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-13529c7e560502ecdf81a435ab8a54ef17e4eb7694bf57a48706b6a45274371a | Under what conditions does a set \( (X, \tau) \) with the finite-closed topology satisfy the second axiom of countability? | [
"X \\text{ is countable}"
] | [
"Under what conditions does a set \\( (X, \\tau) \\) with the finite-closed topology satisfy the second axiom of countability?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-768b27987f2b25f94d17d20bcb3c0e97bc52e86a26f5d7509aa7053fb36a34cb | Under what conditions does a system of linear equations with the same number of equations as variables have a unique solution? | [
"\\text{When the determinant of the coefficient matrix is not zero}"
] | [
"Under what conditions does a system of linear equations with the same number of equations as variables have a unique solution?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cb0906055d54a15ef9b1ae047932afa7378369ced1feadf63df10018411acce4 | Under what conditions does a unique matrix $\boldsymbol{A}$ exist given the constraints $\boldsymbol{b}_{1}\boldsymbol{A}_{(n\times n)}=\boldsymbol{b}_{2}$, $\boldsymbol{b}_{1}\boldsymbol{1}_{n}=1$, and $\boldsymbol{b}_{2}\boldsymbol{1}_{n}=1$? | [
"n = 1"
] | [
"Under what conditions does a unique matrix $\\boldsymbol{A}$ exist given the constraints $\\boldsymbol{b}_{1}\\boldsymbol{A}_{(n\\times n)}=\\boldsymbol{b}_{2}$, $\\boldsymbol{b}_{1}\\boldsymbol{1}_{n}=1$, and $\\boldsymbol{b}_{2}\\boldsymbol{1}_{n}=1$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-37640393a8af8d39ce177aee90f755ae705da7a48b7ffa1e8ce8ccd90de71005 | Under what conditions does equality hold in the inequality \[ \cot^{2}\alpha + \cot^{2}\beta + \cot^{2}\gamma \geq 3\left(9 \cdot \frac{R^{2}}{p^{2}} - 1\right)? \] | [
"$\\alpha = \\beta = \\gamma$"
] | [
"Under what conditions does equality hold in the inequality\n\\[\n\\cot^{2}\\alpha + \\cot^{2}\\beta + \\cot^{2}\\gamma \\geq 3\\left(9 \\cdot \\frac{R^{2}}{p^{2}} - 1\\right)?\n\\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-cde5706b7cf20392515db519f9f67f6cb5d3c6f0787ff45f83a039e6c46f3cf7 | Under what conditions does equality hold in the inequality \[ \int_0^a \lvert f'(t)f(t) \rvert \, \mathrm{d}t \le \frac{a}{2} \int_0^a \lvert f'(t) \rvert^2 \, \mathrm{d}t? \] | [
"\\( f(t) = kt \\) for some constant \\( k \\)."
] | [
"Under what conditions does equality hold in the inequality\n\\[ \\int_0^a \\lvert f'(t)f(t) \\rvert \\, \\mathrm{d}t \\le \\frac{a}{2} \\int_0^a \\lvert f'(t) \\rvert^2 \\, \\mathrm{d}t? \\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 2
} | 4 | 0 | 0 |
omr-ws-7b877860f01c7d4579df80fcc4bc30b5f315bdebef6cf211d6820ac9c5413662 | Under what conditions does the Fourier integral fail to converge to \( f(x) \), and to what value does it converge? | [
"The Fourier integral converges to \\( \\frac{f(a-) + f(a+)}{2} \\) at points of discontinuity. For the function in (1), it converges to 100 at \\( x = 2 \\)."
] | [
"Under what conditions does the Fourier integral fail to converge to \\( f(x) \\), and to what value does it converge?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-d14c348d3d9596d4ea41acfe0ff8250ae302daac0d45ee9bda20c685af5f7da4 | Under what conditions does the area of the region formed by intersecting four lines perpendicular to the sides of a parallelogram (not a rectangle) at their midpoints equal the area of the parallelogram? | [
"\\text{The parallelogram has angles of } 45^\\circ \\text{ and } 135^\\circ."
] | [
"Under what conditions does the area of the region formed by intersecting four lines perpendicular to the sides of a parallelogram (not a rectangle) at their midpoints equal the area of the parallelogram?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-fbb6357684e7230566aa6cd5483d691d9e0ffc80c43821f4c48ae763a6b5b1be | Under what conditions does the equality \( a - b = \sqrt{4n - 3} \) hold for \( n^2 + 1 = ab \) with \( a, b, n \) being positive integers and \( a > b \)? | [
"\\( n = r^2 + r + 1 \\) for some nonnegative integer \\( r \\)"
] | [
"Under what conditions does the equality \\( a - b = \\sqrt{4n - 3} \\) hold for \\( n^2 + 1 = ab \\) with \\( a, b, n \\) being positive integers and \\( a > b \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 42,
"tir": 10
} | 52 | 0 | 0 |
omr-ws-6e53eb1c79bacd585748ee835982fcb5a3b3620a8fc3bd5fc763891ffa36e375 | Under what conditions does the equality hold between the Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM)? | [
"\\(a_1 = a_2 = \\cdots = a_{n-1} = a_n\\)"
] | [
"Under what conditions does the equality hold between the Arithmetic Mean (AM), Geometric Mean (GM), and Harmonic Mean (HM)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-cba21380f69f4c2cb739df01836e2ef7ff549ef5147bf1a1808297a4f804e366 | Under what conditions does the equality hold in the inequality: \[ a_1b_1 + a_2b_2 + \cdots + a_nb_n \geq a_1x_1 + a_2x_2 + \cdots + a_nx_n, \] where \(a_1 \leq a_2 \leq \cdots \leq a_n\) and \(b_1 \leq b_2 \leq \cdots \leq b_n\) are real numbers, and \((x_1, x_2, \ldots, x_n)\) is a permutation of \((b_1, b_2, \ldots,... | [
"\\( (x_1, x_2, \\ldots, x_n) = (b_1, b_2, \\ldots, b_n) \\)"
] | [
"Under what conditions does the equality hold in the inequality:\n\\[ a_1b_1 + a_2b_2 + \\cdots + a_nb_n \\geq a_1x_1 + a_2x_2 + \\cdots + a_nx_n, \\]\nwhere \\(a_1 \\leq a_2 \\leq \\cdots \\leq a_n\\) and \\(b_1 \\leq b_2 \\leq \\cdots \\leq b_n\\) are real numbers, and \\((x_1, x_2, \\ldots, x_n)\\) is a permutat... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 3
} | 4 | 0 | 0 |
omr-ws-dca2a6b456b32de90aa25f6355e44de7f73517afba791d572c36aa2b4d5306d1 | Under what conditions does the equation $\frac{d\left(\frac{\partial L}{\partial \dot{q}}\right)}{dt} = \frac{\partial \dot{L}}{\partial \dot{q}}$ hold? | [
"\\frac{\\partial L}{\\partial q} = 0"
] | [
"Under what conditions does the equation $\\frac{d\\left(\\frac{\\partial L}{\\partial \\dot{q}}\\right)}{dt} = \\frac{\\partial \\dot{L}}{\\partial \\dot{q}}$ hold?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-baad05bde9c88ff38836a6c379fdecb2642c7dd5ce759507cf5fb674a7801e41 | Under what conditions does the equation $kn + b = Aq^n$ have exactly 3 integer solutions for $n$? Find the relationship between the constants $k$, $b$, $A$, and $q$. | [
"q = -2"
] | [
"Under what conditions does the equation $kn + b = Aq^n$ have exactly 3 integer solutions for $n$? Find the relationship between the constants $k$, $b$, $A$, and $q$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-9fbbdcd8a935b2ee2bf5d7051272a4108166a4dcf78a0c8c234fced189dd74c6 | Under what conditions does the equation $x^2 + my^2 = n$ have integer solutions, where $m$ and $n$ are integers? | [
"\\text{The equation has integer solutions if } n \\text{ is non-negative (for } m > 0\\text{), and for every prime } p \\text{ dividing } m, n \\text{ is a quadratic residue modulo } p, \\text{ with additional congruence conditions modulo 4 or 8 if } m \\text{ is even.}"
] | [
"Under what conditions does the equation $x^2 + my^2 = n$ have integer solutions, where $m$ and $n$ are integers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 6
} | 7 | 0 | 0 |
omr-ws-b17be3a4c0dcae336ff8908821f9d4901552e23bc9ed855926b860b8ff98e0bd | Under what conditions does the equation \( f(x) = 0 \) have 0 real roots, 1 real root, or 2 real roots when \( f \) is convex? | [
"0 real roots if \\( f \\) has a minimum greater than 0 or no minimum with an infimum greater than 0; 1 real root if only one limit is greater than 0; 2 real roots if \\( f \\) takes a negative minimum and has two positive limits at the bounds of its domain."
