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-7b5ac09394d2e3d2d31c559062896f2cb804e706baeeb533859a22c245b8613d | Which of the following expressions is the smallest? - \( a) \ 10 - 3\sqrt{11} \) - \( b) \ 8 - 3\sqrt{7} \) - \( c) \ 5 - 2\sqrt{6} \) - \( d) \ 9 - 4\sqrt{5} \) - \( e) \ 7 - 4\sqrt{3} \) | [
"\\( 10 - 3\\sqrt{11} \\)"
] | [
"Which of the following expressions is the smallest?\n- \\( a) \\ 10 - 3\\sqrt{11} \\)\n- \\( b) \\ 8 - 3\\sqrt{7} \\)\n- \\( c) \\ 5 - 2\\sqrt{6} \\)\n- \\( d) \\ 9 - 4\\sqrt{5} \\)\n- \\( e) \\ 7 - 4\\sqrt{3} \\)"
] | [
"c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-a4d82833f24bd13ee047b02c25a8fcfab10d1294d328061ef2e249128718507d | Which of the following has the greatest value? A: $2^{90}$ B: $3^{72}$ C: $5^{45}$ D: $11^{27}$ E: $13^{18}$ | [
"B: \\(3^{72}\\)"
] | [
"Which of the following has the greatest value?\nA: $2^{90}$\nB: $3^{72}$\nC: $5^{45}$\nD: $11^{27}$\nE: $13^{18}$"
] | [
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-fdf324199c07d7a1bd3230d54c51c2b26923ce3e4dc0390c72a9dc099f419278 | Which of the following is a factor of \(12a^2 + 13ab - 35b^2\)? a) \(4a + 5b\) b) \(3a - 7b\) c) \(5a - 4b\) d) \(7a + 3b\) e) None of these | [
"None of these"
] | [
"Which of the following is a factor of \\(12a^2 + 13ab - 35b^2\\)?\n\na) \\(4a + 5b\\)\n\nb) \\(3a - 7b\\)\n\nc) \\(5a - 4b\\)\n\nd) \\(7a + 3b\\)\n\ne) None of these"
] | [
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-fc45c2683c1fe331d1d2491601e566f6a0b9b8689f459aaf12ab83cfd64ce0f8 | Which of the following is closest to $\sqrt{65}-\sqrt{63}$? | [
"0.13"
] | [
"Which of the following is closest to $\\sqrt{65}-\\sqrt{63}$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-e2e494ed2bf92560faaa4f29c8095043db200d30a84abfce5e23c24c1f984e55 | Which of the following is the negation of the statement "P implies (Q and R)"? - P and (Q implies not R) - P and not (Q or R) - (Not P) and (Q implies not R) - (Not P) and (Q implies R) | [
"P and (Q implies not R)"
] | [
"Which of the following is the negation of the statement \"P implies (Q and R)\"?\n- P and (Q implies not R)\n- P and not (Q or R)\n- (Not P) and (Q implies not R)\n- (Not P) and (Q implies R)"
] | [
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-d12dc1f3f1a0717f38993311864182d0abbe1f99231022a07662bf08bdf438d6 | Which of the following is the set of all perfect squares that can be written as the sum of three odd composite numbers? \[ \textbf{a)}\ \{(2k + 1)^2 : k \geq 0\} \] \[ \textbf{b)}\ \{(4k + 3)^2 : k \geq 1\} \] \[ \textbf{c)}\ \{(2k + 1)^2 : k \geq 3\} \] \[ \textbf{d)}\ \{(4k + 1)^2 : k \geq 2\} \] \[ \textbf{e)}\ \tex... | [
"\\bf c"
] | [
"Which of the following is the set of all perfect squares that can be written as the sum of three odd composite numbers?\n\n\\[\n\\textbf{a)}\\ \\{(2k + 1)^2 : k \\geq 0\\}\n\\]\n\\[\n\\textbf{b)}\\ \\{(4k + 3)^2 : k \\geq 1\\}\n\\]\n\\[\n\\textbf{c)}\\ \\{(2k + 1)^2 : k \\geq 3\\}\n\\]\n\\[\n\\textbf{d)}\\ \\{(4k ... | [
"c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-eb08b2048906d7f02ca3ca09d63cf3395dc039926b6229bb07b85d1c9f2f099c | Which of the following is the value of $\sqrt{\log_2{6}+\log_3{6}}?$ $\textbf{(A) } 1 \qquad\textbf{(B) } \sqrt{\log_5{6}} \qquad\textbf{(C) } 2 \qquad\textbf{(D) } \sqrt{\log_2{3}}+\sqrt{\log_3{2}} \qquad\textbf{(E) } \sqrt{\log_2{6}}+\sqrt{\log_3{6}}$ | [
"\\(\\sqrt{\\log_2{3}}+\\sqrt{\\log_3{2}}\\)"
] | [
"Which of the following is the value of $\\sqrt{\\log_2{6}+\\log_3{6}}?$\n\n$\\textbf{(A) } 1 \\qquad\\textbf{(B) } \\sqrt{\\log_5{6}} \\qquad\\textbf{(C) } 2 \\qquad\\textbf{(D) } \\sqrt{\\log_2{3}}+\\sqrt{\\log_3{2}} \\qquad\\textbf{(E) } \\sqrt{\\log_2{6}}+\\sqrt{\\log_3{6}}$"
] | [
"c5_contests_amp_programs"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-5970b26bcaee62751ed5ba04f8b02041b18c87e407463c4a1172907c8821a8c6 | Which of the following logarithmic expressions is the least? \[ \log_4 260, \log_2 18, \log_7 2300, \log_5 630, \log_3 200 \] | [
"\\(\\log_7 2300\\)"
] | [
"Which of the following logarithmic expressions is the least?\n\\[\n\\log_4 260, \\log_2 18, \\log_7 2300, \\log_5 630, \\log_3 200\n\\]"
] | [
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-b4dfec69303b0ef98cc73eefa733aa54f70e45582b5784730d8e872bb0877739 | Which of the following matrices is guaranteed to equal $(A+B)^2$? \[A^2+2AB+B^2,\quad A(A+B)+B(A+B),\quad (A+B)(A+B),\quad A^2+AB+BA+B^2\] | [
"\\((A+B)(A+B)\\)"
] | [
"Which of the following matrices is guaranteed to equal $(A+B)^2$?\n\\[A^2+2AB+B^2,\\quad A(A+B)+B(A+B),\\quad (A+B)(A+B),\\quad A^2+AB+BA+B^2\\]"
] | [
"c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-0a363c237ba8aaf4db26553743836a10d96e72f0db56944e5e39ff176caf7f15 | Which of the following numbers is greater than \(24^{100}\)? A. \(48^{75}\) B. \(5^{200}\) C. \(3^{200}\) D. \(2^{400}\) E. \(72^{50}\) | [
"B"
] | [
"Which of the following numbers is greater than \\(24^{100}\\)?\n\nA. \\(48^{75}\\) \nB. \\(5^{200}\\) \nC. \\(3^{200}\\) \nD. \\(2^{400}\\) \nE. \\(72^{50}\\)"
] | [
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-91ae17ffc14046aa25c7f372a846df4629caf434f6a7382f02b13babd3b270d8 | Which of the following numbers is greater, $ \int_{0}^{1}\sqrt[7]{1-x^{4}}\,dx$ or $ \int_{0}^{1}\sqrt[4]{1-x^{7}}\,dx$? | [
"The integrals are equal."
