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-65f5c0b8ddd88896d435c095faca90219ece2313f0efdc7b2232f588a53ad29d | What is the smallest $n \in \mathbb{N}$ such that for all $a, b, c \in \mathbb{C}$ and for all $p \in \{1, 2, \ldots, n\}$, if $a^p + b^p + c^p = 0$, then $a = b = c = 0$? | [
"3"
] | [
"What is the smallest $n \\in \\mathbb{N}$ such that for all $a, b, c \\in \\mathbb{C}$ and for all $p \\in \\{1, 2, \\ldots, n\\}$, if $a^p + b^p + c^p = 0$, then $a = b = c = 0$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-01b164b8fcaa30494e07f4459d3588e57b84cdd35c16901b99c73e3493ff3e69 | What is the smallest $n$ greater than $1$ such that there exists a palindromic $n$-string, where an $n$-string is defined as a string of digits formed by writing the numbers $1, 2, \ldots, n$ in some order? | [
"19"
] | [
"What is the smallest $n$ greater than $1$ such that there exists a palindromic $n$-string, where an $n$-string is defined as a string of digits formed by writing the numbers $1, 2, \\ldots, n$ in some order?"
] | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 5
} | 5 | 0 | 0 |
omr-ws-181c81ed9449ee8fe2649a153d4e9baaa3b730e2aa0979f9a4f8d31eba7fad47 | What is the smallest $n$ greater than $1$ such that there exists a palindromic $n-$string, where an $n-$string is defined as a string of digits formed by writing the numbers $1, 2, \ldots, n$ in some order (in base 10)? | [
"19"
] | [
"What is the smallest $n$ greater than $1$ such that there exists a palindromic $n-$string, where an $n-$string is defined as a string of digits formed by writing the numbers $1, 2, \\ldots, n$ in some order (in base 10)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 6
} | 6 | 0 | 0 |
omr-ws-2f7ba6efdfa39b3160a1217e44230564ce10c316273d9876bd05885da87d539e | What is the smallest $n$ such that $1$ can be written as the sum of $n$ special numbers, where a positive real number is called special if its decimal representation consists entirely of the digits $0$ and $7$? | [
"8"
] | [
"What is the smallest $n$ such that $1$ can be written as the sum of $n$ special numbers, where a positive real number is called special if its decimal representation consists entirely of the digits $0$ and $7$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c10af8ae4c5dcef6b73ef284714752bb226ab5105b5946cb1e238117647d7597 | What is the smallest $n$ such that $999999n = 111...111$? | [
"5"
] | [
"What is the smallest $n$ such that $999999n = 111...111$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b13953b575b368e0e10d5ef1b85630c4b5f3222accfac2463cd9d9488c8389db | What is the smallest $n$ such that an $n \times n$ square can be divided into both $40 \times 40$ and $49 \times 49$ squares, with both types of squares present? | [
"3920"
] | [
"What is the smallest $n$ such that an $n \\times n$ square can be divided into both $40 \\times 40$ and $49 \\times 49$ squares, with both types of squares present?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-c76a217fe4dfca35e2e996831d7e5023fd18aecef7445243a05f56e5cf3283a6 | What is the smallest $n$ such that we can pick nine distinct positive integers from $1$ to $n$ and place them on a 3x3 board so that the product of all rows and columns is the same? | [
"15"
] | [
"What is the smallest $n$ such that we can pick nine distinct positive integers from $1$ to $n$ and place them on a 3x3 board so that the product of all rows and columns is the same?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 8,
"tir": 2
} | 11 | 0 | 0 |
omr-ws-760c77e88d2cafe5be159dd8c81d4839f32056047017e0121dd41b7858f06f60 | What is the smallest $n>0$ for which $n^2-79n+1601$ is not prime? | [
"80"
] | [
"What is the smallest $n>0$ for which $n^2-79n+1601$ is not prime?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-65b0942bb146f71abc62b9ffd4c3ebcfa66cf7a4d904a48e9ff7e84693dd2f78 | What is the smallest $n>0$ for which $n^2-n+41$ is not prime? | [
"41"
] | [
"What is the smallest $n>0$ for which $n^2-n+41$ is not prime?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-5a61914186835d13c560dd83272df3b9349f308cc214e2558d643cbe03375af7 | What is the smallest (greater than one) number of acute triangles into which a triangle can be cut? | [
"7"
] | [
"What is the smallest (greater than one) number of acute triangles into which a triangle can be cut?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-13845cfd3791690ee6375839790e130c4bf9d68e129c0395e8f4a41d0296f723 | What is the smallest (greatest) number of cells that can be needed to draw a figure with a perimeter of $10$? | [
"4"
] | [
"What is the smallest (greatest) number of cells that can be needed to draw a figure with a perimeter of $10$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-1461d0f3a9b264fbe9d6c51b186d55d3bd22d20ac2313ca2dafcbfef29d49979 | What is the smallest (minimum) value of $n$ such that $(n+1)!$ has four more digits than $n!$? | [
