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-c60ee7f0a3e259094a8bac1e23e477123dad336f6a81546c9cbe2290f44a3b8e | What is the correct upper-bound for the height of an AVL tree with \( n \) nodes? Some sources suggest it is \( 1.44 \log_2(n+2) - 0.328 \), while others suggest it is \( 1.44 \log_2(n+2) - 1.328 \). | [
"1.44 \\log_2(n+2) - 1.328"
] | [
"What is the correct upper-bound for the height of an AVL tree with \\( n \\) nodes? Some sources suggest it is \\( 1.44 \\log_2(n+2) - 0.328 \\), while others suggest it is \\( 1.44 \\log_2(n+2) - 1.328 \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-95a8b1bf8f522f5aac1f2320977362830661cb21d5dc13822ecae46adb1d7185 | What is the correct way to compute the integral $\int \frac{x^2-x}{x}\,dx$? | [
"\\frac{1}{2}x^2 - x + C"
] | [
"What is the correct way to compute the integral $\\int \\frac{x^2-x}{x}\\,dx$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fe718ae3c4d219b20795b2de338002113827536250e84c97c25e2cf0481fb865 | What is the correspondence (# of elements in A compared to # of elements in B) between the set of positive integers, $1, 2, 3, \ldots, n$ and the set of even integers, $2, 4, 6, \ldots, 2n$ (both sets are infinite)? | [
"\\text{The sets have the same number of elements.}"
] | [
"What is the correspondence (# of elements in A compared to # of elements in B) between the set of positive integers, $1, 2, 3, \\ldots, n$ and the set of even integers, $2, 4, 6, \\ldots, 2n$ (both sets are infinite)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-6f0b556db4b99418a90ee9d67c2848d5bdd5fdb27d332fc3e9761f995e339b34 | What is the cosine of the angle formed by two adjacent faces of a regular tetrahedron? | [
"\\(\\frac{1}{3}\\)"
] | [
"What is the cosine of the angle formed by two adjacent faces of a regular tetrahedron?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-08d27bebdaf5faf9609a717df15f4cc427e5ba869a9a1b471626474209341421 | What is the covariance matrix of the random variables \(\zeta\) and \(\varphi\) given by: \[ \zeta = c_1 + a_{11}x + a_{12}y \] \[ \varphi = c_2 + a_{12}x + a_{22}y \] where \(x\) and \(y\) are jointly normal random variables with mean 0 and covariance matrix \(\Sigma\)? | [
"\\(A \\Sigma A^T\\)"
] | [
"What is the covariance matrix of the random variables \\(\\zeta\\) and \\(\\varphi\\) given by:\n\n\\[\n\\zeta = c_1 + a_{11}x + a_{12}y\n\\]\n\\[\n\\varphi = c_2 + a_{12}x + a_{22}y\n\\]\n\nwhere \\(x\\) and \\(y\\) are jointly normal random variables with mean 0 and covariance matrix \\(\\Sigma\\)?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 27
} | 41 | 0 | 0 |
omr-ws-5276bcb386fbac44f433b7fef41e234a46b3b0a3de9ec483b48bfb43e219898a | What is the cube of the square of the second smallest prime number? | [
"729"
] | [
"What is the cube of the square of the second smallest prime number?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-87ca05457f5adecd7b90d25d90f3f70a30d085470a2ff07d9f4680a6924151dd | What is the cumulative distribution function $F(t)$ of counting measure $\mu$? Given: \[ F(t) = \begin{cases} \mu([0,t]) & \text{if } t \ge 0 \\ -\mu((t,0)) & \text{if } t < 0 \end{cases} \] Determine the expression for $F(t)$. | [
"\\( F(t) = \\begin{cases} \n\\lfloor t \\rfloor + 1 & \\text{if } t \\ge 0 \\\\ \n-\\lfloor t \\rfloor & \\text{if } t < 0 \n\\end{cases} \\)"
] | [
"What is the cumulative distribution function $F(t)$ of counting measure $\\mu$?\n\nGiven:\n\\[ F(t) = \\begin{cases} \n\\mu([0,t]) & \\text{if } t \\ge 0 \\\\ \n-\\mu((t,0)) & \\text{if } t < 0 \n\\end{cases} \\]\n\nDetermine the expression for $F(t)$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 2
} | 3 | 0 | 0 |
omr-ws-15718d9273b372bb3d86f8a58559d1cf81962f44dd98168b6c5330326572666a | What is the curve defined by the polar equation \( p = \frac{1}{1 - \cos\theta + \sin\theta} \)? | [
"Hyperbola"
] | [
"What is the curve defined by the polar equation \\( p = \\frac{1}{1 - \\cos\\theta + \\sin\\theta} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-275b2ae27986eae9b0c074243cf72f81135cf4b79eeac797a00744c87a06248d | What is the decomposition of $\mathbb{Z}_{18} \oplus \mathbb{Z}_{24} \oplus \mathbb{Z}_{30} \oplus \mathbb{Z}_{45}$ into p-primary modules? | [
"\\((9\\mathbb{Z}/18\\mathbb{Z} \\oplus 3\\mathbb{Z}/24\\mathbb{Z} \\oplus 15\\mathbb{Z}/30\\mathbb{Z} \\oplus 0/45\\mathbb{Z}) \\oplus (2\\mathbb{Z}/18\\mathbb{Z} \\oplus 8\\mathbb{Z}/24\\mathbb{Z} \\oplus 10\\mathbb{Z}/30\\mathbb{Z} \\oplus 5/45\\mathbb{Z}) \\oplus (0/18\\mathbb{Z} \\oplus 0\\mathbb{Z}/24\\mathbb... | [
"What is the decomposition of $\\mathbb{Z}_{18} \\oplus \\mathbb{Z}_{24} \\oplus \\mathbb{Z}_{30} \\oplus \\mathbb{Z}_{45}$ into p-primary modules?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 1,
"tir": 14
} | 27 | 0 | 0 |
omr-ws-030be8ae2f65361fdf045bbe3007a6a9a4c3fba73c837a17aed5d6577429a634 | What is the definite integral of \(\frac{1}{(d + r - r \cos x)^2}\) from \(0\) to \(2\pi\)? | [
"\\(2\\pi \\cdot \\frac{d+r}{d(d+2r)} \\cdot \\frac{1}{\\sqrt{d(d+2r)}}\\)"
