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-00ebd3ea59bc265e5cb58c9219ee579eaf1265ae24c0270de9b4b468009ee69e | Assume that $n$ copies of unit cubes are glued together side by side to form a rectangular solid block. If the number of unit cubes that are completely invisible is $30$, then find the minimum possible value of $n$. | [
"140"
] | [
"Assume that $n$ copies of unit cubes are glued together side by side to form a rectangular solid block. If the number of unit cubes that are completely invisible is $30$, then find the minimum possible value of $n$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-d51173bfa1d22689ca6288f706eb0efde95762d1acc226e2bec9b91574f5ec95 | Assume that $n$ is divisible by 3. If an $n$-gon has $k$ right angles, find the maximum value of $k$ such that $k = \frac{2}{3}n$ and $n = 3m$ for some positive integer $m$. Calculate the value of $k$ when $m = 10$. | [
"20"
] | [
"Assume that $n$ is divisible by 3. If an $n$-gon has $k$ right angles, find the maximum value of $k$ such that $k = \\frac{2}{3}n$ and $n = 3m$ for some positive integer $m$. Calculate the value of $k$ when $m = 10$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-167f5c4ca427b3e555113f368e4d4429872c50f4623d5ce257bcc74feb5a35cf | Assume that $n=2022$. Find the minimum number of training sessions that the club must run, given that each member attends all sessions, each session divides the members into three groups (swimming, cycling, and running), each group has at least one person, and for any two members, there is at least one session where th... | [
"7"
] | [
"Assume that $n=2022$. Find the minimum number of training sessions that the club must run, given that each member attends all sessions, each session divides the members into three groups (swimming, cycling, and running), each group has at least one person, and for any two members, there is at least one session whe... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c1fd8c62b6ccbef6d5473e88480ea2b89841be88f86eb423c1fe79166eb5c8c5 | Assume that $n=9$. The club has already run one training session and will run one more session. How many ways can the group be divided for the second training session, given that each member attends all sessions, each session divides the members into three groups (swimming, cycling, and running), each group has at leas... | [
"216"
] | [
"Assume that $n=9$. The club has already run one training session and will run one more session. How many ways can the group be divided for the second training session, given that each member attends all sessions, each session divides the members into three groups (swimming, cycling, and running), each group has at... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-88bd8fbb205bb19478b93c73aa13e2d5108bf288b3cc828d8758045c9cb260a0 | Assume that $x_1,x_2,\ldots,x_7$ are real numbers such that \[\begin{aligned} x_1+4x_2+9x_3+16x_4+25x_5+36x_6+49x_7 &= 1 \\ 4x_1+9x_2+16x_3+25x_4+36x_5+49x_6+64x_7 &= 12 \\ 9x_1+16x_2+25x_3+36x_4+49x_5+64x_6+81x_7 &= 123. \end{aligned}\]Find the value of $16x_1+25x_2+36x_3+49x_4+64x_5+81x_6+100x_7$. | [
"334"
] | [
"Assume that $x_1,x_2,\\ldots,x_7$ are real numbers such that\n\\[\\begin{aligned} x_1+4x_2+9x_3+16x_4+25x_5+36x_6+49x_7 &= 1 \\\\ \n4x_1+9x_2+16x_3+25x_4+36x_5+49x_6+64x_7 &= 12 \\\\\n9x_1+16x_2+25x_3+36x_4+49x_5+64x_6+81x_7 &= 123. \\end{aligned}\\]Find the value of $16x_1+25x_2+36x_3+49x_4+64x_5+81x_6+100x_7$."
... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-15dd5b51ed927e6af0487b239f910f7cd8b1e827a66fbdaafaab4b7c45a5afd1 | Assume that $|f(x)| \leq 1$ and $|f''(x)| \leq 1$ for all $x$ on an interval of length at least $2$. Let $M$ be the maximum value of $|f'(x)|$ on the interval. Find the smallest possible value of $M$. | [
"1"
] | [
"Assume that $|f(x)| \\leq 1$ and $|f''(x)| \\leq 1$ for all $x$ on an interval of length at least $2$. Let $M$ be the maximum value of $|f'(x)|$ on the interval. Find the smallest possible value of $M$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 6
} | 8 | 0 | 0 |
omr-ws-f3193cd7c9a816f2ee2dadf711a9cf8796f3b063762d971bbc7aad1a3b6cbd8a | Assume that 80% of the children of Harvard alumni go to Harvard, and the rest go to Yale. Of the children of Yale alumni, 40% go to Yale and the remainder are divided equally between Harvard and Dartmouth. Of the children of Dartmouth alumni, 70% go to Dartmouth, 20% to Harvard, and 10% to Yale. Find the probability th... | [
"0.644"
] | [
"Assume that 80% of the children of Harvard alumni go to Harvard, and the rest go to Yale. Of the children of Yale alumni, 40% go to Yale and the remainder are divided equally between Harvard and Dartmouth. Of the children of Dartmouth alumni, 70% go to Dartmouth, 20% to Harvard, and 10% to Yale. Find the probabili... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-dfbaf931674b37142a971bc4de059694138579359a440282a931ac377d68e1a5 | Assume that A is a 1×n table where \( n = 2^k \). We place the numbers 1 through n in the table in an arbitrary order. We perform the following operation 2k times: for each \( i \) from 1 to \( n \), if the \( i \)-th entry is \( s \), we replace the \( i \)-th entry with the number in the \( s \)-th entry. Let \( T \)... | [
"136"
] | [
"Assume that A is a 1×n table where \\( n = 2^k \\). We place the numbers 1 through n in the table in an arbitrary order. We perform the following operation 2k times: for each \\( i \\) from 1 to \\( n \\), if the \\( i \\)-th entry is \\( s \\), we replace the \\( i \\)-th entry with the number in the \\( s \\)-th... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0389734ae04cefbd69a26442144a0315db153a194788962cdd58a7d9a6138c48 | Assume that \( 0 \le \alpha_1, \alpha_2, \ldots, \alpha_n \le \pi \) such that: \[ \sum_{i=1}^n \cos \alpha_i + n \] is odd. Find the minimum possible value of: \[ \sum_{i=1}^n \sin \alpha_i \] for \( n = 5 \). | [
"1"
] | [
"Assume that \\( 0 \\le \\alpha_1, \\alpha_2, \\ldots, \\alpha_n \\le \\pi \\) such that:\n\\[ \\sum_{i=1}^n \\cos \\alpha_i + n \\]\nis odd. Find the minimum possible value of:\n\\[ \\sum_{i=1}^n \\sin \\alpha_i \\]\nfor \\( n = 5 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4aa6a274310a5515ee316b6f8b1e509e8bbc77ee770f362d21db3047de276fce | Assume that \( 0 \leq x_1, x_2, x_3, y_1, y_2, y_3 \leq \frac{\pi}{2} \). Let \( x_1^2 + x_2^2 + x_3^2 = 10 \) and \( y_1^2 + y_2^2 + y_3^2 = 5 \). Calculate the minimum possible value of \( (\cos y_1 + \cos y_2 + \cos y_3) - (\cos x_1 + \cos x_2 + \cos x_3) \). | [
"3\\left( \\cos \\sqrt{\\frac{5}{3}} - \\cos \\sqrt{\\frac{10}{3}} \\right)"
] | [
"Assume that \\( 0 \\leq x_1, x_2, x_3, y_1, y_2, y_3 \\leq \\frac{\\pi}{2} \\). Let \\( x_1^2 + x_2^2 + x_3^2 = 10 \\) and \\( y_1^2 + y_2^2 + y_3^2 = 5 \\). Calculate the minimum possible value of \\( (\\cos y_1 + \\cos y_2 + \\cos y_3) - (\\cos x_1 + \\cos x_2 + \\cos x_3) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 3
} | 4 | 0 | 0 |
omr-ws-ded2f0307d15e48eb46a5e74e55117d34d3fb29bcdeedd2f2c25f412895d4f6b | Assume that \( A \) and \( B \) are two \( 4 \times 4 \) complex matrices such that \( A^2 = I \) and \( B^2 = I \). Find the dimension of the largest subspace \( V \) of \( \mathbb{C}^4 \) that is invariant under both \( A \) and \( B \). | [
"2"
] | [
"Assume that \\( A \\) and \\( B \\) are two \\( 4 \\times 4 \\) complex matrices such that \\( A^2 = I \\) and \\( B^2 = I \\). Find the dimension of the largest subspace \\( V \\) of \\( \\mathbb{C}^4 \\) that is invariant under both \\( A \\) and \\( B \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-aeeb263f0d87caa26d0d7b5a50809833191a9e1c7c1bcaf7bdfc59085073f1b9 | Assume that \( C \) is a convex subset of \( \mathbb{R}^3 \). Suppose that \( C_1, C_2, \dots, C_n \) are translations of \( C \) such that \( C_i \cap C \neq \emptyset \) but \( C_i \cap C_j = \emptyset \) for \( i \neq j \). Let \( f(C) \) be the maximum number of such translations. If \( C \) is a sphere of radius 1... | [
"12"
] | [
