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-8ed7791e794f9dc1a08613299b28cf3f0a25f698c71d0f8235aaf546676c4bc4 | Assume \( f \) is integrable and non-negative on \([a, b]\). If \(\int_a^b f(x) \, dx = 0\), and \( f \) is continuous at \( x = c \) where \( c \in [a, b] \), calculate \( f(c) \). | [
"0"
] | [
"Assume \\( f \\) is integrable and non-negative on \\([a, b]\\). If \\(\\int_a^b f(x) \\, dx = 0\\), and \\( f \\) is continuous at \\( x = c \\) where \\( c \\in [a, b] \\), calculate \\( f(c) \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1a155d5b54ce16ee5b3d3c8229a9ac340147f2522eac4946177eca8e64a0ded9 | Assume \( f(n) \) represents the number of non-negative integral solutions to the equation \( a + b + c + d + e = n \). Calculate the sum \( f(10) + 2f(9) + 3f(8) + 4f(7) + 5f(6) + 6f(5) + 7f(4) + 8f(3) + 9f(2) + 10f(1) + 11f(0) \). | [
"8008"
] | [
"Assume \\( f(n) \\) represents the number of non-negative integral solutions to the equation \\( a + b + c + d + e = n \\). Calculate the sum \\( f(10) + 2f(9) + 3f(8) + 4f(7) + 5f(6) + 6f(5) + 7f(4) + 8f(3) + 9f(2) + 10f(1) + 11f(0) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-fdb83b16fbd2119ac09269f774ce478ffe4df9a6530fd4b28016b27bc002874e | Assume \( f(x) \) and \( g(x) \) are continuous on the interval \([a, b]\) and differentiable on \((a, b)\), and \( g'(x) \neq 0 \) for \( x \in (a, b) \). Given \( f(a) = 1 \), \( f(b) = 3 \), \( g(a) = 2 \), and \( g(b) = 4 \), find the value of \( \frac{f'(\xi)}{g'(\xi)} \) for some \( \xi \in (a, b) \) that satisfi... | [
"1"
] | [
"Assume \\( f(x) \\) and \\( g(x) \\) are continuous on the interval \\([a, b]\\) and differentiable on \\((a, b)\\), and \\( g'(x) \\neq 0 \\) for \\( x \\in (a, b) \\). Given \\( f(a) = 1 \\), \\( f(b) = 3 \\), \\( g(a) = 2 \\), and \\( g(b) = 4 \\), find the value of \\( \\frac{f'(\\xi)}{g'(\\xi)} \\) for some \... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-115f4b7a61e0fe609faa5d5ec337f594342d7883754d13ed186396023300b603 | Assume \( f(x) \) is a continuous function on \([0,1]\) such that \[ f(0) = 0, \quad f(1) = 1, \quad f(x) + f\left(\frac{1}{2}\right) = f\left(\frac{x}{2}\right) + f\left(\frac{1+2x}{2+x}\right). \] Let \( g(x) = f(x) - x \). Calculate the value of \( g\left(\frac{1}{2}\right) \). | [
"-\\dfrac{1}{2}"
] | [
"Assume \\( f(x) \\) is a continuous function on \\([0,1]\\) such that\n\\[ f(0) = 0, \\quad f(1) = 1, \\quad f(x) + f\\left(\\frac{1}{2}\\right) = f\\left(\\frac{x}{2}\\right) + f\\left(\\frac{1+2x}{2+x}\\right). \\]\nLet \\( g(x) = f(x) - x \\). Calculate the value of \\( g\\left(\\frac{1}{2}\\right) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ef2e1d2c300b2ec322de77309de5352c401e55b4b56f449a208185c9c773158d | Assume \( f(x) \) is a continuously differentiable function such that \( f, f' \in L^2(\mathbb{R}) \), i.e., \( \int_{\mathbb{R}} |f(x)|^2 \, dx < +\infty \) and \( \int_{\mathbb{R}} |f'(x)|^2 \, dx < +\infty \). Calculate the value of \( \int_{\mathbb{R}} \left| \frac{f(x+1/100) - f(x)}{1/100} - f'(x) \right|^2 \, dx ... | [
"\\dfrac{3\\sqrt{2\\pi}}{80000}"
] | [
"Assume \\( f(x) \\) is a continuously differentiable function such that \\( f, f' \\in L^2(\\mathbb{R}) \\), i.e., \\( \\int_{\\mathbb{R}} |f(x)|^2 \\, dx < +\\infty \\) and \\( \\int_{\\mathbb{R}} |f'(x)|^2 \\, dx < +\\infty \\). Calculate the value of \\( \\int_{\\mathbb{R}} \\left| \\frac{f(x+1/100) - f(x)}{1/1... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9f93415ce927ad3102a82bac14ba97b1c36a5f3d67d0d1aef332caf9a59b9f6e | Assume \( f(x) \) is a function where \( \mu \) is a root and there are two values of \( x \) where \( f'(x) = 0 \) in the interval \( [\mu, b] \) where \( b \) is a fixed real number. If \( f'(\mu) = f'(b) \), calculate the value of \( \frac{2 \int_{\mu}^{b} f(x) \,dx}{b-\mu} \). | [
"f(b)"
] | [
"Assume \\( f(x) \\) is a function where \\( \\mu \\) is a root and there are two values of \\( x \\) where \\( f'(x) = 0 \\) in the interval \\( [\\mu, b] \\) where \\( b \\) is a fixed real number. If \\( f'(\\mu) = f'(b) \\), calculate the value of \\( \\frac{2 \\int_{\\mu}^{b} f(x) \\,dx}{b-\\mu} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-2e55ac835937c0ba49e3a2c2ff39e2cbfe1c17e5da18efe3bdcc1cd745e40076 | Assume \( f(x) \) is differentiable over \([a, b]\) and \( f'(a) = f'(b) \). Let \( g(c) = f(c) - f(a) - (c - a) f'(c) \). Find the value of \( g(c) \) at the point \( c \in (a, b) \) where \( g'(c) = 0 \). | [
"0"
] | [
"Assume \\( f(x) \\) is differentiable over \\([a, b]\\) and \\( f'(a) = f'(b) \\). Let \\( g(c) = f(c) - f(a) - (c - a) f'(c) \\). Find the value of \\( g(c) \\) at the point \\( c \\in (a, b) \\) where \\( g'(c) = 0 \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-47f00bca224934225f366a5a17bd0affb10fa7eee30d15751ccfa0bfe34f9792 | Assume \( f(x) \in C^1[0,1] \) and \( f(1) - f(0) = 1 \). Find the minimum value of \( \int_0^1 (f'(x))^2 \, dx \). | [
"1"
] | [
"Assume \\( f(x) \\in C^1[0,1] \\) and \\( f(1) - f(0) = 1 \\). Find the minimum value of \\( \\int_0^1 (f'(x))^2 \\, dx \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d9a5bb723d69deaf1690e867e07b8fc103b1b20f027200930fbaec06c34bc97a | Assume \( f(x) \in C^1[0,1] \) and \( f(1) = 0 \). If \( \int_0^1 f^2(x) \, dx = 1 \), find the minimum value of \( \int_0^1 x^2 f'(x)^2 \, dx \). | [
"\\dfrac{1}{2}"
] | [
"Assume \\( f(x) \\in C^1[0,1] \\) and \\( f(1) = 0 \\). If \\( \\int_0^1 f^2(x) \\, dx = 1 \\), find the minimum value of \\( \\int_0^1 x^2 f'(x)^2 \\, dx \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-8a70d4ab065ceb7ffca7244998e9ec5275045ccdbe60101ede5cfeb35abbbb7c | Assume \( f(x) \in C^2[0,1] \) and \( f(0) + f(1) = -\frac{1}{12} \), \( f'\left(\frac{1}{2}\right) = 2a \), \( \max_{0 \le x \le 1} |f''(x)| = 2b \). Calculate the maximum possible value of \( \int_0^1 f(x) \, dx \) in terms of \(a\) and \(b\). | [
"\\dfrac{b}{6} - \\dfrac{1}{24}"
] | [
"Assume \\( f(x) \\in C^2[0,1] \\) and \\( f(0) + f(1) = -\\frac{1}{12} \\), \\( f'\\left(\\frac{1}{2}\\right) = 2a \\), \\( \\max_{0 \\le x \\le 1} |f''(x)| = 2b \\). Calculate the maximum possible value of \\( \\int_0^1 f(x) \\, dx \\) in terms of \\(a\\) and \\(b\\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-dd1452d16d04c712018393d261e73ada5dc6f03e1c6679b155c684eedcf7107d | Assume \( f, g \in C[0,1] \) and \( \int_0^1 f = 1 = \int_0^1 g \). Let \( h(x) = \int_0^x f(t) \, dt \) and \( k(x) = \int_0^x g(t) \, dt \). Find the value of \( b - a \) for the interval \( I = [a, b] \subset [0,1] \) such that \[ \int_I f = \frac{1}{2} = \int_I g. \] | [
"\\dfrac{1}{2}"
] | [
"Assume \\( f, g \\in C[0,1] \\) and \\( \\int_0^1 f = 1 = \\int_0^1 g \\). Let \\( h(x) = \\int_0^x f(t) \\, dt \\) and \\( k(x) = \\int_0^x g(t) \\, dt \\). Find the value of \\( b - a \\) for the interval \\( I = [a, b] \\subset [0,1] \\) such that\n\n\\[\n\\int_I f = \\frac{1}{2} = \\int_I g.\n\\]"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-5d0e893fd251175e29edd4da9c0bb8f5b1a91366de24553c1591afe5b32688e2 | Assume \( f: (0, \infty) \to \mathbb{R} \) is a continuous function such that for any \( x, y > 0 \), \( f(x) = f(y) \) or \( f\left(\frac{x+y}{2}\right) = f(\sqrt{xy}) \). Find \( f \). | [
"\\( f(x) = c \\) for some constant \\( c \\in \\mathbb{R} \\)."
