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-37fd4abdeec82e134468fbd15329bbc0732baff818a83f9ca986ecd2207c6980 | What is the dimension of the space of all $n \times n$ matrices with real entries such that the sum of the entries in the first row and the sum of the diagonal entries are both zero? | [
"\\( n^2 - 2 \\)"
] | [
"What is the dimension of the space of all $n \\times n$ matrices with real entries such that the sum of the entries in the first row and the sum of the diagonal entries are both zero?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
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"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-11a244274d3922de8851ad58c26bea770072c33b3a08784426080f4f1d9a1059 | What is the dimension of the subspace \( U \) of \( \mathbb{R}^7 \), which coincides with the set of solutions of the homogeneous system: \[ x_1 = 0 \] \[ x_3 = 0 \] \[ 4x_5 - 10x_7 = 0 \] \[ 6x_5 - 15x_7 = 0 \] | [
"4"
] | [
"What is the dimension of the subspace \\( U \\) of \\( \\mathbb{R}^7 \\), which coincides with the set of solutions of the homogeneous system:\n\\[ x_1 = 0 \\]\n\\[ x_3 = 0 \\]\n\\[ 4x_5 - 10x_7 = 0 \\]\n\\[ 6x_5 - 15x_7 = 0 \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 16,
"genselect": 3
} | 19 | 0 | 0 |
omr-ws-1e7950aa5a0ab0e1ef08718440609b35ccd514985c1d2fcd8616496c07a710cb | What is the dimension of the vector space $V(100)$ over the field $\mathbb{Q}$ of rational numbers, where $V(100)$ is spanned by the numbers $\log(2), \log(3), \dots, \log(100)$? | [
"25"
] | [
"What is the dimension of the vector space $V(100)$ over the field $\\mathbb{Q}$ of rational numbers, where $V(100)$ is spanned by the numbers $\\log(2), \\log(3), \\dots, \\log(100)$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-882a25c4047d96653ccd0a87f5b102ec7ebcb9da787a40e1d3196c9a14a97e03 | What is the dimension of the vector space $V(n)$ over the field $\mathbb{Q}$ of rational numbers, where $V(n)$ is spanned by the numbers $\log(2), \log(3), \dots, \log(n)$ for a general $n$? | [
"\\(\\pi(n)\\)"
] | [
"What is the dimension of the vector space $V(n)$ over the field $\\mathbb{Q}$ of rational numbers, where $V(n)$ is spanned by the numbers $\\log(2), \\log(3), \\dots, \\log(n)$ for a general $n$?"
] | [
"aops_c7_college_math"
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"cot",
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"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-d962f3d851e237f0bbde732bda993b3fed01197deedc33a4155b4124cb9b7ce9 | What is the dimension of the vector space of all 3-linear alternating functions $D: F^{3 \times 3} \rightarrow F$? | [
"1"
] | [
"What is the dimension of the vector space of all 3-linear alternating functions $D: F^{3 \\times 3} \\rightarrow F$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-767dac6ff2f5a89d6db4a5fd7b14e14b7b74b7c4436c21d0cc3dc89d33f06b98 | What is the dimension of the vector space of all 3-linear functions $D: F^{3 \times 3} \rightarrow F$? | [
"27"
] | [
"What is the dimension of the vector space of all 3-linear functions $D: F^{3 \\times 3} \\rightarrow F$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-099b600548fe06e4c696c3c555e0c8397fbcd8689dd9351f010febb65ad5773b | What is the dimension of the vector space of symmetric polynomials with $k$ variables, where each variable has a partial degree at most $n$, and the coefficients are in $\mathbb{C}$? | [
"\\(\\binom{n+k}{k}\\)"
] | [
"What is the dimension of the vector space of symmetric polynomials with $k$ variables, where each variable has a partial degree at most $n$, and the coefficients are in $\\mathbb{C}$?"
] | [
"aops_c7_college_math"
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"cot",
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"cot": 32,
"tir": 5
} | 37 | 0 | 0 |
omr-ws-0331efb7151860ce8a69f499081a4f71d5461b24610461cd7a51620ad9e21706 | What is the discriminant of $3x^2 - 7x - 12$? | [
"193"
] | [
"What is the discriminant of $3x^2 - 7x - 12$?"
] | [
"MATH_training_set"
] | [
"cot"
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"cot": 16
} | 16 | 0 | 0 |
omr-ws-5baf43e947a9899b143989104c068c57ce1191df75da6a50d3ce3a4c7a46e1ab | What is the discriminant of the quartic equation $x^4 + ax^2 + bx + c = 0$? | [
"256c^3 - 128a^2c^2 + 144ab^2c + 16a^4c - 4a^3b^2 - 27b^4"
] | [
"What is the discriminant of the quartic equation $x^4 + ax^2 + bx + c = 0$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-9cc64a9c91ccbad06e3c028f37f2f3e1900021e1a03dba809ba5aa750d52fbe7 | What is the distance between (-2,4) and (3,-8)? | [
"13"
] | [
"What is the distance between (-2,4) and (3,-8)?"