] | [
"What is the definite integral of \\(\\frac{1}{(d + r - r \\cos x)^2}\\) from \\(0\\) to \\(2\\pi\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-d47a6ba781717860b34d96d93e9b12706efd95081c9668894524e091ff12467d | What is the definite integral of \(\frac{\arctan x}{x}\) between the limits \(a\) and \(\frac{1}{a}\)? | [
"\\(\\frac{\\pi}{2}\\ln(a)\\)"
] | [
"What is the definite integral of \\(\\frac{\\arctan x}{x}\\) between the limits \\(a\\) and \\(\\frac{1}{a}\\)?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c2cd7af10a08ca7165f87e107fc6f45334c3b4973721ff2e9b951801694cfd9e | What is the degree measure of an angle whose measure is double the measure of its complement? | [
"60"
] | [
"What is the degree measure of an angle whose measure is double the measure of its complement?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5e0b1e2d333b9b59e678607db7ecda04f52d748eb521462e91c4886a26a1d064 | What is the degree measure of an interior angle of a regular pentagon? | [
"108\\text{ degrees}"
] | [
"What is the degree measure of an interior angle of a regular pentagon?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d85258edb742739336a94fcb115d5041358dcb6862a1fd9af211a468de8fcefb | What is the degree measure of the angle at each vertex in a star-polygon drawn on a clock face by drawing a chord from each number to the fifth number counted clockwise from that number? | [
"30"
] | [
"What is the degree measure of the angle at each vertex in a star-polygon drawn on a clock face by drawing a chord from each number to the fifth number counted clockwise from that number?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-2a106eabb14131e1314f104f8a6947179d13c30f4c5ec8b5e91171d986c3e8c0 | What is the degree measure of the argument $\theta$ for the roots of the equation $z^6 + z^3 + 1 = 0$ that lie between $90$ degrees and $180$ degrees in the complex plane? | [
"160"
] | [
"What is the degree measure of the argument $\\theta$ for the roots of the equation $z^6 + z^3 + 1 = 0$ that lie between $90$ degrees and $180$ degrees in the complex plane?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-291ac85314bcbdec0fa87ed3eb3d737262fad81cd927a5ec99b2958b72d21354 | What is the degree measure of the smaller angle between the hour hand and the minute hand of a clock at exactly 2:30 p.m. on a 12-hour analog clock? | [
"105^\\circ"
] | [
"What is the degree measure of the smaller angle between the hour hand and the minute hand of a clock at exactly 2:30 p.m. on a 12-hour analog clock?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0923c08e482a2b175180caeb20c93f9d2007c2c9931854c0a21b979a5cd474f4 | What is the degree measure of the smaller angle formed by the hands of a clock at 10 o'clock? | [
"60 degrees"
] | [
"What is the degree measure of the smaller angle formed by the hands of a clock at 10 o'clock?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fb81be2c6600ecee2ea1123a81190c9f79f308489300751cb9ffb72204358baa | What is the degree measure of the supplement of the complement of a 42-degree angle? | [
"132"
] | [
"What is the degree measure of the supplement of the complement of a 42-degree angle?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-82d4db94930695fff08070d2e05b54ba892aa4c8854ad423072a543b76189ab7 | What is the degree of the differential equation $\ln\left(\frac{dy}{dx}\right)^{0.5} = 3x + 4y$? Is it undefined because it is not a polynomial in $\frac{dy}{dx}$? If we take the antilog on both sides, does the degree become one? | [
"undefined"
] | [
"What is the degree of the differential equation $\\ln\\left(\\frac{dy}{dx}\\right)^{0.5} = 3x + 4y$? Is it undefined because it is not a polynomial in $\\frac{dy}{dx}$? If we take the antilog on both sides, does the degree become one?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 7,
"tir": 16
} | 26 | 0 | 0 |
omr-ws-7c45fa52068c718a0c34dd3a530a941d48e69a9bd12e726b596c64203cbfd417 | What is the degree of the differential equation \( e^{\frac{dy}{dx}} = 2 \)? | [
"1"
] | [
"What is the degree of the differential equation \\( e^{\\frac{dy}{dx}} = 2 \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d516e222ea1e01407ef34572e9ce9c6cd83d3b4d87bbb9fa86efd4e9546277f4 | What is the degree of the differential equation satisfying the equation $\sqrt{1-x^2}+\sqrt{1-y^2}=a(x-y)$? | [
"2"
] | [
"What is the degree of the differential equation satisfying the equation $\\sqrt{1-x^2}+\\sqrt{1-y^2}=a(x-y)$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 6
} | 14 | 0 | 0 |
omr-ws-86706ff1ed99bdf7a84300bb1e756efcc23cecd7233846fd0caba57eaefad559 | What is the degree of the differential form \( w(x,y,z) = x^2 \frac{dy}{dz} + y^2 \frac{dz}{dx} - 2x(x+y) \frac{dx}{dy} \)? | [
"2"
] | [
"What is the degree of the differential form \\( w(x,y,z) = x^2 \\frac{dy}{dz} + y^2 \\frac{dz}{dx} - 2x(x+y) \\frac{dx}{dy} \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-ead7fc9aee8d55ec5f4a6a33c6047385c73f5efd144a8ee01412789f8f5379ff | What is the degree of the extension field $\mathbb{Q}(\alpha)$ over $\mathbb{Q}$, where $\alpha$ is a root of the polynomial $x^{10} - 5$? | [