] | [
"Assume \\( f: (0, \\infty) \\to \\mathbb{R} \\) is a continuous function such that for any \\( x, y > 0 \\), \\( f(x) = f(y) \\) or \\( f\\left(\\frac{x+y}{2}\\right) = f(\\sqrt{xy}) \\). Find \\( f \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-6f3ef963c65469b6f926a91fe2db0f1510d91005cbf20f1f36a44ad0cfd4d941 | Assume \( f: G \to \mathbb{C} \) is holomorphic, \( \bar{\delta}(0,1) \subset G \), and \( |f(z)| \le 1 \) for all \( z \in G \). Given that \( f(1/2) = f(i/2) = 0 \), calculate the maximum possible value of \( |f(0)| \). | [
"\\dfrac{1}{4}"
] | [
"Assume \\( f: G \\to \\mathbb{C} \\) is holomorphic, \\( \\bar{\\delta}(0,1) \\subset G \\), and \\( |f(z)| \\le 1 \\) for all \\( z \\in G \\). Given that \\( f(1/2) = f(i/2) = 0 \\), calculate the maximum possible value of \\( |f(0)| \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d3a270934adee1ae00d4959d7d18d9d3a6532738e8851c636b38442510f3dfd9 | Assume \( f: \mathbb{R} \to \mathbb{R} \) is a continuous function on \([0,1]\) and differentiable on \((0,1)\), with \( f(1) = \frac{1}{3} \) and \( f(0) = 0 \). Find the minimum possible value of \( f'(c_1) + f'(c_2) + f'(c_3) \) for distinct \( c_1, c_2, c_3 \in (0,1) \). | [
"1"
] | [
"Assume \\( f: \\mathbb{R} \\to \\mathbb{R} \\) is a continuous function on \\([0,1]\\) and differentiable on \\((0,1)\\), with \\( f(1) = \\frac{1}{3} \\) and \\( f(0) = 0 \\). Find the minimum possible value of \\( f'(c_1) + f'(c_2) + f'(c_3) \\) for distinct \\( c_1, c_2, c_3 \\in (0,1) \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-28f9ef18b7bc3ad6f33c1ff16b4a7c78f45e32e0b80a978a3e2a65b15a332cdb | Assume \( f:[1,+\infty) \mapsto [e,+\infty) \) is continuous and increasing. Let \( I_1 = \int_1^{+\infty} \frac{dx}{f(x)} \) and \( I_2 = \int_1^{+\infty} \frac{dx}{x \ln f(x)} \). Given that \( I_1 \) diverges, calculate the value of \( \lim_{b \to +\infty} \frac{\int_1^{b} \frac{dx}{x \ln f(x)}}{\int_1^{b} \frac{dx}... | [
"0"
] | [
"Assume \\( f:[1,+\\infty) \\mapsto [e,+\\infty) \\) is continuous and increasing. Let \\( I_1 = \\int_1^{+\\infty} \\frac{dx}{f(x)} \\) and \\( I_2 = \\int_1^{+\\infty} \\frac{dx}{x \\ln f(x)} \\). Given that \\( I_1 \\) diverges, calculate the value of \\( \\lim_{b \\to +\\infty} \\frac{\\int_1^{b} \\frac{dx}{x \... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-56f42e583a10b76c787bb4b5c73ff72c748509d001af533da6e32b20f8aaabfd | Assume \( f_i \in L^p \) and \( f_i \to f \) in \( L^p \). Suppose \( f_i \to g \) almost everywhere (a.e.). Calculate the \( L^p \) norm of the difference \( f - g \). | [
"0"
] | [
"Assume \\( f_i \\in L^p \\) and \\( f_i \\to f \\) in \\( L^p \\). Suppose \\( f_i \\to g \\) almost everywhere (a.e.). Calculate the \\( L^p \\) norm of the difference \\( f - g \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e3d50aed2d85499f6f34d88ee128e4b73a85e250b9291c93f28ed0f655c0628e | Assume \( f_i \in L^p \cap L^q \), \( \| f_i - f \|_p \to 0 \), and \( \| f_i - g \|_q \to 0 \). Let \( h(x) = f(x) - g(x) \). Calculate the limit \(\lim_{i \to \infty} \| h \|_r\) where \( r = \min(p, q) \). | [
"0"
] | [
"Assume \\( f_i \\in L^p \\cap L^q \\), \\( \\| f_i - f \\|_p \\to 0 \\), and \\( \\| f_i - g \\|_q \\to 0 \\). Let \\( h(x) = f(x) - g(x) \\). Calculate the limit \\(\\lim_{i \\to \\infty} \\| h \\|_r\\) where \\( r = \\min(p, q) \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3d200fbaa52f76fc0c3653f08c1773f6bdd8e5125e0b3be7cb1f2afe89464d03 | Assume \( f_n \) is a sequence of functions from a metric space \( X \) to \( Y \) such that \( f_n \rightarrow f \) uniformly and each \( f_n \) has an inverse \( g_n \). Suppose \( f \) has an inverse \( g \) which is uniformly continuous on \( Y \). Calculate the value of \( \lim_{n \to \infty} \sup_{y \in Y} |g_n(y... | [
"0"
] | [
"Assume \\( f_n \\) is a sequence of functions from a metric space \\( X \\) to \\( Y \\) such that \\( f_n \\rightarrow f \\) uniformly and each \\( f_n \\) has an inverse \\( g_n \\). Suppose \\( f \\) has an inverse \\( g \\) which is uniformly continuous on \\( Y \\). Calculate the value of \\( \\lim_{n \\to \\... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6aeec7dd6d5d5beeb6b4a13cce09fc98e038a75f5442b07b166a9128114f5119 | Assume \( k \) is a positive integer. Let \( a_0 \) be an integer and define the sequence \( a_n = \frac{a_{n-1} + n^k}{n} \) for all positive integers \( n \). If \( a_3 \) is an integer, find the smallest possible value of \( k \) such that \( k+1 \) is divisible by 3. | [
"2"
] | [
"Assume \\( k \\) is a positive integer. Let \\( a_0 \\) be an integer and define the sequence \\( a_n = \\frac{a_{n-1} + n^k}{n} \\) for all positive integers \\( n \\). If \\( a_3 \\) is an integer, find the smallest possible value of \\( k \\) such that \\( k+1 \\) is divisible by 3."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5dd688410934a001f1744c1d9ecb9cfb0d6b8323d1af9a769549367af98ff86a | Assume \( n > 1 \) is a natural number. Let \( A = \{ y \in \mathbb{R} : y \geq 0, y^n \leq a \} \). Calculate the value of \(\sup (A)\) for \( a = 27 \) and \( n = 3 \). | [
"3"
] | [
"Assume \\( n > 1 \\) is a natural number. Let \\( A = \\{ y \\in \\mathbb{R} : y \\geq 0, y^n \\leq a \\} \\). Calculate the value of \\(\\sup (A)\\) for \\( a = 27 \\) and \\( n = 3 \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fa60ea9cb76c63e4c28961530946b6c9999d6802deebd1c7e5f5c2ad9ab5a3b1 | Assume \( n > 1 \) is a natural number. Let \( A = \{ y \in \mathbb{R} : y \geq 0, y^n \leq a \} \). Given \( y \) such that \( y^n < a \), find the maximum value of \( \epsilon \) such that \( (y + \epsilon)^n < a \) for \( a = 100 \) and \( n = 3 \). | [
"\\sqrt[3]{100} - y"
] | [
"Assume \\( n > 1 \\) is a natural number. Let \\( A = \\{ y \\in \\mathbb{R} : y \\geq 0, y^n \\leq a \\} \\). Given \\( y \\) such that \\( y^n < a \\), find the maximum value of \\( \\epsilon \\) such that \\( (y + \\epsilon)^n < a \\) for \\( a = 100 \\) and \\( n = 3 \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-100cb09957b81297856ac4191396a97eadb1643d5dcb22c5511421c9bb19bcf4 | Assume \( n > 1 \) is a natural number. Let \( A = \{ y \in \mathbb{R} : y \geq 0, y^n \leq a \} \). Given \( y \) such that \( y^n > a \), find the value of \( \epsilon \) in terms of \( y \) and \( a \) such that \( (y - \epsilon)^n > a \) for \( \epsilon > 0 \). Use \( \epsilon = \frac{1}{2} \min \left( y, \frac{y^n... | [
"\\dfrac{y^n - a}{2n y^{n-1}}"
] | [
"Assume \\( n > 1 \\) is a natural number. Let \\( A = \\{ y \\in \\mathbb{R} : y \\geq 0, y^n \\leq a \\} \\). Given \\( y \\) such that \\( y^n > a \\), find the value of \\( \\epsilon \\) in terms of \\( y \\) and \\( a \\) such that \\( (y - \\epsilon)^n > a \\) for \\( \\epsilon > 0 \\). Use \\( \\epsilon = \\... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-05b0945e0e5e439417fbde3e19199e45aed146ccef6d3598df0de93b65b221a9 | Assume \( n \) is a fixed positive integer. Choose \( n \) real numbers \( x_1, \ldots, x_n \in [-1, 1] \) (uniform distribution). Calculate the expected value of \( \operatorname{max} (x_i + \ldots + x_{i + k}) \) where \( 0 \leq k \leq n - i \) and \( 1 \leq i \leq n \). | [
"0"
] | [
"Assume \\( n \\) is a fixed positive integer. Choose \\( n \\) real numbers \\( x_1, \\ldots, x_n \\in [-1, 1] \\) (uniform distribution). Calculate the expected value of \\( \\operatorname{max} (x_i + \\ldots + x_{i + k}) \\) where \\( 0 \\leq k \\leq n - i \\) and \\( 1 \\leq i \\leq n \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-621b37dfac1153afa721d840f711c829e5c406069e9bba382012fc96e8289ea7 | Assume \( n \) is a positive integer and integer \( a \) is the root of the equation \( x^4 + 3ax^2 + 2ax - 2 \times 3^n = 0 \). Find all \( n \) and \( a \) that satisfy the conditions. | [
"\\((1, 1), (2, -3)\\)"
] | [
"Assume \\( n \\) is a positive integer and integer \\( a \\) is the root of the equation \\( x^4 + 3ax^2 + 2ax - 2 \\times 3^n = 0 \\). Find all \\( n \\) and \\( a \\) that satisfy the conditions."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 25
} | 29 | 0 | 0 |