] | [
"Under what conditions does the equation \\( f(x) = 0 \\) have 0 real roots, 1 real root, or 2 real roots when \\( f \\) is convex?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 13
} | 16 | 0 | 0 |
omr-ws-cfc8915297148e108674c73fe166b6cb1dc1504d6b9c3436fa312e256543a09c | Under what conditions does the equation \(\sqrt{a + \sqrt{a^2 - b}} + \sqrt{a - \sqrt{a^2 - b}} = \sqrt{2a + 2\sqrt{b}}\) hold? | [
"\\(a^2 \\geq b \\quad \\text{and} \\quad a \\geq -\\sqrt{b}\\)"
] | [
"Under what conditions does the equation \\(\\sqrt{a + \\sqrt{a^2 - b}} + \\sqrt{a - \\sqrt{a^2 - b}} = \\sqrt{2a + 2\\sqrt{b}}\\) hold?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-282376af32631c481bd15ab65b0cfa69e9c47d41127660726421dee3ac8175f7 | Under what conditions does the exponential function $y=a^x$ intersect the line $y=x$? | [
"\\( (0, e^{1/e}] \\)"
] | [
"Under what conditions does the exponential function $y=a^x$ intersect the line $y=x$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 1,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-a1b577327665371e83ffe051aacfeefd786908ab376fe940ff2645f9ef6a0aa2 | Under what conditions does the following limit hold? \[ \lim_{x \to 0^+} x \sum_{n \geq 1} f(nx) = \int_0^\infty f(t) \, dt \] | [
"f \\text{ is non-negative, decreasing, and } \\int_0^\\infty f(t) \\, dt \\text{ converges}"
] | [
"Under what conditions does the following limit hold?\n\\[ \\lim_{x \\to 0^+} x \\sum_{n \\geq 1} f(nx) = \\int_0^\\infty f(t) \\, dt \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 2
} | 8 | 0 | 0 |
omr-ws-058304a8fcda888b0c907f1e701ce0c5b5fdcc42095c12fb8d6690531adc34f5 | Under what conditions does the graph $g$ with vertices $v_1, \ldots, v_p$ contain a cycle of length $p$? | [
"\\text{Each vertex has degree at least } \\frac{p}{2} \\text{ and } p \\geq 3"
] | [
"Under what conditions does the graph $g$ with vertices $v_1, \\ldots, v_p$ contain a cycle of length $p$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-573ac71e73019d3a598ef9029364e53d0176334fb2a000bf36f2856be1c03a9d | Under what conditions does the inequality $\left(\displaystyle\int_a^b f \right)^\alpha \le \displaystyle\int_a^b f^\alpha$ hold for a continuous real function $f$ on the interval $[a,b]$ and a real number $\alpha$? | [
"(\\alpha \\geq 1 \\text{ and } b - a \\leq 1) \\text{ or } (\\alpha \\leq 1 \\text{ and } b - a \\geq 1)"
] | [
"Under what conditions does the inequality $\\left(\\displaystyle\\int_a^b f \\right)^\\alpha \\le \\displaystyle\\int_a^b f^\\alpha$ hold for a continuous real function $f$ on the interval $[a,b]$ and a real number $\\alpha$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 7
} | 16 | 0 | 0 |
omr-ws-7ecef970facdac498768ce0d7f6016f611a22cec656087d53dfa6c57111e298c | Under what conditions does the system of equations \[ \begin{cases} x + y + mz = a \\ x + my + z = b \\ mx + y + z = c \end{cases} \] have a solution? | [
"\\( a = b = c \\) when \\( m = 1 \\), \\( a + b + c = 0 \\) when \\( m = -2 \\), and no additional conditions when \\( m \\neq 1 \\) and \\( m \\neq -2 \\)."
] | [
"Under what conditions does the system of equations\n\n\\[\n\\begin{cases}\nx + y + mz = a \\\\\nx + my + z = b \\\\\nmx + y + z = c\n\\end{cases}\n\\]\n\nhave a solution?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 33,
"genselect": 7,
"tir": 27
} | 67 | 0 | 0 |
omr-ws-34885307a3149bc1a423123a71637f3927db30eb8a6c8ffeec9576947e3e9a07 | Under what conditions does the system of equations have exactly one solution? \[ \begin{cases} y = \frac{2x}{x - |x|} \\ (x + a)^2 + y + a = 3 \end{cases} \] | [
"a \\in (-2, 1] \\cup \\{2\\}"
] | [
"Under what conditions does the system of equations have exactly one solution?\n\\[\n\\begin{cases}\ny = \\frac{2x}{x - |x|} \\\\\n(x + a)^2 + y + a = 3\n\\end{cases}\n\\]"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-32205f8266cb4595bfc4e15250c84f3eca12894a1504ca6b3be57cf658ceffbb | Under what conditions does there exist a basis such that both matrices \( A \) and \( B \) are in their normal form or in row echelon form, given that \( P(A, B) = 0 \) is a matrix equation with \( A \) and \( B \) being \( n \times n \) matrices? | [
"A \\text{ and } B \\text{ commute}"
] | [
"Under what conditions does there exist a basis such that both matrices \\( A \\) and \\( B \\) are in their normal form or in row echelon form, given that \\( P(A, B) = 0 \\) is a matrix equation with \\( A \\) and \\( B \\) being \\( n \\times n \\) matrices?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-7689fba0fc363738cd26d79286b76578878a37878bb5646dc76ee64c0afb8867 | Under what conditions is $2^p - 1$ a prime number, where $p$ is a prime number? | [
"p \\text{ is a prime number}"
] | [
"Under what conditions is $2^p - 1$ a prime number, where $p$ is a prime number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-946e90b0ac5f02e7bce9acbbf2c8619c8ff4a970ac98603657db3b5a7f75a8b7 | Under what conditions is $\sqrt{a^2+b^2}=a+b$ true, where $a$ and $b$ are real numbers? | [
"It is true if and only if \\( ab=0 \\) and \\( a+b \\geq 0 \\)."
] | [
"Under what conditions is $\\sqrt{a^2+b^2}=a+b$ true, where $a$ and $b$ are real numbers?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-f2b84414ca4eba72d0d037bcded4b0b3ddec0e138e5e4507e0d84fcbcdb0b679 | Under what conditions is $a^b > b^a$ true, and when is $b^a > a^b$ true? | [
"a^b > b^a \\text{ when } \\frac{\\ln a}{a} > \\frac{\\ln b}{b} \\text{ and } b^a > a^b \\text{ when } \\frac{\\ln b}{b} > \\frac{\\ln a}{a}"
] | [
"Under what conditions is $a^b > b^a$ true, and when is $b^a > a^b$ true?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 2
} | 13 | 0 | 0 |
omr-ws-1328a0558237c80b42a31038801d4f4718e40ecba744534c6d91e4faf8627c92 | Under what conditions is \( X^T X \) invertible? | [
"\\( X \\) must be full column rank, i.e., \\( \\text{rank}(X) = n \\)."
] | [
"Under what conditions is \\( X^T X \\) invertible?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-7406e86f63ef4c9a01114e45cd764600ffc216327a46fd4174b052e3e184884c | Under what conditions is \(\sqrt{a^2 + b^2} = a + b\) true, where \(a\) and \(b\) are real numbers? | [
"\\text{At least one of } a \\text{ or } b \\text{ is } 0 \\text{ and the other is non-negative}"
] | [
"Under what conditions is \\(\\sqrt{a^2 + b^2} = a + b\\) true, where \\(a\\) and \\(b\\) are real numbers?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-09a9557474a3922025e113d82f0dea4d5ec42d7ab1c6f44a49294f7786c93b17 | Under what conditions is \(a^x - b^x\) a prime number? | [
"\\(a = b + 1\\) and \\(x\\) is a prime number."
