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-37480de8d1e45213786ff94fb07e94eeefb8acf6bbf5cdcca2655e525f9ad974 | What is the smallest prime number which can be expressed as the sum of two composite numbers? | [
"13"
] | [
"What is the smallest prime number which can be expressed as the sum of two composite numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-ddb882ad04f096c470f3e0f2e8728e821cef5a29e3bae7bbcdd374844bec96bf | What is the smallest prime that cannot be expressed in the form $\frac{pq+1}{p+q}$, where $p$ and $q$ are prime numbers? | [
"73"
] | [
"What is the smallest prime that cannot be expressed in the form $\\frac{pq+1}{p+q}$, where $p$ and $q$ are prime numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 8,
"tir": 11
} | 26 | 0 | 0 |
omr-ws-d5693861a9d2cbbeff43f41651488ee112e84f143f9486bc056458e0d89a7001 | What is the smallest prime whose digits sum to $19$? | [
"199"
] | [
"What is the smallest prime whose digits sum to $19$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b716ebe178127f3632602b729e36adec97d8521a21da28926621feaafe429a81 | What is the smallest radius \( R \) of the sphere \( x^2 + y^2 + z^2 = R^2 \) such that the sphere does not pass through any lattice points? | [
"7"
] | [
"What is the smallest radius \\( R \\) of the sphere \\( x^2 + y^2 + z^2 = R^2 \\) such that the sphere does not pass through any lattice points?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-70d80f02a38f985c043d6585ab2a3c161e40bb4a5e0bb6ded670b2dce2f5535a | What is the smallest real number \( a \) for which the inequality \[ x^5 + y^5 + z^5 \geq 3 \] holds true for all \( x, y, z \geq a \) satisfying \( x + y + z = 3 \)? | [
"\\(-4.04276\\)"
] | [
"What is the smallest real number \\( a \\) for which the inequality\n\\[ x^5 + y^5 + z^5 \\geq 3 \\]\nholds true for all \\( x, y, z \\geq a \\) satisfying \\( x + y + z = 3 \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-bcdadc7521aa713e6e0a5c51cde62e0de16870edf777c73ac3a06db38694e476 | What is the smallest ring $\mathbb{Z}_n$ that contains exactly 1 billion nilpotents? | [
"\\( 10^{10} \\)"
] | [
"What is the smallest ring $\\mathbb{Z}_n$ that contains exactly 1 billion nilpotents?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-2ee0906c568473808b29b1161784ab7b78c3fea1932eb76e4f9744a2e9450c67 | What is the smallest side length BC of triangle ABC if angle BAC is equal to \(\alpha\) and the area of triangle ABC is \(S\)? | [
"\\(2\\sqrt{S \\csc \\alpha}\\)"
] | [
"What is the smallest side length BC of triangle ABC if angle BAC is equal to \\(\\alpha\\) and the area of triangle ABC is \\(S\\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a0f5d3957ebe4a0c2f3c220c8d43d135f61cdbb301a0be2bf5ac928f51eb55bd | What is the smallest side length of a square that can be folded without overlays from exactly 11 squares with side lengths 1, 2, 3, and so on? | [
"10"
] | [
"What is the smallest side length of a square that can be folded without overlays from exactly 11 squares with side lengths 1, 2, 3, and so on?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-12714df7ed6b755402503537fc51574c9c2b5ef4c0afbff52a2dbcac2fe52f38 | What is the smallest six-digit lucky ticket number other than 000000, where a lucky ticket is defined as one whose sum of the first three digits is equal to the sum of the last three digits? | [
"001001"
] | [
"What is the smallest six-digit lucky ticket number other than 000000, where a lucky ticket is defined as one whose sum of the first three digits is equal to the sum of the last three digits?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 5,
"tir": 6
} | 24 | 0 | 0 |
omr-ws-aa83bf56357532f1745d8ca58911b470a2de007db1795992f12e66b520aa240b | What is the smallest size for a square canvas, such that if each point on the canvas is colored either white or red, there will be two points of the same color exactly 1 inch apart? | [
"\\(\\frac{\\sqrt{6}+\\sqrt{2}}{4}\\)"
] | [
"What is the smallest size for a square canvas, such that if each point on the canvas is colored either white or red, there will be two points of the same color exactly 1 inch apart?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-04bf966d41828994620eebb315c16a553fe7251d3043e62d13d09eeedbb7d183 | What is the smallest solution of the equation $x^4-34x^2+225=0$? | [
"-5"
] | [
"What is the smallest solution of the equation $x^4-34x^2+225=0$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-31a998bcda9c8ed2354ae46554f7632d24e8c588925e14f057f5f98e9972fa15 | What is the smallest square frame that can cover a painting of dimensions 40 inches by 100 inches completely? | [
"70\\sqrt{2}"
] | [
"What is the smallest square frame that can cover a painting of dimensions 40 inches by 100 inches completely?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-6de707e431e3c37b949fb9185ae7141b824294d4d9b5172f104670738bac3c21 | What is the smallest square frame that can cover a painting of dimensions 40 inches by 90 inches completely? | [
"90"
] | [
"What is the smallest square frame that can cover a painting of dimensions 40 inches by 90 inches completely?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-54f808efb28d996961fd306076d60d9eccbb1f6316d1c3d136778b92c49e71aa | What is the smallest sum \( x + y + z \) if \( x, y, z \) are positive and \( x < y \) satisfying \( \frac{1}{(x-2008)^2} + \frac{1}{(y-2009)^2} = \frac{1}{(z-2010)^2} \)? | [