"Assume that \\( C \\) is a convex subset of \\( \\mathbb{R}^3 \\). Suppose that \\( C_1, C_2, \\dots, C_n \\) are translations of \\( C \\) such that \\( C_i \\cap C \\neq \\emptyset \\) but \\( C_i \\cap C_j = \\emptyset \\) for \\( i \\neq j \\). Let \\( f(C) \\) be the maximum number of such translations. If \\... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-95530e42649b1b98c5081412758d971179817edc2c1fdd23c3b25e24d59508c7 | Assume that \( D, E, F \) are midpoints of \( BC, AC, AB \) of \( \triangle ABC \). Suppose that \( AD, BE, CF \) intersect the circumcircle of \( \triangle ABC \) at \( A_1, B_1, C_1 \) respectively. \( A_1E \) and \( A_1F \) intersect the circumcircle of \( \triangle ABC \) again at points \( A', A'' \) respectively.... | [
"3"
] | [
"Assume that \\( D, E, F \\) are midpoints of \\( BC, AC, AB \\) of \\( \\triangle ABC \\). Suppose that \\( AD, BE, CF \\) intersect the circumcircle of \\( \\triangle ABC \\) at \\( A_1, B_1, C_1 \\) respectively. \\( A_1E \\) and \\( A_1F \\) intersect the circumcircle of \\( \\triangle ABC \\) again at points \... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-69708187335d827b793d0282d911babe1bdf4c525b7642141028d10fa682c3cb | Assume that \( F \) is a family of real sequences. For every \( (a_n), (b_n) \in F \), it holds that \( \sum a_n b_n = 0 \). Does it follow that \( F \) is countable? | [
"Yes"
] | [
"Assume that \\( F \\) is a family of real sequences. For every \\( (a_n), (b_n) \\in F \\), it holds that \\( \\sum a_n b_n = 0 \\). Does it follow that \\( F \\) is countable?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-f4cd641c58e8542a164c129383280bcf8889e27afcdb5430f4e2d4b054bf73c0 | Assume that \( K \) is a CW complex containing \( m \) \( n \)-simplices \( \sigma_1, \dots, \sigma_m \) such that \( |K| \) is homeomorphic to an \( n \)-dimensional unit disk. If \( m \geq 1 \), let \( f(m) \) be the number of indices \( i \) such that the sub-CW complex generated by \( \sigma_1, \dots, \sigma_{i-1},... | [
"5"
] | [
"Assume that \\( K \\) is a CW complex containing \\( m \\) \\( n \\)-simplices \\( \\sigma_1, \\dots, \\sigma_m \\) such that \\( |K| \\) is homeomorphic to an \\( n \\)-dimensional unit disk. If \\( m \\geq 1 \\), let \\( f(m) \\) be the number of indices \\( i \\) such that the sub-CW complex generated by \\( \\... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-c55a3bea9a5b4735c2580e1c87a07d87d249eefcf1bc4c05f412ac5172afe2e4 | Assume that \( M \) is a compact \( C^r \)-manifold with \( r \geq 1 \), and \( f: M \rightarrow \mathbb{R} \) is differentiable. If \( f \) has exactly two critical points, determine the Euler characteristic \( \chi(M) \). | [
"2"
] | [
"Assume that \\( M \\) is a compact \\( C^r \\)-manifold with \\( r \\geq 1 \\), and \\( f: M \\rightarrow \\mathbb{R} \\) is differentiable. If \\( f \\) has exactly two critical points, determine the Euler characteristic \\( \\chi(M) \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bc0f23a5d7f4196abe1d451ccbb8b5a0cd9ef3449507ea7c71cea0b921982efe | Assume that \( P \) is the Brocard point of triangle \( ABC \), i.e., \( \angle PAB = \angle PBC = \angle PCA \). Let \( D, E, F \) be the feet of the perpendiculars from \( P \) to \( AB, BC, CA \) respectively. Given that \( BD = CE = FA \), calculate the value of \( \frac{AB}{BC} + \frac{BC}{CA} + \frac{CA}{AB} \) i... | [
"3"
] | [
"Assume that \\( P \\) is the Brocard point of triangle \\( ABC \\), i.e., \\( \\angle PAB = \\angle PBC = \\angle PCA \\). Let \\( D, E, F \\) be the feet of the perpendiculars from \\( P \\) to \\( AB, BC, CA \\) respectively. Given that \\( BD = CE = FA \\), calculate the value of \\( \\frac{AB}{BC} + \\frac{BC}... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8de23e61026d5204805317bffacb391f5911aba713d206fecf9753e8ff5c1bff | Assume that \( R \) is a Noetherian ring and \( M \) is a finitely generated \( R \)-module. Let \( p \in \text{supp}_R(M) \). Calculate the minimum possible dimension of \( \text{Hom}_R(M, R/p) \) as a vector space over the field \( R/p \). | [
"1"
] | [
"Assume that \\( R \\) is a Noetherian ring and \\( M \\) is a finitely generated \\( R \\)-module. Let \\( p \\in \\text{supp}_R(M) \\). Calculate the minimum possible dimension of \\( \\text{Hom}_R(M, R/p) \\) as a vector space over the field \\( R/p \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-129c264735de6d6ef649df6fd0194c3b61dc4df840d299d065bb7e0516fc31d8 | Assume that \( R \) is a commutative ring and \( S \) is the set of zero divisors of \( R \). If \( R \) has a finite number of minimal prime ideals \( P_1, \ldots, P_n \), let \( f(R) \) be the number of elements in the set \( \{ x \in R \mid x \in S^{-1}R \setminus (R_{P_1} \times \cdots \times R_{P_n}) \} \). Calcul... | [
"0"
] | [
"Assume that \\( R \\) is a commutative ring and \\( S \\) is the set of zero divisors of \\( R \\). If \\( R \\) has a finite number of minimal prime ideals \\( P_1, \\ldots, P_n \\), let \\( f(R) \\) be the number of elements in the set \\( \\{ x \\in R \\mid x \\in S^{-1}R \\setminus (R_{P_1} \\times \\cdots \\t... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-e559a9bb10b6103b1c5c2bc4655180979873b215cb86f86a7558a79452c3e864 | Assume that \( W \) is a subspace of \( \mathbb{R}^{4} \) generated by the vectors \( x_1 = (1, 0, 1, 0) \), \( x_2 = (0, 1, 1, 4) \), and \( x_3 = (1, -2, 5, -3) \). Find a basis for \( W \) and extend it to a basis for \( \mathbb{R}^{4} \). | [
"\\{ (1, 0, 1, 0), (0, 1, 1, 4), (1, -2, 5, -3), (5, -19, -5, 6) \\}"
] | [
"Assume that \\( W \\) is a subspace of \\( \\mathbb{R}^{4} \\) generated by the vectors \\( x_1 = (1, 0, 1, 0) \\), \\( x_2 = (0, 1, 1, 4) \\), and \\( x_3 = (1, -2, 5, -3) \\). Find a basis for \\( W \\) and extend it to a basis for \\( \\mathbb{R}^{4} \\)."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-94f3a4e1e1283588b649643fc6824b4ef6f47dc0977ac594c6d1ac561d64d59f | Assume that \( a_n > 0 \) and that \( \sum a_n \) diverges. For \( \alpha > 1 \), let \( S_N = \sum_{n=1}^N \frac{a_n}{1 + n^\alpha a_n} \). Find the smallest value of \( \alpha \) such that \( S_N \) converges for all sequences \( \{a_n\} \) satisfying the given conditions. | [
"2"
] | [
"Assume that \\( a_n > 0 \\) and that \\( \\sum a_n \\) diverges. For \\( \\alpha > 1 \\), let \\( S_N = \\sum_{n=1}^N \\frac{a_n}{1 + n^\\alpha a_n} \\). Find the smallest value of \\( \\alpha \\) such that \\( S_N \\) converges for all sequences \\( \\{a_n\\} \\) satisfying the given conditions."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-ff26805aac10e62d311db0506b01b187e6838801c419345e462d2d2ef41d0298 | Assume that \( f : [0,+\infty) \to [0,+\infty) \) is an increasing smooth function with \( f(0) = 0 \). Suppose also that \( tf''(t) + f(t)f'(t) \le 0 \) for all \( t > 0 \). Find the maximum possible value of \( f(1) \). | [
"2"
] | [
"Assume that \\( f : [0,+\\infty) \\to [0,+\\infty) \\) is an increasing smooth function with \\( f(0) = 0 \\). Suppose also that \\( tf''(t) + f(t)f'(t) \\le 0 \\) for all \\( t > 0 \\). Find the maximum possible value of \\( f(1) \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-c0459455dd7242de333c7d27f6102d43e09b7ac3e8f97f0df26d5e6eb456b98d | Assume that \( f : [0,1) \to \mathbb{R} \) is a function such that \( f(x) - x^3 \) and \( f(x) - 3x \) are both increasing functions. Let \( g(x) = f(x) - x^2 - x \). If \( g(x) \) is an increasing function, calculate \( g(0.5) - g(0) \). If \( g(x) \) is not an increasing function, provide the value \( -1 \). | [
"\\dfrac{3}{4}"
] | [
"Assume that \\( f : [0,1) \\to \\mathbb{R} \\) is a function such that \\( f(x) - x^3 \\) and \\( f(x) - 3x \\) are both increasing functions. Let \\( g(x) = f(x) - x^2 - x \\). If \\( g(x) \\) is an increasing function, calculate \\( g(0.5) - g(0) \\). If \\( g(x) \\) is not an increasing function, provide the va... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4592bb0391fa149e903a635be7506b844c704e544db636d292a051cacee6ba2e | Assume that \( f \) and \( g \) are holomorphic functions on an open set \( U \) and \( \overline{D}(0,1) \subset U \). Let \( M \) be the maximum value of \( |f(z)| + |g(z)| \) for \( z \in \overline{D}(0,1) \). If \( z_0 \) is a point on the boundary \( |z_0| = 1 \) such that \( |f(z_0)| + |g(z_0)| = M \), find the v... | [
"4"
] | [
