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-14686c683866b34161dda054c923132a1cbfa05e065a20b686ee18607d7eacd9 | A clock has two hands, a big hand and a small hand. The following operations can be performed: 1) Rotate the big hand by any angle in either direction, which will also rotate the small hand in the same direction but by an angle 12 times smaller. 2) Simultaneously rotate both hands by the same arbitrary angle in the sam... | [
"180"
] | [
"A clock has two hands, a big hand and a small hand. The following operations can be performed:\n1) Rotate the big hand by any angle in either direction, which will also rotate the small hand in the same direction but by an angle 12 times smaller.\n2) Simultaneously rotate both hands by the same arbitrary angle in ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-964bf6260f28ec6d9084b2aa39d299002811f018c1563013a68e73ee28b9cbb7 | A clock loses 3 minutes every hour. If it is set correctly at 1pm, what will the clock read when the correct time is 10am the next day? | [
"8:57"
] | [
"A clock loses 3 minutes every hour. If it is set correctly at 1pm, what will the clock read when the correct time is 10am the next day?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 1,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-827940fd56fe90c4a26cf9a4ac5b9393aab9ad946b3c2b73559b920ea6a41cde | A clock rolls at constant speed on a flat table forever, and a fly is stuck to the minute hand of the clock. At 3:45 pm, the clock is perfectly vertical, and the fly is 10 cm above the table. At 4:00 pm, the clock has rolled exactly once, and the fly is 14 cm above the table. How far above the table is the fly at 5:05 ... | [
"8 cm"
] | [
"A clock rolls at constant speed on a flat table forever, and a fly is stuck to the minute hand of the clock. At 3:45 pm, the clock is perfectly vertical, and the fly is 10 cm above the table. At 4:00 pm, the clock has rolled exactly once, and the fly is 14 cm above the table. How far above the table is the fly at ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-9451b2bb6c7f40fb93abb777f5bb7c3baeb6d2196cf83b7905d3e9d1acbd3cac | A clock's minute hand catches up to the hour hand every 66 minutes. How many minutes is the clock late within an hour? | [
"0.50"
] | [
"A clock's minute hand catches up to the hour hand every 66 minutes. How many minutes is the clock late within an hour?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-6a38f722d6d6c89bbf00800c12040089c45f3ca4f5132a7254300deb39600516 | A clock's minute hand has a length of 4 and its hour hand has a length of 3. What is the distance between the tips of the hands at the moment when the distance between them is increasing most rapidly? | [
"\\(\\sqrt{7}\\)"
] | [
"A clock's minute hand has a length of 4 and its hour hand has a length of 3. What is the distance between the tips of the hands at the moment when the distance between them is increasing most rapidly?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-88d280518f36834a3e968013b08d4076533fe0ba1e5331497f8476eb9d903f38 | A clock's minute hand has length $x$ and its hour hand has length $y$ ($x>y$). What is the distance between the tips at the moment when it is increasing most rapidly? | [
"\\sqrt{x^2 - y^2}"
] | [
"A clock's minute hand has length $x$ and its hour hand has length $y$ ($x>y$). What is the distance between the tips at the moment when it is increasing most rapidly?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6735821e97d69b45b5cd7c6bcf350fcff3c826fe13b8b45443023481c7267f5e | A closed and bounded figure \( \Phi \) has the property that any two points of \( \Phi \) can be connected by a half-circle lying completely in \( \Phi \). Find the figure \( \Phi \). | [
"disk"
] | [
"A closed and bounded figure \\( \\Phi \\) has the property that any two points of \\( \\Phi \\) can be connected by a half-circle lying completely in \\( \\Phi \\). Find the figure \\( \\Phi \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 6
} | 17 | 0 | 0 |
omr-ws-e724a1747d9102e39545affe42ab973b2f5f4a1d3b211fd38978a0ef2f0d5c2c | A closed box with a square base of length \( w \) and height \( h \) is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet. The four corners of the wrapping paper are folded up over the sides and meet at ... | [
"\\(2(w+h)^2\\)"
] | [
"A closed box with a square base of length \\( w \\) and height \\( h \\) is to be wrapped with a square sheet of wrapping paper. The box is centered on the wrapping paper with the vertices of the base lying on the midlines of the square sheet. The four corners of the wrapping paper are folded up over the sides and... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4070f1962ec47f097632afa4f551ad78f3147298874d420aac46a4af416d609d | A closed broken $n$-line $A_1A_2...A_nA_1$ of length $d$ is given. Circles $D_1, D_2, ..., D_n$ have centers $A_1,..., A_n$ respectively. For each $i = 1, 2,..., n$ the circles $D_i$ and $D_{i + 1}$ do not have common internal points (we assume that $D_{n + 1} = D_1$). Let $R$ be the radius of the smallest circle that ... | [
"\\dfrac{d}{2}"
] | [
"A closed broken $n$-line $A_1A_2...A_nA_1$ of length $d$ is given. Circles $D_1, D_2, ..., D_n$ have centers $A_1,..., A_n$ respectively. For each $i = 1, 2,..., n$ the circles $D_i$ and $D_{i + 1}$ do not have common internal points (we assume that $D_{n + 1} = D_1$). Let $R$ be the radius of the smallest circle ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fe104160d7e66e4c696f21b3305886f5433acd647e5680fe7b18e089fecc8f9a | A closed broken line consists of $203$ straight links, with no two links contained in the same line. What is the greatest number of self-intersections such a broken line may have? | [
"20453"
] | [
"A closed broken line consists of $203$ straight links, with no two links contained in the same line. What is the greatest number of self-intersections such a broken line may have?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fb6222115e9fb7ab216bac90e5f4cd12145662a59dc578a6ddd831ac6e4d2aac | A closed conical container of height 12 cm is partially filled with liquid. When the container is inverted, the liquid reaches exactly the same mark as when the vertex was pointed down. What is the depth of the liquid in the first position? | [
"\\(12 - 6 \\sqrt[3]{4}\\)"
] | [
"A closed conical container of height 12 cm is partially filled with liquid. When the container is inverted, the liquid reaches exactly the same mark as when the vertex was pointed down. What is the depth of the liquid in the first position?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-074ddc34ab69bf3f607032c16afafbce57d710035f8b192b885aa62e6a1f30e4 | A closed conical container of height 12 cm is partially filled with liquid. When the container is inverted, the liquid reaches the same mark as before. Determine the depth of the liquid in the first position. | [
"\\( 2\\sqrt[3]{3} \\)"
] | [
"A closed conical container of height 12 cm is partially filled with liquid. When the container is inverted, the liquid reaches the same mark as before. Determine the depth of the liquid in the first position."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-d7fdd4e783e8162b6fd5a819e3b1bb8d006d0b5d5c5ac2c8bc570ff687e5c070 | A closed container in the shape of a rectangular parallelepiped contains $1$ liter of water. If the container rests horizontally on three different sides, the water levels are $2$ cm, $4$ cm, and $5$ cm. Calculate the volume of the parallelepiped. | [
