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-828ce965eadd21fd92b99c0f8ebda320664f06c37a07bf8d8ad8a9ef7eebccc0 | What is the least $m$ such that for any $m$ integers we can choose $6$ integers such that their sum is divisible by $6$? | [
"11"
] | [
"What is the least $m$ such that for any $m$ integers we can choose $6$ integers such that their sum is divisible by $6$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0b52d3f8f366f9496ebc60a7ab479899ef6fb3232ed5e19ba1899428c74d13b7 | What is the least 4-digit number, whose digits are all even, that is a multiple of 24? | [
"2040"
] | [
"What is the least 4-digit number, whose digits are all even, that is a multiple of 24?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-4a314f8420bf3252bbef1ca2984edac25ed8cb8b87850918e03c90fbb2173281 | What is the least \( x > 1 \) that satisfies \[ \frac{\lfloor x \rfloor \cdot \lfloor \sqrt{x} \rfloor}{\lfloor \sqrt[3]{x} \rfloor \cdot \lfloor \sqrt[4]{x} \rfloor} = 300? \] | [
"420"
] | [
"What is the least \\( x > 1 \\) that satisfies\n\\[\n\\frac{\\lfloor x \\rfloor \\cdot \\lfloor \\sqrt{x} \\rfloor}{\\lfloor \\sqrt[3]{x} \\rfloor \\cdot \\lfloor \\sqrt[4]{x} \\rfloor} = 300?\n\\]"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-acb5e6f54065eadcded62cc3537ba456dac3eeebd4d67e37475e3628f2e0e748 | What is the least amount of $1 \times 1$ tiles that can be used to tile a $3 \times 7$ rectangle using $1 \times 4$, $2 \times 2$, and $1 \times 1$ tiles? | [
"5"
] | [
"What is the least amount of $1 \\times 1$ tiles that can be used to tile a $3 \\times 7$ rectangle using $1 \\times 4$, $2 \\times 2$, and $1 \\times 1$ tiles?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 21,
"genselect": 7
} | 28 | 0 | 0 |
omr-ws-ed53badbed88b9008a03f8e011a29778fe3c93f8964b2a606436c6ca9c24fa12 | What is the least amount of $2 \times 6$ paper strips needed to cover a $13 \times 13$ board such that there are no uncovered cells? The strips can overlap and be rotated, and each strip covers exactly $12$ cells of the board. | [
"15"
] | [
"What is the least amount of $2 \\times 6$ paper strips needed to cover a $13 \\times 13$ board such that there are no uncovered cells? The strips can overlap and be rotated, and each strip covers exactly $12$ cells of the board."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-20964a66c2ac5fde4a9692e061a0e126626e00c6ef296841a8009be8d44bdab0 | What is the least amount of money Nancy can spend to get the storage she needs, given she has 105 GB available on her phone and each 256GB memory card costs $10? | [
"40"
] | [
"What is the least amount of money Nancy can spend to get the storage she needs, given she has 105 GB available on her phone and each 256GB memory card costs $10?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-958e4a282a7eb837f4878b0785d47304a4a3cac980b3fe9a9a3b3c34b8440433 | What is the least amount that a customer must overpay to buy 2 bicycles (totaling 200 dollars) if they can only pay with 7 dollar and 17 dollar bills and no change is given? | [
"0"
] | [
"What is the least amount that a customer must overpay to buy 2 bicycles (totaling 200 dollars) if they can only pay with 7 dollar and 17 dollar bills and no change is given?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9d958332bbb28282ae89a6ba19647d155b4cb196778f482883215ef3f7d7aa9f | What is the least and greatest number of edges that a graph with \( n \) vertices, where the degree of every vertex is at least 1, may have? | [
"Least number of edges: \\( n-1 \\), Greatest number of edges: \\( \\frac{n(n-1)}{2} \\)"
] | [
"What is the least and greatest number of edges that a graph with \\( n \\) vertices, where the degree of every vertex is at least 1, may have?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8bfab3d84fff06e5611a43fef09a5b79764108b601fabac37449c0af24a64225 | What is the least common multiple (LCM) of the polynomials $x$ and $x-30$? | [
"x(x - 30)"
] | [
"What is the least common multiple (LCM) of the polynomials $x$ and $x-30$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-45874f97a87fc38389b51d43dfe999540aeb92e99c916c9ac28320bf96389e6a | What is the least common multiple of 12, 18 and 30? | [
"180"
] | [
"What is the least common multiple of 12, 18 and 30?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a54049d8c6916caea355ac57df671e1708a8fcbbb86ca9709ffc94149eb07315 | What is the least common multiple of 135 and 468? | [
"7020"
] | [
"What is the least common multiple of 135 and 468?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0e65c74843509502e5edaf39af34eff06e05c179c2c52b44c073b114a6fdaf8a | What is the least common multiple of 14 and 21? | [
"42"
] | [
"What is the least common multiple of 14 and 21?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b08e921b2484738fe278646c3a9bb3d595792c788a8a0e399253b450f0a8be7f | What is the least common multiple of 3, 4, 6 and 15? | [
"60"
] | [
"What is the least common multiple of 3, 4, 6 and 15?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-02aa9b36bb21ff29297302af1cb5eeb5a427d79c8d03c652183c08ef683496b4 | What is the least common multiple of 6, 8, and 10? | [
"120"
] | [
"What is the least common multiple of 6, 8, and 10?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-933a49be6488644859ac98885f5a97d246181f8230d95ba2bcbdfdc695f15ea4 | What is the least common multiple of the binomial coefficients $\binom{n}{1}, \binom{n}{2}, \ldots, \binom{n}{n-1}$? | [
"\\(\\frac{\\text{lcm}(1, 2, \\ldots, n+1)}{n+1}\\)"
] | [
