaea2c8d3cab559ef2c8e3ffcd45c7db4c1fae52e256b40 | A circular race track consists of two concentric lanes with the inner lane having a radius of 50 meters and the outer lane having a radius of 52 meters. The finishing points on both lanes are labeled $A$ (inner lane) and $B$ (outer lane), lying on the same radius. If the starting point on the inner lane is at $A$ and $... | [
"13.8"
] | [
"A circular race track consists of two concentric lanes with the inner lane having a radius of 50 meters and the outer lane having a radius of 52 meters. The finishing points on both lanes are labeled $A$ (inner lane) and $B$ (outer lane), lying on the same radius. If the starting point on the inner lane is at $A$ ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 6,
"tir": 21
} | 42 | 0 | 0 |
omr-ws-972897865afe6b152913453a76d0e9ebd7711043c03eced0b66bd5ca3185247e | A circular room has a circular rug in the center. A 20-foot pole is placed on the floor so that both ends touch the wall and is tangent to the rug. What is the area of the room that is not covered by the rug? | [
"\\( 100\\pi \\)"
] | [
"A circular room has a circular rug in the center. A 20-foot pole is placed on the floor so that both ends touch the wall and is tangent to the rug. What is the area of the room that is not covered by the rug?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-996472b101283dccd798c79b3f8c16371e93b003e256a56b708667f6192ec40f | A circular sector of radius \( r \) and angle \( \alpha \) has twice the area as a sector of radius \( R \) and angle \( \beta \). Find the ratio \( \frac{\alpha}{\beta} \) if \( R = 2r \). | [
"8"
] | [
"A circular sector of radius \\( r \\) and angle \\( \\alpha \\) has twice the area as a sector of radius \\( R \\) and angle \\( \\beta \\). Find the ratio \\( \\frac{\\alpha}{\\beta} \\) if \\( R = 2r \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-54f51eafa2e93526612e30c0b53662086bb5584f1c45b08a6dbc125a49e2f8bd | A circular sector with center \( O \) and radii \( OA \) and \( OB \) is cut out of a homogenous thin cardboard and hung freely by the point \( A \). Find the angle between \( AO \) and the vertical if \( \angle AOB = \theta \leq \pi \). | [
"\\(\\arctan\\left(\\frac{1-\\cos\\theta}{{3\\over 2}\\theta-\\sin\\theta}\\right)\\)"
] | [
"A circular sector with center \\( O \\) and radii \\( OA \\) and \\( OB \\) is cut out of a homogenous thin cardboard and hung freely by the point \\( A \\). Find the angle between \\( AO \\) and the vertical if \\( \\angle AOB = \\theta \\leq \\pi \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 2,
"tir": 2
} | 6 | 0 | 0 |
omr-ws-8f3946eb06825f11613955546160c6361c68575715c387052ddbd794127826fa | A circular segment is cut from a regular pentagon, with one vertex as the center and a side of the pentagon as the radius of the circle. If the area of the segment is 17, what is the length of each side of the pentagon? | [
"\\(\\sqrt{\\frac{170}{3 \\pi}}\\)"
] | [
"A circular segment is cut from a regular pentagon, with one vertex as the center and a side of the pentagon as the radius of the circle. If the area of the segment is 17, what is the length of each side of the pentagon?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-453976764b1675993a92cadddd8493b4a7975d5e8dc2e2a8da64c0b44c75e75d | A circular sheet of paper with radius of $6$ cm is cut into three congruent sectors. What is the height of the cone in centimeters that can be created by rolling one of the sections until the edges meet? Express your answer in simplest radical form. | [
"4 \\sqrt{2}"
] | [
"A circular sheet of paper with radius of $6$ cm is cut into three congruent sectors. What is the height of the cone in centimeters that can be created by rolling one of the sections until the edges meet? Express your answer in simplest radical form."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c0978ac7587ee289b2b4317acf265eded227852b28ec9b1327ebd7c3397561a2 | A circular spinner for a game has a radius of 10 cm. The probability of winning on one spin of this spinner is $\frac{2}{5}$. What is the area, in sq cm, of the WIN sector? Express your answer in terms of $\pi$. [asy]import graph; draw(Circle((0,0),25),black); draw((0,0)--(7,18),Arrow); draw((0,0)--(0,25)); draw((0,0)-... | [
"40\\pi"
] | [
"A circular spinner for a game has a radius of 10 cm. The probability of winning on one spin of this spinner is $\\frac{2}{5}$. What is the area, in sq cm, of the WIN sector? Express your answer in terms of $\\pi$.\n\n[asy]import graph;\n\ndraw(Circle((0,0),25),black);\n\ndraw((0,0)--(7,18),Arrow);\n\ndraw((0,0)--(... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9096c1f2133eaa783d6e15aa95cdd578b41c04d826e4528ff27bbbbc6ff974bf | A circular spinner for a game has a radius of 5 cm. The probability of winning on one spin of this spinner is $\frac{2}{5}$. What is the area, in sq cm, of the WIN sector? Express your answer in terms of $\pi$. [asy]import graph; draw(Circle((0,0),25),black); draw((0,0)--(7,18),Arrow); draw((0,0)--(0,25)); draw((0,0)--... | [
"10\\pi"
] | [
"A circular spinner for a game has a radius of 5 cm. The probability of winning on one spin of this spinner is $\\frac{2}{5}$. What is the area, in sq cm, of the WIN sector? Express your answer in terms of $\\pi$.\n\n[asy]import graph;\ndraw(Circle((0,0),25),black);\ndraw((0,0)--(7,18),Arrow);\ndraw((0,0)--(0,25));... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8818252887a900b052877dc70fc6a5289caa8d15b91a35cd2b48a5da1929b0ef | A circular spinner has five sections of equal size, each of which is colored either red or blue. Two colorings are considered the same if one can be rotated to yield the other. In how many ways can the spinner be colored? | [
"8"
] | [
"A circular spinner has five sections of equal size, each of which is colored either red or blue. Two colorings are considered the same if one can be rotated to yield the other. In how many ways can the spinner be colored?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8a4e668163272a33adb8a1fd7fef762dc35e9e915d763e793835154bedff6fa3 | A circular spinner has nine sections of equal size, each of which is colored with one of three colors. Two colorings are considered the same if one can be rotated to yield the other. In how many ways can the spinner be colored? | [
"2193"
] | [