omr-ws-495b98a1a0f4827e4d52b860800d07304c8a2ead19ecd7e5f1cea2c027b93842 | Assume \( n \geq 6 \) and that \( a_i \ (1 \leq i \leq n) \) are real numbers with \( \sum_{i=1}^n a_i = 0 \). Let \( S \) be the number of pairs \((i, j) \ (i < j)\) such that \( a_i + a_j \geq 0 \). Find the minimum possible value of \( S \) for \( n = 10 \). | [
"9"
] | [
"Assume \\( n \\geq 6 \\) and that \\( a_i \\ (1 \\leq i \\leq n) \\) are real numbers with \\( \\sum_{i=1}^n a_i = 0 \\). Let \\( S \\) be the number of pairs \\((i, j) \\ (i < j)\\) such that \\( a_i + a_j \\geq 0 \\). Find the minimum possible value of \\( S \\) for \\( n = 10 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-3ebe2d2262d07610600ad05fe8705d84ce1f80ce2045fbaf5c6cc651cc31f21e | Assume \( n \in \mathbb{Z}^* \), \( i, j \in \{1, 2, \ldots, n\} \), \( a(i, j) > 0 \), and \( a(i, j) \times a(j, i) = 1 \). Define \( c(i) = \sum_{k=1}^{n} a(k, i) \). Calculate the maximum possible value of \( \sum_{i=1}^{n} \frac{1}{c(i)} \) for \( n = 5 \). | [
"1"
] | [
"Assume \\( n \\in \\mathbb{Z}^* \\), \\( i, j \\in \\{1, 2, \\ldots, n\\} \\), \\( a(i, j) > 0 \\), and \\( a(i, j) \\times a(j, i) = 1 \\). Define \\( c(i) = \\sum_{k=1}^{n} a(k, i) \\). Calculate the maximum possible value of \\( \\sum_{i=1}^{n} \\frac{1}{c(i)} \\) for \\( n = 5 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-ad46b8e5d5159e2f5ea99affb317677b9c329179331341b87887653bc6b53b30 | Assume \( p \) is a prime number of the form \( 6m + 1 \), where \( m \in \mathbb{N}^* \). Calculate the remainder when \( \prod_{m=1}^{p-1} (m^2 + m + 1) \) is divided by \( p \). | [
"0"
] | [
"Assume \\( p \\) is a prime number of the form \\( 6m + 1 \\), where \\( m \\in \\mathbb{N}^* \\). Calculate the remainder when \\( \\prod_{m=1}^{p-1} (m^2 + m + 1) \\) is divided by \\( p \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4dd7843216534be962c0468079178cf9fcbc87e5722d265f963a4bb50c6d16b8 | Assume \( p \) is a prime number. If there is a positive integer \( a \) such that \( p! \mid (a^p + 1) \), find the greatest common divisor \( \text{gcd}(a+1, \frac{a^p+1}{a+1}) \) when \( p = 5 \) and \( a = 30 \). | [
"1"
] | [
"Assume \\( p \\) is a prime number. If there is a positive integer \\( a \\) such that \\( p! \\mid (a^p + 1) \\), find the greatest common divisor \\( \\text{gcd}(a+1, \\frac{a^p+1}{a+1}) \\) when \\( p = 5 \\) and \\( a = 30 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-677f2323575028500f8e841905283f1512ea23da1dd8f99e68012e4d5b473622 | Assume \( p \) is a prime number. If there is a positive integer \( a \) such that \( p! \mid (a^p + 1) \), find the smallest possible value of \( \frac{a^p+1}{a+1} \) when \( p = 5 \). | [
"198862805"
] | [
"Assume \\( p \\) is a prime number. If there is a positive integer \\( a \\) such that \\( p! \\mid (a^p + 1) \\), find the smallest possible value of \\( \\frac{a^p+1}{a+1} \\) when \\( p = 5 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-604dd21fc96c366a6d58533ff26f454bf77f54149dec26727c236f3e2e8d17c5 | Assume \( p \) is a prime number. If there is a positive integer \( a \) such that \( p! \mid (a^p + 1) \), find the smallest possible value of \( a \) for \( p = 5 \). | [
"119"
] | [
"Assume \\( p \\) is a prime number. If there is a positive integer \\( a \\) such that \\( p! \\mid (a^p + 1) \\), find the smallest possible value of \\( a \\) for \\( p = 5 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-23f22882b8a1c6d4ebcc005c64aa435568adc12e7dd8a71ebafe29abcfaf20b8 | Assume \( p \ge 5 \) is a prime. Calculate the remainder when \(\sum_{n=1}^{p^2-1} \binom{2n}{n}\) is divided by \( p^2 \). | [
"0"
] | [
"Assume \\( p \\ge 5 \\) is a prime. Calculate the remainder when \\(\\sum_{n=1}^{p^2-1} \\binom{2n}{n}\\) is divided by \\( p^2 \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-73fd310262f610e1caee94270627f2ffb749e9246fb5940d26778241b4b4bf92 | Assume \( u \in C^2(\mathbb{R}^d) \) with \( d \geq 3 \) satisfies \[ \Delta _{\mathbb{R}^d} u = g \in C^1_c (\mathbb{R}^d). \] If \( u(x) \in O(|x|^{1-d}) \) as \( |x| \rightarrow \infty \), calculate the value of \( \int _{\mathbb{R}^d} g \, dx \). | [
"0"
] | [
"Assume \\( u \\in C^2(\\mathbb{R}^d) \\) with \\( d \\geq 3 \\) satisfies \n\\[ \\Delta _{\\mathbb{R}^d} u = g \\in C^1_c (\\mathbb{R}^d). \\]\nIf \\( u(x) \\in O(|x|^{1-d}) \\) as \\( |x| \\rightarrow \\infty \\), calculate the value of \\( \\int _{\\mathbb{R}^d} g \\, dx \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2ac1df537f4b155b0c15a487ea8c25319287df18d8598a7fce929c45c703ea97 | Assume \( u \in C^2(\mathbb{R}^d) \) with \( d \geq 3 \) satisfies \[ \Delta _{\mathbb{R}^d} u = g \in C^1_c (\mathbb{R}^d). \] Let \( u(x) \in O(|x|^{2-d}) \) as \( |x| \rightarrow \infty \). Calculate the value of \[ \partial _j \partial _k u(0) \] for \( j = k = 1 \) when \( g(x) = e^{-|x|^2} \). | [
"\\dfrac{1}{d}"
] | [
"Assume \\( u \\in C^2(\\mathbb{R}^d) \\) with \\( d \\geq 3 \\) satisfies \n\\[ \\Delta _{\\mathbb{R}^d} u = g \\in C^1_c (\\mathbb{R}^d). \\]\nLet \\( u(x) \\in O(|x|^{2-d}) \\) as \\( |x| \\rightarrow \\infty \\). Calculate the value of \n\\[ \\partial _j \\partial _k u(0) \\]\nfor \\( j = k = 1 \\) when \\( g(x... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-d8a217d01c9ad1b124693185f48a7fe66be28eb76f75cc392959b575b1056ae9 | Assume \( x > 1 \), \( y > 1 \), and \( x^x y^y = z^z \). If \( p \) is a prime such that \( p \mid x \), find the smallest possible value of \( y \) when \( x = p \) and \( z = p^2 \). | [
"4"
] | [
"Assume \\( x > 1 \\), \\( y > 1 \\), and \\( x^x y^y = z^z \\). If \\( p \\) is a prime such that \\( p \\mid x \\), find the smallest possible value of \\( y \\) when \\( x = p \\) and \\( z = p^2 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8d2ee8fe1250ad640807d047ff7e6ea6492e768f88ce51b81a84cacf3cd744b5 | Assume \( x \) and \( y \) are both odd integers that satisfy the equation \( x^2 - d y^2 = \pm 4 \). Form the rational numbers \( u \) and \( v \) such that: \[ u + v \sqrt{d} = \left( \frac{x + y \sqrt{d}}{2} \right)^3. \] Calculate the value of \( u^2 - d v^2 \) for \( x = 7 \), \( y = 3 \), and \( d = 5 \). | [
"1"
] | [
"Assume \\( x \\) and \\( y \\) are both odd integers that satisfy the equation \\( x^2 - d y^2 = \\pm 4 \\). Form the rational numbers \\( u \\) and \\( v \\) such that:\n\\[ u + v \\sqrt{d} = \\left( \\frac{x + y \\sqrt{d}}{2} \\right)^3. \\]\nCalculate the value of \\( u^2 - d v^2 \\) for \\( x = 7 \\), \\( y = ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e717b34cf87824b7ca2d772f1298d8a28fb8030234927ba4723c220e2a900829 | Assume \( x \) and \( y \) are different real numbers such that there are four consecutive positive integers \( n \) for which \( \frac{x^n - y^n}{x - y} \) is an integer. Find the smallest positive integer \( k \) such that \( \frac{x^k - y^k}{x - y} \) is guaranteed to be an integer for any such \( x \) and \( y \). | [
"1"
] | [
"Assume \\( x \\) and \\( y \\) are different real numbers such that there are four consecutive positive integers \\( n \\) for which \\( \\frac{x^n - y^n}{x - y} \\) is an integer. Find the smallest positive integer \\( k \\) such that \\( \\frac{x^k - y^k}{x - y} \\) is guaranteed to be an integer for any such \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 8
} | 11 | 0 | 0 |
omr-ws-4d52229ccac7c7f71dd5d361dd4d887fd835ef8a87fa73deae419421593c7afb | Assume \( x, y, z > 0 \) and \( xyz = x + y + z + 2 \). Let \( S = x + y + z \). Find the maximum value of \( 2(\sqrt{xy} + \sqrt{yz} + \sqrt{zx}) - S \). | [
"6"
] | [
"Assume \\( x, y, z > 0 \\) and \\( xyz = x + y + z + 2 \\). Let \\( S = x + y + z \\). Find the maximum value of \\( 2(\\sqrt{xy} + \\sqrt{yz} + \\sqrt{zx}) - S \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8b79155ea5f3d99fd846e02ce74ac8fd5287704e98750bd8984433091d48ffa1 | Assume \( x^2 + dy^2 = p \) where \( p \) is a prime number and \( d \ge 2 \). Find the value of \( x + y \) for the unique pair \((x, y)\) that satisfies the equation when \( p = 13 \) and \( d = 3 \). | [
"3"
] | [