] | [
"MATH_training_set"
] | [
"cot"
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"cot": 16
} | 16 | 0 | 0 |
omr-ws-a881af6d55c178c55b0ad18766fa9ed6491836b2e0e089dedf2f8e779845636e | What is the distance between the center of the circle with equation $x^2+y^2=-4x+6y-12$ and the point $(1,7)$? | [
"5"
] | [
"What is the distance between the center of the circle with equation $x^2+y^2=-4x+6y-12$ and the point $(1,7)$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cfa5555b0644a6a956b9be8de3e94acde5556a7ab475f88758f10fff30cd20ab | What is the distance between the center of the circle with equation $x^2+y^2=2x+4y-1$ and the point $(13,7)$? | [
"13"
] | [
"What is the distance between the center of the circle with equation $x^2+y^2=2x+4y-1$ and the point $(13,7)$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e1d1ab1a1b60da9cb6cde5a9e91d4108932ec6d8119b142644da862e3b9ecb50 | What is the distance between the foci of the hyperbola \(xy = 2\)? | [
"\\(4\\sqrt{2}\\)"
] | [
"What is the distance between the foci of the hyperbola \\(xy = 2\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-4da46a120c2129776c81b12152497e2778021f27dd63f6cd82fdbf6efcf0da57 | What is the distance between the foci of the hyperbola given by the equation \(xy = 1\)? | [
"\\(2\\sqrt{2}\\)"
] | [
"What is the distance between the foci of the hyperbola given by the equation \\(xy = 1\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e4673f660cd997095a0eaadb9252b30b1b6e4316eded0edb3dbb069c984c2019 | What is the distance between the foci of the rectangular hyperbola defined by \(xy = 2\)? | [
"\\(4\\sqrt{2}\\)"
] | [
"What is the distance between the foci of the rectangular hyperbola defined by \\(xy = 2\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-d37ced9ebc90472b8a1bd9bb649eec565cbb275ab2f96cd280c041dedf22edef | What is the distance between the incenter and orthocenter of triangle ABC, where the sides have lengths AB=7, BC=8, and AC=9? | [
"\\dfrac{\\sqrt{30}}{5}"
] | [
"What is the distance between the incenter and orthocenter of triangle ABC, where the sides have lengths AB=7, BC=8, and AC=9?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-c93d805d21528a6f38929c4b37f6a011ce2e5dff92761a44711abac197d9cf92 | What is the distance between the incenter and the centroid of a triangle whose side lengths are $2021$, $2022$, $2023$? | [
"\\dfrac{1}{3}"
] | [
"What is the distance between the incenter and the centroid of a triangle whose side lengths are $2021$, $2022$, $2023$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 7
} | 20 | 0 | 0 |
omr-ws-278d3dbdf69a3ef6b21c9ccee05caee81f8fece6d57f4fbc5ce6ac24ac10cefe | What is the distance between the inner and outer Soddy Circles of the Malfatti Circles inside a triangle with sides 13 cm, 14 cm, and 15 cm? | [
"\\( \\frac{672\\sqrt{37}}{1727} \\)"
] | [
"What is the distance between the inner and outer Soddy Circles of the Malfatti Circles inside a triangle with sides 13 cm, 14 cm, and 15 cm?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e9521d1fdca48288d12cfb4044e59350d49996acb12fd76037fe1f375e88ca22 | What is the distance between the points $\frac{2eh}{e+h}$ and $e$ in the complex plane? | [
"\\dfrac{|e(e - h)|}{|e + h|}"
] | [
"What is the distance between the points $\\frac{2eh}{e+h}$ and $e$ in the complex plane?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-73fc7e8c1153b33176f2791cfae8b3597c0fb3519290db6ba4e2a14a4884cfb6 | What is the distance between the tip of the hour hand and the tip of the minute hand when the clock reads 12:30, given that the hour hand has a length of 6 centimeters and the minute hand has a length of 12 centimeters? | [
"\\(6 \\sqrt{5 + \\sqrt{6} + \\sqrt{2}}\\)"
] | [
"What is the distance between the tip of the hour hand and the tip of the minute hand when the clock reads 12:30, given that the hour hand has a length of 6 centimeters and the minute hand has a length of 12 centimeters?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 8,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-c1835d31e09619af9939ddd9e3f438aafd9bf5f85ef1aeee73ffda5b0f95ed72 | What is the distance between the two $x$-intercepts of the parabola that passes through the vertices $A, B, C,$ and $D$ of a regular hexagon $ABCDEF$ with side length $2$, where points $E$ and $F$ are on the $x$-axis? | [
"\\( 2\\sqrt{7} \\)"
] | [
"What is the distance between the two $x$-intercepts of the parabola that passes through the vertices $A, B, C,$ and $D$ of a regular hexagon $ABCDEF$ with side length $2$, where points $E$ and $F$ are on the $x$-axis?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 43,
"tir": 21
} | 64 | 0 | 0 |
omr-ws-216b9cf24bf5977e1b680e043bec3b2753129c0372831ef6f659ceab0ad3b631 | What is the distance between the two foci of the hyperbola \(xy = 2\)? | [
"4"
] | [
"What is the distance between the two foci of the hyperbola \\(xy = 2\\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b073927c4f8e5f43f95d19732da5ae7c73b4c9abe9babfd576e4d187cb17871d | What is the distance from the origin to the graph of $y=\sqrt{6x-x^{2}-9}$? | [
"3"
] | [
"What is the distance from the origin to the graph of $y=\\sqrt{6x-x^{2}-9}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7ed2d02901c13ee78831bf826836527d854b957946eace8657793391237a35cb | What is the distance on a Cartesian coordinate plane from $(1, -1)$ to $(7, 7)?$ | [
"10"
] | [
"What is the distance on a Cartesian coordinate plane from $(1, -1)$ to $(7, 7)?$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-61d1869e1b2ecbac89a08e23e5840ee3bf6fe3b28320bdf44c7440cfbab67189 | What is the distance traveled by a building in Tampa, Florida (at a latitude of about $28^\circ$ North), in one day (in space) as a result of the Earth's rotation? The equatorial radius of the Earth is $6.38 \times 10^6$ m. | [
"\\(35749787.7 \\text{ m}\\)"
] | [
"What is the distance traveled by a building in Tampa, Florida (at a latitude of about $28^\\circ$ North), in one day (in space) as a result of the Earth's rotation? The equatorial radius of the Earth is $6.38 \\times 10^6$ m."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-09a122ae14f8199c09bbcfb3e03439361b9580e9a09397ce37c3b60119630c7f | What is the distance, in units, between the points $(-3, -4)$ and $(4, -5)$? Express your answer in simplest radical form. | [
"5\\sqrt{2}"
] | [
"What is the distance, in units, between the points $(-3, -4)$ and $(4, -5)$? Express your answer in simplest radical form."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-821ccc334950b50cd504622b356a4bd2026c60b52aaa61f2ebde39eaa28372ea | What is the distance, in units, between the points $(3, -2)$ and $(7, 5)$? | [
"\\sqrt{65}"
] | [
"What is the distance, in units, between the points $(3, -2)$ and $(7, 5)$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9833cc03eaf59bd748296d9b964aac8cdf399562ad89d41be912f75f14cc4898 | What is the distribution of $(B_5 - B_3)^2$, where $B_t$ is standard Brownian motion? ($B_t$ is normally distributed with mean $0$ and variance $t$). | [
"\\( 2 \\chi^2(1) \\)"
] | [
"What is the distribution of $(B_5 - B_3)^2$, where $B_t$ is standard Brownian motion? ($B_t$ is normally distributed with mean $0$ and variance $t$)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-394c4369eb677b0c99808a205481c4e7eced5bd0ee0a02837961a3036c1e7042 | What is the distribution of $(\sqrt{2}Z)^2$, where $Z$ is a standard normal random variable? | [
"\\( 2\\chi^2_1 \\)"
] | [
"What is the distribution of $(\\sqrt{2}Z)^2$, where $Z$ is a standard normal random variable?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 5,
"tir": 17
} | 32 | 0 | 0 |
omr-ws-fbcb01d95bea4c097724715230cc70dfd1cb37c3dc0cc767c20bc62f07483bad | What is the distribution of $L_T$, where $T = \inf\{t : B_t = 1\}$ and $(L_t)$ is the local time of a standard Brownian motion $(B_t)$ at $0$? | [
"\\text{Exp}\\left(\\dfrac{1}{2}\\right)"
] | [
"What is the distribution of $L_T$, where $T = \\inf\\{t : B_t = 1\\}$ and $(L_t)$ is the local time of a standard Brownian motion $(B_t)$ at $0$?"