] | [
"Under what conditions is \\(a^x - b^x\\) a prime number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 3
} | 5 | 0 | 0 |
omr-ws-1e47afdeaf0cbafe722bd9af27ad382b082f8ec307988ff057fa57e369879540 | Under what conditions is a prime number \( p \) congruent to 5 modulo 6? | [
"\\( p > 3 \\) and \\( p \\not\\equiv 1 \\pmod{6} \\)"
] | [
"Under what conditions is a prime number \\( p \\) congruent to 5 modulo 6?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 5,
"tir": 27
} | 47 | 0 | 0 |
omr-ws-8202fa7010aed064b15865b858a0e7be83fd5b06cb6fafebbeec223ff3db88eb | Under what conditions is it possible to construct a line \( r \) parallel to \( PQ \) such that the segment of \( r \) that intersects both circles \( p \) and \( q \) is twice the distance \( PQ \)? Construct this line using only Euclidean tools. | [
"\\( P \\in q \\), \\( Q \\in p \\), \\( PQ \\leq \\frac{1}{3}(R_p + R_q) \\)"
] | [
"Under what conditions is it possible to construct a line \\( r \\) parallel to \\( PQ \\) such that the segment of \\( r \\) that intersects both circles \\( p \\) and \\( q \\) is twice the distance \\( PQ \\)? Construct this line using only Euclidean tools."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-06ff37e5ab7ec22d9da098d7cbda2c640e4b8d2c06330d9ade83b7c517c4a786 | Under what conditions is it possible to partition the terms of the sum $\sum_{i=1}^{n}{i^2}$ into 3 separate sums such that those three separate sums have equal values? Provide an example of such a partition if it exists. | [
"n \\equiv 0, 4, \\text{ or } 8 \\pmod{9}"
] | [
"Under what conditions is it possible to partition the terms of the sum $\\sum_{i=1}^{n}{i^2}$ into 3 separate sums such that those three separate sums have equal values? Provide an example of such a partition if it exists."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 5
} | 20 | 0 | 0 |
omr-ws-d5bdc5e0e0f304ec2345da75d7525966c187543a6e2bf5df397101e112822452 | Under what conditions is it possible to transfer all chips to a single vertex in a finite number of steps for an $n$-gon where each vertex $j$ has $a_j = j$ chips, and at each step, you may select two vertices and transfer a chip from each vertex to a neighboring vertex in opposite directions? | [
"n \\equiv 1 \\pmod{3}"
] | [
"Under what conditions is it possible to transfer all chips to a single vertex in a finite number of steps for an $n$-gon where each vertex $j$ has $a_j = j$ chips, and at each step, you may select two vertices and transfer a chip from each vertex to a neighboring vertex in opposite directions?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-27db0eb2b5401e669a178f8c853b129186ee86e430ba96e1cd21dfae33c5370d | Under what conditions is it true that $\arg\left(z_1 z_2\right) = \arg(z_1) + \arg(z_2)$? | [
"\\(\\arg(z_1 z_2) = \\arg(z_1) + \\arg(z_2) \\pmod{2\\pi}\\)"
] | [
"Under what conditions is it true that $\\arg\\left(z_1 z_2\\right) = \\arg(z_1) + \\arg(z_2)$?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 6
} | 6 | 0 | 0 |
omr-ws-72c307c7589e6b02859b089d37efad7c0f2c8e26aa17b74fc8a422626248a82d | Under what conditions is it valid to interchange the differentiation and integration operators in the expression \( f'(a) = \int_0^\infty \frac{\sin x - x \cos(ax)}{x^2} \, dx \) for the function \( f(a) = \int_0^\infty \frac{a \sin(x) - \sin(ax)}{x^2} \, dx \)? | [
"a > 0"
] | [
"Under what conditions is it valid to interchange the differentiation and integration operators in the expression \\( f'(a) = \\int_0^\\infty \\frac{\\sin x - x \\cos(ax)}{x^2} \\, dx \\) for the function \\( f(a) = \\int_0^\\infty \\frac{a \\sin(x) - \\sin(ax)}{x^2} \\, dx \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-7a5175fd66c20e493e287e67125a293d1f146f4b1189b568ceb0371407c40987 | Under what conditions is it valid to take the reciprocal of both sides of the equation \(\frac{1}{x} = \frac{m+p}{pm}\) to obtain \( x = \frac{pm}{m+p} \)? | [
"\\( x \\neq 0 \\), \\( m \\neq -p \\), \\( p \\neq 0 \\), \\( m \\neq 0 \\)"
] | [
"Under what conditions is it valid to take the reciprocal of both sides of the equation \\(\\frac{1}{x} = \\frac{m+p}{pm}\\) to obtain \\( x = \\frac{pm}{m+p} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-7549726ee24fe951f938f718730d7ed76e8424191ec4760e07c2532060bdff5e | Under what conditions is the block matrix $\begin{bmatrix}A & B \\C & D \end{bmatrix}$ singular when each of the submatrices $A$, $B$, $C$, and $D$ are singular? | [
"\\ker(A) \\cap \\ker(C) \\neq \\{0\\} \\text{ or } \\ker(B) \\cap \\ker(D) \\neq \\{0\\}"
] | [
"Under what conditions is the block matrix $\\begin{bmatrix}A & B \\\\C & D \\end{bmatrix}$ singular when each of the submatrices $A$, $B$, $C$, and $D$ are singular?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 3
} | 12 | 0 | 0 |
omr-ws-aec33d6e49d7c00a8d961d2ffb58c8a089cd9114c8110dedc4908b0bea7c0613 | Under what conditions is the inequality \(\frac{a^n + b^n}{2} \geq \left(\frac{a + b}{2}\right)^n\) true? | [
"\\(n \\in (-\\infty, 0] \\cup [1, \\infty)\\)"
] | [
"Under what conditions is the inequality \\(\\frac{a^n + b^n}{2} \\geq \\left(\\frac{a + b}{2}\\right)^n\\) true?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 38,
"genselect": 4,
"tir": 15
} | 57 | 0 | 0 |
omr-ws-575e53834611c705cd7b0b8e681037a98541c6b2c8423334ad8b31e369fcafd4 | Under what conditions is the matrix $(a_{i}a_{j})_{1\leqslant i,j\leqslant n} \in M_{n}(\mathbf{F})$ diagonalizable, where the field $\mathbf{F}$ is either $\mathbf{C}$ or $\mathbf{R}$? | [
"The matrix is diagonalizable over \\(\\mathbb{R}\\) but not necessarily over \\(\\mathbb{C}\\)."
] | [
"Under what conditions is the matrix $(a_{i}a_{j})_{1\\leqslant i,j\\leqslant n} \\in M_{n}(\\mathbf{F})$ diagonalizable, where the field $\\mathbf{F}$ is either $\\mathbf{C}$ or $\\mathbf{R}$?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-03d566394d6563bbac9a3a33ccd76a904e47c5fd0d12f38e3b67ea41b71145cf | Under what conditions is the polynomial \( P(X) = (X - a_1)(X - a_2) \cdots (X - a_n) \) irreducible, where \( a_1, a_2, \ldots, a_n \) are distinct integers? | [
"n = 1"
] | [
"Under what conditions is the polynomial \\( P(X) = (X - a_1)(X - a_2) \\cdots (X - a_n) \\) irreducible, where \\( a_1, a_2, \\ldots, a_n \\) are distinct integers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c26865a82c713149955c360bc7ed70554bf67e9e755bc7e5c5de0084013e5a19 | Under what conditions is the quadratic form $q'Aq \geq p'Ap$ given that $A$ is an $n \times n$ symmetric matrix with non-negative elements ($a_{ij} \geq 0$) and $p$ and $q$ are probability vectors ($p_i \geq 0$, $\sum p_i = 1$)? | [
"(q - p)'A(q + p) \\geq 0"
] | [
"Under what conditions is the quadratic form $q'Aq \\geq p'Ap$ given that $A$ is an $n \\times n$ symmetric matrix with non-negative elements ($a_{ij} \\geq 0$) and $p$ and $q$ are probability vectors ($p_i \\geq 0$, $\\sum p_i = 1$)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-6569645b8f6e4962dcfe2fcd6957957ab43e612e55b3ab068ee6278a6bc9df56 | Under what conditions is the quotient of two periodic functions $f$ and $g$ also periodic? | [
"\\text{The ratio of their periods is a rational number.}"
] | [
"Under what conditions is the quotient of two periodic functions $f$ and $g$ also periodic?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-cbcb86c639b147837fafaf643b527cb9c6e11d4654b78722e95e0fe614e2f22d | Under what conditions is the solution $(a, b, c)$ unique for the equation $3(a^2 + b^2 + c^2) = \alpha + 2(ab + ac + bc)$? | [
"\\(\\alpha = 3a^2\\) for some real number \\(a\\)."
