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-59d9419383a89d0f68edf14a05b08bdbbc2df6abda00e6980fe3929f5a4c1cde | Are there positive integers \(a, b, c, d, e\) with \((a, b) \neq (c, d)\) such that \[ a^2 + b^2 = c^2 + d^2 = e^2? \] If so, find the smallest sum of \(a + b + c + d + e\) for which this holds. | [
"91"
] | [
"Are there positive integers \\(a, b, c, d, e\\) with \\((a, b) \\neq (c, d)\\) such that\n\\[ a^2 + b^2 = c^2 + d^2 = e^2? \\]\nIf so, find the smallest sum of \\(a + b + c + d + e\\) for which this holds."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 6,
"tir": 6
} | 26 | 0 | 0 |
omr-ws-60343da9ac4c40874bb7c92524a02357fb493b8eb46759c44ba166492d81857e | Are there prime numbers \( p, q, r \) such that \( p^2 + q^3 = r^4 \)? | [
"No"
] | [
"Are there prime numbers \\( p, q, r \\) such that \\( p^2 + q^3 = r^4 \\)?"
] | [
"c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-ec141d4c74812b771f73fe8a7a18f46b2ae943d7beeeb21bba0515d9962a2100 | Are there specific methods to solve the integral $\displaystyle\int\frac{dx}{Ax^4+ Bx^2 + C}$ where $A, B, C$ are real constants and $A \neq 0$? | [
"\\text{Yes}"
] | [
"Are there specific methods to solve the integral $\\displaystyle\\int\\frac{dx}{Ax^4+ Bx^2 + C}$ where $A, B, C$ are real constants and $A \\neq 0$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6fd9ce19ffc0613fb657c0ebfcebda896e5a3baa026a51a55b0af10e80f721a5 | Are there values of \(k > \frac{2}{3}\) for which the inequality \[ a^2 + b^2 + c^2 + d^2 + 8 \cdot \left(\frac{abc + abd + acd + bcd}{4}\right)^k \geq 2(ab + bc + cd + da + ac + bd) \] holds for all \(a, b, c, d \geq 0\) such that \(a + b + c + d = 4\)? If so, find all such values of \(k\). | [
"\\left( \\frac{2}{3}, \\frac{\\ln \\frac{2}{3}}{\\ln \\frac{16}{27}} \\right]"
] | [
"Are there values of \\(k > \\frac{2}{3}\\) for which the inequality\n\\[ a^2 + b^2 + c^2 + d^2 + 8 \\cdot \\left(\\frac{abc + abd + acd + bcd}{4}\\right)^k \\geq 2(ab + bc + cd + da + ac + bd) \\]\nholds for all \\(a, b, c, d \\geq 0\\) such that \\(a + b + c + d = 4\\)? If so, find all such values of \\(k\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-7238c7b367c8204123f1fa06a4500b8ec6bcc968d3dd6b4c43b8db08659f7bca | Ares multiplies two integers which differ by $9$. Grace multiplies two integers which differ by $6$. They obtain the same product $T$. Determine all possible values of $T$. | [
"\\( T \\in \\{112, 0, -8\\} \\)"
] | [
"Ares multiplies two integers which differ by $9$. Grace multiplies two integers which differ by $6$. They obtain the same product $T$. Determine all possible values of $T$."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 25
} | 29 | 0 | 0 |
omr-ws-5305648e6df432f4efe3845a4b40040d9bce1ba6ca7756a67558fe9d1ed98e71 | Ari and Beri play a game using a deck of $2020$ cards with exactly one card with each number from $1$ to $2020$. Ari gets a card with a number $a$ and removes it from the deck. Beri sees the card, chooses another card from the deck with a number $b$ and removes it from the deck. Then Beri writes on the board exactly on... | [
"2020"
] | [
"Ari and Beri play a game using a deck of $2020$ cards with exactly one card with each number from $1$ to $2020$. Ari gets a card with a number $a$ and removes it from the deck. Beri sees the card, chooses another card from the deck with a number $b$ and removes it from the deck. Then Beri writes on the board exact... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-5fab8c0c09ce3503d1f180107b72b58d8fe2a0ea11a2f47f38419229955670fd | Ari and Beri play a game with a deck of 2020 cards, each labeled with a unique integer from 1 to 2020. Ari chooses a card with number \(a\) and removes it from the deck. Beri then chooses a card with number \(b\) and removes it from the deck. Beri then selects one of the trinomials \(x^2 - ax + b\) or \(x^2 - bx + a\).... | [
"4038"
] | [
"Ari and Beri play a game with a deck of 2020 cards, each labeled with a unique integer from 1 to 2020. Ari chooses a card with number \\(a\\) and removes it from the deck. Beri then chooses a card with number \\(b\\) and removes it from the deck. Beri then selects one of the trinomials \\(x^2 - ax + b\\) or \\(x^2... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-afac73ab82f0dda25dca14bdf2bfcabb6f59c6efb2cd894e017e6269d3b5d5b7 | Ari chooses 7 balls at random from \( n \) balls numbered 1 to \( n \). If the probability that no two of the drawn balls have consecutive numbers equals the probability of exactly one pair of consecutive numbers in the chosen balls, find \( n \). | [
"54"
] | [
"Ari chooses 7 balls at random from \\( n \\) balls numbered 1 to \\( n \\). If the probability that no two of the drawn balls have consecutive numbers equals the probability of exactly one pair of consecutive numbers in the chosen balls, find \\( n \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 2,
"tir": 2
} | 6 | 0 | 0 |
omr-ws-6a57792b751194a9235bf805aac6f9b5e652952148e046febb8162d82511fc9a | Ari the Archer is shooting at a target consisting of 100 concentric rings, each of width 1, so that the total radius of the target is 100. The point value of a given ring is equal to its area. Given that Ari hits any point on the target uniformly at random, what is his expected score? | [
"\\(\\frac{13433 \\pi}{100}\\)"
] | [
"Ari the Archer is shooting at a target consisting of 100 concentric rings, each of width 1, so that the total radius of the target is 100. The point value of a given ring is equal to its area. Given that Ari hits any point on the target uniformly at random, what is his expected score?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f05e7613df75b2aa4d60302879e1f999e82b3858ca2299e972e85228ce0ff498 | Ari the ant visits each vertex of a convex octagon exactly once in some order, crawling in straight lines from vertex to vertex. What is the probability that his path self-intersects? | [
"\\dfrac{313}{315}"
] | [
"Ari the ant visits each vertex of a convex octagon exactly once in some order, crawling in straight lines from vertex to vertex. What is the probability that his path self-intersects?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 2
} | 9 | 0 | 0 |
omr-ws-989440faba0c71c9c13f446f92793303dd354c033a41f8835be7dcc36b4a84c8 | Ari, Jonah, and Helen play a game where they roll a die and keep a running product of the rolls. They continue rolling until the product is divisible by either 8, 9, or both. If the final product is divisible by 8 but not 9, Jonah wins. If the final product is divisible by 9 but not 8, Ari wins. If the final product is... | [
"\\(\\frac{5}{72}\\)"
] | [
"Ari, Jonah, and Helen play a game where they roll a die and keep a running product of the rolls. They continue rolling until the product is divisible by either 8, 9, or both. If the final product is divisible by 8 but not 9, Jonah wins. If the final product is divisible by 9 but not 8, Ari wins. If the final produ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ff55d4addb655e48a69496479276739dee8530d34bc409b16a8b0eae27aac008 | Ariana chooses 6 points \( A, B, C, D, E, F \) uniformly at random on the circumference of a circle. The probability that the line segments \( \overline{AB}, \overline{CD}, \overline{EF} \) do not intersect is \( \frac{p}{q} \), where \( \gcd(p, q) = 1 \). Find \( p+q \). | [
"4"
] | [
"Ariana chooses 6 points \\( A, B, C, D, E, F \\) uniformly at random on the circumference of a circle. The probability that the line segments \\( \\overline{AB}, \\overline{CD}, \\overline{EF} \\) do not intersect is \\( \\frac{p}{q} \\), where \\( \\gcd(p, q) = 1 \\). Find \\( p+q \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-ba126e7df1e1b1623935261b9ca112c7a6523afa2083ab9186398fa5d9376438 | Ariel the frog is on the top left square of an $8 \times 10$ grid of squares. Ariel can jump from any square on the grid to any adjacent square, including diagonally adjacent squares. What is the minimum number of jumps required so that Ariel reaches the bottom right corner? | [
"10"
] | [
"Ariel the frog is on the top left square of an $8 \\times 10$ grid of squares. Ariel can jump from any square on the grid to any adjacent square, including diagonally adjacent squares. What is the minimum number of jumps required so that Ariel reaches the bottom right corner?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0bf197c4ae48a6c43c425afce58196d8c558ebd34ccc11b5cabecf975530c17e | Arjun and Beth play a game in which they take turns removing one brick of two adjacent bricks from one "wall" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. Arjun plays first, and the player who removes the last brick wins. For which starting configuration i... | [
"\\text{When the XOR of the Grundy numbers of all walls is zero}"
] | [
"Arjun and Beth play a game in which they take turns removing one brick of two adjacent bricks from one \"wall\" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. Arjun plays first, and the player who removes the last brick wins. For which starting configur... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 7
} | 16 | 0 | 0 |
omr-ws-0a5f76857f0796fe823788f00a7bd94de2ebf307c5f97a196c5eaaa5b1acd101 | Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one "wall" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes $4$ and $2$ can be changed into any of the following by one move... | [
"(2, 2)"
] | [
"Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one \"wall\" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes $4$ and $2$ can be changed into any of the following by o... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 7
} | 15 | 0 | 0 |
omr-ws-bd0078109b37bd5c1661b272002b56d96074894d3c7986bc3981e73c28456adb | Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one "wall" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes $4$ and $2$ can be changed into any of the following by one move... | [