] | [
"aops_c7_college_math"
] | [
"cot",
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] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-f3be15c52bc87005c74f6d407befd5166f803fd10aa4194324b112874b4d1140 | What is the distribution of $L_\tau$, where $\tau = \inf\{t : |B_t| = 1\}$ and $(L_t)$ is the local time of a standard Brownian motion $(B_t)$ at $0$? | [
"\\text{Exp}(1)"
] | [
"What is the distribution of $L_\\tau$, where $\\tau = \\inf\\{t : |B_t| = 1\\}$ and $(L_t)$ is the local time of a standard Brownian motion $(B_t)$ at $0$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9c004d1057d2fd8b001bd4f3dd13603a28f166a6ac7ee9ed08a89dec334eb226 | What is the distribution of leading digits of the powers of 2 in base 3? | [
"\\{1: \\log_3 2, 2: 1 - \\log_3 2\\}"
] | [
"What is the distribution of leading digits of the powers of 2 in base 3?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-fe0818d8e8b41a140b0a40f63f5c8a101d539380594f3ec08c8bfce37ce993a6 | What is the distribution of primes of the form $2^n + 1$? Specifically, how many such primes are there less than a given number $x$? | [
"5"
] | [
"What is the distribution of primes of the form $2^n + 1$? Specifically, how many such primes are there less than a given number $x$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-1f1106c9b9b2d4378e34662aea33f1fb4a3413dca938a0d036529724d977c755 | What is the domain of $\sqrt{\arccos(x) - \lfloor x \rfloor}$? | [
"$(-\\infty, 2\\pi-3]$"
] | [
"What is the domain of $\\sqrt{\\arccos(x) - \\lfloor x \\rfloor}$?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9adce17a9b08c7fc618a5410b551c49742646b7f59d055d744010657a718d92b | What is the domain of $a$ such that the polynomial $x^4 + 4x^3 + ax^2 + 4x + 1 = 0$ has all real roots? | [
"\\( (-\\infty, -10] \\cup \\{6\\} \\)"
] | [
"What is the domain of $a$ such that the polynomial $x^4 + 4x^3 + ax^2 + 4x + 1 = 0$ has all real roots?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-4a1ff66a0f6ec8dea5d6f86c30debd9eac2e45cb0cf78be70dfd9ff4fb571d0e | What is the domain of $y=\dfrac{x^3-27}{x+27}$? (Express your answer using interval notation.) | [
"(-\\infty,-27)\\cup(-27,\\infty)"
] | [
"What is the domain of $y=\\dfrac{x^3-27}{x+27}$? (Express your answer using interval notation.)"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-84d7b73036dad421240589e0e8c01243b48c3b501973521e303844f163284083 | What is the domain of \( f(g(h(x))) \) in either set notation or interval notation, given the functions: \[ h(x) = \frac{1}{x} - 1 \] \[ g(x) = \frac{1}{1 - x} \] \[ f(x) = \frac{1}{x + 1} \] | [
"\\( (-\\infty, 0) \\cup (0, \\frac{1}{3}) \\cup (\\frac{1}{3}, \\frac{1}{2}) \\cup (\\frac{1}{2}, \\infty) \\)"
] | [
"What is the domain of \\( f(g(h(x))) \\) in either set notation or interval notation, given the functions:\n\\[ h(x) = \\frac{1}{x} - 1 \\]\n\\[ g(x) = \\frac{1}{1 - x} \\]\n\\[ f(x) = \\frac{1}{x + 1} \\]"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 26
} | 41 | 0 | 0 |
omr-ws-3c4f5e17556f4fc2d2d17fe2bbc0eaabd0e9922a00ba39c1db4718f6d571d48c | What is the domain of the equation \( y^2 = \frac{x^3 - 4x^2 + 3x}{x^2 - 4} \)? | [
"\\( -2 < x \\le 0, \\ 1 \\le x < 2, \\ 3 \\le x \\)"
] | [
"What is the domain of the equation \\( y^2 = \\frac{x^3 - 4x^2 + 3x}{x^2 - 4} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
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] | {
"cot": 14,
"genselect": 8,
"tir": 30
} | 52 | 0 | 0 |
omr-ws-fc4e2067e6bfa3e5600db08bada7f935cc31d4c6ca523727d98d5f3185300f2c | What is the domain of the function $$\ell(y) = \frac{1}{(y-2)+(y-8)}~?$$ Express your answer in interval notation. | [
"(-\\infty,5)\\cup (5,\\infty)"
] | [
"What is the domain of the function $$\\ell(y) = \\frac{1}{(y-2)+(y-8)}~?$$ Express your answer in interval notation."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a082a599d8c8cf51f00852de11401ca0e2337daf4e3ef51f9fa094f0fccd3135 | What is the domain of the function $$f(t) = \frac{1}{(t-1)^2+(t+1)^2}~?$$ Express your answer in interval notation. | [
"(-\\infty,\\infty)"
] | [
"What is the domain of the function $$f(t) = \\frac{1}{(t-1)^2+(t+1)^2}~?$$ Express your answer in interval notation."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5b0399b74f8ebdd75afedee388c823a9aef6539079f893af0cfb40363d18dba7 | What is the domain of the function $$j(x) = \frac{1}{x+8} + \frac{1}{x^2+8} + \frac{1}{x^3+8}~?$$Express your answer as a union of intervals. | [
"(-\\infty,-8)\\cup (-8,-2)\\cup (-2,\\infty)"
] | [
"What is the domain of the function $$j(x) = \\frac{1}{x+8} + \\frac{1}{x^2+8} + \\frac{1}{x^3+8}~?$$Express your answer as a union of intervals."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fd8871ab93f0a72a9dab169c189b96980a7e5ba54ea392571e8166205f17a38b | What is the domain of the function $$u(x) = \frac{1}{\sqrt x}~?$$ Express your answer in interval notation. | [
"(0,\\infty)"
] | [
"What is the domain of the function $$u(x) = \\frac{1}{\\sqrt x}~?$$ Express your answer in interval notation."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e48d0c4dd7a8e467d19edc4839239598cdfb7b6b1633362fff1ceb3640828c89 | What is the domain of the function $$w(x) = \sqrt{x-2} + \sqrt[3]{x-3}~?$$ Express your answer in interval notation. | [
"[2,\\infty)"
] | [