"Assume that \\( f \\) and \\( g \\) are holomorphic functions on an open set \\( U \\) and \\( \\overline{D}(0,1) \\subset U \\). Let \\( M \\) be the maximum value of \\( |f(z)| + |g(z)| \\) for \\( z \\in \\overline{D}(0,1) \\). If \\( z_0 \\) is a point on the boundary \\( |z_0| = 1 \\) such that \\( |f(z_0)| +... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1bd6cf2f60a606010c14587db63ff13c9cc56a4efbbb143aa7d2f1d0292a1667 | Assume that \( f \) is a continuous real function defined on the interval \((a, b)\) such that \[ f\left(\frac{x+y}{2}\right) \leq \frac{f(x) + f(y)}{2} \] for all \( x, y \in (a, b) \). If \( f(1) = 1 \) and \( f(3) = 9 \), find the maximum possible value of \( f(2) \). | [
"5"
] | [
"Assume that \\( f \\) is a continuous real function defined on the interval \\((a, b)\\) such that\n\\[ f\\left(\\frac{x+y}{2}\\right) \\leq \\frac{f(x) + f(y)}{2} \\]\nfor all \\( x, y \\in (a, b) \\). If \\( f(1) = 1 \\) and \\( f(3) = 9 \\), find the maximum possible value of \\( f(2) \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-22c99ca18023225b88023508020c7ea6d928120b3c07e9f0e4798c6c13f635ca | Assume that \( f \) is a nonzero continuous function on the interval \([0, 1]\). Define \[ a_n = \int_0^1 [f(x)]^{2n} \text{ for } n=1,2,3,\cdots. \] Let \( L = \lim_{n \to \infty} \left(\frac{a_{n+1}}{a_n}\right) \). If \( f(x) = x^2 + 1 \), compute the value of \( L \). | [
"4"
] | [
"Assume that \\( f \\) is a nonzero continuous function on the interval \\([0, 1]\\). Define \n\\[ a_n = \\int_0^1 [f(x)]^{2n} \\text{ for } n=1,2,3,\\cdots. \\]\nLet \\( L = \\lim_{n \\to \\infty} \\left(\\frac{a_{n+1}}{a_n}\\right) \\).\nIf \\( f(x) = x^2 + 1 \\), compute the value of \\( L \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-992a58fc0893d1ce35e8bd394d84acae42fc7a3e586abc935a463778fef672d3 | Assume that \( f \) is an entire function and \( |f(z)| < 1 \) for all \( z \in \mathbb{C} \). If \( f \) is not constant, find the value of \( \int_{|z|=1} f(z) \, dz \). | [
"0"
] | [
"Assume that \\( f \\) is an entire function and \\( |f(z)| < 1 \\) for all \\( z \\in \\mathbb{C} \\). If \\( f \\) is not constant, find the value of \\( \\int_{|z|=1} f(z) \\, dz \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2f3d9cb4c5429131c6af4710c58692ab602a038fba2845fa3f5d61061df9a3b4 | Assume that \( f \) is an entire function such that \( \text{Im}(f(z)) > 0 \) for all \( z \in \mathbb{C} \). Let \( g(z) = e^{-if(z)} \). Find the value of \( g(1) + g(i) \). | [
"0"
] | [
"Assume that \\( f \\) is an entire function such that \\( \\text{Im}(f(z)) > 0 \\) for all \\( z \\in \\mathbb{C} \\). Let \\( g(z) = e^{-if(z)} \\). Find the value of \\( g(1) + g(i) \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-75f0a6c4c49be0f9a0c298392b3d64cd8d8954bda1695390c1f24e9841cad5b9 | Assume that \( f \) is an increasing function and \( a > 0 \) is a real number. Let \( g \) be the inverse function of \( f \). Define \( h(x) = f(x) + g(x) \) for \( x \in [0, a] \). Find the minimum value of \( h(x) \) over the interval \( [0, a] \). | [
"0"
] | [
"Assume that \\( f \\) is an increasing function and \\( a > 0 \\) is a real number. Let \\( g \\) be the inverse function of \\( f \\). Define \\( h(x) = f(x) + g(x) \\) for \\( x \\in [0, a] \\). Find the minimum value of \\( h(x) \\) over the interval \\( [0, a] \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 2
} | 8 | 0 | 0 |
omr-ws-2ddd4d0d4e95fe3935d86a0195e6c202186a626a6b55557a0c8be6ea66e086e5 | Assume that \( f \) is continuous on \([0,1]\), \( f(0) = 0 \), and \( f'(0) \) exists. Calculate the value of the Lebesgue integral \(\int_0^1 f(x) x^{-3/2} \, dx\) given that \( f(x) = x^2 \). | [
"\\dfrac{2}{3}"
] | [
"Assume that \\( f \\) is continuous on \\([0,1]\\), \\( f(0) = 0 \\), and \\( f'(0) \\) exists. Calculate the value of the Lebesgue integral \\(\\int_0^1 f(x) x^{-3/2} \\, dx\\) given that \\( f(x) = x^2 \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0eaee7af10977a7e862e9efc87cced390812d5d81aa962dfd724478b5798a912 | Assume that \( f \) is continuous on \([a, b]\) and differentiable on \((a, b)\), and \( f(a) = f(b) = 0 \). For \( r = 3 \), find the number of possible values of \( c \in (a, b) \) such that \( f(c) \cdot 3 = f'(c) \). | [
"1"
] | [
"Assume that \\( f \\) is continuous on \\([a, b]\\) and differentiable on \\((a, b)\\), and \\( f(a) = f(b) = 0 \\). For \\( r = 3 \\), find the number of possible values of \\( c \\in (a, b) \\) such that \\( f(c) \\cdot 3 = f'(c) \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0b302eba8a1ed03aa918d8ceb0d5cd0122e96be2cb8ff413b397f2cd4c3890c6 | Assume that \( f \) is continuous on \([a, b]\) and is \( k \) times differentiable at \( a \) with \( f^{(i)}(a) = 0 \) for \( i = 1, 2, \dots, k-1 \) and \( f^{(k)}(a) \neq 0 \). If \( c_x \in (a, x) \) such that \( \int_a^x f(t) \, dt = f(c_x) (x - a) \), then find the value of \( \lim_{x \to a} \frac{c_x - a}{x - a... | [
"\\left( \\dfrac{1}{k+1} \\right)^{1/k}"
] | [
"Assume that \\( f \\) is continuous on \\([a, b]\\) and is \\( k \\) times differentiable at \\( a \\) with \\( f^{(i)}(a) = 0 \\) for \\( i = 1, 2, \\dots, k-1 \\) and \\( f^{(k)}(a) \\neq 0 \\). If \\( c_x \\in (a, x) \\) such that \\( \\int_a^x f(t) \\, dt = f(c_x) (x - a) \\), then find the value of \\( \\lim_... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1e69d7dc9fcdf93d20e1999022500bd83095e58297220a315bb029ada6860b45 | Assume that \( f \) is continuous on \([a, b]\) and is \( k \) times differentiable at \( a \) with \( f^{(i)}(a+) = 0 \) for \( 1 \leq i \leq k-1 \) and \( f^{(k)}(a) \neq 0 \). If \( c_x \in (a, x) \) is such that \( \int_{a}^{x} f(t) \, dt = f(c_x)(x-a) \), then find the value of \( \lim_{x \to a+} \frac{c_x - a}{x ... | [
"\\dfrac{1}{\\sqrt[k]{k+1}}"
] | [
"Assume that \\( f \\) is continuous on \\([a, b]\\) and is \\( k \\) times differentiable at \\( a \\) with \\( f^{(i)}(a+) = 0 \\) for \\( 1 \\leq i \\leq k-1 \\) and \\( f^{(k)}(a) \\neq 0 \\). If \\( c_x \\in (a, x) \\) is such that \\( \\int_{a}^{x} f(t) \\, dt = f(c_x)(x-a) \\), then find the value of \\( \\l... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dd60415fdec90542b579ae8a014d030d7a9ab71c2768e11ed35f425602a2ae81 | Assume that \( f \) is continuous on \([a, b]\), \( a > 0 \), and differentiable on \((a, b)\). If \( \frac{f(a)}{a} = \frac{f(b)}{b} \), then find the value of \( x \) in \((a, b)\) such that \( xf'(x) = f(x) \), given \( f(x) = x^3 - 3x^2 + 2x \), \( a = 1 \), and \( b = 2 \). | [
"\\dfrac{3}{2}"
] | [
"Assume that \\( f \\) is continuous on \\([a, b]\\), \\( a > 0 \\), and differentiable on \\((a, b)\\). If \\( \\frac{f(a)}{a} = \\frac{f(b)}{b} \\), then find the value of \\( x \\) in \\((a, b)\\) such that \\( xf'(x) = f(x) \\), given \\( f(x) = x^3 - 3x^2 + 2x \\), \\( a = 1 \\), and \\( b = 2 \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-937de9a5e9b2b70833a1fd17c2c1b7117500c6961eef065c13d6c1ac1dbfd0e9 | Assume that \( f \) is defined on \( (0, 1) \) and for any strictly increasing sequence \( x_n \) with \( \lim_{n \to \infty} x_n = 1 \), the limit \( \lim_{n \to \infty} f(x_n) = L_n \) exists. Let \( S \) be the set of all possible values of \( L_n \). If \( S \) contains exactly one element, find the value of \( \li... | [
"L"
] | [
"Assume that \\( f \\) is defined on \\( (0, 1) \\) and for any strictly increasing sequence \\( x_n \\) with \\( \\lim_{n \\to \\infty} x_n = 1 \\), the limit \\( \\lim_{n \\to \\infty} f(x_n) = L_n \\) exists. Let \\( S \\) be the set of all possible values of \\( L_n \\). If \\( S \\) contains exactly one elemen... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-25b377478381c64d13cd73f96de15beb2c11da1203152a906f305bed53c8debc | Assume that \( f \) is differentiable on the interval \((a, b)\) and has a local maximum at \( c \) within \((a, b)\). If \( f(c) = 5 \) and \( f'(x) = 3x^2 - 6x + k \), find the value of \( k \). | [
"-9"
] | [
"Assume that \\( f \\) is differentiable on the interval \\((a, b)\\) and has a local maximum at \\( c \\) within \\((a, b)\\). If \\( f(c) = 5 \\) and \\( f'(x) = 3x^2 - 6x + k \\), find the value of \\( k \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-5b838d6d72b983ae17e914966d18e505a94c9cfbc6a11fbed2dd7ea46a80074f | Assume that \( f \) is integrable on \([a,b]\). Given \( f(x) = x^2 \), calculate the value of \[ \left| \int_a^b f(x) \, dx - (b-a)f(a) \right| \] for \( a = 1 \) and \( b = 3 \). | [