"(2, 2)"
] | [
"Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one \"wall\" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes $4$ and $2$ can be changed into any of the following by o... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 5
} | 17 | 0 | 0 |
omr-ws-ef805534d09783f1ec1912fbddb71d0663217e8dc822e334b7af97356c80a001 | Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one "wall" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes 4 and 2 can be changed into any of the following by one move: (3... | [
"[2, 2]"
] | [
"Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one \"wall\" among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes 4 and 2 can be changed into any of the following by one m... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-4e3e3bf689a0bf5231ef161c8c20fbc5bc9180151af6a4d77e2210cdbd96c513 | Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one “wall” among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes $4$ and $2$ can be changed into any of the following by one move... | [
"(2, 2)"
] | [
"Arjun and Beth play a game in which they take turns removing one brick or two adjacent bricks from one “wall” among a set of several walls of bricks, with gaps possibly creating new walls. The walls are one brick tall. For example, a set of walls of sizes $4$ and $2$ can be changed into any of the following by one... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-44cdfa3c7acb373949eca69208f96fb1f0290053380ecb29994dfa69f006df62 | Arka and Alexheinis play a game on a $2010 \times 2010$ array. Arka starts by writing a real number in any blank square, and then Alexheinis writes a real number in any remaining blank square. The game continues until all squares are filled. Let $D$ be the determinant of the resultant matrix. If Alexheinis follows an o... | [
"2010 \\times 2009"
] | [
"Arka and Alexheinis play a game on a $2010 \\times 2010$ array. Arka starts by writing a real number in any blank square, and then Alexheinis writes a real number in any remaining blank square. The game continues until all squares are filled. Let $D$ be the determinant of the resultant matrix. If Alexheinis follow... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 2
} | 9 | 0 | 0 |
omr-ws-e0cd84f0e3fd3a5138f8143ce4cb2d309ce4eda5aa8703a75991823a9515643f | Arman drew 4 horizontal and 4 vertical straight lines, forming several rectangles. Among these, four "corner" rectangles are labeled $A$, $B$, $C$, and $D$. The areas of rectangles $A$, $D$, and $C$ are $4$ cm$^2$, $14$ cm$^2$, and $7$ cm$^2$ respectively. Find the area of rectangle $B$. | [
"2"
] | [
"Arman drew 4 horizontal and 4 vertical straight lines, forming several rectangles. Among these, four \"corner\" rectangles are labeled $A$, $B$, $C$, and $D$. The areas of rectangles $A$, $D$, and $C$ are $4$ cm$^2$, $14$ cm$^2$, and $7$ cm$^2$ respectively. Find the area of rectangle $B$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7b2cc4e0b952a316e0fa9326f795e53fb5efea731bc5516ba11e6674361c391a | Armando and Mauricio play Schwarz, a turn-based game on a page of paper marked with 202020 points placed arbitrarily such that no three points are aligned. On each turn, a player must select two pairs of points that have not been previously joined and join each of them with a line segment. If a triangle is formed, the ... | [
"Armando"
] | [
"Armando and Mauricio play Schwarz, a turn-based game on a page of paper marked with 202020 points placed arbitrarily such that no three points are aligned. On each turn, a player must select two pairs of points that have not been previously joined and join each of them with a line segment. If a triangle is formed,... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 5
} | 18 | 0 | 0 |
omr-ws-5a54e520495a2b4b5c67fa23ef7b113f000f18f8b7f5682c3ba2d7e44836a7a2 | Armando has a convex polyhedron with only triangular faces. On each face, he draws exactly two median lines of the triangle. Let $P$ be the polyhedron with $V$ vertices, $E$ edges, and $F$ faces. Find the sum of all possible values of $V$ if Armando can choose a way of drawing so that every vertex of the polyhedron wil... | [
""
] | [
"Armando has a convex polyhedron with only triangular faces. On each face, he draws exactly two median lines of the triangle. Let $P$ be the polyhedron with $V$ vertices, $E$ edges, and $F$ faces. Find the sum of all possible values of $V$ if Armando can choose a way of drawing so that every vertex of the polyhedro... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 1 |
omr-ws-84b29b4135f045d1642addcb29e11432dbd59e72579eb1132577d2d79a09ebcb | Arnaldo and Bernaldo play a game where they alternate saying natural numbers, and the winner is the one who says \( 0 \). In each turn except the first, the possible moves are determined from the previous number \( n \) in the following way: write \[ n = \sum_{m \in O_n} 2^m; \] the valid numbers are the elements \( m ... | [
"0"
] | [
"Arnaldo and Bernaldo play a game where they alternate saying natural numbers, and the winner is the one who says \\( 0 \\). In each turn except the first, the possible moves are determined from the previous number \\( n \\) in the following way: write\n\\[\nn = \\sum_{m \\in O_n} 2^m;\n\\]\nthe valid numbers are t... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 1
} | 14 | 0 | 0 |
omr-ws-ef46cda42afc29986e0d538fde2dafec94717dee946ec24c7feacbc64491d981 | Arnaldo and Bernardo play a game on an $n \times n$ board. Arnaldo places boats on his board, with each boat occupying all $n$ cells of a row or a column, and boats cannot overlap or share a side. Bernardo marks $m$ cells on his board representing shots. After marking, Arnaldo reveals which of these cells are occupied ... | [
"\\(\\left\\lceil \\frac{4n}{3} \\right\\rceil + (n \\mod 3)\\)"
] | [
"Arnaldo and Bernardo play a game on an $n \\times n$ board. Arnaldo places boats on his board, with each boat occupying all $n$ cells of a row or a column, and boats cannot overlap or share a side. Bernardo marks $m$ cells on his board representing shots. After marking, Arnaldo reveals which of these cells are occ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6623d1edc81f1778468d62ad0bfbb183d235d3d67dfb0ce2b49c8e14abb6dcd0 | Arnaldo selects a nonnegative integer \( a \) and Bernaldo selects a nonnegative integer \( b \). Cernaldo writes the numbers \( 5 \), \( 8 \), and \( 15 \) on the board, one of which is the sum \( a + b \). Arnaldo and Bernaldo, individually, write on different slips of paper whether they know which of the numbers on ... | [
"3"
] | [
"Arnaldo selects a nonnegative integer \\( a \\) and Bernaldo selects a nonnegative integer \\( b \\). Cernaldo writes the numbers \\( 5 \\), \\( 8 \\), and \\( 15 \\) on the board, one of which is the sum \\( a + b \\). Arnaldo and Bernaldo, individually, write on different slips of paper whether they know which o... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-de4cdd4c71db97ae6b0277c26642fa57532ef67751f6fc73a08ce9e8148863b4 | Arnav and Barnav play a game where Arnav starts by writing a positive integer $n$ on a whiteboard. They alternate turns, with Barnav starting, and each player replaces $n$ with another integer $m$ such that $m \geq n$ and $\gcd(m, n) = 1$. The first player to write a five-digit integer loses. Define a function $f(n)$ s... | [
"9999"
] | [
"Arnav and Barnav play a game where Arnav starts by writing a positive integer $n$ on a whiteboard. They alternate turns, with Barnav starting, and each player replaces $n$ with another integer $m$ such that $m \\geq n$ and $\\gcd(m, n) = 1$. The first player to write a five-digit integer loses. Define a function $... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d2ad1fbaf43316e3f37cf9e68f5a8b94d1a8cac0eadfdecf8b68b142a73398c1 | Arnav the ant is on the outside of a cylindrical glass, halfway up the glass. The glass is 8 inches tall and has a radius of 2 inches. Arnav wants to walk to a point on the inside of the glass that is diametrically opposite the point he is currently at. What is the shortest distance he can walk? | [
"\\(2\\pi\\)"
] | [
"Arnav the ant is on the outside of a cylindrical glass, halfway up the glass. The glass is 8 inches tall and has a radius of 2 inches. Arnav wants to walk to a point on the inside of the glass that is diametrically opposite the point he is currently at. What is the shortest distance he can walk?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-d6acf876a09a418d925fe330e8ee32a8b052b074c07e474dba90d81503aaa7d1 | Arnav the ant is on the outside of a cylindrical glass, halfway up the glass. The glass is 8 inches tall and has a radius of 2 inches. Arnav wants to walk to a point on the inside of the glass that is diametrically opposite the point he is now at. What is the shortest distance Arnav must travel? | [
"\\(\\sqrt{4\\pi^2+64}\\)"
] | [
"Arnav the ant is on the outside of a cylindrical glass, halfway up the glass. The glass is 8 inches tall and has a radius of 2 inches. Arnav wants to walk to a point on the inside of the glass that is diametrically opposite the point he is now at. What is the shortest distance Arnav must travel?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 14
} | 37 | 0 | 0 |
omr-ws-166f3a4a0f3109bcefbed0dfc1cbf2574d6941b0ab3f1ff513fa4da762e526a9 | Arnim and Brentano play a game with a vase containing 1500 candies and a sack of spare candies under the table. They take turns, with Arnim starting first. On each turn, a player can either eat 7 candies from the vase or take 6 candies from under the table and add them to the vase. A player cannot take candies from und... | [
"Arnim has a winning strategy",
"Brentano has a winning strategy."