"20"
] | [
"What is the degree of the extension field $\\mathbb{Q}(\\alpha)$ over $\\mathbb{Q}$, where $\\alpha$ is a root of the polynomial $x^{10} - 5$?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e0e5ee07a337107f90987f56b820617322f862fba82324c826d53982597f651a | What is the degree of the extension of $\mathbb{Z}_2$ with the zeroes of the polynomial $p(x) = x^3 + x^2 + 1$? | [
"6"
] | [
"What is the degree of the extension of $\\mathbb{Z}_2$ with the zeroes of the polynomial $p(x) = x^3 + x^2 + 1$?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ff7b96d4d4b1d5c8274c2f75511eba7a20334776c165c2bdc3c4091fae343cc6 | What is the degree of the minimal polynomial of the matrix \[ \begin{pmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1\\ 0 & 0 & 1 & 0 \end{pmatrix} \]? | [
"4"
] | [
"What is the degree of the minimal polynomial of the matrix \\[ \\begin{pmatrix}\n0 & 1 & 0 & 0 \\\\ \n1 & 0 & 1 & 0 \\\\ \n0 & 1 & 0 & 1\\\\ \n0 & 0 & 1 & 0\n\\end{pmatrix} \\]?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-7523be13ae016148871b168c3cf932bf04fcfeef7a47c8a8a1c31e459a10b88e | What is the degree of the polynomial $(3x^2 +11)^{12}$? | [
"24"
] | [
"What is the degree of the polynomial $(3x^2 +11)^{12}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-70f1a5ac377177d6c8b7da930625137d4fabc3c2827f5e7aa717b11fa6ce70a2 | What is the degree of the polynomial $(4 +5x^3 +100 +2\pi x^4 + \sqrt{10}x^4 +9)$? | [
"4"
] | [
"What is the degree of the polynomial $(4 +5x^3 +100 +2\\pi x^4 + \\sqrt{10}x^4 +9)$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-52fe560a1732320de523cab55ccee4b88075f6f76b46aed8a1ebb9121f974cae | What is the degree of the polynomial $(x^4+ax^7+bx+c)(x^3+dx^2+e)(x+f)$, where letters $a$ through $f$ are all nonzero constants? | [
"11"
] | [
"What is the degree of the polynomial $(x^4+ax^7+bx+c)(x^3+dx^2+e)(x+f)$, where letters $a$ through $f$ are all nonzero constants?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-32f415551d61eea02852d78382e4b37f3aee29c98846337cad8c658858c983e5 | What is the degree of the quotient polynomial from the operation $[(x^2 + x + 1)^5 - (x^{10} + 2)] \div (x^3 + 1)$? | [
"6"
] | [
"What is the degree of the quotient polynomial from the operation $[(x^2 + x + 1)^5 - (x^{10} + 2)] \\div (x^3 + 1)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c43e3995d57e59185ee75b569bf527eb2fe5846278c92197be089f3d937bf333 | What is the degree of the remainder when $x^{2007} - 1$ is divided by $(x^2 + 1)(x^2 + x + 1)$? | [
"3"
] | [
"What is the degree of the remainder when $x^{2007} - 1$ is divided by $(x^2 + 1)(x^2 + x + 1)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-103972213c748e032c979ea77a4caae3e80f4d797dcd38b10fe08588b3c7b9bd | What is the degree of the remainder when \( x^{2007} - 1 \) is divided by \( (x^2 + 1)(x^2 + x + 1) \)? | [
"3"
] | [
"What is the degree of the remainder when \\( x^{2007} - 1 \\) is divided by \\( (x^2 + 1)(x^2 + x + 1) \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-ee286c297728951676eb35a82b2be376078986a8f54b1d634bf0a5b63b3645f4 | What is the degree of the splitting field \( K \) of \( A(x) = x^5 - x^4 + x^3 - x^2 + x - 1 \) over \( \mathbb{Q} \)? | [
"2"
] | [
"What is the degree of the splitting field \\( K \\) of \\( A(x) = x^5 - x^4 + x^3 - x^2 + x - 1 \\) over \\( \\mathbb{Q} \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-0947c0e46ba9aa46a5bc7c01f2f77503171016678868556108e13c228e5e3cbe | What is the degree of the splitting field of $x^{6}-7$ over the finite field of 3 elements? | [
"1"
] | [
"What is the degree of the splitting field of $x^{6}-7$ over the finite field of 3 elements?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9bdb2799e4312481fc2eb47cafa3677145c124d0fa128ba8e83f4d7ae7efe451 | What is the denominator of the irreducible fraction resulting from the rationalization of \( \frac{1}{6 \cdot \sqrt{50 - 5 \sqrt{75}}} - \sqrt{128 - 16 \sqrt{48}} \)? | [
"60"
] | [
"What is the denominator of the irreducible fraction resulting from the rationalization of \\( \\frac{1}{6 \\cdot \\sqrt{50 - 5 \\sqrt{75}}} - \\sqrt{128 - 16 \\sqrt{48}} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-c114052772d9b5f2aea2e871fd6b7d1f29d0a66536eb174583671f8621828f9d | What is the density function of the x-coordinate when a point is picked randomly and uniformly from the perimeter of a unit circle? | [
"\\(\\frac{1}{\\pi \\sqrt{1 - x^2}}\\)"
] | [
"What is the density function of the x-coordinate when a point is picked randomly and uniformly from the perimeter of a unit circle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 13
} | 30 | 0 | 0 |
omr-ws-2a5da2871d1bd49c98182593e22787891b7830f326de0c620772d58c76dd149d | What is the density of integers that contain only primes of the form $r \pmod{n}$ among all integers, where $(r, n) = 1$? Are they positive or not? | [
"0"
] | [