"What is the least common multiple of the binomial coefficients $\\binom{n}{1}, \\binom{n}{2}, \\ldots, \\binom{n}{n-1}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-fd67c981b61bb3e43a06a6d59cfb3cdfb47761e6a6b53b4b2c171a75c023b22f | What is the least common multiple of the numbers 1332 and 888? | [
"2664"
] | [
"What is the least common multiple of the numbers 1332 and 888?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-15fcec0bcbfbebef0877e122a136bb4a149b4cb4297a2895318b9a5604eb07b7 | What is the least degree of a polynomial with integer coefficients that has $\cos 12^\circ$ as one of its roots? | [
"5"
] | [
"What is the least degree of a polynomial with integer coefficients that has $\\cos 12^\\circ$ as one of its roots?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6b92b7f72b186d3ec76d25bd8c97378982759ce3faf84c00791ba4c82adb6cd1 | What is the least five-digit positive integer which is congruent to 5 (mod 15)? | [
"10,\\!010"
] | [
"What is the least five-digit positive integer which is congruent to 5 (mod 15)?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b0592a9c43227923035bbecb54010d54c0aa409b324038dd379533b0431929c6 | What is the least four-digit positive integer, with all different digits, that is divisible by each of its digits? | [
"1236"
] | [
"What is the least four-digit positive integer, with all different digits, that is divisible by each of its digits?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c407f6c1b8f65455a853c983737c9e0dc7e21359c7ff8c52833c6011e1f73f5b | What is the least four-digit whole number that is both a perfect square and a perfect cube? | [
"4096"
] | [
"What is the least four-digit whole number that is both a perfect square and a perfect cube?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-af60ec9ad877ac060e40a23554bac581bb9a81010c813369b888a135386dab80 | What is the least integer $n$ for which $x_1^3 + x_2^3 + x_3^3 + \cdots + x_n^3 = 2002^{2002}$, where $x_1, x_2, \ldots, x_n$ are positive integers? | [
"4"
] | [
"What is the least integer $n$ for which $x_1^3 + x_2^3 + x_3^3 + \\cdots + x_n^3 = 2002^{2002}$, where $x_1, x_2, \\ldots, x_n$ are positive integers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-6e5c38998f622719f00159c76ae7215b907d3d6935a541d28cf5fff9141b562c | What is the least integer \( N \) greater than 2004 for which the positive difference between the square of the number that is five more than \( N \) and the square of \( N \) is also a perfect square? | [
"2100"
] | [
"What is the least integer \\( N \\) greater than 2004 for which the positive difference between the square of the number that is five more than \\( N \\) and the square of \\( N \\) is also a perfect square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-21d4ddceedef2a9dbea60a35bf786d9710b48b9ef6d5434135b2e1da82e3b99a | What is the least integer \( k \) for which \( 12k = 5n^3 \)? | [
"90"
] | [
"What is the least integer \\( k \\) for which \\( 12k = 5n^3 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cc708b34ea5e57f2a201ef2ca2edb0834847bc87d5acb07cb65c1b329199cb5b | What is the least integer \( n > 1 \) such that in every \( n \)-element subset of the first 100 positive integers, at least one number is a multiple of another? | [
"51"
] | [
"What is the least integer \\( n > 1 \\) such that in every \\( n \\)-element subset of the first 100 positive integers, at least one number is a multiple of another?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-445bf21bd7edfa17bed323192b632a9f14548251526529f6b3c31c899af63546 | What is the least integer \( n > 2003 \) such that \( 5^n + n^5 \) is a multiple of 11? | [
"None of the preceding"
] | [
"What is the least integer \\( n > 2003 \\) such that \\( 5^n + n^5 \\) is a multiple of 11?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c8336c9394c488a36a3cedf2752e9d825ff53d249bacda95bd8497fefd2ffd0b | What is the least integer \( n \geq 100 \) such that \( 77 \) divides \( 1 + 2 + 2^2 + 2^3 + \dots + 2^n \)? | [
"119"
] | [
"What is the least integer \\( n \\geq 100 \\) such that \\( 77 \\) divides \\( 1 + 2 + 2^2 + 2^3 + \\dots + 2^n \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-c3edf9a07081aaab4571ea4a029dca80ed010d6ed8b0aaf805d0a0174d447025 | What is the least integer \(n \geq 100\) such that \(77\) divides \(1 + 2 + 2^2 + 2^3 + \dots + 2^n\)? | [
"119"
] | [
"What is the least integer \\(n \\geq 100\\) such that \\(77\\) divides \\(1 + 2 + 2^2 + 2^3 + \\dots + 2^n\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-758561a9a41cd60c5d7627ed0b7b504f6cefab5d4f0d70bd8292c2397803a59c | What is the least integer greater than $\sqrt{300}$? | [
"18"
] | [
"What is the least integer greater than $\\sqrt{300}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9e3d80664aa17a80aeafb324c1cb50882da14e243d591c92cb75411adc96c146 | What is the least integer value of $x$ such that $\lvert2x+ 7\rvert\le 16$? | [
"-11"
] | [
"What is the least integer value of $x$ such that $\\lvert2x+ 7\\rvert\\le 16$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bb61aff428c6d738846541ef93b67afd0e365380a5177bdc6648585ea5d92cbc | What is the least integer value of \( k \) so that the equation \( x(k - x) = 4 \) has no real solution? | [
"-3"
] | [
"What is the least integer value of \\( k \\) so that the equation \\( x(k - x) = 4 \\) has no real solution?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-01401f9f1e2320b41b4506d6600d840e987f3626cdaf0794ac827c311e8f1ae2 | What is the least integer whose square is 48 more than its double? | [
