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-696f45b2c44247ff56b3ceb9e31346ba8368187f5b5d67708c68c769fc6e35bf | There is a committee composed of eight women and two men. When they meet, they sit in a row---the women in indistinguishable rocking chairs and the men on indistinguishable stools. How many distinct ways are there for me to arrange the eight chairs and two stools for a meeting? | [
"45"
] | [
"There is a committee composed of eight women and two men. When they meet, they sit in a row---the women in indistinguishable rocking chairs and the men on indistinguishable stools. How many distinct ways are there for me to arrange the eight chairs and two stools for a meeting?"
] | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-f2a37355412ee9cf07017a3174bfbba85dcf39becbbf25aff7db6b77a341cdbd | There is a complex number $ z$ with imaginary part $ 164$ and a positive integer $ n$ such that \[ \frac {z}{z + n} = 4i. \]Find $ n$. | [
"697"
] | [
"There is a complex number $ z$ with imaginary part $ 164$ and a positive integer $ n$ such that \n\\[ \\frac {z}{z + n} = 4i.\n\\]Find $ n$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-819d600f2d9c0053e01361250ffa760c7e2b282f25c1e31fdbc2e56ee9c59177 | There is a complex number $z$ with imaginary part $164$ and a positive integer $n$ such that $$\frac {z}{z + n} = 4i.$$ Find $n$. | [
"697"
] | [
"There is a complex number $z$ with imaginary part $164$ and a positive integer $n$ such that \n$$\\frac {z}{z + n} = 4i.$$ \nFind $n$."
] | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-dd24203e9ff0b491adfe3cacb86ed9b53f6714868b26f98a7677e33749b6e579 | There is a complex number \( z \) with imaginary part 164 and a positive integer \( n \) such that \( \frac{z}{z+n} = 4i \). Find \( n \). | [
"697"
] | [
"There is a complex number \\( z \\) with imaginary part 164 and a positive integer \\( n \\) such that \\( \\frac{z}{z+n} = 4i \\). Find \\( n \\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-20369c3fc0269cef7b17d8087e2b20897b5e3e74298958bed12b5576a63a0003 | There is a cube $n \times n \times n$. We call good the lines parallel to the edges and passing through the centers of the cubes. What is the least number of good lines that must be drawn so that any other good line intersects with one of the lines drawn? | [
"3"
] | [
"There is a cube $n \\times n \\times n$. We call good the lines parallel to the edges and passing through the centers of the cubes. What is the least number of good lines that must be drawn so that any other good line intersects with one of the lines drawn?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c32ba0baa6dc2773d31843c6657ef87882f20a1eb73d09edd7f1341f7c64972d | There is a cube with side length 1 and two regular octahedrons: one octahedron is inside the cube and the centers of the faces of the cube are the vertices of the octahedron; the other octahedron is outside the cube and the centers of the faces of this octahedron are the vertices of the cube. What is the ratio of the s... | [
"3"
] | [
"There is a cube with side length 1 and two regular octahedrons: one octahedron is inside the cube and the centers of the faces of the cube are the vertices of the octahedron; the other octahedron is outside the cube and the centers of the faces of this octahedron are the vertices of the cube. What is the ratio of ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-40e44300a1b10e8dcb7a5fba1c450457f11bc3763c246a76ea5d74b5039ee75b | There is a deck of $n$ cards, each of which is independently either red or black with equal probability. You start with $\$1$, and you make a sequence of $n$ even money bets on successive cards. You can bet any fraction of your money on the card being either red or black, the card is then revealed, and you win if you a... | [
"\\left( \\dfrac{2}{m} \\right)^n"
] | [
"There is a deck of $n$ cards, each of which is independently either red or black with equal probability. You start with $\\$1$, and you make a sequence of $n$ even money bets on successive cards. You can bet any fraction of your money on the card being either red or black, the card is then revealed, and you win if... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-c1349d5e27a0fcbb5f547a97d9361e43a8b1574daf7aeaf90114f0f00dbb8461 | There is a deck of $n$ cards, each of which is independently either red or black with equal probability. You start with $\$1$, and you make a sequence of $n$ even money bets on successive cards. You can bet any fraction of your money on the card being either red or black, the card is then revealed, and you win if you a... | [
"2^{n - 2k}"
] | [
"There is a deck of $n$ cards, each of which is independently either red or black with equal probability. You start with $\\$1$, and you make a sequence of $n$ even money bets on successive cards. You can bet any fraction of your money on the card being either red or black, the card is then revealed, and you win if... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-fb2194dd4288547c8bfab9b8c7ce1ba3d70fe7e29592af5fea160bff71046635 | There is a fence consisting of $n$ planks arranged in a line. Each plank is painted with one of 100 available colors. For any two distinct colors $i$ and $j$, there is a plank with color $i$ located to the left of a (not necessarily adjacent) plank with color $j$. Determine the minimum possible value of $n$. | [
"199"
] | [
"There is a fence consisting of $n$ planks arranged in a line. Each plank is painted with one of 100 available colors. For any two distinct colors $i$ and $j$, there is a plank with color $i$ located to the left of a (not necessarily adjacent) plank with color $j$. Determine the minimum possible value of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 44,
"genselect": 8,
"tir": 3
} | 55 | 0 | 0 |
omr-ws-1fd74a65369fc5bdb87eebfe6bf54da7e6d75da5c4f1141d400fb2d004bb13c6 | There is a figure consisting of 6 cells. We write positive integers from 1 to 6 on the cells such that: - Each positive integer is written once. - The GCD between the positive integers written on any two adjacent cells is 1. Find the number of ways to write the integers on the cells. | [
"72"
] | [
"There is a figure consisting of 6 cells. We write positive integers from 1 to 6 on the cells such that:\n- Each positive integer is written once.\n- The GCD between the positive integers written on any two adjacent cells is 1.\nFind the number of ways to write the integers on the cells."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 1
} | 11 | 0 | 0 |
omr-ws-4872935941b3a702dcd4fafbde258ee6a6a39939f776f32f57600d983a9e33e2 | There is a formula for \(A_{n,2}\) along the same lines as the formula for \(A_{n,1}\). Find that number \(C\) so that \(A_{n,2}\) grows like \(C^n\). | [
"4"
] | [
"There is a formula for \\(A_{n,2}\\) along the same lines as the formula for \\(A_{n,1}\\). Find that number \\(C\\) so that \\(A_{n,2}\\) grows like \\(C^n\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-cafeda5ae50819094334c8feee55b99337a1f8ba0dd6911a008da5a5a95f0328 | There is a four-digit year such that when the first two digits are split from the last two digits, the double product of these two two-digit numbers equals the original year. What is this year? | [
"1352"
] | [
"There is a four-digit year such that when the first two digits are split from the last two digits, the double product of these two two-digit numbers equals the original year. What is this year?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-61e7a63ab1e306d04f8c883da84ae0294d4d1b4ec628bb61edba014fdd00023c | There is a function \( f(x) \) which goes through the point \( (1,2) \). For any two real numbers \( a \) and \( b \), the points \( (a, f(a) + a) \), \( (b, f(b) + b) \), and \( (0, ab) \) lie on the same line. Find the function \( y = f(x) \). | [
"\\( f(x) = -x^2 + 3x \\)"
] | [
"There is a function \\( f(x) \\) which goes through the point \\( (1,2) \\). For any two real numbers \\( a \\) and \\( b \\), the points \\( (a, f(a) + a) \\), \\( (b, f(b) + b) \\), and \\( (0, ab) \\) lie on the same line. Find the function \\( y = f(x) \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-d7854105d98763ebe596f3d01638549fabbc2278eb98b7aeb9db005ae428d326 | There is a function \(p(x)\) such that \(p(x)^2 + p(x^2) = 2x^2\) for all real \(x\). If \(p(1) \neq 1\), what is the sum of all possible values of \(p(10)\)? | [
"-121"
] | [
