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-4661738355677dc1d3edaa289163b5b1cf66a6b21ebee5ec464d43a965e79f9a | What is the maximum value of \( f(n) = |s_{1}-s_{2}| + |s_{2}-s_{3}| + \cdots + |s_{n-1}-s_{n}| \) over all permutations \((s_{1}, s_{2}, \ldots, s_{n})\) of \((1, 2, \ldots, n)\)? | [
"\\(\\frac{n(n-1)}{2}\\)"
] | [
"What is the maximum value of \\( f(n) = |s_{1}-s_{2}| + |s_{2}-s_{3}| + \\cdots + |s_{n-1}-s_{n}| \\) over all permutations \\((s_{1}, s_{2}, \\ldots, s_{n})\\) of \\((1, 2, \\ldots, n)\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-4599a48125be482cf2c9d81185b3c4602e35ba50f57b973ecd7a40d6c602c4b3 | What is the maximum value of \( k \) such that some initial configuration cycles through \( k \) distinct configurations and returns to the original configuration for the first time after \( k \) days? | [
"6"
] | [
"What is the maximum value of \\( k \\) such that some initial configuration cycles through \\( k \\) distinct configurations and returns to the original configuration for the first time after \\( k \\) days?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-38591533c5b887433bb310069cc564edb444c7bcf1271013c74236303a63ecdf | What is the maximum value of \( m^2 + n^2 \) if \( m, n \in \{1, 2, 3, \ldots, 1981\} \) and \( (m^2 - mn - n^2)^2 = 1 \)? | [
"3524578"
] | [
"What is the maximum value of \\( m^2 + n^2 \\) if \\( m, n \\in \\{1, 2, 3, \\ldots, 1981\\} \\) and \\( (m^2 - mn - n^2)^2 = 1 \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 28
} | 51 | 0 | 0 |
omr-ws-0b944ccc6c4fdc0db5b83ed237ae223e0e6e470279e409169782a89bb63d75e1 | What is the maximum value of \( n \) for which the inequality \((a^n + b^n + c^n)^2 \ge 3(a^{2n-1}b + b^{2n-1}c + c^{2n-1}a)\) holds for all \( a, b, c \ge 0 \)? | [
"2"
] | [
"What is the maximum value of \\( n \\) for which the inequality \\((a^n + b^n + c^n)^2 \\ge 3(a^{2n-1}b + b^{2n-1}c + c^{2n-1}a)\\) holds for all \\( a, b, c \\ge 0 \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 1
} | 14 | 0 | 0 |
omr-ws-ee5872bcd956819d18df3bcb906842861ccaf82eeef9568e8f70c0a356259264 | What is the maximum value of \( n \) such that \( 2n + 1 \) divides \( 100 + n^2 \)? | [
"200"
] | [
"What is the maximum value of \\( n \\) such that \\( 2n + 1 \\) divides \\( 100 + n^2 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-dedd489f80ae519a2ed0fc311d6bbf15adcb4e2f4d9662c8fed04565c32f9ddb | What is the maximum value of \( n \) such that \( 5^{64} \equiv 1 \pmod{2^n} \)? | [
"8"
] | [
"What is the maximum value of \\( n \\) such that \\( 5^{64} \\equiv 1 \\pmod{2^n} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-a844d45b4f14afc38e6b17d5d549ed3cdb84f37f43b6c4f6c4b52c8afaf5f6be | What is the maximum value of \( n \) such that \( n \) knights and \( n \) rooks can be placed on a standard chessboard so that every knight attacks at most 1 knight, every rook attacks at most 1 rook, no rooks attack any knights, and no knights attack any rooks? | [
"16"
] | [
"What is the maximum value of \\( n \\) such that \\( n \\) knights and \\( n \\) rooks can be placed on a standard chessboard so that every knight attacks at most 1 knight, every rook attacks at most 1 rook, no rooks attack any knights, and no knights attack any rooks?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-2fa1a6d16a3c6e797f6d422698c18a72a29eb46ebf1b4fc89193239d220417e9 | What is the maximum value of \( n \) such that any cube can be divided into \( n \) smaller cubes? | [
"19"
] | [
"What is the maximum value of \\( n \\) such that any cube can be divided into \\( n \\) smaller cubes?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bd722fc000c57450df99ee52b779d7c19cc655479e2f2438633742a32e57580a | What is the maximum value of \( n \) such that the combination of \( n \) line segments (each of length 1) forms a line segment of length 28, with the condition that each line segment has at least one point not shared with any of the other \( n-1 \) line segments? | [
"55"
] | [
"What is the maximum value of \\( n \\) such that the combination of \\( n \\) line segments (each of length 1) forms a line segment of length 28, with the condition that each line segment has at least one point not shared with any of the other \\( n-1 \\) line segments?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-b2626bef1f62c9b8dc5edaaf8640b4f218631b2d8c9cfaa66b2947e0d8c72a7f | What is the maximum value of \( x + 2y \) given that \( \sqrt{x + 2} + \sqrt{y - 5} = 6 \)? | [
"80"
] | [
"What is the maximum value of \\( x + 2y \\) given that \\( \\sqrt{x + 2} + \\sqrt{y - 5} = 6 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-5342c5b8a67c86b90f57a162f518998f322ed46b3e187e76628ca8d15afaa4ad | What is the maximum value of \( x \) for which there exist real numbers \( x \) and \( y \) that satisfy the inequalities \( -10 \le x + y \le 4 \) and \( x^2 + y^2 - 36(x + y) \le 2xy \)? | [