"Assume \\( x^2 + dy^2 = p \\) where \\( p \\) is a prime number and \\( d \\ge 2 \\). Find the value of \\( x + y \\) for the unique pair \\((x, y)\\) that satisfies the equation when \\( p = 13 \\) and \\( d = 3 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cad21d8d7244a01493ada7479f2436094b000fd310b3bc52a5d081226bd53312 | Assume \( x^2 + y^2 = p \) where \( p \) is a prime number. Find the sum of all possible values of \( x + y \) for \( p = 17 \). | [
"0"
] | [
"Assume \\( x^2 + y^2 = p \\) where \\( p \\) is a prime number. Find the sum of all possible values of \\( x + y \\) for \\( p = 17 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-53661497db5f7c84744921efbcb837c8467a6243a39aa1a05e96fbd7b1b722a4 | Assume \( x_0 = 2 \) and \( x_n = x_{n-1}^2 - x_{n-1} + 1 \). Calculate the greatest common divisor of \( x_{10} \) and \( x_{11} \). | [
"1"
] | [
"Assume \\( x_0 = 2 \\) and \\( x_n = x_{n-1}^2 - x_{n-1} + 1 \\). Calculate the greatest common divisor of \\( x_{10} \\) and \\( x_{11} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0ccc7167d0303c5be1a53b1024eb48691f75175a0a0f0cb720128755546005a8 | Assume \( x_0 \in (a, b) \) and \( f(x) \) is differentiable on \( (a, x_0) \) and \( (x_0, b) \). Given that the left-hand and right-hand limits of the derivative of \( f(x) \) at \( x_0 \) are both equal to \( L \), calculate the value of \( f'(x_0) \). | [
"L"
] | [
"Assume \\( x_0 \\in (a, b) \\) and \\( f(x) \\) is differentiable on \\( (a, x_0) \\) and \\( (x_0, b) \\). Given that the left-hand and right-hand limits of the derivative of \\( f(x) \\) at \\( x_0 \\) are both equal to \\( L \\), calculate the value of \\( f'(x_0) \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2ae1c3ecfbf0d3fea413706d72bf405689717d8aadb2256252f7248151d5cdc3 | Assume \( x_1 \) and \( x_2 \) are roots in the interval \([0, \frac{\pi}{2}]\) of the equations \( \sin(\cos x) = x \) and \( \cos(\sin x) = x \) respectively. Calculate the value of \( x_2 - x_1 \). | [
"0.073"
] | [
"Assume \\( x_1 \\) and \\( x_2 \\) are roots in the interval \\([0, \\frac{\\pi}{2}]\\) of the equations \\( \\sin(\\cos x) = x \\) and \\( \\cos(\\sin x) = x \\) respectively. Calculate the value of \\( x_2 - x_1 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-e911e34ec80886bbb6b19a038c711044793e5445f87f166453a64b1237cbb753 | Assume \( x_1, x_2, \ldots, x_n \) are real numbers such that \( |x_i| \ge 1 \) for all \( 1 \le i \le n \). Let \( I = [a, a+1) \) be an interval of length 1 and \( S = \left\{ \sum_{i=1}^n a_i x_i \mid a_i = 0, 1 \right\} \). Calculate the maximum possible value of \( |S \cap I| \) for \( n = 8 \). | [
"70"
] | [
"Assume \\( x_1, x_2, \\ldots, x_n \\) are real numbers such that \\( |x_i| \\ge 1 \\) for all \\( 1 \\le i \\le n \\). Let \\( I = [a, a+1) \\) be an interval of length 1 and \\( S = \\left\\{ \\sum_{i=1}^n a_i x_i \\mid a_i = 0, 1 \\right\\} \\). Calculate the maximum possible value of \\( |S \\cap I| \\) for \\(... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6a645e25f93a465b4c68150bd2a153c332a7396523da98cc57438b2b8b5f4e64 | Assume \( x_n \to 0 \) and \( \sum x_n \) diverges. For a given real number \( x \), let \( g: \mathbb{N} \to \mathbb{N} \) be a permutation such that \( \sum_{n=1}^{\infty} x_{g(n)} = x \). Define \( S_k = \sum_{n=1}^{k} x_{g(n)} \). Find the smallest positive integer \( k \) such that \( |S_k - x| < 0.01 \) for \( x ... | [
"1000"
] | [
"Assume \\( x_n \\to 0 \\) and \\( \\sum x_n \\) diverges. For a given real number \\( x \\), let \\( g: \\mathbb{N} \\to \\mathbb{N} \\) be a permutation such that \\( \\sum_{n=1}^{\\infty} x_{g(n)} = x \\). Define \\( S_k = \\sum_{n=1}^{k} x_{g(n)} \\). Find the smallest positive integer \\( k \\) such that \\( |... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 3
} | 10 | 0 | 0 |
omr-ws-e6fa9ef87a4b53a573271001fb4b50c3e82d9fcf5103cfac40091aa5c3b993b8 | Assume \((1 + x)^{p-2} = 1 + a_1 x + a_2 x^2 + \cdots + a_{p-2} x^{p-2}\) where \(p\) is an odd prime number. Calculate the remainder when \(a_1 + 2 + a_2 - 3 + a_3 + 4 + \cdots + a_{p-3} - (p-2) + a_{p-2} + (p-1)\) is divided by \(p\). | [
"0"
] | [
"Assume \\((1 + x)^{p-2} = 1 + a_1 x + a_2 x^2 + \\cdots + a_{p-2} x^{p-2}\\) where \\(p\\) is an odd prime number. Calculate the remainder when \\(a_1 + 2 + a_2 - 3 + a_3 + 4 + \\cdots + a_{p-3} - (p-2) + a_{p-2} + (p-1)\\) is divided by \\(p\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-49eb8d920d07f44c702853597e3cf6499fdde8e695d458125153ff5e00b4edad | Assume \(0 < \alpha < \frac{\pi}{2}\) and \(\cos\left(\alpha + \frac{\pi}{6}\right) = \frac{4}{5}\). Determine the smallest integer \(R\) such that \(100\sin\left(2\alpha + \frac{\pi}{12}\right) < R\). | [
"49"
] | [
"Assume \\(0 < \\alpha < \\frac{\\pi}{2}\\) and \\(\\cos\\left(\\alpha + \\frac{\\pi}{6}\\right) = \\frac{4}{5}\\). Determine the smallest integer \\(R\\) such that \\(100\\sin\\left(2\\alpha + \\frac{\\pi}{12}\\right) < R\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-1c37d2cf02c193c8bf9bebd11bfad66f661846fe6567c2341536bc470203a8eb | Assume \(0 < \beta < \gamma \leq 1\). Given a function \(u\) such that \(\|u\|_{C^{0,\beta}(U)} = 2\) and \(\|u\|_{C^{0,1}(U)} = 3\), calculate the maximum possible value of \(\|u\|_{C^{0,\gamma}(U)}\). | [
"2^{\\frac{1 - \\gamma}{1 - \\beta}} \\cdot 3^{\\frac{\\gamma - \\beta}{1 - \\beta}}"
] | [
"Assume \\(0 < \\beta < \\gamma \\leq 1\\). Given a function \\(u\\) such that \\(\\|u\\|_{C^{0,\\beta}(U)} = 2\\) and \\(\\|u\\|_{C^{0,1}(U)} = 3\\), calculate the maximum possible value of \\(\\|u\\|_{C^{0,\\gamma}(U)}\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-25fd2fe41ded082ab29cae4e978862b748a5217b81a29d8ce89eaa37e294ee02 | Assume \(1 \le p < \infty\). Suppose \(f_k, f \in L^p\) such that \(f_k \to f\) almost everywhere (a.e.) and \(\|f_k\|_p \to \|f\|_p\). Calculate the limit \(\lim_{k \to \infty} \|f_k - f\|_p^p\). | [
"0"
] | [
"Assume \\(1 \\le p < \\infty\\). Suppose \\(f_k, f \\in L^p\\) such that \\(f_k \\to f\\) almost everywhere (a.e.) and \\(\\|f_k\\|_p \\to \\|f\\|_p\\). Calculate the limit \\(\\lim_{k \\to \\infty} \\|f_k - f\\|_p^p\\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d978ef07d7bb47af875a4bf2fb4190327180bc28402d2fff5c20e82bf3e581f9 | Assume \(ABC\) is an isosceles triangle with \(AB = AC\). Suppose \(P\) is a point on the extension of side \(BC\). Points \(X\) and \(Y\) are on \(AB\) and \(AC\) respectively such that \(PX \parallel AC\) and \(PY \parallel AB\). Let \(T\) be the midpoint of arc \(BC\) not containing \(A\). If \(PT\) intersects \(XY\... | [
"1"
] | [
"Assume \\(ABC\\) is an isosceles triangle with \\(AB = AC\\). Suppose \\(P\\) is a point on the extension of side \\(BC\\). Points \\(X\\) and \\(Y\\) are on \\(AB\\) and \\(AC\\) respectively such that \\(PX \\parallel AC\\) and \\(PY \\parallel AB\\). Let \\(T\\) be the midpoint of arc \\(BC\\) not containing \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-5ecfb2e6264400f5364c24a083c39d41b261f59a67060c0ec28350fa0f289d54 | Assume \(ABC\) is an isosceles triangle with \(AB = AC\). Suppose \(P\) is a point on the extension of side \(BC\). Points \(X\) and \(Y\) are on \(AB\) and \(AC\) respectively such that: \[ PX \parallel AC \quad \text{and} \quad PY \parallel AB. \] Also, \(T\) is the midpoint of arc \(BC\). Let \(Z\) be the intersecti... | [
"\\dfrac{31\\sqrt{85}}{34}"
] | [
"Assume \\(ABC\\) is an isosceles triangle with \\(AB = AC\\). Suppose \\(P\\) is a point on the extension of side \\(BC\\). Points \\(X\\) and \\(Y\\) are on \\(AB\\) and \\(AC\\) respectively such that:\n\\[\nPX \\parallel AC \\quad \\text{and} \\quad PY \\parallel AB.\n\\]\nAlso, \\(T\\) is the midpoint of arc \... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c217ee2dcab888752ee35ce47756f3a5bbf8e03c8c6ba55d8922d53777de469d | Assume \(ABC\) is an isosceles triangle with \(AB = AC\). Suppose \(P\) is a point on the extension of side \(BC\). Points \(X\) and \(Y\) are on \(AB\) and \(AC\) respectively such that: \[PX \parallel AC \quad \text{and} \quad PY \parallel AB.\] Also, \(T\) is the midpoint of arc \(BC\) not containing \(A\). Let \(PT... | [