"-6"
] | [
"What is the least integer whose square is 48 more than its double?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6155462e8b9fedee89f5aa0d5023a90de2d7566d34bd3725cbd034ea4f36ac80 | What is the least non-negative remainder when $22^7$ is divided by $123$? | [
"52"
] | [
"What is the least non-negative remainder when $22^7$ is divided by $123$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-cd744f596f5264ea16caf21577cffd5e246db1d4f575a523a039c64ddd09789b | What is the least nonzero positive integer that can be made by adding and subtracting the numbers 1027 and 1079? | [
"13"
] | [
"What is the least nonzero positive integer that can be made by adding and subtracting the numbers 1027 and 1079?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 21
} | 36 | 0 | 0 |
omr-ws-32ab2c6be9a32edb1c2fc85c6dbad5d220d206d4ae17d16e56d648e519bbe25a | What is the least number $N$ of 4-digit numbers compiled from the digits $1, 2, 3, 4, 5, 6, 7, 8$ you need to choose such that for any two different digits, both of these digits are in at least one but not more than two of the chosen $N$ numbers? | [
"6"
] | [
"What is the least number $N$ of 4-digit numbers compiled from the digits $1, 2, 3, 4, 5, 6, 7, 8$ you need to choose such that for any two different digits, both of these digits are in at least one but not more than two of the chosen $N$ numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-ad44d4cfcb925c5320b925477c98644574d5420cdbcec7183d7665f6aa1fb57e | What is the least number $N$ of 4-digit numbers compiled from the digits $1, 2, 3, 4, 5, 6, 7, 8$ you need to choose such that for any two different digits, both of these digits are in at least one of the chosen $N$ numbers? | [
"6"
] | [
"What is the least number $N$ of 4-digit numbers compiled from the digits $1, 2, 3, 4, 5, 6, 7, 8$ you need to choose such that for any two different digits, both of these digits are in at least one of the chosen $N$ numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 6
} | 21 | 0 | 0 |
omr-ws-7f04ccbad514ea25bd65c7f1f319fcaf511cfd9008c8c4c9b4dae0f5778ac53a | What is the least number \( x \) such that \( 2^x + 5^x \) is divisible by 13, and \( x \) is greater than 100? | [
"103"
] | [
"What is the least number \\( x \\) such that \\( 2^x + 5^x \\) is divisible by 13, and \\( x \\) is greater than 100?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 2
} | 14 | 0 | 0 |
omr-ws-b4fcd6c704acc94f38af866c54fa724fa6634d2ef753c56e8ce9dbabf5ee066c | What is the least number of bags Jake needs to separate 2021 balls into, such that he can obtain any number of balls by picking up some bags? | [
"11"
] | [
"What is the least number of bags Jake needs to separate 2021 balls into, such that he can obtain any number of balls by picking up some bags?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7948f5b1639c9d50b1be70d8de96b938f86673d8f60a3cb02937131d3abf3b7e | What is the least number of beads that can be arranged on a rectangular cardboard that is 8 cm long and 6 cm wide so that there are at least two beads that are less than √10 cm apart? | [
"17"
] | [
"What is the least number of beads that can be arranged on a rectangular cardboard that is 8 cm long and 6 cm wide so that there are at least two beads that are less than √10 cm apart?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ab6ada179757ff45fd7b80ad8e4908a9a1f4433d62b9fb75cffd55cddbac2c8d | What is the least number of cells Peter should mark on a 5 × 5 board to prevent Basil from covering all marked cells with three-cell corners, where the corners must be inside the board and not overlap? | [
"4"
] | [
"What is the least number of cells Peter should mark on a 5 × 5 board to prevent Basil from covering all marked cells with three-cell corners, where the corners must be inside the board and not overlap?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-9af8dfa478bd250d150723461706962a588c6ecf6022b2ded657c33f2feb02d6 | What is the least number of colors needed for pearl necklaces such that any two people have one pearl of the same color if and only if they are friends, for an island with \( n \) aborigenes where each pair is either friends or foes? | [
"\\(\\left\\lfloor\\frac{n^2}{4}\\right\\rfloor\\)"
] | [
"What is the least number of colors needed for pearl necklaces such that any two people have one pearl of the same color if and only if they are friends, for an island with \\( n \\) aborigenes where each pair is either friends or foes?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 30,
"genselect": 5,
"tir": 12
} | 47 | 0 | 0 |
omr-ws-68d0508c2fe33244093cd51938dbef0740ef7aeca522dcdddcb49aa851ed2c6a | What is the least number of colors needed to color the cells of an infinite table such that each p-cross (a figure formed by a unit square and 4 rectangles of dimensions $(p-1) \times 1$ on its sides) covers cells, no two of which are the same color? | [
"\\(4p - 3\\)"
] | [
"What is the least number of colors needed to color the cells of an infinite table such that each p-cross (a figure formed by a unit square and 4 rectangles of dimensions $(p-1) \\times 1$ on its sides) covers cells, no two of which are the same color?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 2
} | 14 | 0 | 0 |
omr-ws-51122648b5ff225d2e44d7ea237afdb6555c862f773012172c4884bac0682a25 | What is the least number of colors needed to color the sides of a unit square such that no two points which are of unit length apart have the same color? | [
"4"
] | [