"8"
] | [
"What is the maximum value of \\( x \\) for which there exist real numbers \\( x \\) and \\( y \\) that satisfy the inequalities \\( -10 \\le x + y \\le 4 \\) and \\( x^2 + y^2 - 36(x + y) \\le 2xy \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4c7ff5e4b04afa5293c7992956a758a87abdeabf8cd45f23f07508703c6780b8 | What is the maximum value of \( x \sin x \) where \( 0 \leq x < 2\pi \)? | [
"\\(\\frac{(2.0287578381)^2}{\\sqrt{1 + (2.0287578381)^2}}\\)"
] | [
"What is the maximum value of \\( x \\sin x \\) where \\( 0 \\leq x < 2\\pi \\)?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-5f834dd5763b880f890eb4e7cdb13e87f5182c4e8030a66ebec7929d0af2d819 | What is the maximum value of \( |\sin (x-y) - \sin (x-z) + \sin (y-z)| \) for all real numbers \( x, y, z \)? Additionally, determine the values of \( x, y, z \) that achieve this maximum. | [
"\\(\\frac{3\\sqrt{3}}{2}\\)"
] | [
"What is the maximum value of \\( |\\sin (x-y) - \\sin (x-z) + \\sin (y-z)| \\) for all real numbers \\( x, y, z \\)? Additionally, determine the values of \\( x, y, z \\) that achieve this maximum."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-9ca09e5dda09fa11c960f2dd2416770a3f620efcaa33a3284d24cf1a55cc989c | What is the maximum value of \( |\sin(a-b)\sin(b-c)\sin(c-a)| \)? | [
"\\(\\frac{3\\sqrt{3}}{8}\\)"
] | [
"What is the maximum value of \\( |\\sin(a-b)\\sin(b-c)\\sin(c-a)| \\)?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-67b9f0c5588fea39616d4675b5112f693210fa275488c1bb0a051ba6b2f84a62 | What is the maximum value of \((\sqrt{x} + \sqrt{y})^2\), where \(27x^2 + y^2 = 27\)? Find the solution without using calculus (tangent line, derivative, sin, cos, etc.). | [
"8"
] | [
"What is the maximum value of \\((\\sqrt{x} + \\sqrt{y})^2\\), where \\(27x^2 + y^2 = 27\\)? Find the solution without using calculus (tangent line, derivative, sin, cos, etc.)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 3
} | 24 | 0 | 0 |
omr-ws-4281b8c6a70856db65ff7da7204d295600e58778929efc2b9424e240260c8bf5 | What is the maximum value of \((x-y)(y-z)(z-x)\) over the interval \([0,2]^3\)? | [
"2"
] | [
"What is the maximum value of \\((x-y)(y-z)(z-x)\\) over the interval \\([0,2]^3\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ca470ae9046b2b8d51ac56b471e7d074729cdb7816f63ccfbb61eaa0afc7f9f3 | What is the maximum value of \(\frac{4x - 9y}{2}\) that satisfies the equation of the conic \(x^2 + 9y^2 - 4x + 6y + 4 = 0\)? | [
"8"
] | [
"What is the maximum value of \\(\\frac{4x - 9y}{2}\\) that satisfies the equation of the conic \\(x^2 + 9y^2 - 4x + 6y + 4 = 0\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-ccce6e6cf7eea9b5e9051320d739efa22a2a59eaae90fb8f9660fb092632f351 | What is the maximum value of \[ \frac{2x-1}{x^3}+\frac{2y-1}{y^3}+\frac{2z-1}{z^3} \] when \( x, y, z \in \mathbb{R} \), \( x, y, z > \frac{1}{2} \), and \( x + y + z = 3 \)? | [
"3"
] | [
"What is the maximum value of \n\n\\[\n\\frac{2x-1}{x^3}+\\frac{2y-1}{y^3}+\\frac{2z-1}{z^3}\n\\]\n\nwhen \\( x, y, z \\in \\mathbb{R} \\), \\( x, y, z > \\frac{1}{2} \\), and \\( x + y + z = 3 \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-eb15a4eb91ae279b64672f8fae668c03635a7366d903db2ba8079595be753315 | What is the maximum value of \[ \sqrt{|x_1 - x_2|} + \sqrt{|x_2 - x_3|} + \sqrt{|x_3 - x_4|} + \cdots + \sqrt{|x_{n-1} - x_n|} + \sqrt{|x_n - x_1|} \] where \(0 \leq x_i \leq 1\) for \(i = 1, 2, 3, \ldots, n\) and \(n\) is an odd integer greater than or equal to 3? | [
"\\(n\\)"
] | [
"What is the maximum value of \n\\[\n\\sqrt{|x_1 - x_2|} + \\sqrt{|x_2 - x_3|} + \\sqrt{|x_3 - x_4|} + \\cdots + \\sqrt{|x_{n-1} - x_n|} + \\sqrt{|x_n - x_1|}\n\\]\nwhere \\(0 \\leq x_i \\leq 1\\) for \\(i = 1, 2, 3, \\ldots, n\\) and \\(n\\) is an odd integer greater than or equal to 3?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f629d690d38fe5fc28ee42e328b663ce3782af1339cc5066907eab4fbc646046 | What is the maximum value of a $3 \times 3$ determinant whose elements are in the set $\{0, 1, 2, 3\}$? | [
"54"
] | [
"What is the maximum value of a $3 \\times 3$ determinant whose elements are in the set $\\{0, 1, 2, 3\\}$?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-8d444c0988a39c62d2bc7e70229916f4ccb85b5b0025537c59c364f8c1116891 | What is the maximum value of a determinant of an \( n \times n \) matrix where all elements are either 0 or 1? | [
"\\left\\lfloor \\frac{(n+1)^{(n+1)/2}}{2^n} \\right\\rfloor",
"\\left\\lfloor \\frac{(n+1)^{\\frac{n+1}{2}}}{2^n} \\right\\rfloor"
] | [
"What is the maximum value of a determinant of an \\( n \\times n \\) matrix where all elements are either 0 or 1?"
