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-a29e07079a6b31f77ba8102dae6f79e8409f35e00b9708e7481d0ba978d6bf50 | What are the nth Pythagorean triplets? Consider using odd numbers to find a pattern. | [
"\\( (2n - 1, 2n^2 - 2n, 2n^2 - 2n + 1) \\)"
] | [
"What are the nth Pythagorean triplets? Consider using odd numbers to find a pattern."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0d8da55e10924e19108cd2cfaf22140894993e7cd1893def1167dea12100c1b4 | What are the numbers \(\left\langle\begin{matrix}n\\ m\end{matrix}\right\rangle\) typically called? | [
"Eulerian numbers"
] | [
"What are the numbers \\(\\left\\langle\\begin{matrix}n\\\\ m\\end{matrix}\\right\\rangle\\) typically called?"
] | [
"c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-29fa21aa0b3f568a96048f9784e2bbcecc6b859636b89b8cea878fcf3c8dd6c5 | What are the odds of drawing a red A card in the opening 7 cards? | [
"56:165"
] | [
"What are the odds of drawing a red A card in the opening 7 cards?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fb60edd1521661f89e97736979fd153eb33f53e0e67b7a70ca3f9577f217a027 | What are the odds of drawing a red card in the opening 7 cards? | [
"11131:55"
] | [
"What are the odds of drawing a red card in the opening 7 cards?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 5
} | 19 | 0 | 0 |
omr-ws-3942fba3431bfc4a2fc0959a41736f08841fd35c6e0a01abce086a365dc034dc | What are the odds of guessing a 7-digit code where numbers can be repeated? | [
"\\(\\frac{1}{10^7}\\) or \\(1/10000000\\)"
] | [
"What are the odds of guessing a 7-digit code where numbers can be repeated?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 3,
"tir": 9
} | 15 | 0 | 0 |
omr-ws-0802d28e400e613646c43cc1c730a12c650ba64b97a53372791ae9f9825f2f55 | What are the odds that if a coin with a 60% chance of landing heads is flipped endlessly, at some point during the flipping there will be a total of three more tails than heads? | [
"\\(\\frac{8}{27}\\)"
] | [
"What are the odds that if a coin with a 60% chance of landing heads is flipped endlessly, at some point during the flipping there will be a total of three more tails than heads?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-67e178ea789a90aa198ab966aba3bcf220c3c04bdc036135c326fe1704e8d15b | What are the only 3 positive integers that cannot be the length of the shortest side of a right triangle whose sides all have integral lengths? | [
"1, 2, 4",
"1, 2, and 4."
] | [
"What are the only 3 positive integers that cannot be the length of the shortest side of a right triangle whose sides all have integral lengths?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 14
} | 14 | 0 | 0 |
omr-ws-6f3f063e692cb17207ef67d691547688a1b7904e9a8d010e3524baed44a79847 | What are the only three positive integers that cannot be the shortest side of a right triangle? | [
"1, 2, 4"
] | [
"What are the only three positive integers that cannot be the shortest side of a right triangle?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 11
} | 11 | 0 | 0 |
omr-ws-de10d65560e3ee09b727934d3e46ddc0f81b5cfc564a54d0d5e1f673b857a3d4 | What are the perimeters of the two plots, given that each plot has an area of $400$ m$^2$ and the total length of the wall in front of the two plots is $36$ m? | [
"80 m and 82 m"
] | [
"What are the perimeters of the two plots, given that each plot has an area of $400$ m$^2$ and the total length of the wall in front of the two plots is $36$ m?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 17
} | 17 | 0 | 0 |
omr-ws-8b54f2c9841dd8352e2fdca12f8b2031738151280134e80d33af945ab221fc6d | What are the planar polygons whose area and perimeter are equal? | [
"Any polygon with an incircle radius of 2."
] | [
"What are the planar polygons whose area and perimeter are equal?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 16,
"genselect": 7,
"tir": 2
} | 25 | 0 | 0 |
omr-ws-914e492de9c0ab2966d0566fb284b09ec351e41cb3f64422d07cf5800117fd07 | What are the polynomials \( f \in K[X] \) such that if \( f(A) \) and \( f(B) \) commute, then \( A \) and \( B \) also commute? | [
"f(X) = aX + b"
] | [
"What are the polynomials \\( f \\in K[X] \\) such that if \\( f(A) \\) and \\( f(B) \\) commute, then \\( A \\) and \\( B \\) also commute?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-c93845504ebc6834dd408882fedb8ebe77e38131ffe6e4802f3a57b7d1637f1e | What are the positive integer numbers that can be obtained in exactly 2007 distinct ways as the sum of two or more consecutive positive integers? What is the smallest of all such numbers? Example: The number 9 can be written in exactly two such distinct ways: $9 = 4 + 5$ $9 = 2 + 3 + 4.$ | [
"3^{250} \\times 5 \\times 7 \\times 11"
] | [
"What are the positive integer numbers that can be obtained in exactly 2007 distinct ways as the sum of two or more consecutive positive integers? What is the smallest of all such numbers?\n\nExample: The number 9 can be written in exactly two such distinct ways:\n$9 = 4 + 5$\n$9 = 2 + 3 + 4.$"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-c6e93ec30b0a54d3ee62a5d37fd25f4a6c061fef2e7a1cc7b3cf01268b4c98bd | What are the positive integer values of \( n \) that make \( 2^n - 1 \) divisible by 7? | [
"\\( n = 3k \\) for \\( k \\in \\mathbb{Z}_{\\geq 0} \\)"
] | [
"What are the positive integer values of \\( n \\) that make \\( 2^n - 1 \\) divisible by 7?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 24
} | 39 | 0 | 0 |
omr-ws-e0f42ae7cea990a6584138b4837d0853d5f3c55653de11bd94fec0fbb8c20836 | What are the possible Galois groups over $\mathbb{Q}$ of an irreducible quartic polynomial $f(x)$ whose discriminant is negative? | [
"D_4"
] | [
