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-4b87a8a4bf88ea189fb34976bf49a372dd7a2fdaf8252aadf0599160e3b39e92 | What are the values of \( x \), \( y \), and \( z \) that will yield an integer value for \( w \) in the equation \[ w = \frac{y + \sqrt{y^2 - 4xz}}{2x}? \] | [
"z = wy - xw^2"
] | [
"What are the values of \\( x \\), \\( y \\), and \\( z \\) that will yield an integer value for \\( w \\) in the equation\n\\[ w = \\frac{y + \\sqrt{y^2 - 4xz}}{2x}? \\]"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 4
} | 14 | 0 | 0 |
omr-ws-eae9d549fec5e9e149eeb1f829cd38ecbea1146d8758418a658f1f865692d241 | What are the values of a and b if neither a nor b are 2 and there are 65 unattainable scores in a game where you can score either a points or b points? | [
"14"
] | [
"What are the values of a and b if neither a nor b are 2 and there are 65 unattainable scores in a game where you can score either a points or b points?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-2b409ba8b328322ea7db7bc1dec7407f9a7b5eca589085b491743bed032bc4f0 | What are the values of the highest common factor (HCF) and the least common multiple (LCM) of the numbers 33333... (30 times) and 222... (40 times)? | [
"HCF = 1111111111, LCM = 666666666733333333339999999999999999999933333333326666666666"
] | [
"What are the values of the highest common factor (HCF) and the least common multiple (LCM) of the numbers 33333... (30 times) and 222... (40 times)?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 11
} | 11 | 0 | 0 |
omr-ws-e51bfd5cf8ecdda3121446c37076be4a0e0a746e9bb700802905c81e6217bafb | What are the values of the moduli of each of the values of $(2i)^{(2i)}$? | [
"\\(\\{ e^{(4k-1)\\pi} : k \\in \\mathbb{Z}\\}\\)"
] | [
"What are the values of the moduli of each of the values of $(2i)^{(2i)}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 7,
"tir": 28
} | 45 | 0 | 0 |
omr-ws-1e1a9f8ca2d5d043ce19683d62bcdfa41171436ae81260af75abd3a0424ea586 | What are the various ways to find $\int_{-\infty}^{\infty} \frac{\cos x}{x^2+1} \, dx$? | [
"$\\frac{\\pi}{e}$"
] | [
"What are the various ways to find $\\int_{-\\infty}^{\\infty} \\frac{\\cos x}{x^2+1} \\, dx$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-fc9882d77c474f5337fd8ed21401e05fe94030a856ec3264aa65a1c132e0b05a | What are the zeros of $T_{n}(x)$? At what numbers does $T_{n}(x)$ have local maximum and minimum values? | [
"\\cos\\left(\\frac{k\\pi}{n}\\right)"
] | [
"What are the zeros of $T_{n}(x)$? At what numbers does $T_{n}(x)$ have local maximum and minimum values?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-8646d0d4d0cc9a44e51520bdbaf0c766377985a7012d32f5ec72d64b07366f90 | What are the zeros of the equation \(\frac{(x-a)^2}{(x-a)}\)? | [
"no zeros"
] | [
"What are the zeros of the equation \\(\\frac{(x-a)^2}{(x-a)}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-36e04faab8b822e4a5b6c9a586d466d846abe8a4cec352170ae677fd9a4ae799 | What are the zeros of the function \( k(x) = 2x^3 - 3x^2 - 10x + 15 \)? | [
"\\( x = \\pm i \\sqrt{5}, x = \\frac{3}{2} \\)"
] | [
"What are the zeros of the function \\( k(x) = 2x^3 - 3x^2 - 10x + 15 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-58b4ee82679dc7e610008236ff413ba80ab4a7f171afece4bc4554e0bebac2d2 | What are the zeros of the function \( y = -\sqrt{9 - x^2} \)? Problem 4a: Describe in words the locus of a point which moves so as to trace out a path represented by the equation \( x^2 + y^2 - 2x + 7y + 2 = 0 \). Problem 4b: Describe in words the locus of a point which moves so as to trace out a path represented by th... | [
"\\( x = 3 \\) and \\( x = -3 \\)"
] | [
"What are the zeros of the function \\( y = -\\sqrt{9 - x^2} \\)?\n\nProblem 4a: Describe in words the locus of a point which moves so as to trace out a path represented by the equation \\( x^2 + y^2 - 2x + 7y + 2 = 0 \\).\n\nProblem 4b: Describe in words the locus of a point which moves so as to trace out a path r... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 18
} | 18 | 0 | 0 |
omr-ws-d965d9cbb64ebd45b9b2f7705f6c782bce28ff4cb39957013e75e7167720026e | What can be concluded about $a$, $m$, and $n$ if $a^n - 1$ divides $a^m + 1$? | [
"a = 2, \\, n = 2, \\, \\text{and } m \\text{ is odd}"
] | [
"What can be concluded about $a$, $m$, and $n$ if $a^n - 1$ divides $a^m + 1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-ce956bf17ed36d4151da25b61b3638812de52a72317ec06c87735f5a6372992f | What can be concluded about the product of $\cos\left(\frac{r\pi}{2n}\right)$ as $r$ varies from $n+1$ to $2n-1$? | [
"(-1)^{n-1} \\frac{\\sqrt{n}}{2^{n-1}}"
] | [
"What can be concluded about the product of $\\cos\\left(\\frac{r\\pi}{2n}\\right)$ as $r$ varies from $n+1$ to $2n-1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-57d543065110b9ad643f3c13ae363e2c00175d03b8a51dea86fd4daf23341d00 | What can be concluded if \( f(g(x)) + f(g^{-1}(x)) = x \), where \( g^{-1}(x) \) is the inverse function of \( g(x) \)? | [
"f(x) = \\dfrac{x}{2}"
] | [
"What can be concluded if \\( f(g(x)) + f(g^{-1}(x)) = x \\), where \\( g^{-1}(x) \\) is the inverse function of \\( g(x) \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-570367315d3a866892c736e8f2ded6dbd13eafec90d0d7e42691a191d8cdc3cc | What can be deduced when a prime greater than 2 is expressed as a sum of two squares? | [
"\\( p \\equiv 1 \\pmod{4} \\)"
] | [
"What can be deduced when a prime greater than 2 is expressed as a sum of two squares?"