"6074"
] | [
"What is the smallest sum \\( x + y + z \\) if \\( x, y, z \\) are positive and \\( x < y \\) satisfying \\( \\frac{1}{(x-2008)^2} + \\frac{1}{(y-2009)^2} = \\frac{1}{(z-2010)^2} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-f62e31eb9d7ceb159485067644fc1c3f4e928da029840af39155fe343ed276dc | What is the smallest sum of digits that $5^n + 6^n + 2022^n$ can take, where $n$ is a positive integer? | [
"8"
] | [
"What is the smallest sum of digits that $5^n + 6^n + 2022^n$ can take, where $n$ is a positive integer?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-ad2a175b3d2433112f141ce334d47ac4983f4df9c7610434cea5bdd26b3a2876 | What is the smallest sum of money that cannot be made using 10 or fewer coins? The allowed coins are 1 cent, 5 cents, 10 cents, 25 cents, 1 dollar, and 2 dollars. Give the answer as a decimal. | [
"3.99"
] | [
"What is the smallest sum of money that cannot be made using 10 or fewer coins? The allowed coins are 1 cent, 5 cents, 10 cents, 25 cents, 1 dollar, and 2 dollars. Give the answer as a decimal."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 1
} | 10 | 0 | 0 |
omr-ws-a248ef7cc60194d3a184ea3ca72fc36e01dcd9c34402cde1458f2262732e3750 | What is the smallest term in the sequence 121, 1221, 12221, ... that is a multiple of 1001? | [
"122221"
] | [
"What is the smallest term in the sequence 121, 1221, 12221, ... that is a multiple of 1001?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-08f66b8210b7be4fc360bde983997d6c7d8b0f468e43395c7b20d0d25f766ad0 | What is the smallest term of the Fibonacci sequence ($F_{1}=1, F_{2}=1$) that is divisible by $10^4$? | [
"\\( F_{7500} \\)"
] | [
"What is the smallest term of the Fibonacci sequence ($F_{1}=1, F_{2}=1$) that is divisible by $10^4$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 35,
"genselect": 8,
"tir": 17
} | 60 | 0 | 0 |
omr-ws-3b77b1a05bdbd633241c5a3b08c16ddb4d4f3cfb3eba7eeca5e9d5e37cea69af | What is the smallest three-digit integer that can be expressed as the sum of a positive integer power of $2$ and a positive integer power of $3$ in two different ways? | [
"259"
] | [
"What is the smallest three-digit integer that can be expressed as the sum of a positive integer power of $2$ and a positive integer power of $3$ in two different ways?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-657bd6af6527437d0cfbbb049101fbaeae9b50eee57ede8177138a18b494153f | What is the smallest three-digit multiple of 13? | [
"104"
] | [
"What is the smallest three-digit multiple of 13?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d7961a276969c84f4404aecbe8c5bd24f92727366cf0a3c34680b2f5fd18029d | What is the smallest three-digit number in Pascal's triangle? | [
"100"
] | [
"What is the smallest three-digit number in Pascal's triangle?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-110a86de1b616a30eb9fe83e20879a4690ba9f5af41bd812547ee895e479d3c7 | What is the smallest two-digit integer $n$ such that switching its digits and then adding 3 results in $2n$? | [
"12"
] | [
"What is the smallest two-digit integer $n$ such that switching its digits and then adding 3 results in $2n$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9fa5e43eb554d94fe02e57057596f8d017f5c9f457b0ec1655c7fab6e82fb179 | What is the smallest value of $N$ that guarantees it will be possible to travel between any two of the 2004 settlements on Mars, given that no two settlements have more than one tunnel connecting them? | [
"2003004"
] | [
"What is the smallest value of $N$ that guarantees it will be possible to travel between any two of the 2004 settlements on Mars, given that no two settlements have more than one tunnel connecting them?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-70e5f193b717707de1f41152d69256b62fcaaff3534f16807b1cd18b38df6345 | What is the smallest value of $\frac{x^2 + 361}{x}$? | [
"38"
] | [
"What is the smallest value of $\\frac{x^2 + 361}{x}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-ea564e199d0f87a6f5a6d41420a7ce3badda57e423a30950afd39f250642a0ed | What is the smallest value of $k$ for which the inequality \[ ad - bc + yz - xt + (a + c)(y + t) - (b + d)(x + z) \leq k \left( \sqrt{a^2 + b^2} + \sqrt{c^2 + d^2} + \sqrt{x^2 + y^2} + \sqrt{z^2 + t^2} \right)^2 \] holds for any 8 real numbers $a, b, c, d, x, y, z, t$? | [
"\\dfrac{1}{4}"
] | [
"What is the smallest value of $k$ for which the inequality\n\\[\nad - bc + yz - xt + (a + c)(y + t) - (b + d)(x + z) \\leq k \\left( \\sqrt{a^2 + b^2} + \\sqrt{c^2 + d^2} + \\sqrt{x^2 + y^2} + \\sqrt{z^2 + t^2} \\right)^2\n\\]\nholds for any 8 real numbers $a, b, c, d, x, y, z, t$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 4
} | 16 | 0 | 0 |
omr-ws-29fb1047174a901af471fbb9872e08035b11bfdac91501e80b3111fc172c05c9 | What is the smallest value of $n$ for a polygon with $n$ sides to have the sum of its internal angles greater than $2012^\circ$? | [
"14"
] | [