"There is a function \\(p(x)\\) such that \\(p(x)^2 + p(x^2) = 2x^2\\) for all real \\(x\\). If \\(p(1) \\neq 1\\), what is the sum of all possible values of \\(p(10)\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-f62e00b4c3a024d867b820dbb8daac5a405b19b65467a73a4422b2712ff6e667 | There is a grandfather, 6 sons and daughters, and 4 grandchildren. They are to be seated in a row. Two grandchildren want to be seated at both ends, and the grandfather refuses to sit next to either of the grandchildren at the ends. Find the number of possible seating arrangements. | [
"362880"
] | [
"There is a grandfather, 6 sons and daughters, and 4 grandchildren. They are to be seated in a row. Two grandchildren want to be seated at both ends, and the grandfather refuses to sit next to either of the grandchildren at the ends. Find the number of possible seating arrangements."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b24382414861e28a0bff6ca82064efbdcde3838e47049ac1ee262fd2a5ba6d5e | There is a graph with 30 vertices. If any 26 of its vertices along with their outgoing edges are deleted, the remaining graph is a connected graph with 4 vertices. What is the smallest number of edges in the initial graph with 30 vertices? | [
"405"
] | [
"There is a graph with 30 vertices. If any 26 of its vertices along with their outgoing edges are deleted, the remaining graph is a connected graph with 4 vertices. What is the smallest number of edges in the initial graph with 30 vertices?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 2
} | 11 | 0 | 0 |
omr-ws-23c331dc33e712c7d01677f8d7a39738e0816ad00b225a88c85955b5a67e19b5 | There is a group of $2n$ people. Each person has no more than $n-1$ enemies. Determine the minimum number of distinct seating arrangements around a round table such that any two neighbors are friends, modulo $10^9 + 7$. | [
"2"
] | [
"There is a group of $2n$ people. Each person has no more than $n-1$ enemies. Determine the minimum number of distinct seating arrangements around a round table such that any two neighbors are friends, modulo $10^9 + 7$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-3aa38e755fbc33f48bfea1ff3ec7b2c4847c73e7e493ae918e708c973bdd84b0 | There is a house with $N$ floors and $n$ identical eggs. An egg will break if dropped from the $d$-th floor or higher, but will not break if dropped from a lower floor. What is the minimum number of attempts required to determine the value of $d$ in any case? | [
"\\min \\left\\{ m \\in \\mathbb{N} \\mid \\sum_{k=1}^{n} \\binom{m}{k} \\geq N \\right\\}"
] | [
"There is a house with $N$ floors and $n$ identical eggs. An egg will break if dropped from the $d$-th floor or higher, but will not break if dropped from a lower floor. What is the minimum number of attempts required to determine the value of $d$ in any case?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 1
} | 15 | 0 | 0 |
omr-ws-4ddf4f58e08419c65e709c715541fe7108a50a3aa010d32b48aa58d2800484f3 | There is a house with $N$ floors and 2 identical eggs. An egg will break if dropped from the $d$-th floor or higher, but will not break if dropped from a lower floor. What is the minimum number of attempts required to determine the value of $d$ in the worst case? | [
"\\left\\lceil \\frac{\\sqrt{8N + 1} - 1}{2} \\right\\rceil"
] | [
"There is a house with $N$ floors and 2 identical eggs. An egg will break if dropped from the $d$-th floor or higher, but will not break if dropped from a lower floor. What is the minimum number of attempts required to determine the value of $d$ in the worst case?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-aea9f4da198eba27410d76d4298ee8c1a3b82e6fdc11332894850d3912a3fca1 | There is a house with $n$ floors. Each floor can be painted red, blue, or green. No two red floors can be adjacent, and the top floor must be red. How many possible ways are there to paint the floors? | [
"\\(\\frac{\\sqrt{3}(1+\\sqrt{3})^{n} - \\sqrt{3}(1-\\sqrt{3})^{n}}{6}\\)"
] | [
"There is a house with $n$ floors. Each floor can be painted red, blue, or green. No two red floors can be adjacent, and the top floor must be red. How many possible ways are there to paint the floors?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 3,
"tir": 13
} | 25 | 0 | 0 |
omr-ws-0fe6d259c15c0d0b98984ac6c801f8f791b72d0bcfd754e010d43786ce911572 | There is a key ring with $N$ keys, one of which is the correct key to a lock. A person tries to open the lock by selecting keys at random and discarding the ones already tried. What is the probability that the lock is opened on the $N^{th}$ trial? | [
"\\(\\frac{1}{N}\\)"
] | [
"There is a key ring with $N$ keys, one of which is the correct key to a lock. A person tries to open the lock by selecting keys at random and discarding the ones already tried. What is the probability that the lock is opened on the $N^{th}$ trial?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-f678b34d88c6ffee454f2dbd3cda12791fb38c181c422711b3694949c44f92fc | There is a light at the top of a tower. Initially, the angle of elevation to the light is $15^\circ$. After moving 200 meters closer to the tower, the angle of elevation becomes $75^\circ$. Determine the height of the tower. | [
"\\(\\frac{100\\sqrt{3}}{3}\\)"
] | [
"There is a light at the top of a tower. Initially, the angle of elevation to the light is $15^\\circ$. After moving 200 meters closer to the tower, the angle of elevation becomes $75^\\circ$. Determine the height of the tower."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-c77650484bbe16d0d33c749b3c471eac1e5c51bde78ca895982988cf1a210058 | There is a line of lockers numbered 1 to 1024, initially all closed. A man walks down the line, opens 1, then alternately skips and opens each closed locker (so he opens 1, 3, 5, ..., 1023). At the end of the line he walks back, opens the first closed locker, then alternately skips and opens each closed locker (so he o... | [
"342"
] | [
"There is a line of lockers numbered 1 to 1024, initially all closed. A man walks down the line, opens 1, then alternately skips and opens each closed locker (so he opens 1, 3, 5, ..., 1023). At the end of the line he walks back, opens the first closed locker, then alternately skips and opens each closed locker (so... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 3,
"tir": 1
} | 5 | 0 | 0 |
omr-ws-044c28c70d049fac92a9dd3d39a824788abf4b7386522d60324181f043f58647 | There is a line segment consisting of five line segments $AB, BC, CD, DE,$ and $EA$, with $10$ intersection points. Find the number of ways to write $1$ or $2$ at each of the $10$ vertices so that the sum of the four numbers written on each line segment $AB, BC, CD, DE,$ and $EA$ is the same. | [
"4"
] | [
"There is a line segment consisting of five line segments $AB, BC, CD, DE,$ and $EA$, with $10$ intersection points. Find the number of ways to write $1$ or $2$ at each of the $10$ vertices so that the sum of the four numbers written on each line segment $AB, BC, CD, DE,$ and $EA$ is the same."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-b3e72ed5212d86696847d7466dfe841ae7c1092953ab1234dce382ad5eeb53db | There is a list of seven numbers. The average of the first four numbers is 5 and the average of the last four numbers is 8. If the average of all seven numbers is 46/7, what is the common number to both sets of four numbers? | [
"6"
] | [
"There is a list of seven numbers. The average of the first four numbers is 5 and the average of the last four numbers is 8. If the average of all seven numbers is 46/7, what is the common number to both sets of four numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c842c567c79de96060ee9bc6544ad5b565c8aeeb484edb4d9e9f3c8f0f2cb2e5 | There is a luck machine that costs 1 dollar to play each time. With a 0.1% probability, you win 70 dollars, and with a 99.9% probability, you win nothing. If you play 50 times and get nothing, the owner gives you 50 dollars back. What is the best strategy for the player, and how many times should one play the machine? | [
"50"
] | [
"There is a luck machine that costs 1 dollar to play each time. With a 0.1% probability, you win 70 dollars, and with a 99.9% probability, you win nothing. If you play 50 times and get nothing, the owner gives you 50 dollars back. What is the best strategy for the player, and how many times should one play the mach... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-8bd69b4b6440c2f9e0fe5e48f29a2c1678da8f4beae320af02efc22761fa02af | There is a machine with 8 toys in it that each cost between 25 cents and 2 dollars, with each toy being 25 cents more expensive than the next most expensive one. Each time Sam presses the big red button on the machine, the machine randomly selects one of the remaining toys and gives Sam the option to buy it. If Sam has... | [