] | [
"Under what conditions is the solution $(a, b, c)$ unique for the equation $3(a^2 + b^2 + c^2) = \\alpha + 2(ab + ac + bc)$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cd5abf4fc89ee64d7649ead04718f9e84b1be418bccc0090a5f001a8642e9704 | Under what conditions is the statement "The limit of the sum of two functions is the sum of the limits of the functions" not correct? | [
"The statement is not correct when either \\(\\lim f\\) or \\(\\lim g\\) does not exist."
] | [
"Under what conditions is the statement \"The limit of the sum of two functions is the sum of the limits of the functions\" not correct?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 16
} | 30 | 0 | 0 |
omr-ws-6af18b1d17f83f741c962e30c65e9c978db61168c5d2e046b1ec546635f79b3e | Under what conditions is the sum $(AD) + (BC)$ maximized, given that four points $A, B, C, D$ move in space such that $(AB) = (AC) = (DB) = (DC) = 1$? | [
"The sum \\(AD + BC\\) is maximized when \\(A, B, C, D\\) form a regular tetrahedron with each edge length equal to 1."
] | [
"Under what conditions is the sum $(AD) + (BC)$ maximized, given that four points $A, B, C, D$ move in space such that $(AB) = (AC) = (DB) = (DC) = 1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 20
} | 33 | 0 | 0 |
omr-ws-fa82d31d6c4720838a3f43736103313f072ff08ae8593e99b0af9e9900f408ce | Under what conditions must the square $n$-by-$n$ matrix $A$ be the zero matrix if the matrix equation $Ax=0$ holds for an $n$-by-$1$ vector $x$? | [
"n = 1 \\text{ and } x \\neq 0"
] | [
"Under what conditions must the square $n$-by-$n$ matrix $A$ be the zero matrix if the matrix equation $Ax=0$ holds for an $n$-by-$1$ vector $x$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-9c5c2da68ca6e0acd68f7118d7c84e108640214c67a16331af5ea67e970d9843 | Under what conditions of the initial triangle \(ABC\) do the semicircles \(\alpha, \beta, \gamma\) (with diameters \(AB\), \(BC\), and \(AC\) respectively) intersect the incircle \(\omega\) a total of 6 times? | [
"The triangle \\(ABC\\) must be acute."
] | [
"Under what conditions of the initial triangle \\(ABC\\) do the semicircles \\(\\alpha, \\beta, \\gamma\\) (with diameters \\(AB\\), \\(BC\\), and \\(AC\\) respectively) intersect the incircle \\(\\omega\\) a total of 6 times?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 1,
"tir": 3
} | 10 | 0 | 0 |
omr-ws-c7616b591f3172ddac82ff18d97b8ec26a92db46a852b61b66cb85516bf2fd89 | Under what conditions on $\alpha$ and $\beta$ can we deduce that a continuous function $f:(0,\infty) \to \mathbb{R}$ is a constant function if we know $f(x) = f(\alpha x) = f(\beta x)$ for all $x > 0$? | [
"\\dfrac{\\log \\alpha}{\\log \\beta} \\notin \\mathbb{Q}"
] | [
"Under what conditions on $\\alpha$ and $\\beta$ can we deduce that a continuous function $f:(0,\\infty) \\to \\mathbb{R}$ is a constant function if we know $f(x) = f(\\alpha x) = f(\\beta x)$ for all $x > 0$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-d96bf8adb836d44fdc7b0bedf4000dfe8d78137ba440296aa71eae2f8d021853 | Under what conditions on $\alpha$ and $\beta$ does the function \[ f(x) = \left\{1+|x|^\alpha\right\} \left\{1+|\log|x||^\beta\right\}^{-1}, \quad x \in \mathbb{R}^n \] belong to $L^p(\mathbb{R}^n)$? | [
"\\text{There are no such } \\alpha \\text{ and } \\beta"
] | [
"Under what conditions on $\\alpha$ and $\\beta$ does the function \n\\[ f(x) = \\left\\{1+|x|^\\alpha\\right\\} \\left\\{1+|\\log|x||^\\beta\\right\\}^{-1}, \\quad x \\in \\mathbb{R}^n \\]\nbelong to $L^p(\\mathbb{R}^n)$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-0038b5a8377cc8b6e92726201ae8f4bd708f7fc0b5a24c2e65f0202e5bbdd74f | Under what conditions on $f$ does there exist a constant $C$ such that \[ \int \int_{\mathbb{D}} |f|^{2} \ln \frac{1}{|z|} \, dxdy \leq C ||f||^{2}_{L^{2}}? \] | [
"f \\text{ is analytic in the unit disk}"
] | [
"Under what conditions on $f$ does there exist a constant $C$ such that\n\\[\n\\int \\int_{\\mathbb{D}} |f|^{2} \\ln \\frac{1}{|z|} \\, dxdy \\leq C ||f||^{2}_{L^{2}}?\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-8ec71336980ce23bce17c4d52f9c92779c89f3825da24c34e9dca575cca1b651 | Under what conditions on $f$ is the integral $\int_{-\infty}^{\infty} f(\delta(x)) \, dx$ meaningful, and what would its value be in those conditions? | [
"f(1)"
] | [
"Under what conditions on $f$ is the integral $\\int_{-\\infty}^{\\infty} f(\\delta(x)) \\, dx$ meaningful, and what would its value be in those conditions?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-5d620b5b2ae41b94a4e70fc6f9e29dca68f6777a8d2c01031842f84f373e43f9 | Under what conditions on $f:\mathbb{R}\to\mathbb{R}$ does the functional equation $f(x+y)=f(x)+f(y)\;\forall x,y\in\mathbb{R}$ imply that $f$ is a linear function? | [
"Continuity at some point, Differentiability, Monotonicity, Bounded on some interval."