] | [
"Which of the following numbers is greater, $ \\int_{0}^{1}\\sqrt[7]{1-x^{4}}\\,dx$ or $ \\int_{0}^{1}\\sqrt[4]{1-x^{7}}\\,dx$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 8,
"tir": 25
} | 44 | 0 | 0 |
omr-ws-e12f32994907e8bad0f20c6d59af888ddf2f261a8825b71355338f8000976d07 | Which of the following open covers of the open interval $(0,1)$ admit a finite subcover? $(b)$ $\bigg\{\bigg(\frac{1}{n},1-\frac{1}{n+1}\bigg) \ \bigg| \ n\in \mathbb{N}\bigg\}$. | [
"No (or \"Not possible\")"
] | [
"Which of the following open covers of the open interval $(0,1)$ admit a finite subcover?\n$(b)$ $\\bigg\\{\\bigg(\\frac{1}{n},1-\\frac{1}{n+1}\\bigg) \\ \\bigg| \\ n\\in \\mathbb{N}\\bigg\\}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-67d23168b67acd0d329c32eecadcdabc903756edcd2e8bfc37ef3ebf6b87f551 | Which of the following open covers of the open interval $(0,1)$ admit a finite subcover? $(c)$ $\bigg\{\bigg(\sin^2\bigg(\frac{n\pi}{100}\bigg),\cos^2\bigg(\frac{n\pi}{100}\bigg)\bigg) \ \bigg| \ n\in \mathbb{N}\bigg\}$. | [
"(c) admits a finite subcover."
] | [
"Which of the following open covers of the open interval $(0,1)$ admit a finite subcover?\n$(c)$ $\\bigg\\{\\bigg(\\sin^2\\bigg(\\frac{n\\pi}{100}\\bigg),\\cos^2\\bigg(\\frac{n\\pi}{100}\\bigg)\\bigg) \\ \\bigg| \\ n\\in \\mathbb{N}\\bigg\\}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 18
} | 41 | 0 | 0 |
omr-ws-3222d76979d4cd40887d82a9b076b85cb6d1c709573986eecb07a9c54e65fb86 | Which of the following open covers of the open interval $(0,1)$ admit a finite subcover? $(d)$ $\bigg\{\bigg(\frac{1}{2}e^{-n},1-\frac{1}{(n+1)^2}\bigg) \ \bigg| \ n\in \mathbb{N}\bigg\}$. | [
"impossible"
] | [
"Which of the following open covers of the open interval $(0,1)$ admit a finite subcover?\n$(d)$ $\\bigg\\{\\bigg(\\frac{1}{2}e^{-n},1-\\frac{1}{(n+1)^2}\\bigg) \\ \\bigg| \\ n\\in \\mathbb{N}\\bigg\\}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 1,
"tir": 3
} | 7 | 0 | 0 |
omr-ws-ddfcf4f5bf9e5655b3634ab28ae39bda7c369a8ae812a973579d843ee2d8367e | Which of the following prime numbers is $2021_b - 221_b$ not divisible by 3? | [
"3"
] | [
"Which of the following prime numbers is $2021_b - 221_b$ not divisible by 3?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-366d8707fdad730adfd4577e513d1d2fb4721177222b9f046a437910ad37c116 | Which of the following subsets of $\mathbb{R}^2$ are connected? a) $\mathbb{R}^2 - \{\mathbb{Q} \times \mathbb{Q}\}$ b) $\{(x, \sin \frac{1}{x}) \in \mathbb{R}^2 \mid 0 < x < \infty\} \cup \{(0,0)\}$ c) $\{(x,y) \in \mathbb{R}^2 \mid xy = 1\} \cup \{(x,y) \in \mathbb{R}^2 \mid y = 0\}$ | [
"a) is connected, b) and c) are not connected."
] | [
"Which of the following subsets of $\\mathbb{R}^2$ are connected?\na) $\\mathbb{R}^2 - \\{\\mathbb{Q} \\times \\mathbb{Q}\\}$\nb) $\\{(x, \\sin \\frac{1}{x}) \\in \\mathbb{R}^2 \\mid 0 < x < \\infty\\} \\cup \\{(0,0)\\}$\nc) $\\{(x,y) \\in \\mathbb{R}^2 \\mid xy = 1\\} \\cup \\{(x,y) \\in \\mathbb{R}^2 \\mid y = 0\... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ab9c9c99c1d7ae853e0315a4015c2008dabfbfa52230f966ecb76f6c35e5bd0d | Which of the following two finite positive numbers (where \( 2 \) is repeated infinitely many times in each) is larger: \[ \sqrt{2 + \sqrt{2 + \sqrt{2 + \cdots}}} \] or \[ \sqrt{2 \sqrt{2 \sqrt{2 \cdots}}} \] | [
"Neither is larger. Both are equal to 2."
] | [
"Which of the following two finite positive numbers (where \\( 2 \\) is repeated infinitely many times in each) is larger:\n\\[\n\\sqrt{2 + \\sqrt{2 + \\sqrt{2 + \\cdots}}}\n\\]\nor\n\\[\n\\sqrt{2 \\sqrt{2 \\sqrt{2 \\cdots}}}\n\\]"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-48ab0aa35babba4af63db5cb47d3971d9724923d72df884be651aa8f1f0bfe40 | Which of the numbers $1, 2, \ldots, 1983$ has the largest number of divisors? | [
"\\( 2^4 \\times 3 \\times 5 \\times 7 \\)"
] | [
"Which of the numbers $1, 2, \\ldots, 1983$ has the largest number of divisors?"
] | [
"c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-b65f2675f87afb13e9a2a2f92503199ac676b4ebaf41b2d85c31bedff96448e0 | Which of the numbers $2^{2006!}$ and $2^{2006}!$ is greater? | [
"\\(2^{2006!}\\) is greater."