"A circular spinner has nine sections of equal size, each of which is colored with one of three colors. Two colorings are considered the same if one can be rotated to yield the other. In how many ways can the spinner be colored?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-be59e3792066434d9c1e5f59fc74a965c27cd3fdcabdd03bb483bbb9503068a0 | A circular spinner has seven sections of equal size, each of which is colored either red or blue. Two colorings are considered the same if one can be rotated to yield the other. In how many ways can the spinner be colored? | [
"18"
] | [
"A circular spinner has seven sections of equal size, each of which is colored either red or blue. Two colorings are considered the same if one can be rotated to yield the other. In how many ways can the spinner be colored?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f214d727428574481c5bef0fbe48e3732b7b37fc2d75208251594461f0d925ff | A circular stained-glass window contains four smaller circles of the same size placed symmetrically around the center of the large circle. The regions inside the large circle but outside the smaller circles are colored blue. The regions where the smaller circles overlap are colored green, and the total area of green gl... | [
"400"
] | [
"A circular stained-glass window contains four smaller circles of the same size placed symmetrically around the center of the large circle. The regions inside the large circle but outside the smaller circles are colored blue. The regions where the smaller circles overlap are colored green, and the total area of gre... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f369da3e3b2fea23a5d3d92d7e4788ecd3c805e3480dd37cda352d48d21aae3d | A circular swimming pool has a depth \( D \) and a radius that equals five times its depth. A rectangular pool has the same depth as the circular pool. Its width is four feet more than three times its depth and its length is two feet less than six times its depth. Find the ratio of the rectangular pool's volume to the ... | [
"\\(\\frac{18D^3 + 18D^2 - 8D}{25D^3\\pi}\\)"
] | [
"A circular swimming pool has a depth \\( D \\) and a radius that equals five times its depth. A rectangular pool has the same depth as the circular pool. Its width is four feet more than three times its depth and its length is two feet less than six times its depth. Find the ratio of the rectangular pool's volume ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-455e84ad283fd5188b601caab9042ce30f2b94eab8a6da438484f4407a513500 | A circular swimming pool has a diameter of $12$ ft, its sides are $4$ ft high, and the depth of the water is $3$ ft. How much work (in ft-lb) is required to pump all of the water out over the side? Use a diagram with variables. (Hint: Water weighs $62.5 \frac{\text{lb}}{\text{ft}^3}$) | [
"\\( 16848 \\pi \\) ft-lb or \\( 16875 \\pi \\) ft-lb"
] | [
"A circular swimming pool has a diameter of $12$ ft, its sides are $4$ ft high, and the depth of the water is $3$ ft. How much work (in ft-lb) is required to pump all of the water out over the side? Use a diagram with variables. (Hint: Water weighs $62.5 \\frac{\\text{lb}}{\\text{ft}^3}$)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-b76e00c921f5cddd4d85d87a6b2134c26917f782713177f4fede911bddc2d9e7 | A circular table has 9 chairs, and 4 people can sit randomly. What is the probability that no two people sit next to each other? | [
"\\(\\frac{2}{63}\\)"
] | [
"A circular table has 9 chairs, and 4 people can sit randomly. What is the probability that no two people sit next to each other?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dee0156e5f556f7e15fe6602ed0bf4f00180aa76a31cd6842288b8a0c49b75a7 | A circular table has exactly 60 chairs around it. What is the smallest number of people, \( N \), that must be seated such that the next person to be seated must sit next to someone? | [
"20"
] | [
"A circular table has exactly 60 chairs around it. What is the smallest number of people, \\( N \\), that must be seated such that the next person to be seated must sit next to someone?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-c9a554ac7e13578a53ea04dc453eeee884b1d57d6b793cfc6f10c0b2d6bdb3ac | A circular table has exactly sixty chairs around it. There are $N$ people seated at this table in such a way that the next person to be seated must sit next to someone. What is the smallest possible value of $N$? | [
"20"
] | [
"A circular table has exactly sixty chairs around it. There are $N$ people seated at this table in such a way that the next person to be seated must sit next to someone. What is the smallest possible value of $N$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b3f5879b0ba987cdf12fc5def6b206b71df9136e96c9be6eef368036854e0df7 | A circular target with a radius of $12$ cm was hit by $19$ shots. What is the maximum possible distance, in centimeters, between any two shots if the distance between at least two of the hits is less than $7$ cm? | [
"24"
] | [
"A circular target with a radius of $12$ cm was hit by $19$ shots. What is the maximum possible distance, in centimeters, between any two shots if the distance between at least two of the hits is less than $7$ cm?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1197b4715b528561f350f9cd2681c81ad9a5d5c9996c9e91832c0a371bc855b3 | A circular tower has 16 doors, each hiding Captain Flint's gold, numbered from 1 to 16 in a clockwise direction. There are 16 pirates, each with a key numbered from 1 to 16. The key with number \( n \) opens the door with number \( m \) if and only if \( n \mid m \). Jim Hawkins can rotate the tower to position the doo... | [
"3"
] | [
"A circular tower has 16 doors, each hiding Captain Flint's gold, numbered from 1 to 16 in a clockwise direction. There are 16 pirates, each with a key numbered from 1 to 16. The key with number \\( n \\) opens the door with number \\( m \\) if and only if \\( n \\mid m \\). Jim Hawkins can rotate the tower to posi... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-e280fa9918a745380ccc725207cd8fda6d85b1f9d9f3eb309ba7880850f12372 | A circular track has a diameter of 2 miles. A man runs at a constant speed of 10 miles per hour and attempts to run 10 laps. How far will the man be from the starting line, rounded to the nearest foot, 30 minutes after he has begun? | [
"6320"
] | [
"A circular track has a diameter of 2 miles. A man runs at a constant speed of 10 miles per hour and attempts to run 10 laps. How far will the man be from the starting line, rounded to the nearest foot, 30 minutes after he has begun?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-ce71b83b902a9f9e2eab08380c5889071a30e8404bed236b9456564cd11caf24 | A circular wheel with radius 4 feet is lowered into a river. If the current causes the wheel to rotate at 10 RPM, what is the speed of the current? | [
"251.3274 feet per minute."