] | [
"Arnim and Brentano play a game with a vase containing 1500 candies and a sack of spare candies under the table. They take turns, with Arnim starting first. On each turn, a player can either eat 7 candies from the vase or take 6 candies from under the table and add them to the vase. A player cannot take candies fro... | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 8
} | 8 | 0 | 0 |
omr-ws-5aca4175121ab9b6ffb85a3a94d2c565a4e4bdab2270ce5922e562b10e7aaf98 | Arnold and Barnold are playing a game with a pile of sticks, starting with Arnold. Each turn, a player can remove either 7 or 13 sticks. If there are fewer than 7 sticks at the start of a player's turn, they lose. Both players play optimally. Find the largest number of sticks under 200 where Barnold has a winning strat... | [
"186"
] | [
"Arnold and Barnold are playing a game with a pile of sticks, starting with Arnold. Each turn, a player can remove either 7 or 13 sticks. If there are fewer than 7 sticks at the start of a player's turn, they lose. Both players play optimally. Find the largest number of sticks under 200 where Barnold has a winning ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-6d6e6b220b188414c7a69d044bdfb986d38c6f9a16c8e0ef46e91019f524b569 | Arnold and Boris are given a sufficiently long paper strip, one with the letter $A$ written and the other with the letter $B$. Every minute, one of them writes the word written on the other strip on his own strip, either to the left or to the right of the word already present. After 1440 minutes, let $P$ be the word on... | [
"2"
] | [
"Arnold and Boris are given a sufficiently long paper strip, one with the letter $A$ written and the other with the letter $B$. Every minute, one of them writes the word written on the other strip on his own strip, either to the left or to the right of the word already present. After 1440 minutes, let $P$ be the wo... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-e9abcee20fd110fcce0644a956c5a5fa425ba2c1a1baeee10380e3bace2ea518 | Arnold has plates weighing $5$, $15$, $25$, $35$, or $45$ pounds. He lifts a barbell consisting of a $45$-pound bar and any number of these plates, totaling $600$ pounds according to Vlad's mistaken observation. Vlad mistook each plate for the next heavier plate, and Arnold was actually lifting $470$ pounds. How many p... | [
"13"
] | [
"Arnold has plates weighing $5$, $15$, $25$, $35$, or $45$ pounds. He lifts a barbell consisting of a $45$-pound bar and any number of these plates, totaling $600$ pounds according to Vlad's mistaken observation. Vlad mistook each plate for the next heavier plate, and Arnold was actually lifting $470$ pounds. How m... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 5
} | 20 | 0 | 0 |
omr-ws-14813350fe36459e40d0fa0dfa5b2b52156e4cce631dd2730d57defe44109cc8 | Arnold is studying the prevalence of three health risk factors, denoted by A, B, and C, within a population of men. For each of the three factors, the probability that a randomly selected man in the population has only this risk factor (and none of the others) is 0.1. For any two of the three factors, the probability t... | [
"76"
] | [
"Arnold is studying the prevalence of three health risk factors, denoted by A, B, and C, within a population of men. For each of the three factors, the probability that a randomly selected man in the population has only this risk factor (and none of the others) is 0.1. For any two of the three factors, the probabil... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3008a54fd8c0277f74f040ad36afd931b3856233c843435bab2c29c09bbec566 | Arnold is studying the prevalence of three health risk factors, denoted by A, B, and C, within a population of men. For each of the three factors, the probability that a randomly selected man in the population has only this risk factor (and none of the others) is 0.1. For any two of the three factors, the probability t... | [
"76"
] | [
"Arnold is studying the prevalence of three health risk factors, denoted by A, B, and C, within a population of men. For each of the three factors, the probability that a randomly selected man in the population has only this risk factor (and none of the others) is 0.1. For any two of the three factors, the probabil... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-ee8d22d7275d29649a67a8a53453a284a5bea99841b61bc514dfcfdc244762c5 | Arnold selects a number \( n \) such that the sum of its digits \( S(n) = 2017 \). Each turn, Bernard gives Arnold a number \( x \), and Arnold announces \( S(|n - x|) \). What is the minimum number of turns Bernard needs to determine \( n \)? | [
"2"
] | [
"Arnold selects a number \\( n \\) such that the sum of its digits \\( S(n) = 2017 \\). Each turn, Bernard gives Arnold a number \\( x \\), and Arnold announces \\( S(|n - x|) \\). What is the minimum number of turns Bernard needs to determine \\( n \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-3f32c6140c4a51f9b9800be22e8b5cd95c93185c494789bfc17ec65eb63f174d | Around a circle are seated \( n \) men and \( n-1 \) women. Find the number of distinct pairs of men such that between them there are exactly \( n \) people. | [
"n-2"
] | [
"Around a circle are seated \\( n \\) men and \\( n-1 \\) women. Find the number of distinct pairs of men such that between them there are exactly \\( n \\) people."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-1bdfa24dbacaa65604f3433e817e323ed88de6af6e34df574c43a3e1cc659ab2 | Around a circle are written all of the positive integers from \( 1 \) to \( n \), \( n \ge 2 \), in such a way that any two adjacent integers have at least one digit in common in their decimal expressions. Find the smallest \( n \) for which this is possible. | [
"45"
] | [
"Around a circle are written all of the positive integers from \\( 1 \\) to \\( n \\), \\( n \\ge 2 \\), in such a way that any two adjacent integers have at least one digit in common in their decimal expressions. Find the smallest \\( n \\) for which this is possible."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5bc3de5728c9a410f6abb6f882661e1cae80ed19a4d8727daf16dfed2a39f34a | Around a circle, 64 equally spaced points are placed, resulting in 2016 possible chords. What is the maximum number of chords that can be drawn such that each chord does not cross more than one other chord? | [
"32"
] | [
"Around a circle, 64 equally spaced points are placed, resulting in 2016 possible chords. What is the maximum number of chords that can be drawn such that each chord does not cross more than one other chord?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-b673d435708162fb35e9a0380d7aa2917197a97133e0699d1a843ae63a949820 | Around a circle, 64 equally spaced points are placed. Chords are drawn such that each chord cannot intersect more than one other chord. What is the maximum number of chords that can be drawn? | [
"32"
] | [
"Around a circle, 64 equally spaced points are placed. Chords are drawn such that each chord cannot intersect more than one other chord. What is the maximum number of chords that can be drawn?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 6,
"tir": 1
} | 21 | 0 | 0 |
omr-ws-a872400a625ea949ed016e82dfc47482c548749973b13a7813ec1acc83191913 | Around a circle, mark 21 evenly spaced points in cyclic order. Color \( n \) of these points red and the rest blue so that no two red points are the same distance apart. Determine the maximum possible value of \( n \). | [
"5"
] | [
"Around a circle, mark 21 evenly spaced points in cyclic order. Color \\( n \\) of these points red and the rest blue so that no two red points are the same distance apart. Determine the maximum possible value of \\( n \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-5486ceb10a2ea7418494b4e350af93f2962145fab9746e708771737afcf73ce3 | Around a circle, n numbers (each 0 or 1) are written, with one underlined. In each round, either the underlined number can be changed or the underline can be moved to an adjacent number. Find the least number of rounds needed to change any initial configuration to any other. | [