"What is the least number of colors needed to color the sides of a unit square such that no two points which are of unit length apart have the same color?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 6
} | 14 | 0 | 0 |
omr-ws-9cd3eff9460842addc522862e2c18240637216e366ff6586987d8f616dcec0a9 | What is the least number of colors needed to paint 1989 equal circles placed on a table without overlap such that any two tangential circles are colored differently? | [
"4"
] | [
"What is the least number of colors needed to paint 1989 equal circles placed on a table without overlap such that any two tangential circles are colored differently?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-99304ef5dd061da7d49935685719e40627372b6c7becd3f1d73addeb65aa731a | What is the least number of colours required to colour the integers $1, 2, \ldots, 2^n - 1$ such that for any set of consecutive integers taken from this set, there will always be a colour that colours exactly one of them? That is, for all integers $i, j$ such that $1 \leq i \leq j \leq 2^n - 1$, there will be a colour... | [
"n"
] | [
"What is the least number of colours required to colour the integers $1, 2, \\ldots, 2^n - 1$ such that for any set of consecutive integers taken from this set, there will always be a colour that colours exactly one of them? That is, for all integers $i, j$ such that $1 \\leq i \\leq j \\leq 2^n - 1$, there will be... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-794ff76fe210f3e692230a528678c80bf678a7839adede2fd1db1013d1364a51 | What is the least number of concerts that need to be scheduled so that every pair of 12 musicians each plays for the other at least once? | [
"11"
] | [
"What is the least number of concerts that need to be scheduled so that every pair of 12 musicians each plays for the other at least once?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-be3580f1dad6805ec6d587dbaa5f0e05063423d42015111545ee05d45517bb9c | What is the least number of convex $5$-gons that must be attached to each other so that a convex $2011$-gon is obtained? | [
"670"
] | [
"What is the least number of convex $5$-gons that must be attached to each other so that a convex $2011$-gon is obtained?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 10,
"tir": 5
} | 15 | 0 | 0 |
omr-ws-fe33e15e604db849e0053333df7147c0c4f704df912e0f41d0af86efb3b0e6b4 | What is the least number of cuts for the rectangle $3 \times 5$ cut into $15$ unit squares, if after cutting the piece you can impose, turn over, but you can not bend? | [
"5"
] | [
"What is the least number of cuts for the rectangle $3 \\times 5$ cut into $15$ unit squares, if after cutting the piece you can impose, turn over, but you can not bend?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 5
} | 20 | 0 | 0 |
omr-ws-bc74a94bd0ee474e763ed8206db82e40775ea25056b9e72abb11ccb22e656226 | What is the least number of cuts that must be made to a square sheet of paper so that among the resulting parts there are exactly 100 decagons? | [
"1699"
] | [
"What is the least number of cuts that must be made to a square sheet of paper so that among the resulting parts there are exactly 100 decagons?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2c0ecfbc02a4c65f22e0a0bb938626bf58a4c73e43d056af2d857b1e350c9627 | What is the least number of different crayons needed to color the interiors of all the unit squares of an 8x8 checkerboard such that any two squares sharing a side or a vertex are colored with different crayons? | [
"4"
] | [
"What is the least number of different crayons needed to color the interiors of all the unit squares of an 8x8 checkerboard such that any two squares sharing a side or a vertex are colored with different crayons?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-6cebb96c76f4e647438d5bb79a667f29357258f1bbb31104674c2b73c88f16ba | What is the least number of disks of radius $\frac{1}{2}$ required to cover a disk of radius 1? | [
"7"
] | [
"What is the least number of disks of radius $\\frac{1}{2}$ required to cover a disk of radius 1?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-e0aa34405cd010e593110ef03e913944f378aee6949b957bf4e3a8740068070d | What is the least number of distinct elements in set $A$ such that the minimum number in $A$ is $1$, the maximum number is $100$, and every number (except $1$) in set $A$ can be represented as the sum of two elements from $A$ (not necessarily distinct)? | [
"9"
] | [
"What is the least number of distinct elements in set $A$ such that the minimum number in $A$ is $1$, the maximum number is $100$, and every number (except $1$) in set $A$ can be represented as the sum of two elements from $A$ (not necessarily distinct)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-907a44f19400fc94d7cd43ca149750db6340e87712baac44b891d6471d8368ed | What is the least number of distinct points we must put at the boundary or in the interior of a regular hexagon with side length $1$ to ensure that the area of a nondegenerate triangle formed by some three of these points is less than $\tfrac{\sqrt{3}}{16}$? | [
"20"
] | [
"What is the least number of distinct points we must put at the boundary or in the interior of a regular hexagon with side length $1$ to ensure that the area of a nondegenerate triangle formed by some three of these points is less than $\\tfrac{\\sqrt{3}}{16}$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c3c5884d28a7cf515ca14c863c7bbc62e4d2ee03b005bbc5bd15a2440db18211 | What is the least number of edges in a connected $n$-vertex graph such that any edge belongs to a triangle? | [