"\\dfrac{20}{3}"
] | [
"Assume that \\( f \\) is integrable on \\([a,b]\\). Given \\( f(x) = x^2 \\), calculate the value of \\[ \\left| \\int_a^b f(x) \\, dx - (b-a)f(a) \\right| \\] for \\( a = 1 \\) and \\( b = 3 \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-889da9739f67eb7e9c5cd05503d41aa65c3c4ad3d25f21dbf5b52940c485b2d6 | Assume that \( f \) is twice differentiable and \( f'' \) is continuous on \( (0, \infty) \). Suppose \[ \lim_{x \to \infty} x f(x) = 0 \quad \text{and} \quad \lim_{x \to \infty} x f''(x) = 0. \] Let \( g(x) = x f'(x) \). Find the value of \( \lim_{x \to \infty} g(x) \). | [
"0"
] | [
"Assume that \\( f \\) is twice differentiable and \\( f'' \\) is continuous on \\( (0, \\infty) \\). Suppose \n\\[\n\\lim_{x \\to \\infty} x f(x) = 0 \\quad \\text{and} \\quad \\lim_{x \\to \\infty} x f''(x) = 0.\n\\]\nLet \\( g(x) = x f'(x) \\). Find the value of \\( \\lim_{x \\to \\infty} g(x) \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-776232f3e3c474dfce52a5c3df43cb56a15a43ad4e2d27fa8c9f1c88462c001b | Assume that \( f \) is twice differentiable on \( \mathbb{R} \) and \( f, f', f'' \) are strictly increasing on \( \mathbb{R} \). For fixed \( a, b \) with \( a \leq b \), let \( x \to \xi(x) \), \( x > 0 \), be defined by the mean value theorem, that is, \[ \frac{f(b+x) - f(a-x)}{b-a+2x} = f'(\xi(x)). \] Calculate the... | [
"\\dfrac{e^3 + 5}{3(e^3 - 1)}"
] | [
"Assume that \\( f \\) is twice differentiable on \\( \\mathbb{R} \\) and \\( f, f', f'' \\) are strictly increasing on \\( \\mathbb{R} \\). For fixed \\( a, b \\) with \\( a \\leq b \\), let \\( x \\to \\xi(x) \\), \\( x > 0 \\), be defined by the mean value theorem, that is, \n\\[\n\\frac{f(b+x) - f(a-x)}{b-a+2x}... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9a0802276fe04b0a867c9659e008f7def6367cea99bdf8365f5d6ebc5f261a03 | Assume that \( f \) is univalent and analytic in \( |z| < 1 \) and is of the form \( f(z) = z + c_2 z^2 + \dots \). Let \( r = \frac{1}{2} \). Calculate the value of \( |f(z)| \) on \( |z| = r \) when \( f(z) = \frac{z}{(1 + e^{i\alpha}z)^2} \) for some real number \( \alpha \). Express your answer as a simplified frac... | [
"2"
] | [
"Assume that \\( f \\) is univalent and analytic in \\( |z| < 1 \\) and is of the form \\( f(z) = z + c_2 z^2 + \\dots \\). Let \\( r = \\frac{1}{2} \\). Calculate the value of \\( |f(z)| \\) on \\( |z| = r \\) when \\( f(z) = \\frac{z}{(1 + e^{i\\alpha}z)^2} \\) for some real number \\( \\alpha \\). Express your a... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-7ac631c006790f466b4ff6a7bf4bc653d28f61107499ac671ea80a4257496de6 | Assume that \( f \in R(\alpha) \) (i.e., is Riemann-Stieltjes integrable) on \([a,b]\). Let \(\{P_n\}\) be a sequence of polynomials such that: \[ \lim_{n\to\infty} \int_a^b |f - P_n| \, d\alpha = 0 \] Given \( f(x) = x^2 \) and \( \alpha(x) = x^3 \) on \([0,1]\), find the smallest integer \( N \) such that for all \( ... | [
"2"
] | [
"Assume that \\( f \\in R(\\alpha) \\) (i.e., is Riemann-Stieltjes integrable) on \\([a,b]\\). Let \\(\\{P_n\\}\\) be a sequence of polynomials such that:\n\n\\[ \\lim_{n\\to\\infty} \\int_a^b |f - P_n| \\, d\\alpha = 0 \\]\n\nGiven \\( f(x) = x^2 \\) and \\( \\alpha(x) = x^3 \\) on \\([0,1]\\), find the smallest i... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-efe4e753516f25c0b3c3b81e858daa9e35d4fa16f9e95b76f7b90e9ec298ee3a | Assume that \( f(x) = 8x^3 + ax^2 + bx - 1 \). Find \( a \) and \( b \) such that \( f\left(\frac{1}{2}\right) = 0 \) and the order of convergence of Newton's method for finding the root \( x = \frac{1}{2} \) with suitable \( x_0 \) is exactly one. | [
"b = 6"
] | [
"Assume that \\( f(x) = 8x^3 + ax^2 + bx - 1 \\). Find \\( a \\) and \\( b \\) such that \\( f\\left(\\frac{1}{2}\\right) = 0 \\) and the order of convergence of Newton's method for finding the root \\( x = \\frac{1}{2} \\) with suitable \\( x_0 \\) is exactly one."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-50779a703ef9850d160e540ad6684bc848f7869ec1fce5b822cc8c6d0dd87b78 | Assume that \( f(x) \) and \( g(x) \) are continuous functions on \( \mathbb{R} \) and \( f(g(x)) = g(f(x)) \) for all \( x \in \mathbb{R} \). Given that \( f(x) = g(x) \) has no real roots, calculate the number of real roots of the equation \( f(f(x)) = g(g(x)) \). | [
"0"
] | [
"Assume that \\( f(x) \\) and \\( g(x) \\) are continuous functions on \\( \\mathbb{R} \\) and \\( f(g(x)) = g(f(x)) \\) for all \\( x \\in \\mathbb{R} \\). Given that \\( f(x) = g(x) \\) has no real roots, calculate the number of real roots of the equation \\( f(f(x)) = g(g(x)) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f7e1a8f564b45bb44096af66518c2935b29de05d13c1be8f4be6049009b0bb49 | Assume that \( f(x) \) is an increasing function and \( n \in \mathbb{N} \). Let \( S_n = \frac{1}{n} \sum_{k=1}^n f\left(\frac{k}{n}\right) \) and \( I = \int_0^1 f(x) \, dx \). Calculate the value of \( S_5 - I \) for the function \( f(x) = x^2 \). | [
"\\dfrac{8}{75}"
] | [
"Assume that \\( f(x) \\) is an increasing function and \\( n \\in \\mathbb{N} \\). Let \\( S_n = \\frac{1}{n} \\sum_{k=1}^n f\\left(\\frac{k}{n}\\right) \\) and \\( I = \\int_0^1 f(x) \\, dx \\). Calculate the value of \\( S_5 - I \\) for the function \\( f(x) = x^2 \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5649a323a42901f3dd6b59a1a89f1b95947193ef5763869a865c8b6599bae95d | Assume that \( f(x) \in C^\infty (-a,a) \), and the sequence \( f^{(n)}(x) \) (which means the \( n \)-th derivative) is uniformly convergent to \( \phi(x) \) for \( x \in (-a,a) \). If \( f(x) = e^x \), find the value of \( C \) such that \( \phi(x) = Ce^x \). | [
"1"
] | [
"Assume that \\( f(x) \\in C^\\infty (-a,a) \\), and the sequence \\( f^{(n)}(x) \\) (which means the \\( n \\)-th derivative) is uniformly convergent to \\( \\phi(x) \\) for \\( x \\in (-a,a) \\). If \\( f(x) = e^x \\), find the value of \\( C \\) such that \\( \\phi(x) = Ce^x \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3785c37e9ff718e922284e1f7170ecc69b3a525676feb12f6fe452ca592ca6ab | Assume that \( f(x,y) \) is \( C^1 \) and the function \( g(x,y) = f(x,y) - \frac{1}{2} (x^2 - y^2 - 2xy) \) attains a local minimum at \((0,0)\). Could the \( 2 \times 2 \) identity matrix be the Hessian of \( f \) at the origin? | [
"No"
] | [
"Assume that \\( f(x,y) \\) is \\( C^1 \\) and the function \\( g(x,y) = f(x,y) - \\frac{1}{2} (x^2 - y^2 - 2xy) \\) attains a local minimum at \\((0,0)\\). Could the \\( 2 \\times 2 \\) identity matrix be the Hessian of \\( f \\) at the origin?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-cd1254273f4cfb55e48c69fc9f8b7fed733aa8f274a9cb231a5e5d13fff6a01e | Assume that \( f, g: [0, 1] \rightarrow [0,1] \) are continuous functions satisfying \( f(g(x)) = g(f(x)) \) for all \( x \in [0, 1] \). Let \( h(x) = f(x) - g(x) \). Find the number of distinct points \( x_0 \in [0,1] \) such that \( h(x_0) = 0 \). | [
"1"
] | [
"Assume that \\( f, g: [0, 1] \\rightarrow [0,1] \\) are continuous functions satisfying \\( f(g(x)) = g(f(x)) \\) for all \\( x \\in [0, 1] \\). Let \\( h(x) = f(x) - g(x) \\). Find the number of distinct points \\( x_0 \\in [0,1] \\) such that \\( h(x_0) = 0 \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-90f7bda7c7d9a0965d3e276aa620a829601752a3cd16e74a660032111435b2ac | Assume that \( f, g: [0,1] \rightarrow [0,1] \) are two continuous functions such that \( f(g(x)) = g(f(x)) \) for all \( x \in [0,1] \). Let \( h(x) = f(x) - g(x) \). Find the minimum number of zeros that \( h(x) \) must have in the interval \( [0,1] \). | [
"1"
] | [
"Assume that \\( f, g: [0,1] \\rightarrow [0,1] \\) are two continuous functions such that \\( f(g(x)) = g(f(x)) \\) for all \\( x \\in [0,1] \\). Let \\( h(x) = f(x) - g(x) \\). Find the minimum number of zeros that \\( h(x) \\) must have in the interval \\( [0,1] \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-61944ec6cf803307ac1565b2b9bb45183f2196ec840899f8f2e8457cadd6bba9 | Assume that \( f: (0,+\infty) \mapsto \mathbb{R} \) is a bounded and differentiable function such that \( f(x)f'(x) \geq \cos x \) for all \( x > 0 \). Let \( M \) be the maximum possible value of \( |f(x)| \) for \( x > 0 \). Find the smallest possible value of \( M \). | [