"\\(2n - 1\\)"
] | [
"Around a circle, n numbers (each 0 or 1) are written, with one underlined. In each round, either the underlined number can be changed or the underline can be moved to an adjacent number. Find the least number of rounds needed to change any initial configuration to any other."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 2,
"tir": 2
} | 12 | 0 | 0 |
omr-ws-748e48187062f14e66a53058943614c457061649c35d81df14d057018b6600ea | Around a circle, place 64 equally spaced points. How many chords can be drawn between these points such that no chord crosses more than one other chord? | [
"156"
] | [
"Around a circle, place 64 equally spaced points. How many chords can be drawn between these points such that no chord crosses more than one other chord?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-98f33cb408f324b12b53bc4199b5cf17794bac865d645f58fdae6e947654b269 | Around a circular table sit 12 people, and on the table there are 28 vases. Two people can see each other if and only if there is no vase directly between them. Calculate the minimum number of pairs of people who can see each other. | [
"6"
] | [
"Around a circular table sit 12 people, and on the table there are 28 vases. Two people can see each other if and only if there is no vase directly between them. Calculate the minimum number of pairs of people who can see each other."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-0f83a9a1cc9ae7ee05d00a28fbf47848ec54df16edf9e3097968584f050e3fd8 | Around a circular table, an even number of persons are seated for a discussion. After a break, they rearrange themselves in a different seating order. Let $N$ be the number of pairs of people such that the number of participants sitting between them before and after the break is the same. Find the minimum possible valu... | [
"10"
] | [
"Around a circular table, an even number of persons are seated for a discussion. After a break, they rearrange themselves in a different seating order. Let $N$ be the number of pairs of people such that the number of participants sitting between them before and after the break is the same. Find the minimum possible... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-c14bbd415a0559e7691e4e564c211fa13b054c0be42bbbc52de65818614033ba | Around a circular table, there are 2005 knights, and one of them has \( z = 2005 \) coins. In each step, a knight with more than one coin gives one coin to each of his neighbors. Can the delivery be finished if \( z = 2005 \)? | [
"Yes, the delivery can be finished if \\( z = 2005 \\)."
] | [
"Around a circular table, there are 2005 knights, and one of them has \\( z = 2005 \\) coins. In each step, a knight with more than one coin gives one coin to each of his neighbors. Can the delivery be finished if \\( z = 2005 \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 21
} | 41 | 0 | 0 |
omr-ws-72f972bff1f5556487c75f8d817c6b2481142c4d004ae6a5c808c73cb1c7a388 | Around a circular table, there are 2005 knights, and one of them has \( z \leq 2005 \) coins. In each step, a knight with more than one coin gives one coin to each of his neighbors. Let \( f(z) \) be the maximum number of steps required for the process to stop when \( z < 2005 \). Find \( f(1000) \). | [
"1005000"
] | [
"Around a circular table, there are 2005 knights, and one of them has \\( z \\leq 2005 \\) coins. In each step, a knight with more than one coin gives one coin to each of his neighbors. Let \\( f(z) \\) be the maximum number of steps required for the process to stop when \\( z < 2005 \\). Find \\( f(1000) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-54f9a9aac9c0b27d8356e350422aa8140aadf843a0748281164282f94dd7323c | Around a circumference are written 2012 numbers, each of which is equal to 1 or -1. If there are no 10 consecutive numbers that sum to 0, find all possible values of the sum of the 2012 numbers. | [
"All even numbers \\( S \\) such that \\( 404 \\le |S| \\le 2012 \\)"
] | [
"Around a circumference are written 2012 numbers, each of which is equal to 1 or -1. If there are no 10 consecutive numbers that sum to 0, find all possible values of the sum of the 2012 numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 5,
"tir": 3
} | 12 | 0 | 0 |
omr-ws-b3fd4492f222b8e7dbca4c448712a663f07f16067c0f370a47e447ac7a73cb38 | Around a round table sit $2n$ Peruvians, $2n$ Bolivians, and $2n$ Ecuadorians. If it is requested that all those who have as neighbors, to their right and to their left, people of the same nationality, find the maximum number of people who can stand up. | [
"\\(6n-4\\)"
] | [
"Around a round table sit $2n$ Peruvians, $2n$ Bolivians, and $2n$ Ecuadorians. If it is requested that all those who have as neighbors, to their right and to their left, people of the same nationality, find the maximum number of people who can stand up."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-297aab4587e31fee2cb30065fd36b7fee6f277db9e7c23e5ad550d45733473e5 | Around a round table, people $P_1, P_2, \ldots, P_{2013}$ are seated in a clockwise order. Each person starts with a certain amount of coins (possibly none), with a total of 10,000 coins. Starting with $P_1$ and proceeding in clockwise order, each person does the following on their turn: - If they have an even number o... | [
"4050156"
] | [
"Around a round table, people $P_1, P_2, \\ldots, P_{2013}$ are seated in a clockwise order. Each person starts with a certain amount of coins (possibly none), with a total of 10,000 coins. Starting with $P_1$ and proceeding in clockwise order, each person does the following on their turn:\n- If they have an even n... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-577ce8c30eb376d48ea4ff6eec70d7470b0a132b268d7fa9473cba841b0485c4 | Around an acute-angled triangle \(ABC\) with \(AB < BC < AC\), a circle \(\omega\) with center \(O\) is circumscribed. Denote \(I\) as the incenter of the triangle, and \(M\) as the midpoint of side \(BC\). Let \(Q\) be the point symmetric to \(I\) with respect to \(M\). The ray \(OM\) intersects the circle \(\omega\) ... | [
"90^\\circ"
] | [
"Around an acute-angled triangle \\(ABC\\) with \\(AB < BC < AC\\), a circle \\(\\omega\\) with center \\(O\\) is circumscribed. Denote \\(I\\) as the incenter of the triangle, and \\(M\\) as the midpoint of side \\(BC\\). Let \\(Q\\) be the point symmetric to \\(I\\) with respect to \\(M\\). The ray \\(OM\\) inter... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-33a81a2395d83461004f92b71ef58c5e0d595886ed6f2cfa240da1622d8235b9 | Around square $ABCD$, circumscribe a circle $\gamma$. Points $M$ and $N$ are selected on sides $BC$ and $CD$ respectively such that $\angle MAN = 45^\circ$. Lines $AM$ and $AN$ intersect circle $\gamma$ again at points $P$ and $Q$ respectively. If $AB = 1$, calculate the length of segment $PQ$. | [
"1"
] | [
"Around square $ABCD$, circumscribe a circle $\\gamma$. Points $M$ and $N$ are selected on sides $BC$ and $CD$ respectively such that $\\angle MAN = 45^\\circ$. Lines $AM$ and $AN$ intersect circle $\\gamma$ again at points $P$ and $Q$ respectively. If $AB = 1$, calculate the length of segment $PQ$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c1e6c52f89a3ecb0accb062ddeeb0b70d3a2b82a70c03112329a48febb0f0efb | Around the acute-angled triangle \(ABC\) (\(AC > CB\)), a circle is circumscribed, and the point \(N\) is the midpoint of the arc \(ACB\) of this circle. Let the points \(A_1\) and \(B_1\) be the feet of the perpendiculars from points \(A\) and \(B\) respectively to the line \(NC\) (segment \(NC\) lies inside the segme... | [
"1"
] | [