"1042"
] | [
"What is the smallest (minimum) value of $n$ such that $(n+1)!$ has four more digits than $n!$?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 20
} | 20 | 0 | 0 |
omr-ws-8fd8ab94a4530414a9f2ea65a1abb5376371c0401bcee8c84e5b4707c0d9598c | What is the smallest 6-digit number (without repeated digits) that can be divisible by 12 and 16? | [
"102384"
] | [
"What is the smallest 6-digit number (without repeated digits) that can be divisible by 12 and 16?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 26
} | 48 | 0 | 0 |
omr-ws-0c9cf55bcde09c84fc7a45f54424a123ea5291a201899e53f8d536d133a786a5 | What is the smallest \( n > 1 \) for which the average of the first \( n \) (non-zero) squares is a square? | [
"337"
] | [
"What is the smallest \\( n > 1 \\) for which the average of the first \\( n \\) (non-zero) squares is a square?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-b861a5b785f3bc996fecc7ce28985b76b7fdc2ce88520e0a91a67b5556178c35 | What is the smallest \( n > 5 \) such that the angle trisectors of a regular \( n \)-gon intersect other vertices of the \( n \)-gon? | [
"8"
] | [
"What is the smallest \\( n > 5 \\) such that the angle trisectors of a regular \\( n \\)-gon intersect other vertices of the \\( n \\)-gon?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-7fa9993e944a18c9f125dd166240a5956ea47f56ecff47fcd2eaf7abe0afd9fd | What is the smallest \( n \) such that 1 can be written as a sum of \( n \) special numbers, where a special number has a decimal representation consisting entirely of the digits 0 and 7? | [
"8"
] | [
"What is the smallest \\( n \\) such that 1 can be written as a sum of \\( n \\) special numbers, where a special number has a decimal representation consisting entirely of the digits 0 and 7?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-814f04970ea1b2854d8831cdbf068ad65fcb8803c0d04fe8e7116e80522abd24 | What is the smallest \( n \) such that 1 can be written as the sum of \( n \) special numbers, where a special number has a decimal representation consisting entirely of the digits 0 and 7? | [
"8"
] | [
"What is the smallest \\( n \\) such that 1 can be written as the sum of \\( n \\) special numbers, where a special number has a decimal representation consisting entirely of the digits 0 and 7?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-6c719919bf035849c2801c7de13b5da18e073d683e4dc6521e4997b8f390fd01 | What is the smallest \( n \) such that \( 1 \) can be written as a sum of \( n \) special numbers, where a special number has a decimal representation consisting entirely of the digits \( 0 \) and \( 7 \)? | [
"8"
] | [
"What is the smallest \\( n \\) such that \\( 1 \\) can be written as a sum of \\( n \\) special numbers, where a special number has a decimal representation consisting entirely of the digits \\( 0 \\) and \\( 7 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-ee3ab7248ab86f8230c85b731314e7835ef7faefebb59a1553b064a1a4e22d07 | What is the smallest amount of money that cannot be paid using only 6 dollar bills and 11 dollar bills? | [
"1"
] | [
"What is the smallest amount of money that cannot be paid using only 6 dollar bills and 11 dollar bills?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-7476f078edd618f582e25066b405b43aecdfe5f12eadd9454fedb0950d902f48 | What is the smallest amount of ribbon that should be used to decorate a 5 m high cylindrical pole with a diameter of 15 cm, if 50 cm of ribbon is used for each turn around the pole? | [
"\\( 10 \\times \\sqrt{2500 + 225\\pi^2} \\)"
] | [
"What is the smallest amount of ribbon that should be used to decorate a 5 m high cylindrical pole with a diameter of 15 cm, if 50 cm of ribbon is used for each turn around the pole?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7c26536b44f5457f26b387301d49896c554732d4ca1e99a553373f25752962b8 | What is the smallest amount of time in which it is possible to get $n$ adult worms starting with one adult worm, given that an adult worm can be cut into two parts which each grow at a speed of one meter per hour until they reach one meter in length? | [
"\\( 1 - \\frac{1}{2^{n-1}} \\)"
] | [
"What is the smallest amount of time in which it is possible to get $n$ adult worms starting with one adult worm, given that an adult worm can be cut into two parts which each grow at a speed of one meter per hour until they reach one meter in length?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-77062dddd5a7b2ba1831557bc531f8ad4e008759550fb84fa396349c7482366a | What is the smallest amount that can be paid using coins with values 33 and 60? | [
"3"
] | [
"What is the smallest amount that can be paid using coins with values 33 and 60?"