"5000 cm\\(^3\\)"
] | [
"A closed container in the shape of a rectangular parallelepiped contains $1$ liter of water. If the container rests horizontally on three different sides, the water levels are $2$ cm, $4$ cm, and $5$ cm. Calculate the volume of the parallelepiped."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 14
} | 29 | 0 | 0 |
omr-ws-dd2a863f93772c71f868cf42d63a802fa2aafb2b86315fd65d0c045b4ae6c299 | A closed five-link broken line is inscribed in a sphere of diameter $1$, with each link having length $\ell$. What is the largest possible value of $\ell$? | [
"\\(\\frac{\\sqrt{10 - 2\\sqrt{5}}}{4}\\)"
] | [
"A closed five-link broken line is inscribed in a sphere of diameter $1$, with each link having length $\\ell$. What is the largest possible value of $\\ell$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 8
} | 8 | 0 | 0 |
omr-ws-d90ae205b4ca2df32180c96dd1b86476e5ff3ba3cc77ec92c75737779a177590 | A closed interval $I$ is defined as "boring" if it contains no "ridiculous numbers" and is contained within an interval $[b, c]$ where $b$ and $c$ are both "ridiculous numbers." A number $a$ is "ridiculous" if $1$ is ridiculous and if $a$ is ridiculous, then $\sqrt{a}$ and $1 + \sqrt{a}$ are also ridiculous. The smalle... | [
"9"
] | [
"A closed interval $I$ is defined as \"boring\" if it contains no \"ridiculous numbers\" and is contained within an interval $[b, c]$ where $b$ and $c$ are both \"ridiculous numbers.\" A number $a$ is \"ridiculous\" if $1$ is ridiculous and if $a$ is ridiculous, then $\\sqrt{a}$ and $1 + \\sqrt{a}$ are also ridicul... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-ad49cf81b22d4b4eb6e4390aacb4698dbeb4671fb0af70ccf7562fa0080e431d | A closed non-self-intersecting polygonal chain is drawn through the centers of some squares on an $8 \times 8$ chessboard. Each link of the chain connects the centers of adjacent squares either horizontally, vertically, or diagonally. Let $B$ be the total area of black squares within the interior of the polygon, and $W... | [
"0"
] | [
"A closed non-self-intersecting polygonal chain is drawn through the centers of some squares on an $8 \\times 8$ chessboard. Each link of the chain connects the centers of adjacent squares either horizontally, vertically, or diagonally. Let $B$ be the total area of black squares within the interior of the polygon, ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-0b09089035b05b98d347ac960eb3c62b4dbd6bea296adc1b598168fb97a66f11 | A closed polyline passes through all faces of a cube with side length $1$. Let $L$ be the length of the shortest such polyline. Find the value of $L^2$. | [
"18"
] | [
"A closed polyline passes through all faces of a cube with side length $1$. Let $L$ be the length of the shortest such polyline. Find the value of $L^2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-c3ca7a3af5310756ca2c44eff8ebb4ba17d171760b583e15560eab79ba946f01 | A closed rectangular box has dimensions \( x \), \( y \), and \( z \) with a volume of 10 cubic feet. The cost of material for each of the six sides is as follows: the bottom costs 1 cent per square foot, the top costs 2 cents per square foot, the front and back faces each cost 3 cents per square foot, and the left and... | [
"\\( x = 2\\sqrt[3]{10}, y = \\sqrt[3]{10}, z = \\frac{\\sqrt[3]{10}}{2} \\)"
] | [
"A closed rectangular box has dimensions \\( x \\), \\( y \\), and \\( z \\) with a volume of 10 cubic feet. The cost of material for each of the six sides is as follows: the bottom costs 1 cent per square foot, the top costs 2 cents per square foot, the front and back faces each cost 3 cents per square foot, and t... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 6,
"tir": 20
} | 29 | 0 | 0 |
omr-ws-b395b6888dd95358d8e586e10c3e232e50cedef1137f693fc617d315c3902d5a | A closed rectangular box with a square base has a total surface area of 16 square feet. If the length of a main diagonal of the box is 3 feet, what are the dimensions of the box? | [
"\\(2 \\times 2 \\times 1\\) or \\(\\frac{4}{3} \\times \\frac{4}{3} \\times \\frac{7}{3}\\)"
] | [
"A closed rectangular box with a square base has a total surface area of 16 square feet. If the length of a main diagonal of the box is 3 feet, what are the dimensions of the box?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 8,
"tir": 25
} | 33 | 0 | 0 |
omr-ws-2e4c16d6a50dd1a219aabff3e816ef3075a83f003a43d68b222a64d41a466c55 | A closed rectangular box with a volume of 16 cubic feet is made from two kinds of materials. The top and bottom are made of material costing 10 cents per square foot, and the sides are made of material costing 5 cents per square foot. What are the dimensions of the box so that the cost of the materials is minimized? | [
"2 \\text{ ft} \\times 2 \\text{ ft} \\times 4 \\text{ ft}"
] | [
"A closed rectangular box with a volume of 16 cubic feet is made from two kinds of materials. The top and bottom are made of material costing 10 cents per square foot, and the sides are made of material costing 5 cents per square foot. What are the dimensions of the box so that the cost of the materials is minimize... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-d29d236e631a92d824661ad7186806acd5a96d52541d6720103f8ccddc6b4c89 | A closed rectangular jewelry box has outer dimensions of 6 inches, 4 inches, and 3 inches. The box has a uniform thickness \(x\) and an inside volume of 30 cubic inches. Find \(x\). | [
"\\dfrac{1}{2}"
] | [
"A closed rectangular jewelry box has outer dimensions of 6 inches, 4 inches, and 3 inches. The box has a uniform thickness \\(x\\) and an inside volume of 30 cubic inches. Find \\(x\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d887a306f37002d1e8f7baed905187b28bc07c50829cb6ff6fa7fc6b9d0cee48 | A closed, non-self-intersecting broken line \( L \) is drawn over a \((2n+1) \times (2n+1)\) chessboard such that the set of \( L \)'s vertices coincides with the set of the vertices of the board’s squares and every edge in \( L \) is a side of some board square. All board squares lying in the interior of \( L \) are c... | [
"0"
] | [
"A closed, non-self-intersecting broken line \\( L \\) is drawn over a \\((2n+1) \\times (2n+1)\\) chessboard such that the set of \\( L \\)'s vertices coincides with the set of the vertices of the board’s squares and every edge in \\( L \\) is a side of some board square. All board squares lying in the interior of... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-8cacb22858c402392648b8c18b596e7e676a8184ce6b154e1a23319313a23afe | A closet contains \( n \) distinct pairs of shoes. If \( 2r \) shoes are selected at random (\( 2r < n \)), find the probability that there is no complete pair in the selection. | [
"\\(\\frac{2^{2r} \\binom{n}{2r}}{\\binom{2n}{2r}}\\)"
] | [
"A closet contains \\( n \\) distinct pairs of shoes. If \\( 2r \\) shoes are selected at random (\\( 2r < n \\)), find the probability that there is no complete pair in the selection."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-8962cfd806d757220079a7ba57bebe08a2499810a903ec8affd0b71290e957ce | A cloth bag contains a solid pool ball. A second pool ball is chosen at random with probability $p$ of being a solid and $1-p$ of being a stripe. The second ball is added to the bag, the bag is shaken, and a ball is drawn at random, which turns out to be a solid. What is the probability that the ball remaining in the b... | [
"$\\frac{2p}{p + 1}$"
] | [