"\\dfrac{8}{17}"
] | [
"Assume \\(ABC\\) is an isosceles triangle with \\(AB = AC\\). Suppose \\(P\\) is a point on the extension of side \\(BC\\). Points \\(X\\) and \\(Y\\) are on \\(AB\\) and \\(AC\\) respectively such that:\n\\[PX \\parallel AC \\quad \\text{and} \\quad PY \\parallel AB.\\]\nAlso, \\(T\\) is the midpoint of arc \\(BC... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-4a1956f2a5f372b593c747f13c6b92ee6d34f3972aa477e2ea4fa9c386804817 | Assume \(V\), \(W\), and \(Z\) are finite-dimensional vector spaces. Let \(T \in L(V, W)\) and \(U \in L(W, Z)\). If \(\dim V = 10\), \(\dim W = 8\), and \(\dim Z = 5\), and \(\dim \ker U = 3\) and \(\dim \ker T = 4\), calculate the maximum possible value of \(\dim \ker UT\). | [
"7"
] | [
"Assume \\(V\\), \\(W\\), and \\(Z\\) are finite-dimensional vector spaces. Let \\(T \\in L(V, W)\\) and \\(U \\in L(W, Z)\\). If \\(\\dim V = 10\\), \\(\\dim W = 8\\), and \\(\\dim Z = 5\\), and \\(\\dim \\ker U = 3\\) and \\(\\dim \\ker T = 4\\), calculate the maximum possible value of \\(\\dim \\ker UT\\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4eae9f5c987933bb662a3aaccea87161148b3af24f4cd0201f9798df412cc042 | Assume \(\frac{a+b-c}{3c} = \frac{p}{q}\), where \(p\) and \(q\) are coprime, and denote by \(d\) the \(\gcd\) of \(p(3p+2q)\) and \(q(2p+q)\). Compute \(a\), \(b\), \(c\) in terms of \(p\), \(q\), \(d\). | [
"c = \\dfrac{q}{d}"
] | [
"Assume \\(\\frac{a+b-c}{3c} = \\frac{p}{q}\\), where \\(p\\) and \\(q\\) are coprime, and denote by \\(d\\) the \\(\\gcd\\) of \\(p(3p+2q)\\) and \\(q(2p+q)\\). Compute \\(a\\), \\(b\\), \\(c\\) in terms of \\(p\\), \\(q\\), \\(d\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-1122ca3378189ad247745414afefd7e61a57d9a4c6ddb9e7117df2bdc082527e | Assume \(a, b, c > 0\). Calculate the minimum value of the expression: \[ \frac{\left( \sum \sqrt{bc(b+c)} \right)^2 (a+b+c)}{(ab+bc+ca)^2} \] | [
"4"
] | [
"Assume \\(a, b, c > 0\\). Calculate the minimum value of the expression:\n\\[\n\\frac{\\left( \\sum \\sqrt{bc(b+c)} \\right)^2 (a+b+c)}{(ab+bc+ca)^2}\n\\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-7314e9005457c60c7ddac04c6c6ba187b6473d0a2afecfd5e10df56d3e2aff17 | Assume \(a, b, c\) are positive numbers such that \(a + b + c = 1\). Calculate the maximum value of the expression \(4(ab + bc + ca) - 9abc\). | [
"1"
] | [
"Assume \\(a, b, c\\) are positive numbers such that \\(a + b + c = 1\\). Calculate the maximum value of the expression \\(4(ab + bc + ca) - 9abc\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4dbd06fd1d6f2114291f85229d7f80ffb3195d41cc1767598166312d7a508c73 | Assume \(a, b, c\) are real numbers such that \(a + b + c = ab + bc + ca\). Calculate the minimum value of the expression \[ \frac{a}{a^2 + 1} + \frac{b}{b^2 + 1} + \frac{c}{c^2 + 1}. \] | [
"-\\dfrac{1}{2}"
] | [
"Assume \\(a, b, c\\) are real numbers such that \\(a + b + c = ab + bc + ca\\). Calculate the minimum value of the expression\n\\[\n\\frac{a}{a^2 + 1} + \\frac{b}{b^2 + 1} + \\frac{c}{c^2 + 1}.\n\\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-047f742e5b8eecd1897e2c9a8ec1398e99862b122fab4e3b7c15201da339a46a | Assume \(a, b, x, y > 0\) and \[3ax^2 + \frac{5}{2}ax^{\frac{3}{2}} \le 3by^2 + \frac{5}{2}by^{\frac{3}{2}} + 6axy + 5ax^{\frac{1}{2}}y + 6bxy + \frac{15}{4}bxy^{\frac{1}{2}}.\] Find the unique positive root \(l\) of the equation \(l^4 - 2l^3 + l^2 - 2l + 1 = 0\) such that \(x \le ly\). Calculate the value of \(l^2 + l... | [
"no_valid_answer_found"
] | [
"Assume \\(a, b, x, y > 0\\) and \n\\[3ax^2 + \\frac{5}{2}ax^{\\frac{3}{2}} \\le 3by^2 + \\frac{5}{2}by^{\\frac{3}{2}} + 6axy + 5ax^{\\frac{1}{2}}y + 6bxy + \\frac{15}{4}bxy^{\\frac{1}{2}}.\\]\nFind the unique positive root \\(l\\) of the equation \\(l^4 - 2l^3 + l^2 - 2l + 1 = 0\\) such that \\(x \\le ly\\). Calcu... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-40be56400dc31c7a0179fa906480b89a223965c68490b05677d36f337db46277 | Assume \(a\) and \(b\) are positive integers such that \(p = \frac{b}{4} \sqrt{\frac{2a - b}{2a + b}}\) is a prime. Find the maximum value of \(p\). | [
"5"
] | [
"Assume \\(a\\) and \\(b\\) are positive integers such that \\(p = \\frac{b}{4} \\sqrt{\\frac{2a - b}{2a + b}}\\) is a prime. Find the maximum value of \\(p\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-38715303b75e304a2dc453cac53e018fc20b102936e3ff9d02a0885083932899 | Assume \(a\) and \(b\) are real numbers such that \(0 < a < b\), and let \(\textsf{K}_{a,b}\) be the set of all non-negative, monotonically decreasing functions \(f\) satisfying \(\int_a^b f(t) \, dt = 1\) and \(af(a) = bf(b)\) on the interval \([a, b]\). Find the value of \(\sup \left\{ \int_a^b \max\{f(t), g(t)\} \, ... | [
"\\dfrac{2\\sqrt{b}}{\\sqrt{a} + \\sqrt{b}}"
] | [
"Assume \\(a\\) and \\(b\\) are real numbers such that \\(0 < a < b\\), and let \\(\\textsf{K}_{a,b}\\) be the set of all non-negative, monotonically decreasing functions \\(f\\) satisfying \\(\\int_a^b f(t) \\, dt = 1\\) and \\(af(a) = bf(b)\\) on the interval \\([a, b]\\). Find the value of \\(\\sup \\left\\{ \\i... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-c6716832aacbe22b12bd582078d89e2b5a99a79613645a139d3364d9fe64e8fe | Assume \(a\) and \(b\) are strictly positive real numbers such that \(a + b = ab\). Calculate the minimum value of the expression \(\frac{a}{b^2 + 4} + \frac{b}{a^2 + 4}\). | [
"\\dfrac{1}{2}"
] | [
"Assume \\(a\\) and \\(b\\) are strictly positive real numbers such that \\(a + b = ab\\). Calculate the minimum value of the expression \\(\\frac{a}{b^2 + 4} + \\frac{b}{a^2 + 4}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-277f0b8c298f87e9028d8f18607d8570fc23c1c8d5c481d413fdaad4671712c4 | Assume \(a_i > 0\), \(x_i > 0\), and \(\sum_{1 \leq i < j \leq n} x_i x_j = \frac{(n-1)n}{2}\). Calculate the minimum possible value of \(\sum_{i=1}^n \left(\frac{a_i}{\sum_{j \neq i} a_j} \sum_{j \neq i} x_j \right)\) for \(n = 5\). | [
"5"
] | [
"Assume \\(a_i > 0\\), \\(x_i > 0\\), and \\(\\sum_{1 \\leq i < j \\leq n} x_i x_j = \\frac{(n-1)n}{2}\\). Calculate the minimum possible value of \\(\\sum_{i=1}^n \\left(\\frac{a_i}{\\sum_{j \\neq i} a_j} \\sum_{j \\neq i} x_j \\right)\\) for \\(n = 5\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-557f02d24a6d57851ca9066acfcf4f04ec16ea5646f538aeed498acef37bddaa | Assume \(n\) is a positive integer. Consider sequences \(a_0, a_1, \ldots, a_n\) for which \(a_i \in \{1, 2, \ldots, n\}\) for all \(i\) and \(a_n = a_0\). (a) Suppose \(n\) is odd. Find the number of such sequences if \(a_i - a_{i-1} \not\equiv i \pmod{n}\) for all \(i = 1, 2, \ldots, n\). (b) Suppose \(n\) is an odd ... | [
"(n-1)^n - (n-1)"
] | [
"Assume \\(n\\) is a positive integer. Consider sequences \\(a_0, a_1, \\ldots, a_n\\) for which \\(a_i \\in \\{1, 2, \\ldots, n\\}\\) for all \\(i\\) and \\(a_n = a_0\\).\n\n(a) Suppose \\(n\\) is odd. Find the number of such sequences if \\(a_i - a_{i-1} \\not\\equiv i \\pmod{n}\\) for all \\(i = 1, 2, \\ldots, n... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-35f05f6a96ef04d408b713cab1f6e1af5646661604d1696a4a5f244c6b84bc31 | Assume \(x, y, z\) are positive real numbers such that \(x + y + z = 3\). Calculate the minimum value of the expression \(\sum{\frac{x^{3}}{y^{3}+8}} - \frac{1}{9} - \frac{2}{27}(xy + yz + zx)\). | [
"0"
] | [
"Assume \\(x, y, z\\) are positive real numbers such that \\(x + y + z = 3\\). Calculate the minimum value of the expression \\(\\sum{\\frac{x^{3}}{y^{3}+8}} - \\frac{1}{9} - \\frac{2}{27}(xy + yz + zx)\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d39d58a958eba9fd2dcdc7fbc2888f3c72c9441e20d57662ab49fda1e2d44213 | Assume a Gaussian random variable with mean = 1 and variance = 1. We create \(N\) samples \(\chi_1, \ldots, \chi_N\) with this distribution and try to calculate the average (which is 1) by using the arithmetic average \(\mu = \frac{\chi_1 + \ldots + \chi_N}{N}\). a) \(\mu\) is a random variable. Find the mean and the v... | [
"- Mean of \\(\\mu\\): 1\n- Variance of \\(\\mu\\): \\(\\frac{1}{N}\\)\n- For one decimal digit accuracy, \\(N > 100\\)."