"\\dfrac{6}{7}"
] | [
"There is a machine with 8 toys in it that each cost between 25 cents and 2 dollars, with each toy being 25 cents more expensive than the next most expensive one. Each time Sam presses the big red button on the machine, the machine randomly selects one of the remaining toys and gives Sam the option to buy it. If Sa... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-7c7281f5b6e4b9033639818c3fa64e5821f00e30adb5a9d6338f7291829c9d05 | There is a meeting with 40 seats attended by 16 invited guests. Each invited guest must occupy at least 1 chair. Determine the number of arrangements. | [
"\\(\\binom{25}{16} \\times 16!\\)"
] | [
"There is a meeting with 40 seats attended by 16 invited guests. Each invited guest must occupy at least 1 chair. Determine the number of arrangements."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-74134d11e1ee03c220ea2a60c1af9019aa8a83aa54dec1d7e2a409b566d06b31 | There is a monster swimming in a straight line with speed \( t \) where \( t \in \mathbb{N} \), beneath the surface of an infinitely long sea. The monster starts directly below an integer point on an infinitely long number line at time zero. You can explode a mine at any integer point at the end of each hour, starting ... | [
"\\text{Detonate at position } x_0 + s \\cdot m \\text{ for each pair } (s, x_0) \\text{ enumerated over all integers } s \\neq 0 \\text{ and } x_0 \\text{, ensuring eventual coverage.}"
] | [
"There is a monster swimming in a straight line with speed \\( t \\) where \\( t \\in \\mathbb{N} \\), beneath the surface of an infinitely long sea. The monster starts directly below an integer point on an infinitely long number line at time zero. You can explode a mine at any integer point at the end of each hour... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 8
} | 9 | 0 | 0 |
omr-ws-512443b5a22dbe64417fccee4d3e5a0f3efa371f02ba3af31be081ebb5b0440e | There is a natural number greater than 1. The product of all its divisors (including 1 and the number itself) is equal to the fifth power of this number. How many divisors does this number have? | [
"10"
] | [
"There is a natural number greater than 1. The product of all its divisors (including 1 and the number itself) is equal to the fifth power of this number. How many divisors does this number have?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-2f7e7d2386eb7e2148420f1c0b41370881d9f2fcff8539c25b8ff54837fda13d | There is a number $1$ or $-1$ at all vertices of a cube, and on all faces of the cube, the number written there is the product of the numbers on the $4$ edges of that face. Find all possible values of the sum of those $14$ numbers. | [
"\\( 14, 6, 2, -2, -6, -10 \\)"
] | [
"There is a number $1$ or $-1$ at all vertices of a cube, and on all faces of the cube, the number written there is the product of the numbers on the $4$ edges of that face. Find all possible values of the sum of those $14$ numbers."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 10
} | 10 | 0 | 0 |
omr-ws-8c24be011b2160f1084189118a76679fbb101175e9e69553504673ed516149c7 | There is a number of windmills such that: - Adding one more windmill allows the total to be counted as a multiple of 12. - Adding two more windmills allows the total to be counted as a multiple of 13. - Adding three more windmills allows the total to be counted as a multiple of 14. How many windmills are there? | [
"1103"
] | [
"There is a number of windmills such that:\n- Adding one more windmill allows the total to be counted as a multiple of 12.\n- Adding two more windmills allows the total to be counted as a multiple of 13.\n- Adding three more windmills allows the total to be counted as a multiple of 14.\nHow many windmills are there... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-c7d321af06101159ab88b97bb6bdabbf54e4885ee2b078fa3c9972bd26a7836f | There is a number that ends with 2. If you move the number 2 to the beginning of the number, the new number is twice the original number. What is the original number? | [
"105263157894736842"
] | [
"There is a number that ends with 2. If you move the number 2 to the beginning of the number, the new number is twice the original number. What is the original number?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-0253af5be645c3d9e65deda679974d17574c3a1b72bf22133fbe863b11e67890 | There is a one-acre field in the shape of a right triangle with sides of length \( x \) and \( y \). At the midpoint of each side, there is a post. Tethered to the posts on each side is a sheep, and tethered to the post on the hypotenuse is a dog. Each animal has a rope just long enough to reach the two adjacent vertic... | [
"1"
] | [
"There is a one-acre field in the shape of a right triangle with sides of length \\( x \\) and \\( y \\). At the midpoint of each side, there is a post. Tethered to the posts on each side is a sheep, and tethered to the post on the hypotenuse is a dog. Each animal has a rope just long enough to reach the two adjace... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-3318264712b948b6e0d75c7849de26ba7bcec3a6149afa1583866cec696a83da | There is a party with $12k$ people where each person has exactly $3k+6$ friends. Additionally, every pair of people has the same number of friends in common. Find all natural numbers $k$ for which such a party is possible, and provide an example for that value of $k$. | [
"\\( k = 3 \\)"
] | [
"There is a party with $12k$ people where each person has exactly $3k+6$ friends. Additionally, every pair of people has the same number of friends in common. Find all natural numbers $k$ for which such a party is possible, and provide an example for that value of $k$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8a492d47612d4e5c0fec7ed86f9e02a11823156602f51312c78455cdcf82f81d | There is a pentagon $ABCDE$ such that: - $ACE$ is an equilateral triangle. - $AB = BC = CD = DE$. - $\angle ABC = \angle CDE = 120^\circ$. a) Construct the figure. b) If $AE = 1$, find the length of $BD$. | [
"1"
] | [
"There is a pentagon $ABCDE$ such that:\n- $ACE$ is an equilateral triangle.\n- $AB = BC = CD = DE$.\n- $\\angle ABC = \\angle CDE = 120^\\circ$.\na) Construct the figure.\nb) If $AE = 1$, find the length of $BD$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-01c898061a0db4c5ee45687d732d58334661736c42ddc89cb9b3a5b41e629747 | There is a perfect square $n$ where the units digit of $n$ is $i$. If $S$ be the sum of all perfect squares $n \leq 10^{12}$ such that $\overline{ni}$ is also a perfect square, find the last 3 digits of $S$. | [
"848"
] | [
"There is a perfect square $n$ where the units digit of $n$ is $i$. If $S$ be the sum of all perfect squares $n \\leq 10^{12}$ such that $\\overline{ni}$ is also a perfect square, find the last 3 digits of $S$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-9bcd1759ec373cc0546d77e043c1ebe7513c36ecc4fa5b88a5e38bae89308561 | There is a piggy bank with 100 coins (nickels, dimes, and quarters) whose total value is $8.35. What is the difference between the largest and smallest number of dimes that could be in the bank? | [
"64"
] | [
"There is a piggy bank with 100 coins (nickels, dimes, and quarters) whose total value is $8.35. What is the difference between the largest and smallest number of dimes that could be in the bank?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-7d1157afaf5cc6641efd063af53a827be8841938e1b955208a1ab1e5ce50e75a | There is a pile of $1234 \dots 19961997$ balls. A move is defined as follows: If the number of balls in the pile is a multiple of 10, remove $\frac{9}{10}$ of the balls. If not, add the smallest possible number of balls such that the number of balls in the pile is a multiple of 10. How many moves are needed to reduce t... | [
"12"
] | [
"There is a pile of $1234 \\dots 19961997$ balls. A move is defined as follows: If the number of balls in the pile is a multiple of 10, remove $\\frac{9}{10}$ of the balls. If not, add the smallest possible number of balls such that the number of balls in the pile is a multiple of 10. How many moves are needed to r... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-fed20e13561e99cad38663df1fba502f42646b25e1f1e2ff5ecdf8d204538738 | There is a pile of 1000 matches. Two players take turns, each can take 1 to 5 matches per turn, and each player can take 6 matches at most 10 times during the game. The player who takes the last match wins. Determine which player has a winning strategy. | [
"The second player has a winning strategy."