] | [
"c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-f6786b57d6d1411957a8be7997bfba625d10f8906aa8d24cb050e750d20ddb54 | What can be said about a polynomial \( P(x) \) that satisfies \( P(\cos x) = P(\cos (x - \theta)) \) for some \( \theta \), for every \( x \in [0, \pi] \)? | [
"\\( P(x) = \\sum_{k=0}^{\\infty} a_k T_k(x) \\)"
] | [
"What can be said about a polynomial \\( P(x) \\) that satisfies \\( P(\\cos x) = P(\\cos (x - \\theta)) \\) for some \\( \\theta \\), for every \\( x \\in [0, \\pi] \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-3a8d2d9955277bb87ef97251d08b2aa992ebbecaf1e73f0ee6fa6d7e2fdd31ba | What can be said about the Jordan normal form of a linear operator $\phi$ such that $\phi^{3} = \phi^{4}$? | [
"The Jordan normal form of \\(\\phi\\) consists of Jordan blocks of size at most 3 for eigenvalue \\(0\\) and Jordan blocks of size 1 for eigenvalue \\(1\\)."
] | [
"What can be said about the Jordan normal form of a linear operator $\\phi$ such that $\\phi^{3} = \\phi^{4}$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-cbb160c744e5fb9e1ea1b3ab342afeb416f5c0a1253558813add7718da0ca8dc | What can be said about the behavior of \( O\left(\frac{1}{x^k}\right) \) when \( k \le 1 \)? Specifically, do the standard convergence tests fail in this case, and why can't we assume \( O(1) \) has a limit? | [
"The standard convergence tests can fail when \\( k \\le 1 \\) because \\( O(1) \\) does not guarantee a limit."
] | [
"What can be said about the behavior of \\( O\\left(\\frac{1}{x^k}\\right) \\) when \\( k \\le 1 \\)? Specifically, do the standard convergence tests fail in this case, and why can't we assume \\( O(1) \\) has a limit?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 20
} | 34 | 0 | 0 |
omr-ws-273f98e7d48491d71fe11a596b2e0f9a9c53293e3e0957e51dfd1bbe1894264f | What can be said about the continuity of \( f: \mathbb{R}^m \to \mathbb{R}^n \) if its graph, considered as a subset of \( \mathbb{R}^{m+n} \), is both closed and connected? | [
"f \\text{ is continuous}"
] | [
"What can be said about the continuity of \\( f: \\mathbb{R}^m \\to \\mathbb{R}^n \\) if its graph, considered as a subset of \\( \\mathbb{R}^{m+n} \\), is both closed and connected?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-12a8081ad0d4ebd09d8f196ea943545264010f1f5cf9acabd94a16b1b38186d5 | What can be said about the inequality between $\binom{2n}{n}$ and $\frac{\prod_{p \le n} p}{\pi(n)}$? | [
"\\binom{2n}{n} > \\frac{\\prod_{p \\leq n} p}{\\pi(n)}"
] | [
"What can be said about the inequality between $\\binom{2n}{n}$ and $\\frac{\\prod_{p \\le n} p}{\\pi(n)}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8bfe34658eda5c6f272e0817aba421522d14528fd85a0ef8d93d78b1949ada61 | What can be said about the number of functions $f$ from $\{1, \ldots, n\}^2 \to \{1, \ldots, n\}^2$ satisfying: \[ ||f(k, m)-(k,m)|| = 1 \] and \[ f \text{ is a bijection} \] for $n > 4$? | [
"0"
] | [
"What can be said about the number of functions $f$ from $\\{1, \\ldots, n\\}^2 \\to \\{1, \\ldots, n\\}^2$ satisfying:\n\\[\n||f(k, m)-(k,m)|| = 1\n\\]\nand\n\\[\nf \\text{ is a bijection}\n\\]\nfor $n > 4$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a4aa35e307945abafd1159cfd9dae75087d7621f27389499bea9382556c12577 | What can be said about the roots of the equation $4x^{4} - 2x + c = 0$ if $c = \frac{3}{4}$? | [
"The equation \\(4x^4 - 2x + \\frac{3}{4} = 0\\) has a double root at \\(x = \\frac{1}{2}\\)."
] | [
"What can be said about the roots of the equation $4x^{4} - 2x + c = 0$ if $c = \\frac{3}{4}$?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 32,
"genselect": 7,
"tir": 28
} | 67 | 0 | 0 |
omr-ws-969027de124dfe9b39a931591ff6c24915995eb5ee93dafa352b4f1b921508b5 | What can be said about the sequence ${a_n}$ if it converges and each $a_n$ is an integer? | [
"\\text{The sequence is eventually constant}"
] | [
"What can be said about the sequence ${a_n}$ if it converges and each $a_n$ is an integer?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-239dfff290dee06eefa56c039122a0de0e298cccd09dcea99a59ea653ceeb2fe | What can be said about the structure of groups of order \( p^2q^2 \) up to isomorphism? | [
"\\text{All groups of order } p^2 q^2 \\text{ are solvable and either abelian (four types) or non-abelian semidirect products under specific congruence conditions.}"
] | [
"What can be said about the structure of groups of order \\( p^2q^2 \\) up to isomorphism?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9e2b0963abac6a5a627e362249d677b6e3f5ade736e95d063983b127dc673370 | What can be said about the vectors \( u \) and \( v \) if the projection of \( u \) onto \( v \) equals 0? | [
"The vectors \\( u \\) and \\( v \\) are mutually perpendicular."
] | [
"What can be said about the vectors \\( u \\) and \\( v \\) if the projection of \\( u \\) onto \\( v \\) equals 0?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-c6acbd90d7336fe45876ee989ce19d78f8e4bffe1394ae62aaa86ddfb7cf2748 | What can be said about two random maps \( X \) and \( Y \) from \( (\Omega, \mathbb{F}, \mathcal{P}) \) to \( \mathbb{C} \) such that \( X \) is independent of \( Y \) and \( X + Y \) is independent of \( X - Y \)? | [
"X \\text{ and } Y \\text{ are independent complex normal random variables with equal variances}"
] | [
"What can be said about two random maps \\( X \\) and \\( Y \\) from \\( (\\Omega, \\mathbb{F}, \\mathcal{P}) \\) to \\( \\mathbb{C} \\) such that \\( X \\) is independent of \\( Y \\) and \\( X + Y \\) is independent of \\( X - Y \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6583c6dce21f25970abcefbd99d2ed5621eccb04a45ae4209eec403db7a97adc | What can be the angle between the hour and minute hands of a clock if it is known that its value has not changed after 30 minutes? | [
"\\( 97.5 \\)"
] | [
"What can be the angle between the hour and minute hands of a clock if it is known that its value has not changed after 30 minutes?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-1a77b49dfac2e2c9f86db572ef0b2c5daf3f2ad94bea51e6c3a96d8883e99d60 | What can the degree of the polynomial \( P \) be if for all real \( x \), \( P(\sin x) + P(\cos x) = 1 \)? | [
"\\( 0 \\) or any positive even number."