"\\( f(n) = \\begin{cases} \n3k - 1 & \\text{if } n = 2k \\\\\n3k & \\text{if } n = 2k + 1 \n\\end{cases} \\)"
] | [
"What is the least number of edges in a connected $n$-vertex graph such that any edge belongs to a triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 30,
"genselect": 8,
"tir": 16
} | 54 | 0 | 0 |
omr-ws-c308cd3523eea5b4dcd60ff1fa3b8b4544f6003fbdbeb05de6ca0f262a233103 | What is the least number of eggs in Brahmagupta's basket such that when the eggs are taken out 2 at a time, there is 1 egg left over; when taken out 3 at a time, there are 2 eggs left over; when taken out 4, 5, and 6 at a time, there are 3, 4, and 5 eggs left over, respectively; and when taken out 7 at a time, there ar... | [
"119"
] | [
"What is the least number of eggs in Brahmagupta's basket such that when the eggs are taken out 2 at a time, there is 1 egg left over; when taken out 3 at a time, there are 2 eggs left over; when taken out 4, 5, and 6 at a time, there are 3, 4, and 5 eggs left over, respectively; and when taken out 7 at a time, the... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-e3ca818e905ae7b661659df10927731a77fcbd07225ce131ebc3510f88cbdf1c | What is the least number of entries that must be altered in the array \[ \begin{bmatrix} 4 & 9 & 2 \\ 8 & 1 & 6 \\ 3 & 5 & 7 \end{bmatrix} \] to make all six row and column sums different from one another? | [
"4"
] | [
"What is the least number of entries that must be altered in the array\n\\[\n\\begin{bmatrix} 4 & 9 & 2 \\\\ 8 & 1 & 6 \\\\ 3 & 5 & 7 \\end{bmatrix}\n\\]\nto make all six row and column sums different from one another?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-7e88130d19b2e4bc6f187cfb0883359c36bb00b97bbf1456a1a7de1c24bea172 | What is the least number of frames in an $8 \times 8$ grid that must be removed so that the rest of the grid can be covered with rectangles of $3 \times 3$? | [
"10"
] | [
"What is the least number of frames in an $8 \\times 8$ grid that must be removed so that the rest of the grid can be covered with rectangles of $3 \\times 3$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 3
} | 11 | 0 | 0 |
omr-ws-93e17b3bc038feb0810b509c05c165e62ab27320e579d33101cf600ffdd15dc5 | What is the least number of lines needed to partition the plane into at least 2010 regions using three families of parallel lines? | [
"77"
] | [
"What is the least number of lines needed to partition the plane into at least 2010 regions using three families of parallel lines?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-c8568d4a0a407c11dc3893ef373c19f3f210b4230c2b7da7297fcb044ee1e4c1 | What is the least number of lines that must be drawn to determine if an invisible point lies inside a square (not outside and not on the boundary)? | [
"4"
] | [
"What is the least number of lines that must be drawn to determine if an invisible point lies inside a square (not outside and not on the boundary)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-9c14144608f85e566274b4cb56ab00121e28321f0f8e96969b4a6ddc9aeedeb0 | What is the least number of minutes after 8:00 that the lesser angle between the vertical and the hour hand equals the lesser angle between the vertical and the minute hand? | [
"\\dfrac{240}{13}"
] | [
"What is the least number of minutes after 8:00 that the lesser angle between the vertical and the hour hand equals the lesser angle between the vertical and the minute hand?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-64104cf4f469adfb07b7729e0232d0e1b71affc94e938b28567df932b704c209 | What is the least number of monominoes that can be placed on an $8 \times 8$ board such that a T-Pentomino can no longer be fitted onto the board? | [
"8"
] | [
"What is the least number of monominoes that can be placed on an $8 \\times 8$ board such that a T-Pentomino can no longer be fitted onto the board?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 2
} | 9 | 0 | 0 |
omr-ws-cf76faf4c01094cdcb8fcf075e2df6a7267a0456c5889ae98bdb4d16d5ecca3f | What is the least number of moves it takes a knight to get from one corner of an $n \times n$ chessboard, where $n \ge 4$, to the diagonally opposite corner? | [
"\\(2 \\left\\lfloor \\dfrac{n+1}{3} \\right\\rfloor\\)"
] | [
"What is the least number of moves it takes a knight to get from one corner of an $n \\times n$ chessboard, where $n \\ge 4$, to the diagonally opposite corner?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 3,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-7b026ff47d46a1497b83dacce2da0d0ba8f2a766485b86b92e0eb1a91becc1f1 | What is the least number of moves necessary to ensure that we have found the correct combination for a lock consisting of 6 rotating discs, each with digits 0 through 9, starting from the position 000000, given that none of the combinations 000000, 111111, 222222, ..., 999999 is correct? | [
"\\(10^6 - 1\\)"
] | [
"What is the least number of moves necessary to ensure that we have found the correct combination for a lock consisting of 6 rotating discs, each with digits 0 through 9, starting from the position 000000, given that none of the combinations 000000, 111111, 222222, ..., 999999 is correct?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ac453f2277adc43ae2143ae7614c5faf046980dc9456acc7e8dc28852b69e0ac | What is the least number of moves necessary to ensure that we have found the correct combination for a lock consisting of 6 rotating discs, each with digits 0 through 9, starting from the position 000000? | [