] | [
"A circular wheel with radius 4 feet is lowered into a river. If the current causes the wheel to rotate at 10 RPM, what is the speed of the current?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-5e838712d55a7b58c6ba45ee28311e410ac2a68f81f318dd2d910259ae3a832d | A circumference \( \alpha \) intersects with circumference \( \beta \) in points \( A \) and \( B \). There is a tangent line to both circumferences \( \alpha \) and \( \beta \) which intersects them in points \( C \) and \( D \) respectively. Points \( C \), \( D \), and \( B \) (with \( B \) being closer to the tange... | [
"6\\sqrt{6}"
] | [
"A circumference \\( \\alpha \\) intersects with circumference \\( \\beta \\) in points \\( A \\) and \\( B \\). There is a tangent line to both circumferences \\( \\alpha \\) and \\( \\beta \\) which intersects them in points \\( C \\) and \\( D \\) respectively. Points \\( C \\), \\( D \\), and \\( B \\) (with \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-9f8979255374c4af0fbc1636aa08461a4ca34d6b8d6f05920054ac4a888c780b | A circumference has three points painted red $A$, $B$, and $C$ in a clockwise direction. We will paint another $1996$ points red as follows: We traverse the circumference clockwise, starting from $C$. We go through a painted point ($A$) and paint $P_1$ at the midpoint of the arc $AB$. We continue to traverse the circum... | [
"665"
] | [
"A circumference has three points painted red $A$, $B$, and $C$ in a clockwise direction. We will paint another $1996$ points red as follows: We traverse the circumference clockwise, starting from $C$. We go through a painted point ($A$) and paint $P_1$ at the midpoint of the arc $AB$. We continue to traverse the c... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-12f10712f6b5b41c098bbbfb2eab174c2eb51895a6e6234022e70cf0a958d5a2 | A circumference of length $999$ is subdivided by points equispaced in arcs of length $1$. On this circumference are placed $d$ arcs of lengths $1, 2, \dots, d$ such that each arc has two of these points as its endpoints. It is known that no arc contains another. What is the maximum value of $d$? | [
"31"
] | [
"A circumference of length $999$ is subdivided by points equispaced in arcs of length $1$. On this circumference are placed $d$ arcs of lengths $1, 2, \\dots, d$ such that each arc has two of these points as its endpoints. It is known that no arc contains another. What is the maximum value of $d$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-072589851676f0402d5faf8784ce762c433998cc1e032e48763e3d2438e079d1 | A circumference was divided into $n$ equal parts. On each of these parts, one number from $1$ to $n$ was placed such that the distance between consecutive numbers is always the same. Numbers $11$, $4$, and $17$ were in consecutive positions. In how many parts was the circumference divided? | [
"20"
] | [
"A circumference was divided into $n$ equal parts. On each of these parts, one number from $1$ to $n$ was placed such that the distance between consecutive numbers is always the same. Numbers $11$, $4$, and $17$ were in consecutive positions. In how many parts was the circumference divided?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-3b5b04c2c75543534cac8e5b2f3217701a8cff8249d8394e59d8cd8deb32b259 | A circumference was divided into \( n \) equal parts. Numbers from \( 1 \) to \( n \) were placed on these parts such that the distance between consecutive numbers is always the same. Numbers \( 11 \), \( 4 \), and \( 17 \) were in consecutive positions. In how many parts was the circumference divided? | [
"20"
] | [
"A circumference was divided into \\( n \\) equal parts. Numbers from \\( 1 \\) to \\( n \\) were placed on these parts such that the distance between consecutive numbers is always the same. Numbers \\( 11 \\), \\( 4 \\), and \\( 17 \\) were in consecutive positions. In how many parts was the circumference divided?... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5dacee0042bfd533be08ba370446609f8254d2244223f02866cd1d30a579f4a5 | A circumscribable trapezoid \(ABCD\) has \(AB \parallel CD\) and \(AD \perp AB\). Given that \(AB = 3CD\) and the area of the trapezoid is 4, find the radius of the inscribed circle. | [
"\\dfrac{\\sqrt{3}}{2}"
] | [
"A circumscribable trapezoid \\(ABCD\\) has \\(AB \\parallel CD\\) and \\(AD \\perp AB\\). Given that \\(AB = 3CD\\) and the area of the trapezoid is 4, find the radius of the inscribed circle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b0ecebfe54b8f73d9708241ebc93470252d009025193da3d4e47acbe59ad2511 | A circumscribed pyramid $ABCDS$ is given. The opposite sidelines of its base meet at points $P$ and $Q$ such that $A$ and $B$ lie on segments $PD$ and $PC$ respectively. The inscribed sphere touches faces $ABS$ and $BCS$ at points $K$ and $L$. If $PK$ and $QL$ are coplanar, find the ratio of the lengths $BD$ to $DC$ wh... | [
"2"
] | [
"A circumscribed pyramid $ABCDS$ is given. The opposite sidelines of its base meet at points $P$ and $Q$ such that $A$ and $B$ lie on segments $PD$ and $PC$ respectively. The inscribed sphere touches faces $ABS$ and $BCS$ at points $K$ and $L$. If $PK$ and $QL$ are coplanar, find the ratio of the lengths $BD$ to $D... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bd1a60ab8bf542c897db547a75c9dbb82744e39e55bdf7b86dd85195586bb224 | A circumscribed pyramid ABCDS is given. The opposite sidelines of its base meet at points P and Q such that A and B lie on segments PD and PC, respectively. The inscribed sphere touches faces ABS and BCS at points K and L. If PK and QL are coplanar, calculate the length of the segment BD given that AB = 6, BC = 8, and ... | [
"no_valid_answer_found"
] | [
"A circumscribed pyramid ABCDS is given. The opposite sidelines of its base meet at points P and Q such that A and B lie on segments PD and PC, respectively. The inscribed sphere touches faces ABS and BCS at points K and L. If PK and QL are coplanar, calculate the length of the segment BD given that AB = 6, BC = 8,... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-88fba10b5805127e2822ad676d26219a03f94078ae7065bbd6a8b47e0cc35ad5 | A citizen is to vote yes or no on each of eight issues this election. She is allowed to abstain from voting on some of the issues, but she must vote on at least five. How many possible ways can she vote? | [