] | [
"aops_c6_high_school_olympiads",
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 18,
"genselect": 7
} | 25 | 0 | 0 |
omr-ws-127dec954f7b0d31ebae531406a93f70495fb3bc4e4e57dfc58a5958a8ae8591 | What is the maximum value of the difference between the largest real root and the smallest real root of the equation system \[\begin{array}{rcl} ax^2 + bx+ c &=& 0 \\ bx^2 + cx+ a &=& 0 \\ cx^2 + ax+ b &=& 0 \end{array}\], where at least one of the reals $a,b,c$ is non-zero? | [
"0"
] | [
"What is the maximum value of the difference between the largest real root and the smallest real root of the equation system \\[\\begin{array}{rcl}\nax^2 + bx+ c &=& 0 \\\\\nbx^2 + cx+ a &=& 0 \\\\\ncx^2 + ax+ b &=& 0\n\\end{array}\\], where at least one of the reals $a,b,c$ is non-zero?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-0c5eb93e6343e2c5bc5adcf43870fce92623b463ee3fda2e13eed904c1a0d05e | What is the maximum value of the equation $(7-x)^4(2+x)^5$ when $x$ is between -2 and 7? | [
"800000"
] | [
"What is the maximum value of the equation $(7-x)^4(2+x)^5$ when $x$ is between -2 and 7?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-892114de8de1a59495ce791d9772784c093052cc2577961e7b4925d50b12e70c | What is the maximum value of the expression $(x+2)(x+1)x + 19$? | [
"19 + \\dfrac{2\\sqrt{3}}{9}"
] | [
"What is the maximum value of the expression $(x+2)(x+1)x + 19$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9afe652464c0687f23b30ff60d66155e4f6af94bda53c337c4674b31c9f9fd9b | What is the maximum value of the expression $-5r^2 + 40r - 12$ for real $r$? | [
"68"
] | [
"What is the maximum value of the expression $-5r^2 + 40r - 12$ for real $r$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8128be9b9d685e5b5caa21adf3ffcf67814e4f310f211a028f4ac9250dfd4de8 | What is the maximum value of the expression $\left(\sqrt{x}+1\right)^2+\left(2\sqrt{x}+4\right)^2 + \left(\sqrt{42-5x}+5\right)^2$ for $x \in \mathbb{R}$ such that $0 \leq x \leq \frac{42}{5}$? | [
"84 + \\dfrac{4\\sqrt{10815}}{5}"
] | [
"What is the maximum value of the expression $\\left(\\sqrt{x}+1\\right)^2+\\left(2\\sqrt{x}+4\\right)^2 + \\left(\\sqrt{42-5x}+5\\right)^2$ for $x \\in \\mathbb{R}$ such that $0 \\leq x \\leq \\frac{42}{5}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-4fa7128a9dae1659328315be32abd2e656e2f847242b1c709d6647ff854a1040 | What is the maximum value of the expression $\left|\frac{a - b}{5 - ab}\right|$, where $a$ and $b$ are complex numbers such that $|a| \le 1$, $|b| \le 1$. Given that the answer be written as $\frac{p}{q}$ in simplest form, find $p + q$. | [
"3"
] | [
"What is the maximum value of the expression $\\left|\\frac{a - b}{5 - ab}\\right|$, where $a$ and $b$ are complex numbers such that $|a| \\le 1$, $|b| \\le 1$. Given that the answer be written as $\\frac{p}{q}$ in simplest form, find $p + q$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-30c27d9ef2ca476df4c923e216db1bfc652635a181d9516516dd3866895b26b7 | What is the maximum value of the expression $a^2 + b^2 + c^2$ given that $a + b + c = 5$ and $a, b, c$ are the lengths of a triangle? | [
"\\(\\frac{25}{2}\\)"
] | [
"What is the maximum value of the expression $a^2 + b^2 + c^2$ given that $a + b + c = 5$ and $a, b, c$ are the lengths of a triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-5734b6d413e11bf2fa863ba0ec32b9fb766a0141925ff3cc7093dc9f49c32072 | What is the maximum value of the expression $a^2 + b^2 + c^2$ given that $a + b + c = 5$ and $ab, bc, ca$ are the lengths of a triangle? | [
"\\(\\frac{25}{2}(3 - \\sqrt{5})\\)"
] | [
"What is the maximum value of the expression $a^2 + b^2 + c^2$ given that $a + b + c = 5$ and $ab, bc, ca$ are the lengths of a triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8acd9857c4a43dc94967b5c9250b6fae71e02b953605ef4f12c136e05fc0309b | What is the maximum value of the function $f(x) = \frac{(1 + x)^{0.3}}{1 + x^{0.3}}$ in the interval $[0,1]$? | [
"1"
] | [
"What is the maximum value of the function $f(x) = \\frac{(1 + x)^{0.3}}{1 + x^{0.3}}$ in the interval $[0,1]$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-24f6f16e3de51b44b8ab525dc3f0039e7d0443fa8a99828d83e3172e474b75c0 | What is the maximum value of the function \( \sin(4x) + \cos(5x) \)? | [
"\\dfrac{\\sqrt{41}}{5}"
] | [
"What is the maximum value of the function \\( \\sin(4x) + \\cos(5x) \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-2f8434532e9b0de14f588d06717b5b675eaaea048d7f704da102242fa8b07f06 | What is the maximum value of the function \( f(x) = x^4 - 6x^2 + 8x - 3 \)? | [
"-27"
] | [
"What is the maximum value of the function \\( f(x) = x^4 - 6x^2 + 8x - 3 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-072584ceee0341d77e78d48f996c76faf010214360ed98cb6895300791043a8a | What is the maximum value of the function \( f(x, y) = x^2 + y^2 \) subject to the constraint \( y - x^2 + 1 = 0 \)? | [
"1"
] | [
"What is the maximum value of the function \\( f(x, y) = x^2 + y^2 \\) subject to the constraint \\( y - x^2 + 1 = 0 \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-07aea57ae50804db62d57d93609b4593a8425fa57f786729b140d7e0b741cd7b | What is the maximum value of the function \( f(z) = |z^3 - z + 2| \) on the unit circle \( |z| = 1 \)? | [
"\\(\\sqrt{13}\\)"