] | [
"What can the degree of the polynomial \\( P \\) be if for all real \\( x \\), \\( P(\\sin x) + P(\\cos x) = 1 \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 3
} | 10 | 0 | 0 |
omr-ws-3ad9cd3af6b556f5bf052ecfe731146cb57b5e5b798b22c8d05ef7de2667abde | What can the product \( ab \) not be equal to such that \( f(x) = \frac{2x + a}{bx - 2} \) and \( f(x) = f^{-1}(x) \) for all \( x \) in the domain of \( f \)? | [
"-4"
] | [
"What can the product \\( ab \\) not be equal to such that \\( f(x) = \\frac{2x + a}{bx - 2} \\) and \\( f(x) = f^{-1}(x) \\) for all \\( x \\) in the domain of \\( f \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-91511fd51a21a78f245bb704406dd36ac95b42b08bf1d24ac726f2f5a8b259f2 | What can the product \( ab \) not be equal to such that \( f(x) = \frac{2x + a}{bx - 2} \) and \( f^{-1}(x) = f(x) \) for all \( x \) in the domain of \( f \)? | [
"4"
] | [
"What can the product \\( ab \\) not be equal to such that \\( f(x) = \\frac{2x + a}{bx - 2} \\) and \\( f^{-1}(x) = f(x) \\) for all \\( x \\) in the domain of \\( f \\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-20ce5e5d4980bf5d7a788c610dea29813e6c7428b74087e73cb981561b00c8fe | What can we say about a matrix $A \in M_n(\mathbb{C})$ such that $\text{Ker}(A + A^*) = \text{Ker}(A) \cap \text{Ker}(A^*)$? | [
"A \\text{ is the direct sum of a zero matrix and a matrix with invertible Hermitian part}"
] | [
"What can we say about a matrix $A \\in M_n(\\mathbb{C})$ such that $\\text{Ker}(A + A^*) = \\text{Ker}(A) \\cap \\text{Ker}(A^*)$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 8
} | 16 | 0 | 0 |
omr-ws-636e733701a86d95eca8eff97b61b0a84e8ff05617bbf71af058ff9de8767bb5 | What can we say about the continuity of the function $x^{\frac{1}{x}}$ at the point $x = 0$? | [
"Not continuous at 0."
] | [
"What can we say about the continuity of the function $x^{\\frac{1}{x}}$ at the point $x = 0$?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 5,
"tir": 27
} | 47 | 0 | 0 |
omr-ws-08dd35fa28feed5ea13857e7d6664a24472c7363b2e3c74c88fd02ae54311559 | What can we say about the continuity of the function \( f(x) = -1 + \frac{\ln x}{x-1} \) at the point \( x = 1 \)? | [
"The function has a removable discontinuity at \\( x = 1 \\)."
] | [
"What can we say about the continuity of the function \\( f(x) = -1 + \\frac{\\ln x}{x-1} \\) at the point \\( x = 1 \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-d24d25347ed1fe232fe0884ce92a0d5a0212a38b498a88ee3eae07229619665c | What can we say about the fractional part of $ (n+n^{\frac{1}{r}})^{\frac{1}{r}}$, where $r > 1$ is a positive integer? | [
"0"
] | [
"What can we say about the fractional part of $ (n+n^{\\frac{1}{r}})^{\\frac{1}{r}}$, where $r > 1$ is a positive integer?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-a267d482c4074a460418a6d6fd46596ca490f257568c971e80f41de07891f537 | What can we say about the limit $\lim_{n\to\infty} ( t^n \cdot \sin n)$ where $t > 1$ is a fixed number? | [
"The limit does not exist."
] | [
"What can we say about the limit $\\lim_{n\\to\\infty} ( t^n \\cdot \\sin n)$ where $t > 1$ is a fixed number?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-fc0034a93e35d4649f479a394040be3be6456317fc4fe86fa9c865e53870b513 | What can we say about the quotient ring $\frac{\mathbb{R}[x,y]}{I}$ where $I=\left \langle 3x+7y-11,8x+y-9 \right \rangle$? | [
"\\(\\{0\\}\\)"
] | [
"What can we say about the quotient ring $\\frac{\\mathbb{R}[x,y]}{I}$ where $I=\\left \\langle 3x+7y-11,8x+y-9 \\right \\rangle$?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9ea130a7dd70b98ed89c052616deccbe4f09b388d3dff59989cc27e83d0da0fd | What can we say about the quotient ring $\frac{\mathbb{R}[x]}{J}$ where $J=\left \langle 3x^{5}+5x^{4}+2x^{3}-3x^{2}-6x,8x^{2}+3x \right \rangle$? | [
"\\(\\mathbb{R}\\)"
] | [
"What can we say about the quotient ring $\\frac{\\mathbb{R}[x]}{J}$ where $J=\\left \\langle 3x^{5}+5x^{4}+2x^{3}-3x^{2}-6x,8x^{2}+3x \\right \\rangle$?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-b718a0b971b3fd804c2d2a04b1362b6b6e9f69b14bced93af567d131292f9cdf | What can we say about two matrices $A, B \in M_n(\mathbb{K})$ such that $AB = BA = A$? | [
"B = I + N \\text{ where } AN = NA = 0"
] | [
"What can we say about two matrices $A, B \\in M_n(\\mathbb{K})$ such that $AB = BA = A$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 4
} | 15 | 0 | 0 |
omr-ws-6611eae3ec98c5526b2cf5b8f568910bfb4cd7d21b9b24f9217776356efcc381 | What can we state about the roots of the equation $x^m + 1 = 0$ in a similar way to the roots of $x^n - 1 = 0$? Is -1 the only real root? | [
"The roots of \\( x^m + 1 = 0 \\) are \\( e^{\\frac{(2k+1)\\pi i}{m}} \\) for \\( 0 \\leq k \\leq m-1 \\), and -1 is the only real root."