] | [
"There is a pile of 1000 matches. Two players take turns, each can take 1 to 5 matches per turn, and each player can take 6 matches at most 10 times during the game. The player who takes the last match wins. Determine which player has a winning strategy."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 20
} | 25 | 0 | 0 |
omr-ws-7acd1217eba650268907063d7ca31c7af7b90c4701aab885b5ba3544a005efa7 | There is a pile of 129 coins on a table, all unbiased except for one which has heads on both sides. Bob chooses a coin at random and tosses it eight times, and it comes up heads every time. What is the probability that it will come up heads the ninth time as well? | [
"\\(\\frac{5}{6}\\)"
] | [
"There is a pile of 129 coins on a table, all unbiased except for one which has heads on both sides. Bob chooses a coin at random and tosses it eight times, and it comes up heads every time. What is the probability that it will come up heads the ninth time as well?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 63
} | 95 | 0 | 0 |
omr-ws-3cccf8addb2a04dc43990866c04ca3016c3c84459cc295d4f823409f12afaf24 | There is a pile of 129 coins on a table, all unbiased except for one which has heads on both sides. Bob chooses a coin at random and tosses it in the air 8 times. The coin comes up heads every time. What is the probability that it will come up heads the ninth time as well? | [
"\\(\\frac{5}{6}\\)"
] | [
"There is a pile of 129 coins on a table, all unbiased except for one which has heads on both sides. Bob chooses a coin at random and tosses it in the air 8 times. The coin comes up heads every time. What is the probability that it will come up heads the ninth time as well?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-cc8a244a7a9edd7be56845a5161cb0997e905f826afba87d7207fc0edac03e3d | There is a pile of eggs. Joan's count was off by 1 in the 1’s place, Tom's count was off by 1 in the 10’s place, and Raoul's count was off by 1 in the 100’s place. Sasha, Jose, Peter, and Morris all counted the eggs correctly. When these seven people added their counts together, the sum was 3162. How many eggs were in ... | [
"439"
] | [
"There is a pile of eggs. Joan's count was off by 1 in the 1’s place, Tom's count was off by 1 in the 10’s place, and Raoul's count was off by 1 in the 100’s place. Sasha, Jose, Peter, and Morris all counted the eggs correctly. When these seven people added their counts together, the sum was 3162. How many eggs wer... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-09cfc4e934ff9683a927d1972d8f36f9e07f6b346e919eaa00a365047ccb611b | There is a pile of marbles and a group of kids. The first kid takes $a$ marbles and the $n$-th part of the remainder, the second kid takes $2a$ marbles and the $n$-th part of the remainder, and so on. In the end, there are no marbles left and all the kids have the same number of marbles. Determine the number of marbles... | [
"\\( k = n - 1 \\) and \\( x = a(n-1)^2 \\)"
] | [
"There is a pile of marbles and a group of kids. The first kid takes $a$ marbles and the $n$-th part of the remainder, the second kid takes $2a$ marbles and the $n$-th part of the remainder, and so on. In the end, there are no marbles left and all the kids have the same number of marbles. Determine the number of ma... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 7
} | 7 | 0 | 0 |
omr-ws-383d9be5d029566d15a380636fad70a2877320815d4cd51ac615a0e020f8f9d0 | There is a pile with 15 coins on a table. At each step, Pedro chooses one of the piles with more than 1 coin and divides it into two piles with at least 1 coin each, writing the product of the sizes of the original pile and the two new piles on the board. He continues until there are 15 piles with 1 coin each. Determin... | [
"1120"
] | [
"There is a pile with 15 coins on a table. At each step, Pedro chooses one of the piles with more than 1 coin and divides it into two piles with at least 1 coin each, writing the product of the sizes of the original pile and the two new piles on the board. He continues until there are 15 piles with 1 coin each. Det... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-7328fbc6b4de4cad4a4d332a61eab0c26efdef16214d04619c47e977d79dccbe | There is a pile with 2022 rocks. Ana and Beto play by turns, starting with Ana. In each turn, if there are $n$ rocks in the pile, the player can remove $S(n)$ rocks or $n-S(n)$ rocks, where $S(n)$ is the sum of the digits of $n$. The person who removes the last rock wins. Determine which of the two players has a winnin... | [
"Ana has a winning strategy."
] | [
"There is a pile with 2022 rocks. Ana and Beto play by turns, starting with Ana. In each turn, if there are $n$ rocks in the pile, the player can remove $S(n)$ rocks or $n-S(n)$ rocks, where $S(n)$ is the sum of the digits of $n$. The person who removes the last rock wins. Determine which of the two players has a w... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 21
} | 44 | 0 | 0 |
omr-ws-3d8b0d68be536e6704147e84f8dff1f6a44dcbcb493c78fa1def19fd40909818 | There is a point $C$ marked on a segment $AB$. Distinct points $X$ and $Y$ are such that the triangles $ABX$ and $ACY$ are equilateral. Denote the midpoints of $YC$ by $K$ and the midpoints of $BX$ by $M$. It turns out that the triangle $AMK$ is a right triangle. Find the quotient $AC/BC$. | [
"1"
] | [
"There is a point $C$ marked on a segment $AB$. Distinct points $X$ and $Y$ are such that the triangles $ABX$ and $ACY$ are equilateral. Denote the midpoints of $YC$ by $K$ and the midpoints of $BX$ by $M$. It turns out that the triangle $AMK$ is a right triangle. Find the quotient $AC/BC$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 47
} | 47 | 0 | 0 |
omr-ws-29d8716c93da466b58a501b2f24b5e15cb907a2732e0484d36ce7134d9bc82e7 | There is a point $O$ and another point $P_0$ is 29 cm away from $O$. Point $P_1$ is such that $\angle{OP_0P_1}$ is a right angle and $P_1$ is 3 cm away from $P_0$. Point $P_2$ is 3 cm away from $P_1$ and $\angle{OP_1P_2}$ is also a right angle. This pattern continues with $\angle{OP_nP_{n+1}}$ being a right angle and t... | [
"35"
] | [
"There is a point $O$ and another point $P_0$ is 29 cm away from $O$. Point $P_1$ is such that $\\angle{OP_0P_1}$ is a right angle and $P_1$ is 3 cm away from $P_0$. Point $P_2$ is 3 cm away from $P_1$ and $\\angle{OP_1P_2}$ is also a right angle. This pattern continues with $\\angle{OP_nP_{n+1}}$ being a right ang... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-11bc5b935ba1c5be8ec7e40412927458fe36af926bf9fe88e46220dea9fcf487 | There is a point P inside a square ABCD such that AP = 5 cm, BP = 8 cm, and CP = 13 cm. Find the area of square ABCD. | [
"153"
] | [
"There is a point P inside a square ABCD such that AP = 5 cm, BP = 8 cm, and CP = 13 cm. Find the area of square ABCD."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-e47af6d240016a189c8ccfffd0a6d5983098a9bfe2fa56eedaa248aaadce072e | There is a point inside an equilateral triangle which is at a distance of 1, 2, and 3 from the three sides. What is the area of the triangle? | [
"12\\sqrt{3}"
] | [
"There is a point inside an equilateral triangle which is at a distance of 1, 2, and 3 from the three sides. What is the area of the triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f6ddcc79933a826437b52f766982de73e7a9420d828477c7e0ea79fc3834f188 | There is a polynomial $P(x)$ with integer coefficients such that $$P(x)=\frac{(x^{2310}-1)^6}{(x^{105}-1)(x^{70}-1)(x^{42}-1)(x^{30}-1)}$$ holds for every $0<x<1.$ Find the coefficient of $x^{2022}$ in $P(x)$. | [
"220"
] | [
"There is a polynomial $P(x)$ with integer coefficients such that $$P(x)=\\frac{(x^{2310}-1)^6}{(x^{105}-1)(x^{70}-1)(x^{42}-1)(x^{30}-1)}$$ holds for every $0<x<1.$ Find the coefficient of $x^{2022}$ in $P(x)$."
] | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 44,
"tir": 17
} | 61 | 0 | 0 |
omr-ws-8594daecc74a84f0e881f4cd9de2e1c65e605cbdce04c4e52b4a786e69d99eac | There is a positive integer $N$, with two base-ten digits, such that $N$ has its digits reversed when expressed in base $16$. What is $N$? | [
"53"
] | [
"There is a positive integer $N$, with two base-ten digits, such that $N$ has its digits reversed when expressed in base $16$. What is $N$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-14a1ab716b47282af633ded48bf108e5de5db484cadc068c8c5dc7045c34616f | There is a positive integer $n$ such that $n^4$ is of the form $\overline{1AABAAB}$, where $A$ and $B$ are digits. Find $n$. | [
"34"
] | [
"There is a positive integer $n$ such that $n^4$ is of the form $\\overline{1AABAAB}$, where $A$ and $B$ are digits. Find $n$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-91ba599094de7e3e8d249fec12d76742783b2fa626e527c9ef896f8fe4701441 | There is a positive integer called $n$. On the blackboard, each number from 1 to $n$ is written either in red or white. The sum of any two integers written in the same color must not be a square number. What is the largest possible value for $n$? | [
"14"
] | [
"There is a positive integer called $n$. On the blackboard, each number from 1 to $n$ is written either in red or white. The sum of any two integers written in the same color must not be a square number. What is the largest possible value for $n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 10
} | 32 | 0 | 0 |
omr-ws-b605ad11b43fab5e2cde8f35728c7120a32789da3983ae84844c01e52f60241b | There is a positive real number $x$ not equal to either $\tfrac{1}{20}$ or $\tfrac{1}{2}$ such that $$\log_{20x} (22x)=\log_{2x} (202x).$$ The value $\log_{20x} (22x)$ can be written as $\log_{10} (\tfrac{m}{n})$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | [
"112"
] | [
"There is a positive real number $x$ not equal to either $\\tfrac{1}{20}$ or $\\tfrac{1}{2}$ such that $$\\log_{20x} (22x)=\\log_{2x} (202x).$$ The value $\\log_{20x} (22x)$ can be written as $\\log_{10} (\\tfrac{m}{n})$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$."
] | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-49ec4aa723aadf7e767c22a45301d3a62cfeffa40e20e0555360bec299003fe3 | There is a positive real number \( x \) not equal to either \( \frac{1}{20} \) or \( \frac{1}{2} \) such that \[ \log_{20x} (22x) = \log_{2x} (202x). \] The value \( \log_{20x} (22x) \) can be written as \( \log_{10} \left( \frac{m}{n} \right) \), where \( m \) and \( n \) are relatively prime positive integers. Find \... | [
"112"
] | [
"There is a positive real number \\( x \\) not equal to either \\( \\frac{1}{20} \\) or \\( \\frac{1}{2} \\) such that \n\\[ \\log_{20x} (22x) = \\log_{2x} (202x). \\]\nThe value \\( \\log_{20x} (22x) \\) can be written as \\( \\log_{10} \\left( \\frac{m}{n} \\right) \\), where \\( m \\) and \\( n \\) are relativel... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 23
} | 71 | 0 | 0 |
omr-ws-d8fe8d3740699353d71880a4f39de3789d2d8a919b2a286e61f69319e924edf8 | There is a prime number $p$ such that $16p+1$ is the cube of a positive integer. Find $p$. | [
"307"
] | [
"There is a prime number $p$ such that $16p+1$ is the cube of a positive integer. Find $p$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7f58f12f591180da1e0a9f56ef9586726176f03b020bea46319cba38e02d69e4 | There is a prime number \(p\) such that \(16p + 1\) is the cube of a positive integer. Find \(p\). | [
"307"
] | [
"There is a prime number \\(p\\) such that \\(16p + 1\\) is the cube of a positive integer. Find \\(p\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-c892a1930d1062671487022b6e4aaca94156237ed1117fbd2687604ab2e72d38 | There is a prime number which is a factor of every sum of three consecutive integers. What is the number? | [
"3"
] | [
"There is a prime number which is a factor of every sum of three consecutive integers. What is the number?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5b23608357ba00cad33a4f4fac7b634fdd9aa704aa0bb0d88f392abe805ab1dc | There is a raffle with 12 tickets and two prizes of \$9 and \$3. Persons A, B, and C each hold one ticket and have probabilities of telling the truth of 1/2, 2/3, and 3/5, respectively. A and B each tell person D that they have the ticket to the \$9 prize, and C tells D that they have the ticket to the \$3 prize. D hol... | [
"6.1333"
] | [
"There is a raffle with 12 tickets and two prizes of \\$9 and \\$3. Persons A, B, and C each hold one ticket and have probabilities of telling the truth of 1/2, 2/3, and 3/5, respectively. A and B each tell person D that they have the ticket to the \\$9 prize, and C tells D that they have the ticket to the \\$3 pri... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7de700e9993140f7cd866d0bd5173cb5b4fae2965c72f3e8081a759802b87196 | There is a real number \( x \) between 0 and \( \frac{\pi}{2} \) such that \[ \frac{\sin^3 x + \cos^3 x}{\sin^5 x + \cos^5 x} = \frac{12}{11}. \] Then \( \sin x + \cos x = \frac{\sqrt{m}}{n} \), where \( m \) and \( n \) are positive integers, and \( m \) is not divisible by the square of any prime. Find \( m+n \). | [
"8"
] | [
"There is a real number \\( x \\) between 0 and \\( \\frac{\\pi}{2} \\) such that\n\\[\n\\frac{\\sin^3 x + \\cos^3 x}{\\sin^5 x + \\cos^5 x} = \\frac{12}{11}.\n\\]\nThen \\( \\sin x + \\cos x = \\frac{\\sqrt{m}}{n} \\), where \\( m \\) and \\( n \\) are positive integers, and \\( m \\) is not divisible by the squar... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-46e2bc9f50d02cf7c92b8d5a73d08de961e81465dcad9086bbd065a95d48817e | There is a real number written in each unit square of an \(N \times N\) board (\(N \ge 2\)), such that each number is equal to the sum of the real numbers written in adjacent unit squares. Given that some of the numbers are not equal to 0, find all possible values of \(N\). | [
"\\(N = 5k - 1\\) or \\(N = 6k - 1\\) for integer \\(k\\)."
] | [
"There is a real number written in each unit square of an \\(N \\times N\\) board (\\(N \\ge 2\\)), such that each number is equal to the sum of the real numbers written in adjacent unit squares. Given that some of the numbers are not equal to 0, find all possible values of \\(N\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 8,
"tir": 9
} | 18 | 0 | 0 |
omr-ws-68dfa76d0615ad755eee70f4333a54a016b0997f5f9e59bdd1b9738d93e0da0e | There is a real number written on every field of an $n \times n$ chessboard. The sum of all numbers in any "cross" (the union of a row and a column) is at least $a$. What is the smallest possible sum of all numbers on the board? | [
"\\(\\frac{n^2 a}{2n-1}\\)"
] | [
"There is a real number written on every field of an $n \\times n$ chessboard. The sum of all numbers in any \"cross\" (the union of a row and a column) is at least $a$. What is the smallest possible sum of all numbers on the board?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-c717d3798000ea1eaec5bbea8513c53594478816b26af432e4d614625b9ef929 | There is a rectangle \(ABCD\). Point \(G\) is on \(AB\) such that point \(G\) is closer to \(B\) than to \(A\), but not on point \(A\) or \(B\). Points \(E\) and \(F\) are on the sides \(DG\) and \(CG\) respectively, such that \(3DE = 4EG\) and \(3CF = 4FG\). Points \(H\) and \(I\) are on side \(EF\), point \(J\) is on... | [
"73553 \\frac{5}{11}"
] | [
"There is a rectangle \\(ABCD\\). Point \\(G\\) is on \\(AB\\) such that point \\(G\\) is closer to \\(B\\) than to \\(A\\), but not on point \\(A\\) or \\(B\\). Points \\(E\\) and \\(F\\) are on the sides \\(DG\\) and \\(CG\\) respectively, such that \\(3DE = 4EG\\) and \\(3CF = 4FG\\). Points \\(H\\) and \\(I\\) ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 7,
"tir": 5
} | 15 | 0 | 0 |
omr-ws-97b646c311d9ebd55b76d2a7312b0f043910932b87fe91fda49fec30e482ec0a | There is a red block and a blue block of equal size. The red block is 100 degrees and the blue block is 0 degrees. You can cut blocks into pieces and put pieces together. Pieces that are put together will reach thermal equilibrium. In the end, you reassemble the red and blue blocks. How hot can the blue block get in th... | [
"\\( 100\\left( 1 - \\frac{1}{e} \\right) \\) or approximately \\( 63.21 \\) degrees."