"What are the possible Galois groups over $\\mathbb{Q}$ of an irreducible quartic polynomial $f(x)$ whose discriminant is negative?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-bab414ae19f001e13b0c08104b2f5ec3c2cd07c3a042d0934cbba22851c6b8dc | What are the possible distances between a point $P$ and the origin, if $P$ is $5$ units away from the point $(0, 3)$ on the $y$-axis? | [
"\\([2, 8]\\)"
] | [
"What are the possible distances between a point $P$ and the origin, if $P$ is $5$ units away from the point $(0, 3)$ on the $y$-axis?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-18004d76b813e49764aae548a32daad3357760a682264c3243260e4bab8402e5 | What are the possible greatest common divisors (GCDs) of the expressions $y^2 + 2 - 2y$ and $y^2 + 2 + 2y$? | [
"1, 2"
] | [
"What are the possible greatest common divisors (GCDs) of the expressions $y^2 + 2 - 2y$ and $y^2 + 2 + 2y$?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 4,
"tir": 21
} | 27 | 0 | 0 |
omr-ws-8f8ab0f598a0c7d3908776ad0701e75eb07658870cc6ddff4fda03af7097f342 | What are the possible lengths of horizontal line segments that start and finish on the graph of $x^3 - x$? | [
"\\( (0, 2] \\)"
] | [
"What are the possible lengths of horizontal line segments that start and finish on the graph of $x^3 - x$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 35,
"genselect": 8,
"tir": 5
} | 48 | 0 | 0 |
omr-ws-66098642098713353c997153f2b4e35c92a0a8a6116ba6f20e4a1daa60d35f55 | What are the possible lengths of horizontal line segments that start and finish on the graph of \( y = x^3 - x \)? | [
"\\(0\\) to \\(2\\)"
] | [
"What are the possible lengths of horizontal line segments that start and finish on the graph of \\( y = x^3 - x \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 36,
"genselect": 8,
"tir": 4
} | 48 | 0 | 0 |
omr-ws-65e7329a3486a451e2063da0aaa7ff5b0f9aef42ecf144bc6c60ed03e08c2cdb | What are the possible multiples that can arise when a natural number is an integral multiple of its reverse, as in the example 8712 = 4 x 2178? | [
"9"
] | [
"What are the possible multiples that can arise when a natural number is an integral multiple of its reverse, as in the example 8712 = 4 x 2178?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-a286ccdae68d3d3d995b80ec58fa191843b797399827bbd9df2275dc6d14f2c5 | What are the possible numbers of common external tangents that two congruent nonconcentric circles in the same plane can have? | [
"2, 3, 4"
] | [
"What are the possible numbers of common external tangents that two congruent nonconcentric circles in the same plane can have?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a9577ce74aa09e0aff8ec987a0566521d13bdd66c6a66f8a786ee03c77ec7510 | What are the possible numbers of elements of order 406 in a group \( G \) having a different number of elements of order 2? | [
"0"
] | [
"What are the possible numbers of elements of order 406 in a group \\( G \\) having a different number of elements of order 2?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-d9afc92eabb68d497bcff230da5eb8afd0b7e58967ced5559668ca85aeca1eb9 | What are the possible orders of elements of the group $G$, where $G$ is the set of all invertible $4 \times 4$ matrices over the integers mod 2? | [
"\\( 1, 2, 3, 4, 5, 6, 7, 15 \\)"
] | [
"What are the possible orders of elements of the group $G$, where $G$ is the set of all invertible $4 \\times 4$ matrices over the integers mod 2?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 2
} | 4 | 0 | 0 |
omr-ws-879c93923117bf248c7348568699300f2597cb681922d31e69744941c7387b55 | What are the possible positions of the midpoints $U$ of $BA$ and $V$ of $AC$ as $\angle OPA$ varies, given two concentric circles with radii $R$ and $r$ ($R > r$), a fixed point $P$ on the smaller circle, and points $B$ and $C$ on the larger circle such that $B, P, C$ are collinear and $BC$ is perpendicular to $AP$? | [
"\\frac{R}{2}"
] | [
"What are the possible positions of the midpoints $U$ of $BA$ and $V$ of $AC$ as $\\angle OPA$ varies, given two concentric circles with radii $R$ and $r$ ($R > r$), a fixed point $P$ on the smaller circle, and points $B$ and $C$ on the larger circle such that $B, P, C$ are collinear and $BC$ is perpendicular to $A... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6061de0a77019c3886d581e124522312cc3ca32c68c08553d506290c6ae10236 | What are the possible remainders of \( x^4 \) modulo 25, where \( x \) is a natural number less than 25? Can this result be extended to any number \( x \) and any number other than 25? | [
"21"
] | [
"What are the possible remainders of \\( x^4 \\) modulo 25, where \\( x \\) is a natural number less than 25? Can this result be extended to any number \\( x \\) and any number other than 25?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-495095aabf6997a78ae59cb856129b00d70bbdf83f60d9713f0d4e93fa306cc1 | What are the possible remainders when each of the primes \(a_1, a_2, \ldots, a_p\) is divided by \(p\)? | [
"\\(1, 2, \\ldots, p-1\\)"
] | [
"What are the possible remainders when each of the primes \\(a_1, a_2, \\ldots, a_p\\) is divided by \\(p\\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-aa7329266bc8d94ac879e1865fe37fe6a8bc08ab3d38a8d26a876f10e5884951 | What are the possible values for $\det(A)$ if $A$ is an idempotent matrix, i.e., $A^2 = A$? | [
"0 or 1"
] | [
"What are the possible values for $\\det(A)$ if $A$ is an idempotent matrix, i.e., $A^2 = A$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 29
} | 40 | 0 | 0 |
omr-ws-9fe9da2ac0fcda7642d45a293336f115d7ecd6becdba93b5d6cfbf0de8ada6c1 | What are the possible values for the minimum degree among the irreducible factors of a polynomial of degree 5? | [
"1"
] | [