"What is the density of integers that contain only primes of the form $r \\pmod{n}$ among all integers, where $(r, n) = 1$? Are they positive or not?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 43,
"genselect": 8
} | 51 | 0 | 0 |
omr-ws-83e5d32d0709cf6b750e1e03f5759a991a97d5ac33c4a92a372824d877fda7f9 | What is the density of positive integers that can be expressed as $N/\tau(N)$, where $\tau(N)$ denotes the number of divisors of $N$? | [
"0"
] | [
"What is the density of positive integers that can be expressed as $N/\\tau(N)$, where $\\tau(N)$ denotes the number of divisors of $N$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 5
} | 19 | 0 | 0 |
omr-ws-c67943468947b4aeb8f4bca3a34348eb7d52caa13a954afd1d8e92da4cca5241 | What is the density of primes congruent to $r \pmod{n}$ among all primes, where $(r, n) = 1$? | [
"\\(\\frac{1}{\\phi(n)}\\)"
] | [
"What is the density of primes congruent to $r \\pmod{n}$ among all primes, where $(r, n) = 1$?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-087d9a736a049a332bc358a2042780c9cbfda08d0adaa0ec071b5757ef429c32 | What is the derivative of $(-1)^x$ with respect to $x$? | [
"i\\pi (-1)^x"
] | [
"What is the derivative of $(-1)^x$ with respect to $x$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-84afb165981d8efcf87000688a4e38fa8b4392bd853cfee0aa253063e2b47f44 | What is the derivative of $\lfloor{x}\rfloor$ with respect to $x$? | [
"\\( \\frac{d\\lfloor x\\rfloor}{dx} = 0 \\) if \\( x \\not\\in \\mathbb{Z} \\), and undefined if \\( x \\in \\mathbb{Z} \\)."
] | [
"What is the derivative of $\\lfloor{x}\\rfloor$ with respect to $x$?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 26
} | 26 | 0 | 0 |
omr-ws-a4c43db14d3c90a6d6cfdd5b09e470d08682e3385b881148c8d21ece1a2c7475 | What is the derivative of $\tan^{-1}\sqrt{\frac{1-x}{1+x}}$? | [
"\\(\\frac{-1}{2\\sqrt{1-x^2}}\\)"
] | [
"What is the derivative of $\\tan^{-1}\\sqrt{\\frac{1-x}{1+x}}$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 31
} | 45 | 0 | 0 |
omr-ws-55eb1c70ef3155e220a425d903c7293633b02f8e1a85241f9b03eb0ffe1584b0 | What is the derivative of $x!$? | [
"x! \\, \\psi(x+1)"
] | [
"What is the derivative of $x!$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-82fd205f136b1bf279517b3fc1cdf6cb31714ca2ea63fd70f76bf9a89a16920f | What is the derivative of \( \log_{\sqrt{x}} (2x-1) \)? | [
"\\(\\frac{4x \\ln x - 4x \\ln(2x-1) + 2 \\ln(2x-1)}{x(2x-1)(\\ln x)^2}\\)"
] | [
"What is the derivative of \\( \\log_{\\sqrt{x}} (2x-1) \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-f0bf54f8423c4d09f95d1efdb877baaed5f0c1bccfbc0780261c5c615b3db568 | What is the derivative of \( \log_{f(x)} (x) \)? | [
"\\dfrac{f(x) \\ln(f(x)) - x f'(x) \\ln(x)}{x f(x) (\\ln(f(x)))^2}"
] | [
"What is the derivative of \\( \\log_{f(x)} (x) \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ead372789c7be9c76000d44866d4150fc7bd77436d7cfd3addaa70ab6a7e79d4 | What is the derivative of \( \sin^2(x) - \cos^2(x) \)? | [
"2\\sin(2x)"
] | [
"What is the derivative of \\( \\sin^2(x) - \\cos^2(x) \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-082e310fbae25d753a65ef17f272f708d2aade5cb08fe20fd15663868e29f7cc | What is the derivative of \( n! \)? | [
"n! \\psi(n+1)"
] | [
"What is the derivative of \\( n! \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1479e3c2f4dd46eb9588f8dbe293b84c021145ce136f8cf0a1c478bea81225c8 | What is the derivative of the Dirac delta function? | [
"\\delta'(x)"
] | [
"What is the derivative of the Dirac delta function?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-2e902ee77111c84bc9189b732af02deb3953c867a77aa5e8413c21acdf42f9c7 | What is the derivative of the factorial function $f(x) = x!$? (Here, the factorial function refers to the gamma function, which is defined for all positive real numbers.) | [
"\\(\\Gamma(x+1) \\psi(x+1)\\)"
] | [
"What is the derivative of the factorial function $f(x) = x!$? (Here, the factorial function refers to the gamma function, which is defined for all positive real numbers.)"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-f3c5b79fda2f9015733a613cb9ddeb5155e3490f287e9f6900050bcfa538f3e7 | What is the derivative of the function \( (\sin x^2)^{\sqrt{3x}} \)? | [
"\\((\\sin x^{2})^{\\sqrt{3x}} \\cdot \\left[\\sqrt{3x} \\cdot\\frac{\\cos x^{2}\\cdot 2x}{\\sin x^{2}}+\\ln (\\sin x^{2})\\cdot \\frac{3}{2\\sqrt{3x}}\\right]\\)"
] | [
"What is the derivative of the function \\( (\\sin x^2)^{\\sqrt{3x}} \\)?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-51ee1042386c231e5d99a7f358928bff49150653fc476e456fb337787301795e | What is the derivative of the function \( t(x) = g(x)^{f(x)} \)? | [
"\\( g(x)^{f(x)} \\left( \\frac{f(x)g'(x)}{g(x)} + f'(x) \\ln g(x) \\right) \\)"
] | [
"What is the derivative of the function \\( t(x) = g(x)^{f(x)} \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-b60eda3cfbd942a137ff247b08db13cd768d222a738275803880170226823ab3 | What is the derivative of the integral with multiple variables involved in the bounds? $$\frac{d}{dx}\int_{x}^{y}x^2dx$$ | [
