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-71cd9a7ec01237984f3f37b9e8c8c8c62592f3b770e3f891c1b149e6e5ffbf92 | We draw a regular polygon \( R \) with 2002 vertices and all its diagonals. How many triangles are formed where all the vertices are vertices of \( R \), but none of the sides are sides of \( R \)? | [
"1331332002"
] | [
"We draw a regular polygon \\( R \\) with 2002 vertices and all its diagonals. How many triangles are formed where all the vertices are vertices of \\( R \\), but none of the sides are sides of \\( R \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-7aae2e7b1358ffecbd3b8719d7f84917d35d56d027b006f045732a972b267086 | We draw cards, one at a time, at random and successively from an ordinary deck of $52$ cards with replacement. What is the probability that an ace appears before a face card? | [
"\\dfrac{1}{4}"
] | [
"We draw cards, one at a time, at random and successively from an ordinary deck of $52$ cards with replacement. What is the probability that an ace appears before a face card?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-532de7ce056637f01d355d1a2a558245857132b3352d642a7ca23295caa808b6 | We draw two lines \((\ell_1)\) and \((\ell_2)\) through the orthocenter \(H\) of triangle \(ABC\) such that each line divides the triangle into two figures of equal area and equal perimeters. Find the angles of the triangle \(ABC\). | [
"\\(60^\\circ\\)"
] | [
"We draw two lines \\((\\ell_1)\\) and \\((\\ell_2)\\) through the orthocenter \\(H\\) of triangle \\(ABC\\) such that each line divides the triangle into two figures of equal area and equal perimeters. Find the angles of the triangle \\(ABC\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f7b5a20e34a5affd24669747639a09173302a0fbf2e623926dc51498a7bc1d1c | We drew several straight lines on the plane and marked all of their intersection points. If one point is marked on one of the drawn lines, three points on another, and five points on a third, calculate the sum of all possible numbers of lines that could have been drawn. | [
"6"
] | [
"We drew several straight lines on the plane and marked all of their intersection points. If one point is marked on one of the drawn lines, three points on another, and five points on a third, calculate the sum of all possible numbers of lines that could have been drawn."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-e069eaa69c87632670db010d581bedac3d765c0f6720af9c3563a1100dac2e25 | We fish 1000 fish, mark them, and release them back into the lake. We then fish another 1000 fish. Calculate \( p(n) \), the probability of finding 100 marked fish, where \( n \) is the total number of fish in the lake (assuming \( n > 1900 \)). | [
"p(n) = \\dfrac{\\dbinom{1000}{100} \\dbinom{n - 1000}{900}}{\\dbinom{n}{1000}}"
] | [
"We fish 1000 fish, mark them, and release them back into the lake. We then fish another 1000 fish. Calculate \\( p(n) \\), the probability of finding 100 marked fish, where \\( n \\) is the total number of fish in the lake (assuming \\( n > 1900 \\))."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-75c34a85491400e5c2e21744cb7773edfbe0f5fc17a87e0e3ab0bdcdeff7f41d | We flip a fair coin 10 times. What is the probability that we get heads in at least 8 of the 10 flips? | [
"\\dfrac{7}{128}"
] | [
"We flip a fair coin 10 times. What is the probability that we get heads in at least 8 of the 10 flips?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-30b28555dd369927d5168d41222f0512fa806979336c2b1d6fd982802196518c | We flip a fair coin 10 times. What is the probability that we get heads in exactly 8 of the 10 flips? | [
"\\dfrac{45}{1024}"
] | [
"We flip a fair coin 10 times. What is the probability that we get heads in exactly 8 of the 10 flips?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-18e2fcbd3ed6fa6271fd8d66587aa272dcbec30e962f5b1e09ac278e5f8336bd | We flip a regular coin $2007$ times and record the outcome: heads or tails. The probability of getting heads or tails each time is $\frac{1}{2}$. Let $p_1$ be the probability that the number of heads recorded at the end is twice the number of tails and $p_2$ be the probability that the number of tails recorded at the e... | [
"1"
] | [
"We flip a regular coin $2007$ times and record the outcome: heads or tails. The probability of getting heads or tails each time is $\\frac{1}{2}$. Let $p_1$ be the probability that the number of heads recorded at the end is twice the number of tails and $p_2$ be the probability that the number of tails recorded at... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6ff081079604e68940681cbabf0c89af5fc8462ebf11dfe1df90555203df0740 | We flip fair coins until you decide to stop. When you stop, assuming we have flipped \( n \) times and \( x \) of them came up heads, you receive \( \frac{x}{n} \) dollars. What is your expected value? | [
"\\dfrac{1}{2}"
] | [
"We flip fair coins until you decide to stop. When you stop, assuming we have flipped \\( n \\) times and \\( x \\) of them came up heads, you receive \\( \\frac{x}{n} \\) dollars. What is your expected value?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 1
} | 12 | 0 | 0 |
omr-ws-853c79eb65b0e8fb49ff8c0994868d22170540fb5f88d08535f306fdbbef77c6 | We form a 16-card deck by taking the Jacks, Queens, Kings, and Aces from a standard deck of cards. David partitions this deck into a positive number of groups such that each group contains at least 2 cards. He then puts all the cards in each group on a key-chain such that for each key-chain, no two consecutive cards sh... | [
"721"
] | [
"We form a 16-card deck by taking the Jacks, Queens, Kings, and Aces from a standard deck of cards. David partitions this deck into a positive number of groups such that each group contains at least 2 cards. He then puts all the cards in each group on a key-chain such that for each key-chain, no two consecutive car... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-581774c3e1b2b8f0d47a9d984e295c5f13115669c5032c13831377ee9481b20e | We form a word using only letters from $\{A, B, C\}$ and we cannot have $A$ next to $C$. Find the number of 8-letter words that can be formed. | [
"1393"
] | [
"We form a word using only letters from $\\{A, B, C\\}$ and we cannot have $A$ next to $C$. Find the number of 8-letter words that can be formed."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-afcd1808393a12edd8428d7b444152514df0de4eaf53b26aab6d5d5342856109 | We generate a sequence of numbers as follows. Let $x_1=1$ and for $i\ge1$, choose $x_{i+1}$ at random from the positive integers from 1 to $n-x_i$, inclusive ($n\ge3$). On average, what fraction of the terms in the sequence will be equal to 1? Leave the answer in terms of $n$. | [
"\\(\\frac{2}{n}\\)"
] | [
"We generate a sequence of numbers as follows. Let $x_1=1$ and for $i\\ge1$, choose $x_{i+1}$ at random from the positive integers from 1 to $n-x_i$, inclusive ($n\\ge3$). On average, what fraction of the terms in the sequence will be equal to 1? Leave the answer in terms of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 21
} | 68 | 0 | 0 |
omr-ws-5ed814b59030ea012479e5bc79603cf85a30fa1e1394b8f0840735c52d2e54f4 | We give 6 different balls to 2 children. Each child gets at least one ball. What is the probability that one of the children receives more than 2 balls more than the other? | [
"\\(\\frac{15}{31}\\)"
] | [
"We give 6 different balls to 2 children. Each child gets at least one ball. What is the probability that one of the children receives more than 2 balls more than the other?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-25bba7f4b550b3452283f3fa92f43ed21607589fe19efab512a704227e71fe6c | We have $20,000 that must be invested among 4 possible mutual funds. Each investment must be in units of $1,000, and there are minimal investments required for each fund: $2,000, $2,000, $3,000, and $4,000. How many different investment strategies are available if an investment must be made in each mutual fund? | [
"220"
] | [
"We have $20,000 that must be invested among 4 possible mutual funds. Each investment must be in units of $1,000, and there are minimal investments required for each fund: $2,000, $2,000, $3,000, and $4,000. How many different investment strategies are available if an investment must be made in each mutual fund?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 5
} | 20 | 0 | 0 |