"What are the possible values for the minimum degree among the irreducible factors of a polynomial of degree 5?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d0b82429a5b892e606f9a169750a6bda4628f321a4be338e3b47646ab6f3a8ba | What are the possible values of $\alpha$ such that the equation \[ \big[x\big]^{\big[x + \alpha\big]} = \big[x + \alpha\big]^{\big[x\big]} \] has real solutions? (Where $\big[.\big]$ is the greatest integer function) | [
"\\(\\alpha \\in (-3, -1) \\cup (-1, 1) \\cup (1, 3)\\)"
] | [
"What are the possible values of $\\alpha$ such that the equation \n\\[\n\\big[x\\big]^{\\big[x + \\alpha\\big]} = \\big[x + \\alpha\\big]^{\\big[x\\big]}\n\\]\nhas real solutions? (Where $\\big[.\\big]$ is the greatest integer function)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 23,
"genselect": 7,
"tir": 8
} | 38 | 0 | 0 |
omr-ws-cda71fa6e1d7d9105920fe99cc968c2b5719a1a858c5d05eb8c2711f589f91fe | What are the possible values of $\gcd(a_n, a_{n+1})$ for the sequence $a_n = n^3 + 2$ as $n$ varies? | [
"\\{1, 109\\}"
] | [
"What are the possible values of $\\gcd(a_n, a_{n+1})$ for the sequence $a_n = n^3 + 2$ as $n$ varies?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 8,
"tir": 21
} | 31 | 0 | 0 |
omr-ws-40f8269d567aea7269ff8931e8cac632f2bc92a7eca8219994c7ad0f804f6c24 | What are the possible values of the expression \(\sum_{1 \leq i < j \leq n} \frac{1}{n-1 + x_i x_j}\) when \(n\) is fixed and \(x_1, \ldots, x_n > 0\) have the sum of their inverses equal to 1? | [
"\\dfrac{n(n-1)}{2(n^2 + n - 1)}"
] | [
"What are the possible values of the expression \\(\\sum_{1 \\leq i < j \\leq n} \\frac{1}{n-1 + x_i x_j}\\) when \\(n\\) is fixed and \\(x_1, \\ldots, x_n > 0\\) have the sum of their inverses equal to 1?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-3e01b473b00d6fc54af890c1608a350e700b22e0dc35022cc1b54ee882654f68 | What are the possible values of the expression \[ \frac{x_1}{x_2 + 2(n-1)x_1 + x_3} + \frac{x_2}{x_3 + 2(n-1)x_2 + x_4} + \cdots + \frac{x_n}{x_1 + 2(n-1)x_n + x_2} \] when \( x_1, x_2, \ldots, x_n > 0 \)? | [
"\\(\\left(\\frac{1}{2(n-1)}, \\frac{1}{2}\\right]\\)"
] | [
"What are the possible values of the expression\n\\[\n\\frac{x_1}{x_2 + 2(n-1)x_1 + x_3} + \\frac{x_2}{x_3 + 2(n-1)x_2 + x_4} + \\cdots + \\frac{x_n}{x_1 + 2(n-1)x_n + x_2}\n\\]\nwhen \\( x_1, x_2, \\ldots, x_n > 0 \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-c32b0a51a070ac3acdec80e84ba1f7dab21fa745da2f132797fe92b3c5d39c87 | What are the prime ideals in $\mathbb{Z}[x]$? | [
"The prime ideals in \\(\\mathbb{Z}[x]\\) are \\((0)\\), \\(p\\mathbb{Z}[X]\\) for a prime \\(p\\), \\((p, F)\\) for a prime \\(p\\) and an irreducible polynomial \\(F\\) in \\(\\mathbb{Z}_p[X]\\), and \\((F)\\) for an irreducible polynomial \\(F\\) of positive degree with content 1."
] | [
"What are the prime ideals in $\\mathbb{Z}[x]$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 16
} | 22 | 0 | 0 |
omr-ws-d73f7760344b5b0127da6aa77c2015e049225a2d4de44d4cd602005b63251a48 | What are the prime ideals of the ring $ R$ of continuous real-valued functions on $ [0,1]$? | [
"M_x \\text{ for each } x \\in [0,1]"
] | [
"What are the prime ideals of the ring $ R$ of continuous real-valued functions on $ [0,1]$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-9d1e92dee1e2f4498fb8420535ce19d3a87c0a78a1d1ac1ff311ed4b7cbcb92e | What are the prime ideals of the ring $A_1 \times A_2$, where $A_1$ and $A_2$ are rings? | [
"\\( P_1 \\times A_2 \\) or \\( A_1 \\times P_2 \\) where \\( P_1 \\) is a prime ideal of \\( A_1 \\) and \\( P_2 \\) is a prime ideal of \\( A_2 \\)."
] | [
"What are the prime ideals of the ring $A_1 \\times A_2$, where $A_1$ and $A_2$ are rings?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 5
} | 19 | 0 | 0 |
omr-ws-056c327424d9ec3c90495fc646dc1e016cfbbcff106697f071eba3d4b526b5fe | What are the prime numbers \( p > 0 \) for which the decimal expansion of \( \frac{1}{p} \) is purely periodic with a period of 5 digits? | [
"41 and 271"
] | [
"What are the prime numbers \\( p > 0 \\) for which the decimal expansion of \\( \\frac{1}{p} \\) is purely periodic with a period of 5 digits?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 23
} | 24 | 0 | 0 |
omr-ws-9814cbcb8e716c9e7b53111be2f6219dc589522298f12a11a47eb9466a0f2e52 | What are the primes in $\mathbb{Z}[\sqrt{-2}]$? | [
"\\sqrt{-2}"
] | [
"What are the primes in $\\mathbb{Z}[\\sqrt{-2}]$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-dc37d5a444a9f2df3572842ca4b53f89b9d2ea35f13b696d18bf34d6aad68251 | What are the primes of the form $x^2 + 2y^2$? | [
"p = 2 \\text{ or } p \\equiv 1, 3 \\pmod{8}"
] | [
"What are the primes of the form $x^2 + 2y^2$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-aace5ccc4c6af72af214fe40bee9f2e8049c808e131eeba3e805120e70aff2f6 | What are the proceeds of a $9,500 note at a 12% discount rate for 8 months? Round your answer to the nearest cent. | [
"8740.00"
] | [
"What are the proceeds of a $9,500 note at a 12% discount rate for 8 months? Round your answer to the nearest cent."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6634813b4052be2c916740e7653ed6d8d845b9fd37d727978074d03cb3130277 | What are the properties of a convex hexagon such that each diagonal cuts off a triangle whose area is not less than $\frac{1}{6}$ of the area of the hexagon? | [
"\\text{The hexagon is regular.}"
] | [
"What are the properties of a convex hexagon such that each diagonal cuts off a triangle whose area is not less than $\\frac{1}{6}$ of the area of the hexagon?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-3c9ed154e2fad00147bcb513e15e35fd9ed9969f3f3f8a011727dd02a5b9bd11 | What are the properties of quadrilateral \(ABCD\) when \(AB^2 \cdot CD^2 + BC^2 \cdot AD^2 = AC^2 \cdot BD^2\)? | [
"Two opposite angles add up to \\(90^\\circ\\) and the other two opposite angles add up to \\(270^\\circ\\)."