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-71e6fc65b9e6cb8007b6f3ccdb6105a4dced239aa3a52abe52c2b15385775365 | What is the smallest amount that may be invested at interest rate $i$, compounded annually, in order to withdraw $1$ dollar at the end of the first year, $4$ dollars at the end of the second year, $\ldots$, $n^2$ dollars at the end of the $n$-th year, in perpetuity? | [
"\\dfrac{(1 + i)(2 + i)}{i^3}"
] | [
"What is the smallest amount that may be invested at interest rate $i$, compounded annually, in order to withdraw $1$ dollar at the end of the first year, $4$ dollars at the end of the second year, $\\ldots$, $n^2$ dollars at the end of the $n$-th year, in perpetuity?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 3
} | 18 | 0 | 0 |
omr-ws-54ef2d658f3f0c4d276d885e1585863a1191b47ff27493c5a387e2be6d1932bf | What is the smallest angle between the tangent lines to the functions \( y = (x+1)^2 \) and \( y = -3 \ln \left( \frac{x}{e^x} \right) \)? | [
"0"
] | [
"What is the smallest angle between the tangent lines to the functions \\( y = (x+1)^2 \\) and \\( y = -3 \\ln \\left( \\frac{x}{e^x} \\right) \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-624bebf9f4ec45f287a479af6192d324a4c2a510c92c90ea073ffa1795f06ab2 | What is the smallest area of a triangle where one side is the space diagonal of a cube with side length $a$, and the other two sides lie on the surface of the cube? | [
"\\(\\frac{a^2\\sqrt{6}}{4}\\)"
] | [
"What is the smallest area of a triangle where one side is the space diagonal of a cube with side length $a$, and the other two sides lie on the surface of the cube?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-0de2b6254759eb8ef6ecc60b245f84980055aabb76389ee0ac049a9c47cb99e3 | What is the smallest area that a rectangle can have that wraps around a regular tetrahedron with edge $1$? | [
"\\sqrt{3}"
] | [
"What is the smallest area that a rectangle can have that wraps around a regular tetrahedron with edge $1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 4
} | 14 | 0 | 0 |
omr-ws-91bf3e8de083a0ab664308209cf48d7908613634b8bad024541fc5409a8eb94f | What is the smallest area that a triangle can have if the lengths of all sides are integers? | [
"6"
] | [
"What is the smallest area that a triangle can have if the lengths of all sides are integers?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-34b5978dbe52dffe5cf77d7961307f1e80baea0a0cd581357cbb16a556ea545e | What is the smallest base $b$ for which the fraction $\frac{5445469}{5445468}$ (numbers in base 10) has a finite number of digits after the decimal point when expressed in base $b$? | [
"42"
] | [
"What is the smallest base $b$ for which the fraction $\\frac{5445469}{5445468}$ (numbers in base 10) has a finite number of digits after the decimal point when expressed in base $b$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-614a37b49da74e01108ec150b707c6897bf671bdd2a86eadf2975464dd0a74bd | What is the smallest base-10 integer that can be represented as $12_a$ in one base and $21_b$ in another base, where $a$ and $b$ are any bases larger than 2? | [
"7"
] | [
"What is the smallest base-10 integer that can be represented as $12_a$ in one base and $21_b$ in another base, where $a$ and $b$ are any bases larger than 2?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2cd7a22b2324fca0fee9d2c1f721a494f216654a847e970e16908caae28a23b5 | What is the smallest base-10 integer that can be represented as $AA_5$ and $BB_7$, where $A$ and $B$ are valid digits in their respective bases? | [
"24_{10}"
] | [
"What is the smallest base-10 integer that can be represented as $AA_5$ and $BB_7$, where $A$ and $B$ are valid digits in their respective bases?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-319eba72b550984628078c688c0bf43d3ba8906dee29a986efa47fd29ec44851 | What is the smallest circle that can enclose a set of points such that the maximum distance between any two points is \( n \)? | [
"\\dfrac{n}{2}"
] | [
"What is the smallest circle that can enclose a set of points such that the maximum distance between any two points is \\( n \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-fa51a8713ac7bd3ec16be72478c8191726453967b5aab5b9b8dafca8ac7df380 | What is the smallest combined number of folds and cuts needed to divide a $1 \times n$ rectangle into $n$ $1 \times 1$ squares? | [
"\\(n\\)"
] | [
"What is the smallest combined number of folds and cuts needed to divide a $1 \\times n$ rectangle into $n$ $1 \\times 1$ squares?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9baa67dc25806c2521dd657213e3f39c688d9f9a0952baceefb4a6fb5634e474 | What is the smallest composite number \( n \) such that there is a unique group of order \( n \)? | [
"15"
] | [
"What is the smallest composite number \\( n \\) such that there is a unique group of order \\( n \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-17ca28eb2da11187699c6740587261659c1f05ba5bf45504c309633353aa8b4b | What is the smallest composite value of \( n^2 - n - 1 \) if \( n \) is a prime number? | [
"155"
] | [