omr-ws-9a3f991914178c07b415172ca1b51de09fca62564fc78ead0fa3859303564d64 | We have $2021$ colors and $2021$ chips of each color, placed in a row. A chip $F$ is called "bad" if there is an odd number of chips with a different color both to the left and to the right of $F$. Determine the minimum possible number of bad chips. | [
"1010"
] | [
"We have $2021$ colors and $2021$ chips of each color, placed in a row. A chip $F$ is called \"bad\" if there is an odd number of chips with a different color both to the left and to the right of $F$. Determine the minimum possible number of bad chips."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 7,
"tir": 4
} | 22 | 0 | 0 |
omr-ws-69b969bcf31ed36248b575b3df66ad3ff0cc0429d9094af60a88dbf0bdcedef0 | We have $2^m$ cards, each with the number $1$ written on it. In each step, we choose two cards with numbers $a$ and $b$ on them, and replace both with the number $a+b$. After performing this operation $m \cdot 2^{m-1}$ times, calculate the minimum possible sum of the numbers on the cards. | [
"4^m"
] | [
"We have $2^m$ cards, each with the number $1$ written on it. In each step, we choose two cards with numbers $a$ and $b$ on them, and replace both with the number $a+b$. After performing this operation $m \\cdot 2^{m-1}$ times, calculate the minimum possible sum of the numbers on the cards."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-811074276d70feef30d69bfe41ba5de3095855786759f5fd9d4f5378feaf51ff | We have $2^m$ sheets of paper, each with the number $1$ written on it. In each step, we choose two distinct sheets with numbers $a$ and $b$, erase these numbers, and write $a+b$ on both sheets. After $m2^{m-1}$ steps, calculate the minimum possible sum of the numbers on all the sheets. | [
"4^m"
] | [
"We have $2^m$ sheets of paper, each with the number $1$ written on it. In each step, we choose two distinct sheets with numbers $a$ and $b$, erase these numbers, and write $a+b$ on both sheets. After $m2^{m-1}$ steps, calculate the minimum possible sum of the numbers on all the sheets."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b1523b7d9dba714981854a07d21c38ea4cbca371250dddae263716fe72c83468 | We have $2^m$ sheets of paper, each with the number $1$ written on it. In each step, we choose two distinct sheets with numbers $a$ and $b$, erase these numbers, and write the number $a + b$ on both sheets. After performing $m2^{m-1}$ steps, let $S$ be the sum of the numbers on all the sheets. Find the minimum possible... | [
"1024"
] | [
"We have $2^m$ sheets of paper, each with the number $1$ written on it. In each step, we choose two distinct sheets with numbers $a$ and $b$, erase these numbers, and write the number $a + b$ on both sheets. After performing $m2^{m-1}$ steps, let $S$ be the sum of the numbers on all the sheets. Find the minimum pos... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8b23b4d8391fd48010dcff440f4bd572efb6b5e97adf5f62e646b2a76cf960b4 | We have $2^m$ sheets of paper, each with the number $1$ written on it. In each step, we choose two sheets, delete the numbers $a$ and $b$ from them, and write the number $a + b$ on both sheets. After $m \cdot 2^{m-1}$ steps, calculate the minimum possible sum of the numbers on all the sheets when $m = 4$. | [
"256"
] | [
"We have $2^m$ sheets of paper, each with the number $1$ written on it. In each step, we choose two sheets, delete the numbers $a$ and $b$ from them, and write the number $a + b$ on both sheets. After $m \\cdot 2^{m-1}$ steps, calculate the minimum possible sum of the numbers on all the sheets when $m = 4$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-44ec0e1fc79906b0d5130a244b324e533631b44bb57691313388ff2e7199471a | We have $2n$ batteries $(n>2)$, with $n$ full batteries and $n$ empty ones. What is the smallest number of tries needed to ensure that we have at least 2 full batteries to make a lamp work? | [
"n + 2"
] | [
"We have $2n$ batteries $(n>2)$, with $n$ full batteries and $n$ empty ones. What is the smallest number of tries needed to ensure that we have at least 2 full batteries to make a lamp work?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 3
} | 17 | 0 | 0 |
omr-ws-d062887e5d0998a99427f6057526abd57ee7e1883fb89e040bf0432698ec8581 | We have $2n+1$ batteries $(n>2)$, with $n+1$ full batteries and $n$ empty ones. What is the smallest number of tries needed to ensure that we have at least 2 full batteries to make a lamp work? | [
"\\( n+2 \\)",
"n + 2"
] | [
"We have $2n+1$ batteries $(n>2)$, with $n+1$ full batteries and $n$ empty ones. What is the smallest number of tries needed to ensure that we have at least 2 full batteries to make a lamp work?"
] | [
"aops_c6_high_school_olympiads",
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 56,
"genselect": 7
} | 63 | 0 | 0 |
omr-ws-635c1f25d36e9b058dbf025242c9545e7dfb9ee24fb6efd975953800e6d20d90 | We have $80$ black squares in a $17 \times 17$ table. At each moment, we can paint black a row or a column that has more than half of its squares black. Determine the maximum number of squares that can be painted black following these rules. | [
"208"
] | [
"We have $80$ black squares in a $17 \\times 17$ table. At each moment, we can paint black a row or a column that has more than half of its squares black. Determine the maximum number of squares that can be painted black following these rules."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-b2edf15fa4cce15794080b14e6b2fe81e10aff1afb85a257eb7d119de195d590 | We have $\dfrac{n(n+1)}{2}$ stones distributed in $k$ piles. In each move, we take one stone from each pile and form a new pile with these stones. After a sufficient number of moves, the configuration stabilizes with $n$ piles. Determine the sum of the number of stones in the piles that contain an odd number of stones ... | [
"\\left\\lceil \\dfrac{n}{2} \\right\\rceil^2"
] | [
"We have $\\dfrac{n(n+1)}{2}$ stones distributed in $k$ piles. In each move, we take one stone from each pile and form a new pile with these stones. After a sufficient number of moves, the configuration stabilizes with $n$ piles. Determine the sum of the number of stones in the piles that contain an odd number of s... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ade07c8834a5e746ec5c0098b5b1b0d196ceaf6e255b6cc7dd8a94e5b1c305f9 | We have $\frac{n(n+1)}{2}$ stones distributed in $k$ piles. In each move, we take one stone from each pile and form a new pile with these stones (if a pile has only one stone, it vanishes after the stone is removed). After performing the moves until the configuration stabilizes, calculate the sum of the squares of the ... | [
"385"
] | [
"We have $\\frac{n(n+1)}{2}$ stones distributed in $k$ piles. In each move, we take one stone from each pile and form a new pile with these stones (if a pile has only one stone, it vanishes after the stone is removed). After performing the moves until the configuration stabilizes, calculate the sum of the squares o... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-48a06754a361f4bf442fc3871035582ad667a5748b8d8b2d14b73cb71560dded | We have $\frac{n(n+1)}{2}$ stones distributed in $k$ piles. In each move, we take one stone from each pile and form a new pile with these stones (if a pile has only one stone, it vanishes after the stone is removed). After performing the moves until the configuration stabilizes, let $S$ be the sum of the number of ston... | [
"55"
] | [
"We have $\\frac{n(n+1)}{2}$ stones distributed in $k$ piles. In each move, we take one stone from each pile and form a new pile with these stones (if a pile has only one stone, it vanishes after the stone is removed). After performing the moves until the configuration stabilizes, let $S$ be the sum of the number o... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4f4bea288f81e70074b969f88ff9832a4e85566c69f6a3c592d11b7d0b65c651 | We have $\frac{n(n+1)}{2}$ stones distributed in $k$ piles. In each move, we take one stone from each pile and form a new pile with these stones. (If a pile has $0$ stones, it vanishes.) After a sufficient number of moves, the configuration stabilizes to $n$ piles with a specific number of stones in each pile. Let $S(n... | [
"55"
] | [
"We have $\\frac{n(n+1)}{2}$ stones distributed in $k$ piles. In each move, we take one stone from each pile and form a new pile with these stones. (If a pile has $0$ stones, it vanishes.) After a sufficient number of moves, the configuration stabilizes to $n$ piles with a specific number of stones in each pile. Le... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-471d38e4bb854c2d98162c24f7a19582d7469d6cca10d59a7261f36e41202180 | We have $a \geq 2$ boxes. Initially, each box contains some number of balls. For each positive integer $n$, after $n$ seconds, we add a ball to every box that contains a number of balls that is a multiple of $n$. Determine the maximum number of seconds $T$ for which it is possible to distribute the balls initially so t... | [