"What is the domain of the function $$w(x) = \\sqrt{x-2} + \\sqrt[3]{x-3}~?$$ Express your answer in interval notation."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e290bf2a455b06b96e51b0eb4d5b4610bbd0b74b6e91d0ead75d9a4e8fa764d9 | What is the domain of the function $f(x)=\frac{x+6}{\sqrt{x^2-3x-4}}$? | [
"(-\\infty, -1) \\cup (4, \\infty)"
] | [
"What is the domain of the function $f(x)=\\frac{x+6}{\\sqrt{x^2-3x-4}}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-97582c18cf47bb6274d2063dcb60bdabc8d65949957f152db9031e5918473186 | What is the domain of the function $f(x)=\log_2(\log_3(\log_4(\log_5x)))$? | [
"(625, \\infty)"
] | [
"What is the domain of the function $f(x)=\\log_2(\\log_3(\\log_4(\\log_5x)))$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0cc17defa8e3ea02649e0fabd2b80ed72a40eff2fb321bc98f0e06e877b9a1f4 | What is the domain of the function $f(x)=\log_2(\log_3(\log_4x))$? | [
"(4, \\infty)"
] | [
"What is the domain of the function $f(x)=\\log_2(\\log_3(\\log_4x))$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0e7de2e71096d4809ef0ae90c41d1176a11247cb17f189d313a1d4ba1e123cc6 | What is the domain of the function $g(x) = \frac{3x+1}{x+8}$ ? Enter your answer using interval notation. | [
"(-\\infty, -8) \\cup (-8, \\infty)"
] | [
"What is the domain of the function $g(x) = \\frac{3x+1}{x+8}$ ? Enter your answer using interval notation."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7534c3cb579943f4a414668e0be05772e8826856bc7984a43b421a2ab6823c74 | What is the domain of the function \( f(x) = (2x - 3)^{x^2 - 9x + 20} \) where \( f(x) \) is an exponential function? | [
"\\( x > \\frac{3}{2} \\) and \\( x \\neq 2 \\)"
] | [
"What is the domain of the function \\( f(x) = (2x - 3)^{x^2 - 9x + 20} \\) where \\( f(x) \\) is an exponential function?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
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"cot": 6,
"genselect": 8,
"tir": 3
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omr-ws-63851aabfcee1460dd44e9ffef142abf5ebc54dc523ac248ac5ca7a0397f8201 | What is the domain of the function \( f(x) = \tan(2x - \pi/4) \)? | [
"\\( x \\neq \\frac{3\\pi}{8} + \\frac{k\\pi}{2} \\) for any integer \\( k \\)"
] | [
"What is the domain of the function \\( f(x) = \\tan(2x - \\pi/4) \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-326917b59fd27ee366ff60bf59fb9cf97265b3f1176d057a6a26c89a26fd833d | What is the domain of the function \( f(x) = x^x \)? | [
"\\( x > 0 \\)"
] | [
"What is the domain of the function \\( f(x) = x^x \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-6353e262976e7421f7f0876d4fc532f1a2072373ca174c91c873e5a5ffbe3e39 | What is the domain of the function \(y = \sqrt{\frac{x - 1}{x - 2\{x\}}}\), where \(\{x\}\) denotes the fractional part of \(x\)? | [
"\\( (-\\infty, -1] \\cup (0, \\infty) \\)"
] | [
"What is the domain of the function \\(y = \\sqrt{\\frac{x - 1}{x - 2\\{x\\}}}\\), where \\(\\{x\\}\\) denotes the fractional part of \\(x\\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
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"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ae832f713b0535d7009d59a24eebd4697d03b4b0733bf20dcded45d4995edf1d | What is the domain of the given piecewise function? $$ f(x) = \begin{cases} 3x+1 ~ & \text{if}~ x < 3 \\ 5 &\text{if } 3 \leq x \leq 8 \\x^3 - 1 & \text{if}~x \geq 9 \end{cases} $$ | [
"$(-\\infty, 8] \\cup [9, \\infty)$"
] | [
"What is the domain of the given piecewise function?\n$$ f(x) = \\begin{cases} 3x+1 ~ & \\text{if}~ x < 3 \\\\ 5 &\\text{if } 3 \\leq x \\leq 8 \\\\x^3 - 1 & \\text{if}~x \\geq 9 \\end{cases} $$"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-0eddb0ce76d91d71a3f42a741880268e42f9edde884206286aca3274e91c566a | What is the double derivative of the function $(x + \frac{1}{x})^{10}$? | [
"\\( 90 \\left( x + \\frac{1}{x} \\right)^8 \\left( 1 - \\frac{1}{x^2} \\right)^2 + 10 \\left( x + \\frac{1}{x} \\right)^9 \\left( \\frac{2}{x^3} \\right) \\)"
] | [
"What is the double derivative of the function $(x + \\frac{1}{x})^{10}$?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d3fd8307bff5fc09f4d887ecddafd6f0218f8fd0cbe4593629ee64bbba2f6c00 | What is the dual of \( T \), the order 12 rotational subgroup of \( T_d \), the tetrahedral group? The dual group is defined as the group of all homomorphisms from \( T \) to \( \mu(\mathbb{C}) \). | [
"\\( \\mathbb{Z}/3\\mathbb{Z} \\)"
] | [
"What is the dual of \\( T \\), the order 12 rotational subgroup of \\( T_d \\), the tetrahedral group? The dual group is defined as the group of all homomorphisms from \\( T \\) to \\( \\mu(\\mathbb{C}) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-b398de1a455e0208c705eadf8de0db7e09ba8e8fe8ea98b6efe7c22840c208c3 | What is the dual space of the Hardy space in the case $1 < p < \infty$? | [
"\\( L^q \\) where \\( \\frac{1}{p} + \\frac{1}{q} = 1 \\)"
] | [
"What is the dual space of the Hardy space in the case $1 < p < \\infty$?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-eb9684dfe886c16f340b65d89686d5242916836848be79b92266d13713fe1289 | What is the easiest way to clear all radicals from the denominator of the expression \( x = \frac{1}{a + b\sqrt{2} + c\sqrt{3} + d\sqrt{6}} \)? | [
"Multiply the numerator and denominator by the conjugates to clear all radicals from the denominator."