] | [
"Under what conditions on $f:\\mathbb{R}\\to\\mathbb{R}$ does the functional equation $f(x+y)=f(x)+f(y)\\;\\forall x,y\\in\\mathbb{R}$ imply that $f$ is a linear function?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 12
} | 12 | 0 | 0 |
omr-ws-0db4d869f130fec10bc02ac65df5e4fe430051e26ec06f73a66f6ad10157236f | Under what conditions on $q$ is the polynomial $X^2 + X + 1$ irreducible over the finite field $F_q$? | [
"\\( 3 \\nmid (q-1) \\)"
] | [
"Under what conditions on $q$ is the polynomial $X^2 + X + 1$ irreducible over the finite field $F_q$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-faec308419fe637ce75514c880b45e63ce531c7945aa54975c37c3b32a40fb33 | Under what conditions on \( n \) is it possible for a monkey to fill an \( n \times n \) grid with the numbers 1 through \( n^2 \) such that the set of products of the numbers in each horizontal row is the same as the set of products of the numbers in each vertical column? | [
"1"
] | [
"Under what conditions on \\( n \\) is it possible for a monkey to fill an \\( n \\times n \\) grid with the numbers 1 through \\( n^2 \\) such that the set of products of the numbers in each horizontal row is the same as the set of products of the numbers in each vertical column?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-d4304052c119661b29523793bfb164856fa8706020d61b822ee2ad50d5f0538f | Under what conditions on \( n \) is it possible for a monkey to fill an \( n \times n \) grid with the numbers 1, 2, ..., \( n^2 \) such that the set of products of the numbers in each horizontal row is the same as the set of products of the numbers in each vertical column? | [
"1"
] | [
"Under what conditions on \\( n \\) is it possible for a monkey to fill an \\( n \\times n \\) grid with the numbers 1, 2, ..., \\( n^2 \\) such that the set of products of the numbers in each horizontal row is the same as the set of products of the numbers in each vertical column?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-5c9dd451db3100d43b827078c09f1969c7cc40a33575d9c8aec261e3b2b5088e | Under what conditions on \( q \) (in terms of the eigenvalues of \( A \)), does the iteration formula \( X_{n+1} = X_n + q(AX_n - I) \) converge to \( A^{-1} \)? | [
"\\( - \\frac{2}{\\lambda_{\\max}} < q < 0 \\)"
] | [
"Under what conditions on \\( q \\) (in terms of the eigenvalues of \\( A \\)), does the iteration formula \\( X_{n+1} = X_n + q(AX_n - I) \\) converge to \\( A^{-1} \\)?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f82ff113132462bfe102fa67296ed8c3d99ed842a03ef94caa7d567e1e6bc561 | Under what conditions on \(a\) and \(b\) (real numbers) does the inequality \((1+x)^a \cdot \left(1+\frac{1}{x}\right)^b \geq 4\) hold? | [
"a \\geq 1 \\text{ and } b \\geq 1"
] | [
"Under what conditions on \\(a\\) and \\(b\\) (real numbers) does the inequality \\((1+x)^a \\cdot \\left(1+\\frac{1}{x}\\right)^b \\geq 4\\) hold?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-14d02845c96d3e1491609ada5eb4e2eee3f460b24415512b1c5acc9145c368e5 | Under what conditions on \(a\), \(b\), and \(c\) do there exist a triangle such that \(a\), \(b\), and \(c\) are its altitudes? | [
"\\( \\frac{1}{a} + \\frac{1}{b} > \\frac{1}{c} \\), \\( \\frac{1}{a} + \\frac{1}{c} > \\frac{1}{b} \\), and \\( \\frac{1}{b} + \\frac{1}{c} > \\frac{1}{a} \\)"
] | [
"Under what conditions on \\(a\\), \\(b\\), and \\(c\\) do there exist a triangle such that \\(a\\), \\(b\\), and \\(c\\) are its altitudes?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-af62baf48f8986bf081c352ba1816b52595dec06c94e38590a486e29f3b60845 | Under what conditions on \(a\), \(b\), and \(c\) do there exist a triangle such that \(a\), \(b\), and \(c\) are its angle bisectors? | [
"No conditions."
] | [
"Under what conditions on \\(a\\), \\(b\\), and \\(c\\) do there exist a triangle such that \\(a\\), \\(b\\), and \\(c\\) are its angle bisectors?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c5cfbdeb2aa9c932ee2c8a75c278049b1c0636f080921eb95570b81d992517e5 | Under what conditions on \(a\), \(b\), and \(c\) do there exist a triangle such that \(a\), \(b\), and \(c\) are its medians? | [
"\\(a\\), \\(b\\), and \\(c\\) must satisfy the triangle inequality."
] | [
"Under what conditions on \\(a\\), \\(b\\), and \\(c\\) do there exist a triangle such that \\(a\\), \\(b\\), and \\(c\\) are its medians?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 12
} | 33 | 0 | 0 |
omr-ws-4a935f6e2be053b003202a3f42a6d3d1ee7965940a0707ed5bd71cbc61df3c85 | Under what conditions on \(a\), \(b\), and \(c\) does the equation \(ax + by + cz = 1\) have a solution, given that \(\gcd(a, b, c) = 1\)? What is a general method for finding a solution when one exists? | [
"The equation \\( ax + by + cz = 1 \\) has a solution if \\(\\gcd(a, b, c) = 1\\). The general method involves using the Extended Euclidean Algorithm."
] | [
"Under what conditions on \\(a\\), \\(b\\), and \\(c\\) does the equation \\(ax + by + cz = 1\\) have a solution, given that \\(\\gcd(a, b, c) = 1\\)? What is a general method for finding a solution when one exists?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 2,
"tir": 23
} | 33 | 0 | 0 |
omr-ws-b815b2988742fb6fb186e3eb76e66e47a728b3341ac126c3f5ee90a6a09f2818 | Under what conditions on a continuous function \( f: \mathbb{R}^m \rightarrow \mathbb{R}^n \) (where \( m < n \)) is the Lebesgue measure of the image of \( f \) equal to zero? | [
"\\( d > \\frac{m}{n} \\)"
] | [
"Under what conditions on a continuous function \\( f: \\mathbb{R}^m \\rightarrow \\mathbb{R}^n \\) (where \\( m < n \\)) is the Lebesgue measure of the image of \\( f \\) equal to zero?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-4bd60d039e0736b68bbdb971d40e0d1c34d82b4e557a29381d4e2a873191ce5c | Under what conditions on an increasing function $f$ defined on the interval [0,1] are the metric spaces ($X$, $d$) and ($X$, $e$) topologically equivalent, where $d(x, y) = |x - y|$ and $e(x, y) = |f(x) - f(y)|$? | [
"f \\text{ is continuous and strictly increasing}"
] | [
"Under what conditions on an increasing function $f$ defined on the interval [0,1] are the metric spaces ($X$, $d$) and ($X$, $e$) topologically equivalent, where $d(x, y) = |x - y|$ and $e(x, y) = |f(x) - f(y)|$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-572b75ddae4ce7c3116c9b4bad77d0274495fc2425ea50561351c0631f3ae2ae | Under what conditions on integers \(x\) and \(y\) does the expression \(\frac{1}{x} - \frac{1}{y}\) equal the reciprocal of an integer, i.e., if \(\frac{1}{x} - \frac{1}{y} = \frac{1}{z}\), under what conditions will \(z\) be an integer? | [
"y - x \\mid x^2"
] | [
"Under what conditions on integers \\(x\\) and \\(y\\) does the expression \\(\\frac{1}{x} - \\frac{1}{y}\\) equal the reciprocal of an integer, i.e., if \\(\\frac{1}{x} - \\frac{1}{y} = \\frac{1}{z}\\), under what conditions will \\(z\\) be an integer?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 5
} | 17 | 0 | 0 |
omr-ws-7185dd6137797644bacdf572c5d94d1295f54c410aa1ad6f8423a6272ba2678c | Under what conditions on positive integers \(a\) and \(b\) does \((n^a + 1, n^b + 1)\) divide \(n^{\gcd(a, b)} + 1\)? | [
"The condition is always satisfied for any positive integers \\(a\\) and \\(b\\)."