] | [
"What are the properties of quadrilateral \\(ABCD\\) when \\(AB^2 \\cdot CD^2 + BC^2 \\cdot AD^2 = AC^2 \\cdot BD^2\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-020d9503b7b55480975ec4cd4be14371a05a63441d6f14da1e84b50452daae47 | What are the real factors of \( x^2 + 4 \)? | [
"Non-existent."
] | [
"What are the real factors of \\( x^2 + 4 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 5,
"tir": 21
} | 38 | 0 | 0 |
omr-ws-60b5927629c633369dad255afe05dac83d547babfee2ecb5f36c6e6689c9c95c | What are the real matrices \( A \) of size \( n \) such that \( A^T A A = I_n \)? | [
"I_n"
] | [
"What are the real matrices \\( A \\) of size \\( n \\) such that \\( A^T A A = I_n \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-501a099a650f4d2388383ad38a2e0b24dc8b274599439197a28c1dfadf3f5b24 | What are the real solutions to the equation \(2^x = x^2\)? | [
"\\(2, 4, -2\\frac{{\\text{LambertW} \\left( \\frac{\\ln \\left( 2 \\right)}{2} \\right)}}{\\ln \\left( 2 \\right)}\\) \n\nNote: The third solution is expressed using the Lambert W-function, which is not an elementary function."
] | [
"What are the real solutions to the equation \\(2^x = x^2\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 16
} | 18 | 0 | 0 |
omr-ws-c0d89f0db793ed822098d3a8236586803a62d163a98e6dfb9534eaf193b132ce | What are the representations of $e$ and $\pi$ in the most well-known and simple continued fractions, integrals, and infinite sums? | [
"\\pi"
] | [
"What are the representations of $e$ and $\\pi$ in the most well-known and simple continued fractions, integrals, and infinite sums?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-883d3925be3ab9b8f394d35fbb9551528c04cd2c8f52856069d38c40574f19fd | What are the restrictions on the functions \( f(x) \) and \( g(x) \) for the equation \[ \int_a^b \bigg(f(x) + g(x)\bigg) \, dx = \int_a^b f(x) \, dx + \int_a^b g(x) \, dx \] to hold true? | [
"f \\text{ and } g \\text{ must be integrable on } [a, b]"
] | [
"What are the restrictions on the functions \\( f(x) \\) and \\( g(x) \\) for the equation\n\\[ \\int_a^b \\bigg(f(x) + g(x)\\bigg) \\, dx = \\int_a^b f(x) \\, dx + \\int_a^b g(x) \\, dx \\]\nto hold true?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-810ee75311adba3184de695bdd832ad7c8c57185e6a843be0c87061288d90ad7 | What are the rightmost three digits of $5^{1993}$? | [
"125"
] | [
"What are the rightmost three digits of $5^{1993}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0877d2d679c6affd88a196e339ae36544650eaa15e2e55cfd1ee2b8dcd684dc8 | What are the rightmost three digits of the positive integer $2016^{{2015}^{2014}}$? | [
"376"
] | [
"What are the rightmost three digits of the positive integer $2016^{{2015}^{2014}}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-55a418f4e7c945dd4f1d69f532cb053cde67b2e2b60881e290409991ef4bb58d | What are the roots of the equation $ \cos\frac{\pi}{1+\sqrt{x}}+\cos{\frac{{\pi }}{1+2x}}=0$ on the segment $[-2013, 2013]$? | [
"\\( \\frac{1}{\\sqrt[3]{4}} \\)"
] | [
"What are the roots of the equation $ \\cos\\frac{\\pi}{1+\\sqrt{x}}+\\cos{\\frac{{\\pi }}{1+2x}}=0$ on the segment $[-2013, 2013]$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 15
} | 30 | 0 | 0 |
omr-ws-ab6eefe64c5297e06b1a97755bd08f92cdae95afa7f1706e74fec32001f703cb | What are the roots of the equation $3x^7 - 3x^6 + 1 = 0$? | [
"-0.758"
] | [
"What are the roots of the equation $3x^7 - 3x^6 + 1 = 0$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-b03881435819d246d7e40c951ffc8853d56d4c884ca02afd48c4d796a3c06e02 | What are the roots of the equation $x^4 - 4x - 1 = 0$? | [
"\\frac{-\\sqrt{2} \\pm i\\sqrt{4\\sqrt{2} + 2}}{2}"
] | [
"What are the roots of the equation $x^4 - 4x - 1 = 0$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-8ff9f60d0f83123d2741f55260ad415fc786937c1bd786f6027ba3aa5cf5d620 | What are the roots of the equation \( x(x^2 + 8x + 16)(4 - x) = 0 \)? | [
"0, 4, -4, -4"
] | [
"What are the roots of the equation \\( x(x^2 + 8x + 16)(4 - x) = 0 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 3,
"tir": 20
} | 34 | 0 | 0 |
omr-ws-abbfb438bd01c050149cbd49baba7bfc7b45e7a0173a5876c548faea94b8d449 | What are the roots of the equation \( x^9 = -1 \)? | [
"\\(\\cos\\left(\\frac{\\pi}{9}+\\frac{2k\\pi}{9}\\right) + i \\cdot \\sin\\left(\\frac{\\pi}{9}+\\frac{2k\\pi}{9}\\right) \\quad \\text{for} \\quad k=1,2,3,4,5,6,7,8.\\)"
] | [
"What are the roots of the equation \\( x^9 = -1 \\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 8
} | 8 | 0 | 0 |
omr-ws-664c1c8333182d71ec9f408a04e4409015b5f098ee63e44abafe9602413eb89e | What are the roots of the equation \(x^3 - 3x^2 + 1 = 0\)? | [
"\\( -\\frac{1}{2} \\sec 20^\\circ, \\frac{1}{2} \\sec 40^\\circ, \\frac{1}{2} \\sec 80^\\circ \\)"
] | [
"What are the roots of the equation \\(x^3 - 3x^2 + 1 = 0\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-8c24664b5639f64c767a389a20f694a0cbedd4f1398d94cae3c9e1b2cd7c0925 | What are the roots of the equation \(x^n + x^{n-1} + x^{n-2} + \cdots + x + 1 = 0\)? | [
"\\( e^{2k\\pi i / (n+1)} \\ \\ \\text{for} \\ \\ k=1,2,\\ldots, n \\)"
] | [
"What are the roots of the equation \\(x^n + x^{n-1} + x^{n-2} + \\cdots + x + 1 = 0\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-6f016ad74c5f0e82c11b707904fd99a2e14cb3e4424ce2e4359c8334f65b8cd8 | What are the roots of the equation \(x^{11} + x^{10} + \cdots + x + 1 = 0\)? | [