"\\(\\binom{8}{5}\\cdot 2^{5} + \\binom{8}{6}\\cdot 2^{6} +\\binom{8}{7}\\cdot 2^{7} +\\binom{8}{8}\\cdot 2^{8}\\)"
] | [
"A citizen is to vote yes or no on each of eight issues this election. She is allowed to abstain from voting on some of the issues, but she must vote on at least five. How many possible ways can she vote?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 5,
"tir": 12
} | 24 | 0 | 0 |
omr-ws-3b3252813b89c0480aa8fce790b15f891e3a58d9c50429b8ec53fb6578d6006e | A city council is working with a hotel developer to build several hotels along its beach-front. The developer has three prototypes: a convention-style hotel with 500 rooms costing $100 million, a vacation-style hotel with 200 rooms costing $20 million, and a small hotel with 50 rooms costing $4 million. The city counci... | [
"0"
] | [
"A city council is working with a hotel developer to build several hotels along its beach-front. The developer has three prototypes: a convention-style hotel with 500 rooms costing $100 million, a vacation-style hotel with 200 rooms costing $20 million, and a small hotel with 50 rooms costing $4 million. The city c... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-9b16713f464bf6296c65f198e9def6924f0cadd67722d0fb6c4277a9569db548 | A city has $4$ horizontal and $n \geq 3$ vertical boulevards which intersect at $4n$ crossroads. The crossroads divide every horizontal boulevard into $n-1$ streets and every vertical boulevard into $3$ streets. The mayor of the city decides to close the minimum possible number of crossroads so that the city doesn't ha... | [
"no_valid_answer_found"
] | [
"A city has $4$ horizontal and $n \\geq 3$ vertical boulevards which intersect at $4n$ crossroads. The crossroads divide every horizontal boulevard into $n-1$ streets and every vertical boulevard into $3$ streets. The mayor of the city decides to close the minimum possible number of crossroads so that the city does... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b78c993bab3e0ecfbf5d7117f7fd8f3c5e3fe2892df4245d9741b7c44b52536e | A city has a system of bus routes with the following properties: - Each route has exactly 11 bus stops. - It is possible to travel between any two bus stops without changing routes. - Any two bus routes have exactly one bus stop in common. What is the number of bus routes in the city? | [
"111"
] | [
"A city has a system of bus routes with the following properties:\n- Each route has exactly 11 bus stops.\n- It is possible to travel between any two bus stops without changing routes.\n- Any two bus routes have exactly one bus stop in common.\nWhat is the number of bus routes in the city?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-d6a44f8457d562b5edcf0c55b1b57469480c6466eaca418992364f2f3fa67976 | A city is built around a circular lake with a radius of 1 mile. The population density of the city is given by \( f(r) \) people per square mile, where \( r \) is the distance from the center of the lake in miles. Calculate the number of people who live within 1 mile of the lake if \( f(r) = 3r \). Express your answer ... | [
"14\\pi"
] | [
"A city is built around a circular lake with a radius of 1 mile. The population density of the city is given by \\( f(r) \\) people per square mile, where \\( r \\) is the distance from the center of the lake in miles. Calculate the number of people who live within 1 mile of the lake if \\( f(r) = 3r \\). Express y... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3beaccf6a146dd8c7bb00b0d9f94256d837e3ec32316f47b16fa1a978c675ec9 | A city is built around a circular lake with a radius of 1 mile. The population density of the city is given by \( f(r) \) people per square mile, where \( r \) is the distance from the center of the lake in miles. Determine the number of people who live within 1 mile of the lake. | [
"\\( 2\\pi \\cdot \\left( \\int_{1}^{2} rf(r) \\, dr \\right) \\)"
] | [
"A city is built around a circular lake with a radius of 1 mile. The population density of the city is given by \\( f(r) \\) people per square mile, where \\( r \\) is the distance from the center of the lake in miles. Determine the number of people who live within 1 mile of the lake."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-58de644ff1bde5a29ba9a870456e83449ffa3a9a33c2c6fd44c6f52c0f7c170b | A city is laid out as an \( n \times n \) grid. A person travels from the southwest corner to the northeast corner each morning, moving only north or east, and returns from the northeast corner to the southwest corner each evening, moving only south or west. Each day, the person chooses their path such that the total l... | [
"2n(n + 1)"
] | [
"A city is laid out as an \\( n \\times n \\) grid. A person travels from the southwest corner to the northeast corner each morning, moving only north or east, and returns from the northeast corner to the southwest corner each evening, moving only south or west. Each day, the person chooses their path such that the... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-a33684ee43d8f2f5b4ef2efed83304a0e4b3f07a3289b139df82cf93dd96292f | A city is represented as an $n \times n$ square grid divided into $1 \times 1$ blocks. A man walks from the south-west corner to the north-east corner and then returns to the south-west corner, moving only north or east on the way to the north-east corner and only south or west on the way back. Each time, he chooses hi... | [
"2n(n + 1)"
] | [