"What is the smallest composite value of \\( n^2 - n - 1 \\) if \\( n \\) is a prime number?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-a76a38f5fdec5175ad9b260315fc8713df1f04f067371645fdc0d651f6b7f3c7 | What is the smallest constant replacing $27$ in the bound for the cardinality of any finite subgroup of $GL_n(\mathbb{Z})$ that can be obtained using elementary arguments? | [
"48"
] | [
"What is the smallest constant replacing $27$ in the bound for the cardinality of any finite subgroup of $GL_n(\\mathbb{Z})$ that can be obtained using elementary arguments?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 3
} | 18 | 0 | 0 |
omr-ws-6a6334dc86f35d175613ff5cee66756f2a9d0387f213efe53f3dae3d3c1996b5 | What is the smallest cube that can contain nine open and pairwise disjoint spheres of radius 1? | [
"\\(2 + 4 \\cdot \\frac{\\sqrt{3}}{3}\\)"
] | [
"What is the smallest cube that can contain nine open and pairwise disjoint spheres of radius 1?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 23,
"genselect": 8,
"tir": 16
} | 47 | 0 | 0 |
omr-ws-db5e22946f70f4bffb3ed5dae8b2571d1a26ceef8dbe17c73e59f9509b8f4128 | What is the smallest degree of a monic polynomial \( p(x) \) with integer coefficients such that \( p(n) \) is a multiple of 2021 for every positive integer \( n \)? | [
"47"
] | [
"What is the smallest degree of a monic polynomial \\( p(x) \\) with integer coefficients such that \\( p(n) \\) is a multiple of 2021 for every positive integer \\( n \\)?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-b8688e37c0cdd1a6e9ba5de0d9af32bb062bc47ca2df530c4e5df98bd9e6fb1e | What is the smallest degree of a polynomial with real coefficients that has the roots $2w$, $2+3w$, $2w^2$, $-1-3w$, and $2-w-w^2$, where $w \neq 1$ is a cube root of unity? | [
"5"
] | [
"What is the smallest degree of a polynomial with real coefficients that has the roots $2w$, $2+3w$, $2w^2$, $-1-3w$, and $2-w-w^2$, where $w \\neq 1$ is a cube root of unity?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-b2f67c1190b5894a9a19ac815e94468e3ba5a4a46112faa3a352a02ab5a9e59c | What is the smallest degree possible for a polynomial with integer coefficients that has $5/2$ and $3 + 2i$ as roots? | [
"3"
] | [
"What is the smallest degree possible for a polynomial with integer coefficients that has $5/2$ and $3 + 2i$ as roots?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-6db6a45d40a56d1e6c07091c89d0ad3be2ee1fb7b62f2fa95e4802097de4212c | What is the smallest digit that is never found in the units place of an even number? | [
"1"
] | [
"What is the smallest digit that is never found in the units place of an even number?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-516028601dbee70773b7bc7b70c7ff9490fa1f4d3ea766cb4efd4e081a53ae13 | What is the smallest distance a beetle must crawl to lick all the honey on the edges of a cube with edge length $1$, if the beetle starts from a vertex and can only crawl along the edges? | [
"15"
] | [
"What is the smallest distance a beetle must crawl to lick all the honey on the edges of a cube with edge length $1$, if the beetle starts from a vertex and can only crawl along the edges?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 8,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-c0e362146ed51ad9166d2de65a773bca032d7b18b3cc27594a6f5a700efdfc95 | What is the smallest downhill number greater than 86432? | [
"86510"
] | [
"What is the smallest downhill number greater than 86432?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 8,
"tir": 21
} | 33 | 0 | 0 |
omr-ws-90a0d72cfcffacf30a7a615a1fe8355ec3227ff27d1081260503de3e67d82957 | What is the smallest factorial that ends in exactly 200 zeros? | [
"805"
] | [
"What is the smallest factorial that ends in exactly 200 zeros?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-d978e65cabdf9631ca713f126fdf59fd467a55f74015f10ec92c7b0a02e54da2 | What is the smallest four-digit number that has exactly 14 positive divisors (including 1 and itself), such that the units digit of one of its prime divisors is 3? | [
"1458"
] | [
"What is the smallest four-digit number that has exactly 14 positive divisors (including 1 and itself), such that the units digit of one of its prime divisors is 3?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-9358b00ca01b4b61b1ff9ffc8959a5334b1b18f8698f5ef1a677d8ad12164597 | What is the smallest four-digit number that is divisible by $33$? | [
"1023"
] | [
"What is the smallest four-digit number that is divisible by $33$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-74fda9065ae637974cd2490663689a68371d9807628136aa22c5757860294619 | What is the smallest four-digit number that is the product of prime numbers \(A\), \(BC\), and \(DE\) where \(A\), \(B\), \(C\), \(D\), and \(E\) are distinct digits? | [
"1222"
] | [
"What is the smallest four-digit number that is the product of prime numbers \\(A\\), \\(BC\\), and \\(DE\\) where \\(A\\), \\(B\\), \\(C\\), \\(D\\), and \\(E\\) are distinct digits?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-2cd5cd4e5e2c2e74d7c398db9f14f44ff75026042140b2ad224c086b6076a4ce | What is the smallest four-digit palindrome that is divisible by 4? (A palindrome is a number that reads the same forwards and backwards, like 61216.) | [
"2112"
] | [