"\\(\\frac{\\sqrt{3}}{2} + \\frac{i}{2}, \\frac{1}{2} + \\frac{i\\sqrt{3}}{2}, i, -\\frac{1}{2} + \\frac{i\\sqrt{3}}{2}, -\\frac{\\sqrt{3}}{2} + \\frac{i}{2}, -1, -\\frac{\\sqrt{3}}{2} - \\frac{i}{2}, -\\frac{1}{2} - \\frac{i\\sqrt{3}}{2}, -i, \\frac{1}{2} - \\frac{i\\sqrt{3}}{2}, \\frac{\\sqrt{3}}{2} - \\frac{i}{2... | [
"What are the roots of the equation \\(x^{11} + x^{10} + \\cdots + x + 1 = 0\\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 5,
"tir": 27
} | 45 | 0 | 0 |
omr-ws-edb649f0dc1b45c856db325d11c9b2734092df5dadb2538aa2ec1ba44e8025d5 | What are the roots of the irreducible polynomial $f(x) = x^4 + x + 1$ over $\mathbb{F}_2$? | [
"\\(\\rho, \\rho^2, \\rho + 1, \\rho^2 + 1\\)"
] | [
"What are the roots of the irreducible polynomial $f(x) = x^4 + x + 1$ over $\\mathbb{F}_2$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 1,
"tir": 17
} | 21 | 0 | 0 |
omr-ws-6f68f2fcbed5c6327cc5a4d5225c28e692a1326e91efaa3c5dd0e1ad56bd8ac0 | What are the roots of the polynomial \(x^3 + x + 1\)? | [
"\\(\\frac{2}{t} - \\frac{t}{6}, \\quad \\frac{t}{12} - \\frac{1}{t} \\pm \\frac{\\sqrt{3}}{2} \\left( \\frac{t}{6} + \\frac{2}{t} \\right)i\\) with \\( t = \\sqrt[3]{108 + 12\\sqrt{93}} \\)"
] | [
"What are the roots of the polynomial \\(x^3 + x + 1\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 4,
"tir": 1
} | 10 | 0 | 0 |
omr-ws-1d2fd3c83ac11c20278c95bb23ed2f05d6e2bc988443a4c6219ad1b517ffce3c | What are the roots of the polynomial equation $(x)_n + (x+n+1)_n = 0$, where $(z)_k = z(z+1)\cdots(z+k-1)$ and $k, n \in \{1, 2, \ldots\}$? | [
"-n"
] | [
"What are the roots of the polynomial equation $(x)_n + (x+n+1)_n = 0$, where $(z)_k = z(z+1)\\cdots(z+k-1)$ and $k, n \\in \\{1, 2, \\ldots\\}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2fb71e130baad66b5d9b16279377f979c9940ae25b2e138480c5c6090c766c19 | What are the roots of the quadratic equation \(x^2 + 5 = 0\)? How do imaginary numbers function in this context? | [
"\\( \\sqrt{5}i \\) and \\( -\\sqrt{5}i \\)"
] | [
"What are the roots of the quadratic equation \\(x^2 + 5 = 0\\)? How do imaginary numbers function in this context?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-180c5cce49a8e8af8aa4a08915701c928d2e263d04607a7598984ee3e2efb53e | What are the semi-axes of the ellipsoid that contains two unit spheres in its interior and has the minimum surface area? | [
"\\dfrac{2}{\\sqrt{3}}"
] | [
"What are the semi-axes of the ellipsoid that contains two unit spheres in its interior and has the minimum surface area?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-97e7950a9ca1030f992f4f9abfca4cb60d2109a30944b8d2646f7cea62bc56c1 | What are the semi-axes of the ellipsoid that contains two unit spheres in its interior and has the minimum volume? | [
"\\dfrac{2}{\\sqrt{3}}"
] | [
"What are the semi-axes of the ellipsoid that contains two unit spheres in its interior and has the minimum volume?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-c288e62f0817a7804cdbfadfa3be3dca7463e331276b43b5fc3c6a90710b986c | What are the semimajor and semiminor axes of the ellipse that contains two unit circles in its interior and has the minimum arclength? | [
"\\dfrac{2\\sqrt{3}}{3}"
] | [
"What are the semimajor and semiminor axes of the ellipse that contains two unit circles in its interior and has the minimum arclength?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-edd65cd5bf681542e9dbadcd04a4f74291dad3dcfb3bfa885adf37176cc32642 | What are the semimajor and semiminor axes of the ellipse that contains two unit circles in its interior and has the minimum area? | [
"\\( a = \\frac{3\\sqrt{2}}{2} \\), \\( b = \\frac{\\sqrt{6}}{2} \\)"
] | [
"What are the semimajor and semiminor axes of the ellipse that contains two unit circles in its interior and has the minimum area?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-57598794e47390baaf0345e4fb4a05ebb287527562c510f3df5d47fbd9539b52 | What are the seven cyclic subgroups of $D_4$? | [
"\\( <1>, <\\rho^2>, <\\sigma>, <\\sigma \\rho>, <\\sigma \\rho^2>, <\\sigma \\rho^3>, <\\rho> \\)"
] | [
"What are the seven cyclic subgroups of $D_4$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 20
} | 25 | 0 | 0 |
omr-ws-05fc356e29ac114966228b7e560843a190ef52cc0814253a123bd1cb1cd6e1b6 | What are the side length ratios for a 15-75-90 right triangle? | [
"\\(1: \\sqrt{3}+2: \\sqrt{2}+\\sqrt{6}\\)"
] | [
"What are the side length ratios for a 15-75-90 right triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-99c3c8b9f3f0c59caa382cf4523f713b569ba23068019eac0e776ee9854229f6 | What are the side length relations of a triangle such that the median AD is divided into three equal parts by the incircle? | [
"\\( b = 0.5a, c = 1.3a \\)"
] | [
"What are the side length relations of a triangle such that the median AD is divided into three equal parts by the incircle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-96ca19590931ba85b5a32fb8dcb7722cc6f8b5203ccccdd83a0d25b2ea1ad67d | What are the sign and units digit of the product of all the odd negative integers strictly greater than $-2015$? | [
"It is a negative number ending with a 5."