] | [
"What is the maximum value of the function \\( f(z) = |z^3 - z + 2| \\) on the unit circle \\( |z| = 1 \\)?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-e42a41ea262517964f4ae745dd404b3fb9960fdbeeb2569792f506592fbb40b3 | What is the maximum value of the integral $\int_a^b \left(\frac{3}{4} - x - x^2\right) \, dx$ over all possible values of $a$ and $b$ where $a < b$? | [
"\\(\\frac{4}{3}\\)"
] | [
"What is the maximum value of the integral $\\int_a^b \\left(\\frac{3}{4} - x - x^2\\right) \\, dx$ over all possible values of $a$ and $b$ where $a < b$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-3070ec24f9835ea7966f417700fa2f467404314a147836984783eef874db7d05 | What is the maximum value of the product \(X_1 \times X_2 \times \cdots \times X_n\) given that \(X_1 + X_2 + \cdots + X_n = 2002\) and \(1 \leq n \leq 2002\)? | [
"\\(2^{2}3^{666}\\)"
] | [
"What is the maximum value of the product \\(X_1 \\times X_2 \\times \\cdots \\times X_n\\) given that \\(X_1 + X_2 + \\cdots + X_n = 2002\\) and \\(1 \\leq n \\leq 2002\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 11
} | 15 | 0 | 0 |
omr-ws-fffc831328d50c920ab6dc48e9703828a077bd5039f55de6788130e4d8fd1283 | What is the maximum value of the product of 10 natural numbers whose sum is 2012? | [
"\\(201^8 \\cdot 202^2\\)"
] | [
"What is the maximum value of the product of 10 natural numbers whose sum is 2012?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 2
} | 50 | 0 | 0 |
omr-ws-a96af337f46a4879ac83c6456edb4002f8bf281662e6171629a5370715d495ff | What is the maximum value of the product of positive integers whose sum is 20? | [
"1458"
] | [
"What is the maximum value of the product of positive integers whose sum is 20?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-863bd5150b0a62e2effdb627e070ab95972b7d1245ad4c7891ddf56da5e5b8cd | What is the maximum value of the quotient of the areas of a triangle and its incircle? When do we get this value? | [
"\\(\\frac{\\pi}{\\sqrt{27}}\\)"
] | [
"What is the maximum value of the quotient of the areas of a triangle and its incircle? When do we get this value?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-0816d3b2d1f593428ecd4b7d1a52829250f0e1494fb2cee2867f696e946b989c | What is the maximum value of the ratio of the inradius \( r \) to the semiperimeter \( p \) of a triangle? Is there an inequality for \( \frac{r}{p} \)? | [
"\\(\\frac{\\sqrt{3}}{9}\\)"
] | [
"What is the maximum value of the ratio of the inradius \\( r \\) to the semiperimeter \\( p \\) of a triangle? Is there an inequality for \\( \\frac{r}{p} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-9a5e60593f26f40308eb20d851344cf45ed69d6fdfd893f8b2bc6c6c19187f8e | What is the maximum value of the sum of the areas of two non-overlapping circles that can be drawn inside a unit square? | [
"\\frac{(9 - 6\\sqrt{2})\\pi}{2}"
] | [
"What is the maximum value of the sum of the areas of two non-overlapping circles that can be drawn inside a unit square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-8e8bf897b3dba2554d90420b5cf3493b39650432564a5641a602c15497631512 | What is the maximum value that can be obtained by splitting 1994 into an arbitrary number of real numbers and taking their product? | [
"\\(\\left( \\frac{1994}{e} \\right)^{\\frac{1994}{e}}\\)"
] | [
"What is the maximum value that can be obtained by splitting 1994 into an arbitrary number of real numbers and taking their product?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-636fcc61191cf5cc314998fb177f67db8e6e804067f5c7f7dfad6dd580a78231 | What is the maximum variance that a random variable $X$, distributed in the interval $[0,1]$, may have? | [
"\\(\\frac{1}{4}\\)"
] | [
"What is the maximum variance that a random variable $X$, distributed in the interval $[0,1]$, may have?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-837853fd2d9d73da352ad35dfdc552facd6f56f99bcff1c7598c1f82be9ffc21 | What is the maximum vertical height \( h_a \) of triangle \( ABC \) given its perimeter is 19 meters and the lateral height \( h_b \) is 7 meters? | [
"4.20599"
] | [
"What is the maximum vertical height \\( h_a \\) of triangle \\( ABC \\) given its perimeter is 19 meters and the lateral height \\( h_b \\) is 7 meters?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 7
} | 7 | 0 | 0 |
omr-ws-3c1c297ad41c0a1a4c0c1af73f1afe0d4de28f56b463322c388418f6232585fa | What is the maximum volume attainable for a cone made from a circular sheet of paper with radius \( r \) by cutting out a sector and gluing the cut edges of the remaining piece together? | [
"\\dfrac{2\\sqrt{3}}{27} \\pi r^3"
] | [
"What is the maximum volume attainable for a cone made from a circular sheet of paper with radius \\( r \\) by cutting out a sector and gluing the cut edges of the remaining piece together?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-7ea5b46a7845cfc94529e24f4aae89ed0f1209e766db1ee302f1feec3e98e176 | What is the maximum volume of a box that can be created from a piece of cardboard with an area of 2700 cm², where the box has an overlapping cover on the front, left, and right sides, and the area of the overlap is the same as the size of the overlapped sides? | [
"2250\\sqrt{6}"
] | [