"What is the smallest four-digit palindrome that is divisible by 4? (A palindrome is a number that reads the same forwards and backwards, like 61216.)"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1c527bbc7eea084fcaa8e91a75309940b076a39d0c52c7b1f5fe5b1b6d8394e4 | What is the smallest four-digit positive integer that is divisible by 47? | [
"1034"
] | [
"What is the smallest four-digit positive integer that is divisible by 47?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b73441a36b6cdd8212c7166dd3f2eeacaa49468a5d2a73c91ae80ac9b49fd518 | What is the smallest ideal in $M_{2}(\mathbb{Z})$ containing the matrix $\left[\begin{array}{cc}1 & 0\\ 0 & 0\end{array}\right]$? | [
"\\( M_{2}(\\mathbb{Z}) \\)"
] | [
"What is the smallest ideal in $M_{2}(\\mathbb{Z})$ containing the matrix $\\left[\\begin{array}{cc}1 & 0\\\\ 0 & 0\\end{array}\\right]$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-d435874c92ad0f838c1a1d10bbcd48c0d56c832d27b01297fb6e3ebf94f47a10 | What is the smallest integer $N$ satisfying $(m+\frac{1}{2021})^2 < N < (m+\frac{1}{2020})^2$ for some positive integer $m$? | [
"1020101"
] | [
"What is the smallest integer $N$ satisfying $(m+\\frac{1}{2021})^2 < N < (m+\\frac{1}{2020})^2$ for some positive integer $m$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 8,
"tir": 12
} | 32 | 0 | 0 |
omr-ws-923750f11989607b8395e95a5aea21a902930ddf20753795aba5cf4944947d69 | What is the smallest integer $b > 3$ for which the base $b$ number $23_b$ is a perfect square? | [
"11"
] | [
"What is the smallest integer $b > 3$ for which the base $b$ number $23_b$ is a perfect square?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f1bd627c4127fb06cd36b76a1c1aa63ba2e7474c60e9e8df0c47b46f02b5786b | What is the smallest integer $k$ such that $k>1$ and $k$ has remainder $1$ when divided by any of $17,$ $6,$ and $2?$ | [
"103"
] | [
"What is the smallest integer $k$ such that $k>1$ and $k$ has remainder $1$ when divided by any of $17,$ $6,$ and $2?$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cd4b68e0ca1db3b09cfc6abe3608bb2557c60d94ed63b87ae840ea8e21c782fc | What is the smallest integer $m$ such that $m^2 + 7m + 89$ is a multiple of $77$? | [
"18"
] | [
"What is the smallest integer $m$ such that $m^2 + 7m + 89$ is a multiple of $77$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 4
} | 27 | 0 | 0 |
omr-ws-3586bfb6295ed620f9d8c6c91777b9fb9998ab41def356c94810f3928a3945e6 | What is the smallest integer $n > 1$ such that $p \mid (n^{p-1} - 1)$ for all integers $2 \leq p \leq 10$? | [
"361"
] | [
"What is the smallest integer $n > 1$ such that $p \\mid (n^{p-1} - 1)$ for all integers $2 \\leq p \\leq 10$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-7cbc2645b1032520209cec5c9b9bea8e23aba56fbfa6bd54c19b8402c6bc041a | What is the smallest integer $n$ for which $\frac{n-6}{5n+17}$ is non-zero and reducible? | [
"53"
] | [
"What is the smallest integer $n$ for which $\\frac{n-6}{5n+17}$ is non-zero and reducible?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 1
} | 8 | 0 | 0 |
omr-ws-8e3b4da6578f5f50c84aa9d5c43c09ab55f1632e89c13bfc805908f1eb6f851c | What is the smallest integer $n$ for which $n!$ terminates in exactly 66 zeros? | [
"270"
] | [
"What is the smallest integer $n$ for which $n!$ terminates in exactly 66 zeros?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-8a6239c6f587ee403735f411fb1b5addb3ec71a3bb122536f25faa43527238fc | What is the smallest integer $n$ for which any subset of $\{1,2,3,\ldots,20\}$ of size $n$ must contain two numbers that differ by $8$? | [
"13"
] | [
"What is the smallest integer $n$ for which any subset of $\\{1,2,3,\\ldots,20\\}$ of size $n$ must contain two numbers that differ by $8$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-5ec6be697c1c72c8aa6352ba5cf79b6aab7ef142fec5cf84066b8deb7d3467cc | What is the smallest integer $n$ such that the number of $n$-digit palindromes that are divisible by $4$ is at least $2014$? | [
"9"
] | [
"What is the smallest integer $n$ such that the number of $n$-digit palindromes that are divisible by $4$ is at least $2014$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-8010825e079001c598743f91182e9cef4c54ca5a1eac2570e69d35d94b13f77c | What is the smallest integer $n$, greater than $1$, such that $n^{-1}\pmod{1050}$ is defined? | [
"11"
] | [
"What is the smallest integer $n$, greater than $1$, such that $n^{-1}\\pmod{1050}$ is defined?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a4dfd4dc3ea8300b9145c1aa06a8a07a304ece6079193ae20d36efa7ed3dfc05 | What is the smallest integer $x$ for which $x<2x-7$ ? | [
"8"
] | [
"What is the smallest integer $x$ for which $x<2x-7$ ?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cd5aff1154943cea564e8b5880e8caea78c882c9260d129d7b254a257c4aa7b3 | What is the smallest integer $x$ larger than $1$ such that $x^2$ ends in the same three digits as $x$ does? | [
"376"
] | [
"What is the smallest integer $x$ larger than $1$ such that $x^2$ ends in the same three digits as $x$ does?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-565b8fecda4fbaee064f577a8dc6298d0edc4928dc059f969088d26c6b9a05f4 | What is the smallest integer $x$ that is greater than 1 million and for which the sum of $2^x$ and $3^x$ is divisible by $7$? | [
"1000005"
] | [