"7"
] | [
"We have $a \\geq 2$ boxes. Initially, each box contains some number of balls. For each positive integer $n$, after $n$ seconds, we add a ball to every box that contains a number of balls that is a multiple of $n$. Determine the maximum number of seconds $T$ for which it is possible to distribute the balls initiall... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-2626ea25d6d673e9c672cf6cc12e6a2f7ee09b65fd58c56643f5b977ed4204c5 | We have $n$ bags each containing 100 coins. All bags have 10 gram coins except one which has 9 gram coins. Using a balance that can weigh up to 1 kilogram, what is the minimum number of weighings required to identify the bag with the 9 gram coins? | [
"\\(1\\) for \\(1 \\leq n \\leq 14\\), \\(2\\) for \\(15 \\leq n \\leq 60\\), \\(3\\) for \\(61 \\leq n \\leq 140\\), \\(k + 3\\) for \\(100k + 40 \\leq n \\leq 100k + 140\\)"
] | [
"We have $n$ bags each containing 100 coins. All bags have 10 gram coins except one which has 9 gram coins. Using a balance that can weigh up to 1 kilogram, what is the minimum number of weighings required to identify the bag with the 9 gram coins?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cc1c8dc703ece24037ecf6111711f5d281545010edb293908a4bda2cb0fb23eb | We have $n$ blue squares and $n$ red squares in the plane. Each blue square has a common point with all the red squares, and vice versa. Is it always true that there is a common point in $n$ squares? If not, what is the configuration of the squares? | [
"\\text{No}"
] | [
"We have $n$ blue squares and $n$ red squares in the plane. Each blue square has a common point with all the red squares, and vice versa. Is it always true that there is a common point in $n$ squares? If not, what is the configuration of the squares?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-833e7521deb54620abde529ff9b0fb2b532edf9e0f82a078dd6685a099a54c8f | We have $n$ boxes in a row, each initially containing numbers from $1$ to $n$ in order. In each step, the number in box $1$ is replaced by the number from box $2$, the number in box $n$ is replaced by the number from box $n-1$, and for $2 \leq i \leq n-1$, the number in box $i$ is replaced by the sum of the numbers fro... | [
"\\( n \\) is any odd positive integer."
] | [
"We have $n$ boxes in a row, each initially containing numbers from $1$ to $n$ in order. In each step, the number in box $1$ is replaced by the number from box $2$, the number in box $n$ is replaced by the number from box $n-1$, and for $2 \\leq i \\leq n-1$, the number in box $i$ is replaced by the sum of the numb... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 39,
"genselect": 5,
"tir": 17
} | 61 | 0 | 0 |
omr-ws-22220042f4f157d95ded9f0b480831831b1223d4223165a1af31369fcc06a1a8 | We have $n$ guinea pigs placed on the vertices of a regular polygon with $n$ sides inscribed in a circumference, one guinea pig in each vertex. Each guinea pig has a direction assigned, either "clockwise" or "anti-clockwise," and a distinct velocity between $1 \text{ km/h}$ and $n \text{ km/h}$. They start moving along... | [
"\\frac{n(n-1)}{2}"
] | [
"We have $n$ guinea pigs placed on the vertices of a regular polygon with $n$ sides inscribed in a circumference, one guinea pig in each vertex. Each guinea pig has a direction assigned, either \"clockwise\" or \"anti-clockwise,\" and a distinct velocity between $1 \\text{ km/h}$ and $n \\text{ km/h}$. They start m... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d4b4fcf1de40833eec1e71ea6aa956debaa976fb22d2e7b392b80ce9b15d636f | We have $n$ kilograms of watermelons, where each watermelon weighs an integer amount between 1 and 7 kilograms. There are 11 farmers, and each farmer can lift up to 30 kilograms. In all possible situations, the 11 farmers can lift the total $n$ kilograms of watermelons. What is the maximum value of $n$? | [
"310"
] | [
"We have $n$ kilograms of watermelons, where each watermelon weighs an integer amount between 1 and 7 kilograms. There are 11 farmers, and each farmer can lift up to 30 kilograms. In all possible situations, the 11 farmers can lift the total $n$ kilograms of watermelons. What is the maximum value of $n$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-ed5dd3689af4c2f71d137756e4721786513fc74e877724377d47d64e1223bee9 | We have $n$ numbers, each consisting of four digits where each digit is either 0 or 1. For each pair of numbers, there must be at least two positions where the corresponding digits are different. Evaluate the maximum value of $n$. | [
"8"
] | [
"We have $n$ numbers, each consisting of four digits where each digit is either 0 or 1. For each pair of numbers, there must be at least two positions where the corresponding digits are different. Evaluate the maximum value of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a8d32060ff6a30509ab038761b572cbe3a9c9d6d13a59f5bb33e67f913a400ca | We have $n$ people with different heights. Form $n!$ queues. Compute the expected number of people who can see something other than the back of the head of the person in front of them, where $n \in \{4,\ldots,30\}$. | [
"H_n"
] | [
"We have $n$ people with different heights. Form $n!$ queues. Compute the expected number of people who can see something other than the back of the head of the person in front of them, where $n \\in \\{4,\\ldots,30\\}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 3
} | 14 | 0 | 0 |
omr-ws-f389a7afc8cdd648808493c482af8cc84e6fcfe307d68fc9aec8d271af0dc65e | We have $n(n+1)/2$ stones distributed in $k$ piles. In each move, we take one stone from each pile and form a new pile with these stones (if a pile has only one stone, it vanishes after the stone is removed). After a sufficient number of moves, the configuration stabilizes with $n$ piles. Find the sum of the number of ... | [
"55"
] | [
"We have $n(n+1)/2$ stones distributed in $k$ piles. In each move, we take one stone from each pile and form a new pile with these stones (if a pile has only one stone, it vanishes after the stone is removed). After a sufficient number of moves, the configuration stabilizes with $n$ piles. Find the sum of the numbe... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8bc804e6239295dc6128d27a19a52797a94edd5ba37ee5068f5453c643293775 | We have $n^m$ sheets of paper, each with the number $1$ written on it. In each step, we choose two distinct sheets and replace the numbers $x_1, x_2, \ldots, x_n$ on $n$ sheets with the number $x_1 + x_2 + \ldots + x_n$ on those $n$ sheets. After $m \cdot n^{m-1}$ steps, calculate the minimum possible sum of the number... | [
"n^{2m}"
] | [
"We have $n^m$ sheets of paper, each with the number $1$ written on it. In each step, we choose two distinct sheets and replace the numbers $x_1, x_2, \\ldots, x_n$ on $n$ sheets with the number $x_1 + x_2 + \\ldots + x_n$ on those $n$ sheets. After $m \\cdot n^{m-1}$ steps, calculate the minimum possible sum of th... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-f732533e441c3950ecb470a54c7603c4e2533b3578c121ce8598188dfb28deff | We have 1,000,000 rectangles with sides parallel to the coordinate axes such that any 100 rectangles do not have a common point. Find the maximum number of disjoint rectangles that can always be selected from these 1,000,000 rectangles. | [
"10000"
] | [
"We have 1,000,000 rectangles with sides parallel to the coordinate axes such that any 100 rectangles do not have a common point. Find the maximum number of disjoint rectangles that can always be selected from these 1,000,000 rectangles."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-a9e2c50482e30aae72abf276edf01795ac9865ae6a6becff0ad0324e5aaa47ee | We have 10 identical tiles as shown. The tiles can be rotated, but not flipped over. A $7 \times 7$ board should be covered with these tiles so that exactly one unit square is covered by two tiles and all other fields by one tile. Designate all unit squares that can be covered with two tiles. | [
"(4,4)"
] | [