] | [
"What is the easiest way to clear all radicals from the denominator of the expression \\( x = \\frac{1}{a + b\\sqrt{2} + c\\sqrt{3} + d\\sqrt{6}} \\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 9
} | 12 | 0 | 0 |
omr-ws-c22f32575eef10caf908aa877eb9d3ad5491e394887cd1f71a41c7d8672c6d1c | What is the easiest, least messy way to approximate the eigenvalues and eigenvectors of the following matrix? \[ \begin{pmatrix} 4 & 2 & 0 & 0 & \cdots & 0 & 0 & 0 \\ 2 & 5 & 2 & 0 & \cdots & 0 & 0 & 0 \\ 0 & 2 & 5 & 2 & \cdots & 0 & 0 & 0 \\ \vdots & \vdots & \vdots & \vdots & \ddots & \vdots & \vdots & \vdots \\ 0 & ... | [
"\\text{Use the QR algorithm for symmetric tridiagonal matrices.}"
] | [
"What is the easiest, least messy way to approximate the eigenvalues and eigenvectors of the following matrix?\n\\[\n\\begin{pmatrix}\n4 & 2 & 0 & 0 & \\cdots & 0 & 0 & 0 \\\\\n2 & 5 & 2 & 0 & \\cdots & 0 & 0 & 0 \\\\\n0 & 2 & 5 & 2 & \\cdots & 0 & 0 & 0 \\\\\n\\vdots & \\vdots & \\vdots & \\vdots & \\ddots & \\vdo... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-0745311e08e104a98eae83985fad51a6dc585ccd042d290c58aad82a374e3c6a | What is the edge length of the largest cube that can be placed inside a regular tetrahedron with edge length 1? | [
"\\(\\frac{\\sqrt{6}}{\\sqrt{6} + 2\\sqrt{2} + 3}\\)"
] | [
"What is the edge length of the largest cube that can be placed inside a regular tetrahedron with edge length 1?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1d4e8714869e66c737371d40d7ca2e51b1b30ab5bed5d31473011a70e76654b9 | What is the edge size of the largest cube that fits between two concentric spheres with radii 2 and 4? | [
"\\(\\frac{8}{\\sqrt{3}}\\)"
] | [
"What is the edge size of the largest cube that fits between two concentric spheres with radii 2 and 4?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-9c79eb0c2d5cc4383220184888f3d35281b287eb901634b5a84e4ac286a08593 | What is the effective rate of $18,600 invested for one year at 7.5%, compounded semiannually? Round your answer to the nearest hundredth. | [
"7.64%"
] | [
"What is the effective rate of $18,600 invested for one year at 7.5%, compounded semiannually? Round your answer to the nearest hundredth."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-90edd5eeb062cd452eabe4f1d06313f69166123f4bd526935f74942b8515dff6 | What is the eighth term in the arithmetic sequence $\frac 23, 1, \frac 43, \dots$? Express your answer in simplest form. | [
"3"
] | [
"What is the eighth term in the arithmetic sequence $\\frac 23, 1, \\frac 43, \\dots$? Express your answer in simplest form."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bb9e472393fd51d97bce9bd8688b28d41f4ea32f5eb43088eb330048d4852916 | What is the eleventh digit after the decimal point of \((1+\sqrt{2})^{1001}\)? | [
"0"
] | [
"What is the eleventh digit after the decimal point of \\((1+\\sqrt{2})^{1001}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-0c344b50384ee504f8557424ea5364fbd94f8afbd0db1fcd4e4b64a317442607 | What is the enclosed area between the graph of $y=\lfloor 10x \rfloor + \sqrt{1-x^2}$ in the interval $[0,1]$ and the $x$ axis? | [
"\\(\\frac{18 + \\pi}{4}\\)"
] | [
"What is the enclosed area between the graph of $y=\\lfloor 10x \\rfloor + \\sqrt{1-x^2}$ in the interval $[0,1]$ and the $x$ axis?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-3886ec2a7aeb36d0668df72c85515a8e33c256d5f8b1daccc6fac1a1a9957b91 | What is the equality case in the Cauchy-Schwarz inequality? | [
"\\text{The vectors are linearly dependent.}"
] | [
"What is the equality case in the Cauchy-Schwarz inequality?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b6c5f5ba717a7d2c36315dc3a329764cdbc35d08c0f23e44e210d98f5a3213c3 | What is the equation (in barycentric coordinates) of the internal and external angle bisectors of one of the angles in a triangle (angles at vertices A, B, or C)? | [
"\\( yc = zb \\) (internal) and \\( -yc = zb \\) (external)"
] | [
"What is the equation (in barycentric coordinates) of the internal and external angle bisectors of one of the angles in a triangle (angles at vertices A, B, or C)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 1,
"tir": 3
} | 9 | 0 | 0 |
omr-ws-22c95e7850864f3510bfae710478669ecd49711e49790063e4a2ca8e2c68401f | What is the equation for a graph that looks like a square (or diamond) using absolute value? | [
"\\( |x| + |y| = k \\)"
] | [
"What is the equation for a graph that looks like a square (or diamond) using absolute value?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 7
} | 15 | 0 | 0 |
omr-ws-ae90edc5be4c51b60ba0f5cc57059b0e584937f7c10ebd93266e8c45470fda20 | What is the equation for the Euler line of a triangle $\triangle ABC$ with vertices $A(x_1, y_1)$, $B(x_2, y_2)$, and $C(x_3, y_3)$? | [