"2"
] | [
"Assume that \\( f: (0,+\\infty) \\mapsto \\mathbb{R} \\) is a bounded and differentiable function such that \\( f(x)f'(x) \\geq \\cos x \\) for all \\( x > 0 \\). Let \\( M \\) be the maximum possible value of \\( |f(x)| \\) for \\( x > 0 \\). Find the smallest possible value of \\( M \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f1364b6e740e3b9b85cd8000adb85a22ff745ae845eaa6db49247f5a857eb299 | Assume that \( f: [0,1] \rightarrow \mathbb{R} \) is a differentiable function for which there exist \( a, b \in (0,1) \) such that \( \int_0^a f(t) \, dt = \int_b^1 f(t) \, dt = 0 \). Let \( M = \sup \{ |f'(x)| : x \in [0,1] \} \). Calculate the maximum possible value of \( \left| \int_0^1 f(t) \, dt \right| \) given ... | [
"\\dfrac{M}{18}"
] | [
"Assume that \\( f: [0,1] \\rightarrow \\mathbb{R} \\) is a differentiable function for which there exist \\( a, b \\in (0,1) \\) such that \\( \\int_0^a f(t) \\, dt = \\int_b^1 f(t) \\, dt = 0 \\). Let \\( M = \\sup \\{ |f'(x)| : x \\in [0,1] \\} \\). Calculate the maximum possible value of \\( \\left| \\int_0^1 f... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7b4cc79fc0c2b3335f090eade63ba2ff9b4292fd11aa0415287fed0932b0b535 | Assume that \( f: [0,2] \to \mathbb{R} \) is a continuous function and \( f(0) = f(2) \). Define \( g(x) = f(x+1) - f(x) \) for \( x \in [0,1] \). Calculate the value of \( \int_0^1 g(x) \, dx \). | [
"no_valid_answer_found"
] | [
"Assume that \\( f: [0,2] \\to \\mathbb{R} \\) is a continuous function and \\( f(0) = f(2) \\). Define \\( g(x) = f(x+1) - f(x) \\) for \\( x \\in [0,1] \\). Calculate the value of \\( \\int_0^1 g(x) \\, dx \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b4976cd604b970370fa7042e7f56ece39d8ffda96e265ef018ceb50e31bfdef1 | Assume that \( f: \mathbb{R}^2 \to \mathbb{R} \) is continuous and differentiable, with partial derivatives \( f_x \) and \( f_y \). Let \( u = -y f_x + x f_y \) and \( I(a) = \iint_{D_a} u(x,y) \, \mathrm{d}x \, \mathrm{d}y \), where \( D_a = \{(x,y) : (x-2\cos a)^2 + (y-2\sin a)^2 \le 1\} \). Find the value of \( I(a... | [
"0"
] | [
"Assume that \\( f: \\mathbb{R}^2 \\to \\mathbb{R} \\) is continuous and differentiable, with partial derivatives \\( f_x \\) and \\( f_y \\). Let \\( u = -y f_x + x f_y \\) and \\( I(a) = \\iint_{D_a} u(x,y) \\, \\mathrm{d}x \\, \\mathrm{d}y \\), where \\( D_a = \\{(x,y) : (x-2\\cos a)^2 + (y-2\\sin a)^2 \\le 1\\}... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7dda686d056d461d0b2f6ae7a2af5dba8e111a0b94be390b213b919da191659b | Assume that \( f: \mathbb{R}^2 \to \mathbb{R} \) is differentiable with \( a(x,y) \) and \( b(x,y) \) as its partial derivatives with respect to \( x \) and \( y \), respectively. There exists a constant \( C > 0 \) such that for all \( |h| + |k| \le 1 \) and for all \( x, y \in \mathbb{R} \), \[ \left| f(x+h, y+k) - f... | [
"\\left( \\dfrac{0.01}{2C} \\right)^2"
] | [
"Assume that \\( f: \\mathbb{R}^2 \\to \\mathbb{R} \\) is differentiable with \\( a(x,y) \\) and \\( b(x,y) \\) as its partial derivatives with respect to \\( x \\) and \\( y \\), respectively. There exists a constant \\( C > 0 \\) such that for all \\( |h| + |k| \\le 1 \\) and for all \\( x, y \\in \\mathbb{R} \\)... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-cefff83daf9b344727b4faf4bd10f314fbbcde30146357134dcb47a11a888b9a | Assume that \( f: \mathbb{R}^n \rightarrow \mathbb{R}^n \) is \( C^1 \) and the derivative at each point is invertible. If the norm of \( Df_x^{-1} \) for all \( x \) is greater than \( c > 0 \), find the minimum value of \( c \) such that \( f \) is guaranteed to be injective. | [
"0"
] | [
"Assume that \\( f: \\mathbb{R}^n \\rightarrow \\mathbb{R}^n \\) is \\( C^1 \\) and the derivative at each point is invertible. If the norm of \\( Df_x^{-1} \\) for all \\( x \\) is greater than \\( c > 0 \\), find the minimum value of \\( c \\) such that \\( f \\) is guaranteed to be injective."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-cfc4e04f1fc47fbf29cb15919faa2dfe0633dc7abb4e4366ba503ddc74a9fb75 | Assume that \( f:(a,\infty) \to \mathbb{R} \). Let \( T \) be a given positive real number. Define the sequence \( \{x_n\} \) such that \( \lim_{n \to \infty} x_n = \infty \) and \( \lim_{n \to \infty} (f(T + x_n) - f(x_n)) = 0 \). Calculate the limit \(\lim_{n \to \infty} \frac{f(x_n + T) - f(x_n)}{T}\). | [
"0"
] | [
"Assume that \\( f:(a,\\infty) \\to \\mathbb{R} \\). Let \\( T \\) be a given positive real number. Define the sequence \\( \\{x_n\\} \\) such that \\( \\lim_{n \\to \\infty} x_n = \\infty \\) and \\( \\lim_{n \\to \\infty} (f(T + x_n) - f(x_n)) = 0 \\). Calculate the limit \\(\\lim_{n \\to \\infty} \\frac{f(x_n + ... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cf7d290740365fed3c929f04137a6510b0ce861a6f23921d2c08cd75fd1e7d28 | Assume that \( f:[a,\infty) \to \mathbb{R} \) is continuous and periodic with period \( T > 0 \), and \( g:[a,\infty) \to \mathbb{R} \) is monotonic with \( \lim\limits_{x \to \infty} g(x) = 0 \). Given that \( \int_a^{a+T} f(x) \, dx = 0 \), calculate the value of the integral \( \int_0^\infty f(x) g(x) \, dx \) if \(... | [
"0"
] | [
"Assume that \\( f:[a,\\infty) \\to \\mathbb{R} \\) is continuous and periodic with period \\( T > 0 \\), and \\( g:[a,\\infty) \\to \\mathbb{R} \\) is monotonic with \\( \\lim\\limits_{x \\to \\infty} g(x) = 0 \\). Given that \\( \\int_a^{a+T} f(x) \\, dx = 0 \\), calculate the value of the integral \\( \\int_0^\\... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 8
} | 11 | 0 | 0 |
omr-ws-b2fe704485df41bb1dfab1a39b4750e64ca66bf054877db17e11ad9481113b6c | Assume that \( g(x) \) is a polynomial with integer coefficients. The function \( f: \mathbb{Z} \rightarrow \mathbb{Z} \) satisfies the following conditions for all \( m, n \in \mathbb{Z} \): \[ \begin{cases} f(m) + f(f(n)) = -f(f(m+n)) - n \\ g(n) = g(f(n)) \end{cases} \] Find \( g(x) \). | [
"0"
] | [
"Assume that \\( g(x) \\) is a polynomial with integer coefficients. The function \\( f: \\mathbb{Z} \\rightarrow \\mathbb{Z} \\) satisfies the following conditions for all \\( m, n \\in \\mathbb{Z} \\):\n\\[ \\begin{cases} \nf(m) + f(f(n)) = -f(f(m+n)) - n \\\\\ng(n) = g(f(n)) \n\\end{cases} \\]\nFind \\( g(x) \\)... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 3
} | 15 | 0 | 0 |
omr-ws-aeda77ae3cdd4dad7b30beb8fe49b5032afd7545a54eb16d801c5823faaa6df4 | Assume that \( m, n, r \) are positive integers such that \((2+\sqrt{3})^{2r-1} = 1 + m + n\sqrt{3}\). Calculate the value of \( m \) when \( r = 5 \). | [
"70225"
] | [
"Assume that \\( m, n, r \\) are positive integers such that \\((2+\\sqrt{3})^{2r-1} = 1 + m + n\\sqrt{3}\\). Calculate the value of \\( m \\) when \\( r = 5 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ef94d04ea49a5f1b36d7081bb60b84d8a09b67cece363ed0041e8d20732eedb3 | Assume that \( n \) copies of unit cubes are glued together side by side to form a rectangular solid block. If the number of unit cubes that are completely invisible is 30, then what is the minimum possible value of \( n \)? | [
"140"
] | [
"Assume that \\( n \\) copies of unit cubes are glued together side by side to form a rectangular solid block. If the number of unit cubes that are completely invisible is 30, then what is the minimum possible value of \\( n \\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 16
} | 48 | 0 | 0 |
omr-ws-49b44d0eddc6168029fcb5398110c98556f9935ef5aa32e9b8874fba24a9b15a | Assume that \( n \) is a positive odd number. Let \( A = \{ a \in \mathbb{Z} : (a, n) = 1, 1 \le a \le \frac{n}{4} \} \). Find all \( n \) such that the cardinality of \( A \) is odd. | [
"n = p^k \\text{ where } p \\equiv 5 \\text{ or } 7 \\pmod{8} \\text{ and } k \\geq 1"
] | [
"Assume that \\( n \\) is a positive odd number. Let \\( A = \\{ a \\in \\mathbb{Z} : (a, n) = 1, 1 \\le a \\le \\frac{n}{4} \\} \\). Find all \\( n \\) such that the cardinality of \\( A \\) is odd."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 3