"Around the acute-angled triangle \\(ABC\\) (\\(AC > CB\\)), a circle is circumscribed, and the point \\(N\\) is the midpoint of the arc \\(ACB\\) of this circle. Let the points \\(A_1\\) and \\(B_1\\) be the feet of the perpendiculars from points \\(A\\) and \\(B\\) respectively to the line \\(NC\\) (segment \\(NC... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-6544a41a533e1f53472a31f7e6dccb933f6da215afdcf6e2fd1eb7be11f70cce | Around the convex quadrilateral \(ABCD\), three rectangles are circumscribed. It is known that two of these rectangles are squares. Let \(s_1\) and \(s_2\) be the side lengths of these two squares. If the third circumscribed rectangle has sides of length \(a\) and \(b\), find the value of \(\frac{a}{b}\) given that \(s... | [
"\\dfrac{5}{7}"
] | [
"Around the convex quadrilateral \\(ABCD\\), three rectangles are circumscribed. It is known that two of these rectangles are squares. Let \\(s_1\\) and \\(s_2\\) be the side lengths of these two squares. If the third circumscribed rectangle has sides of length \\(a\\) and \\(b\\), find the value of \\(\\frac{a}{b}... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-e8cba60c6b4a267c500afba3416f58ac113ab97eba8bdadc881ad1b4bed6a462 | Around triangle \(ABC\) with \(\angle B = 60^\circ\), a circle is circumscribed. The tangents to the circle at points \(A\) and \(C\) intersect at point \(B_1\). Points \(A_0\) and \(C_0\) are marked on the rays \(AB\) and \(CB\) respectively such that \(AA_0 = AC = CC_0\). If \(AC = 1\), calculate the length of segmen... | [
"\\dfrac{\\sqrt{6} - \\sqrt{2}}{2}"
] | [
"Around triangle \\(ABC\\) with \\(\\angle B = 60^\\circ\\), a circle is circumscribed. The tangents to the circle at points \\(A\\) and \\(C\\) intersect at point \\(B_1\\). Points \\(A_0\\) and \\(C_0\\) are marked on the rays \\(AB\\) and \\(CB\\) respectively such that \\(AA_0 = AC = CC_0\\). If \\(AC = 1\\), c... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-433b44babd019722cef4e60f5557b2b87fd24fdfe3046ead585fd36ef34a2371 | Around triangle \(ABC\) with an acute angle \(C\) is circumscribed a circle. On the arc \(AB\) (which does not contain point \(C\)), point \(D\) is chosen. Point \(D'\) is the reflection of point \(D\) with respect to line \(AB\). The lines \(AD'\) and \(BD'\) intersect segments \(BC\) and \(AC\) at points \(E\) and \(... | [
"0"
] | [
"Around triangle \\(ABC\\) with an acute angle \\(C\\) is circumscribed a circle. On the arc \\(AB\\) (which does not contain point \\(C\\)), point \\(D\\) is chosen. Point \\(D'\\) is the reflection of point \\(D\\) with respect to line \\(AB\\). The lines \\(AD'\\) and \\(BD'\\) intersect segments \\(BC\\) and \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-711959793c2ec0b30c252c7f85dcc6facc8044ffdfb29fc4a7fc65cd452509d6 | Around triangle \(ABC\), a circle \(\omega\) is circumscribed, and \(I\) is the incenter (intersection point of the angle bisectors). Line \(CI\) intersects \(\omega\) again at point \(P\). The circle with diameter \(IP\) intersects \(AI\), \(BI\), and \(\omega\) again at points \(M\), \(N\), and \(K\) respectively. Li... | [
"120"
] | [
"Around triangle \\(ABC\\), a circle \\(\\omega\\) is circumscribed, and \\(I\\) is the incenter (intersection point of the angle bisectors). Line \\(CI\\) intersects \\(\\omega\\) again at point \\(P\\). The circle with diameter \\(IP\\) intersects \\(AI\\), \\(BI\\), and \\(\\omega\\) again at points \\(M\\), \\(... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-2ff8b1d2a6cc7ecf255f39bf06f998f05351ae0ba05249b0c64afd8c4d8e3340 | Around triangle \(ABC\), a circle is circumscribed, and a tangent \(t\) is drawn at vertex \(C\). A line \(p\) parallel to this tangent intersects lines \(BC\) and \(AC\) at points \(D\) and \(E\), respectively. If \(\angle BAC = 30^\circ\) and \(\angle ABC = 45^\circ\), calculate the measure of \(\angle ADE\). | [
"30^\\circ"
] | [
"Around triangle \\(ABC\\), a circle is circumscribed, and a tangent \\(t\\) is drawn at vertex \\(C\\). A line \\(p\\) parallel to this tangent intersects lines \\(BC\\) and \\(AC\\) at points \\(D\\) and \\(E\\), respectively. If \\(\\angle BAC = 30^\\circ\\) and \\(\\angle ABC = 45^\\circ\\), calculate the measu... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-62cbe088881354d60e7e0801defa7315710848d7b5941cfdc3a17e9b8c0f9a67 | Arpit is hanging 12 rows of lights on his Christmas tree. If any row of lights is defective, the Christmas tree will not be lit. If the tree is not lit, how many subsets of rows of lights can be broken for the lights to not work? | [
"4095"
] | [
"Arpit is hanging 12 rows of lights on his Christmas tree. If any row of lights is defective, the Christmas tree will not be lit. If the tree is not lit, how many subsets of rows of lights can be broken for the lights to not work?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 6
} | 21 | 0 | 0 |
omr-ws-26ae3016396fc1efe82ffca5d84a97538cd9e96403369738e01cdf66f2afd183 | Arrange $2010$ digits, each of them equal to $1$, $2$, or $3$, in a circle. For each positive integer $k$, it's known that in any block of $3k$ consecutive digits, each of the digits appears at most $k+10$ times. Let $N$ be the number of distinct blocks of consecutive digits that contain exactly the same number of $1$s... | [
"30"
] | [
"Arrange $2010$ digits, each of them equal to $1$, $2$, or $3$, in a circle. For each positive integer $k$, it's known that in any block of $3k$ consecutive digits, each of the digits appears at most $k+10$ times. Let $N$ be the number of distinct blocks of consecutive digits that contain exactly the same number of... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-14c66be0c7ecd8d7e08d6e232446efe52a47a36d3c6d571c250cf83e3ed9a84e | Arrange $2n$ biologists of $2n$ different heights in two rows of $n$ each, such that heights increase from left to right in each row and each person in the back row is taller than the person directly in front of them. Give a formula in terms of $n$ for the number of possible arrangements. | [
"\\(\\frac{(2n)!}{(n+1)!n!}\\)"
] | [
"Arrange $2n$ biologists of $2n$ different heights in two rows of $n$ each, such that heights increase from left to right in each row and each person in the back row is taller than the person directly in front of them. Give a formula in terms of $n$ for the number of possible arrangements."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 10
} | 25 | 0 | 0 |
omr-ws-b5ebef02a8c43410b7f551b16555143e7c70237fbfad6d09014751253fc0c2b1 | Arrange 12 cards (4 Jacks, 4 Queens, and 4 Kings) in a row at random. What is the probability that no 2 cards of the same suit are all together? | [
"\\dfrac{6959}{7700}"
] | [
"Arrange 12 cards (4 Jacks, 4 Queens, and 4 Kings) in a row at random. What is the probability that no 2 cards of the same suit are all together?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-77c44c368174a88e46687db47dc2576c4d55ab7c2d0401fe269372b14aa6531f | Arrange 12 cards (4 Jacks, 4 Queens, and 4 Kings) in a row at random. What is the probability that no 3 cards of the same rank are all together? | [
"\\dfrac{244}{385}"
] | [
"Arrange 12 cards (4 Jacks, 4 Queens, and 4 Kings) in a row at random. What is the probability that no 3 cards of the same rank are all together?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 8
} | 11 | 0 | 0 |
omr-ws-6acbdcda5cc90963b63c110333a97a609e2fbe6fa2ff2fe16a0f3ee6f04425c9 | Arrange 12 cards (4 Jacks, 4 Queens, and 4 Kings) in a row at random. What is the probability that no 3 cards of the same suit are all together? | [
"\\dfrac{12861}{15400}"
] | [
"Arrange 12 cards (4 Jacks, 4 Queens, and 4 Kings) in a row at random. What is the probability that no 3 cards of the same suit are all together?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 45,
"genselect": 8
} | 53 | 0 | 0 |