"A cloth bag contains a solid pool ball. A second pool ball is chosen at random with probability $p$ of being a solid and $1-p$ of being a stripe. The second ball is added to the bag, the bag is shaken, and a ball is drawn at random, which turns out to be a solid. What is the probability that the ball remaining in ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-245dccc7950284bf66652f85a1b7b43549a12cd7f5b6f526980ee80a7932907a | A clothing manufacturer has 100m of silk and 180m of wool. To make a suit requires 2m of silk and 2m of wool, and to make a dress requires 1m of silk and 3m of wool. If the profit on a suit is \$108 and the profit on a dress is \$60, how many suits and dresses should the manufacturer make to maximize the profit? | [
"5880"
] | [
"A clothing manufacturer has 100m of silk and 180m of wool. To make a suit requires 2m of silk and 2m of wool, and to make a dress requires 1m of silk and 3m of wool. If the profit on a suit is \\$108 and the profit on a dress is \\$60, how many suits and dresses should the manufacturer make to maximize the profit?... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fd82ba6220b0ea0e31509f0ac961f569aa63e7b5846f4cce42572f637a765822 | A clown is two steps from the end of a trampoline and will pull 10 balls, 5 red and 5 yellow. Removing a red ball moves the clown one step forward, and removing a yellow ball moves the clown one step back. How many ways can the clown pull the balls so that the clown does not fall off the trampoline? | [
"132"
] | [
"A clown is two steps from the end of a trampoline and will pull 10 balls, 5 red and 5 yellow. Removing a red ball moves the clown one step forward, and removing a yellow ball moves the clown one step back. How many ways can the clown pull the balls so that the clown does not fall off the trampoline?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 27,
"genselect": 7,
"tir": 14
} | 48 | 0 | 0 |
omr-ws-8e14de96cb4fc3bd0e7c72d2ef9bcc2efb5324543f9dc9f4706186d7fb762e4b | A clown walks around the edge of a 10 x 3 rectangle (float) at 2 m/s in a clockwise direction while the float moves forward at 1 m/s. What is the general equation describing the clown's path relative to the ground? | [
"\n\\begin{cases}\nx(t) = t \\\\\ny(t) = 26 - 2t + 26k\n\\end{cases}\n"
] | [
"A clown walks around the edge of a 10 x 3 rectangle (float) at 2 m/s in a clockwise direction while the float moves forward at 1 m/s. What is the general equation describing the clown's path relative to the ground?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0e498637a885c6dd3d1ca963a2728ad93fdb32ba9f1cc5cd6473ef31d422a03e | A club consisting of $11$ men and $12$ women needs to choose a committee from among its members so that the number of women on the committee is one more than the number of men on the committee. The committee could have as few as $1$ member or as many as $23$ members. Let $N$ be the number of such committees that can be... | [
"081"
] | [
"A club consisting of $11$ men and $12$ women needs to choose a committee from among its members so that the number of women on the committee is one more than the number of men on the committee. The committee could have as few as $1$ member or as many as $23$ members. Let $N$ be the number of such committees that c... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-4c23c48ec4eeea777a04fe4f1f33313399113191fee4d3a60ba87a243d3e90ef | A club has $2012$ members. Except for its founders, each person must be nominated by exactly one existing member before joining. Every member is issued a membership card. For administrative purposes, the membership card of each non-founding member must be of the same color as that of the person who nominated them, and ... | [
"50"
] | [
"A club has $2012$ members. Except for its founders, each person must be nominated by exactly one existing member before joining. Every member is issued a membership card. For administrative purposes, the membership card of each non-founding member must be of the same color as that of the person who nominated them,... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-99fc5e5a977fd207383d2baaf43bb5868e512c42fc35b8f2fc46ad2a0eb0fad1 | A club has $5$ members from each of $3$ different schools, for a total of $15$ members. How many possible ways are there to arrange a presidency meeting under the following conditions: i. The club must choose one of the $3$ schools at which to host the meeting, and ii. The host school sends $2$ representatives to the m... | [
"750"
] | [
"A club has $5$ members from each of $3$ different schools, for a total of $15$ members. How many possible ways are there to arrange a presidency meeting under the following conditions:\n\ni. The club must choose one of the $3$ schools at which to host the meeting, and\n\nii. The host school sends $2$ representativ... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ed4cfc0a3e7519c4bba761488ee62181b30cf7f7ae82a3830a6e4eb373586878 | A club has $n$ members and is divided into $10$ smaller clubs, each with $7$ members. For any two random members, they will join no more than one club together. Find the minimum value of $n$. | [
"22"
] | [
"A club has $n$ members and is divided into $10$ smaller clubs, each with $7$ members. For any two random members, they will join no more than one club together. Find the minimum value of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-79508ba2c0686df66d0b6ee75790f1cb7c349bf4a3e5fc3e7c6c4df454b799a5 | A club has $n$ members and is organized into 12 topics, with each topic having 24 members. Any two topics have no more than 1 common member. Find the minimum value of $n$. | [
"222"
] | [
"A club has $n$ members and is organized into 12 topics, with each topic having 24 members. Any two topics have no more than 1 common member. Find the minimum value of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-f6e938242c5d85b7e1516627637280c0ad9f9c8bea4e71e35623f69b8374582a | A club has 10 members, 5 boys and 5 girls. Two of the members are chosen at random. What is the probability that they are both girls? | [
"\\dfrac{2}{9}"
] | [
"A club has 10 members, 5 boys and 5 girls. Two of the members are chosen at random. What is the probability that they are both girls?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c8f2cdd310df134ca4dd94dac4e821d35febf212b8f850ce21510242dce6a985 | A club has 12 members, with 3 members from each of the years 10, 11, 12, and 13. In each of the year 10 and year 12 classes, one member takes Computing and two take Physics. In each of the year 11 and year 13 classes, two members take Computing and one takes Physics. Find the number of ways to form a committee of 6 mem... | [
"592"
] | [
"A club has 12 members, with 3 members from each of the years 10, 11, 12, and 13. In each of the year 10 and year 12 classes, one member takes Computing and two take Physics. In each of the year 11 and year 13 classes, two members take Computing and one takes Physics. Find the number of ways to form a committee of ... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 38,
"genselect": 8,
"tir": 7
} | 53 | 0 | 0 |
omr-ws-4a599236db36a7bb3c1b74d216df311b22b8a2d482ca4292356f6d72a273d255 | A club has 15 members and needs to choose 2 members to be co-presidents. In how many ways can the club choose its co-presidents? | [
"105"
] | [
"A club has 15 members and needs to choose 2 members to be co-presidents. In how many ways can the club choose its co-presidents?",
"A club has 15 members and needs to choose 2 members to be co-presidents. In how many ways can the club choose its co-presidents?"