"0"
] | [
"What is the derivative of the integral with multiple variables involved in the bounds? $$\\frac{d}{dx}\\int_{x}^{y}x^2dx$$"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ea93917d9eff74369a3d9bb741b28a33ccfbe11a97cf6ddd618e53bce2fed684 | What is the derivative of the integral with multiple variables involved in the bounds? $$\frac{d}{dx}\int_{y}^{2y}x^2dx$$ | [
"0"
] | [
"What is the derivative of the integral with multiple variables involved in the bounds? $$\\frac{d}{dx}\\int_{y}^{2y}x^2dx$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-1a87889fa0686380ea70306bd84f3287db97105a66cece98dc5b274c1e7a4bf6 | What is the derivative of the integral with one constant and one variable bound? $$\frac{d}{dx}\int_{2}^{x}x^2dx$$ | [
"\\( x^2 \\)"
] | [
"What is the derivative of the integral with one constant and one variable bound? $$\\frac{d}{dx}\\int_{2}^{x}x^2dx$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 3,
"tir": 17
} | 25 | 0 | 0 |
omr-ws-fc2d603db90878f48af1281289071bd99cf6823474ed1431b59054976cd0d5c0 | What is the derivative of the integral with one constant and one variable bound? $$\frac{d}{dx}\int_{x}^{5}x^2dx$$ | [
"\\(-x^2\\)"
] | [
"What is the derivative of the integral with one constant and one variable bound? $$\\frac{d}{dx}\\int_{x}^{5}x^2dx$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 7,
"tir": 19
} | 38 | 0 | 0 |
omr-ws-1ec64de770a72b8373792d01991df8d00ba05d4d289a4870b88ee0ee416120bc | What is the derivative of the integral with variable bounds involving a variable expression? $$\frac{d}{dx}\int_{x}^{2x}x^2dx$$ | [
"\\( 7x^2 \\)"
] | [
"What is the derivative of the integral with variable bounds involving a variable expression? $$\\frac{d}{dx}\\int_{x}^{2x}x^2dx$$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 1,
"tir": 8
} | 15 | 0 | 0 |
omr-ws-3550de3eba9da1f5f877edca219ce2a3e8db23842233e25b263edc20a7862c18 | What is the determinant of the 1x1 matrix A = (-2)? | [
"-2"
] | [
"What is the determinant of the 1x1 matrix A = (-2)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-69f29764ea3e37c5e64cbc50762177cdf813e42f7e640242428077b353d34470 | What is the determinant of the matrix \(\begin{pmatrix} x_{1}^{y_{1}} &x_{2}^{y_{1}} &\dots&x_{n}^{y_{1}} \\ x_{1}^{y_{2}} &x_{2}^{y_{2}} &\dots&x_{n}^{y_{2}}\\\vdots &\vdots & \vdots & \vdots \\x_{1}^{y_{n}} &x_{2}^{y_{n}} &\dots&x_{n}^{y_{n}} \end{pmatrix}\), given that \( x_{i} \neq x_{j} \) and \( y_{i} \neq y_{j} ... | [
"\\prod_{1 \\leq i < j \\leq n} (x_j - x_i)"
] | [
"What is the determinant of the matrix \\(\\begin{pmatrix} x_{1}^{y_{1}} &x_{2}^{y_{1}} &\\dots&x_{n}^{y_{1}} \\\\ x_{1}^{y_{2}} &x_{2}^{y_{2}} &\\dots&x_{n}^{y_{2}}\\\\\\vdots &\\vdots & \\vdots & \\vdots \\\\x_{1}^{y_{n}} &x_{2}^{y_{n}} &\\dots&x_{n}^{y_{n}} \\end{pmatrix}\\), given that \\( x_{i} \\neq x_{j} \\)... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 2
} | 14 | 0 | 0 |
omr-ws-60e5124926266625c80f57fcaf5605e471c98d783f2ad5094659b2057292f9f7 | What is the diameter in centimeters of a circle whose area is $100\pi \text{cm}^2$? | [
"20"
] | [
"What is the diameter in centimeters of a circle whose area is $100\\pi \\text{cm}^2$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6aab6ad4badf6d8d75116551210875440f5285e1c9215d875791d35db2f222f2 | What is the diameter of a circle in which a quadrilateral with side lengths 25, 39, 52, and 60 (taken consecutively) is inscribed? | [
"65"
] | [
"What is the diameter of a circle in which a quadrilateral with side lengths 25, 39, 52, and 60 (taken consecutively) is inscribed?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 20
} | 35 | 0 | 0 |
omr-ws-ce6f0cac455e83da10bb031993756d75773665af805d47337b36be8237921f9c | What is the diameter of a circle in which a quadrilateral with sides of lengths 25, 39, 52, and 60 (taken consecutively) is inscribed? | [
"65"
] | [
"What is the diameter of a circle in which a quadrilateral with sides of lengths 25, 39, 52, and 60 (taken consecutively) is inscribed?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 63,
"tir": 88
} | 151 | 0 | 0 |
omr-ws-20672310b779ad3529cac2d5f5adc285616e0f986c132022eadc94db0644216f | What is the diameter of the circle inscribed in triangle $ABC$ if $AB = 11,$ $AC=6,$ and $BC=7$? Express your answer in simplest radical form. | [
"\\sqrt{10}"
] | [
"What is the diameter of the circle inscribed in triangle $ABC$ if $AB = 11,$ $AC=6,$ and $BC=7$? Express your answer in simplest radical form."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3157aaffba7899a78b1cfe7e5187f0281dfec8188ab4b56eb36d5b40d0f31bc9 | What is the diameter of the circle that circumscribes a cyclic quadrilateral with side lengths 80, 150, 102, and 136 in that order? | [
"170"
] | [
"What is the diameter of the circle that circumscribes a cyclic quadrilateral with side lengths 80, 150, 102, and 136 in that order?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 27
} | 49 | 0 | 0 |