omr-ws-446658883315c9b72932bf1596f923a70bde08409bd6f319f8b3e1c16812791b | Arrange 12 cards (4 Jacks, 4 Queens, and 4 Kings) in a row at random. What is the probability that no 4 cards of the same rank are all together? | [
"\\dfrac{782}{825}"
] | [
"Arrange 12 cards (4 Jacks, 4 Queens, and 4 Kings) in a row at random. What is the probability that no 4 cards of the same rank are all together?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 48
} | 48 | 0 | 0 |
omr-ws-c4cfd8e8b7584be04244e6d782802dbd094fdf3231d48936c7e23390e7b50f65 | Arrange 126 squares into a 12 by 10 quilt grid such that: - There are 7 patterns (numbered 1-7), each with 3 colors (white (w), red (r), light blue (b), teal (t), navy (n)). - Each pattern has 6 squares of each of its colors. - No two squares of the same color are adjacent (vertically or horizontally). - No two squares... | [
"\\text{No solution exists due to conflicting constraints.}"
] | [
"Arrange 126 squares into a 12 by 10 quilt grid such that:\n- There are 7 patterns (numbered 1-7), each with 3 colors (white (w), red (r), light blue (b), teal (t), navy (n)).\n- Each pattern has 6 squares of each of its colors.\n- No two squares of the same color are adjacent (vertically or horizontally).\n- No tw... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-8f8dc0c0089d9ed02a29a1ebb7cb392cbf6d9fbff9cf753d7e4e5d98966ca08f | Arrange 126 squares into a 9 by 14 quilt grid such that: - There are 7 patterns (numbered 1-7), each with 3 colors (white (w), red (r), light blue (b), teal (t), navy (n)). - Each pattern has 6 squares of each of its colors. - No two squares of the same color are adjacent (vertically or horizontally). - No two squares ... | [
"The quilt grid is arranged with each of the 7 patterns distributed in a non-repeating, spaced manner, ensuring no same-color adjacency and diagonal constraints are met. Each pattern's colors are placed in a checkerboard-like layout across the grid."
] | [
"Arrange 126 squares into a 9 by 14 quilt grid such that:\n- There are 7 patterns (numbered 1-7), each with 3 colors (white (w), red (r), light blue (b), teal (t), navy (n)).\n- Each pattern has 6 squares of each of its colors.\n- No two squares of the same color are adjacent (vertically or horizontally).\n- No two... | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-610a33c1eff88ce0e056e1431b97c481d45d16003b8ca5c49d78cc56427e4fed | Arrange 126 squares into an 11 by 11 quilt grid such that: - There are 7 patterns (numbered 1-7), each with 3 colors (white (w), red (r), light blue (b), teal (t), navy (n)). - Each pattern has 6 squares of each of its colors. - No two squares of the same color are adjacent (vertically or horizontally). - No two square... | [
"\\text{No solution exists due to conflicting grid size and square count.}"
] | [
"Arrange 126 squares into an 11 by 11 quilt grid such that:\n- There are 7 patterns (numbered 1-7), each with 3 colors (white (w), red (r), light blue (b), teal (t), navy (n)).\n- Each pattern has 6 squares of each of its colors.\n- No two squares of the same color are adjacent (vertically or horizontally).\n- No t... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f208282ccb76c81ecfc5f12701253b36cfb0e06e859b2e927c4c5c269b9447e0 | Arrange 15 schoolgirls into 5 groups of 3 each day for a week so that each pair of girls walks together in a group exactly once. | [
"\n\\begin{aligned}\n&\\text{Day 1: } (1,2,5), (3,11,15), (4,12,13), (6,9,14), (7,8,10) \\\\\n&\\text{Day 2: } (1,3,6), (2,11,14), (4,7,15), (5,9,12), (8,10,13) \\\\\n&\\text{Day 3: } (1,4,8), (2,13,15), (3,5,7), (6,10,12), (9,11,14) \\\\\n&\\text{Day 4: } (1,9,10), (2,3,4), (5,8,14), (6,7,13), (11,12,15) \\\\\n&\\... | [
"Arrange 15 schoolgirls into 5 groups of 3 each day for a week so that each pair of girls walks together in a group exactly once."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-b42ffcb2388bc1b5389acd2f930da47d3738a867742f62155f783a5504c35c8d | Arrange 15 students into 5 rows with 3 students in each row such that no two students are in the same row more than once over 7 days. Determine the number of distinct ways to arrange the students over the weekend. | [
"168"
] | [
"Arrange 15 students into 5 rows with 3 students in each row such that no two students are in the same row more than once over 7 days. Determine the number of distinct ways to arrange the students over the weekend."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 2
} | 8 | 0 | 0 |
omr-ws-a42bc5b63bdee57cba460b12c69bef2ecd575e0974e9cae578de4e86d05d53e3 | Arrange 16 people, conveniently named A, B, ..., P, on a (large) Möbius band consisting of 4 rows and 4 columns such that no two people with adjacent letters sit next to each other. The chairs are labeled and physical restrictions are ignored. How many ways are there to arrange them? | [
"288"
] | [
"Arrange 16 people, conveniently named A, B, ..., P, on a (large) Möbius band consisting of 4 rows and 4 columns such that no two people with adjacent letters sit next to each other. The chairs are labeled and physical restrictions are ignored. How many ways are there to arrange them?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-725b5b4cbd072854d7a604ffdcee84b9a36dce78c6b4de9295d4613fdba0be7d | Arrange 1650 students in 22 rows and 75 columns. It is known that for any two columns, the number of occasions that two students in the same row are of the same sex does not exceed 11. Calculate the maximum possible number of boy students under these conditions. | [
"825"
] | [
"Arrange 1650 students in 22 rows and 75 columns. It is known that for any two columns, the number of occasions that two students in the same row are of the same sex does not exceed 11. Calculate the maximum possible number of boy students under these conditions."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-73eb8826662a2fc26efb73b1fce12beea69b56584be4eef83efa0d005b12a348 | Arrange 1650 students into 22 rows and 75 columns. Suppose that in any two randomly chosen columns, the number of situations where two students in the same row have the same gender is not over 11. Calculate the maximum possible number of boys in this arrangement. | [
"825"
] | [
"Arrange 1650 students into 22 rows and 75 columns. Suppose that in any two randomly chosen columns, the number of situations where two students in the same row have the same gender is not over 11. Calculate the maximum possible number of boys in this arrangement."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-a28908caed2c82a95c02f1691f49a050b0f0721c9e3562822326f9a5494197d5 | Arrange 20 white balls and 20 black balls in a circle. What is the expected number of pairs of balls that are both black? | [
"\\dfrac{380}{39}"
] | [
"Arrange 20 white balls and 20 black balls in a circle. What is the expected number of pairs of balls that are both black?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-00f99e4c0158e2367f63b86f02260e6424119d4666260b7ba30d7f51e599ec93 | Arrange 2024 pebbles in a row, where each pebble is either black or white. For each pair of consecutive pebbles, the number of black pebbles to the right of a white pebble must be less than the number of black pebbles to the left of that white pebble. Find the number of ways to arrange the pebbles. | [
"\\(2^{2022}\\)"
] | [
"Arrange 2024 pebbles in a row, where each pebble is either black or white. For each pair of consecutive pebbles, the number of black pebbles to the right of a white pebble must be less than the number of black pebbles to the left of that white pebble. Find the number of ways to arrange the pebbles."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-53ddaa003be93c2cfd8d824d4d40dc295d11c8d279b6cdff83e073dea5cc9b01 | Arrange 27 numbers such that each number from 1 to 9 appears exactly three times. When placing a number \( n \), you must "jump" \( n \) spaces before placing the next occurrence of \( n \). For example, for number 1, the sequence would be 1 (other number) 1 (other number) 1, and for number 2, the sequence would be 2 (... | [