"We have 10 identical tiles as shown. The tiles can be rotated, but not flipped over. A $7 \\times 7$ board should be covered with these tiles so that exactly one unit square is covered by two tiles and all other fields by one tile. Designate all unit squares that can be covered with two tiles."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-25cbafbf60b1e0cab460938fdf40c49d62f01c64d006fab445385a1124e3805c | We have 10 pegs with diameters of 1, 2, 3, ..., 10 centimeters and 10 holes with diameters of 1, 2, 3, ..., 10 centimeters in a random order. A turn consists of taking the smallest remaining peg and placing it in a random hole that has a diameter at least as large as the peg's diameter. Once a peg is placed in a hole, ... | [
"\\dfrac{7381}{2520}"
] | [
"We have 10 pegs with diameters of 1, 2, 3, ..., 10 centimeters and 10 holes with diameters of 1, 2, 3, ..., 10 centimeters in a random order. A turn consists of taking the smallest remaining peg and placing it in a random hole that has a diameter at least as large as the peg's diameter. Once a peg is placed in a h... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-0c431220be07eecb644d2647bdd49fa006795dc649b150f51692b47e0b74891c | We have 10 points in the plane, with all distances between them distinct. For each point, we mark red the point nearest to it. What is the smallest number of points that can be marked red? | [
"3"
] | [
"We have 10 points in the plane, with all distances between them distinct. For each point, we mark red the point nearest to it. What is the smallest number of points that can be marked red?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-47fc5323ca643529c77201a1b3695796cdc4e6fcf095945af87f80c5cb34fe99 | We have 100 marbles, with 20 White, 20 Red, 30 Black, and 30 Green. Suppose we choose 60 marbles at random with replacement. Simulate the random numbers of the number of marbles for each color. | [
"17"
] | [
"We have 100 marbles, with 20 White, 20 Red, 30 Black, and 30 Green. Suppose we choose 60 marbles at random with replacement. Simulate the random numbers of the number of marbles for each color."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-20cac587f2baa4de4f3b59eb5c742c31f4da5bb43166e5c20c51adb083c6d7fa | We have 100 numbers on a circle such that each number is equal to the sum of its two neighboring numbers minus 1. If \(a_i, a_{i+1}, a_{i+2}\) are consecutive numbers on the circle, then \(a_{i+1} + 1 = a_i + a_{i+2}\). Find the numbers. | [
"1"
] | [
"We have 100 numbers on a circle such that each number is equal to the sum of its two neighboring numbers minus 1. If \\(a_i, a_{i+1}, a_{i+2}\\) are consecutive numbers on the circle, then \\(a_{i+1} + 1 = a_i + a_{i+2}\\). Find the numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-012ecc22aa4b0e6a611726e608a8cadacb1aed27994539847bdec03f83701413 | We have 1000 1s written in a line. We randomly choose a strictly increasing sequence from \(\{1, 2, \ldots, 1000\}\) such that the last term is 1000. If the chosen sequence is \(a_1, a_2, \ldots, a_k\) (where \(k\) is not fixed), then at the \(n\)th step, we choose the first \(a_i\) numbers on the line and change the 1... | [
"501"
] | [
"We have 1000 1s written in a line. We randomly choose a strictly increasing sequence from \\(\\{1, 2, \\ldots, 1000\\}\\) such that the last term is 1000. If the chosen sequence is \\(a_1, a_2, \\ldots, a_k\\) (where \\(k\\) is not fixed), then at the \\(n\\)th step, we choose the first \\(a_i\\) numbers on the li... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2250e9add31f5f7abc09df436c969a1672fc5e2f3e382be67dbff8dccc430829 | We have 111 coins to be placed in the unit cells of an $n \times n$ table such that the number of coins in any two neighboring cells (sharing a common side) differs exactly by 1. What is the maximal value of $n$ for which this is possible? | [
"13"
] | [
"We have 111 coins to be placed in the unit cells of an $n \\times n$ table such that the number of coins in any two neighboring cells (sharing a common side) differs exactly by 1. What is the maximal value of $n$ for which this is possible?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-3237912b22a31ec4055f15075843beddba0534c445ac83ac0a2b3434e8692318 | We have 12 balls, with 11 of them being identical and one being different in weight. Using a balance scale that allows us to weigh two groups of balls at a time, how can we identify the different ball with only 3 weighings? | [
"The different ball can be identified in 3 weighings using the method above."
] | [
"We have 12 balls, with 11 of them being identical and one being different in weight. Using a balance scale that allows us to weigh two groups of balls at a time, how can we identify the different ball with only 3 weighings?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4ccf0cbdc415b4c2d9e70b2878246a9eaad177d467805814b16c3da616f94f11 | We have 14 apples, 3 of which are known to be 'special'. We also have a special machine that takes 3 apples and tells us whether they are all special or not (it will only inform us if all 3 given apples are special). Determine the minimum number of times we must use this machine to guarantee finding at least one of the... | [
"11"
] | [
"We have 14 apples, 3 of which are known to be 'special'. We also have a special machine that takes 3 apples and tells us whether they are all special or not (it will only inform us if all 3 given apples are special). Determine the minimum number of times we must use this machine to guarantee finding at least one o... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-5c363f5140c10bdd46cd0d34584fe6c2443438127e2cee4b4fc1e44eded8abc5 | We have 2 cables and a router that has 4 ports. How many different arrangements of these 2 cables are possible? | [
"12"
] | [
"We have 2 cables and a router that has 4 ports. How many different arrangements of these 2 cables are possible?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 6
} | 21 | 0 | 0 |
omr-ws-e238cf814eb64e9088a3c299d9eb3f245e8b290baa9d0186a7d2abf26c7ed74f | We have 2013 distinct points on a straight line, each colored either blue or black. For each point \( A \), we count the sum of the number of blue points to the left of \( A \) and the number of black points to the right of \( A \). Among these 2013 sums, exactly one number occurs an odd number of times. Find all possi... | [
"1006"
] | [
"We have 2013 distinct points on a straight line, each colored either blue or black. For each point \\( A \\), we count the sum of the number of blue points to the left of \\( A \\) and the number of black points to the right of \\( A \\). Among these 2013 sums, exactly one number occurs an odd number of times. Fin... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 1
} | 12 | 0 | 0 |
omr-ws-664416096ce0c55c56241b70e20204fbf1d3b0919c9a781d7aed4f11a9fec428 | We have 2019 boxes, initially all empty. In one operation, we can add exactly 100 stones to some 100 boxes and exactly one stone to each of several other (perhaps none) boxes. What is the smallest possible number of moves after which all boxes will have the same (positive) number of stones? | [
"2019"
] | [
"We have 2019 boxes, initially all empty. In one operation, we can add exactly 100 stones to some 100 boxes and exactly one stone to each of several other (perhaps none) boxes. What is the smallest possible number of moves after which all boxes will have the same (positive) number of stones?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-240935d577ac010d318b5e8c86ef83c5e16f10ca296137685da81993d56f5d85 | We have 2022 ones written on a board in a line. We randomly choose a strictly increasing sequence from \(\{1, 2, \ldots, 2022\}\) such that the last term is 2022. If the chosen sequence is \(a_1, a_2, \ldots, a_k\) (where \(k\) is not fixed), then at the \(i\)-th step, we choose the first \(a_i\) numbers on the line an... | [
"2023"
] | [
"We have 2022 ones written on a board in a line. We randomly choose a strictly increasing sequence from \\(\\{1, 2, \\ldots, 2022\\}\\) such that the last term is 2022. If the chosen sequence is \\(a_1, a_2, \\ldots, a_k\\) (where \\(k\\) is not fixed), then at the \\(i\\)-th step, we choose the first \\(a_i\\) num... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-87e381718d912113d4a36a1cca2c5b845ad2fd4382fa4b5e32c639ed9956013a | We have 27 shells. We divide them into heaps such that no heap contains a number of shells that is a multiple of 3. What is the maximum possible product of the number of shells in each heap? | [
"10240"
] | [