"A city is represented as an $n \\times n$ square grid divided into $1 \\times 1$ blocks. A man walks from the south-west corner to the north-east corner and then returns to the south-west corner, moving only north or east on the way to the north-east corner and only south or west on the way back. Each time, he cho... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-33a5ea76aa6a66c63035197c10fb3c5b74a27e981f236b2a35dff94cc4aec3e0 | A class contains 3 girls and 4 boys. Every Saturday, 5 students go on a picnic; a different group is sent each week. During the picnic, each person is given a cake by the escorting teacher. After all possible groups of five have gone once, find the total number of cakes received by the girls during the picnics. | [
"45"
] | [
"A class contains 3 girls and 4 boys. Every Saturday, 5 students go on a picnic; a different group is sent each week. During the picnic, each person is given a cake by the escorting teacher. After all possible groups of five have gone once, find the total number of cakes received by the girls during the picnics."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-11882183d14d00a3129959aebe2a68898cbb38c0f4ae031fb0120fb7b4a32332 | A class contains 5 boys and 5 girls. They select seats at random around a circular table that seats 10. What is the probability that some two girls will sit next to one another? | [
"\\(\\frac{125}{126}\\)"
] | [
"A class contains 5 boys and 5 girls. They select seats at random around a circular table that seats 10. What is the probability that some two girls will sit next to one another?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-e435b2e22f0a3615ea61c5d8f64204d28ca55b90806f216882309aec83dde6a7 | A class contains 5 boys and 5 girls. They select seats at random around a circular table that seats ten. What is the probability that some two girls will sit next to one another? | [
"\\(\\frac{125}{126}\\)"
] | [
"A class contains 5 boys and 5 girls. They select seats at random around a circular table that seats ten. What is the probability that some two girls will sit next to one another?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-86fb9f67091ffae4d47dc0345fb54bb60517969eb5434fe84b59ff1abb9c4c50 | A class contains n boys and n girls. If they all randomly select seats around a circular table that seats 2n, what is the probability that some two boys will sit next to one another? | [
"\\(1 - \\frac{2}{(2n-1)!}\\)"
] | [
"A class contains n boys and n girls. If they all randomly select seats around a circular table that seats 2n, what is the probability that some two boys will sit next to one another?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0691c8d735b8eec00adad1aec574fd4bb91412839dd689bc52f018623db56976 | A class has 25 students. The teacher wants to stock $N$ candies and distribute them based on student success in an Olympiad such that those who solve more tasks get more candies (including possibly zero candies for those who solve fewer tasks). What is the smallest value of $N$ that allows this distribution regardless ... | [
"600"
] | [
"A class has 25 students. The teacher wants to stock $N$ candies and distribute them based on student success in an Olympiad such that those who solve more tasks get more candies (including possibly zero candies for those who solve fewer tasks). What is the smallest value of $N$ that allows this distribution regard... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2c2677f4ba6c306f2980e870a4ad15c6efd7bd3d7e1960dbaebb382d133243bf | A class has 9 students. On the first day, the students are arranged into 3 groups of 3 students. On each subsequent day, the students must form new groups of 3 such that no student is in a group with any student they were previously grouped with. How many days can this process continue? | [
"4"
] | [
"A class has 9 students. On the first day, the students are arranged into 3 groups of 3 students. On each subsequent day, the students must form new groups of 3 such that no student is in a group with any student they were previously grouped with. How many days can this process continue?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6cf3519f74f04d56b0b9203b5243d4159339053b70178e0b4d4422a83b777abc | A class has \( n > 1 \) boys and \( n \) girls. For each arrangement \( X \) of the class in a line, let \( f(X) \) be the number of ways to divide the line into two non-empty segments such that the number of boys and girls is equal in each segment. Let \( A \) be the number of arrangements with \( f(X) = 0 \) and \( B... | [
"2"
] | [
"A class has \\( n > 1 \\) boys and \\( n \\) girls. For each arrangement \\( X \\) of the class in a line, let \\( f(X) \\) be the number of ways to divide the line into two non-empty segments such that the number of boys and girls is equal in each segment. Let \\( A \\) be the number of arrangements with \\( f(X)... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-98fd1e4a9f235076995477f0bb0cfa4927033c92a1c29f5d3e5cb29ed8193923 | A class has ten students, and the teacher has twenty distinct stones. The teacher randomly gives each stone to one of the students. The happiness of the class is defined as the product of the number of stones each student has. Find the expected value of the happiness of the class, expressed as $\frac{m}{n}$, where $m$ ... | [
"\\(\\frac{20!}{10^{10} \\cdot 10!}\\)"
] | [
"A class has ten students, and the teacher has twenty distinct stones. The teacher randomly gives each stone to one of the students. The happiness of the class is defined as the product of the number of stones each student has. Find the expected value of the happiness of the class, expressed as $\\frac{m}{n}$, wher... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0b2e588a3d336f2325739cafcfbaf96aa422b7b73ee447245c53fefc4c2534ec | A class has ten students, and the teacher has twenty distinct stones. The teacher randomly gives each stone to one of the students. The happiness of the class is defined as the product of the number of stones each student has. The expected value of the happiness of the class can be expressed as $\frac{m}{n}$, where $m$... | [
"788"
] | [