"What is the maximum volume of a box that can be created from a piece of cardboard with an area of 2700 cm², where the box has an overlapping cover on the front, left, and right sides, and the area of the overlap is the same as the size of the overlapped sides?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-1631856c1bf6554aad31506d774cdc24887779a24d88a19d5995483744ed8c2e | What is the maximum volume of a cone that can be cut out from a sphere? | [
"\\(\\frac{32 \\pi r^3}{81}\\)"
] | [
"What is the maximum volume of a cone that can be cut out from a sphere?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-b6813b9e82e25551173c8a2b01bc654c012026c608080635466a17e0b76038cc | What is the maximum volume of a cylinder inscribed within a cone with base radius $r$ and height $h$? | [
"\\(\\frac{4\\pi r^2 h}{27}\\)"
] | [
"What is the maximum volume of a cylinder inscribed within a cone with base radius $r$ and height $h$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 26
} | 43 | 0 | 0 |
omr-ws-10fcaaca72dace3dbdbcb175f4539b4f96a9dd6e2a0e5c53ce335da057500036 | What is the maximum volume of a tetrahedron if at least four of its edges are at most one unit in length? | [
"\\dfrac{\\sqrt{3}}{12}"
] | [
"What is the maximum volume of a tetrahedron if at least four of its edges are at most one unit in length?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-06503be02a552c978305db1988739150252494d7a708f3536379e00699fcc3c0 | What is the maximum volume of a tetrahedron whose edge midpoints lie on a sphere? | [
"\\(\\frac{8r^3}{3}\\)"
] | [
"What is the maximum volume of a tetrahedron whose edge midpoints lie on a sphere?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 1,
"tir": 2
} | 8 | 0 | 0 |
omr-ws-5d9ad1acd1203dcda6baebf95cb46a1980b8fe4adfc0217eec421cbcea462279 | What is the maximum volume of a tetrahedron with vertices in a given cube (or on its boundary)? | [
"\\(\\frac{1}{6}\\)"
] | [
"What is the maximum volume of a tetrahedron with vertices in a given cube (or on its boundary)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7c100eca622b41e5d2b8a42a01fe0986e495a3c977291ad6fa2eb2700992b374 | What is the mean and variance of the stochastic integral $\int_{0}^{T} sgn(B_t) \ dB_t$, where $sgn(x) = -1$ for $x < 0$, $sgn(0) = 0$, $sgn(x) = 1$ for $x > 0$, and $B_t$ is a Brownian motion with mean $0$ and variance $t$? | [
"Mean = 0, Variance = T"
] | [
"What is the mean and variance of the stochastic integral $\\int_{0}^{T} sgn(B_t) \\ dB_t$, where $sgn(x) = -1$ for $x < 0$, $sgn(0) = 0$, $sgn(x) = 1$ for $x > 0$, and $B_t$ is a Brownian motion with mean $0$ and variance $t$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 13
} | 14 | 0 | 0 |
omr-ws-43805167da441f7443cc647aad12307e7dec89ebae3af10b91ad78d658231b2d | What is the mean distance between two random points in a unit square? | [
"\\(\\frac{2+\\sqrt{2}+5\\ln(1+\\sqrt{2})}{15}\\)",
"\\dfrac{2 + \\sqrt{2} + 5 \\ln(1 + \\sqrt{2})}{15}"
] | [
"What is the mean distance between two random points in a unit square?"
] | [
"aops_c6_high_school_olympiads",
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 24
} | 55 | 0 | 0 |
omr-ws-a0f6dba4509da21222458e11d879c59488c521a11f3c18293464e9ac91363f56 | What is the mean number of total rolls needed to get 4 sixes when a single fair die is rolled repeatedly? | [
"24"
] | [
"What is the mean number of total rolls needed to get 4 sixes when a single fair die is rolled repeatedly?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9fad4dd7b78e455ee58a9e1b321b91793be2156573f807bb8f72d573fed69607 | What is the mean value of the function \( U(x, y) = x^2 - y^2 + x \) on the circle \( |z - i| = 2 \)? | [
"-1"
] | [
"What is the mean value of the function \\( U(x, y) = x^2 - y^2 + x \\) on the circle \\( |z - i| = 2 \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4630ee8e8dcedd5fbe88a35fc80023c8c98dedd9c82cbc82a5dece62c5e54a75 | What is the mean value of the measures of the three interior angles of any triangle? | [
"60^\\circ"
] | [
"What is the mean value of the measures of the three interior angles of any triangle?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-90736ea02677155cc6f392708dc6f7eacb6e9708c7102a7a958ede6d9535a496 | What is the measure \( x \) of the smallest positive angle in radians such that \(\sin(x \text{ radians}) = \sin(x \text{ degrees})\)? Express your answer as a simple fraction involving \(\pi\). | [
"\\(\\frac{180\\pi}{180 + \\pi}\\)"
] | [
"What is the measure \\( x \\) of the smallest positive angle in radians such that \\(\\sin(x \\text{ radians}) = \\sin(x \\text{ degrees})\\)? Express your answer as a simple fraction involving \\(\\pi\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-225cad8ebb1e28cfc17c52c8d88a34ed87ee1089f8b1ea0531dcd2f264f8873a | What is the measure of an angle, in degrees, if its supplement is six times its complement? | [
"72^{\\circ}"
] | [
"What is the measure of an angle, in degrees, if its supplement is six times its complement?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7ea0a0ee5ca8e1dded3e94e02e3728334a53f421db0dd2c9608359dc2071fb27 | What is the measure of the acute angles of a rhombus if its side is the geometric mean of the diagonals? | [
"\\(30\\) degrees or \\(\\frac{\\pi}{6}\\) radians."