"We have 27 shells. We divide them into heaps such that no heap contains a number of shells that is a multiple of 3. What is the maximum possible product of the number of shells in each heap?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-53dab281e5a7a32dcb50ff94fe1cf8f39e1776ed544aa8aac0c306f0d1875c0b | We have 27 unit cubes, each with 4 faces colored orange and the remaining 2 faces sharing a common side. When these cubes are assembled into a larger cube with side length 3, what is the probability that all the faces of the larger cube are orange? | [
"\\dfrac{5^{12}}{2^{34} \\cdot 3^{18}}"
] | [
"We have 27 unit cubes, each with 4 faces colored orange and the remaining 2 faces sharing a common side. When these cubes are assembled into a larger cube with side length 3, what is the probability that all the faces of the larger cube are orange?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 6
} | 14 | 0 | 0 |
omr-ws-222f15481cc6839774a2e1d109e6c78790efe914fb4b2ca282657aa45609c963 | We have 27 weights with masses 1g, 2g, ..., 27g. Using these weights, form groups such that in each group, the mass of the heaviest weight is equal to the sum of the masses of the other weights in the group. Find the number of such groups. | [
"1233"
] | [
"We have 27 weights with masses 1g, 2g, ..., 27g. Using these weights, form groups such that in each group, the mass of the heaviest weight is equal to the sum of the masses of the other weights in the group. Find the number of such groups."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-3e262240237f2725525d0f06a9db294c51edabf5d53f79c11437ffeb1a74b3bc | We have 3 boxes that can each carry a maximum weight of 14kg. We have 15 objects with the following weights: two 1kg objects, five 2kg objects, five 3kg objects, and three 4kg objects. How many different combinations of these objects can be placed into the boxes such that the total weight in each box does not exceed 14... | [
"10"
] | [
"We have 3 boxes that can each carry a maximum weight of 14kg. We have 15 objects with the following weights: two 1kg objects, five 2kg objects, five 3kg objects, and three 4kg objects. How many different combinations of these objects can be placed into the boxes such that the total weight in each box does not exce... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-f639054b0b91bad4f4f94076cc010ce4d2a86a1b165482d52c14f9d62ae0faf0 | We have 36 cards (3 of each card from 3 to A). We divide these cards into 4 distinguishable pockets, each containing 9 cards. Pocket nr 1 has no Ace card. Find the probability that one pocket has exactly 2 Ace cards. The 3 cards having the same number are indistinguishable. | [
"\\dfrac{216}{325}"
] | [
"We have 36 cards (3 of each card from 3 to A). We divide these cards into 4 distinguishable pockets, each containing 9 cards. Pocket nr 1 has no Ace card. Find the probability that one pocket has exactly 2 Ace cards. The 3 cards having the same number are indistinguishable."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-96f74073105ba747ee2b6c7317917bfbe1c65b650c1e455c501a7c1c2bedbd9e | We have 36 cards (3 of each card from 3 to A). We divide these cards into 4 distinguishable pockets, each containing 9 cards. Pocket nr 1 has no Ace card. Find the probability that one pocket has exactly 2 Ace cards. The cards are distinguishable. | [
"\\dfrac{216}{325}"
] | [
"We have 36 cards (3 of each card from 3 to A). We divide these cards into 4 distinguishable pockets, each containing 9 cards. Pocket nr 1 has no Ace card. Find the probability that one pocket has exactly 2 Ace cards. The cards are distinguishable."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-ed414f63ed87787a775f7f40cd21afa93083685e24c182d5e7505566e32fcc26 | We have 4 cubes with faces painted in 4 colors (red, white, blue, green) as follows: - Cube 1: Red, White, White, Red, Blue, Green - Cube 2: White, White, Blue, Red, Blue, Green - Cube 3: Red, White, Green, Blue, Red, Blue - Cube 4: Green, Blue, White, Red, Green, Blue Can we arrange these cubes such that each color is... | [
"Yes"
] | [
"We have 4 cubes with faces painted in 4 colors (red, white, blue, green) as follows:\n- Cube 1: Red, White, White, Red, Blue, Green\n- Cube 2: White, White, Blue, Red, Blue, Green\n- Cube 3: Red, White, Green, Blue, Red, Blue\n- Cube 4: Green, Blue, White, Red, Green, Blue\n\nCan we arrange these cubes such that e... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 1
} | 14 | 0 | 0 |
omr-ws-244db534119372a1ae1592d0bdc573ec1c63d6e2051401419453ba14acac85f3 | We have 4 equal right triangles. In each turn, we can draw the altitude of a right triangle and separate it into 2 new right triangles. After 5 turns, let $N$ be the number of pairs of equal right triangles. Calculate the minimum possible value of $N$. | [
"1"
] | [
"We have 4 equal right triangles. In each turn, we can draw the altitude of a right triangle and separate it into 2 new right triangles. After 5 turns, let $N$ be the number of pairs of equal right triangles. Calculate the minimum possible value of $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-5ec516eff3db3c8c742e04054e37ce31e2ceb1fa51e2fd54380deed2fa0823d6 | We have 4 pencils, each of a different colour (R, Y, G, B), and 4 erasers, each of a different colour (R, Y, G, B). Let \( N \) be the number of ways in which 4 students can be given 1 pencil and 1 eraser each, so that no student gets both items of the same colour. What is the sum of the digits of \( N \)? | [
"9"
] | [
"We have 4 pencils, each of a different colour (R, Y, G, B), and 4 erasers, each of a different colour (R, Y, G, B). Let \\( N \\) be the number of ways in which 4 students can be given 1 pencil and 1 eraser each, so that no student gets both items of the same colour. What is the sum of the digits of \\( N \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e1ca4f65c4a74dc6590296b12484073a7624cc5350fa6efc4f197e6e9c9e83ec | We have 48 fair coins, each showing either heads or tails. We flip all of them at once. We then pick up each coin that shows tails and flip it again, repeating this process until all coins show heads. What is the expected value of the total number of coin flips required? | [
"96"
] | [
"We have 48 fair coins, each showing either heads or tails. We flip all of them at once. We then pick up each coin that shows tails and flip it again, repeating this process until all coins show heads. What is the expected value of the total number of coin flips required?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4db26cb77233032eb58ed1707068c18cb89bec72e24cada1b56a7a608168e709 | We have 5 balls such that any 4 of them can be divided into 2 groups (each containing 2 balls) with equal weight. Let the weights of the balls be represented by $a, b, c, d,$ and $e$. Calculate the value of $\frac{a+b+c+d+e}{5}$. | [
"a"
] | [
"We have 5 balls such that any 4 of them can be divided into 2 groups (each containing 2 balls) with equal weight. Let the weights of the balls be represented by $a, b, c, d,$ and $e$. Calculate the value of $\\frac{a+b+c+d+e}{5}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-76a7e55ed3038e829068a8f20252fcf6174630f844994b8ac4a8bfeb56336746 | We have 50 integers. 27 of them are divisible by 3, 18 by 5, 16 by 6, 11 by 10, 7 by 15, and 4 by 30. Additionally, 4 of the integers are not divisible by 2, 3, or 5. How many of the integers are even? | [
"31"
] | [
"We have 50 integers. 27 of them are divisible by 3, 18 by 5, 16 by 6, 11 by 10, 7 by 15, and 4 by 30. Additionally, 4 of the integers are not divisible by 2, 3, or 5. How many of the integers are even?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2a0b3fb3b5aacc30e6733aeb336b35db4e482941457be4737a46f349c7652375 | We have 5000 - n lamps, out of which 100 - n are broken. If we randomly choose 50 + n lamps, what is the probability that more than n of the chosen lamps are broken? | [
"\\(\\frac{\\sum_{i=1}^{50} \\left( \\binom{100 - n}{n + i} \\cdot \\binom{4900}{50 - i} \\right)}{\\binom{5000 - n}{50 + n}}\\)"
] | [
"We have 5000 - n lamps, out of which 100 - n are broken. If we randomly choose 50 + n lamps, what is the probability that more than n of the chosen lamps are broken?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 35,
"genselect": 8,
"tir": 11
} | 54 | 0 | 0 |
omr-ws-2155a614e7519b1745bd4c90b88f26d4f83e323b899cbc846dcc291fbfc38159 | We have 504 points in space where no three are collinear. We have 6 colors. If any of the possible triangles are chosen, then two of its edges have the same color. Find all possible numbers of edges with the same color. | [
"126756"
] | [