] | [
"MATH_training_set",
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 2
} | 33 | 0 | 0 |
omr-ws-a79488a24bea4f7fda23aa37e178e1a95fb90be62962181e0b5166d728a40bea | A club has 80 members who play golf in foursomes (groups of 4). In each round, 20 foursomes play. The Golf Association requires that each member must play at least once with every other member. What is the minimum number of rounds the club must schedule to ensure that all members play at least once with every other pla... | [
"27"
] | [
"A club has 80 members who play golf in foursomes (groups of 4). In each round, 20 foursomes play. The Golf Association requires that each member must play at least once with every other member. What is the minimum number of rounds the club must schedule to ensure that all members play at least once with every othe... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dbd4ff365ede629565dc14a4396d6676a98ec614151e72fa6b813be2463f1422 | A club of $n$ members is organized into four committees such that each member belongs to exactly two committees and each pair of committees has exactly one member in common. Find all possible values of $n$. | [
"\\( n = 6 \\)"
] | [
"A club of $n$ members is organized into four committees such that each member belongs to exactly two committees and each pair of committees has exactly one member in common. Find all possible values of $n$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-4518de293f5158d43dd83624e38ea09c2656b162a428d8b3a7825770123beee3 | A club wants to induct 3 new members from 20 possible candidates. The probability of each candidate getting in is weighted such that the first candidate has probability $\frac{1}{n}$, the second has probability $\frac{2}{n}$, and so on. Find $n$. | [
"70"
] | [
"A club wants to induct 3 new members from 20 possible candidates. The probability of each candidate getting in is weighted such that the first candidate has probability $\\frac{1}{n}$, the second has probability $\\frac{2}{n}$, and so on. Find $n$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-781780d7ce508f762854386a36c0d0ef882fc4deeda16de2d27fbc8c162a6c6f | A club with 7 members forms three-person committees, but no two committees can have more than one member in common. What is the maximum number of committees that can be formed? | [
"7"
] | [
"A club with 7 members forms three-person committees, but no two committees can have more than one member in common. What is the maximum number of committees that can be formed?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-da5c9e151d0645409cfc5609386f9912bfe60c6f8bf7c26c2cfc483525cbf7f3 | A club with \( x \) members is organized into four committees according to the following rules: - Each member belongs to exactly two committees. - Each pair of committees has exactly one member in common. Determine the value of \( x \). | [
"6"
] | [
"A club with \\( x \\) members is organized into four committees according to the following rules:\n- Each member belongs to exactly two committees.\n- Each pair of committees has exactly one member in common.\nDetermine the value of \\( x \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1bbaebf4a89536c6f98e3962bf8d2c34bb80df1b8ebf8bdc53ee3c4c98c830b5 | A club with \( x \) members is organized into four committees such that each member belongs to exactly two committees and each pair of committees has exactly one member in common. Determine the value of \( x \). | [
"6"
] | [
"A club with \\( x \\) members is organized into four committees such that each member belongs to exactly two committees and each pair of committees has exactly one member in common. Determine the value of \\( x \\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f4f8b720be90fc8e7ca7fe1e482638fabc35dade4eec2ebd32578e6b6d03e021 | A coastal battery sights an enemy cruiser lying one kilometer off the coast and opens fire on it at the rate of one round per minute. After the first shot, the cruiser begins to move away at a speed of 60 kilometers per hour. The probability of a hit is given by \(0.75x^{-2}\), where \(x\) denotes the distance (in kilo... | [
"7"
] | [
"A coastal battery sights an enemy cruiser lying one kilometer off the coast and opens fire on it at the rate of one round per minute. After the first shot, the cruiser begins to move away at a speed of 60 kilometers per hour. The probability of a hit is given by \\(0.75x^{-2}\\), where \\(x\\) denotes the distance... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b954984f5be2c8db8f7553de06922dfeadc15570325ee43b447e77535e197847 | A coastal battery sights an enemy ship 1 kilometre away. The battery fires one shell per minute, and the ship moves directly away from the battery at 1 kilometre per minute. The probability of a shot hitting the ship is \(\frac{0.75}{x^2}\), where \(x\) is the distance in kilometres between the battery and the ship at ... | [
"\\dfrac{2\\sqrt{2}}{3\\pi}"
] | [
"A coastal battery sights an enemy ship 1 kilometre away. The battery fires one shell per minute, and the ship moves directly away from the battery at 1 kilometre per minute. The probability of a shot hitting the ship is \\(\\frac{0.75}{x^2}\\), where \\(x\\) is the distance in kilometres between the battery and th... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5053f85d0905fcc8fc3920d2fd4151abfd0f1f149482fd741f53ab4b8011773e | A coat is covered by 5 patches, each with an area of at least 0.5. Calculate the minimum possible area of intersection between any two patches. | [
"\\dfrac{3}{20}"
] | [
"A coat is covered by 5 patches, each with an area of at least 0.5. Calculate the minimum possible area of intersection between any two patches."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-93b18b89d39185a88c9a525864a9fbb4e9bada3395ccb36611e183eaa9404d82 | A code is developed as follows. Begin by assigning A=1, B=1, ..., Z=26 and space = 0. Then a message without punctuation can be written as a list of numbers \(x_1, y_1, x_2, y_2, \ldots, x_m, y_m\) with a space added at the end (if needed) so that the list has an even number of terms. To code this message, we write a n... | [
"MATH IS EU N"
] | [
"A code is developed as follows. Begin by assigning A=1, B=1, ..., Z=26 and space = 0. Then a message without punctuation can be written as a list of numbers \\(x_1, y_1, x_2, y_2, \\ldots, x_m, y_m\\) with a space added at the end (if needed) so that the list has an even number of terms. To code this message, we w... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 16
} | 31 | 0 | 0 |
omr-ws-3e10d66db51c5eedde8bb81cdb8be3698c6ec13fc9981f60a2eeda3ca2b1bfc5 | A coffee manufacturer wants to create a new blend of coffee that costs $3.90 per pound by mixing two types of coffee that sell for $2.75 and $5.00 per pound, respectively. What amounts of each type of coffee should be blended to obtain the desired mixture, assuming the total weight of the desired blend is 100 pounds? | [
"\\( 48.\\overline{8} \\) pounds of $2.75 coffee and \\( 51.\\overline{1} \\) pounds of $5.00 coffee."
] | [
"A coffee manufacturer wants to create a new blend of coffee that costs $3.90 per pound by mixing two types of coffee that sell for $2.75 and $5.00 per pound, respectively. What amounts of each type of coffee should be blended to obtain the desired mixture, assuming the total weight of the desired blend is 100 poun... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 32
} | 46 | 0 | 0 |
omr-ws-21a4edacdcf5fc7ada7f27d789365a6ac85b7a805d5b564ebc1918f294b78287 | A coffee shop owner has a balance beam scale and wants to purchase standard weights to weigh coffee. She wants to be able to weigh out all integral numbers of ounces up to $n$ using the weights. a) What 3 weights should she buy to maximize $n$ and what is the maximum value of $n$ using 3 weights? b) What 4 weights shou... | [
"a) \\( 1, 3, 9 \\) and \\( n = 13 \\)\nb) \\( 1, 3, 9, 27 \\) and \\( n = 40 \\)\nc) \\( 1, 3, 9, \\ldots, 3^{p-1} \\) and \\( n = \\frac{3^p - 1}{2} \\)"
] | [
"A coffee shop owner has a balance beam scale and wants to purchase standard weights to weigh coffee. She wants to be able to weigh out all integral numbers of ounces up to $n$ using the weights.\na) What 3 weights should she buy to maximize $n$ and what is the maximum value of $n$ using 3 weights?\nb) What 4 weigh... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 29
} | 30 | 0 | 0 |
omr-ws-ef109a4e7011785eea87b2613b30323823da0784f9ff81bf18cb5efcbd917c30 | A coin A of radius 1 is placed on a table. A second coin B of radius 3 is placed externally tangent to A. A third coin C of radius 2 is placed tangent to both A and B. A fourth coin D of radius 2 is placed tangent to A and C but not B. Finally, a coin E is tangent to A, B, and D but not C. What is the radius of coin E? | [
"2.92284345"
] | [
"A coin A of radius 1 is placed on a table. A second coin B of radius 3 is placed externally tangent to A. A third coin C of radius 2 is placed tangent to both A and B. A fourth coin D of radius 2 is placed tangent to A and C but not B. Finally, a coin E is tangent to A, B, and D but not C. What is the radius of co... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-fca6fd8fc1d4f3f7108a814c4db93f6eaa85457ae36063d9df7f3b3fa45a4fc6 | A coin collection consists of nickels, dimes, and quarters. There are 20 coins in the collection with a total value of $2.00, and the number of nickels is three times the number of quarters. Find all possible solutions for the numbers of each type of coin. | [
"\\((0, 20, 0), (3, 16, 1), (6, 12, 2), (9, 8, 3), (12, 4, 4), (15, 0, 5)\\)"
] | [
"A coin collection consists of nickels, dimes, and quarters. There are 20 coins in the collection with a total value of $2.00, and the number of nickels is three times the number of quarters. Find all possible solutions for the numbers of each type of coin."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 5,
"tir": 18
} | 30 | 0 | 0 |
omr-ws-4dc5d7073d4039d1177a4eaa47d859a8e4412171b405bc6511cc1cb05dd85560 | A coin has a $\frac{2}{3}$ chance of rolling heads. It is rolled $1000$ times. What is the probability that the number of heads rolled is an even number? | [
"\\(\\dfrac{3^{1000}+1}{2\\cdot 3^{1000}}\\)"
] | [
"A coin has a $\\frac{2}{3}$ chance of rolling heads. It is rolled $1000$ times. What is the probability that the number of heads rolled is an even number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-43584bfa1315758e04ecaef7cb818b7902d2262ea8d89339d7d4694f6ccfa2c0 | A coin has a probability of $\frac{1}{3}$ for coming up heads and $\frac{2}{3}$ for coming up tails. On average, how many flips of this coin are needed to guarantee that both heads and tails appear at least once? | [
"\\(\\frac{7}{2}\\)"
] | [
"A coin has a probability of $\\frac{1}{3}$ for coming up heads and $\\frac{2}{3}$ for coming up tails. On average, how many flips of this coin are needed to guarantee that both heads and tails appear at least once?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-3be5b320ead08201eb9735c1be7dae6c01ff7eba2a5aee8efb3c9233279526a0 | A coin has the chance of landing heads, tails, or side with a probability of $\frac{1}{3}$ each. When continuously flipping this coin, what is the probability of observing the continuous sequence HMMT before HMT? | [
"\\(\\frac{1}{4}\\)"
] | [
"A coin has the chance of landing heads, tails, or side with a probability of $\\frac{1}{3}$ each. When continuously flipping this coin, what is the probability of observing the continuous sequence HMMT before HMT?"