] | [
"Under what conditions on positive integers \\(a\\) and \\(b\\) does \\((n^a + 1, n^b + 1)\\) divide \\(n^{\\gcd(a, b)} + 1\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 37,
"genselect": 8,
"tir": 21
} | 66 | 0 | 0 |
omr-ws-de4195b374831aea2b6da7149207f23f6c649a7fae8e9760a88c1b5fa7e62bc6 | Under what conditions on subsets \( A \) and \( B \) of \( \mathbb{R}^2 \) does the identity \(\int_{A \cup B} F \, dxdy = \int_{A} F \, dxdy + \int_{B} F \, dxdy\) hold, where \( F \) is a \( C^1 \) function? | [
"\\(A \\cap B = \\emptyset\\) or \\(A \\cap B \\subseteq \\ker F\\)"
] | [
"Under what conditions on subsets \\( A \\) and \\( B \\) of \\( \\mathbb{R}^2 \\) does the identity \\(\\int_{A \\cup B} F \\, dxdy = \\int_{A} F \\, dxdy + \\int_{B} F \\, dxdy\\) hold, where \\( F \\) is a \\( C^1 \\) function?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 6
} | 6 | 0 | 0 |
omr-ws-97ee9b635a50745c540bcebae99cbdcc29cdd47171f6cab715a9368d12808f04 | Under what conditions on the coefficients $\alpha_i, \beta_i, \gamma_i \in \mathbb{R}$ for $i=1,2,3$ is the curve determined by $\alpha \beta = \gamma^2$ a finite number of points? | [
"\\begin{vmatrix} \\alpha_1 & \\beta_1 & \\gamma_1 \\\\ \\alpha_2 & \\beta_2 & \\gamma_2 \\\\ \\alpha_3 & \\beta_3 & \\gamma_3 \\end{vmatrix} = 0 \\text{ and no two of } \\alpha, \\beta, \\gamma \\text{ are proportional}"
] | [
"Under what conditions on the coefficients $\\alpha_i, \\beta_i, \\gamma_i \\in \\mathbb{R}$ for $i=1,2,3$ is the curve determined by $\\alpha \\beta = \\gamma^2$ a finite number of points?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-ab49a4e90c02fc10abcbc69e699e956de644ca4dbe0eee2a4f87a5136a8c8743 | Under what conditions on the coefficients $\alpha_i, \beta_i, \gamma_i \in \mathbb{R}$ for $i=1,2,3$ is the curve determined by $\alpha \beta = \gamma^2$ a hyperbola? | [
"\\((\\alpha_1 \\beta_2 + \\alpha_2 \\beta_1 - 2 \\gamma_1 \\gamma_2)^2 - 4(\\alpha_1 \\beta_1 - \\gamma_1^2)(\\alpha_2 \\beta_2 - \\gamma_2^2) > 0\\)"
] | [
"Under what conditions on the coefficients $\\alpha_i, \\beta_i, \\gamma_i \\in \\mathbb{R}$ for $i=1,2,3$ is the curve determined by $\\alpha \\beta = \\gamma^2$ a hyperbola?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 1,
"tir": 7
} | 11 | 0 | 0 |
omr-ws-5016a90caa397514abbfb894526184b6ca87a4c40b33add28bc2e8e0203e18ab | Under what conditions on the coefficients $\alpha_i, \beta_i, \gamma_i \in \mathbb{R}$ for $i=1,2,3$ is the curve determined by $\alpha \beta = \gamma^2$ a parabola? | [
"(\\alpha_1\\beta_2 + \\alpha_2\\beta_1 - 2\\gamma_1\\gamma_2)^2 = 4(\\alpha_1\\beta_1 - \\gamma_1^2)(\\alpha_2\\beta_2 - \\gamma_2^2)"
] | [
"Under what conditions on the coefficients $\\alpha_i, \\beta_i, \\gamma_i \\in \\mathbb{R}$ for $i=1,2,3$ is the curve determined by $\\alpha \\beta = \\gamma^2$ a parabola?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-9bec6acb1ae2ac45be60d8a713726dd1a951db4c9c049e9e99d84d892ddb5bee | Under what conditions on the coefficients $\alpha_i, \beta_i, \gamma_i \in \mathbb{R}$ for $i=1,2,3$ is the curve determined by $\alpha \beta = \gamma^2$ an ellipse? | [
"(\\alpha_1\\beta_2 + \\alpha_2\\beta_1 - 2\\gamma_1\\gamma_2)^2 - 4(\\alpha_1\\beta_1 - \\gamma_1^2)(\\alpha_2\\beta_2 - \\gamma_2^2) < 0"
] | [
"Under what conditions on the coefficients $\\alpha_i, \\beta_i, \\gamma_i \\in \\mathbb{R}$ for $i=1,2,3$ is the curve determined by $\\alpha \\beta = \\gamma^2$ an ellipse?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-ad0f68f38666697b367a02702591236f24bf4b1872c32fa99b573e554d20fcae | Under what conditions on the coefficients $\alpha_i, \beta_i, \gamma_i \in \mathbb{R}$ for $i=1,2,3$ is the curve determined by $\alpha \beta = \gamma^2$ one line? | [
"(\\alpha_2\\beta_3 + \\alpha_3\\beta_2 - 2\\gamma_2\\gamma_3)^2 = 4(\\alpha_2\\beta_2 - \\gamma_2^2)(\\alpha_3\\beta_3 - \\gamma_3^2)"
] | [
"Under what conditions on the coefficients $\\alpha_i, \\beta_i, \\gamma_i \\in \\mathbb{R}$ for $i=1,2,3$ is the curve determined by $\\alpha \\beta = \\gamma^2$ one line?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-4b3d3ef0348530e8ee3a7faed36e7e98be20ab57156c751a04fa992cffae519a | Under what conditions on the coefficients $\alpha_i, \beta_i, \gamma_i \in \mathbb{R}$ for $i=1,2,3$ is the curve determined by $\alpha \beta = \gamma^2$ two lines? | [
"\\(\\gamma\\) must be a linear combination of \\(\\alpha\\) or \\(\\beta\\)."
] | [
"Under what conditions on the coefficients $\\alpha_i, \\beta_i, \\gamma_i \\in \\mathbb{R}$ for $i=1,2,3$ is the curve determined by $\\alpha \\beta = \\gamma^2$ two lines?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 4,
"tir": 5
} | 12 | 0 | 0 |
omr-ws-1af8cfe8b3b854a724c21788f26fb232e05a78d9db670afc7346d09f9ea15c79 | Under what conditions on the function $f$ does the following inequality hold? Assume that the function $f(x)$ is monotonically decreasing on $[t,\infty)$. \[ \int_t^\infty (x-t)f(t)f(x)dx > \left( \int_t^\infty f(x) dx \right)^2 \] | [
"f \\text{ is not exponentially decreasing}"
] | [
"Under what conditions on the function $f$ does the following inequality hold? Assume that the function $f(x)$ is monotonically decreasing on $[t,\\infty)$.\n\n\\[\n\\int_t^\\infty (x-t)f(t)f(x)dx > \\left( \\int_t^\\infty f(x) dx \\right)^2\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-10c09ae16879deb385eee2715bbbb2ec8f104d4b821253d08c75e08e891e37f9 | Under what conditions on the measure space $(X,\mu)$ is the set \[\left\{f:\int f d\mu=1\right\}\] complete in the complete metric space $L^p(\mu)$ for $0<p<1$ with the metric $d(f,g)=\int |f-g|^pd\mu$? | [
"The set \\(\\{f \\in L^p \\cap L^1 : \\int_X f \\, d\\mu = 1\\}\\) is complete in \\(L^p\\) for \\(0 < p < 1\\) if and only if there do not exist measurable sets \\(A_1, A_2, \\ldots\\) of positive finite measure such that \\(\\mu(A_n) \\to 0\\)."
] | [
"Under what conditions on the measure space $(X,\\mu)$ is the set \\[\\left\\{f:\\int f d\\mu=1\\right\\}\\] complete in the complete metric space $L^p(\\mu)$ for $0<p<1$ with the metric $d(f,g)=\\int |f-g|^pd\\mu$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 2
} | 4 | 0 | 0 |
omr-ws-b1f4e4197e083f379400e7db7ea081ee19123890a1b703f554b7e42c88a2fdec | Under what conditions on the sequence $(x_i)_{i=1}^{n}$ of real numbers, where $n \geq 2$, are the vectors $(\sin x_1, \ldots, \sin x_n)$, $(\cos x_1, \ldots, \cos x_n)$, and $(1, \ldots, 1)$ linearly independent? | [
"At least three different values of \\(x_i \\mod 2\\pi\\)."
] | [
"Under what conditions on the sequence $(x_i)_{i=1}^{n}$ of real numbers, where $n \\geq 2$, are the vectors $(\\sin x_1, \\ldots, \\sin x_n)$, $(\\cos x_1, \\ldots, \\cos x_n)$, and $(1, \\ldots, 1)$ linearly independent?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 5,
"tir": 19
} | 28 | 0 | 0 |
omr-ws-dadc2b04745ab1e68d0f22b391140531f7d06f99c002da95c236a66b43e57543 | Under what conditions on the sequence \(1 = x_1 < x_2 < x_3 < \cdots\) will the limit \[ \lim_{N \to \infty} \left\{ \sum_{n=1}^{N} f(x_n) - \int_1^{x_N} f(x) \, dx \right\} \] converge, given that \(f\) is positive and decreasing on \([1, \infty)\) and \(\int_1^{\infty} f(x) \, dx\) diverges? | [
"The limit converges if \\( |x_n - cn| \\) is bounded for some constant \\( c \\) and \\( f \\) has bounded variation."