"What is the smallest value of $n$ for a polygon with $n$ sides to have the sum of its internal angles greater than $2012^\\circ$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-537cad4ceef77e31073c33b1028625187cb1749cbba67983792721deed9ca988 | What is the smallest value of $n$ for which there are two non-conjugate $n \times n$ matrices that have the same minimal and characteristic polynomials and eigenspaces of equal dimension? | [
"7"
] | [
"What is the smallest value of $n$ for which there are two non-conjugate $n \\times n$ matrices that have the same minimal and characteristic polynomials and eigenspaces of equal dimension?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 7
} | 9 | 0 | 0 |
omr-ws-1e11afb8c4d6486bd46373f5fbddde964a83425cdcd023e8dd5b864bbfb4b95f | What is the smallest value of $n$ such that $9^{(100)} < 3^{(n)}$, where $n^{(1)} = n$, $n^{(2)} = n^n$, $n^{(3)} = n^{n^n}$, etc.? | [
"101"
] | [
"What is the smallest value of $n$ such that $9^{(100)} < 3^{(n)}$, where $n^{(1)} = n$, $n^{(2)} = n^n$, $n^{(3)} = n^{n^n}$, etc.?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 7,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-dc6e9265c52699693abf1de4e6856be0b5948da81994b5bfd0e94ed73de64db4 | What is the smallest value of $n$ such that $\frac{n!}{40^9}$ is an integer? | [
"40"
] | [
"What is the smallest value of $n$ such that $\\frac{n!}{40^9}$ is an integer?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-92665329666f5a020c18d052fdea8c36a0b3db2084692b96b84e3115a58f5c96 | What is the smallest value of $n$ such that a checkered square with side length $n$ can be cut into more than $n$ rectangles of different areas, with none of the parts being a square? Provide an example of such a cutting. | [
"\\( 5 \\)"
] | [
"What is the smallest value of $n$ such that a checkered square with side length $n$ can be cut into more than $n$ rectangles of different areas, with none of the parts being a square? Provide an example of such a cutting."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-1428ce0ea9a6300d767d44ff29e0b6dc1ab1cbe544696d3a6b07a6cef6de0aab | What is the smallest value of $n$ such that a giraffe, starting from a corner on an $n \times n$ chessboard, can visit all the squares of the board? The giraffe moves 4 squares in one direction and then 1 square in a perpendicular direction. | [
"8"
] | [
"What is the smallest value of $n$ such that a giraffe, starting from a corner on an $n \\times n$ chessboard, can visit all the squares of the board? The giraffe moves 4 squares in one direction and then 1 square in a perpendicular direction."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 7
} | 16 | 0 | 0 |
omr-ws-eff9c8c10c0c784374a0291ab6d0fee7835838808a9b8386ffe79ad5c174004e | What is the smallest value of $n$ such that no matter how the numbers from 1 to $n$ are arranged on a circle, there will always exist ten consecutively increasing numbers $A_1 < A_2 < A_3 < \ldots < A_{10}$ forming a convex decagon? | [
"91"
] | [
"What is the smallest value of $n$ such that no matter how the numbers from 1 to $n$ are arranged on a circle, there will always exist ten consecutively increasing numbers $A_1 < A_2 < A_3 < \\ldots < A_{10}$ forming a convex decagon?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-d5dc40b56eded7e46415a28e24edd021ff85c123ea5c90d69754893606522c1a | What is the smallest value of $n$ such that there exists an $n$-gon that can be cut into a triangle, a quadrilateral, ..., and a $2006$-gon? | [
"2011015"
] | [
"What is the smallest value of $n$ such that there exists an $n$-gon that can be cut into a triangle, a quadrilateral, ..., and a $2006$-gon?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ddeec636a4264934cb497ce02c9df1bbadd46f2a77ddb8a18e6274b5cc2dfeb6 | What is the smallest value of $x$ that satisfies the equation $8x^2 - 38x + 35 = 0$? Express your answer as a decimal. | [
"1.25"
] | [
"What is the smallest value of $x$ that satisfies the equation $8x^2 - 38x + 35 = 0$? Express your answer as a decimal."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b10e7f4cba1d989513d4353b44ae4b245f980c3fa2382fa4989c42e51bca98e7 | What is the smallest value of \( N \) for which the fraction \(\frac{N-14}{5N+16}\) is irreducible? | [
"1"
] | [
"What is the smallest value of \\( N \\) for which the fraction \\(\\frac{N-14}{5N+16}\\) is irreducible?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-165e359eb4116ca9e02d88040726a6dcadb730a07e524ac9cdb31fd6297854eb | What is the smallest value of \( N \) that guarantees it will be possible to travel between any two settlements, given that there are 2,004 settlements and no two settlements have more than one tunnel connecting them? | [
"2005004"
] | [
"What is the smallest value of \\( N \\) that guarantees it will be possible to travel between any two settlements, given that there are 2,004 settlements and no two settlements have more than one tunnel connecting them?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 42,
"genselect": 7,
"tir": 4
} | 53 | 0 | 0 |
omr-ws-4dbd4398e216be9b1b98282bf4a4f0e2fe036de99516861adb30269c7de79d1c | What is the smallest value of \( k \) such that there are at least 2004 \( k \)-digit palindromes consisting of the digits 8 and 9, with each palindrome containing exactly one 8 and one 9? | [
"21"
] | [