] | [
"There is a red block and a blue block of equal size. The red block is 100 degrees and the blue block is 0 degrees. You can cut blocks into pieces and put pieces together. Pieces that are put together will reach thermal equilibrium. In the end, you reassemble the red and blue blocks. How hot can the blue block get ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 1,
"tir": 2
} | 13 | 0 | 0 |
omr-ws-7f25000ac58ae6c96f53e064a37e18d6c541cdd72fb0e24ffc7fb281b348227d | There is a round table divided into 4 equal quadrants labeled A, B, C, and D, each containing a cup that is either face-up or face-down. You are blindfolded and can instruct a genie to flip the cups in any chosen quadrants. After each flip, the genie may rotate the table by 0, 90, 180, or 270 degrees. Devise a strategy... | [
"a strategy to ensure all cups are face-up in a finite number of moves for \\( n = 4 \\) is provided in the original problem statement. The specific sequence of moves is:\n\nFlip none, \nflip all, \nflip A and C, \nflip all, \nflip A and B, \nflip all, \nflip A and C, \nflip all, \nflip A, \nflip all, \nflip A and ... | [
"There is a round table divided into 4 equal quadrants labeled A, B, C, and D, each containing a cup that is either face-up or face-down. You are blindfolded and can instruct a genie to flip the cups in any chosen quadrants. After each flip, the genie may rotate the table by 0, 90, 180, or 270 degrees. Devise a str... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 2,
"tir": 2
} | 16 | 0 | 0 |
omr-ws-ab510eb1924e2002ed8eaca74fb9b7d2ef883b8e5e3ff6b7a25cf8a2f86fb390 | There is a row of 100 cells each containing a token. For 1 dollar, it is allowed to interchange two neighboring tokens. Additionally, it is allowed to interchange any two tokens with no charge if there are exactly 3 tokens between them. What is the minimum price for arranging all the tokens in reverse order? | [
"50"
] | [
"There is a row of 100 cells each containing a token. For 1 dollar, it is allowed to interchange two neighboring tokens. Additionally, it is allowed to interchange any two tokens with no charge if there are exactly 3 tokens between them. What is the minimum price for arranging all the tokens in reverse order?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-09da4680d0a99fc4c6bb0bec7d92dc500449bf0b8c5f27a45172c4b90904dc18 | There is a row of 2023 identical boxes, each containing one coin. On each turn, you can open any box and take the coin if it contains one. Afterward, the game director will secretly swap the box you opened with one of its neighboring boxes. Determine the maximum number of coins you can guarantee to collect in finite ti... | [
"1518"
] | [
"There is a row of 2023 identical boxes, each containing one coin. On each turn, you can open any box and take the coin if it contains one. Afterward, the game director will secretly swap the box you opened with one of its neighboring boxes. Determine the maximum number of coins you can guarantee to collect in fini... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-770b5a2ba63a79b15f7e8be63d46f785aaa8400a8ae73249ca54a439baf2facb | There is a safe that can be opened by entering a secret code consisting of $n$ digits, each of which is $0$ or $1$. Initially, $n$ zeros were entered, and the safe is closed (so, all zeros is not the secret code). In one attempt, you can enter an arbitrary sequence of $n$ digits, each of which is $0$ or $1$. If the ent... | [
"\\( n \\)"
] | [
"There is a safe that can be opened by entering a secret code consisting of $n$ digits, each of which is $0$ or $1$. Initially, $n$ zeros were entered, and the safe is closed (so, all zeros is not the secret code). In one attempt, you can enter an arbitrary sequence of $n$ digits, each of which is $0$ or $1$. If th... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-5d4e5982ccbbb2bc861cd87e712b8924c75f1cf30669748eaef00af9e5ad4ba0 | There is a sequence $h_m$ with $h_1 = 5$ and $h_2 = 29$ such that for all $m \geq 1$, $h_m < h_{m+1}$ and $(2, h_m, h_{m+1})$ is a solution to $x^2 + y^2 + z^2 = 3xyz$. Find $h_3, h_4, h_5, h_6, h_7$. | [
"\\( h_3 = 169, h_4 = 985, h_5 = 5741, h_6 = 33461, h_7 = 195025 \\)"
] | [
"There is a sequence $h_m$ with $h_1 = 5$ and $h_2 = 29$ such that for all $m \\geq 1$, $h_m < h_{m+1}$ and $(2, h_m, h_{m+1})$ is a solution to $x^2 + y^2 + z^2 = 3xyz$. Find $h_3, h_4, h_5, h_6, h_7$."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 24
} | 24 | 0 | 0 |
omr-ws-566057ade79a76dccedf3b7cdccfcd9bdeb9137c16688c1c6cdad29e261dbcb7 | There is a sequence of $n$ integers such that the sum of any 3 consecutive members is positive and the sum of any 4 consecutive members is negative. What is the largest possible value of $n$? | [
"5"
] | [
"There is a sequence of $n$ integers such that the sum of any 3 consecutive members is positive and the sum of any 4 consecutive members is negative. What is the largest possible value of $n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-99017aecae73b27074baacd69f748b0bf1ce2ee9427f26f738c0ad9d82d08153 | There is a sequence of numbers $+1$ and $-1$ of length $n$. It is known that the sum of every $10$ neighbouring numbers in the sequence is $0$ and that the sum of every $12$ neighbouring numbers in the sequence is not zero. What is the maximal value of $n$? | [
"15"
] | [
"There is a sequence of numbers $+1$ and $-1$ of length $n$. It is known that the sum of every $10$ neighbouring numbers in the sequence is $0$ and that the sum of every $12$ neighbouring numbers in the sequence is not zero. What is the maximal value of $n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-54835b89fc2613a0e55f1bac0b421d16cdd0fe482f1cb544209917b9c6004920 | There is a sequence of positive numbers $a_1, a_2, a_3, \ldots, a_m$ such that $a_1 = \frac{233}{9}$, $a_2 = 18$, $a_3 = 16$, and $a_m = 0$, which follows the relation $$a_{n+3} = a_{n+1} \left( \frac{a_n}{a_{n+2}} - 1 \right) \,\,\, \text{for} \,\,\, n = 1, 2, 3, \ldots, m.$$ Find the value of $m$. | [
"15"
] | [
"There is a sequence of positive numbers $a_1, a_2, a_3, \\ldots, a_m$ such that $a_1 = \\frac{233}{9}$, $a_2 = 18$, $a_3 = 16$, and $a_m = 0$, which follows the relation $$a_{n+3} = a_{n+1} \\left( \\frac{a_n}{a_{n+2}} - 1 \\right) \\,\\,\\, \\text{for} \\,\\,\\, n = 1, 2, 3, \\ldots, m.$$ Find the value of $m$."
... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-b8ee844b1b5015941ca66fe577d208877a914bdbcea707aa9ba8c6b241c9e18b | There is a sequence of real numbers \(a_1, a_2, \dots\) such that \(a_1=1\), \(a_2=\frac{3}{5}\), and \(\frac{1}{a_n}=\frac{2}{a_{n-1}}-\frac{1}{a_{n-2}}\) for every \(n \geq 3\). The number \(a_{2020}\) is a fraction in the form of \(\frac{p}{q}\) where \(p\) and \(q\) are relatively prime positive integers. Find the ... | [
"4044"
] | [
"There is a sequence of real numbers \\(a_1, a_2, \\dots\\) such that \\(a_1=1\\), \\(a_2=\\frac{3}{5}\\), and \\(\\frac{1}{a_n}=\\frac{2}{a_{n-1}}-\\frac{1}{a_{n-2}}\\) for every \\(n \\geq 3\\). The number \\(a_{2020}\\) is a fraction in the form of \\(\\frac{p}{q}\\) where \\(p\\) and \\(q\\) are relatively prim... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6913a4f3ae5ccce54a22313c164a634c6b573b5c87b28feac84ffaa063226b70 | There is a sequence so that $a_0 = 2$, $a_1 = 3$, and $a_i = a_{i-1} \cdot a_{i-2}$. Find the units digit of $a_{a_{2023}}$. | [
"2"
] | [
"There is a sequence so that $a_0 = 2$, $a_1 = 3$, and $a_i = a_{i-1} \\cdot a_{i-2}$. Find the units digit of $a_{a_{2023}}$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 48
} | 48 | 0 | 0 |
omr-ws-fa53062f404f6673f0099a9f5e7a2ce4575e33e536a20a508093f7288b95c372 | There is a set $S$ of numbers $\{1, 2, 3, \ldots, 20\}$. A subset $X$ is chosen from the subsets of $S$ such that the sum of the elements of $X$ is 25. If an element of $X$ is picked at random, what is the expected value of that element? | [
"5"
] | [
"There is a set $S$ of numbers $\\{1, 2, 3, \\ldots, 20\\}$. A subset $X$ is chosen from the subsets of $S$ such that the sum of the elements of $X$ is 25. If an element of $X$ is picked at random, what is the expected value of that element?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 1
} | 14 | 0 | 0 |
omr-ws-5aabf3dedf11b5da533feb820a5138fea0767c76197ebff2386863d045ed0d86 | There is a set S of numbers {1, 2, 3, ..., 20}. A subset X is chosen from S such that the sum of the elements of X is 25. If an element of X is chosen at random, what is the expected value of that element? | [
"5"
] | [
"There is a set S of numbers {1, 2, 3, ..., 20}. A subset X is chosen from S such that the sum of the elements of X is 25. If an element of X is chosen at random, what is the expected value of that element?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-9dafcfe2947f5578e6d71d725d526b30251b5c31ef797a350802e2b5f6f7e477 | There is a set of $n$ coins with distinct integer weights $w_1, w_2, \ldots, w_n$. It is known that if any coin with weight $w_k$, where $1 \leq k \leq n$, is removed from the set, the remaining coins can be split into two groups of the same weight. Find all $n$ for which such a set of coins exists. | [
"\\( n \\) is any odd number greater than 5."
] | [
"There is a set of $n$ coins with distinct integer weights $w_1, w_2, \\ldots, w_n$. It is known that if any coin with weight $w_k$, where $1 \\leq k \\leq n$, is removed from the set, the remaining coins can be split into two groups of the same weight. Find all $n$ for which such a set of coins exists."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-bfdd419d173c253d9ed8c6efb07731d28fe1b61711a218226501e4750b584145 | There is a set of 1000 switches, each of which has four positions, called $A, B, C$, and $D$. When the position of any switch changes, it is only from $A$ to $B$, from $B$ to $C$, from $C$ to $D$, or from $D$ to $A$. Initially each switch is in position $A$. The switches are labeled with the 1000 different integers $(2... | [
"650"
] | [
"There is a set of 1000 switches, each of which has four positions, called $A, B, C$, and $D$. When the position of any switch changes, it is only from $A$ to $B$, from $B$ to $C$, from $C$ to $D$, or from $D$ to $A$. Initially each switch is in position $A$. The switches are labeled with the 1000 different inte... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-2c6bb7551845d18c8e5a40559d5344b77b5e67a670b32b5445e4564ea21df7f6 | There is a set of 1000 switches, each of which has four positions, called $A, B, C$, and $D$. When the position of any switch changes, it is only from $A$ to $B$, from $B$ to $C$, from $C$ to $D$, or from $D$ to $A$. Initially each switch is in position $A$. The switches are labeled with the 1000 different integers $(2... | [
"650"
] | [
"There is a set of 1000 switches, each of which has four positions, called $A, B, C$, and $D$. When the position of any switch changes, it is only from $A$ to $B$, from $B$ to $C$, from $C$ to $D$, or from $D$ to $A$. Initially each switch is in position $A$. The switches are labeled with the 1000 different integer... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-13dbede731581fa732779300ef17999e984789a24d225ab23c9d3d193fef2f99 | There is a set of 1000 switches, each of which has four positions, called $A, B, C,$ and $D.$ When the position of any switch changes, it is only from $A$ to $B,$ from $B$ to $C,$ from $C$ to $D,$ or from $D$ to $A.$ Initially each switch is in position $A.$ The switches are labeled with the 1000 different integers $2^... | [
"650"
] | [
"There is a set of 1000 switches, each of which has four positions, called $A, B, C,$ and $D.$ When the position of any switch changes, it is only from $A$ to $B,$ from $B$ to $C,$ from $C$ to $D,$ or from $D$ to $A.$ Initially each switch is in position $A.$ The switches are labeled with the 1000 different integer... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-b137299ddd9ede0dab22ab8e1c8d2195ef5594294a0cebcc965be2a9308a33c1 | There is a set of 1000 switches, each of which has four positions, called A, B, C, and D. When the position of any switch changes, it is only from A to B, from B to C, from C to D, or from D to A. Initially, each switch is in position A. The switches are labeled with the 1000 different integers \((2^x)(3^y)(5^z)\), whe... | [
"650"
] | [
"There is a set of 1000 switches, each of which has four positions, called A, B, C, and D. When the position of any switch changes, it is only from A to B, from B to C, from C to D, or from D to A. Initially, each switch is in position A. The switches are labeled with the 1000 different integers \\((2^x)(3^y)(5^z)\... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-b44e4cba54b11b257d6d7179d7acf0c5ca4d1069c75f9e680065a182794b16ea | There is a set of 1000 switches, each of which has four positions, called A, B, C, and D. When the position of any switch changes, it is only from A to B, from B to C, from C to D, or from D to A. Initially, each switch is in position A. The switches are labeled with the 1000 different integers \(2^x \cdot 3^y \cdot 5^... | [
"650"
] | [
"There is a set of 1000 switches, each of which has four positions, called A, B, C, and D. When the position of any switch changes, it is only from A to B, from B to C, from C to D, or from D to A. Initially, each switch is in position A. The switches are labeled with the 1000 different integers \\(2^x \\cdot 3^y \... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c796c3197e4962394a0e02d099ad2c2bdf11a428c76cb12963f5451ed2c3b430 | There is a set of 1998 different natural numbers such that none of them can be represented as the sum of several other numbers in this set. What is the smallest value that the largest of these numbers can take? | [
"\\(2^{1997}\\)"
] | [
"There is a set of 1998 different natural numbers such that none of them can be represented as the sum of several other numbers in this set. What is the smallest value that the largest of these numbers can take?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 1
} | 3 | 0 | 0 |
omr-ws-3ec8bf9d26cd28cbb457de478d0cde4ea368ab86e423c2d8207f37b4c6a3e6e9 | There is a set of 2024 cards. Each card has two sides, each colored in one of three colors — red, blue, or white, with the two sides being different colors. A card is considered to be lying beautifully if at least one of the following conditions is met: its upper side is red, or its underside is blue. Initially, exactl... | [
"1874"
] | [
"There is a set of 2024 cards. Each card has two sides, each colored in one of three colors — red, blue, or white, with the two sides being different colors. A card is considered to be lying beautifully if at least one of the following conditions is met: its upper side is red, or its underside is blue. Initially, e... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f93d3e61a8c98fc12d99e0f9b552f52fe647c3cc7b75294b24b0111312682e0c | There is a set of 50 cards. Each card has two sides, each colored in one of three colors — red, blue, or white, with the two sides being different colors. A card is said to "lie beautifully" if at least one of the following conditions is met: its upper side is red, or its underside is blue. Initially, exactly 25 cards ... | [
"25"
] | [
"There is a set of 50 cards. Each card has two sides, each colored in one of three colors — red, blue, or white, with the two sides being different colors. A card is said to \"lie beautifully\" if at least one of the following conditions is met: its upper side is red, or its underside is blue. Initially, exactly 25... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-e3b40f7f7caa88dda170bafe0adecd43cb908d48d654232eb8723c5f6bd87b68 | There is a set of 6 parallel lines and another set of six parallel lines, where these two sets of lines are not parallel with each other. If Blythe adds 6 more lines, not necessarily parallel with each other, find the maximum number of triangles that could be made. | [