] | [
"Assume a Gaussian random variable with mean = 1 and variance = 1. We create \\(N\\) samples \\(\\chi_1, \\ldots, \\chi_N\\) with this distribution and try to calculate the average (which is 1) by using the arithmetic average \\(\\mu = \\frac{\\chi_1 + \\ldots + \\chi_N}{N}\\).\na) \\(\\mu\\) is a random variable. ... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 17
} | 19 | 0 | 0 |
omr-ws-ff13ae572d368102e6d1f80502634d79875d625d82da14e32a10c40b7ede47ef | Assume a positive sequence $\{x_n\}$ satisfies $$x_{n+1}=x_n+\frac{n}{x_1+x_2+\cdots+x_n}.$$ Find $\lim_{n \to \infty}\frac{x_n^2}{n}$. | [
"3"
] | [
"Assume a positive sequence $\\{x_n\\}$ satisfies $$x_{n+1}=x_n+\\frac{n}{x_1+x_2+\\cdots+x_n}.$$ Find $\\lim_{n \\to \\infty}\\frac{x_n^2}{n}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 1
} | 15 | 0 | 0 |
omr-ws-35e71ce812d198880e8e244828f8bd4671f5bbf7b0792e8a49423a78cc9a6f15 | Assume a random variable \(\psi = \chi_1 + \chi_2 + \ldots + \chi_N\) where the random variables \(\chi_i\) are independent with probability density functions \(f_i(\chi)\). The number of summands \(N\) is also a random variable with a-priori probabilities \(P\{N=n\} = \pi_n\). a) Calculate the density probability \(f_... | [
"f_\\psi(y) = (1 - p) e^{-(1 - p) y} u(y)"
] | [
"Assume a random variable \\(\\psi = \\chi_1 + \\chi_2 + \\ldots + \\chi_N\\) where the random variables \\(\\chi_i\\) are independent with probability density functions \\(f_i(\\chi)\\). The number of summands \\(N\\) is also a random variable with a-priori probabilities \\(P\\{N=n\\} = \\pi_n\\).\na) Calculate th... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-6f6ad20410ceb373283635c794cdbab27c4b4b0e69a2826295f412694aa3ced2 | Assume a solution of the form \( u(x, t) = X(x)T(t) \) to the modified diffusion equation \( u_t - Du_{xx} - au = 0 \). Find the expression for \( X(x) \) and \( T(t) \), and then determine the value of \( k \) if \( X(x) = e^{kx} \) and \( T(t) = e^{rt} \). | [
"\\pm \\sqrt{\\dfrac{r - a}{D}}"
] | [
"Assume a solution of the form \\( u(x, t) = X(x)T(t) \\) to the modified diffusion equation \\( u_t - Du_{xx} - au = 0 \\). Find the expression for \\( X(x) \\) and \\( T(t) \\), and then determine the value of \\( k \\) if \\( X(x) = e^{kx} \\) and \\( T(t) = e^{rt} \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0ff53e700ac5b2bb135a918a6dac50c0073ca72e1af8652f63fe60418542dd9a | Assume a spacecraft is launched vertically upwards from the Earth and at height \( y = h > 0 \) when it ran out of gas has the speed \( v_0 > 0 \) at time \( t = 0 \). Ignoring all forces except Earth's gravitational force, Newton's second law gives us the equation: \[ \frac{d^2y}{dt^2} = -\frac{gR^2}{(y+R)^2}, \] wher... | [
"\\sqrt{\\dfrac{2gR^2}{R + h}}"
] | [
"Assume a spacecraft is launched vertically upwards from the Earth and at height \\( y = h > 0 \\) when it ran out of gas has the speed \\( v_0 > 0 \\) at time \\( t = 0 \\). Ignoring all forces except Earth's gravitational force, Newton's second law gives us the equation:\n\\[ \\frac{d^2y}{dt^2} = -\\frac{gR^2}{(y... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ee9ce09aacb7d21db1934ee29d12a5eb41940eeaa7228dd5528aea89b7de1abb | Assume a triangle \(ABC\). Points \(D\), \(E\), \(X\), \(Y\), \(Z\), \(T\), \(M\), and \(N\) are such that \(ACDE\), \(ABXY\), \(BCZT\), and \(ZTMN\) are squares. Let \(O\) be the circumcenter of square \(ZTMN\). Calculate the ratio of the area of triangle \(AOE\) to the area of triangle \(YZT\). | [
"\\dfrac{1}{2}"
] | [
"Assume a triangle \\(ABC\\). Points \\(D\\), \\(E\\), \\(X\\), \\(Y\\), \\(Z\\), \\(T\\), \\(M\\), and \\(N\\) are such that \\(ACDE\\), \\(ABXY\\), \\(BCZT\\), and \\(ZTMN\\) are squares. Let \\(O\\) be the circumcenter of square \\(ZTMN\\). Calculate the ratio of the area of triangle \\(AOE\\) to the area of tri... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-dd683d127446c57983d76418d704db0fb762e0469baf5d85db7dbe95e22df00a | Assume each letter in the following formula represents a unique digit. Find the number represented by the digits DAMON in the equation: \[ \text{NOMAD} \times 4 = \text{DAMON} \] | [
"87912"
] | [
"Assume each letter in the following formula represents a unique digit. Find the number represented by the digits DAMON in the equation:\n\n\\[ \\text{NOMAD} \\times 4 = \\text{DAMON} \\]"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-54b2a27ffea68568baa456aab961867596bf5ea017f6768888595913fdc164c2 | Assume integer \( m \geq 2 \). There are \( 3m \) people in a meeting, and any two of them either shake hands with each other once or not. We call the meeting "$n$-interesting" if there exists \( n \) (where \( n \leq 3m-1 \)) people such that the number of handshakes each of these \( n \) people makes with the other \... | [
"\\( 2m+1 \\)"
] | [
"Assume integer \\( m \\geq 2 \\). There are \\( 3m \\) people in a meeting, and any two of them either shake hands with each other once or not. We call the meeting \"$n$-interesting\" if there exists \\( n \\) (where \\( n \\leq 3m-1 \\)) people such that the number of handshakes each of these \\( n \\) people mak... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 25,
"genselect": 4
} | 29 | 0 | 0 |
omr-ws-7d5a5aa79c4530d2c84548e5309b94b533707269652b3db9acb6e134655dbafc | Assume natural number \( n \ge 2 \). Amin and Ali take turns playing the following game: In each step, the player whose turn has come chooses an index \( i \) from the set \(\{0, 1, \cdots, n\}\), such that none of the two players had chosen this index in the previous turns; also, this player chooses a nonzero rational... | [
"\\( n \\) is even."