] | [
"What can we state about the roots of the equation $x^m + 1 = 0$ in a similar way to the roots of $x^n - 1 = 0$? Is -1 the only real root?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-d721e1a187d14f99bdd434cc403bde7fbade31f7f5a272e28af16400b019fc03 | What can you say about \( n \) if the value of \( \phi(n) \) is a prime number? | [
"6",
"\\( n = 3, 4, 6 \\)"
] | [
"What can you say about \\( n \\) if the value of \\( \\phi(n) \\) is a prime number?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 27
} | 44 | 0 | 0 |
omr-ws-52a5a1913612d439f1fd37f3970eb411fb5137ff2e2d4e81ac306b337406ff41 | What can you say about \( n \) if the value of \( \phi(n) \) is the square of a prime number? | [
"12",
"\\( n = 5, 8, 10, 12 \\)"
] | [
"What can you say about \\( n \\) if the value of \\( \\phi(n) \\) is the square of a prime number?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 2,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-e323348e72497aab5426d63bf77c4f330b55c80a4a9dbdb998d2126440c43a1a | What can you say about matrices \( A \in M(n, \mathbb{R}) \) such that there exists an orthogonal matrix \( P \) for which \( ^{t}PAP = ^{t}A \)? | [
"A \\text{ is a normal matrix}"
] | [
"What can you say about matrices \\( A \\in M(n, \\mathbb{R}) \\) such that there exists an orthogonal matrix \\( P \\) for which \\( ^{t}PAP = ^{t}A \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 3
} | 11 | 0 | 0 |
omr-ws-179d42d090290518aac8fde182a22d4b75ce09980ffff98625be7817cd7f7f64 | What can you say about the integral \(\int_{-1}^{1}\frac{t^{2n}(1-t^{2})^{2n}}{(1+it)^{3n+1}}dt\)? | [
"\\text{The integral is a real number.}"
] | [
"What can you say about the integral \\(\\int_{-1}^{1}\\frac{t^{2n}(1-t^{2})^{2n}}{(1+it)^{3n+1}}dt\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 5
} | 17 | 0 | 0 |
omr-ws-4aa579ce95e01d46b54ce3505bc1467cdcc3b0e21ed42555cb73129c9a2e005c | What can you say about the series $\sum_{n=2}^{\infty} \frac{\sin(n)}{\log(n)}$? | [
"\\text{The series converges conditionally.}"
] | [
"What can you say about the series $\\sum_{n=2}^{\\infty} \\frac{\\sin(n)}{\\log(n)}$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fc98dbeab28e7909599f8a58a62fd8f4ac1fed27eb8f0eb8a93a22cb7ff5698a | What combination of three semicircles placed inside a triangle with sides 13 cm, 14 cm, and 15 cm will produce the maximum total area of the semicircles? | [
"24\\pi"
] | [
"What combination of three semicircles placed inside a triangle with sides 13 cm, 14 cm, and 15 cm will produce the maximum total area of the semicircles?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-6cfc538213211ce7a523c7a6ecfd18026cd89c86611a1515a10fea76906f56fc | What common fraction (that is, a fraction reduced to its lowest terms) is equivalent to $.3\overline{25}$? | [
"\\frac{161}{495}"
] | [
"What common fraction (that is, a fraction reduced to its lowest terms) is equivalent to $.3\\overline{25}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9fb71eb124f472b092c73df2dcb4cef48f18b03a0ccefdfc0579fca3e2209d66 | What common fraction is equivalent to $0.01020304...$? | [
"\\(\\frac{100}{9801}\\)"
] | [
"What common fraction is equivalent to $0.01020304...$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 30
} | 51 | 0 | 0 |
omr-ws-1bd16792acc9145ecddb474245918d1680d8d4ab8eeeb3b7389cb57fca7c6b07 | What common fraction is equivalent to $\frac{\frac{1}{1^3}+\frac{1}{3^3}+\frac{1}{5^3}+\frac{1}{7^3}+...}{\frac{1}{1^3}+\frac{1}{2^3}+\frac{1}{3^3}+\frac{1}{4^3}+...}$? | [
"\\(\\frac{7}{8}\\)"
] | [
"What common fraction is equivalent to $\\frac{\\frac{1}{1^3}+\\frac{1}{3^3}+\\frac{1}{5^3}+\\frac{1}{7^3}+...}{\\frac{1}{1^3}+\\frac{1}{2^3}+\\frac{1}{3^3}+\\frac{1}{4^3}+...}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-ffee8c2fdb31792fdcc0f6fb71685103f84ee022c62b3d701c86252fa50813af | What common fraction is exactly half-way between $\frac{2}{3}$ and $\frac{4}{5}$? | [
"\\frac{11}{15}"
] | [
"What common fraction is exactly half-way between $\\frac{2}{3}$ and $\\frac{4}{5}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9881be1dbf54c8d5b8a4c00dfd999b93580e654926ce7c7ececb70946b9166d9 | What complex numbers (if any) are the roots of some polynomial with positive coefficients? | [
"Any complex number not on the nonnegative real line."
] | [
"What complex numbers (if any) are the roots of some polynomial with positive coefficients?"
] | [
"c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-7333fe1a0266ca4c891ba6d022bf9044cc352101b86e1f2b9e9d0ffd54caa356 | What conclusions can be drawn about the ideal of the ring \(R[x,y]\) generated by the polynomials \(f(x,y) = 4x^9 + 7y^{16} - 1\) and \(g(x,y) = 5x^6 + 9y^9 - 1\)? | [
"(f, g) \\neq R[x, y]"
] | [
"What conclusions can be drawn about the ideal of the ring \\(R[x,y]\\) generated by the polynomials \\(f(x,y) = 4x^9 + 7y^{16} - 1\\) and \\(g(x,y) = 5x^6 + 9y^9 - 1\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 4
} | 15 | 0 | 0 |
omr-ws-4abca6dc6d80eeddb2687214fe06199107e96392479e4b9f4cd323fef29344ff | What condition is necessary so that \[ \lim_{x\to\infty}\left(\int_{a}^{b}f(x,y) \ dy\right) = \int_{a}^{b}\lim_{x\to\infty}f(x,y) \ dy \]? | [
"Uniform convergence, Dominated Convergence Theorem, or Eventual Equiintegrability."