"9 7 5 3 1 8 6 4 2 1 3 5 7 9 2 4 6 8 1 3 5 7 9 2 4 6 8"
] | [
"Arrange 27 numbers such that each number from 1 to 9 appears exactly three times. When placing a number \\( n \\), you must \"jump\" \\( n \\) spaces before placing the next occurrence of \\( n \\). For example, for number 1, the sequence would be 1 (other number) 1 (other number) 1, and for number 2, the sequence... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-bb6a09b862d2f84da1d8ba9482b7b533851a78487a2dbd74343f5720b32cf888 | Arrange 8 biologists of 8 different heights in two rows for a photograph. Each row must have 4 biologists. Height must increase from left to right in each row. Each person in the back must be taller than the person directly in front of him. How many different arrangements are possible? | [
"14"
] | [
"Arrange 8 biologists of 8 different heights in two rows for a photograph. Each row must have 4 biologists. Height must increase from left to right in each row. Each person in the back must be taller than the person directly in front of him. How many different arrangements are possible?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-65915f0b1999830df2cbabeee7e51af9c58bc47bf5f7f45c37ea8aed9b9e6da4 | Arrange 8 biologists of 8 different heights in two rows of 4 each, such that heights increase from left to right in each row and each person in the back row is taller than the person directly in front of them. How many different arrangements are possible? | [
"14"
] | [
"Arrange 8 biologists of 8 different heights in two rows of 4 each, such that heights increase from left to right in each row and each person in the back row is taller than the person directly in front of them. How many different arrangements are possible?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5c088efad4655d7a78b8565802636d30848c6dc354cf188e5601ba30aa538e5a | Arrange 98 cards numbered from 1 to 98 in a sequence such that the absolute difference between any two consecutive numbers is greater than 48. Determine the number of ways to make such a sequence. | [
"2"
] | [
"Arrange 98 cards numbered from 1 to 98 in a sequence such that the absolute difference between any two consecutive numbers is greater than 48. Determine the number of ways to make such a sequence."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 18,
"genselect": 7
} | 25 | 0 | 0 |
omr-ws-52cb34c2c785baf8272a1cb9c48b19d04e83aab3a783390e63f71e721d9633f8 | Arrange a schedule of five evenings for 16 friends meeting for dinner at four tables (4 diners per table) so that each person dines exactly once with every other person. | [
"\n\\begin{aligned}\n&\\text{Evening 1: } \\{0,1,2,3\\}, \\{4,5,6,7\\}, \\{8,9,10,11\\}, \\{12,13,14,15\\} \\\\\n&\\text{Evening 2: } \\{0,4,8,12\\}, \\{1,5,9,13\\}, \\{2,6,10,14\\}, \\{3,7,11,15\\} \\\\\n&\\text{Evening 3: } \\{0,5,10,15\\}, \\{1,4,11,14\\}, \\{2,7,8,13\\}, \\{3,6,9,12\\} \\\\\n&\\text{Evening 4: ... | [
"Arrange a schedule of five evenings for 16 friends meeting for dinner at four tables (4 diners per table) so that each person dines exactly once with every other person."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-3260a7d53e39457266cc21ce70ad73fd9d2a8929ff83e6077e7d749f136ce251 | Arrange all six-digit numbers with all different digits such that each digit (except the first) is less than the digit to its right. If these numbers are ordered from smallest to largest, what is the number in the 45th position? | [
"136425"
] | [
"Arrange all six-digit numbers with all different digits such that each digit (except the first) is less than the digit to its right. If these numbers are ordered from smallest to largest, what is the number in the 45th position?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-75f13f6e718aecd16927abef65d5bca27f07edcb71c9467659f1e10429959283 | Arrange all square-free positive integers in ascending order $a_1, a_2, a_3, \ldots, a_n, \ldots$. Find the smallest positive integer $n$ such that $a_{n+1} - a_n = 2020$. | [
"2020"
] | [
"Arrange all square-free positive integers in ascending order $a_1, a_2, a_3, \\ldots, a_n, \\ldots$. Find the smallest positive integer $n$ such that $a_{n+1} - a_n = 2020$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-e23a67debec6d0c8e19d770ba1e6bcb76528ff2294c53e564d326f4e22c7e1ef | Arrange arithmetic operations between the numbers to obtain 20: \[ 9 \ast 9 \ast 9 \ast 9 = 20 \] | [
"9 + 9 + \\frac{9 + 9}{9}"
] | [
"Arrange arithmetic operations between the numbers to obtain 20:\n\\[ 9 \\ast 9 \\ast 9 \\ast 9 = 20 \\]"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-e1380471f8bfc5f84719eae4d22bc7a68764a92a99c9569f0dfea2a9f22ce55b | Arrange as many points as possible on the plane so that any three points do not lie on one straight line and are the vertices of an isosceles triangle. | [
"5"
] | [
"Arrange as many points as possible on the plane so that any three points do not lie on one straight line and are the vertices of an isosceles triangle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 1
} | 4 | 0 | 0 |
omr-ws-0fc189a07ca9712f985046dce8938e6a262a01cdf7e64742fef11c4b5677e7b1 | Arrange boxes of sizes 1x1, 1x2, 1x3, and 1x4 to create a perfect square. | [
"4"
] | [
"Arrange boxes of sizes 1x1, 1x2, 1x3, and 1x4 to create a perfect square."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-aa8b34f214c7d47051a93ed4043503559f84e893a92698b826c82379f484469c | Arrange dominoes of size 1x2 into an m x 2n table (with n ≥ m) such that no two dominoes form a 2x2 square. Given that the last row of the table has exactly n dominoes, find the maximum number of dominoes that can be placed in the table. | [
"mn - \\left\\lfloor \\dfrac{m}{2} \\right\\rfloor"
] | [
"Arrange dominoes of size 1x2 into an m x 2n table (with n ≥ m) such that no two dominoes form a 2x2 square. Given that the last row of the table has exactly n dominoes, find the maximum number of dominoes that can be placed in the table."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-cc6e2cb5b0426edfa41585425202280370247047c3be8f7771c661468aa599d1 | Arrange eight of the nine numbers \(2, 3, 4, 7, 10, 11, 12, 13, 15\) in the vacant squares of the \(3 \times 4\) array so that the arithmetic average of the numbers in each row and in each column is the same integer. Specify which one of the nine numbers must be left out when completing the array. [asy] defaultpen(line... | [
"10"
] | [
"Arrange eight of the nine numbers \\(2, 3, 4, 7, 10, 11, 12, 13, 15\\) in the vacant squares of the \\(3 \\times 4\\) array so that the arithmetic average of the numbers in each row and in each column is the same integer. Specify which one of the nine numbers must be left out when completing the array.\n\n[asy]\nd... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 48
} | 48 | 0 | 0 |
omr-ws-73f65227a22f2bb6d2e3db180e898b8d6b93e658c7dea07ee711afaad90be111 | Arrange four identical coins on a table so that a fifth coin of the same size can be placed on the table touching all four coins. No other objects or measuring instruments are allowed. | [
"\\text{Arrange the four coins in a square with each coin spaced } 2r\\sqrt{2} \\text{ apart, then place the fifth coin at the center.}"
] | [
"Arrange four identical coins on a table so that a fifth coin of the same size can be placed on the table touching all four coins. No other objects or measuring instruments are allowed."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 7
} | 8 | 0 | 0 |
omr-ws-b13f551bdf9cbddb784b00608de19a2fc2a3fc223c5310faf902551410e22517 | Arrange nine of the numbers $4, 5, 6, 7, 8, 12, 13, 16, 18, 19$ into the empty cells of the following table: \[ \begin{array} {|c|c|c|} \cline{1-3} 10 & & \\ \cline{1-3} & & 9 \\ \cline{1-3} & 3 & \\ \cline{1-3} 11 & & 17 \\ \cline{1-3} & 20 & \\ \cline{1-3} \end{array} \] such that each row sums to the same number, an... | [