] | [
"What are the sign and units digit of the product of all the odd negative integers strictly greater than $-2015$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 3,
"tir": 19
} | 36 | 0 | 0 |
omr-ws-d646123621ac8424a1d4a02aba061d2764778659a681976e781876dd8b9a73c2 | What are the slopes of the tangents from the origin to the circle $(x-3)^2+(y-4)^2=4$? | [
"\\(\\frac{12 + 2\\sqrt{21}}{5}\\) and \\(\\frac{12 - 2\\sqrt{21}}{5}\\)"
] | [
"What are the slopes of the tangents from the origin to the circle $(x-3)^2+(y-4)^2=4$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 24
} | 30 | 0 | 0 |
omr-ws-eaf64903f7e9c83acbf5942f4f514cce3e8508a709bf0735242a18f8ce0cc4b8 | What are the smallest possible values of \(a\), \(b\), and \(c\) so that the number 1997abc is divisible by 2, 3, 5, and 11? | [
"\\(a = 1\\), \\(b = 6\\), \\(c = 0\\)"
] | [
"What are the smallest possible values of \\(a\\), \\(b\\), and \\(c\\) so that the number 1997abc is divisible by 2, 3, 5, and 11?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 24
} | 39 | 0 | 0 |
omr-ws-7a527566ca2a21b5af6c50f4f2106a0052499c987e4d95278b048c0d50e9599c | What are the solutions of a multiplicative function $f \colon \mathbb{N} \to \mathbb{N}$ such that $f(mn) = f(m)f(n)$? | [
"\\( f(n) = \\prod_{k=1}^s f(p_k)^{e_k} \\) where \\( n = \\prod_{k=1}^s p_k^{e_k} \\) and \\( f \\) can be defined arbitrarily on the primes."
] | [
"What are the solutions of a multiplicative function $f \\colon \\mathbb{N} \\to \\mathbb{N}$ such that $f(mn) = f(m)f(n)$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 28,
"genselect": 8,
"tir": 16
} | 52 | 0 | 0 |
omr-ws-f5b1079b12bdb8dce87161fd33350ab317eb0f922a6db90dacc58582e3ca7609 | What are the solutions of the differential equation $f'(x) = f^{-1}(x)$? | [
"For \\( x > 0 \\):\n\\[ f(x) = \\left(\\frac{1 + \\sqrt{5}}{2}\\right)^{\\frac{1 - \\sqrt{5}}{2}} x^{\\frac{1 + \\sqrt{5}}{2}} \\]\n\nFor \\( x < 0 \\):\n\\[ f(x) = -\\left(\\frac{-1 + \\sqrt{5}}{2}\\right)^{\\frac{1 + \\sqrt{5}}{2}} (-x)^{\\frac{1 - \\sqrt{5}}{2}} \\]"
] | [
"What are the solutions of the differential equation $f'(x) = f^{-1}(x)$?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bd1ddc1881f61cb5bdd31d73dc88592d60995410dfc1f4822b3b54635dfc09b0 | What are the solutions of the equation $x^2 - 64 = 0$ in the complex numbers? | [
"\\( x = -8, 8 \\)"
] | [
"What are the solutions of the equation $x^2 - 64 = 0$ in the complex numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 27
} | 32 | 0 | 0 |
omr-ws-8f9578d88f2ef937590ed69944c371e3167038f9132d7773116af5ee73c9a800 | What are the solutions of the equation $x^3 + 2x^2t + 3xt^2 + 4t^3 = 0$? | [
"\\( t = -x \\left( \\sqrt[3]{\\frac{5}{64} + \\frac{5}{24 \\sqrt{6}}} + \\sqrt[3]{\\frac{5}{64} - \\frac{5}{24 \\sqrt{6}}} + \\frac{1}{4} \\right) \\)"
] | [
"What are the solutions of the equation $x^3 + 2x^2t + 3xt^2 + 4t^3 = 0$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-854b5e044737b21f23907cee4f9720bd11c99bf19d073051bbd09f6c51e52d0f | What are the solutions of the equation \( t^{r^k} = t \) for \( t \) in the finite field \( F_q \), where \( F_q \) is a finite field with \( q = r^f \) and \( k < f \)? | [
"\\( \\mathbb{F}_{r^{gcd(k,f)}} \\)"
] | [
"What are the solutions of the equation \\( t^{r^k} = t \\) for \\( t \\) in the finite field \\( F_q \\), where \\( F_q \\) is a finite field with \\( q = r^f \\) and \\( k < f \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 12
} | 33 | 0 | 0 |
omr-ws-2903181f353bd501d097b8dd9cc69fe25e6805b5fcc4af751c55e0caf2dd8a66 | What are the solutions to the equation $\sqrt{n} \leq \delta(n)$, where $\delta(n)$ is the number of divisors of $n$? Is the number of solutions finite? | [
"The solutions are finite and listed above."