"What is the smallest value of \\( k \\) such that there are at least 2004 \\( k \\)-digit palindromes consisting of the digits 8 and 9, with each palindrome containing exactly one 8 and one 9?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-d781f31108651a1856ccd84551186634ac8d554b219a48fba44fe72c9c3efb43 | What is the smallest value of \( k \) such that there are at least 2004 k-digit palindromes consisting only of B's and 9's, with at least one of each digit? | [
"11"
] | [
"What is the smallest value of \\( k \\) such that there are at least 2004 k-digit palindromes consisting only of B's and 9's, with at least one of each digit?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-90f978b8c5c0a1241595f0270c3012495b89edb4ce1289e3ad2d00d02325a6fb | What is the smallest value of \( k \) such that there are at least 2004 k-digit palindromes consisting only of the digits 8 and 9, with each palindrome containing at least one of each digit? | [
"21"
] | [
"What is the smallest value of \\( k \\) such that there are at least 2004 k-digit palindromes consisting only of the digits 8 and 9, with each palindrome containing at least one of each digit?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-a06cb7bf36e12244edfd460f34f18237f67dac1ffd5b71764a94fbda73b29a67 | What is the smallest value of \( k \) such that there is no integer solution \( n \) to \( \left\lfloor \frac{n^2}{36} \right\rfloor = k \)? | [
"15"
] | [
"What is the smallest value of \\( k \\) such that there is no integer solution \\( n \\) to \\( \\left\\lfloor \\frac{n^2}{36} \\right\\rfloor = k \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 13
} | 35 | 0 | 0 |
omr-ws-09b679220901ed7f855d592c6d03918a4eb04ee6673e1b83da2935b4653d5705 | What is the smallest value of \( n \) so that if the outer two layers of unit cubes are removed from an \( n \times n \times n \) cubic block, more than half the original unit cubes will still remain? | [
"20"
] | [
"What is the smallest value of \\( n \\) so that if the outer two layers of unit cubes are removed from an \\( n \\times n \\times n \\) cubic block, more than half the original unit cubes will still remain?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 12
} | 27 | 0 | 0 |
omr-ws-f93edeaec6542d438a4efa424d44b0667e622d9abee04ded341984fddb8149f0 | What is the smallest value of \( n \) such that \( (1 + 2^{\frac{1}{2}})^n \) is within \( 10^{-9} \) of a whole number? | [
"24"
] | [
"What is the smallest value of \\( n \\) such that \\( (1 + 2^{\\frac{1}{2}})^n \\) is within \\( 10^{-9} \\) of a whole number?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d5ab0ed2881e4c4fc6d8a0e7864cc6e395ad8108cd7b7697068f4bb9d3903c42 | What is the smallest value of \( n \) such that \( 3^{2^n} - 1 \) is divisible by \( 2^{2005} \)? | [
"2003"
] | [
"What is the smallest value of \\( n \\) such that \\( 3^{2^n} - 1 \\) is divisible by \\( 2^{2005} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-ea324762081e43a27d3006898f5bee525376b82acb445e9b9174f80344c0e6c5 | What is the smallest value of \( n \) such that every set of \( n \) integers contains at least two elements whose sum or difference is exactly divisible by 17? | [
"10"
] | [
"What is the smallest value of \\( n \\) such that every set of \\( n \\) integers contains at least two elements whose sum or difference is exactly divisible by 17?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ea4bc0f225b0c0288e3a742b8c5a0b18fa5ad49dbb592cc005cabb58c0db6c13 | What is the smallest value of \( n \) such that no matter how \( n \) points are placed inside or on the surface of a cube with a side length of 16 units, there are at least two points which are closer than 14 units apart? | [
"9"
] | [
"What is the smallest value of \\( n \\) such that no matter how \\( n \\) points are placed inside or on the surface of a cube with a side length of 16 units, there are at least two points which are closer than 14 units apart?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-c903166f2104c57829a595edda1730341175c728f49bec00505d884df5414157 | What is the smallest value of \( n \) such that the triangle with vertices \((0,0)\), \((1,n)\), and \((n,1)\) contains at least 1000 points with integer coordinates inside it? | [
"46"
] | [
"What is the smallest value of \\( n \\) such that the triangle with vertices \\((0,0)\\), \\((1,n)\\), and \\((n,1)\\) contains at least 1000 points with integer coordinates inside it?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-66b37789443c0319911a023e792ff1cf6ce5d975aad6b1ce5c632ec0a88329d0 | What is the smallest value of the expression $\sqrt{x^2 + 1} + \sqrt{(y - x)^2 + 4} + \sqrt{(z - y)^2 + 1} + \sqrt{(10 - z)^2 + 9}$, where $x$, $y$, and $z$ are arbitrary real numbers? | [
"13"
] | [
"What is the smallest value of the expression $\\sqrt{x^2 + 1} + \\sqrt{(y - x)^2 + 4} + \\sqrt{(z - y)^2 + 1} + \\sqrt{(10 - z)^2 + 9}$, where $x$, $y$, and $z$ are arbitrary real numbers?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a7a8a6b02b0289aaedb9b6e6ddfe9687b11ab7f0f6be0de9dafee016bbf4a898 | What is the smallest value of the ratio of the lengths of the largest side of a triangle to the radius of its inscribed circle? | [
"2\\sqrt{3}"
] | [