"\\(\\begin{tabular}{|c|c|c|} \\hline 10 & 19 & 4 \\\\ \\hline 16 & 8 & 9 \\\\ \\hline 12 & 3 & 18 \\\\ \\hline 11 & 5 & 17 \\\\ \\hline 6 & 20 & 7 \\\\ \\hline \\end{tabular}\\)"
] | [
"Arrange nine of the numbers $4, 5, 6, 7, 8, 12, 13, 16, 18, 19$ into the empty cells of the following table:\n\\[\n\\begin{array} {|c|c|c|}\n\\cline{1-3}\n10 & & \\\\ \\cline{1-3}\n& & 9 \\\\ \\cline{1-3}\n& 3 & \\\\ \\cline{1-3}\n11 & & 17 \\\\ \\cline{1-3}\n& 20 & \\\\ \\cline{1-3}\n\\end{array}\n\\]\nsuch that ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 18,
"genselect": 6,
"tir": 14
} | 38 | 0 | 0 |
omr-ws-08a685dc94b0f87add1142dc5aa9785c33664f0f7f60b41ddc20c5e644a00278 | Arrange pawns on an 8x8 chessboard such that: (1) Each 2x1 or 1x2 rectangle has at least 1 pawn. (2) Each 7x1 or 1x7 rectangle has at least 2 adjacent pawns. What is the minimum number of pawns required for such an arrangement? | [
"28"
] | [
"Arrange pawns on an 8x8 chessboard such that:\n(1) Each 2x1 or 1x2 rectangle has at least 1 pawn.\n(2) Each 7x1 or 1x7 rectangle has at least 2 adjacent pawns.\nWhat is the minimum number of pawns required for such an arrangement?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-de121b001c5040443bd42b2ca97df769fcb0345b55d5f34335c64e4a5ce2d595 | Arrange seven anti-aircraft installations so that among any three of them, there are two installations with a distance of exactly $10$ kilometers between them. | [
"\\text{Three installations form an equilateral triangle and four form a regular tetrahedron, all edges 10 km}"
] | [
"Arrange seven anti-aircraft installations so that among any three of them, there are two installations with a distance of exactly $10$ kilometers between them."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 3
} | 4 | 0 | 0 |
omr-ws-b157dadc9c989340424ba29bd4920c0301bbc7c80bc0051e9bcc8d2ab171e06b | Arrange the cards numbered 1, 3, 5, ..., 59 such that the following procedure is satisfied: - If the topmost card is moved to the bottom of the pile, the card numbered 1 will be on top. Remove this card. - Move the next 3 cards to the bottom of the pile, and the card numbered 3 will be on top. Remove this card. - Repea... | [
"59, 1, 55, 57, 51, 53, 47, 49, 43, 45, 39, 41, 35, 37, 31, 33, 27, 29, 23, 25, 19, 21, 15, 17, 11, 13, 7, 9, 3, 5"
] | [
"Arrange the cards numbered 1, 3, 5, ..., 59 such that the following procedure is satisfied:\n- If the topmost card is moved to the bottom of the pile, the card numbered 1 will be on top. Remove this card.\n- Move the next 3 cards to the bottom of the pile, and the card numbered 3 will be on top. Remove this card.\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c563f119fafd4c2d8f89cebc0a51c7373a94310e0b32f13e2404499a2472500c | Arrange the digits $1, 1, 2, 2, 3, 3$ as a six-digit number in which the $1$s are separated by one digit, the $2$s are separated by two digits, and the $3$s are separated by three digits. Find both possible arrangements. | [
"$312132$ and $231213$"
] | [
"Arrange the digits $1, 1, 2, 2, 3, 3$ as a six-digit number in which the $1$s are separated by one digit, the $2$s are separated by two digits, and the $3$s are separated by three digits. Find both possible arrangements."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 32
} | 32 | 0 | 0 |
omr-ws-30683d2f2b70da0ad6dc8ee29c94736d5dfed03a98ee1fe127b6b7ba83e14fc2 | Arrange the digits 0 through 9 so that the first digit is divisible by 1, the first two digits are divisible by 2, the first three digits are divisible by 3, and so on, until the first nine digits are divisible by 9 and the ten-digit number is divisible by 10. | [
"3816547290"
] | [
"Arrange the digits 0 through 9 so that the first digit is divisible by 1, the first two digits are divisible by 2, the first three digits are divisible by 3, and so on, until the first nine digits are divisible by 9 and the ten-digit number is divisible by 10."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 31
} | 31 | 0 | 0 |
omr-ws-6094ed4293e90750c9d3f37f53fc02ab69392fd7b63028c6d53a105c6a799cbb | Arrange the digits 0 through 9, using each exactly once, to form five two-digit numbers whose product is as large as possible. For example, $10 \times 23 \times 45 \times 67 \times 89 = 61717050$. | [
"1785641760"
] | [
"Arrange the digits 0 through 9, using each exactly once, to form five two-digit numbers whose product is as large as possible. For example, $10 \\times 23 \\times 45 \\times 67 \\times 89 = 61717050$."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 6,
"tir": 22
} | 32 | 0 | 0 |
omr-ws-ab5e02e821ce42f0bacbc00ccb023bed0fbb8115729fed32513cf748630eed7d | Arrange the digits 1 through 9 in sequence without repetition such that any two consecutive digits form a number divisible by 7 or 13. | [
"784913526"
] | [
"Arrange the digits 1 through 9 in sequence without repetition such that any two consecutive digits form a number divisible by 7 or 13."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-541373d6c7718a916a778e189eea489cc20c54f957b057bd2f9ae240e85ea89a | Arrange the digits 1 through 9 into the fractions $\frac{c}{ab} + \frac{f}{de} + \frac{i}{gh} = 1$ such that each digit is used exactly once. | [
"\\frac{9}{12} + \\frac{5}{34} + \\frac{7}{68} = 1"
] | [
"Arrange the digits 1 through 9 into the fractions $\\frac{c}{ab} + \\frac{f}{de} + \\frac{i}{gh} = 1$ such that each digit is used exactly once."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-589f4ff46539ae87fa0a18c0cd60016465629fffe3f9092d16ecca2f7695fc09 | Arrange the expression $\left (x^\frac{1}{2} + \frac{1}{2.x^\frac{1}{4}}\right)^n$ in decreasing power of $x$. Suppose the coefficients of the first three terms form an arithmetic progression. Then the number of terms in the expansion having integer powers of $x$ is. | [
"3"
] | [
"Arrange the expression $\\left (x^\\frac{1}{2} + \\frac{1}{2.x^\\frac{1}{4}}\\right)^n$ in decreasing power of $x$. Suppose the coefficients of the first three terms form an arithmetic progression. Then the number of terms in the expansion having integer powers of $x$ is."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-96d3db571605a9a7ce9951fdd2c6cc2439daf6198ab93aebf5054517ec45907d | Arrange the expressions \(a = (\sin x)^{\sin x}\), \(b = (\sin x)^{\cos x}\), \(c = (\cos x)^{\sin x}\), and \(d = (\cos x)^{\cos x}\) in increasing order, where \(x \in \left(0, \frac{\pi}{4}\right)\). | [
"\\(b < a < d < c\\)"
] | [
"Arrange the expressions \\(a = (\\sin x)^{\\sin x}\\), \\(b = (\\sin x)^{\\cos x}\\), \\(c = (\\cos x)^{\\sin x}\\), and \\(d = (\\cos x)^{\\cos x}\\) in increasing order, where \\(x \\in \\left(0, \\frac{\\pi}{4}\\right)\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-97fadf07da9e20a169990e17ceab2620bd2015c03c51dd6e47415fa807b35e89 | Arrange the first 20 natural numbers around a circle. The sum of the 20 positive differences between adjacent numbers should be at least 38. | [
"38"
] | [
"Arrange the first 20 natural numbers around a circle. The sum of the 20 positive differences between adjacent numbers should be at least 38."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 4,
"tir": 3
} | 20 | 0 | 0 |
omr-ws-881ac4bf6a1dab402fe65864243b263b9d9b0f82fb902ae9b966d5df5e93156d | Arrange the following expressions in descending order without using a calculator: \[ a = \sqrt{11} + \sqrt{3} \] \[ b = \sqrt{12} + \sqrt{2} \] \[ c = \sqrt{6} + \sqrt{4} \] | [
"a > b > c"
] | [
"Arrange the following expressions in descending order without using a calculator:\n\\[ a = \\sqrt{11} + \\sqrt{3} \\]\n\\[ b = \\sqrt{12} + \\sqrt{2} \\]\n\\[ c = \\sqrt{6} + \\sqrt{4} \\]"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 1
} | 14 | 0 | 0 |
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