"A class has ten students, and the teacher has twenty distinct stones. The teacher randomly gives each stone to one of the students. The happiness of the class is defined as the product of the number of stones each student has. The expected value of the happiness of the class can be expressed as $\\frac{m}{n}$, whe... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 29
} | 51 | 0 | 0 |
omr-ws-5de6703f736802a625718352daa124e2e93f46369175a6aaceb8021e5972cd18 | A class meets 31 times. In each meeting, the professor divides the students into working groups of 5 so that every two students are in the same group exactly once. How many students are there in the class? | [
"125 students"
] | [
"A class meets 31 times. In each meeting, the professor divides the students into working groups of 5 so that every two students are in the same group exactly once. How many students are there in the class?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 11
} | 26 | 0 | 0 |
omr-ws-6d3f03ad008cc47e4435ba4de5904528ea731fb5fb29acbe8f0e90805754e12a | A class of 10 students is to be randomly divided into two groups A and B. If every student belongs to exactly one group and the number of students in group A is more than the number of students in group B, determine the number of ways to form such groups. | [
"386"
] | [
"A class of 10 students is to be randomly divided into two groups A and B. If every student belongs to exactly one group and the number of students in group A is more than the number of students in group B, determine the number of ways to form such groups."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 1,
"tir": 1
} | 14 | 0 | 0 |
omr-ws-e1201c42bb45f7723c8c24328f569f81febc115ddb29f6653de926ad673f3b6e | A class of 10 students wants to determine the class president by drawing slips of paper from a box. One of the students, Bob, puts a slip of paper with his name into the box. Each other student has a \(\frac{1}{2}\) probability of putting a slip of paper with their own name into the box and a \(\frac{1}{2}\) probabilit... | [
"\\dfrac{5120}{1023}"
] | [
"A class of 10 students wants to determine the class president by drawing slips of paper from a box. One of the students, Bob, puts a slip of paper with his name into the box. Each other student has a \\(\\frac{1}{2}\\) probability of putting a slip of paper with their own name into the box and a \\(\\frac{1}{2}\\)... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5c061c47ace67ec554cb5415a5178a55eb39bcc93e7a7cb3c7b7dc9082ad2057 | A class of 18 students is seated randomly around 3 tables with 6 chairs each. How many different seating arrangements are there if two arrangements are considered the same when the same group of students sit at each table (regardless of the seating arrangement at each table)? | [
"28561"
] | [
"A class of 18 students is seated randomly around 3 tables with 6 chairs each. How many different seating arrangements are there if two arrangements are considered the same when the same group of students sit at each table (regardless of the seating arrangement at each table)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ad1731425af944a3733efa319d5eb5799b6cd8ab02ba5833ddffbdc0d5bf97ae | A class of 2023 students at a fictional Anchorage Inner Mathematics Academy each solves a set of problems corresponding to their student number and the two adjacent numbers (mod 2023). Students with ID numbers divisible by 10, 100, and 1000 solve additional problems as specified. The teacher aims to split the students ... | [
"2023"
] | [
"A class of 2023 students at a fictional Anchorage Inner Mathematics Academy each solves a set of problems corresponding to their student number and the two adjacent numbers (mod 2023). Students with ID numbers divisible by 10, 100, and 1000 solve additional problems as specified. The teacher aims to split the stud... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-7f7c82030856f8116cd855c877423b10d77d0d286da5587e2d3ac5423bc61f84 | A class of 218 students takes a test. Each student's score is an integer from 0 to 100, inclusive. Compute the greatest possible difference between the mean and the median scores. | [
"\\(\\frac{5400}{109}\\)"
] | [
"A class of 218 students takes a test. Each student's score is an integer from 0 to 100, inclusive. Compute the greatest possible difference between the mean and the median scores."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-894e406c96ee893c003fa7d35fe1e47a0417f68080f380982af8cc7b903fd328 | A class of 25 students and 2 teachers collect money for charity. Each of the teachers donates 0, 3, or 6 coins, and each student donates 0 or 1 coin. Let \( a_n \) be the number of ways in which \( n \) coins can be collected. a) Find the generating function \( G_{\bar{a}}(x) \) for the sequence \( \bar{a} \). b) Deter... | [
"5601355"
] | [
"A class of 25 students and 2 teachers collect money for charity. Each of the teachers donates 0, 3, or 6 coins, and each student donates 0 or 1 coin. Let \\( a_n \\) be the number of ways in which \\( n \\) coins can be collected.\n\na) Find the generating function \\( G_{\\bar{a}}(x) \\) for the sequence \\( \\ba... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7819ba32f669cc692b665198d5d65e7d281da295e77e50663e1278712476f23c | A class of 27 students is to be assigned into 3 groups, \(A\), \(B\), and \(C\). A student may belong to more than one group or none at all. The conditions are: - If a student \(X\) belongs to group \(A\), then \(X\) must also belong to group \(C\). - If a student \(X\) belongs to group \(B\), then \(X\) must also belo... | [
"\\( 18014398509481984 \\)"
] | [
"A class of 27 students is to be assigned into 3 groups, \\(A\\), \\(B\\), and \\(C\\). A student may belong to more than one group or none at all. The conditions are:\n- If a student \\(X\\) belongs to group \\(A\\), then \\(X\\) must also belong to group \\(C\\).\n- If a student \\(X\\) belongs to group \\(B\\), ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-7fbd20019b60e80578068d98c4301f3a6ecbcbe56bed8ad0b6ba9ce4b12595a2 | A class of 50 students took an exam with 4 questions. At least 1 of any 40 students gave exactly 3 correct answers, at least 2 of any 40 gave exactly 2 correct answers, and at least 3 of any 40 gave exactly 1 correct answer. At least 4 of any 40 students gave exactly 4 wrong answers. What is the least number of student... | [