"What is the smallest integer $x$ that is greater than 1 million and for which the sum of $2^x$ and $3^x$ is divisible by $7$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-8d637f6e83da4048849570e381e1c686005627a5e4689184e7e2cee3d6019712 | What is the smallest integer \( a > 1 \) such that there exists a positive integer \( n \) for which \( a^{2^n} \equiv 1 \pmod{2015} \)? | [
"92"
] | [
"What is the smallest integer \\( a > 1 \\) such that there exists a positive integer \\( n \\) for which \\( a^{2^n} \\equiv 1 \\pmod{2015} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 7,
"tir": 11
} | 23 | 0 | 0 |
omr-ws-9171e5d20648547ced67e3082761c1efebb757f0ce42349079ffc232179d47e9 | What is the smallest integer \( n > 0 \) such that for any integer \( m \) in the range \( 1, 2, 3, \ldots, 1992 \), we can always find an integral multiple of \( \frac{1}{n} \) in the open interval \( \left( \frac{m}{1993}, \frac{m+1}{1994} \right) \)? | [
"3987"
] | [
"What is the smallest integer \\( n > 0 \\) such that for any integer \\( m \\) in the range \\( 1, 2, 3, \\ldots, 1992 \\), we can always find an integral multiple of \\( \\frac{1}{n} \\) in the open interval \\( \\left( \\frac{m}{1993}, \\frac{m+1}{1994} \\right) \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 8,
"tir": 10
} | 29 | 0 | 0 |
omr-ws-8ca5b0529f1510fd604c631ccba7a0901b2cc3cd787edfd030f56273c22fdeff | What is the smallest integer \( n > 1 \) for which the average of the first \( n \) squares is a perfect square? | [
"337"
] | [
"What is the smallest integer \\( n > 1 \\) for which the average of the first \\( n \\) squares is a perfect square?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-fd7daa35b11d4352f8d8741f23bcc498254bbd2aa408d1c1aba1c3621e2b8260 | What is the smallest integer \( n \) such that 1 can be written as a sum of \( n \) special numbers, where a special number has a decimal representation consisting entirely of the digits 0 and 7? | [
"8"
] | [
"What is the smallest integer \\( n \\) such that 1 can be written as a sum of \\( n \\) special numbers, where a special number has a decimal representation consisting entirely of the digits 0 and 7?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-736778c20004951eeb1f82ac0f02f1f767336d39c751fa616048bec645febc58 | What is the smallest integer \( n \) such that \( G \) is a non-abelian group of order \( n \) and \( \mathrm{Aut}(G) \) is abelian? | [
"64"
] | [
"What is the smallest integer \\( n \\) such that \\( G \\) is a non-abelian group of order \\( n \\) and \\( \\mathrm{Aut}(G) \\) is abelian?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 4
} | 5 | 0 | 0 |
omr-ws-6bc5b56f0edcf3f6d4ddcdcbbfbb16e455e0ae755d01c271d423e43a270486dc | What is the smallest integer \( n \) such that \( \frac{n}{2} \) is a square of an integer, \( \frac{n}{3} \) is a cube of an integer, \( \frac{n}{5} \) is a fifth power of an integer, and \( \frac{n}{11} \) is an eleventh power of an integer? | [
"\\( 2^{165} \\cdot 3^{330} \\cdot 5^{66} \\cdot 11^{30} \\)"
] | [
"What is the smallest integer \\( n \\) such that \\( \\frac{n}{2} \\) is a square of an integer, \\( \\frac{n}{3} \\) is a cube of an integer, \\( \\frac{n}{5} \\) is a fifth power of an integer, and \\( \\frac{n}{11} \\) is an eleventh power of an integer?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f469fa695a16a7298687f1eef2f231808a9d594ad437cec160bdbe5d2f7dfb0b | What is the smallest integer \( n \) such that \( n! \) is divisible by \( 2009^{29} \)? | [
"1189"
] | [
"What is the smallest integer \\( n \\) such that \\( n! \\) is divisible by \\( 2009^{29} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-64c31e4a5065461a3b70e8d5f27ca78183a92ab16baf7aa40699515d96945bdd | What is the smallest integer \( x \) that is greater than 1 million and for which the sum of \( 2^x \) and \( 3^x \) is divisible by 7? | [
"1000005"
] | [
"What is the smallest integer \\( x \\) that is greater than 1 million and for which the sum of \\( 2^x \\) and \\( 3^x \\) is divisible by 7?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 27
} | 59 | 0 | 0 |
omr-ws-ca2e09db31e9c94b7b41ad8318bc4edfa772ab62f88dd470bb6d545fde0e8a32 | What is the smallest integer angle (in degrees) that can be constructed with a straightedge and compass? | [
"1"
] | [
"What is the smallest integer angle (in degrees) that can be constructed with a straightedge and compass?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8a23e425ef995127bfd0254359892947b4be1c48044e280bb48bfd288e84e9ec | What is the smallest integer divisible by 7 that is greater than \(100^{100}\)? | [
"\\(100^{100} + 5\\)"
] | [
"What is the smallest integer divisible by 7 that is greater than \\(100^{100}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 24
} | 39 | 0 | 0 |
omr-ws-fe5eb37d62c3e145d714e68e7dc3806460870a03f9208ed4d2fb7c6f35bda0d4 | What is the smallest integer greater than $2010$ that is divisible by $2$ but not by $4$, divisible by $3$ but not by $9$, and divisible by $5$ but not by $25$? | [
"2130"
] | [
"What is the smallest integer greater than $2010$ that is divisible by $2$ but not by $4$, divisible by $3$ but not by $9$, and divisible by $5$ but not by $25$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 10
} | 25 | 0 | 0 |