"\\left(y - \\frac{y_1 + y_2 + y_3}{3}\\right) = \\left(\\frac{O_y - \\frac{y_1 + y_2 + y_3}{3}}{O_x - \\frac{x_1 + x_2 + x_3}{3}}\\right)\\left(x - \\frac{x_1 + x_2 + x_3}{3}\\right)"
] | [
"What is the equation for the Euler line of a triangle $\\triangle ABC$ with vertices $A(x_1, y_1)$, $B(x_2, y_2)$, and $C(x_3, y_3)$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-ecd626286cc8bc509ec4981563255e1022704b82645f440736651ebff6d9f614 | What is the equation in polar coordinates of the equiangular spiral which starts at the origin and goes clockwise? | [
"\\( r = ae^{b\\theta} \\) where \\( a \\) and \\( b \\) are arbitrary constants and \\( b < 0 \\)."
] | [
"What is the equation in polar coordinates of the equiangular spiral which starts at the origin and goes clockwise?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-6ebecd24fbc8e73d4af69ba7fde1ff4c971c43c256e107a152f0a146ba6bb57e | What is the equation of a hyperbola with vertices $(2, -1)$ and $(2, 7)$ and asymptotes whose slopes are $\pm \frac{4}{7}$? | [
"\\frac{(y - 3)^2}{16} - \\frac{(x - 2)^2}{49} = 1"
] | [
"What is the equation of a hyperbola with vertices $(2, -1)$ and $(2, 7)$ and asymptotes whose slopes are $\\pm \\frac{4}{7}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7a4ec66cb9e58dbf50ed390550763b49af4aa19691dc663fe704ae6fbc64506e | What is the equation of a hyperbola with vertices $(3, -4)$ and $(7, -4)$ and foci at $(5 \pm \sqrt{29}, -4)$? | [
"\\dfrac{(x - 5)^2}{4} - \\dfrac{(y + 4)^2}{25} = 1"
] | [
"What is the equation of a hyperbola with vertices $(3, -4)$ and $(7, -4)$ and foci at $(5 \\pm \\sqrt{29}, -4)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cce5438b2a94690f27e4b9c4aca3a98085945cc85bd5221a3075c6d6e3084ede | What is the equation of a line parallel to the line \(2x - 3y = 7\) and that contains the point \((-6, -5)\)? | [
"\\(2x - 3y = 3\\)"
] | [
"What is the equation of a line parallel to the line \\(2x - 3y = 7\\) and that contains the point \\((-6, -5)\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-fbcd8d7f3064d09ba340858ac8d659315dda5c97f81a0f975315c6cfc0ca0b56 | What is the equation of a line perpendicular to the line represented by \( y = -\frac{1}{2}x - 5 \) and passing through the point (6, -4)? | [
"\\( y = 2x - 16 \\)"
] | [
"What is the equation of a line perpendicular to the line represented by \\( y = -\\frac{1}{2}x - 5 \\) and passing through the point (6, -4)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-ba68438802f118cc1e7a7450b1fb65462100100e9dc48e280ca6e077bd73cf32 | What is the equation of a line that passes through the points (1,0,0,0) and (2,2,2,2) in four-dimensional space? | [
"\\(\\frac{w-1}{1} = \\frac{x}{2} = \\frac{y}{2} = \\frac{z}{2}\\)"
] | [
"What is the equation of a line that passes through the points (1,0,0,0) and (2,2,2,2) in four-dimensional space?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 1,
"tir": 20
} | 35 | 0 | 0 |
omr-ws-391ced7bc29b51d4fcbbcd6bcaeb3d47e4cd948ce22760e140659bc449c993e4 | What is the equation of an ellipse that has been rotated $45^{\circ}$ about its center? | [
"\\frac{(x + y)^2}{2a^2} + \\frac{(x - y)^2}{2b^2} = 1"
] | [
"What is the equation of an ellipse that has been rotated $45^{\\circ}$ about its center?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ac5b331602cf37c291c082445930265c4a15d559bd8ece64c08e73f2db253752 | What is the equation of the circumcircle of triangle $XYZ$ in trilinear coordinates, given the points $X$, $Y$, and $Z$? Provide an example by determining the circumcircle of triangle $I_aI_bI_c$. | [
"a\\alpha + b\\beta + c\\gamma = 0"
] | [
"What is the equation of the circumcircle of triangle $XYZ$ in trilinear coordinates, given the points $X$, $Y$, and $Z$? Provide an example by determining the circumcircle of triangle $I_aI_bI_c$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-fd939ff6e42031028d4869cbf5b3e5ef4bd4525179b6971fb9f9f8dca671d0ac | What is the equation of the cone whose surface makes a $75^\circ$ angle with the $x$-axis? | [
"y^2 + z^2 = (7 + 4\\sqrt{3})x^2"
] | [
"What is the equation of the cone whose surface makes a $75^\\circ$ angle with the $x$-axis?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-843faea67e7cff9154cf2a6905a33e2b0b7f7d0d443b0c79bca75151e7752cab | What is the equation of the graph of \( \log(x) = y \) after it is rotated 90 degrees counterclockwise? | [
"\\( y = 10^{-x} \\)"
] | [
"What is the equation of the graph of \\( \\log(x) = y \\) after it is rotated 90 degrees counterclockwise?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 5,
"tir": 1
} | 19 | 0 | 0 |
omr-ws-237789bf05bb239d54e769fa5be17f3fd4f4aff9daf7b541afd092c92b6f03e9 | What is the equation of the graph of the function that is translated 9 units down and 4 units to the left of \( f(x) = x^2 \) and then vertically dilated by a factor of \( \frac{1}{2} \)? | [