"24"
] | [
"A class of 50 students took an exam with 4 questions. At least 1 of any 40 students gave exactly 3 correct answers, at least 2 of any 40 gave exactly 2 correct answers, and at least 3 of any 40 gave exactly 1 correct answer. At least 4 of any 40 students gave exactly 4 wrong answers. What is the least number of st... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 48
} | 48 | 0 | 0 |
omr-ws-2ef156e11cf8d9a5959823e835b5f9cfdbc0694e176cf15f89f88c4fd737b05d | A class of seven students is doing a Secret Santa, where each student contributes a gift. One student is absent, and the teacher randomly assigns each of the six present students to a gift that is not their own. If $P$ is the probability that the seventh student is left with his own gift once he arrives, find $\lfloor ... | [
"368"
] | [
"A class of seven students is doing a Secret Santa, where each student contributes a gift. One student is absent, and the teacher randomly assigns each of the six present students to a gift that is not their own. If $P$ is the probability that the seventh student is left with his own gift once he arrives, find $\\l... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-322bdeaafe48bcdc5eb2770f99416d5e4e0c443fba713bfd593d4775df0ef36f | A class of six students has to split into two indistinguishable teams of three people. Compute the number of distinct team arrangements that can result. | [
"10"
] | [
"A class of six students has to split into two indistinguishable teams of three people. Compute the number of distinct team arrangements that can result."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2dc13927a0b4f2f542dad34f8f0a085c314301a8ae0e223a3b35a0a73f091617 | A class of students is lined up in order of height with the tallest person in the front. At any stage, a student who is not in the first or second position is permitted to move 2 places towards the front. For what size class is it possible that the students could end up in the reverse order of height? | [
"n \\equiv 0 \\text{ or } 1 \\pmod{4}"
] | [
"A class of students is lined up in order of height with the tallest person in the front. At any stage, a student who is not in the first or second position is permitted to move 2 places towards the front. For what size class is it possible that the students could end up in the reverse order of height?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 2
} | 9 | 0 | 0 |
omr-ws-c53e72774383db4b241c2083773cda4c16e67d3cea757deb6b60986fe89f83c6 | A class of students took a test, each receiving an integer grade from \(0\) to \(10\) inclusive. If the average score of the top \(25\) students was \(9\frac{1}{5}\), and the average score of all the students was \(6\frac{2}{7}\), what is the smallest possible number of students? | [
"42"
] | [
"A class of students took a test, each receiving an integer grade from \\(0\\) to \\(10\\) inclusive. If the average score of the top \\(25\\) students was \\(9\\frac{1}{5}\\), and the average score of all the students was \\(6\\frac{2}{7}\\), what is the smallest possible number of students?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-3339a077396bde5dcbe95351793026b1464042290c93394d178467e12f995620 | A class with 8 students is divided into different groups. Each group has exactly 3 students. Any two groups have at most one student in common. Each student participates in some groups. What is the maximum number of groups? | [
"8"
] | [
"A class with 8 students is divided into different groups. Each group has exactly 3 students. Any two groups have at most one student in common. Each student participates in some groups. What is the maximum number of groups?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-4b72ad2780be455bef5ea98ddf7d2db883cec75e1ce4f1c3aa33cfc6e9ed2020 | A clay sphere with radius 13 has a circular hole of radius 5 drilled through its center. The remaining bead is molded into a new sphere. What is the radius of this new sphere? | [
"12"
] | [
"A clay sphere with radius 13 has a circular hole of radius 5 drilled through its center. The remaining bead is molded into a new sphere. What is the radius of this new sphere?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-adc0b2b8beee609b93db178a9ba5600ee983ecc082173af0ef0e5bc39d45b1fc | A clerk counts $q$ quarters, $d$ dimes, $n$ nickels, and $c$ cents. Later, it is discovered that $x$ nickels were counted as quarters and $x$ dimes were counted as cents. To correct the total obtained, what adjustment must the clerk make? | [
"-11x"
] | [
"A clerk counts $q$ quarters, $d$ dimes, $n$ nickels, and $c$ cents. Later, it is discovered that $x$ nickels were counted as quarters and $x$ dimes were counted as cents. To correct the total obtained, what adjustment must the clerk make?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a3af15a5c9f882be2ebd87f8f7f3e2624ae3c539d43e105b2239e855506cf878 | A clerk earns $8.80 an hour, plus a commission of 4% of sales. If she works a 40 hour week and sells $3600 worth of goods, what does she earn that week? | [
"496"
] | [
"A clerk earns $8.80 an hour, plus a commission of 4% of sales. If she works a 40 hour week and sells $3600 worth of goods, what does she earn that week?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b5a243b7b2fa1efa5e22a915a8a03e86d4c198d1ebf2ff1a71ab1aaca6685e0c | A cliff is $300$ meters high. Two raindrops start falling from the top of the cliff. The second raindrop starts falling when the first raindrop has already fallen $0.001$ millimeters. Calculate the distance between the two raindrops at the moment the first raindrop hits the ground, accurate to within $0.1$ mm. Neglect ... | [
"0.001 mm"
] | [