] | [
"Which of the numbers $2^{2006!}$ and $2^{2006}!$ is greater?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-51adbd025d0a94f25f11c4ce30dc770db0d2291124995fc4b50b7d1a7ef8ebf6 | Which of the numbers is greater? \[ A = \binom{F_{1000}}{F_{998}+1}, \quad B = \binom{F_{1000}}{F_{999}+1} \] where \( F_n \) denotes the Fibonacci sequence. | [
"\\( A > B \\)"
] | [
"Which of the numbers is greater?\n\\[ A = \\binom{F_{1000}}{F_{998}+1}, \\quad B = \\binom{F_{1000}}{F_{999}+1} \\]\nwhere \\( F_n \\) denotes the Fibonacci sequence."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-2f458c32d57a32fa2fce3afd43706030be45994365a9a3a67f25b1e5068c2080 | Which of the two expressions, \((1 + x^2 - x^3)^{100}\) or \((1 - x^2 + x^3)^{100}\), has the larger coefficient of \(x^{20}\)? | [
"\\((1 + x^2 - x^3)^{100}\\) has the larger coefficient of \\(x^{20}\\)."
] | [
"Which of the two expressions, \\((1 + x^2 - x^3)^{100}\\) or \\((1 - x^2 + x^3)^{100}\\), has the larger coefficient of \\(x^{20}\\)?"
] | [
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-330ff09e1ee2c167b201e320a1683f2c78e6e64e58235ef4e32f3f8bf079efbc | Which of the two numbers $99^n+100^n$ and $101^n$ is greater (here $n$ is a positive integer)? | [
"The number $99^n+100^n$ is greater than $101^n$ for $n\\le 48$ and is smaller than $101^n$ for $n>48$"
] | [
"Which of the two numbers $99^n+100^n$ and $101^n$ is greater (here $n$ is a positive integer)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4f5fda0d42289a711fe8bef8ee6c7bf1c9e997b8ef921c41842c110610d4879b | Which of the two numbers below is greater? $B$ (see Problem 214(a)) and $C=4^{4^{4^{.^{. ^{. ^4}}}}}$ where the expression of $C$ involves 999 fours. In this problem by an expression of the form $a_1^{a_2^{a_3^{.^{. ^{. ^{a_{n-1}^{a_n}}}}}}}$ is always meant the number $a_1^{\Bigg(a_2^{\Big(a_3^{.^{. ^{. ^{\left(a_{n-1... | [
"The first of the numbers is smaller"
] | [
"Which of the two numbers below is greater?\n$B$ (see Problem 214(a)) and $C=4^{4^{4^{.^{. ^{. ^4}}}}}$ where the expression of $C$ involves 999 fours. In this problem by an expression of the form $a_1^{a_2^{a_3^{.^{. ^{. ^{a_{n-1}^{a_n}}}}}}}$ is always meant the number $a_1^{\\Bigg(a_2^{\\Big(a_3^{.^{. ^{. ^{\\le... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-92ddc3216cbcc5909d525594f210364bb22e95b8e555cce4c90004c066297305 | Which of the two numbers is greater, $\frac{1 + 13^{2002}}{1 + 13^{2003}}$ or $\frac{1 + 13^{2003}}{1 + 13^{2004}}$? Could you generalize this by replacing $13$ with a positive $x$ and $2002$ with an integer $n$? For what positive value of $x$ can the two fractions coincide? | [
"\\(\\frac{1 + 13^{2002}}{1 + 13^{2003}}\\) is greater, and the fractions coincide when \\( x = 1 \\)."
] | [
"Which of the two numbers is greater, $\\frac{1 + 13^{2002}}{1 + 13^{2003}}$ or $\\frac{1 + 13^{2003}}{1 + 13^{2004}}$? Could you generalize this by replacing $13$ with a positive $x$ and $2002$ with an integer $n$? For what positive value of $x$ can the two fractions coincide?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 16
} | 16 | 0 | 0 |
omr-ws-4531e406547f95ade5174105f0831c20ed74993fb6d3789736e6f98d997a4e5b | Which of the two numbers is greater: $100!$ or $10^{150}$? | [
"\\( 100! \\) is greater than \\( 10^{150} \\)."
] | [
"Which of the two numbers is greater: $100!$ or $10^{150}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 48
} | 48 | 0 | 0 |
omr-ws-501b80c803949b07785115c9cd263b785ae12ecc9ddefe25825130b317fda22c | Which of the two polynomials \((1 + x^2 - x^3)^{1000}\) or \((1 - x^2 + x^3)^{1000}\) has a larger coefficient of \(x^{20}\)? | [
"\\((1 + x^2 - x^3)^{1000}\\) has a larger coefficient of \\(x^{20}\\)."
] | [
"Which of the two polynomials \\((1 + x^2 - x^3)^{1000}\\) or \\((1 - x^2 + x^3)^{1000}\\) has a larger coefficient of \\(x^{20}\\)?"
] | [
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-6fa7725cffa78605a693820823139a6203b63ee9a924ea6a4fd6dbf144caccb5 | Which of the two triangles has a multiplicative progression? The smallest angle of one triangle is three times the smallest angle of the other triangle, and the sum of the two largest angles is 240°. Determine the values of the angles in circular units (radians). | [
"\\(\\frac{\\pi}{25}, (0.02\\sqrt{97}-0.02)\\pi, (0.98-0.02\\sqrt{97})\\pi\\)"
] | [
"Which of the two triangles has a multiplicative progression? The smallest angle of one triangle is three times the smallest angle of the other triangle, and the sum of the two largest angles is 240°. Determine the values of the angles in circular units (radians)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2bc49f16b3e45b1c2695447be855b431cc93437ef524a8ace8a81a5908add6e5 | Which of the two-digit pairs 07, 29, 41, 63, or 85 can be the last two digits of the square of an integer? | [
"29, 41"
] | [
"Which of the two-digit pairs 07, 29, 41, 63, or 85 can be the last two digits of the square of an integer?"
] | [
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-fd7f15ab5a20fabf5f155a4a4173dae10b9918c9597e647890241bfffb1b8866 | Which one does not divide the numbers of \( 500 \)-subset of a set with \( 1000 \) elements? | [
"11"
] | [
"Which one does not divide the numbers of \\( 500 \\)-subset of a set with \\( 1000 \\) elements?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6b891273da28a01a220107106d2d1c6ee3e52c1ffd8aa837020b25117f894b55 | Which one is taller, a regular tetrahedron or a regular octahedron, both with edge length 1, when placed on one side? By how much? | [
"\\(\\frac{\\sqrt{6} - \\sqrt{2}}{\\sqrt{3}}\\)"
] | [
"Which one is taller, a regular tetrahedron or a regular octahedron, both with edge length 1, when placed on one side? By how much?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-33eb7b124599dc312e7dce7dac5394e26db6b239835faa89ca90a983b609498b | Which one of the following numbers is nearest in value to the quantity $\sqrt{101} - 10$? | [
"$\\frac{1}{20}$"
] | [
"Which one of the following numbers is nearest in value to the quantity $\\sqrt{101} - 10$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-f13cda9e135bbc60e9aa59db63571d842f9412bfd70e48a1c835b82a5bd02137 | Which ordinals cannot be represented as the sum of two smaller ordinals? | [
"\\(\\omega^\\beta\\) where \\(\\beta\\) is an ordinal."