"\\( f(x) = \\frac{1}{2}(x+4)^2 - 9 \\)"
] | [
"What is the equation of the graph of the function that is translated 9 units down and 4 units to the left of \\( f(x) = x^2 \\) and then vertically dilated by a factor of \\( \\frac{1}{2} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 5,
"tir": 6
} | 25 | 0 | 0 |
omr-ws-83dd1396b372a6a49ea6db14e5fb081a60df8f96e083e16db637ab60b33b8cca | What is the equation of the graph when you rotate \( y = x^2 \) counter-clockwise by \( 15^\circ \)? | [
"(2 + \\sqrt{3})x^2 + 2xy + (2 - \\sqrt{3})y^2 + (\\sqrt{6} - \\sqrt{2})x - (\\sqrt{6} + \\sqrt{2})y = 0"
] | [
"What is the equation of the graph when you rotate \\( y = x^2 \\) counter-clockwise by \\( 15^\\circ \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-410574f11f2e021b601b29b010e536232c75f2080e98d854dcfee0c057789925 | What is the equation of the line parallel to $4x+2y=8$ and passing through the point $(0,1)$? Write the equation in slope-intercept form. | [
"y=-2x+1"
] | [
"What is the equation of the line parallel to $4x+2y=8$ and passing through the point $(0,1)$? Write the equation in slope-intercept form."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-74da82394ddb990f5f9e590e7a28588c61e36fbb3a5c3b0012fd9aa3546d9b79 | What is the equation of the line parallel to the line \(2x - 3y = 7\) and containing the point \((-6, -5)\)? | [
"\\( y = \\frac{2}{3}x - 1 \\)"
] | [
"What is the equation of the line parallel to the line \\(2x - 3y = 7\\) and containing the point \\((-6, -5)\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-d44fd828125e0d5e9d22272b55ba6b3f03650cf3bcc54c00a2c837044104fcac | What is the equation of the line tangent to the intersection of the surface $z = \arctan(xy)$ with the plane $x=2$, at the point $(2,\frac{1}{2}, \frac{\pi}{2})$? | [
"\\( x = 2, \\quad y = \\frac{1}{2} + t, \\quad z = \\frac{\\pi}{2} + t \\)"
] | [
"What is the equation of the line tangent to the intersection of the surface $z = \\arctan(xy)$ with the plane $x=2$, at the point $(2,\\frac{1}{2}, \\frac{\\pi}{2})$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 9
} | 11 | 0 | 0 |
omr-ws-93655afe3ca36a289d72e95e21859cbc6568c94c4a963654128e930814cbb63a | What is the equation of the line that passes through the point $(10, -1)$ and is perpendicular to the curve $y = \frac{1}{4}x^2 - 2$? | [
"\\( y = -\\frac{1}{2}x + 4 \\)"
] | [
"What is the equation of the line that passes through the point $(10, -1)$ and is perpendicular to the curve $y = \\frac{1}{4}x^2 - 2$?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-86badc5223719406643047dc7f6a79d0b2bd737f775a7a01995b773b2b0abbb6 | What is the equation of the locus of points equidistant from points A(3, 1) and B(7, 1)? | [
"\\(x = 5\\)"
] | [
"What is the equation of the locus of points equidistant from points A(3, 1) and B(7, 1)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-bc0b784994b9d1a6352bb1cc6fc772dac9fc3ade1975c846b4b08b465eadbd96 | What is the equation of the oblique asymptote of the graph of $\frac{2x^2+7x+10}{2x+3}$? Enter your answer in the form $y = mx + b.$ | [
"y = x+2"
] | [
"What is the equation of the oblique asymptote of the graph of $\\frac{2x^2+7x+10}{2x+3}$?\n\nEnter your answer in the form $y = mx + b.$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fee4b4a2d36a9e0184bf4b80fff69201c3206a8b5ba640f64fc02747e5c0d814 | What is the equation of the parabola that is rotated $45$ degrees clockwise around the origin from the parabola $y=x^2$? | [
"$(x'-y')^{2}-\\sqrt{2}(x'+y')=0$"
] | [
"What is the equation of the parabola that is rotated $45$ degrees clockwise around the origin from the parabola $y=x^2$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 13
} | 28 | 0 | 0 |
omr-ws-90f21aac3e654bf66f9c3c6ae40535b6912473d36bf66725c99a2bb3dddd6e91 | What is the equation of the parabola that serves as a circumconic of a triangle and has its axis of symmetry parallel to the Euler line of the triangle? | [
"(b^2 - c^2)x + (c^2 - a^2)y + (a^2 - b^2)z = 0"
] | [
"What is the equation of the parabola that serves as a circumconic of a triangle and has its axis of symmetry parallel to the Euler line of the triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 1
} | 11 | 0 | 0 |
omr-ws-b8e201fbc577b26b3f9cda3879ee0cfcae1e2ba8eb165156fefbf6ddb89c16e1 | What is the equation of the plane containing the point (20,0,3) that is parallel to the plane containing the points (2,1,1), (4,4,-1), and (2,-1,2)? | [
"\\( 19x - 10y + 4z = 392 \\)"
] | [
"What is the equation of the plane containing the point (20,0,3) that is parallel to the plane containing the points (2,1,1), (4,4,-1), and (2,-1,2)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-be4455cad651d42b1831ea696e7d99ba686387bc3c0610a343eff5d8a5d73da8 | What is the equivalent of the infinite product $\prod_{\substack{p \leq x \\ p \, \text{prime}}} p$ as $x$ approaches infinity? | [