] | [
"What are the solutions to the equation $\\sqrt{n} \\leq \\delta(n)$, where $\\delta(n)$ is the number of divisors of $n$? Is the number of solutions finite?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 4,
"tir": 22
} | 39 | 0 | 0 |
omr-ws-4bd3b75fd16b2daea8a78bd42fc30aee319cdac8afe809fd95e2e1ad0bbd2fc5 | What are the solutions to the equation $x^x = 1$? | [
"1"
] | [
"What are the solutions to the equation $x^x = 1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ffbc96815cf9185da6a97b7da63ec6a07232737e1850420c404d4743c72ba816 | What are the solutions to the equation \( z^6 + 2z^3 + 2 = 0 \)? | [
"\\( z = \\frac{\\sqrt{2}}{2} e^{i\\frac{(2k\\pi - \\frac{3\\pi}{4})}{3}} \\) for \\( k = 1, 2, 3 \\) and \\( z = \\frac{\\sqrt{2}}{2} e^{i\\frac{(2k\\pi - \\frac{3\\pi}{4})}{3}} \\) for \\( k = 1, 2, 3 \\)"
] | [
"What are the solutions to the equation \\( z^6 + 2z^3 + 2 = 0 \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 30
} | 41 | 0 | 0 |
omr-ws-7ed89734ed99effb23b9092f273aa6c72084110d766a567935050afa020b45a5 | What are the subgroups of $Z_{2012}$ and what are their orders? | [
"Orders: 1, 2, 4, 503, 1006, 2012; Subgroups: \\{0\\}, \\{0, 1006\\}, \\{0, 503, 1006, 1509\\}, \\{0, 4, 8, ..., 2004, 2008\\}, \\{0, 2, 4, ..., 2008, 2010\\}, \\{0, 1, 2, ..., 2010, 2011\\}"
] | [
"What are the subgroups of $Z_{2012}$ and what are their orders?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 22
} | 22 | 0 | 0 |
omr-ws-fe5d213c27b544bfdfaedb9344a90f89f7b143630e6e4ea6e8e5e6549eeaa9ad | What are the subgroups of \((\mathbb{Z}, +)\) and their related conjugate classes? | [
"\\text{All subgroups are } n\\mathbb{Z} \\text{ for } n \\in \\mathbb{N}_0 \\text{, each with conjugacy class } \\{n\\mathbb{Z}\\}"
] | [
"What are the subgroups of \\((\\mathbb{Z}, +)\\) and their related conjugate classes?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d9df8363bbd703eff71025fba1bb415549b63c46c3b30606abbc827fd54747a9 | What are the subgroups of the group $(\mathbb{Z}_3 \times \mathbb{Z}_3, \circ)$, where $\circ$ is the componentwise addition operation defined by $(a,b) \circ (c,d) = (a+c, b+d)$? | [
"\\(\\{(0,0)\\}, \\mathbb{Z}_3 \\times \\mathbb{Z}_3, \\{(0,0),(1,1),(2,2)\\}, \\{(0,0),(0,1),(0,2)\\}, \\{(0,0),(1,0),(2,0)\\}, \\{(0,0),(2,1),(1,2)\\}\\)"
] | [
"What are the subgroups of the group $(\\mathbb{Z}_3 \\times \\mathbb{Z}_3, \\circ)$, where $\\circ$ is the componentwise addition operation defined by $(a,b) \\circ (c,d) = (a+c, b+d)$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 8,
"tir": 27
} | 35 | 0 | 0 |
omr-ws-7170e1bfe784a302512f72d2ff6453d24f89fc42aae10d1369404b569656de0f | What are the subrings of $\mathbb{Z} \times \mathbb{Z}$? | [
"m\\mathbb{Z} \\times n\\mathbb{Z} \\text{ and } \\{(a, b) \\mid a \\equiv b \\pmod{d}\\} \\text{ for some non-negative integers } m, n, d"
] | [
"What are the subrings of $\\mathbb{Z} \\times \\mathbb{Z}$?"
] | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 6
} | 7 | 0 | 0 |
omr-ws-ea289974a75e944f464195634b310d4dab030ab380d04b0905c66e70b92f728e | What are the subrings of the quaternions $\mathbb{H}$ that form an integral domain, excluding the obvious examples $\mathbb{C}$, $\mathbb{R}$, $\mathbb{Q}$, and $\mathbb{Z}$? | [
"\\mathbb{Z}[i]"
] | [
"What are the subrings of the quaternions $\\mathbb{H}$ that form an integral domain, excluding the obvious examples $\\mathbb{C}$, $\\mathbb{R}$, $\\mathbb{Q}$, and $\\mathbb{Z}$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-1fe29963d47cc3c8d9c63e44b7ffb5852848286ebdd89ee49fb11c8b37b6c1b7 | What are the subsets $A$ of the sphere $S^1$ such that there exists a continuous function $f: \mathbb{R}^2 \to \mathbb{R}^2$ with $A$ being the image of the unit closed ball under $f$ and $f$ being the identity on $A$? | [
"compact arcs of the circle."
] | [
"What are the subsets $A$ of the sphere $S^1$ such that there exists a continuous function $f: \\mathbb{R}^2 \\to \\mathbb{R}^2$ with $A$ being the image of the unit closed ball under $f$ and $f$ being the identity on $A$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 7,
"tir": 5
} | 22 | 0 | 0 |
omr-ws-a2d537a8d66d25b429aff104023502067e89d081e138b7832f72cd4a77442ac8 | What are the sufficient and necessary conditions for a sequence $\{a_i\}$ such that $\sum_{i=1}^{\infty} (-1)^i a_i$ is convergent? | [
"\\text{The series converges if and only if } \\lim_{i \\to \\infty} a_i = 0 \\text{ and } \\{a_i\\} \\text{ is eventually non-increasing.}"
] | [
"What are the sufficient and necessary conditions for a sequence $\\{a_i\\}$ such that $\\sum_{i=1}^{\\infty} (-1)^i a_i$ is convergent?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-08d33256f8b56a4b646a381e4cb99666882a20168dfd15da6980d5eaf8c04a3a | What are the sufficient and/or necessary conditions on the coefficients for a monic polynomial to take only positive values on \( \mathbb{R}^+ \) or \( \mathbb{R}^* \)? | [
"\\text{All coefficients are non-negative}"
] | [
"What are the sufficient and/or necessary conditions on the coefficients for a monic polynomial to take only positive values on \\( \\mathbb{R}^+ \\) or \\( \\mathbb{R}^* \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-a62a2205eba70b9d9337ef2edd1fa9920fe3e7c8a8ca574bcabfdf84d563eed1 | What are the sufficient or necessary conditions on the coefficients of a real polynomial of degree \(2n\) such that \(p(x) \ge 0\) for all real numbers \(x\)? | [
"p(x) \\text{ is a sum of squares of real polynomials}"
] | [