omr-ws-03c102569ccb0bcd688902610e86a51c79796a643cb9fdc5efc5550601c4685a | What is the difference between \( e^x \) and \( e^{-x} \)? | [
"\\( e^x - \\frac{1}{e^x} \\)"
] | [
"What is the difference between \\( e^x \\) and \\( e^{-x} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 5,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-085d8317d10211c00865e3980e1656abbab6696aaa44d95ec0c869505f35bef4 | What is the difference between the approximation being near 0, or near 1, 3, etc.? | [
"The difference between the approximation being near 0, or near 1, 3, etc., is that \\(a\\) in the formula \\(f(a+h) \\approx f(a) + f'(a) \\cdot h\\) represents the point around which the approximation is being made."
] | [
"What is the difference between the approximation being near 0, or near 1, 3, etc.?"
] | [
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-1e4ddf58530a6b8748970444d08a5331c4cdc28a0dc9ce3342781846ad0a705c | What is the difference between the distance \( PQ \) and the distance \( QP \) for a line segment with endpoints \( P \) and \( Q \)? | [
"\\( 2PQ \\)"
] | [
"What is the difference between the distance \\( PQ \\) and the distance \\( QP \\) for a line segment with endpoints \\( P \\) and \\( Q \\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dfe0aa2870de04eee98d3dafb81e484841f8ebbf34c3abec74c6965a67cce9c4 | What is the difference between the following three integrations: a.) $\int dx$ b.) $\int\int dx dy$ c.) $\int\int\int dxdydz$ | [
"\\text{a) Length, b) Area, c) Volume}"
] | [
"What is the difference between the following three integrations:\na.) $\\int dx$\nb.) $\\int\\int dx dy$\nc.) $\\int\\int\\int dxdydz$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-a514ad23954ae26ffe5d543bba99828d68420fff3bb75baaf5e6a4180412c70c | What is the difference between the graph of $f(x) = 0.2 (x-3)(x+2)(x+3)$ and $f(x) = (x-3)(x+2)(x+3)$? | [
"The graph of \\( f(x) = 0.2 (x-3)(x+2)(x+3) \\) is a vertical compression by a factor of 5 of the graph of \\( f(x) = (x-3)(x+2)(x+3) \\)."
] | [
"What is the difference between the graph of $f(x) = 0.2 (x-3)(x+2)(x+3)$ and $f(x) = (x-3)(x+2)(x+3)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 3,
"tir": 2
} | 10 | 0 | 0 |
omr-ws-469b8427d9f33ed722c41c7499a2649310266eb694c2540ba01bd874569914a6 | What is the difference between the greatest and least values of the function \( f(x) = (3 - \sqrt{4 - x^2})^2 + (1 + \sqrt{4 - x^2})^3 \)? | [
"\\(\\frac{500}{27}\\)"
] | [
"What is the difference between the greatest and least values of the function \\( f(x) = (3 - \\sqrt{4 - x^2})^2 + (1 + \\sqrt{4 - x^2})^3 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-bc78d85583bc591d60ae1261c8f6cbe6017cfaaa3c526587f0833c806a46a7ef | What is the difference between the highest and lowest root of the equation $\frac{\sqrt{2+x}+\sqrt{2-x}}{\sqrt{2+x}-\sqrt{2-x}}=\frac{2}{x}$? | [
"4"
] | [
"What is the difference between the highest and lowest root of the equation $\\frac{\\sqrt{2+x}+\\sqrt{2-x}}{\\sqrt{2+x}-\\sqrt{2-x}}=\\frac{2}{x}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-66b7603bec26870c2a67a2133894e1abc6ba7f129636b9f5738472d883143684 | What is the difference between the largest and smallest real zeroes of the function $F(x) = 2x^{4}-7x^{3}+2x^{2}+7x+2$? | [
"\\(\\frac{3}{2} + \\sqrt{2}\\)"
] | [
"What is the difference between the largest and smallest real zeroes of the function $F(x) = 2x^{4}-7x^{3}+2x^{2}+7x+2$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-e926bc9a1f0139fa7ad468cd8605e1e6e1562f104b9599149c13cabc87a39521 | What is the difference between the largest and smallest roots of the polynomial $2x^4 - 7x^3 + 2x^2 + 7x + 2$? | [
"\\(\\sqrt{2} + \\frac{3}{2}\\)"
] | [
"What is the difference between the largest and smallest roots of the polynomial $2x^4 - 7x^3 + 2x^2 + 7x + 2$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-008124770ce7aa79e7bd3398ec30ddee1f09b0d50ce4080930ed6e5241a9fe5d | What is the difference between the largest and smallest value for \( \text{lcm}(a, b, c) \), where \( a, b \), and \( c \) are distinct positive integers between 1 and 10, inclusive? | [
"626"
] | [
"What is the difference between the largest and smallest value for \\( \\text{lcm}(a, b, c) \\), where \\( a, b \\), and \\( c \\) are distinct positive integers between 1 and 10, inclusive?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f60368a42c915475324ff5d700a3ddbaec2ba6084f31ed93f0bd02171c2fed60 | What is the difference between the largest and the smallest possible positive integer $n$ such that $n!$ ends in exactly $1001$ zeros? | [
"4"
] | [
"What is the difference between the largest and the smallest possible positive integer $n$ such that $n!$ ends in exactly $1001$ zeros?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 9
} | 24 | 0 | 0 |
omr-ws-8ba849878ec72bd2a026d2c00bb41d5bcf9e714d4558ebac8a4e505fed46967c | What is the difference between the numbers 849 and 3A7 in the duodecimal (base-12) system? | [
"462"
] | [
"What is the difference between the numbers 849 and 3A7 in the duodecimal (base-12) system?"