} | 12 | 0 | 0 |
omr-ws-016ef5d42a1a7106992c842f4b3d41db23f18206d0ab61b52a56354ee7e0e223 | Assume that \( n^\alpha a_n \to 0 \) and \( a_n \to 0 \) as \( n \to \infty \). Given that \( \sum a_n \) diverges, calculate the value of \( \alpha \) for which the series \( \sum \frac{a_n}{1 + n^\alpha a_n} \) converges, assuming \( a_n = \frac{1}{n \log n} \). | [
"\\alpha > 1"
] | [
"Assume that \\( n^\\alpha a_n \\to 0 \\) and \\( a_n \\to 0 \\) as \\( n \\to \\infty \\). Given that \\( \\sum a_n \\) diverges, calculate the value of \\( \\alpha \\) for which the series \\( \\sum \\frac{a_n}{1 + n^\\alpha a_n} \\) converges, assuming \\( a_n = \\frac{1}{n \\log n} \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-233b5758c80bf6353d0e745838b7bf80f5f6a06c88d9d033b28c8d04f25c5d54 | Assume that \( p \) is a prime and \( p^m \) divides \( n! \). Calculate the largest possible value of \( m \) when \( n = 100 \) and \( p = 3 \). | [
"48"
] | [
"Assume that \\( p \\) is a prime and \\( p^m \\) divides \\( n! \\). Calculate the largest possible value of \\( m \\) when \\( n = 100 \\) and \\( p = 3 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e6d957f1ccbd4b5a111cd88729409ddd4c66d23afa62d04be9479133b0583ca4 | Assume that \( p \) is a prime number with \( p \equiv 3 \mod 4 \). Let \( P(p) = \prod_{1 \leq x < y \leq \frac{p-1}{2}} \left(x^2 + y^2\right) \mod p \). Calculate \( P(11) \). | [
"10"
] | [
"Assume that \\( p \\) is a prime number with \\( p \\equiv 3 \\mod 4 \\). Let \\( P(p) = \\prod_{1 \\leq x < y \\leq \\frac{p-1}{2}} \\left(x^2 + y^2\\right) \\mod p \\). Calculate \\( P(11) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cb4e1b22f1a57833a97cda55eb82fd04881f41f8179849c8ac2f1514619011b4 | Assume that \( p \) is a prime. Find the smallest integer \( m > 1 \) such that \( p^m \) does not divide \( 1! + 2! + \cdots + (mp-1)! \) for \( p = 3 \). | [
"3"
] | [
"Assume that \\( p \\) is a prime. Find the smallest integer \\( m > 1 \\) such that \\( p^m \\) does not divide \\( 1! + 2! + \\cdots + (mp-1)! \\) for \\( p = 3 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-32e08db6e4aab201975b8431bb58fea35caa53b1d672e7bcb600b5b1e5228a6b | Assume that \( p \) is an odd prime and \( G \) is a finite simple group with exactly \( 2p + 1 \) Sylow \( p \)-subgroups. Let \( P \) be a Sylow \( p \)-subgroup of \( G \). If \( P \) is abelian, calculate the number of distinct elements of order \( p \) in \( P \). | [
"p^2 - 1"
] | [
"Assume that \\( p \\) is an odd prime and \\( G \\) is a finite simple group with exactly \\( 2p + 1 \\) Sylow \\( p \\)-subgroups. Let \\( P \\) be a Sylow \\( p \\)-subgroup of \\( G \\). If \\( P \\) is abelian, calculate the number of distinct elements of order \\( p \\) in \\( P \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-807b62125acf04c7ce7a8b374398d93754eff4e596a1a24724f7bd50e4d8c6a0 | Assume that \( p, q, a, b, c \) are positive integers such that \( p + q \geqslant c \). Calculate the value of the expression \[ \sum_{x} {p \choose x} {q \choose c - x} (a + p - x)(b + q - c + x) - {p + q \choose c} (a + p + q - c) b, \] where the summation on the right side holds for \( 0 \leqslant x \leqslant p \) ... | [
"252"
] | [
"Assume that \\( p, q, a, b, c \\) are positive integers such that \\( p + q \\geqslant c \\). Calculate the value of the expression\n\\[\n\\sum_{x} {p \\choose x} {q \\choose c - x} (a + p - x)(b + q - c + x) - {p + q \\choose c} (a + p + q - c) b,\n\\]\nwhere the summation on the right side holds for \\( 0 \\leqs... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6cb05a71cccd2d7df457537066279cb278358e95de0a6d64d7a9b84b2591fdb4 | Assume that \( s, t \geq 1 \) and \( \frac{1}{s} + \frac{1}{t} = \frac{\log 3}{\log 2} \). Let \( f(x) = 1 + x + x^2 - (1 + x^s)^{1/s} (1 + x^t)^{1/t} \). Find the maximum value of \( s + t \) such that \( f(x) \geq 0 \) for all \( x \in [0,1] \). | [
"3"
] | [
"Assume that \\( s, t \\geq 1 \\) and \\( \\frac{1}{s} + \\frac{1}{t} = \\frac{\\log 3}{\\log 2} \\). Let \\( f(x) = 1 + x + x^2 - (1 + x^s)^{1/s} (1 + x^t)^{1/t} \\). Find the maximum value of \\( s + t \\) such that \\( f(x) \\geq 0 \\) for all \\( x \\in [0,1] \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-a5e3f9965f02bd932ffcd2872d087e532d6b79ada29ba857e376ef6e8b98f78d | Assume that \( x, y \ge 0 \). Find the maximum value of the expression \( x^2 + y^2 + 1 - \sqrt{(x^3 + y + 1)(y^3 + x + 1)} \). | [
"0"
] | [
"Assume that \\( x, y \\ge 0 \\). Find the maximum value of the expression \\( x^2 + y^2 + 1 - \\sqrt{(x^3 + y + 1)(y^3 + x + 1)} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-117ffe8d3e1e07275b9073f1d22cee56582675d4f0217da5f05e488378d695db | Assume that \( x_0, x_1, \dots, x_n \) are \( n+1 \) distinct real (or complex) numbers, with \( x_i \ne 0 \), \( x_i \ne 1 \), and \( x_i \ne -1 \) for all \( i \). Let \( f(x) = \prod_{i=0}^{n} (x - x_i) \). Calculate the value of: \[ \prod_{i=0}^{n} \frac{x_i^{n+1}}{f'(x_i)} \cdot \frac{f(x_i^{-1})}{1 + x_i} \] for ... | [
"-1"
] | [
"Assume that \\( x_0, x_1, \\dots, x_n \\) are \\( n+1 \\) distinct real (or complex) numbers, with \\( x_i \\ne 0 \\), \\( x_i \\ne 1 \\), and \\( x_i \\ne -1 \\) for all \\( i \\). Let \\( f(x) = \\prod_{i=0}^{n} (x - x_i) \\). Calculate the value of:\n\\[\n\\prod_{i=0}^{n} \\frac{x_i^{n+1}}{f'(x_i)} \\cdot \\fra... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-eee02cee54a23f489aad0b42e5cbb76574bd849384df4fb3135a105ee2aa2e87 | Assume that \( x_1, \ldots, x_n \) are distinct integers greater than 1. Define \( a_{ij} = \sin^2 \left( \frac{j \cdot \pi}{x_i} \right) \). Calculate the determinant of the matrix \( A = (a_{ij}) \) for \( n = 4 \) and \( x_i = i + 1 \) for \( i = 1, 2, 3, 4 \). | [
"\\dfrac{3(5 + \\sqrt{5})}{32}"
] | [
"Assume that \\( x_1, \\ldots, x_n \\) are distinct integers greater than 1. Define \\( a_{ij} = \\sin^2 \\left( \\frac{j \\cdot \\pi}{x_i} \\right) \\). Calculate the determinant of the matrix \\( A = (a_{ij}) \\) for \\( n = 4 \\) and \\( x_i = i + 1 \\) for \\( i = 1, 2, 3, 4 \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-69fa9d40a377d726bee5d0abf70070062481f59a88d90a37f437b0f63399b84d | Assume that \( x_n \ge 0 \) and \( x_{n+1} = \sqrt{2 + x_n} \). Compute \( \lim_{n \rightarrow \infty} 4^n (2 - x_n) \). | [
"\\dfrac{16}{9}"
] | [
"Assume that \\( x_n \\ge 0 \\) and \\( x_{n+1} = \\sqrt{2 + x_n} \\). Compute \\( \\lim_{n \\rightarrow \\infty} 4^n (2 - x_n) \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fc4ec5967d3f9ae8dfa2456063f745702cb1d09a77acb81cd63f15088ee1ab9b | Assume that \( y \) is a differentiable function of \( x \) and that \( x^2 + 2xy^2 = 4 - ye^y \). Find the slope(s) of the curve at \( y = 0 \). | [
"\\(\\pm 4\\)"
] | [
"Assume that \\( y \\) is a differentiable function of \\( x \\) and that \\( x^2 + 2xy^2 = 4 - ye^y \\). Find the slope(s) of the curve at \\( y = 0 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 22
} | 26 | 0 | 0 |
omr-ws-8036796e8a8b737c35e71fb86772a22ea4a4375f617a454351e5762f416fdf4d | Assume that \(-\frac{\pi}{2} \leq u \leq v \leq \frac{\pi}{2}\) and \(u, v \neq -\frac{\pi}{2}, \frac{\pi}{2}\). Let \(f(x) = a \cos x + b \sin 2x + c \cos 3x - x\). Find the minimum value of \(|a| + |b| + |c|\) such that \(f(x)\) has no zero in \([u, v]\) for any \(u, v\) satisfying the given conditions. | [
"2"
] | [
"Assume that \\(-\\frac{\\pi}{2} \\leq u \\leq v \\leq \\frac{\\pi}{2}\\) and \\(u, v \\neq -\\frac{\\pi}{2}, \\frac{\\pi}{2}\\). Let \\(f(x) = a \\cos x + b \\sin 2x + c \\cos 3x - x\\). Find the minimum value of \\(|a| + |b| + |c|\\) such that \\(f(x)\\) has no zero in \\([u, v]\\) for any \\(u, v\\) satisfying t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 6
} | 9 | 0 | 0 |
omr-ws-ffb0628a58fee9b0db7963436f100b0ba5d0c1cb2717506c917ab13956b7b4fc | Assume that \(P(n) = \alpha^n + \beta^n\) can be expressed as a polynomial in \(\alpha\beta\) and \(\alpha + \beta\). Let \(S_n\) denote the sum of its coefficients. Find \(S_n\). | [
"\\(S_n\\) is periodic with period 6, and the values are \\(2, 1, -1, -2, -1, 1\\)."