omr-ws-575b7987b566d6b040a7b3c8e360c41fc855049c61716d35463228faad24daac | What is the smallest integer larger than $(\sqrt{3} + \sqrt{2})^6$? | [
"970"
] | [
"What is the smallest integer larger than $(\\sqrt{3} + \\sqrt{2})^6$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-154ff4575fe0a35bbaa9dc38af36353b2c71574a15af3968797bee9810549b30 | What is the smallest integer multiple of 49 whose digits are all the same? | [
"777777"
] | [
"What is the smallest integer multiple of 49 whose digits are all the same?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 39
} | 71 | 0 | 0 |
omr-ws-d7f3958c7267fc94f0c1c8c2747d8f67c8e91a0e11ef88b4d02fdf879dc880f2 | What is the smallest integer multiple of 9997, other than 9997 itself, which contains only odd digits? | [
"33339995"
] | [
"What is the smallest integer multiple of 9997, other than 9997 itself, which contains only odd digits?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 8,
"tir": 19
} | 29 | 0 | 0 |
omr-ws-bf646a2e171135702226f42cdb234ea101a28141448cf6f1b9e254950adbdb3f | What is the smallest integer number \( n \) such that a cube with edge length \( n \) can be subdivided into 2008 smaller cubes, each with integer edge lengths? How can this subdivision be made? | [
"13"
] | [
"What is the smallest integer number \\( n \\) such that a cube with edge length \\( n \\) can be subdivided into 2008 smaller cubes, each with integer edge lengths? How can this subdivision be made?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dc69f678030c5d7fe8eefbd752522372936b90308c9ec4233dc98c0835f0c869 | What is the smallest integer such that if you rotate the number to the left, you get a number that is exactly one and a half times the original number? (To rotate the number left, take the first digit off the front and append it to the end of the number.) | [
"\\(\\frac{20 \\cdot 10^{16} - 2}{17}\\)"
] | [
"What is the smallest integer such that if you rotate the number to the left, you get a number that is exactly one and a half times the original number? (To rotate the number left, take the first digit off the front and append it to the end of the number.)"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-f2aeba83b3de28b096133717b4de432d91c8fe05e6bf02afc3ff2948af595c79 | What is the smallest integer that can be written in the form $13x + 73y$ for three different pairs $(x, y)$, where $x$ and $y$ are natural numbers? | [
"1984"
] | [
"What is the smallest integer that can be written in the form $13x + 73y$ for three different pairs $(x, y)$, where $x$ and $y$ are natural numbers?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-2e8cbdce1a9426cfc2766062545bd97eb936006a6d4fd70f879659e3151a7145 | What is the smallest integer that has 6 odd divisors and 12 even divisors? | [
"[hide]180[/hide]\n\nTherefore, the smallest integer that has 6 odd divisors and 12 even divisors is 180."
] | [
"What is the smallest integer that has 6 odd divisors and 12 even divisors?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-2950778f42cc4210b2ffb4fba2b9ef28eff3e8613526d08e67f0ff1365ba1a05 | What is the smallest integer that leaves a remainder of 4 when divided by 6 and leaves a remainder of 2 when divided by 9? | [
"\\text{No solution exists}"
] | [
"What is the smallest integer that leaves a remainder of 4 when divided by 6 and leaves a remainder of 2 when divided by 9?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ae623a8b2f87c957502eb3ca06dbe18aa0ac3195aeb767dabc8eb26a050f2271 | What is the smallest integer value $n > 1$ such that $n!$ is a perfect square? | [
"No such \\( n \\) exists."
] | [
"What is the smallest integer value $n > 1$ such that $n!$ is a perfect square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 8,
"tir": 29
} | 49 | 0 | 0 |
omr-ws-1c30e00df14a0b46367d5e3faf36cec722f9570d3b2fa4dfeb5687de39ba5469 | What is the smallest integer value of $k$ such that $2^k > 3^{2011}$? | [
"3188"
] | [
"What is the smallest integer value of $k$ such that $2^k > 3^{2011}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a527c52b05ab5cb580fe9538940e51409ecc995e468cea83084794532937eb48 | What is the smallest integer whose cube ends in 117? | [
"373"
] | [
"What is the smallest integer whose cube ends in 117?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-2580e9f0765ffae6466d5997913ad89392ea344e2bdef47da7f4f3b4b0482c37 | What is the smallest integral value of $k$ such that the equation \[ 2x(kx-4)-x^2+6=0 \] has no real roots? | [
"2"
] | [
"What is the smallest integral value of $k$ such that the equation\n\\[ 2x(kx-4)-x^2+6=0 \\]\nhas no real roots?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3efff83f410ce0a9e937e7923af84551ae59f019d17f6681dcb71dead56d0f2e | What is the smallest integral value of \( k \) such that the equation \( 2x(kx - 4) - x^2 + 6 = 0 \) has no real roots? | [
"2"
] | [
"What is the smallest integral value of \\( k \\) such that the equation \\( 2x(kx - 4) - x^2 + 6 = 0 \\) has no real roots?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-c2626c4fc35e565706757479671906bfe8f59d3767eb41e87332ef2aa1564aed | What is the smallest interior angle sum possible for a triangle on a sphere that has 2 right angles? | [
"180°"
] | [