] | [
"What is the measure of the acute angles of a rhombus if its side is the geometric mean of the diagonals?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-678ca7602afb2217b80f012ae14cb21fcefa7b902607ca1e88b59340216e49b3 | What is the measure of the angle formed by the intersection of the bisectors of the exterior angles at the vertices of the acute angles of a right triangle? | [
"45 degrees"
] | [
"What is the measure of the angle formed by the intersection of the bisectors of the exterior angles at the vertices of the acute angles of a right triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 5,
"tir": 12
} | 26 | 0 | 0 |
omr-ws-9712f11358f52f5f37bcbc239098c946614f361b7295dbaa971dee95fc610938 | What is the measure of the smaller angle between the hands of a 12-hour clock at 12:25 pm, in degrees? Express your answer as a decimal to the nearest tenth. | [
"137.5\\text{ degrees}"
] | [
"What is the measure of the smaller angle between the hands of a 12-hour clock at 12:25 pm, in degrees? Express your answer as a decimal to the nearest tenth."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-358190ea477b00ea618eebe85fefae3eaf18b760f6123e8afee7f412e5a5d3a6 | What is the measure of the smaller angle formed by the vectors \((-5, -2)\) and \((5, 7)\) to the nearest tenth of a degree? | [
"147.3"
] | [
"What is the measure of the smaller angle formed by the vectors \\((-5, -2)\\) and \\((5, 7)\\) to the nearest tenth of a degree?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 1,
"tir": 30
} | 44 | 0 | 0 |
omr-ws-2db006e87c43be5bc244d01c810940f7583dd3d3a222ad25cf38d804e755ace8 | What is the measure of the smaller angle made by the minute hand and the hour hand at 2:48? | [
"152"
] | [
"What is the measure of the smaller angle made by the minute hand and the hour hand at 2:48?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-977ffdcf6480456d11239b3b69188b746beae371e2fa05b4d9a47ef173ca2459 | What is the measure, in degrees, of one interior angle of a regular hexagon? | [
"120"
] | [
"What is the measure, in degrees, of one interior angle of a regular hexagon?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-615a0763e22bfc4ead8c5eef0863c8e6f8deeb47107473803974df2ba9751f64 | What is the measure, in degrees, of the acute angle formed by the hour hand and the minute hand of a 12-hour clock at 6:48? | [
"84"
] | [
"What is the measure, in degrees, of the acute angle formed by the hour hand and the minute hand of a 12-hour clock at 6:48?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-73fc63d59d5ec81ab8037aaea26374cd673fb8aaf8ee0e8788bfd5466c285337 | What is the measure, in degrees, of the acute angle formed by the minute hand and the hour hand on a standard clock when it indicates $9$:$40$? | [
"50"
] | [
"What is the measure, in degrees, of the acute angle formed by the minute hand and the hour hand on a standard clock when it indicates $9$:$40$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-708d2802401d6caf932f285af87c31a420a738e3cd7017b10a991f5658ff32cf | What is the measure, in units, of the hypotenuse of a right triangle with leg lengths of 75 and 100 units? | [
"125"
] | [
"What is the measure, in units, of the hypotenuse of a right triangle with leg lengths of 75 and 100 units?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d0b4b6a3ea103861e3b4e695248725ac36c419855c11f71c8633d7e7bb770e92 | What is the median number of moons per planet? (Include Pluto, although arguments rage on about Pluto's status...) \begin{tabular}{c|c} Planet & $\#$ of Moons\\ \hline Mercury&0\\ Venus &0\\ Earth &1\\ Mars &2\\ Jupiter&16\\ Saturn&23\\ Uranus&15\\ Neptune&2\\ Pluto&5\\ \end{tabular} | [
"2"
] | [
"What is the median number of moons per planet? (Include Pluto, although arguments rage on about Pluto's status...) \\begin{tabular}{c|c}\nPlanet & $\\#$ of Moons\\\\\n\\hline\nMercury&0\\\\\nVenus &0\\\\\nEarth &1\\\\\nMars &2\\\\\nJupiter&16\\\\\nSaturn&23\\\\\nUranus&15\\\\\nNeptune&2\\\\\nPluto&5\\\\\n\\end{tab... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9a88a384ef2ada8a39cca8cdaadd5f0cc32c19b4933222a16602d258219095e0 | What is the median of the first ten positive integers? Express your answer as a decimal to the nearest tenth. | [
"5.5"
] | [
"What is the median of the first ten positive integers? Express your answer as a decimal to the nearest tenth."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5a927a055d61ac25a6c4f335e567642027c649247c6f4e04e89058a54b8d4c18 | What is the median of the following list of 4040 numbers? $$1, 2, 3, \ldots, 2020, 1^2, 2^2, 3^2, \ldots, 2020^2$$ | [
"1976.5"
] | [
"What is the median of the following list of 4040 numbers?\n\n$$1, 2, 3, \\ldots, 2020, 1^2, 2^2, 3^2, \\ldots, 2020^2$$"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-cc54236f07302cdf086724a5874a0226be542892b0cee55e2a08b85187200db2 | What is the median of the smallest element of all nonempty subsets of the set of positive integers less than or equal to 2019? | [
"1"
] | [
"What is the median of the smallest element of all nonempty subsets of the set of positive integers less than or equal to 2019?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-230329818b80e12ee581b389bcd0cedf44c2f87de3d08d24b833c2413a678214 | What is the method to evaluate integrals of the form \[\int\frac{dx}{f(x)\sqrt{g(x)}}\] where \( f(x) \) and \( g(x) \) are linear or quadratic functions of \( x \)? | [
"Euler substitution"
] | [
"What is the method to evaluate integrals of the form \\[\\int\\frac{dx}{f(x)\\sqrt{g(x)}}\\] where \\( f(x) \\) and \\( g(x) \\) are linear or quadratic functions of \\( x \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-4fa0541a8b7f8352a4970aa82d012a72e8bca718e3727592bf35fbd5f5105997 | What is the middle digit of the product $968880726456484032 \cdot 875$? | [
"4"
] | [
"What is the middle digit of the product $968880726456484032 \\cdot 875$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 16,
"genselect": 7
} | 23 | 0 | 0 |
omr-ws-551cc4c01fa6cb8529f08586618df0d80a1297e880f337b369eb595ebc6db277 | What is the middlemost digit of the sequence formed by writing the integers from 1 to 2012 consecutively without spaces in between? | [
"1"
] | [