] | [
"aops_c4_high_school_math",
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 15
} | 47 | 0 | 0 |
omr-ws-4a86980a9283dbd2a09d3030dd75164d7eb91cba8c8da739f5b299b6239fbda2 | What is the difference between the smallest and largest number of intersection points of 12 equilateral triangles, given that each pair of triangles must intersect at least one point, and no three triangles can intersect at the same point? | [
"264"
] | [
"What is the difference between the smallest and largest number of intersection points of 12 equilateral triangles, given that each pair of triangles must intersect at least one point, and no three triangles can intersect at the same point?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 3
} | 4 | 0 | 0 |
omr-ws-57fed718584fd22adb9d11c7242b7299ca64437e6ec84be412efec19653c68a4 | What is the difference between the sum of the first 2003 even counting numbers and the sum of the first 2003 odd counting numbers? | [
"2003"
] | [
"What is the difference between the sum of the first 2003 even counting numbers and the sum of the first 2003 odd counting numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 46
} | 46 | 0 | 0 |
omr-ws-f07bed8b9e4098c01cde470e1172d0592b0a5a72c34ee5d567eaa05be553fbaf | What is the difference between the two values: $2^5$ and $4^3$? | [
"32"
] | [
"What is the difference between the two values: $2^5$ and $4^3$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1d91ba47e3109952194852604465cf149be2a9456fa1cde92588f2501eea8e95 | What is the difference in areas of two circles that have a common external tangent of length 75 units, a common internal tangent of length 13 units, and a distance of 85 units between their centers? | [
"\\(3360\\pi\\)"
] | [
"What is the difference in areas of two circles that have a common external tangent of length 75 units, a common internal tangent of length 13 units, and a distance of 85 units between their centers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-f2edae391db22e4d17850938143f0ce0b5640f1fe2597ec6f3e6f2e28e671cc8 | What is the difference in the number of factors between \(x\) and \(y\) if \(2xy = 120\) and \(x\) is greater than \(121212 \cdot 0^1 + 1^0\)? | [
"11"
] | [
"What is the difference in the number of factors between \\(x\\) and \\(y\\) if \\(2xy = 120\\) and \\(x\\) is greater than \\(121212 \\cdot 0^1 + 1^0\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-26360cfd85dfe2d85bb7c2754130f75624f187adcf55cfacc8549b5ac91e039a | What is the digit at the 50th place from the left of $\left(\sqrt{50}+7\right)^{50}$? | [
"9"
] | [
"What is the digit at the 50th place from the left of $\\left(\\sqrt{50}+7\\right)^{50}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 2,
"tir": 1
} | 13 | 0 | 0 |
omr-ws-6a07ba2180d9a297b23f57ae98b2991aaed59c5d54f35ba25de53ff7911651e0 | What is the digit at the 50th place from the left of \((\sqrt{50} + 7)^{50}\)? | [
"9"
] | [
"What is the digit at the 50th place from the left of \\((\\sqrt{50} + 7)^{50}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 2,
"tir": 1
} | 12 | 0 | 0 |
omr-ws-a86d5ceedc7caf71ffa63f376d4eb722b6789811da95b0940c82205f637302f3 | What is the digit at the tens place of \(2183^{2156}\)? | [
"8"
] | [
"What is the digit at the tens place of \\(2183^{2156}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-dc2963f7e558316992993fcbf40e77253dae962a44d5fe3158580d220687ee82 | What is the digit immediately after the decimal point in the development of \( (1 + \sqrt{2010})^{2010} \)? | [
"9"
] | [
"What is the digit immediately after the decimal point in the development of \\( (1 + \\sqrt{2010})^{2010} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 5
} | 14 | 0 | 0 |
omr-ws-575662da70f56af32eeea81ee04a5c1173e6b4b5676dab453060fcecd0d5d918 | What is the digit in the tens of thousands place of the number $5^{5^{5^{5^{5}}}}$? | [
"0"
] | [
"What is the digit in the tens of thousands place of the number $5^{5^{5^{5^{5}}}}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-ff48cf12d3abc1a80f394631d6999c16a10af8259ad529819e505718f6f4c568 | What is the digit in the tens place of the sum \(1! + 1! + 2! + 3! + 5! + 8! + \ldots + 34!\)? | [
"5"
] | [
"What is the digit in the tens place of the sum \\(1! + 1! + 2! + 3! + 5! + 8! + \\ldots + 34!\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-9b9f0e3d61c8b2e6e0e8a5cc130465e9752ff6808272a2d0006e1d61006828fd | What is the digit in the tens place when $7^{2005}$ is expressed in decimal notation? | [
"0"
] | [
"What is the digit in the tens place when $7^{2005}$ is expressed in decimal notation?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-57e7bef02d7603e30e8536454f759df506622dd200879ec5eed30b17b54dcf44 | What is the digit in the tenth place in the smallest 10-digit number formed using all digits 0 through 9 that is divisible by 11111? | [
"5"
] | [