"\\(10^6 - 1\\)"
] | [
"What is the least number of moves necessary to ensure that we have found the correct combination for a lock consisting of 6 rotating discs, each with digits 0 through 9, starting from the position 000000?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-b716946ff0a18d01ff48ddaa4674cccc0c3bbabcf4e7ca4a1671c265f5cb3677 | What is the least number of moves required for a snickel to get from (0, 0) to (19, 0), where each move consists of 8 units horizontally or vertically followed by 3 units in a perpendicular direction? | [
"11"
] | [
"What is the least number of moves required for a snickel to get from (0, 0) to (19, 0), where each move consists of 8 units horizontally or vertically followed by 3 units in a perpendicular direction?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-4d1ffce70b6a0372fe04d42071f1d4ac4318264558d781105aea59a1c021bc2b | What is the least number of moves required to reverse the pile of 2008 coins initially on a white table, given that at each move, the uppermost coin on a table can be transferred to an empty table or to the top of the pile on a non-empty table? | [
"6022"
] | [
"What is the least number of moves required to reverse the pile of 2008 coins initially on a white table, given that at each move, the uppermost coin on a table can be transferred to an empty table or to the top of the pile on a non-empty table?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d880269eab4af964680537ac3071574da6de735266e76b087e5b4889d0df5675 | What is the least number of moves required to turn all 6 glasses upside down if in each move exactly 4 glasses are turned over? | [
"3"
] | [
"What is the least number of moves required to turn all 6 glasses upside down if in each move exactly 4 glasses are turned over?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-29651bbf7b10b10c8b3790f2314bafeed7611d0a77b6aea4e77ad671857034a9 | What is the least number of odd integers in a sum of 335 pairwise distinct positive integers that equals 100000? | [
"20"
] | [
"What is the least number of odd integers in a sum of 335 pairwise distinct positive integers that equals 100000?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 45,
"tir": 25
} | 70 | 0 | 0 |
omr-ws-b985344f8b5e191180af564cb347bc899d5926ad61c5ea63a5f0d3379702da55 | What is the least number of rooks that can be placed on a $9 \times 9$ chessboard with a standard checkered pattern (white squares at the four corners) so that all the white squares are attacked? (A rook also attacks the square it is on, in addition to every other square in the same row or column.) | [
"5"
] | [
"What is the least number of rooks that can be placed on a $9 \\times 9$ chessboard with a standard checkered pattern (white squares at the four corners) so that all the white squares are attacked? (A rook also attacks the square it is on, in addition to every other square in the same row or column.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 1
} | 21 | 0 | 0 |
omr-ws-bce663c6b6e8ed82303c0352d3c8b4689f4656ece64aee93cf89117bcf7cd375 | What is the least number of rooks that can be placed on a standard $8 \times 8$ chessboard so that all the white squares are attacked? (A rook also attacks the square it is on, in addition to every other square in the same row or column.) | [
"5"
] | [
"What is the least number of rooks that can be placed on a standard $8 \\times 8$ chessboard so that all the white squares are attacked? (A rook also attacks the square it is on, in addition to every other square in the same row or column.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d23db870241b4cd5807ebcbcd0eff1f815144ae7112babb1a86bb6105b6ccd7d | What is the least number of seats that a raft can have so that n monks and n cannibalists can cross a river without any monks being eaten, given that the solution must be possible in the general case for any n? | [
"4"
] | [
"What is the least number of seats that a raft can have so that n monks and n cannibalists can cross a river without any monks being eaten, given that the solution must be possible in the general case for any n?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-40afaf6af77260e5d74229a7f15922d4836491bf34d1876aca4916fd2e3cd8a5 | What is the least number of squares into which a rectangle with dimensions 18 by 19 can be cut? | [
"8"
] | [
"What is the least number of squares into which a rectangle with dimensions 18 by 19 can be cut?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-1b760042d84c4ff603a7aa391f45ce32da8d3c5f21aa938159625909c52b3cb2 | What is the least number of squares of any size needed to tile an $m$ by $n$ rectangle for all positive integers $m$ and $n$? | [
"\\text{Sum of the quotients in the Euclidean algorithm for } m \\text{ and } n"
] | [
"What is the least number of squares of any size needed to tile an $m$ by $n$ rectangle for all positive integers $m$ and $n$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-b2f68f0ade4cf1b73cc97f80fcfeeaaa8baa0442c32ac573f3f55a3960a21add | What is the least number of straight pieces that a given straight stick needs to be randomly broken into, so that there is just greater than a 50% chance of being able to form at least one triangular shape if three of the pieces are chosen among all potential 3-piece combinations? | [
"5"
] | [