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 8
} | 8 | 0 | 0 |
omr-ws-621312cfdc3825f1da7180cc31009f1ac9701b3897d4d9556c2bba4dcedd1b97 | A coin has the chance of landing heads, tails, or side with a probability of $\frac{1}{3}$ each. When continuously flipping this coin, what is the probability of observing the continuous sequence HTMT before HMT? | [
"\\dfrac{1}{4}"
] | [
"A coin has the chance of landing heads, tails, or side with a probability of $\\frac{1}{3}$ each. When continuously flipping this coin, what is the probability of observing the continuous sequence HTMT before HMT?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-09f958749ed29691b40814840b84e3209c56a11aa7559af956f387cfc9cccfda | A coin is altered so that the probability that it lands on heads is less than $\frac{1}{2}$. When the coin is flipped four times, the probability of an equal number of heads and tails is $\frac{1}{6}$. What is the probability that the coin lands on heads? | [
"\\(\\frac{3 - \\sqrt{3}}{6}\\)"
] | [
"A coin is altered so that the probability that it lands on heads is less than $\\frac{1}{2}$. When the coin is flipped four times, the probability of an equal number of heads and tails is $\\frac{1}{6}$. What is the probability that the coin lands on heads?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-b9de891814765b3a4d2113afc69bada6bf92da81c79ebbb321a58e444d0b44a6 | A coin is altered so that the probability that it lands on heads is less than $\frac{1}{2}$. When the coin is flipped four times, the probability of getting an equal number of heads and tails is $\frac{1}{6}$. What is the probability that the coin lands on heads? | [
"\\(\\frac{3-\\sqrt{3}}{6}\\)"
] | [
"A coin is altered so that the probability that it lands on heads is less than $\\frac{1}{2}$. When the coin is flipped four times, the probability of getting an equal number of heads and tails is $\\frac{1}{6}$. What is the probability that the coin lands on heads?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-40abc630a636ab9e386e80845bab1cb30342bebbab2b1f2290c5b633c1daadba | A coin is biased so that it is twice as likely to land heads as tails. A and B are playing a game where the first to toss a head wins. They take alternate tosses. Find the probability that A wins if he tosses the coin first. | [
"\\(\\frac{3}{4}\\)"
] | [
"A coin is biased so that it is twice as likely to land heads as tails. A and B are playing a game where the first to toss a head wins. They take alternate tosses. Find the probability that A wins if he tosses the coin first."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-80c0cc10d2ae8e66a0cd16c0ddf5ab71e6ed63816a44f9ce80869020d666c2e7 | A coin is biased so that it is twice as likely to land heads as tails. A, B, and C are playing a game where the first to toss a head wins. They take alternate tosses. Find the probability that A wins. | [
"\\(\\frac{9}{13}\\)"
] | [
"A coin is biased so that it is twice as likely to land heads as tails. A, B, and C are playing a game where the first to toss a head wins. They take alternate tosses. Find the probability that A wins."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-65fe7cbe5106efd1d472852da665bf2397295883a1c7395994a9342be0b37af5 | A coin is biased so that it is twice as likely to land heads as tails. If $n \geq 2$ people are playing a game where the first to toss a head wins and they take alternate tosses, find the probability that A wins in terms of $n$. | [
"\\(\\frac{2}{3} \\cdot \\frac{1}{1 - \\left( \\frac{1}{3} \\right)^n}\\)"
] | [
"A coin is biased so that it is twice as likely to land heads as tails. If $n \\geq 2$ people are playing a game where the first to toss a head wins and they take alternate tosses, find the probability that A wins in terms of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 5
} | 53 | 0 | 0 |
omr-ws-e9de1bd5d20a6d8dc7725ea841d674a47afe3ab90153345477b4e7a1b3b900c1 | A coin is biased so that the probability of obtaining a head is $p$, where $0 < p < 1$. Two players, A and B, take turns flipping the coin until one of the sequences $HHH$ or $HTH$ occurs. If the sequence $HHH$ occurs first, then A wins. If $HTH$ occurs first, then B wins. For what value of $p$ is the game fair? | [
"\\(\\frac{3 - \\sqrt{5}}{2}\\)"
] | [
"A coin is biased so that the probability of obtaining a head is $p$, where $0 < p < 1$. Two players, A and B, take turns flipping the coin until one of the sequences $HHH$ or $HTH$ occurs. If the sequence $HHH$ occurs first, then A wins. If $HTH$ occurs first, then B wins. For what value of $p$ is the game fair?"
... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5ed38e3f1d7d06f17ed5ff18b813b1d4d898bdcb37d29957767700886e124960 | A coin is biased such that the probability of landing on a digit is $\frac{1}{4}$. If the coin is tossed $n$ times, the probability of getting exactly 2 digits is equal to the probability of getting exactly 3 digits. Find the value of $n$. | [
"11"
] | [
"A coin is biased such that the probability of landing on a digit is $\\frac{1}{4}$. If the coin is tossed $n$ times, the probability of getting exactly 2 digits is equal to the probability of getting exactly 3 digits. Find the value of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-ad7e2d88f8d68e08a0f1dbba27e8ee4771c5399e33d0a42fdc5d7914040555e7 | A coin is flipped 10 times in a row. Given that exactly 7 heads appeared, find the probability that no two consecutive flips were both tails. | [
"\\(\\frac{7}{15}\\)",
"\\dfrac{7}{15}"
] | [
"A coin is flipped 10 times in a row. Given that exactly 7 heads appeared, find the probability that no two consecutive flips were both tails."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 28
} | 76 | 0 | 0 |
omr-ws-e196c017697e20dbe0245b677506706e9ca375f041a1d9b55f415cc66cde9237 | A coin is flipped 1001 times. What is the probability that at least 3 heads will appear consecutively in the sequence of heads and tails? | [
"1 - \\frac{a_{1001}}{2^{1001}}"
] | [
"A coin is flipped 1001 times. What is the probability that at least 3 heads will appear consecutively in the sequence of heads and tails?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-d405b23d5d6adf045f2fa02d1771e1b774885e585f5beb611b99be4510bd0ace | A coin is flipped eight times, and the sequence of heads and tails occurring is recorded. How many distinct sequences are possible? | [
"256"
] | [
"A coin is flipped eight times, and the sequence of heads and tails occurring is recorded. How many distinct sequences are possible?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-71098f8f1c7b68ca76f9b20879102b8a594c5dab4bf187d00a851017d497d0fc | A coin is flipped four times. Let $P$ be the event "More Heads than Tails occurred" and $Q$ be the event "First two results are different." Find the following probabilities: - $Pr[P]$ - $Pr[Q]$ - $Pr[P|Q]$ - $Pr[\bar{Q}|\bar{P}]$ | [
"\\dfrac{5}{11}"
] | [
"A coin is flipped four times. Let $P$ be the event \"More Heads than Tails occurred\" and $Q$ be the event \"First two results are different.\" Find the following probabilities:\n- $Pr[P]$\n- $Pr[Q]$\n- $Pr[P|Q]$\n- $Pr[\\bar{Q}|\\bar{P}]$"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-d78915962f42e839f655e103a24af569c34b24a5d8f307130a645ac6eefa8bd6 | A coin is flipped with a $1/17$ chance of landing heads and a $16/17$ chance of landing tails. What is the expected number of flips until both a head and a tail are flipped? | [
"\\(\\dfrac{273}{16}\\)"
] | [
"A coin is flipped with a $1/17$ chance of landing heads and a $16/17$ chance of landing tails. What is the expected number of flips until both a head and a tail are flipped?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-5c0a0b1d65cc82f1e15c67ac06cfc879135df7b31ba8951337202633d22244e3 | A coin is forged such that its probability of landing heads is a random variable $p$ chosen uniformly on the interval $(0,1)$. John flips the coin and it comes up heads. What is the probability that $p > 0.6$? | [
"0.64"
] | [
"A coin is forged such that its probability of landing heads is a random variable $p$ chosen uniformly on the interval $(0,1)$. John flips the coin and it comes up heads. What is the probability that $p > 0.6$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-8e23c14d9a6d721fd3cc220884fbdaacbf182c05d81be892381c74a7158f061f | A coin is forged such that its probability of landing heads is a random variable $p$ chosen uniformly on the interval $(0,1)$. John flips the coin three times and it comes up heads all three times. What is the probability that $p > 0.6$? | [
"0.8704"
] | [
"A coin is forged such that its probability of landing heads is a random variable $p$ chosen uniformly on the interval $(0,1)$. John flips the coin three times and it comes up heads all three times. What is the probability that $p > 0.6$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-c85228054dff55b1cf986d3594edfe477117caacb60954ad52057938cd6e2925 | A coin is forged such that its probability of landing heads is a random variable $p$ chosen uniformly on the interval $(0,1)$. John flips the coin twice and it comes up heads both times. What is the probability that $p > 0.6$? | [
"0.784"
] | [
"A coin is forged such that its probability of landing heads is a random variable $p$ chosen uniformly on the interval $(0,1)$. John flips the coin twice and it comes up heads both times. What is the probability that $p > 0.6$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-6046d05c5d3dd49ee558502645a24e45fd0c1926a7140e7df62aa1f66d20b68e | A coin is repeatedly tossed, and at each toss comes up heads with probability $p$, the outcomes being independent. What is the expected number of tosses until the end of the first run of $k$ heads in a row? | [
"\\(\\frac{\\sum_{i=0}^{k-1} p^i}{p^k}\\)"
] | [
"A coin is repeatedly tossed, and at each toss comes up heads with probability $p$, the outcomes being independent. What is the expected number of tosses until the end of the first run of $k$ heads in a row?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-3d543aafb69004f1d2fde719016f6c377e780619e5aa2d5b0273e42c9cb98ec3 | A coin is tossed $m+n$ times, where $m > n$. Find the probability of getting exactly $m$ consecutive heads. | [
"\\(\\frac{n+3}{2^{m+2}}\\)"
] | [
"A coin is tossed $m+n$ times, where $m > n$. Find the probability of getting exactly $m$ consecutive heads."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 4,
"tir": 2
} | 10 | 0 | 0 |
omr-ws-230dce26117ea6ac57715ab261b445621d012da99c4b253cb9f2261d35225e81 | A coin is tossed $n$ times, with the probability of showing heads on each toss being $p$. What is the probability of showing heads at least $k$ times in succession, where $k \leq n$? | [
"1 - Q(n, k)"
] | [
"A coin is tossed $n$ times, with the probability of showing heads on each toss being $p$. What is the probability of showing heads at least $k$ times in succession, where $k \\leq n$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-3dad42e8b2f0c7f121b719b0ecf1b3550d750ab704ee9d0f9f62ba8e30a8ee57 | A coin is tossed 10 times. Let $X$ be the number of times 2 heads appear in a row. Find the variance of $X$. | [
"\\dfrac{43}{16}"
] | [
"A coin is tossed 10 times. Let $X$ be the number of times 2 heads appear in a row. Find the variance of $X$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-aaf404d8fc7d15db7664c3c9a4d3b4f6d54bb44292d05e341069ae4babdcb993 | A coin is tossed 3 times and the number of tails obtained is counted. Determine the probability distribution function of the random variable \(X\): \[ F_X(t) = P(\{\omega \in \Omega : X(\omega) \leq t\}), \quad \forall t \in \mathbb{R} \] | [
"\\( F_X(t) = \\begin{cases} \n0 & \\text{if } t < 0 \\\\\n\\frac{1}{8} & \\text{if } 0 \\leq t < 1 \\\\\n\\frac{1}{2} & \\text{if } 1 \\leq t < 2 \\\\\n\\frac{7}{8} & \\text{if } 2 \\leq t < 3 \\\\\n1 & \\text{if } t \\geq 3 \n\\end{cases} \\)"
] | [
"A coin is tossed 3 times and the number of tails obtained is counted. Determine the probability distribution function of the random variable \\(X\\):\n\\[ F_X(t) = P(\\{\\omega \\in \\Omega : X(\\omega) \\leq t\\}), \\quad \\forall t \\in \\mathbb{R} \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-47a46ee619b1e4313ddf46e3ff946c0c7348e033e3bd5a637e675d6e08a4ba24 | A coin is tossed 3 times and the number of tails obtained is counted. Tell what the σ-algebra and probability measure are. | [
"\\sigma\\text{-algebra generated by the number of tails and probability measure } P(X=k) = \\binom{3}{k} \\left(\\frac{1}{2}\\right)^3 \\text{ for } k=0,1,2,3"
] | [
"A coin is tossed 3 times and the number of tails obtained is counted. Tell what the σ-algebra and probability measure are."
] | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 4
} | 5 | 0 | 0 |
omr-ws-99bcbe0c5bd30dcbb12bbcbbaf284de95a4cfdb9c4a4a41526f43f00df315ef6 | A coin is tossed 3 times and the number of tails obtained is counted. Write the probability space associated with the random experiment. | [
"\\(\\Omega = \\{HHH, HHT, HTH, HTT, THH, THT, TTH, TTT\\}\\) with each outcome having a probability of \\(\\frac{1}{8}\\)."