"What are the sufficient or necessary conditions on the coefficients of a real polynomial of degree \\(2n\\) such that \\(p(x) \\ge 0\\) for all real numbers \\(x\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 5
} | 19 | 0 | 0 |
omr-ws-119466352939dc2d8876c9638dd73013f845df9521244759e451b48477054e40 | What are the three largest prime factors of 555,555,555,555? | [
"37, 101, 9901"
] | [
"What are the three largest prime factors of 555,555,555,555?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 26
} | 28 | 0 | 0 |
omr-ws-675bc7b1b60fd222ab5aeb420107ae53ffb4c0db4512b3dd003a7e0e963e3ba9 | What are the three least significant bits in the binary representation of $3^{101} + 2$? | [
"011"
] | [
"What are the three least significant bits in the binary representation of $3^{101} + 2$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4d7a2002a602fd0c8df6fa79faa4d58e7f43ccfa13983cf701a5f879228c54f4 | What are the total number of different integer values the function \( f(x) = \lfloor x \rfloor + \lfloor 2x \rfloor + \left\lfloor \frac{5x}{3} \right\rfloor + \lfloor 3x \rfloor + \lfloor 4x \rfloor \) takes for real numbers \( x \) with \( 0 \leq x \leq 100 \)? | [
"734"
] | [
"What are the total number of different integer values the function \\( f(x) = \\lfloor x \\rfloor + \\lfloor 2x \\rfloor + \\left\\lfloor \\frac{5x}{3} \\right\\rfloor + \\lfloor 3x \\rfloor + \\lfloor 4x \\rfloor \\) takes for real numbers \\( x \\) with \\( 0 \\leq x \\leq 100 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-79139b79924d5741e09a2ed5054d9089545ab662dc9a78ac0dd9cd3502e5149b | What are the two angles formed by the diagonals in a rhombus? | [
"90^\\circ \\text{ and } 90^\\circ"
] | [
"What are the two angles formed by the diagonals in a rhombus?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a973e4d3d50b8a0e6187b0406a57835302417b3e5e13a8338c7f6869e50cbcd9 | What are the two arithmetic means between \( -13 \) and \( 8 \)? | [
"1"
] | [
"What are the two arithmetic means between \\( -13 \\) and \\( 8 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-8bfeec6ee55a1b25d5dd1a501ebbbfc65d4bbe9497af2733e284a2dd62dcc4aa | What are the two different cubes that sum to 1729? | [
"\\(10^3 + 9^3\\) and \\(12^3 + 1^3\\)"
] | [
"What are the two different cubes that sum to 1729?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 27
} | 38 | 0 | 0 |
omr-ws-fdd7c02cb6169cef9aa0a83d6debddb721832382ece97ef82d05a54cab03c25d | What are the two divisors of \( 4^{12} - 1 \) that lie between 60 and 80? | [
"63 and 65"
] | [
"What are the two divisors of \\( 4^{12} - 1 \\) that lie between 60 and 80?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 32
} | 32 | 0 | 0 |
omr-ws-0a75b66181b7617cd7a0306ea0b801f8e6db89358750586ce2d3df525a0dfb87 | What are the unit and tenth digits of $2^{19}$? | [
"88"
] | [
"What are the unit and tenth digits of $2^{19}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 23
} | 37 | 0 | 0 |
omr-ws-778170784113c797bd049670e7c19603aa5dce7cd6b1457775048e13aadedde6 | What are the unit and tenth digits when $2007$ is divided by $9$? | [
"3 and 2"
] | [
"What are the unit and tenth digits when $2007$ is divided by $9$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 25
} | 47 | 0 | 0 |
omr-ws-446210b0dbafab60b6eb475bf766548ec5bb1c930d1f33b49b3935303179c676 | What are the values of $-1^i$ and $-1^{-i}$? | [
"$e^{-\\pi}$ and $e^{\\pi}$"
] | [
"What are the values of $-1^i$ and $-1^{-i}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 17
} | 18 | 0 | 0 |
omr-ws-b9a8ddb8090e548a9f5f2fd77a5b0154680eb8e32a599368a956618ad3ba48c5 | What are the values of $\omega^3$ and $1 + \omega + \omega^2$, where $1, \omega,$ and $\omega^2$ are the three cube roots of unity? | [
"\\(\\omega^3 = 1\\) and \\(1 + \\omega + \\omega^2 = 0\\)"
] | [
"What are the values of $\\omega^3$ and $1 + \\omega + \\omega^2$, where $1, \\omega,$ and $\\omega^2$ are the three cube roots of unity?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 29
} | 29 | 0 | 0 |
omr-ws-04c288c2e7db98c42aa6b261367af57837473929124618563e4d6f04091284e7 | What are the values of $k$ such that the roots of the quadratic equation $2x^2 + kx + 12 = 0$, denoted by $p$ and $q$, satisfy $1 < p < 2$ and $3 < q < 4$? | [
"\\( -11 < k < -10 \\)"
] | [
"What are the values of $k$ such that the roots of the quadratic equation $2x^2 + kx + 12 = 0$, denoted by $p$ and $q$, satisfy $1 < p < 2$ and $3 < q < 4$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 1,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-5f57fe7c4e64775f5c98b1b3d15e7089f7e0c031269df2bf5119cad3c3c42cf1 | What are the values of $n$ when $T_{n}$ is maximum for the sequence defined by $T_{1}= 1$, $T_{2n}= T_{n}$, and $T_{2n+1}=T_{2n}+1$? | [
"\\( 524287, 1048575, 1572863 \\)"
] | [
"What are the values of $n$ when $T_{n}$ is maximum for the sequence defined by $T_{1}= 1$, $T_{2n}= T_{n}$, and $T_{2n+1}=T_{2n}+1$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 3,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-6a18de90703c05218a908ab73c85e913e873f764325108bd2547db166b4b1cbf | What are the values of \( k \) such that there exist different prime numbers \( p_1, p_2, \ldots, p_k \) satisfying: \[ p_1^2 + p_2^2 + \cdots + p_k^2 = 2010 \] | [
"7"
] | [
"What are the values of \\( k \\) such that there exist different prime numbers \\( p_1, p_2, \\ldots, p_k \\) satisfying:\n\\[ p_1^2 + p_2^2 + \\cdots + p_k^2 = 2010 \\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 8,
"tir": 4
} | 15 | 0 | 0 |
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