"What is the smallest value of the ratio of the lengths of the largest side of a triangle to the radius of its inscribed circle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-b5dcf47c27a9296c2ca78754d5133051e1cca9e8175f33a001250c5d19534ccb | What is the smallest value that the expression $$\sqrt{3x-2y-1}+\sqrt{2x+y+2}+\sqrt{3y-x}$$ can take? | [
"\\(\\sqrt{3}\\)"
] | [
"What is the smallest value that the expression $$\\sqrt{3x-2y-1}+\\sqrt{2x+y+2}+\\sqrt{3y-x}$$ can take?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-b5c82b1db87f913e676a3563a523d87d449601a6c913b437f4b9824fa088d52f | What is the smallest value that the expression \[ \sqrt{49 + a^2 - 7a\sqrt{2}} + \sqrt{a^2 + b^2 - ab\sqrt{2}} + \sqrt{50 + b^2 - 10b} \] can have for positive real numbers \(a\) and \(b\)? | [
"13"
] | [
"What is the smallest value that the expression \n\n\\[\n\\sqrt{49 + a^2 - 7a\\sqrt{2}} + \\sqrt{a^2 + b^2 - ab\\sqrt{2}} + \\sqrt{50 + b^2 - 10b}\n\\]\n\ncan have for positive real numbers \\(a\\) and \\(b\\)?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 10,
"tir": 14
} | 35 | 0 | 0 |
omr-ws-05548cea990c93e9c071fbdb8601020a2adf8a4fcea311f53d2d9670770c0246 | What is the smallest value that the function \( f(a, b) = \sqrt{4 + a^2} + \sqrt{4 + (a - b)^2} + \sqrt{1 + (14 - b)^2} \) takes on for real values of \( a \) and \( b \)? | [
"\\(\\sqrt{221}\\)"
] | [
"What is the smallest value that the function \\( f(a, b) = \\sqrt{4 + a^2} + \\sqrt{4 + (a - b)^2} + \\sqrt{1 + (14 - b)^2} \\) takes on for real values of \\( a \\) and \\( b \\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 5,
"tir": 29
} | 49 | 0 | 0 |
omr-ws-46d18dfdfc8d53df15be60e79371c9357ebe631c679aa776753f0654a7b88c5c | What is the smallest whole number $b$ such that 62 can be expressed in base $b$ using only three digits? | [
"4"
] | [
"What is the smallest whole number $b$ such that 62 can be expressed in base $b$ using only three digits?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-916ce1bffead6cb3cbf047879b9ebc8dc8b7827a27679197a88ed32e633e6b09 | What is the smallest whole number larger than the perimeter of any triangle with sides of lengths 5 and 19? | [
"48"
] | [
"What is the smallest whole number larger than the perimeter of any triangle with sides of lengths 5 and 19?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6bfda1394026d655f61f0cdabf04e14c4b851fba98f5c9ee8a4895463220f40c | What is the smallest whole number that has a remainder of 1 when divided by 4, a remainder of 1 when divided by 3, and a remainder of 2 when divided by 5? | [
"37"
] | [
"What is the smallest whole number that has a remainder of 1 when divided by 4, a remainder of 1 when divided by 3, and a remainder of 2 when divided by 5?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bfee7e9d3626c3b92dd592ab70eb251922781af62ba305f2be179d38176c7933 | What is the solid angle subtended by a cone with a semivertical angle $\alpha$? | [
"\\(2(1 - \\cos\\alpha)\\)"
] | [
"What is the solid angle subtended by a cone with a semivertical angle $\\alpha$?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f8c8e533c22fd41f1c21c4e8529f04ea61d45f31340bb1250aaecd676d26fcda | What is the solution of the Ornstein-Uhlenbeck process driven by the following equation: \[dr_{t}=a(b-r_{t})dt+\sigma dW_{t}\] with initial condition \(r_{0}=r\), where \(a, b, \sigma, r\) are constants, \((W_{t})\) is a Wiener process, and \(a>0\), \(b>0\), \(\sigma>0\)? | [
"\\( r_t = r e^{-at} + b(1 - e^{-at}) + \\sigma \\int_{0}^{t} e^{-a(t-s)} dW_s \\)"
] | [
"What is the solution of the Ornstein-Uhlenbeck process driven by the following equation:\n\\[dr_{t}=a(b-r_{t})dt+\\sigma dW_{t}\\]\nwith initial condition \\(r_{0}=r\\), where \\(a, b, \\sigma, r\\) are constants, \\((W_{t})\\) is a Wiener process, and \\(a>0\\), \\(b>0\\), \\(\\sigma>0\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-90507ff9b2fae1284523ab8cf34cd2776807dba0b71d4e5dcf545a3bd0bb11f8 | What is the solution set \( S \) of the inequality \[ \left[(x-1) \cdot (x-2)\right]^{-1} > \left[(x-2) \cdot (x-3)\right]^{-1}? \] | [
"\\( S = (-\\infty, 1) \\cup (2, 3) \\)"
] | [
"What is the solution set \\( S \\) of the inequality\n\\[\n\\left[(x-1) \\cdot (x-2)\\right]^{-1} > \\left[(x-2) \\cdot (x-3)\\right]^{-1}?\n\\]"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-9750559e7c51648b2c9af33df2bf9d6698889d1b375ba089ab5ffbff843d1bf3 | What is the solution set of the equation \( x^{10} = 10^x \)? | [
"\\( x = 10 \\) and another irrational solution."
] | [
"What is the solution set of the equation \\( x^{10} = 10^x \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 1,
"tir": 6
} | 15 | 0 | 0 |
omr-ws-16506887ac4ebef7a6e40b6b1422d9dc29d83e3c5c0ab2f374ab64b32d2b6fc3 | What is the solution to \( (1 + a_1)(1 + a_2)(1 + a_3) \cdots (1 + a_{10}) \) if \( a_1, a_2, \cdots, a_{10} \) are the imaginary solutions to \( x^{11} - 1 = 0 \)? | [
"1"
] | [
"What is the solution to \\( (1 + a_1)(1 + a_2)(1 + a_3) \\cdots (1 + a_{10}) \\) if \\( a_1, a_2, \\cdots, a_{10} \\) are the imaginary solutions to \\( x^{11} - 1 = 0 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c64893f091a2dcec2d1f07950a8a916991c66f46b871b09c1c498c0f923a8e51 | What is the solution to \( x' + x = u(t-t_{0}) \)? | [
"\\( x(t) = \\begin{cases} \n0 & \\text{for } t < t_{0} \\\\\n1 - e^{-(t-t_{0})} & \\text{for } t \\geq t_{0} \n\\end{cases} \\)"