] | [
"Under what conditions on the sequence \\(1 = x_1 < x_2 < x_3 < \\cdots\\) will the limit\n\\[\n\\lim_{N \\to \\infty} \\left\\{ \\sum_{n=1}^{N} f(x_n) - \\int_1^{x_N} f(x) \\, dx \\right\\}\n\\]\nconverge, given that \\(f\\) is positive and decreasing on \\([1, \\infty)\\) and \\(\\int_1^{\\infty} f(x) \\, dx\\) d... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1b4e28592700cdc41b6b243f1241b725f0deeff613da4365e992b7fa8aa7cca0 | Under what conditions will \( p^n \equiv -1 \pmod{k} \) be satisfied for arbitrary integers \( p < k \)? | [
"2"
] | [
"Under what conditions will \\( p^n \\equiv -1 \\pmod{k} \\) be satisfied for arbitrary integers \\( p < k \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 1
} | 11 | 0 | 0 |
omr-ws-1c076f119c2940c7bd59126718b0cb6572d0f0652e0bae92b78a41d9f963d7af | Under what conditions will the system of equations be inconsistent? \[ \begin{cases} 3x + y + z = t \\ x - y + 2z = 1 - t \\ x + 3y - 3z = 1 + t \end{cases} \] | [
"\\( t \\neq \\frac{3}{2} \\)"
] | [
"Under what conditions will the system of equations be inconsistent?\n\\[ \\begin{cases} \n3x + y + z = t \\\\\nx - y + 2z = 1 - t \\\\\nx + 3y - 3z = 1 + t \n\\end{cases} \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-af19280e90537acfc98159929c54c4abc08608746196b9e154bd0a7f2b5e9ca3 | Under what conditions will the system of equations be inconsistent? \[ \begin{cases} tx + y = 0 \\ x + ty - z = 1 \\ -y + tz = 0 \end{cases} \] | [
"\\( t \\neq \\frac{3}{2} \\)"
] | [
"Under what conditions will the system of equations be inconsistent?\n\\[ \\begin{cases} \ntx + y = 0 \\\\\nx + ty - z = 1 \\\\\n-y + tz = 0 \n\\end{cases} \\]"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-39c935de4f085bbd7ad33a7b67da3b5e1e8ef1d6c60de8a577b2b55773f08455 | Under what conditions would the quantity \( I(P\|Q) = \int \log\left( \frac {dP}{dQ}\right) dP = \int \frac {dP} {dQ} \log\left(\frac {dP}{dQ}\right) \) be \( \infty \), given that \( P \ll Q \)? | [
"I(P\\|Q) = \\infty \\text{ if and only if } \\frac{dP}{dQ} \\log \\frac{dP}{dQ} \\text{ is not integrable with respect to } Q."
] | [
"Under what conditions would the quantity \\( I(P\\|Q) = \\int \\log\\left( \\frac {dP}{dQ}\\right) dP = \\int \\frac {dP} {dQ} \\log\\left(\\frac {dP}{dQ}\\right) \\) be \\( \\infty \\), given that \\( P \\ll Q \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-0dfd49a38b93f0e9220cd2dc33d00fc0f7bee586b7b78ea4b93cd2d02e7e1c00 | Under what geometric condition on real harmonic functions \( u \) and \( v \) on a region \( G \) is the function \( uv \) harmonic? | [
"The level curves of \\(u\\) and \\(v\\) are orthogonal."
] | [
"Under what geometric condition on real harmonic functions \\( u \\) and \\( v \\) on a region \\( G \\) is the function \\( uv \\) harmonic?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-2aa3f30d9553173e4507ac4fcc8c4c79c15a2f32382b9daa2a9f05c5012fe4f8 | Under what necessary and sufficient conditions are the multiplicative groups $((\mathbb{Z}/n\mathbb{Z})^{*},.)$ and $((\mathbb{Z}/m\mathbb{Z})^{*},.)$ isomorphic to each other? | [
"\\nu_2(n) = \\nu_2(m) \\text{ and } \\{\\phi(p^k) \\mid p^k \\parallel n \\text{ odd}\\} = \\{\\phi(q^l) \\mid q^l \\parallel m \\text{ odd}\\}"
] | [
"Under what necessary and sufficient conditions are the multiplicative groups $((\\mathbb{Z}/n\\mathbb{Z})^{*},.)$ and $((\\mathbb{Z}/m\\mathbb{Z})^{*},.)$ isomorphic to each other?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-781b73453271ce98a588fa886a55fe94b6693196ec75ef57f734e64daa66ebf0 | Under what necessary and sufficient conditions for the side lengths $a$ and $b$ of a rectangle (with $a < b$) is it possible to place the rectangle in a right-angled coordinate system such that there is no point with integer coordinates in its interior or on its edge? | [
"At least one of $a$ or $b$ must be less than 1."
] | [
"Under what necessary and sufficient conditions for the side lengths $a$ and $b$ of a rectangle (with $a < b$) is it possible to place the rectangle in a right-angled coordinate system such that there is no point with integer coordinates in its interior or on its edge?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-38783ecc7f0d51a5c8c93fa57ff64d186f744aa0ab93b6818d0f435451b99b56 | Under what special conditions does $a^k \equiv b^k \pmod{n}$ imply $a \equiv b \pmod{n}$, where $a, b, n, k$ are integers and $k \neq 0$? | [
"\\( \\gcd(n, k) = 1 \\)"
] | [
"Under what special conditions does $a^k \\equiv b^k \\pmod{n}$ imply $a \\equiv b \\pmod{n}$, where $a, b, n, k$ are integers and $k \\neq 0$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3a9a3838d9da95ae7fb5a0519f45e0f39910874794f6eb8f628b470b412cc31d | Under which (sufficient AND necessary) conditions does $ z $ reach the upper bound $ \sum_{i=1}^n \frac{b_i l_i}{L} $? | [
"b_1 = b_2 = \\cdots = b_n"
] | [
"Under which (sufficient AND necessary) conditions does $ z $ reach the upper bound $ \\sum_{i=1}^n \\frac{b_i l_i}{L} $?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-611cdfdbd8823245c8e08343bf53023e861258920474aecda980d53e058606c4 | Under which condition regarding the quadruplet $(a,b,c,d)$ of integers do we have $\gcd(ax+by,cx+dy) = \gcd(x,y)$ for all integers $x$ and $y$? | [
"\\(ad - bc = \\pm 1\\)"
] | [
"Under which condition regarding the quadruplet $(a,b,c,d)$ of integers do we have $\\gcd(ax+by,cx+dy) = \\gcd(x,y)$ for all integers $x$ and $y$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-5da7748a814359815ba46d18aa3ce0c3ff59e93f0f80087f3e90a76f077418ea | Under which conditions does there exist an equilateral triangle having each vertex in a different disc, given three discs in the plane with radii \( r_1, r_2, r_3 \) and centers \( O_1, O_2, O_3 \) respectively? | [
"\\( O_3 \\) lies in the union of two closed discs of radius \\( r_1 + r_2 + r_3 \\) centered at points \\( \\Omega_1 \\) and \\( \\Omega_2 \\), where each of \\( \\Omega_1 \\) and \\( \\Omega_2 \\) forms an equilateral triangle with \\( O_1 \\) and \\( O_2 \\)."