"What is the smallest interior angle sum possible for a triangle on a sphere that has 2 right angles?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 4,
"tir": 14
} | 33 | 0 | 0 |
omr-ws-f1bcf7e22b0a1c1a2476a473cf4122d828a7ed03d6c3b3c64c9e4dbe32917fe2 | What is the smallest length $x$ such that a regular tetrahedron with side length $L$ can be passed through a loop of twine of length $x$? | [
"\\( 2L \\)"
] | [
"What is the smallest length $x$ such that a regular tetrahedron with side length $L$ can be passed through a loop of twine of length $x$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-ad67751e66244333e21fd1db133db7b9a144af938571b8e6f87414a363cf884d | What is the smallest length of a piecewise smooth curve that starts from the origin and intersects every tangent line of the unit circle? | [
"1 + 2\\pi"
] | [
"What is the smallest length of a piecewise smooth curve that starts from the origin and intersects every tangent line of the unit circle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-4663c97e20ff08b0ce8f583042dd2259a47d16ccb3faeb896b3c4396b8927d40 | What is the smallest length of ribbon needed to wrap around a 2"×10"×20" rectangular prism? | [
"50.338"
] | [
"What is the smallest length of ribbon needed to wrap around a 2\"×10\"×20\" rectangular prism?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2aa862c9f61fb40031247d5ea61dfb57f8ce79e5831a58b504895666e680a2d7 | What is the smallest multiple of $2018$ whose decimal digits start with $1111\cdots$? Answer $0$ if you think such a multiple does not exist. | [
"1111918"
] | [
"What is the smallest multiple of $2018$ whose decimal digits start with $1111\\cdots$? Answer $0$ if you think such a multiple does not exist."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 5,
"tir": 19
} | 28 | 0 | 0 |
omr-ws-64e01f6d0c231bec722ed3a248b2877213ededdea5e0496e1b5f83e5e5e66722 | What is the smallest multiple of 99 whose sum of digits is 99 and that begins and ends with 97? | [
"9750989999997"
] | [
"What is the smallest multiple of 99 whose sum of digits is 99 and that begins and ends with 97?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e9e1b6a913fe8189a9d9e66f2ffb057ecb4b009f00a3c087f5cc5ca1d33ec15f | What is the smallest natural number $a$ for which there are numbers $b$ and $c$ such that the quadratic trinomial $ax^2 + bx + c$ has two different positive roots not exceeding $\frac{1}{1000}$? | [
"1001000"
] | [
"What is the smallest natural number $a$ for which there are numbers $b$ and $c$ such that the quadratic trinomial $ax^2 + bx + c$ has two different positive roots not exceeding $\\frac{1}{1000}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-4c00a7c6a743101273515bd0c8c1e61c180b637ae8fbb96a1ba3d0f1559ac98c | What is the smallest natural number $k$ such that $10^k$ divides a number in the first $100000001$ Fibonacci numbers? | [
"1"
] | [
"What is the smallest natural number $k$ such that $10^k$ divides a number in the first $100000001$ Fibonacci numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-fa8fccd310bcf0d52fcf3ce24170b020da294e2d192eb821b67532bacf020933 | What is the smallest natural number \( n \) having exactly 100 factors? | [
"45360"
] | [
"What is the smallest natural number \\( n \\) having exactly 100 factors?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-934f48bfc09c490b2f953e371addadef6a75a58b53dbbdb8d09028fccdc19c4f | What is the smallest natural number \( n \) such that \( n+3 \) and \( 2020n+1 \) are both perfect squares? | [
"2022"
] | [
"What is the smallest natural number \\( n \\) such that \\( n+3 \\) and \\( 2020n+1 \\) are both perfect squares?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 1
} | 9 | 0 | 0 |
omr-ws-b8e9362cf48eba0e8fd1fa683aa14e8041a0801f08f5d573726e7f3150274c31 | What is the smallest natural number multiple of $5$ such that the sum of its digits is $26$? | [
"3995"
] | [
"What is the smallest natural number multiple of $5$ such that the sum of its digits is $26$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-a58411b8e834590e198f77f254637aef5108d24c5754b680f9f427de6e711de7 | What is the smallest natural number that Vasya can write using only the digits $1$ and $2$ that is divisible by $2016$? | [
"12122222112"
] | [
"What is the smallest natural number that Vasya can write using only the digits $1$ and $2$ that is divisible by $2016$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 7,
"tir": 17
} | 32 | 0 | 0 |
omr-ws-b9cf0f67bfb32992aa9e2039204daa03a97c7df1bc85f5f5a36f0b93f0f05a9f | What is the smallest natural number that can be expressed as the sum of two distinct squares and also as the sum of two distinct cubes? | [
"64"
] | [
"What is the smallest natural number that can be expressed as the sum of two distinct squares and also as the sum of two distinct cubes?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d018a589772221e025511276dd2ab15d8d0e273a5e060ca4f194ec51b3a86615 | What is the smallest natural number that, when concatenated with itself, results in a number that is a multiple of $2015$? | [
"155"
] | [
"What is the smallest natural number that, when concatenated with itself, results in a number that is a multiple of $2015$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
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