] | [
"Assume natural number \\( n \\ge 2 \\). Amin and Ali take turns playing the following game: In each step, the player whose turn has come chooses an index \\( i \\) from the set \\(\\{0, 1, \\cdots, n\\}\\), such that none of the two players had chosen this index in the previous turns; also, this player chooses a n... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 31,
"genselect": 5,
"tir": 6
} | 42 | 0 | 0 |
omr-ws-e873d0c563b8294f9e0217561d1ab86824728d58c51a9f67e9aaeaf4fccf826d | Assume now to observe a counting process $N_t$ associated with the Poisson process at a sequence of equidistant points $(\tau_i)_{i \geq 0}$, where $\tau_i = \tau_{i-1} + \delta$ for $i > 1$ and $\tau_0 = 0$. Derive the maximum likelihood estimator of $\lambda$ using the observations of $N_t$ only and find its asymptot... | [
"\\(\\hat{\\lambda} = \\frac{N_{\\tau_n}}{n \\delta}\\) and \\(\\hat{\\lambda} \\sim \\mathcal{N}\\left(\\lambda, \\frac{\\lambda}{n \\delta}\\right)\\)"
] | [
"Assume now to observe a counting process $N_t$ associated with the Poisson process at a sequence of equidistant points $(\\tau_i)_{i \\geq 0}$, where $\\tau_i = \\tau_{i-1} + \\delta$ for $i > 1$ and $\\tau_0 = 0$. Derive the maximum likelihood estimator of $\\lambda$ using the observations of $N_t$ only and find ... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 1,
"tir": 20
} | 29 | 0 | 0 |
omr-ws-080758c167209ee03e6475b530ded4197e59b986617526ba5e4f4f9ef0b31680 | Assume one year has $365$ days. What is the probability that out of $20$ people chosen at random, two people have a birthday on the same day? | [
"0.411"
] | [
"Assume one year has $365$ days. What is the probability that out of $20$ people chosen at random, two people have a birthday on the same day?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dae999113fbce756e2ad9d5734f81e28bc3f702bd37bbfd19127c0461f0da677 | Assume sets \( X \) and \( Y \) each contain \( p \) non-negative integers, where \( p \) is a prime. Denote \( X+Y = \{x+y \mid x \in X, y \in Y \} \). If \( X+Y = \{0, 1, 2, \ldots, p^2 - 1 \} \), find the sum of all possible values of \( x_0 + y_0 \), where \( x_0 \) is the smallest element in \( X \) and \( y_0 \) ... | [
"0"
] | [
"Assume sets \\( X \\) and \\( Y \\) each contain \\( p \\) non-negative integers, where \\( p \\) is a prime. Denote \\( X+Y = \\{x+y \\mid x \\in X, y \\in Y \\} \\). If \\( X+Y = \\{0, 1, 2, \\ldots, p^2 - 1 \\} \\), find the sum of all possible values of \\( x_0 + y_0 \\), where \\( x_0 \\) is the smallest elem... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-364635231f96e240121ab21f5a5908d05123ff51525b2d41247ce8d6600a7a8d | Assume that $(G,+)$ is an Abelian group with $k$ elements. Let $B$ and $C$ be two subsets of $G$ such that: (1) $0 \in B \cap C$. (2) If $b \in B$ and $c \in C$ satisfy $b + c = 0$, then $b = c = 0$. Given $|B| = 15$ and $|C| = 10$, find the minimum possible value of $|B + C|$ if $k = 20$. | [
"20"
] | [
"Assume that $(G,+)$ is an Abelian group with $k$ elements. Let $B$ and $C$ be two subsets of $G$ such that:\n(1) $0 \\in B \\cap C$.\n(2) If $b \\in B$ and $c \\in C$ satisfy $b + c = 0$, then $b = c = 0$.\nGiven $|B| = 15$ and $|C| = 10$, find the minimum possible value of $|B + C|$ if $k = 20$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-af551c1202693d5f13ea73fdbde6fc67c0aaa5796f0c289fef18e249b7f3efc1 | Assume that $(n,k)=1$ and $t \leq n$, where $n$, $k$, and $t$ are natural numbers. What is the maximum number of $k$-element subsets $S$ of the set $\{1, 2, \ldots, t\}$ such that $n$ divides $\sum_{i \in S} x_i$? Here, the maximum is taken over all sequences $(x_1, \ldots, x_n)$ of integer numbers which are not multip... | [
"\\dbinom{t-1}{k-1}"
] | [
"Assume that $(n,k)=1$ and $t \\leq n$, where $n$, $k$, and $t$ are natural numbers. What is the maximum number of $k$-element subsets $S$ of the set $\\{1, 2, \\ldots, t\\}$ such that $n$ divides $\\sum_{i \\in S} x_i$? Here, the maximum is taken over all sequences $(x_1, \\ldots, x_n)$ of integer numbers which ar... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-0da222a24889f1720b2103bf77f38d0addf9c67d63982e6d0c191e8c68c5f90e | Assume that $(x-1)^{2024} + (x-1)^{2023} + \cdots + (x-1)^2 = a_{2024}x^{2024} + a_{2023}x^{2023} + \cdots + a_1 x + a_0$. If $3 (a_{2024}-a_{2023}+a_{2022}-a_{2021}+\cdots+a_{2}-a_{1}) = 2^m$, find $\lfloor m \rfloor$. | [
"2025"
] | [
"Assume that $(x-1)^{2024} + (x-1)^{2023} + \\cdots + (x-1)^2 = a_{2024}x^{2024} + a_{2023}x^{2023} + \\cdots + a_1 x + a_0$. If $3 (a_{2024}-a_{2023}+a_{2022}-a_{2021}+\\cdots+a_{2}-a_{1}) = 2^m$, find $\\lfloor m \\rfloor$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-ed9fad1a7bd53694407a46a254a2a1715702bc60ccecece8b97e5d41668bc94f | Assume that $A, B, C, D$ are four people in a competition with corresponding powers $a, b, c, d$. The probability for $X$ to defeat $Y$ is $\frac{x}{x+y}$, where $x$ and $y$ are the powers of $X$ and $Y$, respectively. $A$ and $B$ will compete first, then $C$ and $D$ will compete in the second match. The winner of each... | [
"\\(\\frac{a}{a+b} \\cdot \\left( \\frac{c}{c+d} \\cdot \\frac{a}{a+c} + \\frac{d}{c+d} \\cdot \\frac{a}{a+d} \\right)\\)"
] | [
"Assume that $A, B, C, D$ are four people in a competition with corresponding powers $a, b, c, d$. The probability for $X$ to defeat $Y$ is $\\frac{x}{x+y}$, where $x$ and $y$ are the powers of $X$ and $Y$, respectively. $A$ and $B$ will compete first, then $C$ and $D$ will compete in the second match. The winner o... | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-290693b0a0c589e5322681e1a2b1a8610dd14d609d29f1d9b03dd7d64e270937 | Assume that $ABECFD$ is a complete quadrilateral with $AC \cap BD = G$. The Miquel points of the complete quadrilaterals $ABECFD$, $ADFCBG$, and $ABECDG$ are $L$, $M$, and $N$, respectively. Let $LM \cap EG = U$ and $LN \cap FG = V$. Point $H'$ satisfies $H'M \perp FG$ and $H'N \perp EG$. If the perpendicular lines fro... | [
"10"
] | [
"Assume that $ABECFD$ is a complete quadrilateral with $AC \\cap BD = G$. The Miquel points of the complete quadrilaterals $ABECFD$, $ADFCBG$, and $ABECDG$ are $L$, $M$, and $N$, respectively. Let $LM \\cap EG = U$ and $LN \\cap FG = V$. Point $H'$ satisfies $H'M \\perp FG$ and $H'N \\perp EG$. If the perpendicular... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-74274d5efe071815c8c2f817e80cbcf490fcccbc9b6c23177215e7713352610e | Assume that $M$ and $N$ are the midpoints of the edges $AA_1$ and $CC_1$ of the cube $ABCDA_1B_1C_1D_1$, and a point $P$ is taken on the extension of the edge $D_1D$ beyond the point $D$ such that $|DP|=1/2m$. A plane is drawn through the points $M$, $N$ and $P$. Find the area of the section if the edge of the cube is ... | [
"$\\frac{3\\sqrt 3}{4}$"
] | [
"Assume that $M$ and $N$ are the midpoints of the edges $AA_1$ and $CC_1$ of the cube $ABCDA_1B_1C_1D_1$, and a point $P$ is taken on the extension of the edge $D_1D$ beyond the point $D$ such that $|DP|=1/2m$. A plane is drawn through the points $M$, $N$ and $P$. Find the area of the section if the edge of the cub... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-53cdf1cdd30341ed0c27729209d91e376243ae813328e998615ff582c17528ef | Assume that $O$, $I$, and $I_a$ are the circumcenter, incenter, and the excenter corresponding to the edge $BC$ of the triangle $ABC$, respectively. Let $II_a$ intersect the segment $BC$ and the circumcircle $(O)$ at $A_1$ and $M$, respectively. Let $N$ be the midpoint of the arc $MCA$ and $S$, $T$ be the intersections... | [
"2"
] | [
"Assume that $O$, $I$, and $I_a$ are the circumcenter, incenter, and the excenter corresponding to the edge $BC$ of the triangle $ABC$, respectively. Let $II_a$ intersect the segment $BC$ and the circumcircle $(O)$ at $A_1$ and $M$, respectively. Let $N$ be the midpoint of the arc $MCA$ and $S$, $T$ be the intersec... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-177fc0d3cf5f3dae61e4be0c9959952959b9c9854d783673c50f09daf8aaa924 | Assume that $U$ is bounded and there exists a smooth vector field $\alpha$ such that $\alpha\cdot\nu\geq 1$ along $\partial U$, where $\nu$ denotes the outward unit normal. Assume $1\leq p<\infty$. Let $u\in C^{1}(\bar{U})$ and apply the Gauss-Green Theorem to $\int_{\partial U}|u|^{p}\alpha\cdot\nu\ dS$. Determine the... | [
"C = \\sup_{x \\in U} \\left( \\text{div}\\, \\alpha(x) + (p - 1)p^{\\frac{1}{p - 1}} |\\alpha(x)|^{\\frac{p}{p - 1}} \\right)"
] | [
"Assume that $U$ is bounded and there exists a smooth vector field $\\alpha$ such that $\\alpha\\cdot\\nu\\geq 1$ along $\\partial U$, where $\\nu$ denotes the outward unit normal. Assume $1\\leq p<\\infty$. Let $u\\in C^{1}(\\bar{U})$ and apply the Gauss-Green Theorem to $\\int_{\\partial U}|u|^{p}\\alpha\\cdot\\n... | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-3a0995b9e38e0d617d3ac1bbdce9c71b4ad8068d635558ab95468a12b7631dbe | Assume that $X$ and $Y$ are two fixed points that lie in the interior of the circle $(O)$. $H$ is a fixed point which lies on the line $XY$. A line $d$ passes through $H$ and cuts $(O)$ at $M$ and $N$ ($M \in HN$). $MX$ and $NY$ cut $(O)$ at $P$ and $Q$ respectively. Let $T$ be the intersection point of $PQ$ and $XY$. ... | [
