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-60bd3710bffedc9e718aee5881ca65daa7c7259f8fc66ba21851caff54e1e71f | Write the coordinates of the vertex of the quadratic equation \( y = 3x^2 + 24x + 43 \). | [
"(-4, -5)"
] | [
"Write the coordinates of the vertex of the quadratic equation \\( y = 3x^2 + 24x + 43 \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-e531aeaf08beca52e6fc80857b02f040fbcdbc88a8e669db9916ed229bf8c54e | Write the determinant of the matrix \( A = \begin{vmatrix} a & x & x & x \\ x & b & x & x \\ x & x & c & x \\ x & x & x & d \end{vmatrix} \) in the form of a product. | [
"(a - x)(b - x)(c - x)(d - x) \\left(1 + x \\left( \\frac{1}{a - x} + \\frac{1}{b - x} + \\frac{1}{c - x} + \\frac{1}{d - x} \\right) \\right)"
] | [
"Write the determinant of the matrix \\( A = \\begin{vmatrix} a & x & x & x \\\\ x & b & x & x \\\\ x & x & c & x \\\\ x & x & x & d \\end{vmatrix} \\) in the form of a product."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-36e32801a3b1ffe4fe75f17a087ca7363aae378c661ec3b67973d5ebb1643921 | Write the differential equation \( x^{2}y'' + 2xy' - 2y + f(x) = 0 \) in self-adjoint form. | [
"\\( (x^{2}y')' - 2y = -f(x) \\)"
] | [
"Write the differential equation \\( x^{2}y'' + 2xy' - 2y + f(x) = 0 \\) in self-adjoint form."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-8df36207d5630a8b9237d82a79c7c0e94857acc33b3fe0d6ab9b1e6e3491051a | Write the differential equation \((1 - x^2)y'' - 2xy' + 6y = 0\) in self-adjoint form. | [
"\\(\\frac{d}{dx}\\left[(1 - x^2) \\frac{dy}{dx}\\right] + 6y = 0\\)"
] | [
"Write the differential equation \\((1 - x^2)y'' - 2xy' + 6y = 0\\) in self-adjoint form."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 5
} | 19 | 0 | 0 |
omr-ws-bc262f43d9164a8852a79c50282276f872ce391ed98ea8f9bed48ee4e98960f8 | Write the differential of surface area, \( dS \), in terms of \( x \) for the sphere formed by rotating the circle \( x^2 + y^2 = 25 \) around the x-axis. | [
"\\( \\frac{5}{\\sqrt{25 - x^2}} \\, dx \\)"
] | [
"Write the differential of surface area, \\( dS \\), in terms of \\( x \\) for the sphere formed by rotating the circle \\( x^2 + y^2 = 25 \\) around the x-axis."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-36bf3e07b74bc73e02cde6a193e74047e4263cfa288cf079bee653d25e6edd02 | Write the equation \( 5x + 3y = 4 \) in parametric form. That is, express both \( x \) and \( y \) in terms of the same parameter. | [
"\\( x = \\frac{t}{5}, \\quad y = \\frac{4 - t}{3} \\)"
] | [
"Write the equation \\( 5x + 3y = 4 \\) in parametric form. That is, express both \\( x \\) and \\( y \\) in terms of the same parameter."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9a88dc438b622aec6d9c85d1a04ca3083f81e02450449b0c051554485abfc2b2 | Write the equation \( y = -\frac{3}{5}x + 3 \) in intercepts form. | [
"\\dfrac{x}{5} + \\dfrac{y}{3} = 1"
] | [
"Write the equation \\( y = -\\frac{3}{5}x + 3 \\) in intercepts form."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-4ac5856c96186039e21632bb3bbea27fe592250ef9450e6dcd987bb93e8ee1d4 | Write the equation \( y = \frac{3}{2}x + \frac{7}{2} \) in standard form. | [
"\\(3x - 2y = -7\\)"
] | [
"Write the equation \\( y = \\frac{3}{2}x + \\frac{7}{2} \\) in standard form."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d12d750ea5531c12fb2e84dc147c2e47898e0ae3c93cc0a7cb2b8406c5f91376 | Write the equation \( y = x^2 - 10x + 35 \) in vertex form. State the vertex and the axis of symmetry. | [