] | [
"Under which conditions does there exist an equilateral triangle having each vertex in a different disc, given three discs in the plane with radii \\( r_1, r_2, r_3 \\) and centers \\( O_1, O_2, O_3 \\) respectively?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 5,
"tir": 7
} | 26 | 0 | 0 |
omr-ws-3797a550b5711ab5a775f6826d9a0b74afa8d95559e365e896f9f25348210825 | Under which conditions for positive integers \( n_i \) and \( k \) is the following inequality true? \[ \prod_{i=1}^{k} \left(1 - \frac{1}{n_i}\right) > 1 - \frac{1}{m} + \frac{1}{m^2} \] | [
"\\sum_{i=1}^{k} \\frac{1}{n_i} < \\frac{m - 1}{m^2}"
] | [
"Under which conditions for positive integers \\( n_i \\) and \\( k \\) is the following inequality true?\n\n\\[\n\\prod_{i=1}^{k} \\left(1 - \\frac{1}{n_i}\\right) > 1 - \\frac{1}{m} + \\frac{1}{m^2}\n\\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-b79bab5f995b820cda95a8b09f8c7c124e6b0aa8e3bb2904f7f68b6a696fc251 | Under which conditions is $A^{(I)} = \bigoplus_{i \in I} A$ a direct summand of $A^{I} = \prod_{i \in I} A$, where $I$ is an infinite set and $A$ is an abelian group? | [
"A \\text{ is divisible}"
] | [
"Under which conditions is $A^{(I)} = \\bigoplus_{i \\in I} A$ a direct summand of $A^{I} = \\prod_{i \\in I} A$, where $I$ is an infinite set and $A$ is an abelian group?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-4c02ab9f2a120e479fc6b09c85cb329103d60953663b9a88ceeed707ab0bf638 | Under which conditions on the constants \(a\) and \(b\) does the limit \[ \lim_{n \to \infty} \frac{\sinh(an)}{\cosh(bn)} \] exist, and find its value (depending on \(a\) and \(b\)). | [
"\n\\begin{cases}\n0 & \\text{if } |a| < |b|, \\\\\n\\operatorname{sgn}(a) & \\text{if } |a| = |b|.\n\\end{cases}\n",
"The limit exists if \\(a \\leq b\\). The value of the limit is \\(0\\) if \\(a < b\\) and \\(1\\) if \\(a = b\\)."
] | [
"Under which conditions on the constants \\(a\\) and \\(b\\) does the limit\n\\[\n\\lim_{n \\to \\infty} \\frac{\\sinh(an)}{\\cosh(bn)}\n\\]\nexist, and find its value (depending on \\(a\\) and \\(b\\))."
] | [
"aops_c6_high_school_olympiads",
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 4,
"tir": 2
} | 13 | 0 | 0 |
omr-ws-192bc151347d5310ea5ed5161f432046c26304ddf1a6fe85d262dc5d685c6343 | Unique polynomial $u(x)$ is defined as below: \[u(n) = (x_1 + x_2 + x_3 + \cdots + x_{n+1})^{n+2}\] Find the number of terms in polynomial $u(7)$ (when it's expanded). | [
"11440"
] | [
"Unique polynomial $u(x)$ is defined as below:\n\n\\[u(n) = (x_1 + x_2 + x_3 + \\cdots + x_{n+1})^{n+2}\\]\n\nFind the number of terms in polynomial $u(7)$ (when it's expanded)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-90670c3d540e39d38bd7bc00a1dc93e74dc9efcc509e94e3eb8715ed24b76968 | Unit square $ABCD$ is divided into $10^{12}$ smaller squares (not necessarily equal). Calculate the maximum possible sum of the perimeters of all the smaller squares that have at least one common point with diagonal $AC$. | [
"4000000"
] | [
"Unit square $ABCD$ is divided into $10^{12}$ smaller squares (not necessarily equal). Calculate the maximum possible sum of the perimeters of all the smaller squares that have at least one common point with diagonal $AC$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-897213c07d46c61a01e032444a272e0cf13e908ab667fb895760f11674c47e6a | Unit square \( S_1 \) has vertices \( A_1, B_1, C_1, D_1 \). Square \( S_2 \) is drawn with its vertices on \( S_1 \) such that \( A_1A_2 = A_2B_1 \). An infinite number of more squares are drawn in a similar fashion: square \( S_{x+1} \) is drawn with its vertices on square \( S_x \) such that \( A_xA_{x+1} = (2^x - 1... | [
"0"
] | [
"Unit square \\( S_1 \\) has vertices \\( A_1, B_1, C_1, D_1 \\). Square \\( S_2 \\) is drawn with its vertices on \\( S_1 \\) such that \\( A_1A_2 = A_2B_1 \\). An infinite number of more squares are drawn in a similar fashion: square \\( S_{x+1} \\) is drawn with its vertices on square \\( S_x \\) such that \\( A... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-23c9bf8f76b4290c9270f73ef3d3a1b4dd44833eb32d03b49aae6470a35f2d2e | Unit square \( S_1 \), whose vertices are \( A_1, B_1, C_1, D_1 \), is drawn in a plane. Square \( S_2 \), whose vertices are \( A_2, B_2, C_2, D_2 \), is drawn with its vertices on \( S_1 \) such that \( A_1A_2 = A_2B_1 \). An infinite number of more squares are drawn in a similar fashion: Square \( S_{x+1} \), whose ... | [
"\\(\\prod_{k=1}^{\\infty}\\frac{(2^{k}-1)^{2}+1}{2^{2k}}\\)"
] | [
"Unit square \\( S_1 \\), whose vertices are \\( A_1, B_1, C_1, D_1 \\), is drawn in a plane. Square \\( S_2 \\), whose vertices are \\( A_2, B_2, C_2, D_2 \\), is drawn with its vertices on \\( S_1 \\) such that \\( A_1A_2 = A_2B_1 \\). An infinite number of more squares are drawn in a similar fashion: Square \\( ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-9b2585ab75d8ce49172733fb1218b2731b5844630015fbdeb9e0df49740a30a6 | Unless isomorphic, how many simple four-vertex graphs are there? | [
"11"
] | [
"Unless isomorphic, how many simple four-vertex graphs are there?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-06cdc08ec3e37279d7212c6513ae34f2347d158abcaaa987759d21bbfe6dcab8 | Up to rotation, how many different ways are there to color the vertices of a pentagon so that no two adjacent vertices are the same color? Assume you have seven colors to choose from. | [
"1554"
] | [
"Up to rotation, how many different ways are there to color the vertices of a pentagon so that no two adjacent vertices are the same color? Assume you have seven colors to choose from."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7fb479068cc5e048f48af71474b6ad642bff814d317bc1c591b6a111de2fdd43 | Up to rotation, how many distinct necklaces can be made with 4 black beads and 8 white beads? | [
"43"
] | [
"Up to rotation, how many distinct necklaces can be made with 4 black beads and 8 white beads?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 9
} | 57 | 0 | 0 |
omr-ws-bb30dd5f94dba238a2c92dea77cd770fc1b636f5dbfa7cc12197b62b9fdaa951 | Upik throws $n$ dice. He calculates the probability that the sum of the dice is $6$. For what value of $n$ is the probability the greatest? | [
"1"
] | [
"Upik throws $n$ dice. He calculates the probability that the sum of the dice is $6$. For what value of $n$ is the probability the greatest?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1f95684f14a694b1a2e259eb004f3ac71e6b9f6d62b09aba3e89135672c17829 | Uri buys two burgers and a soda for $\$2.10$, and Gen buys a burger and two sodas for $\$2.40$. How many cents does a soda cost? | [
"90"
] | [
"Uri buys two burgers and a soda for $\\$2.10$, and Gen buys a burger and two sodas for $\\$2.40$. How many cents does a soda cost?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ce717df893bb72b494d2f61d9b043604f2a1396d4c350bebd918f274fde7da68 | Uri has 99 empty bags and an unlimited number of balls, each with a weight of the form $3^n$ where $n$ is an integer. Uri distributes a finite number of these balls into the bags such that all bags have the same total weight and no balls are left over. It is known that Uri chose at most $k$ balls of the same weight. Fi... | [
"66"
] | [
"Uri has 99 empty bags and an unlimited number of balls, each with a weight of the form $3^n$ where $n$ is an integer. Uri distributes a finite number of these balls into the bags such that all bags have the same total weight and no balls are left over. It is known that Uri chose at most $k$ balls of the same weigh... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-8b5159077600289a1fa3995c09af057d88a29d6609575064dd460c4b6dac996c | Urn 1 has 5 red balls and 1 blue ball. Urn 2 has 2 red balls and 4 blue balls. A fair die is tossed. If it results in an even number, balls are repeatedly drawn one at a time with replacement from Urn 1. If it is an odd number, balls are repeatedly drawn one at a time with replacement from Urn 2. If the first two draws... | [
"\\(\\frac{7}{10}\\)"
] | [
"Urn 1 has 5 red balls and 1 blue ball. Urn 2 has 2 red balls and 4 blue balls. A fair die is tossed. If it results in an even number, balls are repeatedly drawn one at a time with replacement from Urn 1. If it is an odd number, balls are repeatedly drawn one at a time with replacement from Urn 2. If the first two ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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