"What is the middlemost digit of the sequence formed by writing the integers from 1 to 2012 consecutively without spaces in between?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8ee8e67398cdd40328d89f00321f67010d7b5fc8ba15414dab9f5e1c3be19d81 | What is the midpoint of the segment with endpoints (7,-6) and (-3,4)? | [
"(2,-1)"
] | [
"What is the midpoint of the segment with endpoints (7,-6) and (-3,4)?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-007f6e06886889efba4c65f723a1ea8495559085f0e1119204f7b8efe341b065 | What is the minimal area of a convex quadrilateral with vertices on the grid, no additional grid points on its edges or diagonals, and an area greater than $1$? | [
"2"
] | [
"What is the minimal area of a convex quadrilateral with vertices on the grid, no additional grid points on its edges or diagonals, and an area greater than $1$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6b54a3e507aa05a3c2558e4ee4e726b28893457cbfa2cd1812c5500ab7da12dc | What is the minimal area of a quadrilateral with vertices on the grid, no additional grid points on its edges or diagonals, and an area greater than $1$? | [
"2"
] | [
"What is the minimal area of a quadrilateral with vertices on the grid, no additional grid points on its edges or diagonals, and an area greater than $1$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f6f38a86fa4d31c00543678d18c4c2fb621c5e049ab51de644f392a0aaa70ee3 | What is the minimal area that a convex polygon $P$ with exactly 2020 integral coordinates inside can attain? | [
"2091"
] | [
"What is the minimal area that a convex polygon $P$ with exactly 2020 integral coordinates inside can attain?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-c9df5197b0d7a7cb533e38db69a222194e26aa32114c0ebf27d33624a6e3ff9a | What is the minimal degree of a polynomial \( P \in \mathbb{R}[x] \) such that \( P(3) = P\left(\frac{-1}{2}\right) = P(5-2i) = P(4+i) = P'(4+i) = P(4-i) = 0 \)? | [
"8"
] | [
"What is the minimal degree of a polynomial \\( P \\in \\mathbb{R}[x] \\) such that \\( P(3) = P\\left(\\frac{-1}{2}\\right) = P(5-2i) = P(4+i) = P'(4+i) = P(4-i) = 0 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-92df72c59789b8edd4ffddd5a0439a8c31e3eefba3a5ad995e63e36f32b7141f | What is the minimal degree of a polynomial \( P \in \mathbb{R}[x] \) that satisfies: \[ P(3) = P(-12) = P(5-2i) = P(4+i) = P(4-i) = P'(4+i) \]? | [
"8"
] | [
"What is the minimal degree of a polynomial \\( P \\in \\mathbb{R}[x] \\) that satisfies:\n\\[ P(3) = P(-12) = P(5-2i) = P(4+i) = P(4-i) = P'(4+i) \\]?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 2
} | 13 | 0 | 0 |
omr-ws-c9c9539791939ce9d3f457f8feaab064b482d75697c74dbab682a7e910c9618e | What is the minimal length $t$ of a fishing pole such that Mrs. Ippi can reach any point inside a triangular pond with side lengths 6, 7, and 8 from outside the pond? | [
"\\dfrac{\\sqrt{15}}{2}"
] | [
"What is the minimal length $t$ of a fishing pole such that Mrs. Ippi can reach any point inside a triangular pond with side lengths 6, 7, and 8 from outside the pond?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-72bed79ac9850b961cf2f24ff6f744154fbbd47a0afe5e0099d1b37a3c4d0161 | What is the minimal number $n$ of discontinuities that a real function $f$ which satisfies $f(f(x)) = -x$ can have? | [
"2"
] | [
"What is the minimal number $n$ of discontinuities that a real function $f$ which satisfies $f(f(x)) = -x$ can have?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-4f8b903c1390ac5fd89c510ce5cd960538873989f4ece478c88f3da604056b72 | What is the minimal number \( k < n^2 \) such that for any vector space \( V \) of dimension \( k \), there exists a matrix \( M \) such that \( V \) contains \( C(M) \), where \( C(M) \) is the subspace of matrices that commute with \( M \)? | [
"n^2 - n + 1"
] | [
"What is the minimal number \\( k < n^2 \\) such that for any vector space \\( V \\) of dimension \\( k \\), there exists a matrix \\( M \\) such that \\( V \\) contains \\( C(M) \\), where \\( C(M) \\) is the subspace of matrices that commute with \\( M \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-de976b25b1cacddf80ff0757b0682e5419ff6443403b961e9824fc0118ce49c4 | What is the minimal number of $1 \times 2$ dominoes required to tile a $2008 \times 2010$ rectangle using both $1 \times 2$ dominoes and $2 \times 3$ tetraminoes (with two opposite corners removed)? | [
"672680"
] | [
"What is the minimal number of $1 \\times 2$ dominoes required to tile a $2008 \\times 2010$ rectangle using both $1 \\times 2$ dominoes and $2 \\times 3$ tetraminoes (with two opposite corners removed)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-4d554e271dc885f10046de1655692ff3d71c4ba90153281d2efe98eef38453ba | What is the minimal number of cuts needed to ensure that 17 people each get at least one slice from a round pizza? | [
"6"
] | [
"What is the minimal number of cuts needed to ensure that 17 people each get at least one slice from a round pizza?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-4df7f896a20b6a551782f60bcb353e7b1cf4332058faf079b1d9cdbd74f47a40 | What is the minimal number of different roots of the polynomial $P(x) = x^{2010} + ax^n + 1$, where $a \in \mathbb{R}$ and $n \in \mathbb{N}$ with $n \neq 2010$? | [
"0"
] | [
"What is the minimal number of different roots of the polynomial $P(x) = x^{2010} + ax^n + 1$, where $a \\in \\mathbb{R}$ and $n \\in \\mathbb{N}$ with $n \\neq 2010$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-06df1851cbdcb2e9f1107dba3bdf173f2b6c0e43ff9d149dc974375c88eb973d | What is the minimal number of digits that need to be pressed to ensure that a 4-digit secret code (which can include repeated digits) is entered, without knowing the specific code? | [
"10003"
] | [
"What is the minimal number of digits that need to be pressed to ensure that a 4-digit secret code (which can include repeated digits) is entered, without knowing the specific code?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-575319e1b48012c652206a85c516ed9191c273ef9334d83d6978560c68532966 | What is the minimal number of edges in a connected graph with $n$ vertices such that every edge belongs to a triangle? | [
"\\(\\left\\lceil \\frac{3(n-1)}{2} \\right\\rceil\\)"
] | [
"What is the minimal number of edges in a connected graph with $n$ vertices such that every edge belongs to a triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 22,