] | [
"Assume that \\(P(n) = \\alpha^n + \\beta^n\\) can be expressed as a polynomial in \\(\\alpha\\beta\\) and \\(\\alpha + \\beta\\). Let \\(S_n\\) denote the sum of its coefficients. Find \\(S_n\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-eda6c333b294d792a12e3bec1c9c564db9702227503a9aeeaa600037eba9863e | Assume that \(X\) and \(Y\) are identically distributed from a continuous function \(F\). Calculate \(E\left(\frac{X}{Y}\right) - 1\) and express your answer as a function of \(F\). | [
" \\left( \\int x \\, dF(x) \\right) \\left( \\int \\frac{1}{x} \\, dF(x) \\right) - 1 "
] | [
"Assume that \\(X\\) and \\(Y\\) are identically distributed from a continuous function \\(F\\). Calculate \\(E\\left(\\frac{X}{Y}\\right) - 1\\) and express your answer as a function of \\(F\\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7167d58a9074fadde56d704bf3719f6e50e67165578d9d6c80600673c636921a | Assume that \(\alpha = \beta = \gamma = 90^\circ\) for a trihedral angle \( Sxyz \). Let \( O \in Sz \) be a fixed point such that \( SO = a \), and let \( M, N \) be variable points on \( x, y \) respectively. If \( \theta = \angle SOM + \angle SON + \angle MON \), find the value of \( \theta \). Additionally, if \( I... | [
"180^\\circ"
] | [
"Assume that \\(\\alpha = \\beta = \\gamma = 90^\\circ\\) for a trihedral angle \\( Sxyz \\). Let \\( O \\in Sz \\) be a fixed point such that \\( SO = a \\), and let \\( M, N \\) be variable points on \\( x, y \\) respectively. If \\( \\theta = \\angle SOM + \\angle SON + \\angle MON \\), find the value of \\( \\t... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9de784b42f73af55b239bc51449f6e58fa2513b02e37045cc81e2903248af09e | Assume that \(a < b \le c < d\), \(ad = bc\), and \(\sqrt{d} - \sqrt{a} \le 1\) for some \(a, b, c, d \in \{1, 2, 3, \dots\}\). If \(a\) is a square of an integer, find the smallest possible value of \(d\) when \(a = 4\). | [
"9"
] | [
"Assume that \\(a < b \\le c < d\\), \\(ad = bc\\), and \\(\\sqrt{d} - \\sqrt{a} \\le 1\\) for some \\(a, b, c, d \\in \\{1, 2, 3, \\dots\\}\\). If \\(a\\) is a square of an integer, find the smallest possible value of \\(d\\) when \\(a = 4\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e3440df25c646bbe538e6530fda04d607482ccc73bcede2c1275aad4388c7aab | Assume that \(a, b, c > 0\) and \(\frac{1}{x} + \frac{1}{y} + \frac{1}{z} \leq 3\). Given \(x = a^2\), \(y = b^2\), and \(z = c^2\), calculate the minimum value of the expression \(ab^2c + bc^2a + ca^2b - (a^2 + b^2 + c^2)\). | [
"0"
] | [
"Assume that \\(a, b, c > 0\\) and \\(\\frac{1}{x} + \\frac{1}{y} + \\frac{1}{z} \\leq 3\\). Given \\(x = a^2\\), \\(y = b^2\\), and \\(z = c^2\\), calculate the minimum value of the expression \\(ab^2c + bc^2a + ca^2b - (a^2 + b^2 + c^2)\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-f6db5d4faba57cd5e7a070331a065792b4679a81d0c1038a1e14000bbc78dfd1 | Assume that \(a, b, c, d > 0\). Calculate the minimum possible value of the expression \[ \frac{a^2 - bd}{b + d + 2c} + \frac{b^2 - ca}{c + a + 2d} + \frac{c^2 - db}{d + b + 2a} + \frac{d^2 - ac}{a + c + 2b}. \] | [
"0"
] | [
"Assume that \\(a, b, c, d > 0\\). Calculate the minimum possible value of the expression\n\\[\n\\frac{a^2 - bd}{b + d + 2c} + \\frac{b^2 - ca}{c + a + 2d} + \\frac{c^2 - db}{d + b + 2a} + \\frac{d^2 - ac}{a + c + 2b}.\n\\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-ed8d6178538207bb1d93e33775aa334f085599cea4d65cddcf91635d01ddcfb4 | Assume that \(a, b, c, d\) are positive real numbers such that \((a+c)(b+d) = 4abcd\). Find the greatest value of the real \(q\) for which \[ abcd \geq 1 + q \left( \frac{(a-c)^2}{ac} + \frac{(b-d)^2}{bd} \right). \] | [
"0"
] | [
"Assume that \\(a, b, c, d\\) are positive real numbers such that \\((a+c)(b+d) = 4abcd\\). Find the greatest value of the real \\(q\\) for which\n\\[ abcd \\geq 1 + q \\left( \\frac{(a-c)^2}{ac} + \\frac{(b-d)^2}{bd} \\right). \\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1fe4136008628e4340ee6c9d478d70f54a0a9452097c396534a2f5317d0be844 | Assume that \(a, b, c, d\) are positive real numbers such that \(a^2 + b^2 + c^2 + d^2 = 1\). Calculate the minimum value of the expression \(101(a^3 + b^3 + c^3 + d^3) + 125abcd\). | [
"\\dfrac{933}{16}"
] | [
"Assume that \\(a, b, c, d\\) are positive real numbers such that \\(a^2 + b^2 + c^2 + d^2 = 1\\). Calculate the minimum value of the expression \\(101(a^3 + b^3 + c^3 + d^3) + 125abcd\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-71afc88c2fd5b314b2a20aee9c06d6618b70ae5267c7fcd459ab5604a30a002c | Assume that \(a, b, c,\) and \(d\) are positive integers such that \(a^5 = b^4\), \(c^3 = d^2\), and \(c - a = 19\). Determine \(d - b\). | [
"757"
] | [
"Assume that \\(a, b, c,\\) and \\(d\\) are positive integers such that \\(a^5 = b^4\\), \\(c^3 = d^2\\), and \\(c - a = 19\\). Determine \\(d - b\\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-13ae6b72ca074a170c3b8dab76aaba3c6a325ae72d28e84e6dc6f1e6f7f669b2 | Assume that \(a, b, c\) are positive numbers such that \(a + b + c = 3\). Calculate the minimum value of the expression \[ \frac{a}{b^2 + c^2} + \frac{b}{c^2 + a^2} + \frac{c}{a^2 + b^2}. \] | [
"\\dfrac{4}{3}"
] | [
"Assume that \\(a, b, c\\) are positive numbers such that \\(a + b + c = 3\\). Calculate the minimum value of the expression \n\\[\n\\frac{a}{b^2 + c^2} + \\frac{b}{c^2 + a^2} + \\frac{c}{a^2 + b^2}.\n\\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e6a23aca324169c1351f7eea19059811fba1c0b6d72a3170025885ab258139a9 | Assume that \(a, b, c\) are positive reals that cannot be the sides of a triangle. Calculate the minimum value of the expression \(\frac{(abc)^{2}(a+b+c)^{2}(a+b-c)(a-b+c)(-a+b+c)}{(a^{2}+b^{2}+c^{2})^{3}(a^{2}+b^{2}-c^{2})(a^{2}-b^{2}+c^{2})(-a^{2}+b^{2}+c^{2})}.\) | [
"0"
] | [
"Assume that \\(a, b, c\\) are positive reals that cannot be the sides of a triangle. Calculate the minimum value of the expression \\(\\frac{(abc)^{2}(a+b+c)^{2}(a+b-c)(a-b+c)(-a+b+c)}{(a^{2}+b^{2}+c^{2})^{3}(a^{2}+b^{2}-c^{2})(a^{2}-b^{2}+c^{2})(-a^{2}+b^{2}+c^{2})}.\\)"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c766434c8533398eb5c6ac927f8b9f7910cf0543922b9530ee5c500eafbdf7c6 | Assume that \(a\) and \(b\) are integers and \(n\) is a natural number. If \(2^n a + b\) is a perfect square for every \(n\), find the value of \(a + b\) given that \(b\) is the smallest positive integer such that this condition holds. | [
"1"
] | [
"Assume that \\(a\\) and \\(b\\) are integers and \\(n\\) is a natural number. If \\(2^n a + b\\) is a perfect square for every \\(n\\), find the value of \\(a + b\\) given that \\(b\\) is the smallest positive integer such that this condition holds."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-03f04c421c3557d61d37faf519ba4a370ee3603dc363ee6a21137e9d148ad83f | Assume that \(a\) and \(b\) are integers such that for all sufficiently large primes \(p\), there exists a permutation \(c_1, c_2, \ldots, c_{p-1}\) of the numbers \(1, 2, \ldots, p-1\) such that \[ p \mid ac_{i+1} - bc_i \] for \(i = 1, \ldots, p-2\). Determine the value of \(a^2 + b^2\). | [
"0"
] | [
"Assume that \\(a\\) and \\(b\\) are integers such that for all sufficiently large primes \\(p\\), there exists a permutation \\(c_1, c_2, \\ldots, c_{p-1}\\) of the numbers \\(1, 2, \\ldots, p-1\\) such that \n\\[ p \\mid ac_{i+1} - bc_i \\]\nfor \\(i = 1, \\ldots, p-2\\). Determine the value of \\(a^2 + b^2\\)."
... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-29ad9b4e07cb94a499ad104d71488ac4c296f6ab214dec7cab2b9d5a221e435f | Assume that \(a_n, b_n\) are sequences of distinct integers such that \(a_n\) contains arbitrarily long arithmetic progressions. If \(\sum_{n \geq 1} \frac{1}{1 + |a_n - b_n|}\) diverges, does \(b_n\) also contain arbitrarily long arithmetic progressions? | [
"Yes"
] | [
"Assume that \\(a_n, b_n\\) are sequences of distinct integers such that \\(a_n\\) contains arbitrarily long arithmetic progressions. If \\(\\sum_{n \\geq 1} \\frac{1}{1 + |a_n - b_n|}\\) diverges, does \\(b_n\\) also contain arbitrarily long arithmetic progressions?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b3dc9085bbc24f2a589236615afb83bf653e9ce5eff90a1d146c600258e00dbd | Assume that \(a_n\) is a sequence of positive numbers satisfying \(a_n \ge (1-\delta_n)a_{n-1} + \delta_n a_{\left[\frac{n}{2}\right]}\) for \(n \ge 2\) where \(\delta_n \in (0,1)\) is some sequence. Can we guarantee that there exists a (possibly infinite) limit \(\lim_{n \to \infty} a_n\)? | [
"Yes"
] | [
"Assume that \\(a_n\\) is a sequence of positive numbers satisfying \\(a_n \\ge (1-\\delta_n)a_{n-1} + \\delta_n a_{\\left[\\frac{n}{2}\\right]}\\) for \\(n \\ge 2\\) where \\(\\delta_n \\in (0,1)\\) is some sequence. Can we guarantee that there exists a (possibly infinite) limit \\(\\lim_{n \\to \\infty} a_n\\)?"
... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-78587b9de6a1c86975dd8ed09ec3bbf5d46773e145377cd4070f752de7a61995 | Assume that \(x, y, z \in \mathbb{R^+} \cup \{0\}\) such that \(x^2 + y^2 + z^2 = 3\). Calculate the maximum value of the expression \(\sum \frac{x}{\sqrt{x^2 + y + z}}\). | [
"\\sqrt{3}"
] | [
"Assume that \\(x, y, z \\in \\mathbb{R^+} \\cup \\{0\\}\\) such that \\(x^2 + y^2 + z^2 = 3\\). Calculate the maximum value of the expression \\(\\sum \\frac{x}{\\sqrt{x^2 + y + z}}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8f0a1b82ef49f2beaac196963d7636fa29686337db6e157739e98b76c16bcfc1 | Assume that \(x, y, z\) are positive real numbers such that \(xyz = 4(x + y + z)\) and \(xy + yz + zx = 44\). Find the minimum and maximum value of \(xyz(x + y + z)\). | [
"584 + 8\\sqrt{5}"
] | [
"Assume that \\(x, y, z\\) are positive real numbers such that \\(xyz = 4(x + y + z)\\) and \\(xy + yz + zx = 44\\). Find the minimum and maximum value of \\(xyz(x + y + z)\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f100461f67fba35228513e407e61fea61f5d55996660c3097f2bc88956d61712 | Assume that \(x, y,\) and \(z\) are positive real numbers that satisfy the equations: \[ x + y + xy = 8 \] \[ y + z + yz = 15 \] \[ z + x + zx = 35 \] Determine the value of \(x + y + z + xyz\). | [
"36"
] | [
"Assume that \\(x, y,\\) and \\(z\\) are positive real numbers that satisfy the equations:\n\\[ x + y + xy = 8 \\]\n\\[ y + z + yz = 15 \\]\n\\[ z + x + zx = 35 \\]\nDetermine the value of \\(x + y + z + xyz\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-ee0da1ff55cbb57792e8a0ef1d330ef15788d7b512e1834f78d354fa8d033967 | Assume that a function $f$ is continuous on the closed interval $[0,1]$ and that $0 \leq f(x) \leq 1$ for all $x \in [0,1]$. Let $g(x) = f(x) - x$. Find the value of $g(c)$ where $c \in [0,1]$ is such that $f(c) = c$. | [
"0"
] | [
"Assume that a function $f$ is continuous on the closed interval $[0,1]$ and that $0 \\leq f(x) \\leq 1$ for all $x \\in [0,1]$. Let $g(x) = f(x) - x$. Find the value of $g(c)$ where $c \\in [0,1]$ is such that $f(c) = c$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4f1d614548b9ff0f898a61b18015bf1ce001e28e03f7ce191aef0f1ec33f6a57 | Assume that a man’s profession can be classified as professional, skilled laborer, or unskilled laborer. The transition probabilities are as follows: - Sons of professional men: 80% professional, 10% skilled laborer, 10% unskilled laborer. - Sons of skilled laborers: 60% skilled laborer, 20% professional, 20% unskilled... | [
"\\(\\frac{3}{8}\\)"
] | [
"Assume that a man’s profession can be classified as professional, skilled laborer, or unskilled laborer. The transition probabilities are as follows:\n- Sons of professional men: 80% professional, 10% skilled laborer, 10% unskilled laborer.\n- Sons of skilled laborers: 60% skilled laborer, 20% professional, 20% un... | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-bd242a859a4819b7c48fd4c631ec108d66d73ee43aeedf6166d6748cc02d8a32 | Assume that a new light bulb will burn out after \( t \) hours, where \( t \) is chosen from \([0, \infty)\) with an exponential density \( f(t) = \lambda e^{-\lambda t} \). (a) Assume that \( \lambda = 0.01 \), and find the probability that the bulb will not burn out before \( T \) hours. This probability is often cal... | [
"100 \\ln 2"
] | [
"Assume that a new light bulb will burn out after \\( t \\) hours, where \\( t \\) is chosen from \\([0, \\infty)\\) with an exponential density \\( f(t) = \\lambda e^{-\\lambda t} \\).\n(a) Assume that \\( \\lambda = 0.01 \\), and find the probability that the bulb will not burn out before \\( T \\) hours. This pr... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-22a575e1c8f31ffbcbe27773cfb3267b45ee4249a7b93f144e23ee6208a52825 | Assume that a point $P$ inside the convex quadrilateral $ABCD$ satisfies $\angle APB = \angle DPC = 90^\circ$. Let $\odot O_1$ and $\odot O_2$ be the circumscribed circles of $\triangle APC$ and $\triangle BPD$ respectively. It is known that the line connecting the midpoints $E$ and $F$ of sides $AB$ and $CD$ passes th... | [
"72"
] | [
"Assume that a point $P$ inside the convex quadrilateral $ABCD$ satisfies $\\angle APB = \\angle DPC = 90^\\circ$. Let $\\odot O_1$ and $\\odot O_2$ be the circumscribed circles of $\\triangle APC$ and $\\triangle BPD$ respectively. It is known that the line connecting the midpoints $E$ and $F$ of sides $AB$ and $C... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-fa81a28e6aa7009b6debb129821c3c2eb6ae7eee8c275cde0de6a97e6867a326 | Assume that all the angles of a given triangle \(ABC\) are smaller than \(120^\circ\). Equilateral triangles \(AFB\), \(BDC\), and \(CEA\) are constructed in the exterior of \(\triangle ABC\). The lines \(AD\), \(BE\), and \(CF\) intersect at a point \(S\). If \(SA = 5\), \(SB = 7\), and \(SC = 9\), calculate the value... | [
"42"
] | [
"Assume that all the angles of a given triangle \\(ABC\\) are smaller than \\(120^\\circ\\). Equilateral triangles \\(AFB\\), \\(BDC\\), and \\(CEA\\) are constructed in the exterior of \\(\\triangle ABC\\). The lines \\(AD\\), \\(BE\\), and \\(CF\\) intersect at a point \\(S\\). If \\(SA = 5\\), \\(SB = 7\\), and ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-81229d2e0231a3c3643d75b4af8d4434a2ffbb4fa1b854c5ad66a0ad77e9acc9 | Assume that all the roots of the polynomial \( x^8 + a_7 x^7 + a_6 x^6 + a_5 x^5 + a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + 420 \) are positive integers. What is the maximum number of distinct roots of such a polynomial? | [
"5"
] | [
"Assume that all the roots of the polynomial \\( x^8 + a_7 x^7 + a_6 x^6 + a_5 x^5 + a_4 x^4 + a_3 x^3 + a_2 x^2 + a_1 x + 420 \\) are positive integers. What is the maximum number of distinct roots of such a polynomial?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 18,
"tir": 2
} | 20 | 0 | 0 |
omr-ws-c164341a595f96c6f723e91251091a7618da8b0a52f5c1a63586a7f0ac3774b1 | Assume that arrivals of small aircraft at airports can be modeled by a Poisson process with an average of 1 arrival per hour. What is the probability that one has to wait at least three hours for the arrival of three aircraft? | [
"0.423"
] | [
"Assume that arrivals of small aircraft at airports can be modeled by a Poisson process with an average of 1 arrival per hour. What is the probability that one has to wait at least three hours for the arrival of three aircraft?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 15
} | 15 | 0 | 0 |
omr-ws-e26b84d91457e78d2c2c2dcd881bed1723f81d9f230de3a3ffd7eff0979f294e | Assume that as a cubical bar of soap is used, all edges shrink at a constant rate of $n$ units per day. Starting with a full bar, the soap was used for 6 days and its surface area was cut in half. Starting with a full bar of soap, find exactly the time it could take for the volume to become one-eighth of the original v... | [
"\\(\\sqrt{2} + 2\\)"
] | [
"Assume that as a cubical bar of soap is used, all edges shrink at a constant rate of $n$ units per day. Starting with a full bar, the soap was used for 6 days and its surface area was cut in half. Starting with a full bar of soap, find exactly the time it could take for the volume to become one-eighth of the origi... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-519e38b020b48453e46a7f2d3354b5c1f8aa4fc54ce85ab9e8f1f1b7447a205c | Assume that both the attacking player and the defending player have exactly \( m = 6 \) pieces on each of their territories. Calculate which player has the greater chance of "winning" the entire battle between the territories, if the attacking player is reduced to 1 piece, then he loses; if the defending player is redu... | [
"\\text{defending player}"
] | [
"Assume that both the attacking player and the defending player have exactly \\( m = 6 \\) pieces on each of their territories. Calculate which player has the greater chance of \"winning\" the entire battle between the territories, if the attacking player is reduced to 1 piece, then he loses; if the defending playe... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
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