"What is the least number of straight pieces that a given straight stick needs to be randomly broken into, so that there is just greater than a 50% chance of being able to form at least one triangular shape if three of the pieces are chosen among all potential 3-piece combinations?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-d305e2670d68e6361c2faea89f2ee847dbca698bef4ce75466f7a91ebc63da58 | What is the least number of students needed to guarantee that at least three of them write down two-digit positive integers whose sum of digits are the same? | [
"35"
] | [
"What is the least number of students needed to guarantee that at least three of them write down two-digit positive integers whose sum of digits are the same?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-19536740a91cfd62e74cf31fef545e588f71dd84b4b2fb3969be5a2f2089e61e | What is the least number of two-digit numbers needed to sum up to 2024? | [
"21"
] | [
"What is the least number of two-digit numbers needed to sum up to 2024?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-fb74d80b580ad74ed540727251223225f61dda9582b79ce479899b2390b7b7f5 | What is the least number of vertices that can be painted black in a regular 2016-gon so that there is no acute triangle with all vertices still white? | [
"1007"
] | [
"What is the least number of vertices that can be painted black in a regular 2016-gon so that there is no acute triangle with all vertices still white?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-44341627fe79fcf34352ef4de8a0ebc9ae2eb43da216c1309549816cb24b4e14 | What is the least number of weighings needed to determine the sum of weights of 13 watermelons such that exactly two watermelons are weighed in each weighing? | [
"8"
] | [
"What is the least number of weighings needed to determine the sum of weights of 13 watermelons such that exactly two watermelons are weighed in each weighing?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-8ef9e4396be2cc80355c706e8f26e3ad4cee9087d1b91e8cd48d65355c0635ce | What is the least number of weights in a set that can be divided into either 3, 4, or 5 piles of equal mass? | [
"9"
] | [
"What is the least number of weights in a set that can be divided into either 3, 4, or 5 piles of equal mass?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-9662a8d67ac17e695b5372e326e5f4d296888d457697cb86ed7fed27dd0a5583 | What is the least number of weights in a set that can be divided into either 4, 5, or 6 piles of equal mass? | [
"60"
] | [
"What is the least number of weights in a set that can be divided into either 4, 5, or 6 piles of equal mass?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 2
} | 9 | 0 | 0 |
omr-ws-fa99d74d172b763f7f5e164c73cd9831596bf19e2e8dcd0b88a9730092ab202a | What is the least number that can be expressed as the sum of two positive cubes in two different ways? | [
"1729"
] | [
"What is the least number that can be expressed as the sum of two positive cubes in two different ways?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-61cffe9fa476df3f50c8fb4272daf797d2e99c28c05a4db8047ceea73051ebf2 | What is the least odd prime divisor of \(2019^8 + 1\)? | [
"97"
] | [
"What is the least odd prime divisor of \\(2019^8 + 1\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-b7b1f01bcce6ac328b302331b9674a6704a30eff3489816925c0d4789b7976bc | What is the least odd prime factor of \(2022^{26} + 1\)? | [
"5"
] | [
"What is the least odd prime factor of \\(2022^{26} + 1\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 2
} | 23 | 0 | 0 |
omr-ws-734d56d82fba5065b2e0ce154caa09188ad60d4b606be5895f9382ca43f59650 | What is the least period of the function $\cos(\sin(x))$? | [
"\\(\\pi\\)"
] | [
"What is the least period of the function $\\cos(\\sin(x))$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-f7be6a0c2db682f6b30bf35b60302429712b6d95235ff4597fa955584febb611 | What is the least positive integer $N$ such that $100 \times 102 \times 104 \times 106 + N$ is a perfect square? | [
"16"
] | [
"What is the least positive integer $N$ such that $100 \\times 102 \\times 104 \\times 106 + N$ is a perfect square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 20
} | 43 | 0 | 0 |
omr-ws-d89031e55b00044fa8e082bdcf6b039f740bce43129620c76d86355b4af1db90 | What is the least positive integer $k$ such that $n+k \in S$ for every $n \in S$, where $S = \{n : n3^n + (2n+1)5^n \equiv 0 \pmod{7}\}$? | [
"21"
] | [
"What is the least positive integer $k$ such that $n+k \\in S$ for every $n \\in S$, where $S = \\{n : n3^n + (2n+1)5^n \\equiv 0 \\pmod{7}\\}$?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 40,
"genselect": 7,
"tir": 20
} | 67 | 0 | 0 |
omr-ws-e8cc554869be405d1a07d646e849fddf57f6a64a2cd26afb7562bbc5b3b3b430 | What is the least positive integer $k$ such that, in every convex 1001-gon, the sum of any $k$ diagonals is greater than or equal to the sum of the remaining diagonals? | [
"501"
] | [
"What is the least positive integer $k$ such that, in every convex 1001-gon, the sum of any $k$ diagonals is greater than or equal to the sum of the remaining diagonals?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-ca2ffd0d250715236dd4ed711a5e3fafc39cacffd3cd1707ff4e0643df7d23aa | What is the least positive integer $n$ such that $2023$ is equal to the sum of $n$ perfect squares, i.e., there exist $a_1, a_2, \ldots, a_n$ such that $2023 = a_1^2 + a_2^2 + \ldots + a_n^2$? | [
"4"
] | [
"What is the least positive integer $n$ such that $2023$ is equal to the sum of $n$ perfect squares, i.e., there exist $a_1, a_2, \\ldots, a_n$ such that $2023 = a_1^2 + a_2^2 + \\ldots + a_n^2$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-49b8b559eb6891ebe2c2d841ac11aab57cc94857e203a759cca55ac79a5c39ce | What is the least positive integer $n$ such that $25^n + 16^n$ leaves a remainder of $1$ when divided by $121$? | [