"416"
] | [
"There is a set of 6 parallel lines and another set of six parallel lines, where these two sets of lines are not parallel with each other. If Blythe adds 6 more lines, not necessarily parallel with each other, find the maximum number of triangles that could be made."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d2a3b7da6480aac95257918a9f453cc3829c1efa8faebfc41f40f073c40cdc16 | There is a set of \( k \) stones, and it is known that this set can be divided into 4, 5, and 9 groups of equal weight. Find the least value of \( k \). Two stones can have equal weight, and the weight of a stone is not necessarily an integer. | [
"16"
] | [
"There is a set of \\( k \\) stones, and it is known that this set can be divided into 4, 5, and 9 groups of equal weight. Find the least value of \\( k \\). Two stones can have equal weight, and the weight of a stone is not necessarily an integer."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3f48d68c02cf897b0c7589d5c4597e9435761139318838fff107fc7ef76527b0 | There is a set of five positive integers whose average (mean) is 5, whose median is 5, and whose only mode is 8. What is the difference between the largest and smallest integers in the set? | [
"7"
] | [
"There is a set of five positive integers whose average (mean) is 5, whose median is 5, and whose only mode is 8. What is the difference between the largest and smallest integers in the set?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7964301708bb217b82022c2b5d9eed36b656fe0a93cb99eeb1eb9901a697067b | There is a smallest positive real number $a$ such that there exists a positive real number $b$ such that all the roots of the polynomial $x^3 - ax^2 + bx - a$ are real. For this value of $a$, the value of $b$ is unique. What is this value of $b$? | [
"9"
] | [
"There is a smallest positive real number $a$ such that there exists a positive real number $b$ such that all the roots of the polynomial $x^3 - ax^2 + bx - a$ are real. For this value of $a$, the value of $b$ is unique. What is this value of $b$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ccf00e3af697094b704b931499a27cd9e89932705739536e85b01f5de38a0aa2 | There is a smallest positive real number $a$ such that there exists a positive real number $b$ such that all the roots of the polynomial $x^3-ax^2+bx-a$ are real. In fact, for this value of $a$ the value of $b$ is unique. What is the value of $b?$ | [
"9"
] | [
"There is a smallest positive real number $a$ such that there exists a positive real number $b$ such that all the roots of the polynomial $x^3-ax^2+bx-a$ are real. In fact, for this value of $a$ the value of $b$ is unique. What is the value of $b?$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-555f35a7d32765a55b610283c1457b8fbd2d1fa5d457476465e6782b4fe14c31 | There is a solid composed of equilateral triangles and regular pentagons, with exactly one pentagon and four triangles meeting at each vertex. If $T$ is the number of triangles and $P$ is the number of pentagons, find the value of $T \times P$. | [
"960"
] | [
"There is a solid composed of equilateral triangles and regular pentagons, with exactly one pentagon and four triangles meeting at each vertex. If $T$ is the number of triangles and $P$ is the number of pentagons, find the value of $T \\times P$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-3bff188e1db9ac286302915aa4e2140a0cf1797f37898202e1ef644568e1d971 | There is a solution of sodium chloride of four different concentrations in the laboratory. If we mix the first, the second, and the third solution in the weight ratio $3 : 2 : 1$, we get a 15\% solution. The second, the third, and the fourth solution, taken in equal proportions, give a 24\% solution when mixed, and, fi... | [
"29\\%"
] | [
"There is a solution of sodium chloride of four different concentrations in the laboratory. If we mix the first, the second, and the third solution in the weight ratio $3 : 2 : 1$, we get a 15\\% solution. The second, the third, and the fourth solution, taken in equal proportions, give a 24\\% solution when mixed, ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1251146018066ffb485ff756a22c6127ffa895172f6ad5ed0175ffd1bf8c92c5 | There is a special calculator that, when given an integer \( n \) (\( n > 0 \)), outputs an integer randomly from the set \([0, n-1]\). If you input \( 2021 \) and repeatedly input the output until you get \( 0 \), what is the probability that the numbers \( 9 \), \( 99 \), and \( 999 \) have all been output at least o... | [
"\\dfrac{1}{1000000}"
] | [
"There is a special calculator that, when given an integer \\( n \\) (\\( n > 0 \\)), outputs an integer randomly from the set \\([0, n-1]\\). If you input \\( 2021 \\) and repeatedly input the output until you get \\( 0 \\), what is the probability that the numbers \\( 9 \\), \\( 99 \\), and \\( 999 \\) have all b... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-cd43c0cf4734c3abae951fd619c4bd59b684d9389e46499be73613fa807e6213 | There is a square on the plane with sides parallel to the horizontal and vertical directions. Draw a number of horizontal or vertical line segments in the square, dividing the square into several rectangles, such that any two line segments do not belong to the same line and any two line segments have no common inner po... | [
"kl"
] | [
"There is a square on the plane with sides parallel to the horizontal and vertical directions. Draw a number of horizontal or vertical line segments in the square, dividing the square into several rectangles, such that any two line segments do not belong to the same line and any two line segments have no common inn... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-635798e10aa675d1bbbade1f08df8def4b7b313f6add651c5e8230a64dfdfba4 | There is a store that sells each of the 26 letters. All letters of the same type cost the same amount (i.e., any ‘a’ costs the same as any other ‘a’), but different letters may or may not cost different amounts. For example, the cost of spelling “trade” is the same as the cost of spelling “tread,” even though the cost ... | [
"2024"
] | [
"There is a store that sells each of the 26 letters. All letters of the same type cost the same amount (i.e., any ‘a’ costs the same as any other ‘a’), but different letters may or may not cost different amounts. For example, the cost of spelling “trade” is the same as the cost of spelling “tread,” even though the ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-94307c004d290ba3e4b93ddf67a626352055e9fb643996d5e7944d555a747d6e | There is a table with $n$ rows and $18$ columns. Each cell contains a $0$ or a $1$. The table satisfies the following properties: 1. Every two rows are different. 2. Each row contains exactly $6$ cells that contain $1$. 3. For every three rows, there exists a column so that the intersection of the column with the three... | [
"\\(\\binom{17}{6}\\)"
] | [
"There is a table with $n$ rows and $18$ columns. Each cell contains a $0$ or a $1$. The table satisfies the following properties:\n1. Every two rows are different.\n2. Each row contains exactly $6$ cells that contain $1$.\n3. For every three rows, there exists a column so that the intersection of the column with t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 37,
"tir": 18
} | 55 | 0 | 0 |
omr-ws-cea129dc829c31270b63633b9ea805759e0acf55feb8074740af3cf84d8a232d | There is a table with \( n \) rows and 18 columns, where each cell contains a 0 or a 1. The table satisfies the following properties: - Every two rows are different. - Each row contains exactly 6 cells that contain 1. - For every three rows, there exists a column such that the intersection of the column with the three ... | [
"12,376",
"12376"
] | [
"There is a table with \\( n \\) rows and 18 columns, where each cell contains a 0 or a 1. The table satisfies the following properties:\n- Every two rows are different.\n- Each row contains exactly 6 cells that contain 1.\n- For every three rows, there exists a column such that the intersection of the column with ... | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 23,
"genselect": 16,
"tir": 19
} | 58 | 0 | 0 |
omr-ws-428c93916d936b8480cc91cd0b0a157f292f19f0a3d6207c004b6459e10d3116 | There is a target at which $n$ shots are fired. Every shot hits the target with the probability $p$. How many shots must be fired for the probability of hitting the target to be not less than $P$? | [
"$n>\\ln(1-P)/\\ln(1-p)$, where $n$ is the number of shots fired"
] | [
"There is a target at which $n$ shots are fired. Every shot hits the target with the probability $p$. How many shots must be fired for the probability of hitting the target to be not less than $P$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f0a56a3ae53dde8b982b0320a9ecb72d8a696bfab41a75b2ba86eedec69135c6 | There is a test among the students in each of \( n \) (\( n \geq 3 \)) papers. It is known that for any paper exactly three students got the best score, and in any two papers exactly one student got the best score in both papers. Find the smallest \( n \) such that the above conditions imply that exactly one student go... | [
"7"
] | [
"There is a test among the students in each of \\( n \\) (\\( n \\geq 3 \\)) papers. It is known that for any paper exactly three students got the best score, and in any two papers exactly one student got the best score in both papers. Find the smallest \\( n \\) such that the above conditions imply that exactly on... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 1
} | 8 | 0 | 0 |
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