] | [
"A coin is tossed 3 times and the number of tails obtained is counted. Write the probability space associated with the random experiment."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 1,
"tir": 22
} | 25 | 0 | 0 |
omr-ws-350321c2a7444a886f5df3e2126a4a89f833a33221338969ed4ed194aefc0f19 | A coin is tossed 400 times. Use the normal distribution as an approximation to find the probability of getting heads: a. between 185 to 210 times, b. exactly 205 times, c. more than 176 times or more than 227 times. | [
"0.79263"
] | [
"A coin is tossed 400 times. Use the normal distribution as an approximation to find the probability of getting heads:\na. between 185 to 210 times,\nb. exactly 205 times,\nc. more than 176 times or more than 227 times."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8620f3d946a20d849d42cd0871a7af7e02b755be939a48d42dbdf90d152daf76 | A coin is tossed five times. Determine the probability that heads never occurs twice in a row. | [
"\\(\\frac{13}{32}\\)"
] | [
"A coin is tossed five times. Determine the probability that heads never occurs twice in a row."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-4903218516d60a2e44c5dfd7d4fbf6f5bed80827960bd61c0224f8d403f14e82 | A coin is tossed five times. Determine the probability that neither heads nor tails ever occurs twice in a row. | [
"\\(\\frac{1}{16}\\)"
] | [
"A coin is tossed five times. Determine the probability that neither heads nor tails ever occurs twice in a row."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-f4a2cbf7a5d46955ca7562358295f2f9b2483ed9f92250d9936e978fa264bafe | A coin is tossed independently $n$ times with probability $0<p<1$ of getting heads in any individual toss. Let $X$ denote the difference between the number of heads and the number of tails. Find the discrete density function of $X$. | [
"\\( P(X=j) = {n \\choose \\frac{n+j}{2}} p^{\\frac{n+j}{2}} q^{\\frac{n-j}{2}} \\)"
] | [
"A coin is tossed independently $n$ times with probability $0<p<1$ of getting heads in any individual toss. Let $X$ denote the difference between the number of heads and the number of tails. Find the discrete density function of $X$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-e3f23272e873af357f6048087f3f2cf7ff85b1bec081ec2f2f82374473fa5954 | A coin is tossed independently $n$ times with probability $0<p<1$ of getting heads in any individual toss. Let $X$ denote the difference between the number of heads and the number of tails. Find the expected value $E(X)$ and the variance $VAR(X)$. | [
"\\( E(X) = n(2p-1) \\), \\( VAR(X) = 4p(1-p)n(n+1) \\)"
] | [
"A coin is tossed independently $n$ times with probability $0<p<1$ of getting heads in any individual toss. Let $X$ denote the difference between the number of heads and the number of tails. Find the expected value $E(X)$ and the variance $VAR(X)$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f9c370c6386207f55c20a63b4839450dbf70ba67b0b90e039d2229f1c24fb512 | A coin is tossed repeatedly, with heads appearing with probability $p$ on each toss. Let $X$ be the number of tosses until the first occasion by which three heads have appeared successively. Find the expected value $E(X)$. | [
"14"
] | [
"A coin is tossed repeatedly, with heads appearing with probability $p$ on each toss. Let $X$ be the number of tosses until the first occasion by which three heads have appeared successively. Find the expected value $E(X)$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-06df42512b8afdaf0abf9f7a97d82e29f181a2f940afa81b6a75ac1d84af7355 | A coin is tossed until a head appears or a tail appears 4 times in succession. Find the probability distribution for the number of tosses. | [
"\\( P(X=1) = \\frac{1}{2}, P(X=2) = \\frac{1}{4}, P(X=3) = \\frac{1}{8}, P(X=4) = \\frac{1}{8} \\)"
] | [
"A coin is tossed until a head appears or a tail appears 4 times in succession. Find the probability distribution for the number of tosses."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 8,
"tir": 23
} | 31 | 0 | 0 |
omr-ws-be188ab5af2b7c160cf8a715e36fd8465a419048addb4f5c19ca3b8713d4a17e | A coin is tossed. If it shows a head, toss the coin again; otherwise, throw a dice. Find the probability that the dice shows a prime number given that there is at least one tail. | [
"\\(\\frac{2}{9}\\)"
] | [
"A coin is tossed. If it shows a head, toss the coin again; otherwise, throw a dice. Find the probability that the dice shows a prime number given that there is at least one tail."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-edc227526bb2ae4788337bf3e7e86f01a47dea40dbc24639b64d3099eee7d9e8 | A coin lands on heads with probability $k$ for some fixed $0 \le k \le 1$. The game involves flipping the coin repeatedly, and the player wins if at any point the number of heads exceeds the number of tails. What is the probability that the player is stuck playing the game forever, as a function of $k$? (i.e., let $p_n... | [
"\\[\n\\begin{cases} \n0 & \\text{if } k < 1 \\\\\n1 & \\text{if } k = 1 \n\\end{cases}\n\\]"
] | [
"A coin lands on heads with probability $k$ for some fixed $0 \\le k \\le 1$. The game involves flipping the coin repeatedly, and the player wins if at any point the number of heads exceeds the number of tails. What is the probability that the player is stuck playing the game forever, as a function of $k$? (i.e., l... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-85f9b741aca18343c386d15674b05271d0a6feb1a5d25097597288f104de2114 | A coin of diameter \( d = \frac{1}{2} \) is dropped at random on a Cartesian plane. Find the probability that it will cover a point with integer coordinates. | [
"\\(1 - \\frac{\\pi}{4}\\)"
] | [
"A coin of diameter \\( d = \\frac{1}{2} \\) is dropped at random on a Cartesian plane. Find the probability that it will cover a point with integer coordinates."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-37cd7468f832d7042e2e774718f79700633a6c384b3c8f8aee27d076783726c3 | A coin of radius $1$ is flipped onto a $500 \times 500$ square grid divided into $2500$ equal squares. Circles are inscribed in $n$ of these squares. Let $P_n$ be the probability that the coin lands completely within one of the circles, given that it lands completely within one of the smaller squares. Let $P$ be the pr... | [
"7"
] | [
"A coin of radius $1$ is flipped onto a $500 \\times 500$ square grid divided into $2500$ equal squares. Circles are inscribed in $n$ of these squares. Let $P_n$ be the probability that the coin lands completely within one of the circles, given that it lands completely within one of the smaller squares. Let $P$ be ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-48eff22842199cd1b8bfa89bf8061c8c8a9275dc63b0866ec1549b4fdbf08c9d | A coin that comes up heads with probability \( p \) is continually flipped until the pattern T,T,H appears. Let \( X \) denote the number of flips made. Find \( E[X] \). | [
"\\dfrac{1}{p(1 - p)^2}"
] | [
"A coin that comes up heads with probability \\( p \\) is continually flipped until the pattern T,T,H appears. Let \\( X \\) denote the number of flips made. Find \\( E[X] \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-28d67f45145e114a0bd0a67c8b10181e38a7658e84e764446be034e76d961d84 | A coin with a diameter of 1 cm rolls around outside a regular hexagon with a side length of 1 cm. What is the length of the path traced by the center of the coin after one complete rotation around the hexagon? | [
"6 + \\pi"
] | [
"A coin with a diameter of 1 cm rolls around outside a regular hexagon with a side length of 1 cm. What is the length of the path traced by the center of the coin after one complete rotation around the hexagon?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 37,
"genselect": 7,
"tir": 17
} | 61 | 0 | 0 |
omr-ws-9bc4456126b877d2e9fcf0a81ea63c7e626d1e94b1c177a6a835a1868c4decb1 | A coin with a diameter of 8 units is tossed onto a 5 by 5 grid of squares, each having a side length of 10 units. The coin is in a winning position if no part of it touches or crosses a grid line. Given that the coin lands in a random position so that no part of it is off the grid, what is the probability that it is in... | [
"\\(\\frac{25}{441}\\)"
] | [
"A coin with a diameter of 8 units is tossed onto a 5 by 5 grid of squares, each having a side length of 10 units. The coin is in a winning position if no part of it touches or crosses a grid line. Given that the coin lands in a random position so that no part of it is off the grid, what is the probability that it ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-fdedeb995c93764c22a1b8589dd0101d85d0def59a60cdd2f64465e6589f87f0 | A coin with diameter $d$ is placed randomly flat on a paper that contains a grid of squares, each $d$ units on each side. Find the probability that the coin does not cover a vertex of any square of the grid. | [
"1 - \\dfrac{\\pi}{4}"
] | [
"A coin with diameter $d$ is placed randomly flat on a paper that contains a grid of squares, each $d$ units on each side. Find the probability that the coin does not cover a vertex of any square of the grid."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-da807b9e7913fdb984872f3924c862a185c65f8ed190191f19f114458a222b9a | A coin with probability \( P(\text{Heads}) = p \) is tossed until two successive Tails are obtained. Find the probability that the experiment is completed on the \( n \)-th toss. | [
" \\frac{(1-p)^2}{\\sqrt{4p - 3p^2}} \\left( \\left( \\frac{p + \\sqrt{4p - 3p^2}}{2} \\right)^{n-2} - \\left( \\frac{p - \\sqrt{4p - 3p^2}}{2} \\right)^{n-2} \\right) ",
"(1 - p)^2 \\cdot \\frac{\\left( \\frac{p + \\sqrt{4p - 3p^2}}{2} \\right)^{n-1} - \\left( \\frac{p - \\sqrt{4p - 3p^2}}{2} \\right)^{n-1}}{\\s... | [
"A coin with probability \\( P(\\text{Heads}) = p \\) is tossed until two successive Tails are obtained. Find the probability that the experiment is completed on the \\( n \\)-th toss."