] | [
"Which ordinals cannot be represented as the sum of two smaller ordinals?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 2,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-74c24c2b588f2709047df0fc6bc17fe1674fb46bdc70144e610488fd4e44556e | Which out of the groups $GL(n,\mathbb{R})$ and $Sym(3)$ has the property that every proper subgroup of the group is abelian? | [
"Sym(3)"
] | [
"Which out of the groups $GL(n,\\mathbb{R})$ and $Sym(3)$ has the property that every proper subgroup of the group is abelian?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a83ae2779d016557402f26b836fd98875a146e3c7eda3aca7e86190163044072 | Which pairs $(x, m)$ of integers satisfy the equation $|x^2 - 1| + |x^2 - 4| = mx$? | [
"\\((-1, 3), (1, 3), (5, 9), (-5, 9)\\)"
] | [
"Which pairs $(x, m)$ of integers satisfy the equation $|x^2 - 1| + |x^2 - 4| = mx$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 7
} | 7 | 0 | 0 |
omr-ws-e6039d630389b2e495f228658d4a0822a1d61939972cdeadee134aba27520216 | Which pairs \((a, b)\) of positive integers satisfy \(a^2 - b^7 = 7\)? | [
"\\text{No solution}"
] | [
"Which pairs \\((a, b)\\) of positive integers satisfy \\(a^2 - b^7 = 7\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-b2cb7802e36887d812c28f5fe1f07eaae8e54ce6422643e9d5c18cee194e1d47 | Which pattern requires a larger expected time to occur: HHTT or HTHT? | [
"HTHT"
] | [
"Which pattern requires a larger expected time to occur: HHTT or HTHT?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 6
} | 21 | 0 | 0 |
omr-ws-167ee7eb30f07cc27ac5f9c45083a6eb37db75e383a59cc126a7aa5d7109634f | Which plane containing a body diagonal of a cube with edge length $a$ cuts out a figure with the smallest area from the cube? Calculate the area of such a cut figure. | [
"\\(\\frac{3\\sqrt{3}}{2} a^2\\)"
] | [
"Which plane containing a body diagonal of a cube with edge length $a$ cuts out a figure with the smallest area from the cube? Calculate the area of such a cut figure."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-345606d57ce159db8061720ebecd13ad1b22323961bfe5361147584bbe9c66af | Which planes cut the surface \(xy + yz + xz = 0\) in (1) circles, and (2) parabolas? | [
"ax + by + cz = d \\text{ where } (a + b + c)^2 = 4(ab + bc + ac) \\text{ and } d \\neq 0"
] | [
"Which planes cut the surface \\(xy + yz + xz = 0\\) in (1) circles, and (2) parabolas?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-586f9276ddbfadac2025acf584aea0de9530e4ae179dfc7f054cf784fdc7150d | Which point \( P \) inside a triangle \( ABC \) minimizes the sum \( PA + PB + PC \)? | [
"Fermat point."
] | [
"Which point \\( P \\) inside a triangle \\( ABC \\) minimizes the sum \\( PA + PB + PC \\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 28
} | 50 | 0 | 0 |
omr-ws-8c6c1d7e16faa6cc8ea7ed27456185090f11ab07b5c389dd9fa69fb586444553 | Which point in the plane of a triangle has the minimum sum of distances to the edges of the triangle? | [
"\\text{The vertex opposite the longest side}"
] | [
"Which point in the plane of a triangle has the minimum sum of distances to the edges of the triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 7
} | 16 | 0 | 0 |
omr-ws-b3b1600d0be5a494e260aecfc839538a29f12295d6a9c81a1f11eb070e166e73 | Which point on the circle \((x-5)^2 + (y-4)^2 = 4\) is closest to the circle \((x-1)^2 + (y-1)^2 = 1\)? | [
"\\left( \\dfrac{17}{5}, \\dfrac{14}{5} \\right)"
] | [
"Which point on the circle \\((x-5)^2 + (y-4)^2 = 4\\) is closest to the circle \\((x-1)^2 + (y-1)^2 = 1\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ccb1c8b4b21b42ad3603a3cd19bf7ba6f9449203ff297b32d6a59661a524d72d | Which point on the parabola \( y = x^2 \) is nearest to the point \( (1,0) \)? Find the coordinates to two decimal places. [Hint: Minimize the square of the distance to avoid square roots.] | [
"\\((0.65, 0.42)\\)"
] | [
"Which point on the parabola \\( y = x^2 \\) is nearest to the point \\( (1,0) \\)? Find the coordinates to two decimal places. [Hint: Minimize the square of the distance to avoid square roots.]"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-515bf3e3e5f798ddf1deb0b8d463a8a4aadc886df95cde342ad950996f5c5731 | Which point on the plane $3x + 2y + z = 6$ in $\mathbb{R}^3$ is closest to the origin with respect to the $\|\cdot\|_{1}$ norm? | [
"\\((2, 0, 0)\\)"
] | [
"Which point on the plane $3x + 2y + z = 6$ in $\\mathbb{R}^3$ is closest to the origin with respect to the $\\|\\cdot\\|_{1}$ norm?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-20f6828f3608ea19ddc158870b5929186d0082adf89b7e445222ffb0dc21eaf9 | Which point on the plane $3x + 2y + z = 6$ in $\mathbb{R}^3$ is closest to the origin with respect to the $\|\cdot\|_{2}$ norm? | [
"$\\frac{3}{7}\\sqrt{14}$"
] | [
"Which point on the plane $3x + 2y + z = 6$ in $\\mathbb{R}^3$ is closest to the origin with respect to the $\\|\\cdot\\|_{2}$ norm?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ef24d80fde16b948de6179a4b10c4dabb9706da78af37141706b995c27dfd0d8 | Which point on the plane $3x + 2y + z = 6$ in $\mathbb{R}^3$ is closest to the origin with respect to the $\|\cdot\|_{\infty}$ norm? | [
"\\((1, 1, 1)\\)"
] | [
"Which point on the plane $3x + 2y + z = 6$ in $\\mathbb{R}^3$ is closest to the origin with respect to the $\\|\\cdot\\|_{\\infty}$ norm?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-57355cccd52e9e5251d6b1d6b74b5cd927318565d1c28a95350ed6ceff70152e | Which points $(p, q)$ can be reached from the origin if each move is a diagonal step, consisting of one unit to the right or left followed by one unit either up or down? | [
"\\(p + q\\) is even."