"We have 504 points in space where no three are collinear. We have 6 colors. If any of the possible triangles are chosen, then two of its edges have the same color. Find all possible numbers of edges with the same color."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-866dd1493279b7772dd50eb942b8d6438bec2b7092584066f159146e53d3b6d8 | We have 6 bookshelves. There are 4 math books, 5 science books, and 7 chess books that we need to arrange on the shelves. Each bookshelf has 3 shelves. Each shelf can only contain 1 of any type of book. Each shelf can hold 12 books (it can be any number >7). A shelf can be empty. Find the number of ways this is possibl... | [
" \\binom{21}{3} \\cdot \\binom{22}{4} \\cdot \\binom{24}{6} "
] | [
"We have 6 bookshelves. There are 4 math books, 5 science books, and 7 chess books that we need to arrange on the shelves. Each bookshelf has 3 shelves. Each shelf can only contain 1 of any type of book. Each shelf can hold 12 books (it can be any number >7). A shelf can be empty. Find the number of ways this is po... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ce0ba503f69531c738d12a605bd1361f2132b38cf1220fa245d129ca11db5bde | We have 6 distinct colors and 100 balls for each color, totaling 600 balls. These balls are to be distributed into \( n \) boxes such that any two balls of the same color are in different boxes, and for any two boxes, there is at least one color such that neither box contains a ball of that color. What is the minimum v... | [
"200"
] | [
"We have 6 distinct colors and 100 balls for each color, totaling 600 balls. These balls are to be distributed into \\( n \\) boxes such that any two balls of the same color are in different boxes, and for any two boxes, there is at least one color such that neither box contains a ball of that color. What is the mi... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-17434c1b84f2912b500e6b884d68a57c1602bbccc965e79dd273d6cc52f77b7a | We have 7 balls labeled 1 through 7 and 4 identical boxes. Each box can contain any number of balls from 0 to 7. If we randomly select one box, what is the probability that the box contains all 7 balls? | [
"\\(\\frac{1}{4^6}\\)"
] | [
"We have 7 balls labeled 1 through 7 and 4 identical boxes. Each box can contain any number of balls from 0 to 7. If we randomly select one box, what is the probability that the box contains all 7 balls?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-00803f559b3fc01c2f6db590d6fee910f88239bf96cd29cdee747bb95589b7cb | We have 7 balls labeled 1 through 7 and 4 identical boxes. Each box can contain any number of balls from 0 to 7. If we randomly select one box, what is the probability that the box is empty? | [
"\\(\\frac{7}{10}\\)"
] | [
"We have 7 balls labeled 1 through 7 and 4 identical boxes. Each box can contain any number of balls from 0 to 7. If we randomly select one box, what is the probability that the box is empty?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ab75e7cce16aeeb830b31fa7c6204107cc67e36a4e4153804f27c1b894733e78 | We have 7 different cases and 7 different keys, with each key opening exactly one case. Each key is placed in one of the cases, and all cases are locked. We are allowed to break only one case to take out the key inside and then use that key to open the remaining cases one after the other. How many different ways can we... | [
"720"
] | [
"We have 7 different cases and 7 different keys, with each key opening exactly one case. Each key is placed in one of the cases, and all cases are locked. We are allowed to break only one case to take out the key inside and then use that key to open the remaining cases one after the other. How many different ways c... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-79d367a7bd56cae2db9a869b5c9ed19ab32eb0268b9576d414d342923b7f58ca | We have 7 pairs of keys and locks \((K_1, L_1), (K_2, L_2), \ldots, (K_7, L_7)\). Each key can only open the lock in the same pair. If we randomly re-arrange the pairs of keys and locks, what is the probability that none of \( K_1, K_2, K_3 \) can open the lock in the new pair? | [
"\\(\\frac{67}{105}\\)"
] | [
"We have 7 pairs of keys and locks \\((K_1, L_1), (K_2, L_2), \\ldots, (K_7, L_7)\\). Each key can only open the lock in the same pair. If we randomly re-arrange the pairs of keys and locks, what is the probability that none of \\( K_1, K_2, K_3 \\) can open the lock in the new pair?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-858985634248b9968c23a3cac33a8be525dbbf2af20376a392356baf8a347f81 | We have N electric plants, each with power P [MW] and the probability of failure of 0.1. Each plant is available for 5000 [h]. The goal is to provide at least 150 [MW] of power for 95% of the time, with the constraint that the total installed power must be less than 205 [MW] (i.e., \( N \cdot P < 205 \)). Determine the... | [
"14.64"
] | [
"We have N electric plants, each with power P [MW] and the probability of failure of 0.1. Each plant is available for 5000 [h]. The goal is to provide at least 150 [MW] of power for 95% of the time, with the constraint that the total installed power must be less than 205 [MW] (i.e., \\( N \\cdot P < 205 \\)). Deter... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-0e08f441f7abcb61dfc8f01f4937705d852e43916c216462d99e6601e81a39b2 | We have N trees in a row with one bird on each tree. Every minute, two birds move, one moving k trees to the left and the other moving k trees to the right (where k is any positive integer and can be different between minutes), given that there is a tree for it to move to. Find all N such that it's possible to get all ... | [
"\\( N \\) is odd."
] | [
"We have N trees in a row with one bird on each tree. Every minute, two birds move, one moving k trees to the left and the other moving k trees to the right (where k is any positive integer and can be different between minutes), given that there is a tree for it to move to. Find all N such that it's possible to get... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-77cf57161b600189c72e5473a14d334a32decabe3ebc9b327eb81597e3877bd2 | We have \( n \) balls, numbered 1 to \( n \), and \( n-1 \) bags, numbered 1 to \( n-1 \). We want to place the balls in the bags such that: - The first bag contains at least 2 balls, - The first 2 bags contain at least 3 balls, - ... - The first \( k \) bags contain at least \( k+1 \) balls, - ... In how many ways can... | [
"\\((n-1)^{n-1}\\)"
] | [
"We have \\( n \\) balls, numbered 1 to \\( n \\), and \\( n-1 \\) bags, numbered 1 to \\( n-1 \\). We want to place the balls in the bags such that:\n- The first bag contains at least 2 balls,\n- The first 2 bags contain at least 3 balls,\n- ...\n- The first \\( k \\) bags contain at least \\( k+1 \\) balls,\n- ..... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 35,
"genselect": 7,
"tir": 9
} | 51 | 0 | 0 |
omr-ws-9d27f0b6536dd3a895b3304a5f12ff0f9262b7390033bacc3aa7a0cee3120b1f | We have \( n \) balls. Suppose that balls are tossed into \( b \) bins. Each toss is independent, and each ball is equally likely to end up in any bin. What is the expected number of ball tosses before at least one of the bins contains two balls? | [
"\\sum_{k=0}^{b} \\frac{b!}{(b - k)! \\, b^k}"
] | [
"We have \\( n \\) balls. Suppose that balls are tossed into \\( b \\) bins. Each toss is independent, and each ball is equally likely to end up in any bin. What is the expected number of ball tosses before at least one of the bins contains two balls?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bdb53fb0d360ad213786d7d3eeadd80c6f64ec3be12896c7143e01b78e786443 | We have \( n \) boxes with initial counts of marbles \( x_1, x_2, x_3, \ldots, x_n \). In each operation, we randomly select one box, take one marble from it, and place it into another randomly chosen box (different from the origin). The process continues until one of the boxes becomes empty. On average, how many opera... | [
"\\dfrac{\\sum_{1 \\leq i < j \\leq n} x_i x_j}{n - 1}"
] | [
"We have \\( n \\) boxes with initial counts of marbles \\( x_1, x_2, x_3, \\ldots, x_n \\). In each operation, we randomly select one box, take one marble from it, and place it into another randomly chosen box (different from the origin). The process continues until one of the boxes becomes empty. On average, how ... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-1c314fa0b67434932ad51c1d4e3e2e401d9d652c8288ff9413b0d3dcac701f9d | We have \( n \) numbers, each consisting of six digits where the possible digits are \( 0, 1, 2, \) and \( 3 \). For each pair of numbers, there must be at least two positions where the corresponding digits are different. Evaluate the maximum value of \( n \). | [