"What is the digit in the tenth place in the smallest 10-digit number formed using all digits 0 through 9 that is divisible by 11111?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-2c7e790b431e8ee22e1a4f902c3037e9ea0330a630baf463ca1fa8b06a9983a8 | What is the digit in the thousandths place of the decimal equivalent of $\frac{3}{16}$? | [
"7"
] | [
"What is the digit in the thousandths place of the decimal equivalent of $\\frac{3}{16}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e6b91b2395080c7192fef9062764c0cb2e1f5754ac203051f254ad6dda0e007a | What is the digit in the unit's place of $\sum_{i=1}^{60} i^{60!}$? | [
"6"
] | [
"What is the digit in the unit's place of $\\sum_{i=1}^{60} i^{60!}$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7f069a158bda2f51c44555f8e3f99cc039f7993652e448d000ca174b915be65d | What is the digit in the units place of the greatest 10-digit number formed using all digits 0 through 9 that is divisible by 11111? | [
"0"
] | [
"What is the digit in the units place of the greatest 10-digit number formed using all digits 0 through 9 that is divisible by 11111?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-cc16e3ba8d4e31aa9b0ca1f49c8fcfa95262f28a69054b4c55e017e36611e19b | What is the digit sum of the number of ways to place 45 different types of chocolates in 3 identical bags such that each bag has at least one chocolate? | [
"91"
] | [
"What is the digit sum of the number of ways to place 45 different types of chocolates in 3 identical bags such that each bag has at least one chocolate?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-4487db835bc6f5309505e06e0f37ac581953907650ccf2e02d2a1b770225dcf9 | What is the dihedral angle between two faces of a regular dodecahedron? | [
"\\arccos\\left( -\\frac{\\sqrt{5}}{5} \\right)"
] | [
"What is the dihedral angle between two faces of a regular dodecahedron?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7940c0faa705ce3dd7f05c3e894f2d3c90eca29325031d3f01e51b109253e8c6 | What is the dihedral angle between two faces of a regular icosahedron? | [
"\\arccos\\left(-\\frac{\\sqrt{5}}{3}\\right)"
] | [
"What is the dihedral angle between two faces of a regular icosahedron?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-efc693cf7ec1d8e26e94fba0865db364e733f930c940db8be9483ca6a65e1ec3 | What is the dihedral angle between two faces of a regular octahedron? | [
"\\arccos\\left(-\\frac{1}{3}\\right)"
] | [
"What is the dihedral angle between two faces of a regular octahedron?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f20d56cfe98af6c188c1f52ab1d37db7865a7c8012e595fdb2c8eff70f7efbcd | What is the dihedral angle between two faces of a regular tetrahedron? | [
"\\arccos\\left(\\frac{1}{3}\\right)"
] | [
"What is the dihedral angle between two faces of a regular tetrahedron?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3f5e7d696cd9b3a549f22a35fbc1579f00adda82a77b85ba2f03ea20754a38e8 | What is the dimension of $SO_n$, the space of real orthogonal matrices with determinant $1$? | [
"\\(\\frac{(n-1)n}{2}\\)"
] | [
"What is the dimension of $SO_n$, the space of real orthogonal matrices with determinant $1$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-79948424bfa3685396387fc4a6f54033626ac650fc408c024644a72d526f00dc | What is the dimension of $\mathbb{Q}(\varepsilon_n)$ over $\mathbb{Q}$, where $\varepsilon_n = \exp\left(\frac{2\pi i}{n}\right)$? | [
"\\(\\varphi(n)\\)"
] | [
"What is the dimension of $\\mathbb{Q}(\\varepsilon_n)$ over $\\mathbb{Q}$, where $\\varepsilon_n = \\exp\\left(\\frac{2\\pi i}{n}\\right)$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-0199ff8fea2234616c2806d57d8dc915b55c95ff004436dcdb40222613aed347 | What is the dimension of $\mathbb{R}^n$ over $\mathbb{C}$? Can anything be explicitly stated when $n$ is even or odd? | [
"Dimension is $\\frac{n}{2}$ when $n$ is even, and not defined when $n$ is odd."
] | [
"What is the dimension of $\\mathbb{R}^n$ over $\\mathbb{C}$? Can anything be explicitly stated when $n$ is even or odd?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 6,
"tir": 24
} | 33 | 0 | 0 |
omr-ws-1092fc78cd96a03ddb2d68d33fab1532bf8536f373e5fbe34cd7026b55ddfbca | What is the dimension of the $\mathbb{Q}$-vector-space {$f$:$\mathbb{R}$ $\to$ $\mathbb{R}$ | $f$ is continuous and Image($f$) $\subseteq$ $\mathbb{Q}$}? | [
"1"
] | [
"What is the dimension of the $\\mathbb{Q}$-vector-space {$f$:$\\mathbb{R}$ $\\to$ $\\mathbb{R}$ | $f$ is continuous and Image($f$) $\\subseteq$ $\\mathbb{Q}$}?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-de46cb9330402182348d26fe4fcb7c95928b9ebebf950d25156192e28bb349cc | What is the dimension of the kernel of the linear transformation defined by \( p(x) \mapsto \frac{d}{dx} p(x) + 2xp(x) \)? | [
"0"
] | [
"What is the dimension of the kernel of the linear transformation defined by \\( p(x) \\mapsto \\frac{d}{dx} p(x) + 2xp(x) \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
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