] | [
"Which points $(p, q)$ can be reached from the origin if each move is a diagonal step, consisting of one unit to the right or left followed by one unit either up or down?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 28
} | 45 | 0 | 0 |
omr-ws-cdb9eade88d2052bb332580a9c0a685466f489a7a6089d5d65ea2141823e42c7 | Which points $(p,q)$ can be reached from the origin if each move is a diagonal step, consisting of one unit to the right or left followed by one unit either up or down? | [
"The points \\((p, q)\\) that can be reached from the origin are those for which \\(p + q\\) is even.",
"The points \\((p,q)\\) that can be reached from the origin are those for which \\( p - q \\) is even."
] | [
"Which points $(p,q)$ can be reached from the origin if each move is a diagonal step, consisting of one unit to the right or left followed by one unit either up or down?"
] | [
"aops_c6_high_school_olympiads",
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 29,
"genselect": 2,
"tir": 50
} | 81 | 0 | 0 |
omr-ws-a9b11cbcd5e462e26cb951431548476faf8ac17ab005bd9e84e68b0259d7ce8a | Which points (p,q) can be reached from the origin by moving diagonally, where each move consists of one unit to the right or left followed by one unit either up or down? | [
"Points (p,q) where \\( p + q \\) is even."
] | [
"Which points (p,q) can be reached from the origin by moving diagonally, where each move consists of one unit to the right or left followed by one unit either up or down?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-2ff1492fb64bfef6b498ef66d19a08ba4b476616b216bd4e2d9dc7c8029da850 | Which points \((p, q)\) can be reached from the origin \((0, 0)\) if each move consists of one unit to the right or left followed by one unit either up or down? | [
"\\(2 \\mid (p - q)\\)"
] | [
"Which points \\((p, q)\\) can be reached from the origin \\((0, 0)\\) if each move consists of one unit to the right or left followed by one unit either up or down?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-e0607b9dba262c9ab7aad8ba4529b952087c9744263661c4bc926d1e8e830915 | Which points transform into (1, 2, 0) after rotating around the vector (1, 1, 1) by 90°? | [
"\\left( \\dfrac{3 + 2\\sqrt{3}}{3}, \\dfrac{3 - \\sqrt{3}}{3}, \\dfrac{3 - \\sqrt{3}}{3} \\right)"
] | [
"Which points transform into (1, 2, 0) after rotating around the vector (1, 1, 1) by 90°?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8f9db030df9576dcfa3b71ea4f7a47a48d0619baef15ed2ff209d8a4d095fc12 | Which polygon inscribed in a given circle has the greatest sum of squares of its sides? | [
"\\text{Regular triangle}"
] | [
"Which polygon inscribed in a given circle has the greatest sum of squares of its sides?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-8c84799ef67cc81c8daeec9d1419a60d7665a3a1262ef15ca496bb9e1185b4bc | Which polynomial has the larger coefficient of \(x^{20}\): \((1+x^2-x^3)^{1000}\) or \((1-x^2+x^3)^{1000}\)? | [
"\\((1+x^2-x^3)^{1000}\\) has the larger coefficient of \\(x^{20}\\)."
] | [
"Which polynomial has the larger coefficient of \\(x^{20}\\): \\((1+x^2-x^3)^{1000}\\) or \\((1-x^2+x^3)^{1000}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-48607840ef373e294943cf6f814a02c0c56247e4b4518ca71518536214eac800 | Which polynomials $P \in \mathbb{R}[x]$ satisfy $P(P'(x)) = P'(P(x))$ for all $x \in \mathbb{R}$? | [
"\\frac{1}{n^{n-1}}x^n"
] | [
"Which polynomials $P \\in \\mathbb{R}[x]$ satisfy $P(P'(x)) = P'(P(x))$ for all $x \\in \\mathbb{R}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-47fcaf413cd025f64ecc29ecae020d6cc679f5e70fdf293ad45a136b18ee884f | Which positive integer \( N \) satisfies the condition that there exists at least one \( N \times N \) table such that: 1. Every row is a permutation of the numbers from 1 to \( N \). 2. If \( a_{i,j} = a_{p,q} \), then \( a_{i,j+1} \neq a_{p,q+1} \) for \( 1 \leq j, q \leq N-1 \) and \( 1 \leq i < p \leq N \)? | [
"\\( N = 7, 9, 11 \\)"
] | [
"Which positive integer \\( N \\) satisfies the condition that there exists at least one \\( N \\times N \\) table such that:\n1. Every row is a permutation of the numbers from 1 to \\( N \\).\n2. If \\( a_{i,j} = a_{p,q} \\), then \\( a_{i,j+1} \\neq a_{p,q+1} \\) for \\( 1 \\leq j, q \\leq N-1 \\) and \\( 1 \\leq... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e3f92c80b31166059df7d0852b0108a17333e5fceb64b34f63ea05d11f74fba6 | Which positive integer \( n \) and probabilities \( p_0, p_1, \ldots, p_n \) will maximize the probability that the offspring of a given man go extinct in exactly the tenth generation, given that the number of offspring for every man can be \( 0, 1, \ldots, n \) with probabilities \( p_0, p_1, \ldots, p_n \) independen... | [
"\\dfrac{9}{10}"
] | [
"Which positive integer \\( n \\) and probabilities \\( p_0, p_1, \\ldots, p_n \\) will maximize the probability that the offspring of a given man go extinct in exactly the tenth generation, given that the number of offspring for every man can be \\( 0, 1, \\ldots, n \\) with probabilities \\( p_0, p_1, \\ldots, p_... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-a157502efa6d377a2de4120e0bd9a8a3eed44d16b7fee3579d8bb4b98636b9de | Which positive integer becomes 57 times smaller by removing its first digit? | [
"7125 (or any number of the form \\(7125 \\cdot 10^m\\) where \\(m \\geq 0\\))"
] | [
"Which positive integer becomes 57 times smaller by removing its first digit?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 4,
"tir": 1
} | 13 | 0 | 0 |
omr-ws-4a294023ae318025e85f05c40cc582e7668b0819c993c56a7e37be407f4bdebd | Which positive integer becomes 57 times smaller when the leftmost digit is removed? | [
"7125"
] | [
"Which positive integer becomes 57 times smaller when the leftmost digit is removed?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-5b26d262011fc5c6aadc23094e406f487d25926846ca7cfce88e2ebed317b0f2 | Which positive integer can be represented as $$ \frac {\gcd{(x,y)}+\gcd{(y,z)}}{\gcd{(z,x)}}$$ | [
"Any positive integer \\(n\\)."