] | [
"What is the solution to \\( x' + x = u(t-t_{0}) \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 10
} | 25 | 0 | 0 |
omr-ws-dc8a53d2a976821234ae6dbc3fac9ed9d3a47c0c1f69ef57219bf84a9148fddb | What is the solution to the Cauchy functional equation $f(x+y) = f(x) + f(y)$? | [
"f(x) = kx"
] | [
"What is the solution to the Cauchy functional equation $f(x+y) = f(x) + f(y)$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-198f5edfcec6877775dadaee26ccb4f13529a2de48154920ad7b3f43eceeac3e | What is the solution to the differential equation \( x^2 \, dy - y^2 \, dx + xy^2 (x - y) \, dy = 0 \)? | [
"(y - x) e^{-\\frac{y^2}{2}} = C x y"
] | [
"What is the solution to the differential equation \\( x^2 \\, dy - y^2 \\, dx + xy^2 (x - y) \\, dy = 0 \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 1
} | 6 | 0 | 0 |
omr-ws-e302e4f01fa7083270542a68df81900bd13ded8090a25f9104927a048ecf1d44 | What is the solution to the equation \(a^x = x\)? | [
"-\\frac{W(-\\ln a)}{\\ln a}"
] | [
"What is the solution to the equation \\(a^x = x\\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7d6b95208a35e630dcdacc1cf77342def4d4422ff94b887047898325988694f4 | What is the solution to the integral $\int e^{-x^2} \, dx$? | [
"\\frac{\\sqrt{\\pi}}{2} \\text{erf}(x) + C"
] | [
"What is the solution to the integral $\\int e^{-x^2} \\, dx$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fad76f69ea3427d7b6ee075bea642ebeabe9ee05ae10a48fffe7722929b36310 | What is the solution to the one-dimensional version of the moving sofa problem, i.e., find the sofa with maximum length that can be moved around a right-angled hallway of unit width? | [
"2\\sqrt{2}"
] | [
"What is the solution to the one-dimensional version of the moving sofa problem, i.e., find the sofa with maximum length that can be moved around a right-angled hallway of unit width?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-efc7a25c65df894ae5cd443ba2b2fc40c00ac4049812f5b711a41e826be10b38 | What is the solution to the second-order ODE \( aD^2(x)_t + bD(x)_t = f \) when \( a, b, f \neq 0 \) and \( c = 0 \)? | [
"x(t) = C_1 + C_2 e^{-\\frac{b}{a} t} + \\frac{f}{b} t"
] | [
"What is the solution to the second-order ODE \\( aD^2(x)_t + bD(x)_t = f \\) when \\( a, b, f \\neq 0 \\) and \\( c = 0 \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fd00908fbea6972d23aba45ed691b4d10752d9e348562a0a28283707ea17d9b6 | What is the span of the set of rational points on the unit circle? | [
"\\mathbb{R}^2"
] | [
"What is the span of the set of rational points on the unit circle?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d5558b04f0d20a695ec6f8c9a454e49d8af4b17cab812028d3f68e5a0498a17b | What is the spectral radius of the operator \(iI\), where \(I\) is the identity operator? | [
"1"
] | [
"What is the spectral radius of the operator \\(iI\\), where \\(I\\) is the identity operator?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f79a2f949214010ed559b13b8453f7f9951514a71a4bf091a572b5f4f60298f0 | What is the spectrum of the Fourier-Plancherel transform $ f \to \hat{f}$, which is a unitary operator from $ L^2(\mathbb{R})$ to itself? | [
"\\(\\{1, i, -1, -i\\}\\)"
] | [
"What is the spectrum of the Fourier-Plancherel transform $ f \\to \\hat{f}$, which is a unitary operator from $ L^2(\\mathbb{R})$ to itself?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-2d5802651cc98a6c4a7cd5f58660cf546b0034ab22698817d6ee3428adba1f17 | What is the splitting field of $x^{p^8} - 1$ over the field $\mathbb{Z}/p\mathbb{Z}$? | [
"\\(\\mathbb{F}_p\\)"
] | [
"What is the splitting field of $x^{p^8} - 1$ over the field $\\mathbb{Z}/p\\mathbb{Z}$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-73546b137020d7d672b28fb59e0079dab79699f16d4f9e5943c817625e81c127 | What is the splitting field of the polynomial \(x^2 + 3ix + 1\)? | [
"\\( \\mathbb{Q}(i, \\sqrt{13}) \\)"
] | [
"What is the splitting field of the polynomial \\(x^2 + 3ix + 1\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-16476438e9a3a3911d833f4ce476eee8e620ca6f2f41de30b3c4574c0fe8623d | What is the square of the maximum value of $\frac{20bc + 18bd + 20ad - 18ac}{a^2 + b^2 + c^2 + d^2}$ given that numbers $a, b, c, d$ satisfy the equation $a^2 + b^2 = c^2 + d^2$? | [
"181"
] | [
"What is the square of the maximum value of $\\frac{20bc + 18bd + 20ad - 18ac}{a^2 + b^2 + c^2 + d^2}$ given that numbers $a, b, c, d$ satisfy the equation $a^2 + b^2 = c^2 + d^2$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 3
} | 17 | 0 | 0 |
omr-ws-fb17df35a10da6c24d5dd5954fea7332e017d59a9cc63f1e0cbf66d78c6a2535 | What is the square of the middle digit of the number formed by writing the integers from 1 to 2023 in one long row without spaces between them? | [
"0"
] | [
"What is the square of the middle digit of the number formed by writing the integers from 1 to 2023 in one long row without spaces between them?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7eee2a968f4fde857d489ea3733234f0471af4348fa6b956a8c47f6b6457f7f2 | What is the square of the radius of a circle inscribed in a quadrilateral with sides 45, 63, 60, and 42? | [
"647"
] | [