"\\dfrac{XH}{HY}"
] | [
"Assume that $X$ and $Y$ are two fixed points that lie in the interior of the circle $(O)$. $H$ is a fixed point which lies on the line $XY$. A line $d$ passes through $H$ and cuts $(O)$ at $M$ and $N$ ($M \\in HN$). $MX$ and $NY$ cut $(O)$ at $P$ and $Q$ respectively. Let $T$ be the intersection point of $PQ$ and ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-3ab85f3fe085d8850721c128f9fa8f92c6988754f8197f08cf675677c5c06631 | Assume that $\alpha$ is positive and that $x_n$ is the only root in the interval $\left[n, \infty\right]$ of the equation \[ \frac{1}{x} + \frac{2x}{x^2-1^2} + \frac{2x}{x^2-2^2} + \cdots + \frac{2x}{x^2-n^2} = \alpha. \] Find \[ \lim_{n \to \infty} \left( x_n - \left( n + \frac{1}{2} \right) \left( \frac{1 + e^{-\alph... | [
"0"
] | [
"Assume that $\\alpha$ is positive and that $x_n$ is the only root in the interval $\\left[n, \\infty\\right]$ of the equation \n\\[\n\\frac{1}{x} + \\frac{2x}{x^2-1^2} + \\frac{2x}{x^2-2^2} + \\cdots + \\frac{2x}{x^2-n^2} = \\alpha.\n\\]\nFind \n\\[\n\\lim_{n \\to \\infty} \\left( x_n - \\left( n + \\frac{1}{2} \\... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-d8525b5ba7e3bc2c799d7e375db67cc15b5a0dddd72df1d99ff9fbbd8980efc4 | Assume that $\alpha, n, p$ are positive integers and $p$ is a prime number. Let $m$ be an integer that is one less than a multiple of $p$. Calculate the largest power of $p$ that divides $\sum_{k=0}^{\left[ \frac{n}{p^\alpha} \right]} \binom{n}{p^{\alpha}k} m^k$ for $n = 1000$, $\alpha = 2$, and $p = 5$. | [
"0"
] | [
"Assume that $\\alpha, n, p$ are positive integers and $p$ is a prime number. Let $m$ be an integer that is one less than a multiple of $p$. Calculate the largest power of $p$ that divides $\\sum_{k=0}^{\\left[ \\frac{n}{p^\\alpha} \\right]} \\binom{n}{p^{\\alpha}k} m^k$ for $n = 1000$, $\\alpha = 2$, and $p = 5$."... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b12fb8b8218e6cbe9d0422d47770fff3830f0376a308d4dccc844f608aee0386 | Assume that $\frac{n(n+1)}{3}$ is a perfect square. If $n$ is the smallest positive integer such that $\frac{n(n+1)}{3} = k^2$ for some integer $k$, find the value of $\frac{n}{3} + k^2$. | [
"5"
] | [
"Assume that $\\frac{n(n+1)}{3}$ is a perfect square. If $n$ is the smallest positive integer such that $\\frac{n(n+1)}{3} = k^2$ for some integer $k$, find the value of $\\frac{n}{3} + k^2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dd7587009d94e982e11fd9e0c2bebb3754684990cfb288bd749197e5dfe2f85a | Assume that $\int_0^{\frac{\pi}{2}} |f(x)|\,dx < +\infty$ and $\lim_{x\rightarrow \frac{\pi}{2}-}f(x)=A$. Let $L_n = \frac{\int_0^{\frac{\pi}{2}} \sin^nx f(x)\,dx}{\int_0^{\frac{\pi}{2}} \sin^nx\,dx}$. Calculate the value of $L_{1000}$ given that $A = 2$. | [
"2"
] | [
"Assume that $\\int_0^{\\frac{\\pi}{2}} |f(x)|\\,dx < +\\infty$ and $\\lim_{x\\rightarrow \\frac{\\pi}{2}-}f(x)=A$. Let $L_n = \\frac{\\int_0^{\\frac{\\pi}{2}} \\sin^nx f(x)\\,dx}{\\int_0^{\\frac{\\pi}{2}} \\sin^nx\\,dx}$. Calculate the value of $L_{1000}$ given that $A = 2$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-106ea86e44c9ab2ed9d38efde3bcc998e7c2cd4514cb41884bad4b62b7806c69 | Assume that $\lim_{x\to0}f(x)=L$. If $\lim_{x\to0}f(x^3)=M$, find the value of $M$. | [
"L"
] | [
"Assume that $\\lim_{x\\to0}f(x)=L$. If $\\lim_{x\\to0}f(x^3)=M$, find the value of $M$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-226d852f326a0182188b4ed8982604aabed3072a113526717c04c7137bdfb85a | Assume that $\sum\limits_{n=1}^{\infty} a_n$ diverges ($a_n>0$). Define $S_n=a_1+a_2+...+a_n$. Calculate the value of $\sum\limits_{n=1}^{1000}\dfrac{a_n}{1+a_n}$ given that $a_n = \frac{1}{n}$ for all $n$. | [
"H_{1001} - 1"
] | [
"Assume that $\\sum\\limits_{n=1}^{\\infty} a_n$ diverges ($a_n>0$). Define $S_n=a_1+a_2+...+a_n$. Calculate the value of $\\sum\\limits_{n=1}^{1000}\\dfrac{a_n}{1+a_n}$ given that $a_n = \\frac{1}{n}$ for all $n$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3f004a216343242f105830b0c72f3573ec534a6db526a22d374a5abe447b2e87 | Assume that $\sum\limits_{n=1}^{\infty} a_n$ diverges ($a_n>0$). Define $S_n=a_1+a_2+...+a_n$. Calculate the value of $\sum\limits_{n=1}^{1000}\dfrac{a_n}{S_n}$ given that $a_n = \frac{1}{n}$ for all $n$. | [
"2"
] | [
"Assume that $\\sum\\limits_{n=1}^{\\infty} a_n$ diverges ($a_n>0$). Define $S_n=a_1+a_2+...+a_n$. Calculate the value of $\\sum\\limits_{n=1}^{1000}\\dfrac{a_n}{S_n}$ given that $a_n = \\frac{1}{n}$ for all $n$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-b6740ac42da5873f2a831cf5727450b3e247031ffb252cea48cdcb5b0c31fe05 | Assume that $\triangle ABC$ and $\triangle DEF$ have the same circumcircle and the same inscribed circle. Let $O$ be their circumcenter and $I$ be their incenter. Let $P = BE \cap CF$, $Q = CF \cap AD$, and $R = AD \cap BE$. Let $J$ be the incenter of $\triangle PQR$. If the distance from $O$ to $I$ is 10 units, calcul... | [
"5"
] | [
"Assume that $\\triangle ABC$ and $\\triangle DEF$ have the same circumcircle and the same inscribed circle. Let $O$ be their circumcenter and $I$ be their incenter. Let $P = BE \\cap CF$, $Q = CF \\cap AD$, and $R = AD \\cap BE$. Let $J$ be the incenter of $\\triangle PQR$. If the distance from $O$ to $I$ is 10 un... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-31096a21ec24d1d339a431a4597ed1c3c9f42fb837b38c6b6cec86c46eb254b2 | Assume that $a > 1$ is a root of the equation $x^3 - x - 1 = 0$. Evaluate $\sqrt[3]{3a^2 - 4a} + \sqrt[3]{3a^2 + 4a + 2}$. | [
"2"
] | [
"Assume that $a > 1$ is a root of the equation $x^3 - x - 1 = 0$. Evaluate $\\sqrt[3]{3a^2 - 4a} + \\sqrt[3]{3a^2 + 4a + 2}$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5fb48ade1455517169c98b6cbb033166c76b193ef7e3fffe3cd83f414f59ec16 | 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 $d - b$.",
"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 $d - b$.\n"
] | [
"MATH_training_set",
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 5
} | 36 | 0 | 0 |
omr-ws-93d03f778ab09a6b05e610e1c2cd3342343791d593fa404335889bfddc66ad96 | Assume that $a_i \geq 1$ for $1 \leq i \leq n$. Define $x_0 = 1$ and $x_k = \frac{1}{1 + a_k x_{k-1}}$ for $1 \leq k \leq n$. Let $A = 1 + \sum_{i=1}^n a_i$. Calculate the minimum possible value of $\sum_{i=1}^n x_i$ when $n=5$ and $a_i = i$ for $1 \leq i \leq 5$. | [
""
] | [
"Assume that $a_i \\geq 1$ for $1 \\leq i \\leq n$. Define $x_0 = 1$ and $x_k = \\frac{1}{1 + a_k x_{k-1}}$ for $1 \\leq k \\leq n$. Let $A = 1 + \\sum_{i=1}^n a_i$. Calculate the minimum possible value of $\\sum_{i=1}^n x_i$ when $n=5$ and $a_i = i$ for $1 \\leq i \\leq 5$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 1 |
omr-ws-7e71764bb7e7c468766e5d283d6c5b635bdc2797e8185b3bb37858c9692bb8ac | Assume that $f$ and $g$ are functions for which $f^{-1}(g(x))=5x+3$. Find $g^{-1}(f(-7))$. | [
"-2"
] | [
"Assume that $f$ and $g$ are functions for which $f^{-1}(g(x))=5x+3$. Find $g^{-1}(f(-7))$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-627ceca12ceaebbc90d8fcbf467749f13ea5b885d99a93643bc4a6603845ac37 | Assume that $f$ is defined on natural numbers such that $f(p) = 1$ for a prime $p$ and $f(ab) = af(b) + bf(a)$. Calculate $f(2023)$. | [
"527"
] | [
"Assume that $f$ is defined on natural numbers such that $f(p) = 1$ for a prime $p$ and $f(ab) = af(b) + bf(a)$. Calculate $f(2023)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-619eb93824c193131c593e367e4f016837ea577f5a83a5069860419c5ec916e5 | Assume that $f$ is defined on natural numbers such that $f(p) = 1$ for a prime $p$ and $f(ab) = af(b) + bf(a)$. Find all numbers $n$ such that $f(n) = n$. | [
"p^p"
] | [
"Assume that $f$ is defined on natural numbers such that $f(p) = 1$ for a prime $p$ and $f(ab) = af(b) + bf(a)$. Find all numbers $n$ such that $f(n) = n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-44565f0634cdecb7cf47d2e3fb207c068a34fc190217e99cb206fd5f020ed5cf | Assume that $f(x)$ is continuous on $[0,1]$ and $\int_{0}^{1}xf(x)dx=0$. Let $p$ be the smallest positive number in $(0,1)$ such that $\int_{0}^{p}f(x)dx=0$. Calculate $\int_{p}^{1}xf(x)dx$. | [
"0"
] | [
"Assume that $f(x)$ is continuous on $[0,1]$ and $\\int_{0}^{1}xf(x)dx=0$. Let $p$ be the smallest positive number in $(0,1)$ such that $\\int_{0}^{p}f(x)dx=0$. Calculate $\\int_{p}^{1}xf(x)dx$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8a0fdf954d4f72a085b72a93540485f347fd2c631f9698166654e4e9c4400bc4 | Assume that $n \ge 3$ people with different names sit around a round table. An unordered pair of people, $M$ and $N$, is called dominating if: 1) they do not sit in adjacent seats, 2) on one or both arcs connecting $M$ and $N$ along the table, all people have names coming alphabetically after $M$ and $N$. Determine the... | [
"n - 3"
] | [
"Assume that $n \\ge 3$ people with different names sit around a round table. An unordered pair of people, $M$ and $N$, is called dominating if:\n1) they do not sit in adjacent seats,\n2) on one or both arcs connecting $M$ and $N$ along the table, all people have names coming alphabetically after $M$ and $N$.\n\nDe... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 8
} | 16 | 0 | 0 |
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