] | [
"What condition is necessary so that\n\\[ \\lim_{x\\to\\infty}\\left(\\int_{a}^{b}f(x,y) \\ dy\\right) = \\int_{a}^{b}\\lim_{x\\to\\infty}f(x,y) \\ dy \\]?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 4,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-a06559acc074a74683f811782a387a61cc1f57484ef38b05014d1cfc4f7a6dab | What condition must $p$ satisfy for the equation $x^{2n} + y^{2n} = p$ to have integer solutions? Note: The original post suggests that the condition $2^{2n} \mid (p-1)$ is not sufficient, but it does not provide a complete condition. The problem is to determine the necessary condition for $p$. | [
"p \\equiv 0, 1, \\text{ or } 2 \\pmod{2^{2n}}"
] | [
"What condition must $p$ satisfy for the equation $x^{2n} + y^{2n} = p$ to have integer solutions?\n\nNote: The original post suggests that the condition $2^{2n} \\mid (p-1)$ is not sufficient, but it does not provide a complete condition. The problem is to determine the necessary condition for $p$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-01d6ce9cdb6050f2a80719ce66baa153e467722eb99743d284aee1ce0d51729f | What condition must be satisfied by the coefficients $u, v, w$ if the roots of the polynomial $x^3 - ux^2 + vx - w$ are the sides of a triangle? | [
"\\(uv > 2w\\)"
] | [
"What condition must be satisfied by the coefficients $u, v, w$ if the roots of the polynomial $x^3 - ux^2 + vx - w$ are the sides of a triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2cc2b7dbfd38a41ab4549ecc0a1cedae785bf88965970409017961c32cebf64a | What condition must be satisfied for the points A, B, C, D to be concyclic? | [
"\\(\\frac{c-b}{a-b}:\\frac{c-d}{a-d}\\in\\mathbb R^*\\)"
] | [
"What condition must be satisfied for the points A, B, C, D to be concyclic?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9761f33a2eac6949726a191854f0c86a9d722ec4c9c8a347918a267ac7060c63 | What condition must the angles $\angle A, \angle B, \angle C$ of $\triangle ABC$ fulfill in order that there is a point $P$ in space such that $\angle APB, \angle BPC, \angle CPA$ are right angles? | [
"\\angle A < 90^\\circ, \\angle B < 90^\\circ, \\angle C < 90^\\circ"
] | [
"What condition must the angles $\\angle A, \\angle B, \\angle C$ of $\\triangle ABC$ fulfill in order that there is a point $P$ in space such that $\\angle APB, \\angle BPC, \\angle CPA$ are right angles?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-12956650c7ac3e0612f8cb0287038588442edcb9aa6661786b6312b3fa9a297e | What condition must the coefficients of the equations \(a_1x^2 + b_1x + c_1 = 0\) and \(a_2x^2 + b_2x + c_2 = 0\) satisfy such that they have one common root? | [
"\\((c_1a_2 - a_1c_2)^2 - (a_1b_2 - b_1a_2)(b_1c_2 - c_1b_2) = 0\\)"
] | [
"What condition must the coefficients of the equations \\(a_1x^2 + b_1x + c_1 = 0\\) and \\(a_2x^2 + b_2x + c_2 = 0\\) satisfy such that they have one common root?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-8c67de9458463b82cc4b9d584ac4012ce94bbaea9b4d9f0979510b54b20555f8 | What condition must the eigenvalues of a symmetric real matrix satisfy for there to exist an antisymmetric real matrix whose square is the symmetric matrix? | [
"The eigenvalues of the symmetric real matrix must be non-positive, and the negative eigenvalues must come in pairs (i.e., each non-zero eigenvalue must have even multiplicity)."
] | [
"What condition must the eigenvalues of a symmetric real matrix satisfy for there to exist an antisymmetric real matrix whose square is the symmetric matrix?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-8e26b6bf8dfa92425360759bebf18f8188cd987ed58bfc2d7f1bada2cf24e27e | What condition must the function \( f \) satisfy to be convex on the entire interval \( a \leq x \leq b \), given that \( f \) is convex on \( a \leq x \leq c \) and on \( c \leq x \leq b \)? | [
"\\( f \\) must be continuous at \\( x = c \\)."
] | [
"What condition must the function \\( f \\) satisfy to be convex on the entire interval \\( a \\leq x \\leq b \\), given that \\( f \\) is convex on \\( a \\leq x \\leq c \\) and on \\( c \\leq x \\leq b \\)?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-d8989619614c91107c91d5592bfa1f0bab0863e65e2f597bdd1e13792880bf49 | What condition must the lengths of the sides of a triangle satisfy in order for a triangle composed of its medians to be similar to the original triangle? | [
"The triangle must be equilateral."
] | [
"What condition must the lengths of the sides of a triangle satisfy in order for a triangle composed of its medians to be similar to the original triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-f21d37ffdf1a3805b715566e3b4833c87fee3ec1fa3f599c0016735606c0573e | What condition must the positive numbers $S$ and $p$ satisfy such that there exists a right-angled triangle with area $S$ and perimeter $p$? | [
"\\( p^2 - (12 - 8\\sqrt{2})S \\ge 0 \\)"
] | [
"What condition must the positive numbers $S$ and $p$ satisfy such that there exists a right-angled triangle with area $S$ and perimeter $p$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 2
} | 5 | 0 | 0 |
omr-ws-1eb1eaac486f3c5c990d2326fe8ff54f30da2172571f91a56a6e44416939a9fd | What condition must the sets \( A_1, A_2, \ldots \) of positive integers satisfy such that there exists a function \( f \) from the positive integers to the same set, where \( f^k \) has the image \( A_k \) for all \( k \)? Here, \( f^k \) denotes the \( k \)-th iterate of the function \( f \). | [
"A_1 \\supseteq A_2 \\supseteq A_3 \\supseteq \\cdots \\text{ and each } A_k \\text{ is non-empty}"
] | [
"What condition must the sets \\( A_1, A_2, \\ldots \\) of positive integers satisfy such that there exists a function \\( f \\) from the positive integers to the same set, where \\( f^k \\) has the image \\( A_k \\) for all \\( k \\)? Here, \\( f^k \\) denotes the \\( k \\)-th iterate of the function \\( f \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 5
} | 17 | 0 | 0 |