"64"
] | [
"We have \\( n \\) numbers, each consisting of six digits where the possible digits are \\( 0, 1, 2, \\) and \\( 3 \\). For each pair of numbers, there must be at least two positions where the corresponding digits are different. Evaluate the maximum value of \\( n \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d5a59f0449d80f9f3f642eca7b6871dda9ff538bee3143d35537824c7acc7232 | We have a $2011 \times 2011$ chessboard. A unit square at a corner of the board is removed. Find the least $n$ such that the mutilated chessboard can be partitioned into $n$ squares following the lines of the board. | [
"2011"
] | [
"We have a $2011 \\times 2011$ chessboard. A unit square at a corner of the board is removed. Find the least $n$ such that the mutilated chessboard can be partitioned into $n$ squares following the lines of the board."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-0f1cd203e042eb29c3b5ef1fcd246656f07411ec2da531192e6793dabe0056a0 | We have a $2n \times 2n \times 2n$ cube divided into $8n^3$ unit cubes. Some of these cubes are initially infected. If a non-infected cube has 3 neighbors (sharing a side) that are infected, it becomes infected. What is the minimum number of initially infected cubes required to ensure that the entire cube becomes infec... | [
"\\(4n^2\\)"
] | [
"We have a $2n \\times 2n \\times 2n$ cube divided into $8n^3$ unit cubes. Some of these cubes are initially infected. If a non-infected cube has 3 neighbors (sharing a side) that are infected, it becomes infected. What is the minimum number of initially infected cubes required to ensure that the entire cube become... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 38,
"genselect": 3,
"tir": 4
} | 45 | 0 | 0 |
omr-ws-b34bd00bb1ebfaf9f60afe2e5c8fb4b008b4690faed560374ea1aff7080ee24e | We have a $3 \times 3$ table and the integers from $1$ to $9$. Using the same moves as before (select a row or column, reduce two neighboring numbers by $1$, and increase the third one by $1$), our aim is to create a table with all the same entries, maximizing the value of the entries. What is the highest possible numb... | [
"4"
] | [
"We have a $3 \\times 3$ table and the integers from $1$ to $9$. Using the same moves as before (select a row or column, reduce two neighboring numbers by $1$, and increase the third one by $1$), our aim is to create a table with all the same entries, maximizing the value of the entries. What is the highest possibl... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 4
} | 18 | 0 | 0 |
omr-ws-eb53bc9389262197d220ae318baf68927876c1473c2c9e19dcee376e9624b9e2 | We have a $4 \times 4$ magic square where the sum of neighboring squares (over an edge) is 1. Find the sum of all numbers in the square. \[ \begin{array}{|c|c|c|c|} \hline M & N & O & P\\ \hline I & J & K & L \\ \hline E & F & G & H\\ \hline A & B & C & D\\ \hline \end{array} \] For example: $E+B=1$, $A+F+C=1$, $G+J+L+... | [
"6"
] | [
"We have a $4 \\times 4$ magic square where the sum of neighboring squares (over an edge) is 1. Find the sum of all numbers in the square.\n\\[\n\\begin{array}{|c|c|c|c|}\n\\hline\nM & N & O & P\\\\\n\\hline\nI & J & K & L \\\\\n\\hline\nE & F & G & H\\\\\n\\hline\nA & B & C & D\\\\\n\\hline\n\\end{array}\n\\]\nFor... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 5
} | 17 | 0 | 0 |
omr-ws-2b2f5e990e0170bbe716548437fdb37e21dfc9ee1086d8b06100e480ed9cf12e | We have a $5 \times 5$ square grid where the center square is initially colored. In each step, we can color any neighboring square (sharing a common side) of a square that is already colored. Determine the number of ways, $k$, to color the entire grid. | [
"80313433200"
] | [
"We have a $5 \\times 5$ square grid where the center square is initially colored. In each step, we can color any neighboring square (sharing a common side) of a square that is already colored. Determine the number of ways, $k$, to color the entire grid."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-102fc43bb8e732f2a8c82a4f6f1bc572a9b707a8d4c688e0b4413b4753928e3d | We have a 5x5x5 cube, and we want to color all the 1x1 squares on each side using red, white, or black. Every two adjacent squares (sharing a side) must be different colors. What is the smallest number of black squares needed? | [
"8"
] | [
"We have a 5x5x5 cube, and we want to color all the 1x1 squares on each side using red, white, or black. Every two adjacent squares (sharing a side) must be different colors. What is the smallest number of black squares needed?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-f7098ed00c15c9fecbc57597af5cad313d05dccd64677a7e995306ac6c16d019 | We have a box with 99 candies labeled with the positive integers from 2 to 100. We randomly take one candy. If the number on the candy is prime, we eat it. If the number is composite, we put it back in the box and change its number to the smallest number less than the original number such that their greatest common div... | [
"173"
] | [
"We have a box with 99 candies labeled with the positive integers from 2 to 100. We randomly take one candy. If the number on the candy is prime, we eat it. If the number is composite, we put it back in the box and change its number to the smallest number less than the original number such that their greatest commo... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-3d791e44e216fabaaa0c7b3bb8cfac62d358e2469d738c1c075b4bd345f4ed2c | We have a circle with 1991 points numbered from 1 to 1991. Starting from point 1, we delete every 2nd and 3rd point (for example, the first 4 points to be deleted are 2, 3, 5, 6). We repeat this process until only one point remains. What is the label of the last remaining point? | [
"1894"
] | [
"We have a circle with 1991 points numbered from 1 to 1991. Starting from point 1, we delete every 2nd and 3rd point (for example, the first 4 points to be deleted are 2, 3, 5, 6). We repeat this process until only one point remains. What is the label of the last remaining point?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-ac8ffbda82824a52b5cd390a369929bce0d6bd344f6953216be7f281f592eb83 | We have a circle with radius $r$ and another circle with radius $R$, where $R > r$. Both circles are tangent to each other and to both sides of angle $A$. Find $\sin A$. | [
"\\(\\frac{(R - r) \\cdot 2 \\sqrt{Rr}}{(R + r)^2}\\)"
] | [
"We have a circle with radius $r$ and another circle with radius $R$, where $R > r$. Both circles are tangent to each other and to both sides of angle $A$. Find $\\sin A$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-053e8beabd89b21c4a1b02d518ec1d216ea3f2d455a05a9bf64619bfe1ec2566 | We have a cube with 4 blue faces and 2 red faces. What's the probability that when it is rolled, a blue face will be facing up? | [
"\\frac{2}{3}"
] | [
"We have a cube with 4 blue faces and 2 red faces. What's the probability that when it is rolled, a blue face will be facing up?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-01c43a158ede165249beec4d248a6498048e0d42ad09405f7ee90d627f8d5b1a | We have a deck of 52 playing cards, each with a value from $1$ to $13$. The deck is dealt to four players, with each player receiving 13 cards. How many ways are there to deal the deck such that there exists a round $i$ (where a round consists of dealing one card to each of the four players) where at least one player r... | [
"20366826385239437910994563581940510272786822469946975293800448"
] | [
"We have a deck of 52 playing cards, each with a value from $1$ to $13$. The deck is dealt to four players, with each player receiving 13 cards. How many ways are there to deal the deck such that there exists a round $i$ (where a round consists of dealing one card to each of the four players) where at least one pla... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-f7f31daf50382393429dd801a08ff38cfce9f89f36240040e54b4ea660f53aaa | We have a fair 6-sided die $D_0$ with sides labelled $1,2,3,4,5,6$. Die $D_1$ is constructed by rolling $D_0$ 6 times and using these 6 numbers as the 6 sides of $D_1$. In general, die $D_n$ is constructed by rolling $D_{n-1}$ 6 times and using those 6 values as the new 6 sides of $D_n$. Let $D_k$ be the first die that... | [
"9.656"
] | [