"32"
] | [
"What is the least positive integer $n$ such that $25^n + 16^n$ leaves a remainder of $1$ when divided by $121$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 16
} | 29 | 0 | 0 |
omr-ws-ac9d55e38fb3755b76aef113a1603b0c02c7f53fed549aaabcb26c82c4404efc | What is the least positive integer $n$ such that $4125$ is a factor of $n!$? | [
"15"
] | [
"What is the least positive integer $n$ such that $4125$ is a factor of $n!$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bbde80f5c96313026555b5cb7138da3d25cc2f01fbb337aa8b23908b22527df5 | What is the least positive integer $n$ such that $\overbrace{f(f(\dots f}^{21 \text{ times}}(n)))=2013$ where $f(x)=x+1+\lfloor \sqrt x \rfloor$? ($\lfloor a \rfloor$ denotes the greatest integer not exceeding the real number $a$.) | [
"1178"
] | [
"What is the least positive integer $n$ such that $\\overbrace{f(f(\\dots f}^{21 \\text{ times}}(n)))=2013$ where $f(x)=x+1+\\lfloor \\sqrt x \\rfloor$? ($\\lfloor a \\rfloor$ denotes the greatest integer not exceeding the real number $a$.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 28
} | 41 | 0 | 0 |
omr-ws-5eb6fd22676806799e1c6253c48c175719ed935b83eb1e026b76ef02121e2c63 | What is the least positive integer $n$ such that $n^2 - n$ is divisible by some but not all integer values of $k$ when $1\leq k \leq n$? | [
"5"
] | [
"What is the least positive integer $n$ such that $n^2 - n$ is divisible by some but not all integer values of $k$ when $1\\leq k \\leq n$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d65e1bf51d090519e415f84822919c6765b052c54d75bd994c1333cedd7442a0 | What is the least positive integer \( m \) such that \( m \cdot 2! \cdot 3! \cdot 4! \cdot 5! \cdots 16! \) is a perfect square? | [
"70"
] | [
"What is the least positive integer \\( m \\) such that \\( m \\cdot 2! \\cdot 3! \\cdot 4! \\cdot 5! \\cdots 16! \\) is a perfect square?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-209470b1680dc81a7cb03a1b434327166f389d88948daf601727884c59794df5 | What is the least positive integer \( m \) such that given \( m \) integers between 1 and 2023, inclusive, there must exist two of them \( a \) and \( b \) such that \( 1 < \frac{a}{b} \leq 2 \)? | [
"12"
] | [
"What is the least positive integer \\( m \\) such that given \\( m \\) integers between 1 and 2023, inclusive, there must exist two of them \\( a \\) and \\( b \\) such that \\( 1 < \\frac{a}{b} \\leq 2 \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 2,
"tir": 1
} | 8 | 0 | 0 |
omr-ws-ad467357f091384e560912ce228a140bef47e2a2b05621166fd9f0ab21bcc818 | What is the least positive integer \( n \) such that \( n^2 - n \) is divisible by some but not all integer values of \( k \) when \( 1 \leq k \leq n \)? | [
"5"
] | [
"What is the least positive integer \\( n \\) such that \\( n^2 - n \\) is divisible by some but not all integer values of \\( k \\) when \\( 1 \\leq k \\leq n \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-23ac3c3bdde6d318b1c2055386cca2e78c88ccc3cb9bb52393035892fbfdaeb3 | What is the least positive integer \( n \) such that \(\dfrac{n}{2024}\) is a perfect square, \(\dfrac{n}{2208}\) is a perfect cube, and \(\dfrac{n}{2760}\) is a perfect fifth power? | [
"\\( 2^{23} \\times 3^{16} \\times 5^6 \\times 11^{15} \\times 23 \\)"
] | [
"What is the least positive integer \\( n \\) such that \\(\\dfrac{n}{2024}\\) is a perfect square, \\(\\dfrac{n}{2208}\\) is a perfect cube, and \\(\\dfrac{n}{2760}\\) is a perfect fifth power?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 26
} | 48 | 0 | 0 |
omr-ws-2fd8e740d08c5c1eb9afd1e8ce75a097d14b07789cf50356c42aafd1a09d0d62 | What is the least positive integer \( x \) which has 2016 divisors? | [
"4655851200"
] | [
"What is the least positive integer \\( x \\) which has 2016 divisors?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 6
} | 8 | 0 | 0 |
omr-ws-bdec50e20a960d621d1138916cdcfa6e70a1e283744a9cdbcea0ee26a0d97bba | What is the least positive integer divisible by each of the first eight positive integers? | [
"840"
] | [
"What is the least positive integer divisible by each of the first eight positive integers?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5fd834a30d733c02a03c176cffbb3e792ba7c9a456d38f87fb1778cd7e2a4c55 | What is the least positive integer greater than 1 that leaves a remainder of 1 when divided by each of 2, 3, 4, 5, 6, 7, 8 and 9? | [
"2521"
] | [
"What is the least positive integer greater than 1 that leaves a remainder of 1 when divided by each of 2, 3, 4, 5, 6, 7, 8 and 9?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a28c8cf3a2660fbb164592a570081b71e694d3a22c0c7373d075d7cebe7c1737 | What is the least positive integer that can be added to 412 to yield a multiple of 3? | [
"2"
] | [
"What is the least positive integer that can be added to 412 to yield a multiple of 3?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a957994104809d9a29349731191c0fa41e398def7e5206b04c786d1ebfc05c26 | What is the least positive integer that can be expressed as a sum of 3 or more consecutive positive integers in at least 3 distinct ways? | [
"30"
] | [
"What is the least positive integer that can be expressed as a sum of 3 or more consecutive positive integers in at least 3 distinct ways?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 12
} | 44 | 0 | 0 |
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