omr-ws-88ae3eb5e850d18deea6809cda800522ec65727a0f46881e7b01e1749431f08b | What condition must triangle \(ABC\) satisfy so that the Euler's line \((E)\) is parallel to side \(BC\)? | [
"\\(2 \\cos C \\cos B = \\cos A\\)"
] | [
"What condition must triangle \\(ABC\\) satisfy so that the Euler's line \\((E)\\) is parallel to side \\(BC\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1f9110844d8efe361e864b9027768956e97ba02fdd0d8c1a680eb122c6436ed1 | What condition on \( f \) will ensure that \( f(C) \subseteq D \) if and only if \( C \subseteq g(D) \), where \( g \) is the inverse of \( f \)? | [
"f \\text{ is bijective}"
] | [
"What condition on \\( f \\) will ensure that \\( f(C) \\subseteq D \\) if and only if \\( C \\subseteq g(D) \\), where \\( g \\) is the inverse of \\( f \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6d5019f4e942af18060af327df4fcc3c60071776747ad9964f0897ac7de705d9 | What condition on the radii of three pairwise tangent spheres guarantees the existence of a common tangent plane to the spheres? | [
"r_3\\ge \\frac{r_1r_2}{\\left(\\sqrt {r_1}+\\sqrt {r_2}\\right)^2}"
] | [
"What condition on the radii of three pairwise tangent spheres guarantees the existence of a common tangent plane to the spheres?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8cd42f9dd607c0f5551cc56f82fb51f4937f77c5ca377325a71063f6386cf700 | What condition should the lengths $a$ and $b$ meet to make it possible to unscrew a square nut with side $a$ using a wrench whose hole has the form of a regular hexagon with side $b$? | [
"\\( a = (3 - \\sqrt{3})b \\)"
] | [
"What condition should the lengths $a$ and $b$ meet to make it possible to unscrew a square nut with side $a$ using a wrench whose hole has the form of a regular hexagon with side $b$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8ce4c73a58e008bc7ac71c3059dffb23447c61051ad6ffe42ba26674f5ec85ab | What conditions are necessary and sufficient for one root of the quadratic equation \( ax^2 + bx + c \) to be the square of the other root? | [
"\\(\\left(\\frac{c}{a}\\right)^2 = \\left(\\frac{b^2 - ac}{a(a - b)}\\right)^3\\)"
] | [
"What conditions are necessary and sufficient for one root of the quadratic equation \\( ax^2 + bx + c \\) to be the square of the other root?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-20b681a8141839bf83f4396bf5f2d727d70eb1a69a4956619feb71895f2ff909 | What conditions does the equality case of the inequality $0 \leq Tr(AB) \leq Tr(A)Tr(B)$ imply? | [
"AB = 0 \\text{ or } A \\text{ and } B \\text{ are rank-one and proportional}"
] | [
"What conditions does the equality case of the inequality $0 \\leq Tr(AB) \\leq Tr(A)Tr(B)$ imply?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 5
} | 20 | 0 | 0 |
omr-ws-7ebb7410c7e3418687a6e9ad08a473ea199c9fd25ef5f4d6f87800bbe309f09a | What conditions must $a, b \in \mathbb{C}$ satisfy such that $\lim_{n \to \infty} a^n + b^n \in \mathbb{C}$? | [
"\\(|a| < 1\\) and \\(|b| < 1\\)"
] | [
"What conditions must $a, b \\in \\mathbb{C}$ satisfy such that $\\lim_{n \\to \\infty} a^n + b^n \\in \\mathbb{C}$?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d420f440757a367469ab35cdb4933defba7844f7b86f150e785cc545cfc909b7 | What conditions must $a, b, c \in \mathbb{C}$ satisfy such that $\lim_{n \to \infty} a^n + b^n + c^n \in \mathbb{C}$? | [
"\\(|a| < 1\\) or \\(a = 1\\), \\(|b| < 1\\) or \\(b = 1\\), \\(|c| < 1\\) or \\(c = 1\\)"
] | [
"What conditions must $a, b, c \\in \\mathbb{C}$ satisfy such that $\\lim_{n \\to \\infty} a^n + b^n + c^n \\in \\mathbb{C}$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-bc699ca66c5ee8de207a40a791e08703fe668d26abd0bb108532743feec3330d | What conditions must \( x \) and \( k \) satisfy so that the equation \(\log_{10}(\sec x) = k\) is meaningful? | [
"nra"
] | [
"What conditions must \\( x \\) and \\( k \\) satisfy so that the equation \\(\\log_{10}(\\sec x) = k\\) is meaningful?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ada930112ac5f26bbadbab6abc0938da28c7e8b63daeb07e281bebcade5a251d | What conditions must \(a\) and \(b\) satisfy if the function \(y = x^3 + ax^2 + bx\) is to have a local maximum and a local minimum? | [
"\\( a^2 > 3b \\)"
] | [
"What conditions must \\(a\\) and \\(b\\) satisfy if the function \\(y = x^3 + ax^2 + bx\\) is to have a local maximum and a local minimum?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-309f673e556a6637cbd450b6f216e70fbc90d3ff9105ebbb993dabf7ea031305 | What conditions must a function $f:\mathbb{R}\to\mathbb{R}$ satisfy in order for there to exist a function $g:\mathbb{R}\to\mathbb{R}$ such that $g\circ g=f$? | [
"\\text{No conditions are required; such a function } g \\text{ exists for any } f."
] | [
"What conditions must a function $f:\\mathbb{R}\\to\\mathbb{R}$ satisfy in order for there to exist a function $g:\\mathbb{R}\\to\\mathbb{R}$ such that $g\\circ g=f$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-de72acedee66f4ac9d77ca702b618dd1c20fa9a1e07ec5e10159bc37356f7167 | What conditions must be imposed on the number $d>0$ for the inequality $$0<\frac{d^2+R^2-r^2}{2dR}\le 1$$ to be valid for $R\ge r>0$? | [
"R - r ≤ d ≤ R + r"
] | [
"What conditions must be imposed on the number $d>0$ for the inequality $$0<\\frac{d^2+R^2-r^2}{2dR}\\le 1$$ to be valid for $R\\ge r>0$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-142416345f825927268d364a08e3d7c30dc010b70b7a7b29be956d9e067f82a2 | What conditions must be imposed on the numbers $a$ and $b$ for the equation $$1+\log_b(2\log a-x)\log_x b=\frac{2}{\log_b x}$$ to have at least one solution? Find all the solutions of this equation. | [
"x_{1,2}=\\log a\\pm \\sqrt {\\log^2 a-b^2}\n\nwith conditions:\n- For \\(a \\ge 10^b\\) and \\(\\log a \\ne \\frac{1}{2}(b^2+1)\\), both roots are positive and unequal to unity.\n- For \\(\\log a = \\frac{1}{2}(b^2+1)\\), only take the root \\(x_1 = b^2\\).\n- For \\(a < 10^b\\), the equation has no roots."