"\\( y = (x - 5)^2 + 10 \\), Vertex: \\( (5, 10) \\), Axis of Symmetry: \\( x = 5 \\)"
] | [
"Write the equation \\( y = x^2 - 10x + 35 \\) in vertex form. State the vertex and the axis of symmetry."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 15
} | 17 | 0 | 0 |
omr-ws-23505dc69dcd44712bbc02331f0ec3f869329679cd8155f4548aac87f640457d | Write the equation \((1 - k)(xy)^2 + k(xy^2 + yx^2 + x - y) = k + 1\) in the form \((\text{product}) = (\text{constant})\), where \(k\) is a constant. | [
"(xy - 1)((1 - k)xy + k(x + y)) = k + 1"
] | [
"Write the equation \\((1 - k)(xy)^2 + k(xy^2 + yx^2 + x - y) = k + 1\\) in the form \\((\\text{product}) = (\\text{constant})\\), where \\(k\\) is a constant."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-27ee43b87de0ef8ce9e65eca6ff4eae5c28a2127bf21c16d7b73457751c7549d | Write the equation \((1 - k)xyz + k(xy + xz + x + yz - y - z) = k + 1\) in the form \((\text{product}) = (\text{constant})\), where \(k\) is a constant. | [
"\\((1 - xyz)(xy + xz + yz - xyz + x - y - z - 1)^{-1} = k\\)"
] | [
"Write the equation \\((1 - k)xyz + k(xy + xz + x + yz - y - z) = k + 1\\) in the form \\((\\text{product}) = (\\text{constant})\\), where \\(k\\) is a constant."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-24230a8eae8a435ff371f8c2e47dd8a9ebda2f65966091500b93d824575118b9 | Write the equation \(x^2 + x - y^2 - 3y - 2 = 0\) in its general form and identify whether it represents a circle, parabola, ellipse, or hyperbola. | [
"hyperbola"
] | [
"Write the equation \\(x^2 + x - y^2 - 3y - 2 = 0\\) in its general form and identify whether it represents a circle, parabola, ellipse, or hyperbola."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2ee7c5a3675a8456877295a2bedcccf2397a39b5d477246250f7c5b5c45ae88e | Write the equation for the line through the two extreme points of the graph of the function \( y = \frac{x^2 + x + 2m}{x^2 + m} \). | [
"\\( x - 2m(y - 2) = 0 \\)"
] | [
"Write the equation for the line through the two extreme points of the graph of the function \\( y = \\frac{x^2 + x + 2m}{x^2 + m} \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 8,
"tir": 15
} | 33 | 0 | 0 |
omr-ws-97cce4226f0c95359fbe1a5cbee339107c4f4f3779c209db35c461f5689068c3 | Write the equation for the surface generated by revolving the curve \(4x + 9y^2 = 36\) around the y-axis. | [
"4\\sqrt{x^2 + z^2} + 9y^2 = 36"
] | [
"Write the equation for the surface generated by revolving the curve \\(4x + 9y^2 = 36\\) around the y-axis."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 2
} | 12 | 0 | 0 |
omr-ws-0f3cd2ea3bbf4bdd5c91c7fdf2a2e889d44882f5a1438e13ff1f58c7608fe2fd | Write the equation of $\sqrt{3}x + y - 3 = 0$ using rotated axes so that the line is parallel to one of the rotated axes. What is the distance from the origin to the line? | [
"\\(\\frac{3}{2}\\)"
] | [
"Write the equation of $\\sqrt{3}x + y - 3 = 0$ using rotated axes so that the line is parallel to one of the rotated axes. What is the distance from the origin to the line?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-41edf0f5a15d15a9dd0b1259ee8c600724f710a196c877361fe23ada1babbeb4 | Write the equation of a geometric sequence where \(a_5 + a_2 - a_4 = 111\) and \(a_4 + a_1 - a_3 = 34\). | [
"a_n = \\frac{1336336}{988021} \\left( \\frac{111}{34} \\right)^{n-1}"
] | [