"e^x"
] | [
"What is the equivalent of the infinite product $\\prod_{\\substack{p \\leq x \\\\ p \\, \\text{prime}}} p$ as $x$ approaches infinity?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-21ab5f7b6db1f2f003f65a3ddc9238a035d1c5b84dedee33530bdd007b8304a6 | What is the estimate of $(1+\epsilon)^{\frac{1}{n}}$ using a cubic Taylor Polynomial, where $n$ is a positive integer? | [
"\\(1 + \\frac{\\epsilon}{n} - \\frac{(n-1)\\epsilon^2}{2n^2} + \\frac{(n-1)(2n-1)\\epsilon^3}{6n^3}\\)"
] | [
"What is the estimate of $(1+\\epsilon)^{\\frac{1}{n}}$ using a cubic Taylor Polynomial, where $n$ is a positive integer?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-1c83b066ee32e96609ee3962d350729574b746bd7db5b424bd475fc3899ae0d3 | What is the exact area of the region in the first quadrant defined by the inequality \((x-6)^2 + (y-4)^2 \le 100\)? | [
"\\(\\pi \\left(63 - 2\\sqrt{21}\\right)\\)"
] | [
"What is the exact area of the region in the first quadrant defined by the inequality \\((x-6)^2 + (y-4)^2 \\le 100\\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6e49d18e63b060cc780b8de8edd56085f22911226294f93ed5f02eee39eac370 | What is the exact value of $$\tan^6(20^\circ) + \tan^6(40^\circ) + \tan^6(60^\circ)?$$ | [
"33273"
] | [
"What is the exact value of $$\\tan^6(20^\\circ) + \\tan^6(40^\\circ) + \\tan^6(60^\\circ)?$$"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-280ad2d867f356296135ccab6e8adda3bd22757c32a2ed06d9fb956cca866934 | What is the exact value of $\cos \frac{\pi}{7}$? | [
"\\frac{1}{6} + \\frac{\\sqrt{7}}{3} \\cos\\left( \\frac{1}{3} \\arccos\\left( -\\frac{1}{2\\sqrt{7}} \\right) \\right)"
] | [
"What is the exact value of $\\cos \\frac{\\pi}{7}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-983223641c167adb7eeb8156a0a46a8b183be59a4e0fab3a735e29b15d0ac3ff | What is the exact value of $\frac{x^2}{x^2+y^2+z^2}+\frac{a^2}{a^2+b^2+c^2}$, given that $x+y+z=a+b+c$ and $ax+by+cz=0$? | [
"1"
] | [
"What is the exact value of $\\frac{x^2}{x^2+y^2+z^2}+\\frac{a^2}{a^2+b^2+c^2}$, given that $x+y+z=a+b+c$ and $ax+by+cz=0$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-2e87494b4d1de7fc2a5766c26a546c7fde069122d1f8594f20e67d0a354a45ca | What is the exact value of $\sin \left(-\frac{19\pi^2}{6}\right)$? | [
"\\dfrac{1}{2}"
] | [
"What is the exact value of $\\sin \\left(-\\frac{19\\pi^2}{6}\\right)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-e2d80ab7a3d30becb100148a9471a0254b1f0842dbf038201c9797e4e4646de1 | What is the exact value of $\sin(10^\circ) \cdot \sin(50^\circ) \cdot \sin(70^\circ)$? | [
"\\(\\frac{1}{8}\\)"
] | [
"What is the exact value of $\\sin(10^\\circ) \\cdot \\sin(50^\\circ) \\cdot \\sin(70^\\circ)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-906546e99f6dcd44447319a6b6eb5c574de0094b544532f572ddbde5a170afa4 | What is the exact value of $\sin(\pi/7)$? | [
"\\(\\sqrt{\\frac{7}{12}-\\frac{7^{\\frac{2}{3}}(1+i\\sqrt{3})}{12*2^{\\frac{2}{3}}\\sqrt[3]{-1+3i\\sqrt{3}}}-\\frac{1}{24}(1-i\\sqrt{3})\\sqrt[3]{\\frac{7}{2}(-1+3i\\sqrt{3})}}\\)"
] | [
"What is the exact value of $\\sin(\\pi/7)$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-32f90107d293dbea432394fd1d0cf44056fe68776849b8e7a4baa2f964feb626 | What is the exact value of \( \frac{\cos 1^{\circ} + \cos 2^{\circ} + \ldots + \cos 43^{\circ} + \cos 44^{\circ}}{\sin 1^{\circ} + \sin 2^{\circ} + \ldots + \sin 43^{\circ} + \sin 44^{\circ}} \)? | [
"\\(\\sqrt{2} + 1\\)"
] | [
"What is the exact value of \\( \\frac{\\cos 1^{\\circ} + \\cos 2^{\\circ} + \\ldots + \\cos 43^{\\circ} + \\cos 44^{\\circ}}{\\sin 1^{\\circ} + \\sin 2^{\\circ} + \\ldots + \\sin 43^{\\circ} + \\sin 44^{\\circ}} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-5dfe98df82872e70b9f308430cf453dab73181f897fdad808fc9ee9cfa5177ca | What is the expanded form, written using the sigma symbols for cyclic and symmetric sums, of $(a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n)^2$? | [
"\\sum_{k=0}^{2n} \\left( \\sum_{i=0}^{k} a_i a_{k-i} \\right) x^k"
] | [
"What is the expanded form, written using the sigma symbols for cyclic and symmetric sums, of $(a_0 + a_1 x + a_2 x^2 + \\cdots + a_n x^n)^2$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-41786377ada96b1de582861dfc70017b95e133770ce592aa6a7d9e186bf9cdb6 | What is the expansion of \((x+y+z)^5\)? | [
"x^5 + 5x^4y + 5x^4z + 10x^3y^2 + 20x^3yz + 10x^3z^2 + 10x^2y^3 + 30x^2y^2z + 30x^2yz^2 + 10x^2z^3 + 5xy^4 + 20xy^3z + 30xy^2z^2 + 20xyz^3 + 5xz^4 + y^5 + 5y^4z + 10y^3z^2 + 10y^2z^3 + 5yz^4 + z^5"
] | [
"What is the expansion of \\((x+y+z)^5\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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