] | [
"What conditions must be imposed on the numbers $a$ and $b$ for the equation $$1+\\log_b(2\\log a-x)\\log_x b=\\frac{2}{\\log_b x}$$ to have at least one solution? Find all the solutions of this equation."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0a3266360cd62e77dc8d86dc2cc1d572240b9e105335cabc3fd49a329b8597b7 | What conditions must be imposed on the real coefficients $a_1$, $b_1$, $a_2$, $b_2$, $a_3$ and $b_3$ for the expression $$(a_1+b_1x)^2+(a_2+b_2x)^2+(a_3+b_3x)^2$$ to be the square of a polynomial of the first degree in $x$ with real coefficients? | [
"a_1=\\lambda b_1, a_2=\\lambda b_2, a_3=\\lambda b_3 \\text{ and } b_1^2+b_2^2+b_3^2\\ne 0"
] | [
"What conditions must be imposed on the real coefficients $a_1$, $b_1$, $a_2$, $b_2$, $a_3$ and $b_3$ for the expression $$(a_1+b_1x)^2+(a_2+b_2x)^2+(a_3+b_3x)^2$$ to be the square of a polynomial of the first degree in $x$ with real coefficients?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7ac5e0e784d75e3104d72347dbbc9587f7630c3fb99b0796533d9b631d024b9f | What conditions must be met by a system of linear equations for it to have: a) no solution? b) a unique solution? c) infinitely many solutions? | [
"a) \\(k = -1\\), b) \\(k \\neq -1\\), c) Not possible"
] | [
"What conditions must be met by a system of linear equations for it to have:\na) no solution?\nb) a unique solution?\nc) infinitely many solutions?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9dbee3110fd8b3297b7a93ab02b5099d7d3b3f79075c2ffeb8edddf5439432fa | What conditions must be satisfied by the numbers $a$, $b$ and $c$ for the systems of equations $$\left\{\begin{array}{lll} \sin x+\sin y=2a,\medskip \\ \cos x+\cos y=2b,\medskip \\ \tan x\tan y=c \end{array}\right.$$ to have at least one solutions? | [
"0 < a^2 + b^2 \\le 1 \\text{ and } c = \\frac{(a^2 + b^2)^2 - b^2}{(a^2 + b^2)^2 - a^2}, \\text{ or } a = b = 0 \\text{ and } c \\ge 0"
] | [
"What conditions must be satisfied by the numbers $a$, $b$ and $c$ for the systems of equations $$\\left\\{\\begin{array}{lll} \\sin x+\\sin y=2a,\\medskip \\\\ \\cos x+\\cos y=2b,\\medskip \\\\ \\tan x\\tan y=c \\end{array}\\right.$$ to have at least one solutions?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1c7fd4ed0fb9a90c338147d77b3aa2130083e2a5fcf6b34b31ddef804dd8ad61 | What conditions on \( n \) are required for \( n \) to be expressible as a sum of squares in more than one way? | [
"n \\text{ is divisible by at least two distinct primes congruent to } 1 \\pmod{4} \\text{ or by a prime congruent to } 1 \\pmod{4} \\text{ raised to at least the second power, and all primes congruent to } 3 \\pmod{4} \\text{ in its factorization have even exponents}"
] | [
"What conditions on \\( n \\) are required for \\( n \\) to be expressible as a sum of squares in more than one way?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 8
} | 9 | 0 | 0 |
omr-ws-4eb312239c60f9857e9012e234c3812717b85b8b142329ef969890aba5c76131 | What conditions on \(a, b, c, d\) are necessary and sufficient for the function \(f(x) = \frac{ax + b}{cx + d}\) to coincide with its inverse function \(f^{-1}(x)\)? | [
"\\(a + d = 0\\) or \\(b = c = 0\\) and \\(a = d\\)"
] | [
"What conditions on \\(a, b, c, d\\) are necessary and sufficient for the function \\(f(x) = \\frac{ax + b}{cx + d}\\) to coincide with its inverse function \\(f^{-1}(x)\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 1,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-c8fb57e88aae203514d860b5db8306a015bf324fd63c9449445874f55dfe77b0 | What conditions should the field \( K \) and the integer \( n \) satisfy so that there exists an irreducible polynomial \( x^n - a \in K[x] \) and a cyclic extension \( E \) of \( K \) with degree \( n \)? | [
"\\( K \\) must contain the \\( n \\)-th roots of unity and \\( \\operatorname{char}(K) \\nmid n \\)."
] | [
"What conditions should the field \\( K \\) and the integer \\( n \\) satisfy so that there exists an irreducible polynomial \\( x^n - a \\in K[x] \\) and a cyclic extension \\( E \\) of \\( K \\) with degree \\( n \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-0911eb836f4c44afa608b6c897b98c05602b081d231d6540aaf54cdf9436edd0 | What conditions should vectors \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\) satisfy so that they can be the sides of a triangle? | [
"\\(\\lambda_1 \\mathbf{a} + \\lambda_2 \\mathbf{b} + \\lambda_3 \\mathbf{c} = \\mathbf{0}\\) where \\(\\lambda_1, \\lambda_2, \\lambda_3 \\in \\{-1, 1\\}\\)"
] | [
"What conditions should vectors \\(\\mathbf{a}\\), \\(\\mathbf{b}\\), and \\(\\mathbf{c}\\) satisfy so that they can be the sides of a triangle?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 3
} | 4 | 0 | 0 |
omr-ws-05f9dcac9be2bc61482a9ce0bf6eaa6b46d614f321bc07d4196545bae66441a3 | What constraints must the elements $(R_{ij})$ of the three-dimensional rotation matrix satisfy in order to preserve the length of $\bold{A}$ (for all vectors $\bold{A}$)? | [
"\\( R^T R = I \\)"
] | [
"What constraints must the elements $(R_{ij})$ of the three-dimensional rotation matrix satisfy in order to preserve the length of $\\bold{A}$ (for all vectors $\\bold{A}$)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-2355802a775bdf7d40e261be213a3df3be5806b74a9462ef02fc82e3fd38c6a9 | What could be the defining function for the sequence 3, 4, 6, 12, ...? | [
"\\( a_n = \\frac{a_{n-2}a_{n-1}}{2} \\) for \\( n \\geq 2 \\) with \\( a_0 = 3 \\) and \\( a_1 = 4 \\)"
] | [
"What could be the defining function for the sequence 3, 4, 6, 12, ...?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-03cffe6398704e36f9f606f9f1fba388703a6e66f16d4f998060f0854dab14be | What criteria should matrix \( B = \left[\begin{array}{ccc}a & b & c \\ d & e & f \\ g & h & i\end{array}\right] \) satisfy if we want matrices \( A = \left[\begin{array}{ccc}2 & 0 & 0 \\ 0 & -5 & 0 \\ 0 & 0 & -9\end{array}\right] \) and \( B \) to be congruent over \( \mathbb{Q} \)? (Matrices \( A \) and \( B \) are c... | [
"B \\text{ is symmetric, } \\det(B) = 90k^2 \\text{ for some } k \\in \\mathbb{Q}, \\text{ and } B \\text{ has signature } (1, 2)"
] | [
"What criteria should matrix \\( B = \\left[\\begin{array}{ccc}a & b & c \\\\ d & e & f \\\\ g & h & i\\end{array}\\right] \\) satisfy if we want matrices \\( A = \\left[\\begin{array}{ccc}2 & 0 & 0 \\\\ 0 & -5 & 0 \\\\ 0 & 0 & -9\\end{array}\\right] \\) and \\( B \\) to be congruent over \\( \\mathbb{Q} \\)? (Matr... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-bf2045b9bc6d7a6bb326521c5d7179506f57ca7d46226a22b8ae514211c7fc6e | What day of the week will April 5th, 4021 be, given that April 5th, 2019 is a Friday? | [