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 4
} | 18 | 0 | 0 |
omr-ws-373e8016feea632120024e3a7a02899771af6eec700967b4fc4a571cf3d43eec | A coin with probability \( p \) of heads is tossed \( n \) times. Let \( E \) be the event 'a head is obtained on the first toss' and \( F_k \) the event 'exactly \( k \) heads are obtained'. For which pairs of integers \( (n, k) \) are \( E \) and \( F_k \) independent? | [
"\\( k = np \\)"
] | [
"A coin with probability \\( p \\) of heads is tossed \\( n \\) times. Let \\( E \\) be the event 'a head is obtained on the first toss' and \\( F_k \\) the event 'exactly \\( k \\) heads are obtained'. For which pairs of integers \\( (n, k) \\) are \\( E \\) and \\( F_k \\) independent?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 5,
"tir": 1
} | 7 | 0 | 0 |
omr-ws-bb93553fd0bff54d78759545d02bce72edb13b8a29fd26891dbbdb5b116a7552 | A cold soda with an initial temperature of 43 degrees F. is left outside where the temperature is a steady 101 degrees F. After 4 minutes, the soda's temperature is 48 degrees F. What is the expected temperature of the soda after an additional 8 minutes? | [
"57"
] | [
"A cold soda with an initial temperature of 43 degrees F. is left outside where the temperature is a steady 101 degrees F. After 4 minutes, the soda's temperature is 48 degrees F. What is the expected temperature of the soda after an additional 8 minutes?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-15d0757053bb8cc9a1070ca55547d86603d7ed5587b5c4aaaec080308f716fe7 | A collection \( F \) of distinct subsets of \( X = \{1, 2, \ldots, 300\} \) is called *lovely* if for any three (not necessarily distinct) sets \( A \), \( B \), and \( C \) in \( F \), at most three out of the following eight sets are non-empty: \[ A \cap B \cap C, \ \ \ \overline{A} \cap B \cap C, \ \ \ A \cap \overl... | [
"8"
] | [
"A collection \\( F \\) of distinct subsets of \\( X = \\{1, 2, \\ldots, 300\\} \\) is called *lovely* if for any three (not necessarily distinct) sets \\( A \\), \\( B \\), and \\( C \\) in \\( F \\), at most three out of the following eight sets are non-empty:\n\\[\nA \\cap B \\cap C, \\ \\ \\ \\overline{A} \\cap... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 7
} | 16 | 0 | 0 |
omr-ws-c8cced44b32da5b51474db330a458aec4ffbfa1fd8d4503f8b100a36de9cdbe1 | A collection \( S \) of integers is defined by the following rules: (I) 2 is in \( S \); (II) For every \( x \) in \( S \), \( 3x \) and \( x+7 \) are also in \( S \); (III) no integers except those defined by rules (I) and (II) are in \( S \). What is the smallest integer greater than 2004 which is NOT in \( S \)? | [
"2009"
] | [
"A collection \\( S \\) of integers is defined by the following rules: (I) 2 is in \\( S \\); (II) For every \\( x \\) in \\( S \\), \\( 3x \\) and \\( x+7 \\) are also in \\( S \\); (III) no integers except those defined by rules (I) and (II) are in \\( S \\). What is the smallest integer greater than 2004 which i... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 14
} | 34 | 0 | 0 |
omr-ws-088e140a0b984e0082a256a6889697631f23f09a1e17ba7932ac3ee42b6c5eec | A collection of $2n$ letters contains $2$ each of $n$ different letters. The collection is partitioned into $n$ pairs, each pair containing $2$ letters which may be the same or different. Denote the number of distinct partitions by $u_n$. Find the value of $u_5$ given that $u_1 = 1$ and $u_3 = 3$. | [
"15"
] | [
"A collection of $2n$ letters contains $2$ each of $n$ different letters. The collection is partitioned into $n$ pairs, each pair containing $2$ letters which may be the same or different. Denote the number of distinct partitions by $u_n$. Find the value of $u_5$ given that $u_1 = 1$ and $u_3 = 3$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-cedd1afb17f22c57682637896246b918a40c16005fc39ae53319fe2c334e41ac | A collection of 2000 congruent circles is given on the plane such that no two circles are tangent and each circle meets at least two other circles. Let \( N \) be the number of points that belong to at least two of the circles. Find the smallest possible value of \( N \). | [
"4000"
] | [
"A collection of 2000 congruent circles is given on the plane such that no two circles are tangent and each circle meets at least two other circles. Let \\( N \\) be the number of points that belong to at least two of the circles. Find the smallest possible value of \\( N \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 1
} | 15 | 0 | 0 |
omr-ws-5c820dd2d680d0fa469d5dadc18c23828109410219ed3d91baa57570e4fcbf6d | A collection of 30 volumes is on a book shelf. How many ways are there of arranging the series (a) for volumes 1 and 2 to be side by side; (b) for volumes 3 and 4 not to be side by side? | [
"(a) $2\\cdot 29!$; (b) $28\\cdot 29!$"
] | [
"A collection of 30 volumes is on a book shelf. How many ways are there of arranging the series (a) for volumes 1 and 2 to be side by side; (b) for volumes 3 and 4 not to be side by side?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-650d6d4b802a9d66ac927d135a93dfaf35c7e51fd894ec8ee6e887f98815b8c2 | A collection of 8 cubes consists of one cube with edge - length $k$ for each integer $k, 1 \le k \le 8.$ A tower is to be built using all 8 cubes according to the rules: Any cube may be the bottom cube in the tower. The cube immediately on top of a cube with edge-length $k$ must have edge-length at most $k+2.$ Let $T$ ... | [
"458"
] | [
"A collection of 8 cubes consists of one cube with edge - length $k$ for each integer $k, 1 \\le k \\le 8.$ A tower is to be built using all 8 cubes according to the rules: \nAny cube may be the bottom cube in the tower. The cube immediately on top of a cube with edge-length $k$ must have edge-length at mos... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 21
} | 22 | 0 | 0 |
omr-ws-97cbaa4258b34660945413ddf839ea553a771b59ff6856a7289a083e858653d3 | A collection of 8 cubes consists of one cube with edge-length $k$ for each integer $k, 1 \le k \le 8$. A tower is to be built using all 8 cubes according to the rules: - Any cube may be the bottom cube in the tower. - The cube immediately on top of a cube with edge-length $k$ must have edge-length at most $k+2$. Let $T... | [
"458"
] | [
"A collection of 8 cubes consists of one cube with edge-length $k$ for each integer $k, 1 \\le k \\le 8$. A tower is to be built using all 8 cubes according to the rules:\n- Any cube may be the bottom cube in the tower.\n- The cube immediately on top of a cube with edge-length $k$ must have edge-length at most $k+2... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 2,
"tir": 22
} | 25 | 0 | 0 |
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