] | [
"Which positive integer can be represented as \n$$ \\frac {\\gcd{(x,y)}+\\gcd{(y,z)}}{\\gcd{(z,x)}}$$"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 11
} | 23 | 0 | 0 |
omr-ws-e7dce6c12ac7e37882ae16c9ef3c73247e7197d66ae68682070659f2e93518b3 | Which positive integers $k$ make $k^4 - 2k^2 - 8k - 3$ a perfect square? | [
"3"
] | [
"Which positive integers $k$ make $k^4 - 2k^2 - 8k - 3$ a perfect square?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 19
} | 19 | 0 | 0 |
omr-ws-363939c5f0881c03535c0544e730d4fe564feed1ed26dfe3871f5920c85476cb | Which positive integers $m$ are such that $k^m - 1$ is divisible by $2^m$ for all odd numbers $k \ge 3$? | [
"\\( m = 1, 2, 4 \\)"
] | [
"Which positive integers $m$ are such that $k^m - 1$ is divisible by $2^m$ for all odd numbers $k \\ge 3$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 16
} | 17 | 0 | 0 |
omr-ws-b61f310b935e710f5bda90d88b462157287fab5e3c264ea5476b46bc0a31e1d4 | Which positive integers $n$ have the property that the set $\{1, 2, 3, \dots, n\}$ can be partitioned into two subsets $A$ and $B$ such that the sum of the squares of the elements of $A$ is equal to the sum of the squares of the elements of $B$? | [
"\\( n \\equiv 0, 3 \\pmod{4} \\) and \\( n \\geq 7 \\)"
] | [
"Which positive integers $n$ have the property that the set $\\{1, 2, 3, \\dots, n\\}$ can be partitioned into two subsets $A$ and $B$ such that the sum of the squares of the elements of $A$ is equal to the sum of the squares of the elements of $B$?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 22,
"genselect": 2,
"tir": 30
} | 54 | 0 | 0 |
omr-ws-614bc23bc9179bbd33be5e31c4847c6455f7847354a8b5f6e50f5cd507183f9f | Which positive integers $n$ have the property that the set $\{1, 2, \ldots, n\}$ can be partitioned into two subsets $A$ and $B$ such that the sum of the squares of the elements of $A$ is equal to the sum of the squares of the elements of $B$? | [
"\\( n \\equiv 0 \\pmod{4} \\) or \\( n \\equiv 3 \\pmod{4} \\)"
] | [
"Which positive integers $n$ have the property that the set $\\{1, 2, \\ldots, n\\}$ can be partitioned into two subsets $A$ and $B$ such that the sum of the squares of the elements of $A$ is equal to the sum of the squares of the elements of $B$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 3,
"tir": 18
} | 29 | 0 | 0 |
omr-ws-ab6f4c7bbbf5d58e8a957e5ec0fc81a7763170c8dfe2a5c8b2193a574903fdc6 | Which positive integers $n$ satisfy the equation \[\sum_{i=1}^n \sum_{j=1}^n \left\lfloor \frac{ij}{n+1} \right\rfloor=\frac{n^2(n-1)}{4}?\] | [
"\\( n = p - 1 \\) where \\( p \\) is a prime number."
] | [
"Which positive integers $n$ satisfy the equation \\[\\sum_{i=1}^n \\sum_{j=1}^n \\left\\lfloor \\frac{ij}{n+1} \\right\\rfloor=\\frac{n^2(n-1)}{4}?\\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 7
} | 7 | 0 | 0 |
omr-ws-c529149db07eb6484bc5d2dbab075d6e55dee1cb101f74057e85c046adfec9c6 | Which positive integers \( N \) have the property that \( N^2 \) has a remainder of \( N \) when divided by 100? | [
"\\( 1, 25, 76 \\)"
] | [
"Which positive integers \\( N \\) have the property that \\( N^2 \\) has a remainder of \\( N \\) when divided by 100?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 8
} | 9 | 0 | 0 |
omr-ws-1f539f6875df319fbb31aa01abae9d87acff87e54f0d6cca7defcb622485f5c1 | Which positive integers \(m\) are such that \(k^m - 1\) is divisible by \(2^m\) for all odd \(k \geq 3\)? | [
"4"
] | [
"Which positive integers \\(m\\) are such that \\(k^m - 1\\) is divisible by \\(2^m\\) for all odd \\(k \\geq 3\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-4dbe17870566aac9311c3b6861bc5b79f54fda3d81b8cb2d8ba0a487d2486de3 | Which positive integers are missing in the sequence $\{a_n\}$, where $a_n = n + \left[\sqrt{n}\right] + \left[\sqrt[3]{n}\right]$ for all $n \ge 1$? ($[x]$ denotes the largest integer less than or equal to $x$.) | [
"$p^2 + p - 1 + \\left[\\sqrt[3]{p^2}\\right]$, $p^3 + p - 1 + \\left[\\sqrt{p^3}\\right]$, $p^6 + p^3 + p^2 - 2$ for all $p \\in \\mathbb{N}$"
] | [
"Which positive integers are missing in the sequence $\\{a_n\\}$, where $a_n = n + \\left[\\sqrt{n}\\right] + \\left[\\sqrt[3]{n}\\right]$ for all $n \\ge 1$? ($[x]$ denotes the largest integer less than or equal to $x$.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-0bf30517ced87bde90283cbeff7d8fc9aa6bd55392f61bca7c41cb3ad1d65fe0 | Which positive integers can be expressed in the form $ab(a+b)$ where $a$ and $b$ are positive integers? | [
"\\text{All positive even integers}"
] | [
"Which positive integers can be expressed in the form $ab(a+b)$ where $a$ and $b$ are positive integers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-52cbcbfc8006bebfd016b57c50dc9293160685d31f5a35f999930b3062383201 | Which positive integers can be represented as a difference of two cubes? | [
"\\text{All positive integers not congruent to }4 \\text{ or }5 \\pmod{9}"
] | [
"Which positive integers can be represented as a difference of two cubes?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-f39295290b6e5589d7db4936d890d46c843c5bc7b32b0df29f3bbf7a4274a1f7 | Which positive integers can be written in the form \[\frac{\operatorname{lcm}(x, y) + \operatorname{lcm}(y, z)}{\operatorname{lcm}(x, z)}\] for positive integers $x$, $y$, $z$? | [
"All even positive integers."
] | [
"Which positive integers can be written in the form \\[\\frac{\\operatorname{lcm}(x, y) + \\operatorname{lcm}(y, z)}{\\operatorname{lcm}(x, z)}\\] for positive integers $x$, $y$, $z$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 20,
"tir": 15
} | 35 | 0 | 0 |
omr-ws-68e298a05070611d8dde19f8b19bf76b53e0769018684891864b4ed1ff2dc3d3 | Which positive integers have a unique base 10 over-expansion? | [
"Positive integers without the digit 0 in their standard base 10 expression."
] | [
"Which positive integers have a unique base 10 over-expansion?"