"Write the equation of a geometric sequence where \\(a_5 + a_2 - a_4 = 111\\) and \\(a_4 + a_1 - a_3 = 34\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-20da63563a09dfb775c5e46b21b4c669be6ab26395731214bf0caa9ec2a4079a | Write the equation of a geometric sequence where \(a_5 - a_1 = 15\) and \(a_4 - a_2 = 6\). | [
"\\(a_1 = 1, r = 2\\) or \\(a_1 = -16, r = \\frac{1}{2}\\)"
] | [
"Write the equation of a geometric sequence where \\(a_5 - a_1 = 15\\) and \\(a_4 - a_2 = 6\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 15
} | 19 | 0 | 0 |
omr-ws-e29cf7ece03dba78becd373de309c96943b34e30b49e1d46f9ad34a12d115568 | Write the equation of a horizontal tangent to the graph of the function $$y=e^x+e^{-x}.$$ | [
"$y=2$"
] | [
"Write the equation of a horizontal tangent to the graph of the function $$y=e^x+e^{-x}.$$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f169e82d5d7b13d0a398f11557af9c86ef37c7ac138b80d69d2cde543af658ce | Write the equation of a hyperbola in standard form given its vertices and not its foci. (Note: Specific vertices are not provided, so this is a general problem statement.) | [
"\\dfrac{(y - k)^2}{a^2} - \\dfrac{(x - h)^2}{b^2} = 1"
] | [
"Write the equation of a hyperbola in standard form given its vertices and not its foci. (Note: Specific vertices are not provided, so this is a general problem statement.)"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-f0e06041345e6b361588141614ff664a72e44c724c1265b6b37150c3c6ade9d7 | Write the equation of a line and a plane that intersect at the point (1,4,-1). Given sample answer: Plane equation: \( x + 2y + 5z - 4 = 0 \) and line equation: \([x, y, z] = [-1, 3, 0] + t[2, 1, -1]\). Explain how to derive these equations. | [
"\\( x + 2y + 5z - 4 = 0 \\) and \\([x, y, z] = [-1, 3, 0] + t[2, 1, -1]\\)"
] | [
"Write the equation of a line and a plane that intersect at the point (1,4,-1). \n\nGiven sample answer: Plane equation: \\( x + 2y + 5z - 4 = 0 \\) and line equation: \\([x, y, z] = [-1, 3, 0] + t[2, 1, -1]\\). Explain how to derive these equations."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 19
} | 31 | 0 | 0 |
omr-ws-b8d15601f92f904862b8adf171f6fbb452ba532f8d1cf8843595634c35f35a60 | Write the equation of a sine function with amplitude 3, period $3\pi$, and phase shift $-\frac{1}{3}$. | [
"\\( 3 \\sin \\left( \\frac{2}{3} \\left( x + \\frac{1}{3} \\right) \\right) \\)"
] | [
"Write the equation of a sine function with amplitude 3, period $3\\pi$, and phase shift $-\\frac{1}{3}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-2da36e2dda5aabad6d3d968d646cbb08a6c553780b95dc3288d53d4cc21f2e65 | Write the equation of a straight line which passes through the point $M (2, 1)$ at the angle $\pi /4$ to the straight line $2x+3y+4=0$. | [
"$x-5y+3=0$ or $5x+y-11=0$"
] | [
"Write the equation of a straight line which passes through the point $M (2, 1)$ at the angle $\\pi /4$ to the straight line $2x+3y+4=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-98d4be67c2c62ae75111713d5ee3719685e217a09a0265fd6c30fff7167f8b1e | Write the equation of a straight line, which is parallel to two given straight lines $l_1$ and $l_2$ and is equidistant from $l_1$ and $l_2$, if: (a) $l_1:\ 3x-2y-1=0, l_2:\ \frac{x-1}{2}=\frac{y+3}{3}$; (b) $l_1:\ 3x-15y-1=0, l_2:\ \frac{x+1/2}{5}=\frac{y+1/2}{1}$. | [
"(a) $3x-2y-5=0$; (b) $x-5y-\\frac{7}{6}=0$"
] | [