"What is the square of the radius of a circle inscribed in a quadrilateral with sides 45, 63, 60, and 42?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ffdfe5549f4f4431457b1c0db38d8c5954b5126ca7c299d424e8bf42726a4afe | What is the square root of \( i \)? Is it defined? | [
"\\( \\frac{\\sqrt{2}}{2} + i\\frac{\\sqrt{2}}{2} \\) and \\( -\\frac{\\sqrt{2}}{2} - i\\frac{\\sqrt{2}}{2} \\)"
] | [
"What is the square root of \\( i \\)? Is it defined?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 14
} | 19 | 0 | 0 |
omr-ws-6c64c36fc4ac1eddbb224c7b38f039dae2159aad468d4572ed8e6840cc3ec16d | What is the square root of \(4 + 4i\)? Where \(i = \sqrt{-1}\). | [
"\\pm \\left( \\sqrt{2 + 2\\sqrt{2}} + i\\sqrt{2\\sqrt{2} - 2} \\right)"
] | [
"What is the square root of \\(4 + 4i\\)? Where \\(i = \\sqrt{-1}\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-c12d7dd37c1f68ee8f4d3d835d4e8b31a5f4640d1f920577d465031523a1a87f | What is the square root of a complex number \(a + bi\) in terms of \(a\) and \(b\)? | [
"\\(\\pm \\left( \\sqrt{\\frac{\\sqrt{a^2 + b^2} + a}{2}} + i \\sqrt{\\frac{\\sqrt{a^2 + b^2} - a}{2}} \\right)\\)"
] | [
"What is the square root of a complex number \\(a + bi\\) in terms of \\(a\\) and \\(b\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 2,
"tir": 6
} | 18 | 0 | 0 |
omr-ws-54bd211fc13262e34b5897fa67fa1574b257b4c8bed5dee2c811b9334d30a553 | What is the square root of the imaginary unit \( i \)? | [
"\\(\\frac{\\sqrt{2}}{2} + \\frac{i\\sqrt{2}}{2}\\)"
] | [
"What is the square root of the imaginary unit \\( i \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 2,
"tir": 21
} | 34 | 0 | 0 |
omr-ws-6d1104a2670b27f155f4a9ad73d375e29d6302d096538060f17e29db7772492c | What is the square root of the value of \( (\sqrt{85} + \sqrt{205} + \sqrt{218})(\sqrt{85} + \sqrt{205} - \sqrt{218})(\sqrt{85} - \sqrt{205} + \sqrt{218})(-\sqrt{85} + \sqrt{205} + \sqrt{218}) \)? | [
"254"
] | [
"What is the square root of the value of \\( (\\sqrt{85} + \\sqrt{205} + \\sqrt{218})(\\sqrt{85} + \\sqrt{205} - \\sqrt{218})(\\sqrt{85} - \\sqrt{205} + \\sqrt{218})(-\\sqrt{85} + \\sqrt{205} + \\sqrt{218}) \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-a54a11affb3500a719a703d370c0c7d78f1435adc52179ee133135be039bfcd6 | What is the square root of the volume of a cone with height 5 inches and base radius 3 inches? Round to the nearest tenth if necessary. | [
"\\(\\sqrt{15\\pi}\\)"
] | [
"What is the square root of the volume of a cone with height 5 inches and base radius 3 inches? Round to the nearest tenth if necessary."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 4
} | 7 | 0 | 0 |
omr-ws-26c03a246dfd456be97bbc86eb8907a0b771dc62a1f558e45dcdd61d01956d33 | What is the standard deviation of the data set 11, 11, 16, 6, 10, 12, 15, 7, 11? | [
"3.240"
] | [
"What is the standard deviation of the data set 11, 11, 16, 6, 10, 12, 15, 7, 11?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-d0e4f8204db2f6ad2a1494939e248bc199c82e8ada5039aaacba3a7856f0a59a | What is the standard procedure to solve the Diophantine equation $23x + 62y = 9$? | [
"x = 243 + 62t \\text{ and } y = -90 - 23t \\text{ for all integers } t"
] | [
"What is the standard procedure to solve the Diophantine equation $23x + 62y = 9$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-43e526fe06dfca6f6f0104b6b202c1004200761d7c3e390427c0496a0d53f59f | What is the structure of the Sylow 2-subgroup of the simple group \( L_2(q) \) with \( q \) odd? | [
"Dihedral group \\( \\langle a \\rangle \\rtimes \\langle b \\rangle \\) of order \\( 2^n \\) or \\( 2^{n-1} \\) depending on \\( q \\equiv 1 \\pmod{4} \\) or \\( q \\equiv -1 \\pmod{4} \\)."
] | [
"What is the structure of the Sylow 2-subgroup of the simple group \\( L_2(q) \\) with \\( q \\) odd?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 4
} | 7 | 0 | 0 |
omr-ws-08e7e3471b03455454623fc3b5228288e701bcbe4af210d299030c5873b5abc7 | What is the structure of the group of symmetries of a $2 \times 2 \times 2$ Rubik's cube? | [
"S_4"
] | [
"What is the structure of the group of symmetries of a $2 \\times 2 \\times 2$ Rubik's cube?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 4
} | 11 | 0 | 0 |
omr-ws-cd158b2f370b7332493c0d3608f233ce0474d29c5f2e2404faf4fd0d7a3cee32 | What is the sufficient condition for the function \( K(x,y) \) such that for all \( f \in C[0,1] \), \[ \int_{[0,1]^2} K(x,y) f(x)f(y) \, dx \, dy \ge 0? \] | [
"K(x,y) \\text{ is a positive semi-definite kernel}"
] | [
"What is the sufficient condition for the function \\( K(x,y) \\) such that for all \\( f \\in C[0,1] \\),\n\\[ \\int_{[0,1]^2} K(x,y) f(x)f(y) \\, dx \\, dy \\ge 0? \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 4
} | 16 | 0 | 0 |
omr-ws-ac9a3fe986fd5085363082eb0388a66ea98ab2b1acb5a17f4afa1deb6b82e81a | What is the sum $1^1 + 2^2 + 3^3 + \cdots + n^n$ in terms of $n$? | [
"\\sum_{k=1}^n k^k"
] | [
"What is the sum $1^1 + 2^2 + 3^3 + \\cdots + n^n$ in terms of $n$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7f8e64308e3626c23915af9c738243498a54088ed759ad609fcb437c6dd72716 | What is the sum $\sum_{k=1}^{n} k^k$ in terms of $n$? | [
"\\sum_{k=1}^{n} k^k"
] | [
"What is the sum $\\sum_{k=1}^{n} k^k$ in terms of $n$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1a3161139a61668e08be37d564931535026c58d4aa4d7cd8c7b293c6f8eef609 | What is the sum $\sum_{m=1}^n \sqrt{m}$ without using definite integrals? | [