"A cliff is $300$ meters high. Two raindrops start falling from the top of the cliff. The second raindrop starts falling when the first raindrop has already fallen $0.001$ millimeters. Calculate the distance between the two raindrops at the moment the first raindrop hits the ground, accurate to within $0.1$ mm. Neg... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9f1f4b849de86864ddeef50c0c4767a3d53e376c492cd5e9da17eb4d33ca0b9f | A clock chimes on the hour a number of times equal to the hour and also chimes once every quarter hour. If the clock has just chimed once, what is the longest time you might have to wait to be sure of knowing the correct time? | [
"15 minutes"
] | [
"A clock chimes on the hour a number of times equal to the hour and also chimes once every quarter hour. If the clock has just chimed once, what is the longest time you might have to wait to be sure of knowing the correct time?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-8c90077f737fbe48bf4d191f13ee11c92d2ecf113f6a0c3064b1f16336ce70c5 | A clock chimes once at 30 minutes past each hour and chimes according to the hour on the hour. For example, at 1 PM there is one chime, and at noon and midnight there are 12 chimes. Starting at 11:15 on February 26, 2003, on what date will the 2003rd chime occur? | [
"March 9, 2003"
] | [
"A clock chimes once at 30 minutes past each hour and chimes according to the hour on the hour. For example, at 1 PM there is one chime, and at noon and midnight there are 12 chimes. Starting at 11:15 on February 26, 2003, on what date will the 2003rd chime occur?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 1
} | 11 | 0 | 0 |
omr-ws-0922bfa9205afee9fa0405364b567e53155f670b3ba7a2be096a9d2cec7309bf | A clock chimes once at 30 minutes past each hour and chimes on the hour according to the hour. For example, at 1 PM there is one chime and at noon and midnight there are twelve chimes. Starting at 11:15 AM on February 26, 2003, on what date will the 2003rd chime occur? | [
"March 9",
"March 9, 2003"
] | [
"A clock chimes once at 30 minutes past each hour and chimes on the hour according to the hour. For example, at 1 PM there is one chime and at noon and midnight there are twelve chimes. Starting at 11:15 AM on February 26, 2003, on what date will the 2003rd chime occur?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 19,
"genselect": 3,
"tir": 24
} | 46 | 0 | 0 |
omr-ws-5c18bca4853e7a4add39d29d15b9cb968be2343d8f2ced344257046a6a926d52 | A clock goes back 15 minutes every hour. If the clock shows the correct time at 9:00 a.m., what is the actual time when the clock first shows 10:00 a.m.? | [
"10:20 a.m."
] | [
"A clock goes back 15 minutes every hour. If the clock shows the correct time at 9:00 a.m., what is the actual time when the clock first shows 10:00 a.m.?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-1530d4e91f6df7f8364287ea8cb4faac5d4d61c3e23238390fb70c443026bc7d | A clock has a $10\text{cm}$ long minute hand and a $5\text{cm}$ long hour hand. How fast is the distance between the end of the minute hand and the end of the hour hand changing (in $\frac{\text{cm}}{\text{hr}}$) at $3\text{ o'clock}$? | [
"\\(-\\frac{220\\pi}{\\sqrt{5}} \\frac{\\text{cm}}{\\text{hr}}\\)"
] | [
"A clock has a $10\\text{cm}$ long minute hand and a $5\\text{cm}$ long hour hand. How fast is the distance between the end of the minute hand and the end of the hour hand changing (in $\\frac{\\text{cm}}{\\text{hr}}$) at $3\\text{ o'clock}$?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4dee150cba70e4c5b63c41275525cd282b789b6c9168a4bbed5560d6d608340a | A clock has a radius of 20 meters. At what rate is the distance between the tip of the second hand and the 12 o'clock mark changing when the second hand points to the 4 o'clock position? | [
"\\(\\frac{\\pi}{3}\\)"
] | [
"A clock has a radius of 20 meters. At what rate is the distance between the tip of the second hand and the 12 o'clock mark changing when the second hand points to the 4 o'clock position?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-2273d906e8ada98c86bfd05530f25f157e493cdd96672d3f68bd7ed82a5c0772 | A clock has an hour, minute, and second hand, all of length $1$. Let $T$ be the triangle formed by the ends of these hands. If a time of day is chosen uniformly at random, what is the expected value of the area of $T$? | [
"\\(\\frac{3}{2\\pi}\\)"
] | [
"A clock has an hour, minute, and second hand, all of length $1$. Let $T$ be the triangle formed by the ends of these hands. If a time of day is chosen uniformly at random, what is the expected value of the area of $T$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 6,
"tir": 6
} | 16 | 0 | 0 |
omr-ws-4c7f26ec43b35641f349b7f31cd7d951ab724aeb13fadc5492c3d62e137d00de | A clock has twice the radius of a circular disk. The disk is placed externally tangent to the 12 o'clock point on the clock. The clock starts rotating in place clockwise such that it completes one rotation in 3 seconds. At the same time, the disk starts rotating in place counterclockwise such that it completes one rota... | [
"8"
] | [
"A clock has twice the radius of a circular disk. The disk is placed externally tangent to the 12 o'clock point on the clock. The clock starts rotating in place clockwise such that it completes one rotation in 3 seconds. At the same time, the disk starts rotating in place counterclockwise such that it completes one... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 1
} | 9 | 0 | 0 |
omr-ws-fcecc7ca7ae87809bba042a556aaa2bde7d05134cdde12eb5a7b5cc1213bfd1e | A clock has two hands of equal length. A time on the clock face is called "ambiguous" if one can swap the two hands (minute and hour) and it reads another time. What is the number of "ambiguous" times on the clock? | [
"132"
] | [
"A clock has two hands of equal length. A time on the clock face is called \"ambiguous\" if one can swap the two hands (minute and hour) and it reads another time. What is the number of \"ambiguous\" times on the clock?"
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 5
} | 5 | 0 | 0 |
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