"Write the equation of a straight line, which is parallel to two given straight lines $l_1$ and $l_2$ and is equidistant from $l_1$ and $l_2$, if: (a) $l_1:\\ 3x-2y-1=0, l_2:\\ \\frac{x-1}{2}=\\frac{y+3}{3}$; (b) $l_1:\\ 3x-15y-1=0, l_2:\\ \\frac{x+1/2}{5}=\\frac{y+1/2}{1}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-008b751d3d23e98b2c2d17136097d47eda3b6e11843e30b848a0b4da2504a3b7 | Write the equation of a tangent line to the curve $y=(x+1)/x$, if it is known that the tangent passes through the point $M (a, b)$. How many solutions does the problem have depending on the choice of the point? Find the solutions. | [
"$y=-\\frac{1}{x_0^2}x+\\frac{2}{x_0}+1$, where $x_0=\\frac{2}{b-1}$, if $a=0$, $b\\ne 1$: $x_0=\\frac{a}{2}$ for $a\\ne 0$, $b=1$; $x_0=\\frac{1}{b-1}$ for $a\\ne 0$, $b=1+\\frac{1}{a}$; $x_0=\\frac{-1\\pm \\sqrt {1+a(1-b)}}{1-b}$ for $a>0$, $b\\ne 1$, $b<1+\\frac{1}{a}$ and for $a<0$, $b\\ne 1$, $b>1+\\frac{1}{a}... | [
"Write the equation of a tangent line to the curve $y=(x+1)/x$, if it is known that the tangent passes through the point $M (a, b)$. How many solutions does the problem have depending on the choice of the point? Find the solutions."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3c0e01b830edec5bc4221fa81b6f08d6e9d9301f5fd65c946e150f829659d963 | Write the equation of the cosine function that has 180 degrees rotational symmetry about the point \((\pi/4, 3)\), has an amplitude of 5, and a period of \(\pi\). | [
"\\( y = 5\\cos(2(x - \\pi/4)) + 3 \\)"
] | [
"Write the equation of the cosine function that has 180 degrees rotational symmetry about the point \\((\\pi/4, 3)\\), has an amplitude of 5, and a period of \\(\\pi\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-76306dec7c1120858bcbe97c3b66840d55a4b7466baf4a2470e39c597b7294c5 | Write the equation of the cosine function that has 180 degrees rotational symmetry about the point \((\pi/6, 4)\), has an amplitude of 7, and a period of \(\pi\). | [
"\\( y = 7\\cos(3x) + 4 \\)"
] | [
"Write the equation of the cosine function that has 180 degrees rotational symmetry about the point \\((\\pi/6, 4)\\), has an amplitude of 7, and a period of \\(\\pi\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3e8b309f8c7523d2939866b91a767df219c8c5b6bf0e3e28346c33f31388ca4c | Write the equation of the cosine function that has rotational symmetry about the point \((\frac{\pi}{6}, 4)\), has an amplitude of 7, and a period of \(\pi\). | [
"y = 7 \\cos\\left(2x + \\frac{\\pi}{6}\\right) + 4"
] | [
"Write the equation of the cosine function that has rotational symmetry about the point \\((\\frac{\\pi}{6}, 4)\\), has an amplitude of 7, and a period of \\(\\pi\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 4
} | 17 | 0 | 0 |
omr-ws-67da934c7e9ef018a83b277ba3f26a534d2f4d9132121b1e1179819eba150900 | Write the equation of the ellipse in general form if the foci are $(-1,-1)$ and $(1,1)$ and the minor axis has length $2\sqrt{14}$. | [
"\\( 15x^2 - 2xy + 15y^2 - 224 = 0 \\)"
] | [
"Write the equation of the ellipse in general form if the foci are $(-1,-1)$ and $(1,1)$ and the minor axis has length $2\\sqrt{14}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 18
} | 65 | 0 | 0 |
omr-ws-f1aec841c7702c5e38889fcdc931fe3db47d871e8426ebf6f3a1beeb9fd9647f | Write the equation of the ellipse in general form if the foci are $(5,0)$ and $(0,-5)$ and the major axis has length $12$. | [
"119x^2 - 50xy + 119y^2 - 720x + 720y - 1584 = 0",
"\\( 119x^2 + 119y^2 - 50xy - 720x + 720y = 1584 \\)"
] | [
"Write the equation of the ellipse in general form if the foci are $(5,0)$ and $(0,-5)$ and the major axis has length $12$."