] | [
"aops_c7_college_math"
] | [
"genselect"
] | {
"genselect": 8
} | 8 | 0 | 0 |
omr-ws-5b6f625128627de23c4880b1177e17745c49c5b455853a2b2ff32f496f75103c | Which positive integers have the property that if the final (units) digit is deleted, the original integer is divisible by the resulting integer? | [
"Any multiple of 10 and the numbers 11-19, 22, 24, 26, 28, 33, 36, 39, 44, 48, 55, 66, 77, 88, 99"
] | [
"Which positive integers have the property that if the final (units) digit is deleted, the original integer is divisible by the resulting integer?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 10
} | 12 | 0 | 0 |
omr-ws-f2887c3afa368c8d7a24f5d721d5eabd4b3cfb56536ca1809813761026fc46bf | Which positive integers of the form $6n$ can be represented as a sum of three distinct positive integers $a$, $b$, and $c$ such that $a \mid b$ and $b \mid c$? | [
"All positive integers of the form \\(6n\\) except \\(6, 12, 24\\)."
] | [
"Which positive integers of the form $6n$ can be represented as a sum of three distinct positive integers $a$, $b$, and $c$ such that $a \\mid b$ and $b \\mid c$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 5,
"tir": 14
} | 29 | 0 | 0 |
omr-ws-0492a92d38e6af7bd6fbf61c7d80dd260143bb41713b008d95c11a6c1ad7a5f7 | Which positive integers, when repeated to form a new number, result in a perfect square? Use mathematical methods instead of computational methods. | [
"\\text{No such positive integers exist}"
] | [
"Which positive integers, when repeated to form a new number, result in a perfect square? Use mathematical methods instead of computational methods."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-8e1c303d7f7422bb72041398b58467410da32dc41d35b499f9496bade19531c1 | Which powers of $2$ give $5$ digit numbers? | [
"\\( 2^{14}, 2^{15}, 2^{16} \\)"
] | [
"Which powers of $2$ give $5$ digit numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 6,
"tir": 8
} | 19 | 0 | 0 |
omr-ws-76000ef81505d1e3b7cae1b294b9695bcad974f895a0cb9002faea48d734ef55 | Which pre-tax sales totals are charged exactly $25$ cents sales tax, assuming a tax rate of $8\%$ and that sales tax is rounded to the nearest cent? | [
"[3.07, 3.18]"
] | [
"Which pre-tax sales totals are charged exactly $25$ cents sales tax, assuming a tax rate of $8\\%$ and that sales tax is rounded to the nearest cent?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-faccd1312388d03abc7b3235e43c48d16ed56cde695299878358472f2123090a | Which prime number below one-million can be written as the sum of the most consecutive prime numbers? | [
"997651"
] | [
"Which prime number below one-million can be written as the sum of the most consecutive prime numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e38973d6c06f5f0da3b26f3ba7f59c4a7b71c781405c526636e6f62560857d20 | Which prime numbers are $1$ greater than a square? | [
"401"
] | [
"Which prime numbers are $1$ greater than a square?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-8de25bab575a28e72c25d0b04ebb86b25e9a655a9dc7c5e1868dd15f6f30a299 | Which prime numbers are 1 greater than a square? | [
"401"
] | [
"Which prime numbers are 1 greater than a square?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-e3c666986e7fc0f86b0cf23349d8026a08440d3f07b46e192595c5f80d83399b | Which prime numbers are 1 less than a number to the power of \( n \)? | [
"3"
] | [
"Which prime numbers are 1 less than a number to the power of \\( n \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-b6cbb4b07f00e0d939a57963b8a4f9a048aafdc6fbddf599e456ce42b91ec36c | Which prime numbers can be expressed as the sum of two square numbers? | [
"\\( 2 \\) and primes \\( p \\) such that \\( p \\equiv 1 \\pmod{4} \\)"
] | [
"Which prime numbers can be expressed as the sum of two square numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 29
} | 31 | 0 | 0 |
omr-ws-db4c1f32bc0e30b053afe9fa139e6bebcf403e853b7e0b09d6e10c7d7a5098cb | Which prime numbers can be expressed in the form $3x^2 + y^2$? | [
"\\(p = 3\\) or \\(p \\equiv 1 \\pmod{3}\\)"
] | [
"Which prime numbers can be expressed in the form $3x^2 + y^2$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 24
} | 37 | 0 | 0 |
omr-ws-9bc6ee7398d713556049b026b3e398abcc1d164a1ccee485d8202ff074390c1c | Which prime numbers can be expressed in the form $x^2 + 5y^2$? | [
"\\( p \\equiv 1, 9 \\mod 20 \\)"
] | [
"Which prime numbers can be expressed in the form $x^2 + 5y^2$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 2,
"tir": 4
} | 18 | 0 | 0 |
omr-ws-f47f47101913117fea15db650ff8a8b0116151b52fb9efeb4c37c68edf5e1c50 | Which prime numbers can be expressed in the form $x^2 + y^2$? | [
"\\( p = 2 \\) or \\( p \\equiv 1 \\pmod{4} \\)"
] | [
"Which prime numbers can be expressed in the form $x^2 + y^2$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 8,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-78accfc9689ed431c13809683f50107904bd78a5eb758f0b68a125857f25be45 | Which prime numbers can be represented as $x^2 + xy + 2y^2$ where $x$ and $y$ are integers? | [
"\\( p \\equiv 1, 2, 4, 8, 9, 11 \\pmod{14} \\)"
] | [
"Which prime numbers can be represented as $x^2 + xy + 2y^2$ where $x$ and $y$ are integers?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 15
} | 17 | 0 | 0 |
omr-ws-dfa780822b99b60250cd429d7c4d4c01fcf7996c907cdfc18e0ee7aa1cb8e148 | Which prime numbers can be written as the sum of two perfect squares? | [
"\\( p \\equiv 1 \\pmod{4} \\)"
] | [
"Which prime numbers can be written as the sum of two perfect squares?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 3
} | 14 | 0 | 0 |
omr-ws-10bbdb47981e895ec1ff67a15b4f09342df0e893fc59ff1208a22e59c3b1946d | Which prime numbers can divide \( n^2 + n \) for some integer \( n \)? | [
"Any prime number"
] | [
"Which prime numbers can divide \\( n^2 + n \\) for some integer \\( n \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-bd70d892503d653df2a1bace221489e673758c662ba86399942651d86382020c | Which primes $p > 2$ can be written in the form $4x^2 + 4xy + 5y^2$ using integers $x$ and $y$? | [
"p \\equiv 5 \\pmod{8}"
] | [