"We have a fair 6-sided die $D_0$ with sides labelled $1,2,3,4,5,6$. Die $D_1$ is constructed by rolling $D_0$ 6 times and using these 6 numbers as the 6 sides of $D_1$. In general, die $D_n$ is constructed by rolling $D_{n-1}$ 6 times and using those 6 values as the new 6 sides of $D_n$. Let $D_k$ be the first die... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-ca733414ffec39305b1f1a862599528f542b613cb60d7494f14d3f1b5d0802b2 | We have a four-sided die where each side is labeled with a random number from 1 to 4 (uniform distribution). We throw the die repeatedly and sum the numbers that appear on the top face after each throw. Let \( T \) be the first throw for which the cumulative sum is divisible by 4. Find the expected value of \( T \). | [
"4"
] | [
"We have a four-sided die where each side is labeled with a random number from 1 to 4 (uniform distribution). We throw the die repeatedly and sum the numbers that appear on the top face after each throw. Let \\( T \\) be the first throw for which the cumulative sum is divisible by 4. Find the expected value of \\( ... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-767d492cf48b6406deca950e06b7c6bb92c6c95b0f6324ba8065f8417755dec2 | We have a geometric sequence of 3 terms. If the sum of these terms is 26, and their sum of squares is 364, find the terms of the sequence. | [
"\\(\\frac{26}{7}, \\frac{52}{7}, \\frac{104}{7}\\) or \\(\\frac{104}{7}, \\frac{52}{7}, \\frac{26}{7}\\)"
] | [
"We have a geometric sequence of 3 terms. If the sum of these terms is 26, and their sum of squares is 364, find the terms of the sequence."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6792e028991c6eaa75d5a204348ddca6c4aa4a23c15e579ea0404eeb87f406cf | We have a group of $n$ anime girls. For each pair of anime girls, at least one has sent a message to the other one. For each anime girl $A$, among the anime girls to whom $A$ has sent a message, exactly $\tfrac 14$ of them have sent a message to $A.$ Find the number of possible two-digit values of $n.$ | [
"26"
] | [
"We have a group of $n$ anime girls. For each pair of anime girls, at least one has sent a message to the other one. For each anime girl $A$, among the anime girls to whom $A$ has sent a message, exactly $\\tfrac 14$ of them have sent a message to $A.$ Find the number of possible two-digit values of $n.$"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-d0df1a9a8dd321c37fb512ba703e0cc2a5b7730a3a497806993262c369104940 | We have a large supply of black, white, red, and green hats. We want to give 8 of these hats to 8 students sitting around a round table. Find the number of ways of doing that in each of these cases (assuming for the purposes of this problem that students will not change their places, and that hats of the same color are... | [
"6564"
] | [
"We have a large supply of black, white, red, and green hats. We want to give 8 of these hats to 8 students sitting around a round table. Find the number of ways of doing that in each of these cases (assuming for the purposes of this problem that students will not change their places, and that hats of the same colo... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-5b33e335e216af051894570aed35f26dd36c71c58a8af2c107d92021d76faeec | We have a number greater than $1000$. Perform the following operations on it: 1. Add $9$ to it. 2. Interchange the last two digits of the new number. 3. Interchange the first and third digits of the new number. 4. Double the new number. If the resultant number is equal to the original number, what is the original numbe... | [
"2052"
] | [
"We have a number greater than $1000$. Perform the following operations on it:\n1. Add $9$ to it.\n2. Interchange the last two digits of the new number.\n3. Interchange the first and third digits of the new number.\n4. Double the new number.\n\nIf the resultant number is equal to the original number, what is the or... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 24
} | 46 | 0 | 0 |
omr-ws-853e51e048988a995775d8c8c0e6a0d595220cd4042af63c2fe9d3c8b2a83683 | We have a rectangle of dimensions $x - 2$ by $2x + 5$ such that its area is $8x - 6$. What is the value of $x$? | [
"4"
] | [
"We have a rectangle of dimensions $x - 2$ by $2x + 5$ such that its area is $8x - 6$. What is the value of $x$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-35f3df931db669461701ea54bd1742f3941136825b5cd91d7f023ce06191984b | We have a regular octahedron with one side painted red and the other seven sides painted blue. We throw the octahedron like a die and repaint the top face according to the rule: if it is red, it becomes blue, and if it is blue, it becomes red. We repeat this process 10 times. How many different octahedra can we get aft... | [
"8"
] | [
"We have a regular octahedron with one side painted red and the other seven sides painted blue. We throw the octahedron like a die and repaint the top face according to the rule: if it is red, it becomes blue, and if it is blue, it becomes red. We repeat this process 10 times. How many different octahedra can we ge... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-082127c110cc51fa9bf5d46794be5af25d93b98b12336f247cefe11977f3c929 | We have a row of 203 cells. Initially, the leftmost cell contains 203 tokens, and the rest are empty. On each move, we can either take one token and move it to an adjacent cell (left or right) or take exactly 20 tokens from the same cell and move them all to an adjacent cell (all left or all right). After 2023 moves, e... | [
"13"
] | [
"We have a row of 203 cells. Initially, the leftmost cell contains 203 tokens, and the rest are empty. On each move, we can either take one token and move it to an adjacent cell (left or right) or take exactly 20 tokens from the same cell and move them all to an adjacent cell (all left or all right). After 2023 mov... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-0d08346b3296df446b177dc151722ec854e9b9c29ac3a7617492301e52220b7b | We have a set of $343$ closed jars, each containing blue, yellow, and red marbles with the number of marbles from each color being at least $1$ and at most $7$. No two jars have exactly the same contents. Initially, all jars are with the caps up. To flip a jar means to change its position from cap-up to cap-down or vic... | [
"998"
] | [
"We have a set of $343$ closed jars, each containing blue, yellow, and red marbles with the number of marbles from each color being at least $1$ and at most $7$. No two jars have exactly the same contents. Initially, all jars are with the caps up. To flip a jar means to change its position from cap-up to cap-down o... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 3
} | 10 | 0 | 0 |
omr-ws-8deb58588f497af4209a2b6790c178b80cd7eeac55a8f5620890d90acceaa1b4 | We have a set of two-digit numbers, all of which are distinct, and the number of such numbers is \( n \). The sum of all \( n \) numbers is equal to \( 33n \). What is the maximum value of \( n \)? | [
"47"
] | [
"We have a set of two-digit numbers, all of which are distinct, and the number of such numbers is \\( n \\). The sum of all \\( n \\) numbers is equal to \\( 33n \\). What is the maximum value of \\( n \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-329abb1e52bfdf59e9127302818821ab2a6e3947d0b0f8de4f3c4cad037e0c0b | We have a sheet of paper divided into a $100 \times 100$ grid of unit squares. In some squares, we place right-angled isosceles triangles with leg length 1, such that each triangle occupies half of a unit square and each unit grid segment (including the boundary) is under one leg of a triangle. Find the maximal number ... | [
"2450"
] | [
"We have a sheet of paper divided into a $100 \\times 100$ grid of unit squares. In some squares, we place right-angled isosceles triangles with leg length 1, such that each triangle occupies half of a unit square and each unit grid segment (including the boundary) is under one leg of a triangle. Find the maximal n... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a7a22b4c4f9573b9b2d4c373a3defbffc28de8c622ff26072498172ecd614694 | We have a stack of $n$ books labeled $1, 2, \ldots, n$. In each round, we make $n$ moves where in the $i$-th move, we turn over the top $i$ books as a single unit. Determine the smallest positive integer $k$ such that after $k$ rounds, the books will return to their initial arrangement for $n=10$. | [
"6"
] | [
"We have a stack of $n$ books labeled $1, 2, \\ldots, n$. In each round, we make $n$ moves where in the $i$-th move, we turn over the top $i$ books as a single unit. Determine the smallest positive integer $k$ such that after $k$ rounds, the books will return to their initial arrangement for $n=10$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
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