"\\sum_{m=1}^n \\sqrt{m}"
] | [
"What is the sum $\\sum_{m=1}^n \\sqrt{m}$ without using definite integrals?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-db91f2017b4006571b045afd8f66257535bd1b6d3022cef7f4bc036c6a5d93ac | What is the sum $b_0 + b_1 + \cdots + b_{n-1}$ for the repeating decimal expansion of $\frac{1}{99^2} = 0.\overline{b_{n-1}b_{n-2}\ldots b_2b_1b_0}$, where $n$ is the length of the period of the repeating decimal expansion? | [
"883"
] | [
"What is the sum $b_0 + b_1 + \\cdots + b_{n-1}$ for the repeating decimal expansion of $\\frac{1}{99^2} = 0.\\overline{b_{n-1}b_{n-2}\\ldots b_2b_1b_0}$, where $n$ is the length of the period of the repeating decimal expansion?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-35cf25e7fa5e2169bc6854888e559cf23822702e36d28b324dd8eb02d4858957 | What is the sum $b_0 + b_1 + \cdots + b_{n-1}$ for the repeating decimal expansion of the fraction $\dfrac{1}{99^2}$, where $n$ is the length of the period of the repeating decimal expansion? | [
"883"
] | [
"What is the sum $b_0 + b_1 + \\cdots + b_{n-1}$ for the repeating decimal expansion of the fraction $\\dfrac{1}{99^2}$, where $n$ is the length of the period of the repeating decimal expansion?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 25
} | 39 | 0 | 0 |
omr-ws-b0d647e01e9ca72248bbb5479d797c11a41e395e36d573a9dc69d32071d29ef0 | What is the sum $b_0 + b_1 + \cdots + b_{n-1}$ for the repeating decimal expansion of the fraction $\frac{1}{99^2}$, where $n$ is the length of the period of the repeating decimal expansion? | [
"883"
] | [
"What is the sum $b_0 + b_1 + \\cdots + b_{n-1}$ for the repeating decimal expansion of the fraction $\\frac{1}{99^2}$, where $n$ is the length of the period of the repeating decimal expansion?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 20
} | 43 | 0 | 0 |
omr-ws-485ecaa7c0da36d6b9abdeb99d3a898bffeb70a149369597d6a0af9e470458b8 | What is the sum of $1+2+4+8+16+ \cdots + 1024$? | [
"2047"
] | [
"What is the sum of $1+2+4+8+16+ \\cdots + 1024$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7c7a498a806bc3668dc33ffbc320a8833b3ac9ba8c387e2624e3d26c2e43f5ed | What is the sum of $1234 + 4123 + 3412 + 2341$? | [
"11110"
] | [
"What is the sum of $1234 + 4123 + 3412 + 2341$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-48ad459dd794848bd70f4fe362ea3372ff2900cf9460e2f7124ec68b97bad411 | What is the sum of $3 + 3^2 + 3^{2^2} + 3^{2^3} + \dots + 3^{2^{2006}}$ modulo 11? | [
"10"
] | [
"What is the sum of $3 + 3^2 + 3^{2^2} + 3^{2^3} + \\dots + 3^{2^{2006}}$ modulo 11?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-47ed0eb8fc2757b4d01a83893a030a4dddd56cda5e38caa546ac2ef8c48f2399 | What is the sum of $\left(\dfrac{1}{3}\right) + \left(\dfrac{1}{3}\right)^2 + \left(\dfrac{1}{3}\right)^3 + \left(\dfrac{1}{3}\right)^4$? | [
"\\dfrac{40}{81}"
] | [
"What is the sum of $\\left(\\dfrac{1}{3}\\right) + \\left(\\dfrac{1}{3}\\right)^2 + \\left(\\dfrac{1}{3}\\right)^3 + \\left(\\dfrac{1}{3}\\right)^4$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8ed5e1e95941b67b10732446c7c85d32c357f76d89d9bb0236aeee6882bb60f5 | What is the sum of $\left(\frac{2}{3}\right)^{n+1} \left(\frac{1}{3}\right)^{n-1} \binom{2n-1}{n-1}$ as $n$ goes from 1 to infinity? | [
"\\(\\frac{1}{2}\\)"
] | [
"What is the sum of $\\left(\\frac{2}{3}\\right)^{n+1} \\left(\\frac{1}{3}\\right)^{n-1} \\binom{2n-1}{n-1}$ as $n$ goes from 1 to infinity?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-18b1e4d360ed435dc28c0ab65b6ca2819a5ce32b3f9f3409abd085afd92eade0 | What is the sum of $\sum_{n=1}^{\infty}\frac{\cos(nx)}{n}$? | [
"\\(-\\log\\left( 2 \\bigg| \\sin \\left( \\frac{x}{2} \\right) \\bigg| \\right)\\)"
] | [
"What is the sum of $\\sum_{n=1}^{\\infty}\\frac{\\cos(nx)}{n}$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 5,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-fa6930dc4b3d6b475fc9487e96bbbefec729132468d5470b2e444e0472ba17cc | What is the sum of \( \frac{1}{\sqrt{1} + \sqrt{2}} + \frac{1}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{4}} + \cdots + \frac{1}{\sqrt{99} + \sqrt{100}} \)? | [
"9"
] | [
"What is the sum of \\( \\frac{1}{\\sqrt{1} + \\sqrt{2}} + \\frac{1}{\\sqrt{2} + \\sqrt{3}} + \\frac{1}{\\sqrt{3} + \\sqrt{4}} + \\cdots + \\frac{1}{\\sqrt{99} + \\sqrt{100}} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b69ecfe01ca8b5de1ef885592288b1ec37c55a34b9b7ac0afbd563f12857c398 | What is the sum of \(1^k + 2^k + 3^k + \ldots + n^k\) for any natural numbers \(k\) and \(n\)? | [
"\\frac{1}{k+1} \\sum_{j=0}^k \\binom{k+1}{j} B_j n^{k+1 - j}"
] | [
"What is the sum of \\(1^k + 2^k + 3^k + \\ldots + n^k\\) for any natural numbers \\(k\\) and \\(n\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 3
} | 13 | 0 | 0 |
omr-ws-562ae8330a0553b4f829ed2d2852937a5f42e36adb8578765e51283733a99062 | What is the sum of all $n$ for which $\frac{n}{\phi(n)}$ is maximal for $1 \le n \le 500$? | [
"630"
] | [
"What is the sum of all $n$ for which $\\frac{n}{\\phi(n)}$ is maximal for $1 \\le n \\le 500$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
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