"Which primes $p > 2$ can be written in the form $4x^2 + 4xy + 5y^2$ using integers $x$ and $y$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-145008daad2d9996a47743ade8354d118b8ef434cf94d1182658d1d8299dac47 | Which primes $p > 2$ can be written in the form $x^2 + 16y^2$ using integers $x$ and $y$? | [
"\\( p \\equiv 1 \\pmod{4} \\)"
] | [
"Which primes $p > 2$ can be written in the form $x^2 + 16y^2$ using integers $x$ and $y$?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-793911abec876eef608096736222c28e4c85df06d78f3e927556d4d5c17ecd43 | Which primes \( p \) can be expressed in the form \( x^2 + y^2 \equiv 0 \pmod{p} \)? | [
"p = 2 \\text{ or } p \\equiv 1 \\pmod{4}"
] | [
"Which primes \\( p \\) can be expressed in the form \\( x^2 + y^2 \\equiv 0 \\pmod{p} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fd2990515b0763c7305f290556709f166617dcb1a2f9c7b1ad620bbc2c1259bb | Which primes less than $200$ do not divide $3^n - 2$ for any integer $n$? | [
"11, 13, 37, 41, 59, 61, 67, 73, 83, 103, 107, 109, 131, 151, 157, 179, 181, 193"
] | [
"Which primes less than $200$ do not divide $3^n - 2$ for any integer $n$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4f3693fdbd3dc0b162a30d91a732db8737ec6e5c9b80a2e1de97420a71786d20 | Which quantity has a greater value: $2^\pi$ or $2\sqrt{3^e}$? | [
"2\\sqrt{3^e}"
] | [
"Which quantity has a greater value: $2^\\pi$ or $2\\sqrt{3^e}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a30cd859e9abd8d9515eec13efe1eff1f32eadb8c353c5d54041e1a744aa4e25 | Which radial functions (i.e., functions of the polar coordinate \( r \) but independent of \( \theta \) or any other angular variables) are harmonic (i.e., satisfy the PDE \( \Delta u = 0 \)) in \( n \) dimensions, where \( n \geq 2 \)? | [
"u(r) = \\begin{cases} \nC_1 + C_2 \\ln r & \\text{if } n = 2, \\\\\nC_1 + C_2 r^{2 - n} & \\text{if } n \\geq 3 \n\\end{cases}"
] | [
"Which radial functions (i.e., functions of the polar coordinate \\( r \\) but independent of \\( \\theta \\) or any other angular variables) are harmonic (i.e., satisfy the PDE \\( \\Delta u = 0 \\)) in \\( n \\) dimensions, where \\( n \\geq 2 \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-803802f7cbf9ce9768c76193d5bac82ebabacdbcb722722756915ad72000b2c2 | Which rational numbers $t$ (excluding integers) are such that $3t^3 + 10t^2 - 3t$ is an integer? | [
"\\( t \\in \\frac{-1}{3} + 3\\mathbb{Z} \\)"
] | [
"Which rational numbers $t$ (excluding integers) are such that $3t^3 + 10t^2 - 3t$ is an integer?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-416425a51b3e6418535770fa5193aca28e5a727b3fb4ce60e2b9abadcceb2e06 | Which real functions $f(x,y)$ satisfy the conditions $f(x,0)=x$ and $f(f(x,y),-y)=x$ for all $x,y \in \mathbb{R}$? | [
"\\( f(x, y) = \\begin{cases} \n\\phi_y(x) & \\text{if } y > 0 \\\\\n\\phi_{-y}^{-1}(x) & \\text{if } y < 0 \\\\\nx & \\text{if } y = 0 \n\\end{cases} \\)"
] | [
"Which real functions $f(x,y)$ satisfy the conditions $f(x,0)=x$ and $f(f(x,y),-y)=x$ for all $x,y \\in \\mathbb{R}$?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-c8f422449c82d694e0d66f480916bddf9278ff31224a0830484ba78e6d141bb2 | Which real numbers are chords of all continuous functions \( f \) with domain \([0,1]\) such that \( f(0) = f(1) \)? A chord \( c \) of a function \( g \) is defined as a value such that \( g(x) = g(x + c) \) for some \( x \). | [
"Reciprocals of positive integers."
] | [
"Which real numbers are chords of all continuous functions \\( f \\) with domain \\([0,1]\\) such that \\( f(0) = f(1) \\)? A chord \\( c \\) of a function \\( g \\) is defined as a value such that \\( g(x) = g(x + c) \\) for some \\( x \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 1
} | 8 | 0 | 0 |
omr-ws-664b33d655cc8e39e0b20680b961ea37a8658f270eebe71bcf737481bd733d63 | Which regular n-gons can be inscribed in a noncircular ellipse? | [
"Triangle and square only."
] | [
"Which regular n-gons can be inscribed in a noncircular ellipse?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 2,
"tir": 1
} | 4 | 0 | 0 |
omr-ws-83ae641b1bd18f464ae807256c3bf2eddf2eb9109ecf588bea2662b47b8d82d7 | Which roots of unity are contained in \( \mathbb{Q}(d^{1/2}) \), where \( d \) is a square-free integer? | [
"1, 2, 3, 4, 6"
] | [
"Which roots of unity are contained in \\( \\mathbb{Q}(d^{1/2}) \\), where \\( d \\) is a square-free integer?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-343eeb51d9e699850dd1be64bb751b99847cc5b43b6c889395aa923f7c786e0d | Which sequences of integers \(a_0, a_1, a_2, \ldots\) satisfy that for some positive integers \(i, j,\) and \(k\) (not necessarily distinct), every natural number \(n\) has a unique representation as \(a_i + 2a_j + 4a_k\)? | [
"a_n = 8^n"
] | [
"Which sequences of integers \\(a_0, a_1, a_2, \\ldots\\) satisfy that for some positive integers \\(i, j,\\) and \\(k\\) (not necessarily distinct), every natural number \\(n\\) has a unique representation as \\(a_i + 2a_j + 4a_k\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 6
} | 6 | 0 | 0 |
omr-ws-b6667e1a4169ff241da9dbef1bec916b51ccc58c5cab2c856a246e7caa437ba5 | Which set is "bigger," the power set of the non-negative integers $P(\mathbb{N})$ or the power set of the rational numbers $P(\mathbb{Q})$? | [
"P(\\mathbb{N}) \\text{ and } P(\\mathbb{Q}) \\text{ have the same cardinality}"
] | [
"Which set is \"bigger,\" the power set of the non-negative integers $P(\\mathbb{N})$ or the power set of the rational numbers $P(\\mathbb{Q})$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-44adf69302f2a36f1613cbfb4adde941f9858c4f114379a185b34209ee545606 | Which set is larger? The set of non-negative integers that can be expressed as $x^2 + y^2$, where $x$ and $y$ are integers, or the set of non-negative integers that cannot be expressed as $x^2 + y^2$, where $x$ and $y$ are integers? | [
"Both sets have the same cardinality."
] | [
"Which set is larger? The set of non-negative integers that can be expressed as $x^2 + y^2$, where $x$ and $y$ are integers, or the set of non-negative integers that cannot be expressed as $x^2 + y^2$, where $x$ and $y$ are integers?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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