"Monday"
] | [
"What day of the week will April 5th, 4021 be, given that April 5th, 2019 is a Friday?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-6575c2e54fcbd9a96b0e805aa9702b36afabba5e64efbb52e44b2b921865eea4 | What day of the week will it be 40 days from Tuesday? | [
"\\text{Sunday}"
] | [
"What day of the week will it be 40 days from Tuesday?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c9d0a2a75445d4f7e323f7e8700f687dffe40d280146cbd55cace559ded0c47f | What degree should a Taylor Polynomial have to approximate $ \sqrt{e} $ to within .001, assuming it is centered at 0 and using the function $ f(x) = e^x $ with $ x = \frac{1}{2} $? | [
"4"
] | [
"What degree should a Taylor Polynomial have to approximate $ \\sqrt{e} $ to within .001, assuming it is centered at 0 and using the function $ f(x) = e^x $ with $ x = \\frac{1}{2} $?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-633629d9207908b0a0dedcafd1c3319b35ff3fccbb8bfe8e8a2402255c2f47dc | What digit $A$ will make the number $83A5$ divisible by $9$? | [
"2"
] | [
"What digit $A$ will make the number $83A5$ divisible by $9$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-94d215ab672a301c34992e3f344036d58ae79445a0bbc0310df116d1729bc442 | What digit is immediately to the right of the decimal point in $(\sqrt{3}+\sqrt{2})^{2002}$? | [
"9"
] | [
"What digit is immediately to the right of the decimal point in $(\\sqrt{3}+\\sqrt{2})^{2002}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b03e6e4e3b70efdc7a4aa4e764015f3ad38a699091d216512ebf290b8b0ec7bf | What digit is in the 2005th place after the decimal point in the decimal expansion of the fraction \( \frac{1}{81} \)? | [
"6"
] | [
"What digit is in the 2005th place after the decimal point in the decimal expansion of the fraction \\( \\frac{1}{81} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-4557438e0f9662d8bb1018b8a3b0c2474cc249cd274f062fdddf2718eb61ecd1 | What digit occupies the 206,788th position in the sequence formed by writing all integers beginning with 1 successively? | [
"7"
] | [
"What digit occupies the 206,788th position in the sequence formed by writing all integers beginning with 1 successively?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f60a7225c7aeceb3c7d9871a2cc8f3d4e910d601d2dad49dcb8daafb6219cd98 | What digits can stand at the beginning of the whole numbers which increase three times when these initial digits are carried to the end of the numbers? Find all such numbers. | [
"The smallest possible number is 142857"
] | [
"What digits can stand at the beginning of the whole numbers which increase three times when these initial digits are carried to the end of the numbers? Find all such numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a9f27c06b7620c766c16f1151bd4335fb4a7381577967221bb26abab5ee67d49 | What digits should be put instead of \( x \) and \( y \) in \( 30x0y03 \) in order to make the number divisible by 13? | [
"\\( (8, 1), (5, 2), (2, 3), (9, 5), (6, 6), (3, 7), (0, 8) \\)"
] | [
"What digits should be put instead of \\( x \\) and \\( y \\) in \\( 30x0y03 \\) in order to make the number divisible by 13?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 19
} | 31 | 0 | 0 |
omr-ws-cd847068e7802f5d084b2127aa9db2200a4a3c294ca733616be5021ef496d480 | What do the finite subgroups of $\left(\mathbb{C}\setminus \{0\}, *\right)$ look like? | [
"The finite subgroups of \\(\\left(\\mathbb{C}\\setminus \\{0\\}, *\\right)\\) are cyclic groups consisting of the \\(n\\)th roots of unity for some positive integer \\(n\\)."
] | [
"What do the finite subgroups of $\\left(\\mathbb{C}\\setminus \\{0\\}, *\\right)$ look like?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-480a157b486af0575c408c9f478992e798000aa11d1a4003c1243f631c8620ae | What do we get if the entries in the 100th row of the array are summed with alternating signs, where the first row contains the numbers 1 and 2, and every further entry is the sum of the two neighbors above? | [
"0"
] | [
"What do we get if the entries in the 100th row of the array are summed with alternating signs, where the first row contains the numbers 1 and 2, and every further entry is the sum of the two neighbors above?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-39b79f9149b1b350c25f2e22f02d02d83b0853d10b7959f607d07c3e7d3c1737 | What do you need to multiply the numerator and denominator by to rationalize the expression? $$\frac{9-40\sqrt[3]{6}-6\sqrt[3]{36}}{1-\sqrt[3]{6}-3\sqrt[3]{36}}$$ | [
"17 - 55\\sqrt[3]{6} - 4\\sqrt[3]{36}"
] | [
"What do you need to multiply the numerator and denominator by to rationalize the expression?\n\n$$\\frac{9-40\\sqrt[3]{6}-6\\sqrt[3]{36}}{1-\\sqrt[3]{6}-3\\sqrt[3]{36}}$$"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-ca52e13ca17b45e0727f0ba0f50f6d17f4b510a1e8746af0bd2afc8c21ec9bb4 | What does "uniform on \( B(0,1) \)" mean? | [
"\\( X_i \\) are uniformly distributed over the \\( d \\)-dimensional unit ball \\( B(0,1) \\)."
] | [
"What does \"uniform on \\( B(0,1) \\)\" mean?"
] | [
"c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-69d08262efd5878da42b2f93e8a56a2a879d27139dbf966fe38bc497ddfc4f98 | What does $E$ represent in the expression $E=(\mathbb{Z}/2\mathbb{Z})^I$, where $I$ is a set? | [
"Cartesian product of \\( |I| \\) copies of \\( \\mathbb{Z}/2\\mathbb{Z} \\)"
] | [
"What does $E$ represent in the expression $E=(\\mathbb{Z}/2\\mathbb{Z})^I$, where $I$ is a set?"
] | [
"c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-31b0e378793d2d88b54d45dbaa8c4087e6f1cd394a29684f2589fc694a520aee | What does $\sqrt{21-20i}$ equal? Does it only equal one value, and if so, is it $5-2i$ or $5+2i$? | [
"\\(5 - 2i\\)"
] | [
"What does $\\sqrt{21-20i}$ equal? Does it only equal one value, and if so, is it $5-2i$ or $5+2i$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-8ab40e3c9eccd79b1122c42b24a17d0a4c1217548c8aeadc043fad13aee0a2d4 | What does $\sum_{n=1}^x \frac{\sigma(n)}{n}$ equal? | [
"\\(\\frac{\\pi^2 x}{6} + o(x)\\)"
] | [
"What does $\\sum_{n=1}^x \\frac{\\sigma(n)}{n}$ equal?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 9
} | 9 | 0 | 0 |
omr-ws-b986403b326d58033a8a0f0d267a0c57936905716c36e4aabfeba62a85dda34e | What does $a \in \mathbb{Q}$ equal if $$\sqrt{4-2\sqrt{3}} + a\sqrt{12}$$ is a rational number? | [
"-0.5"
] | [
"What does $a \\in \\mathbb{Q}$ equal if $$\\sqrt{4-2\\sqrt{3}} + a\\sqrt{12}$$ is a rational number?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
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