"genselect": 8,
"tir": 11
} | 41 | 0 | 0 |
omr-ws-d5c2654336cf2eb8b08ba504524ef95acf961998f47c03f208bebb33447a214d | What is the minimal number of folds needed to divide a $1 \times n$ rectangle into $n$ $1 \times 1$ squares with a single cut? | [
"\\(\\left\\lceil \\frac{n-1}{2} \\right\\rceil\\)"
] | [
"What is the minimal number of folds needed to divide a $1 \\times n$ rectangle into $n$ $1 \\times 1$ squares with a single cut?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ef1f99154bdf6a7dcbca7c838ab34353dda6be7a8812514521aa8bf194d691de | What is the minimal number of gardens if $n=6$ for an $n \times n$ lot where each apartment must have a scenic view, meaning that if an apartment is not on the edge, one of the 4 adjacent lots must have a garden? | [
"4"
] | [
"What is the minimal number of gardens if $n=6$ for an $n \\times n$ lot where each apartment must have a scenic view, meaning that if an apartment is not on the edge, one of the 4 adjacent lots must have a garden?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-f089d35709314926b25154d277f539a0929ad29b2cb90dbab38195c1bc277622 | What is the minimal number of generators of the alternating group $A_n$? | [
"2"
] | [
"What is the minimal number of generators of the alternating group $A_n$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3adfa255642cacbe268f0d2c35f886fe4121dfb077583bd5cca9d831607b27f8 | What is the minimal number of matrices $A_1, \ldots, A_k$ such that any $n \times n$ matrix with integer elements and determinant equal to 1 can be written as a product of matrices from $A_1, \ldots, A_k$? | [
"2"
] | [
"What is the minimal number of matrices $A_1, \\ldots, A_k$ such that any $n \\times n$ matrix with integer elements and determinant equal to 1 can be written as a product of matrices from $A_1, \\ldots, A_k$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-6f18acf815e9f3ad0f25a83959b0e3ba76f4c5c22d3f7d45e8a951faf64a65e2 | What is the minimal number of numbers that should be erased from the set $\{1, 2, \ldots, 18\}$ such that the sum of every two remaining numbers is not a perfect square of a positive integer? | [
"9"
] | [
"What is the minimal number of numbers that should be erased from the set $\\{1, 2, \\ldots, 18\\}$ such that the sum of every two remaining numbers is not a perfect square of a positive integer?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 8
} | 18 | 0 | 0 |
omr-ws-de48b6f0b67371fe40ab0fe8235808099567f1f96bd452ae1bff1cd0d563dceb | What is the minimal number of operations needed to repaint a $100 \times 100$ entirely white grid to be entirely black, if in one move we can choose 99 cells from any row or column and change their color? | [
"200"
] | [
"What is the minimal number of operations needed to repaint a $100 \\times 100$ entirely white grid to be entirely black, if in one move we can choose 99 cells from any row or column and change their color?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 18,
"genselect": 3
} | 21 | 0 | 0 |
omr-ws-c45d6364618ac7ba492f1feba6cb257d39bde01cf4f7d6bd3df2e3707e8013a5 | What is the minimal number of pieces of cake one has to cut so that $n$ students can equally partition $m$ identical cakes between themselves? Each cut increases the number of pieces by 1. | [
"\\( m \\times n \\)"
] | [
"What is the minimal number of pieces of cake one has to cut so that $n$ students can equally partition $m$ identical cakes between themselves? Each cut increases the number of pieces by 1."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dd0c352375b2d6917df928edd2d1790a4701bceb2bdef7fc5e9a45ed4dbe015c | What is the minimal number of places where Alice has spent the New Year, given that she has spent the New Year inside every possible hemisphere of Earth except one? Points on the border of a hemisphere do not lie inside the hemisphere. | [
"3"
] | [
"What is the minimal number of places where Alice has spent the New Year, given that she has spent the New Year inside every possible hemisphere of Earth except one? Points on the border of a hemisphere do not lie inside the hemisphere."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 26,
"genselect": 5
} | 31 | 0 | 0 |
omr-ws-0be0ad02c4de5650334a838ddb3e07fdfeff1e34d3111b5c29a7debdf0ad479b | What is the minimal number of projections needed to map one straight line to another on the projective plane? | [
"2"
] | [
"What is the minimal number of projections needed to map one straight line to another on the projective plane?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-db6f32b722c6532b10df615de8f609c7bd203a2edc5bf277ed49037e2ccc3488 | What is the minimal number of rooks needed to attack all boxes in an $n \times n \times n$ cube, where a rook attacks all boxes in the same horizontal, vertical, and height lines? | [
"$\\frac{n^2}{2}$ for even $n$, $\\frac{n^2+1}{2}$ for odd $n$"
] | [
"What is the minimal number of rooks needed to attack all boxes in an $n \\times n \\times n$ cube, where a rook attacks all boxes in the same horizontal, vertical, and height lines?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-8531cc14d033e73788619d32d9f129a4d4fc69562700ced033becc2df997d1ed | What is the minimal number of rooks needed to ensure that all boxes in an $n \times n \times n$ cube are under attack, where a rook attacks all boxes in the same horizontal, vertical, and height lines? | [
"n"
] | [
"What is the minimal number of rooks needed to ensure that all boxes in an $n \\times n \\times n$ cube are under attack, where a rook attacks all boxes in the same horizontal, vertical, and height lines?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-5663cfe9f3b1b2271445e9651b866cd08ab48800bc6df21017fff417220840ba | What is the minimal number of three-cell corners that can be placed on an $8 \times 8$ board so that no additional three-cell corner can be added? Provide an example of such an arrangement. | [
"22"
] | [
"What is the minimal number of three-cell corners that can be placed on an $8 \\times 8$ board so that no additional three-cell corner can be added? Provide an example of such an arrangement."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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