] | [
"aops_c4_high_school_math",
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 30,
"genselect": 12,
"tir": 18
} | 60 | 0 | 0 |
omr-ws-ee632b39e5bb54531ffe5bc1bbe11faeebc3e9f19e00ce9677c83aacdb5073dc | Write the equation of the graph that is the union of \( y = x \) and the region where \( x < 0 \) and \( y < 0 \). | [
"\\((x-y)(x+y + |x| + |y|) = 0\\)"
] | [
"Write the equation of the graph that is the union of \\( y = x \\) and the region where \\( x < 0 \\) and \\( y < 0 \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 6,
"tir": 8
} | 23 | 0 | 0 |
omr-ws-3dca3f8064759ed22baf94f6741636f6dcf37ae798069818d4987f7f6d4adb40 | Write the equation of the inverse of \( f(x) = (x+2)^2 - 9 \) and state whether the inverse is a function. | [
"\\( y = -2 \\pm \\sqrt{x + 9} \\) (not a function)"
] | [
"Write the equation of the inverse of \\( f(x) = (x+2)^2 - 9 \\) and state whether the inverse is a function."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 20
} | 35 | 0 | 0 |
omr-ws-2af57074b5c076196a1a885a0d7b3b075756d11f70c70beb700f9b840768b80a | Write the equation of the line \( l \) that passes through the point \( M(-2, -1, 3) \), is parallel to the plane \( \pi': 5x - y - z - 4 = 0 \), and is perpendicular to the line \( d \) defined by the intersection of the planes \( \pi_1: 3x - y - 5 = 0 \) and \( \pi_2: 2x - z - 2 = 0 \). | [
"\\frac{x + 2}{1} = \\frac{y + 1}{-11} = \\frac{z - 3}{16}"
] | [
"Write the equation of the line \\( l \\) that passes through the point \\( M(-2, -1, 3) \\), is parallel to the plane \\( \\pi': 5x - y - z - 4 = 0 \\), and is perpendicular to the line \\( d \\) defined by the intersection of the planes \\( \\pi_1: 3x - y - 5 = 0 \\) and \\( \\pi_2: 2x - z - 2 = 0 \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bb48c902363aba815cadeba9fbae997caca4ad0d9741351189257ab8818d8c9f | Write the equation of the line tangent to the graph of the function $y = x^4 - x^2 + x$ at least at two points. | [
"\\( y = x - \\frac{1}{4} \\)"
] | [
"Write the equation of the line tangent to the graph of the function $y = x^4 - x^2 + x$ at least at two points."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-fd65c35fb0fd71f438a1cc3bd5d337c4bae8485dcc8d77f8f0580d4db452afe6 | Write the equation of the line that passes through the points A(-11, 12) and B(12, -11) in segment form. | [
"\\(\\begin{pmatrix} x \\\\ y \\end{pmatrix} = \\begin{pmatrix} -11 \\\\ 12 \\end{pmatrix} + t \\begin{pmatrix} 23 \\\\ -23 \\end{pmatrix}\\)"
] | [
"Write the equation of the line that passes through the points A(-11, 12) and B(12, -11) in segment form."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 16
} | 16 | 0 | 0 |
omr-ws-c3d4da9847c1f5ee64aabc31c155d73f211d693d0d2fcb63ad7b804748291176 | Write the equation of the line that passes through the points T(-14, -8) and P(8, 3) in both general and explicit form. | [
"\\( y = \\frac{1}{2}x - 1 \\) and \\( x - 2y - 2 = 0 \\)"
] | [
"Write the equation of the line that passes through the points T(-14, -8) and P(8, 3) in both general and explicit form."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 28
} | 28 | 0 | 0 |
omr-ws-632abf067753986abdb35c0e881c839899e67301ffb0d407508977f437c5abcb | Write the equation of the parabola given the focus at (-3, -2) and the directrix is the line y = -6. Leave the equation in standard form. | [
"\\( (x + 3)^2 = 8(y + 4) \\)"
] | [
"Write the equation of the parabola given the focus at (-3, -2) and the directrix is the line y = -6. Leave the equation in standard form."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-3db224b64ccbf7c47bf60381ceb3c13ee52e85fbd6c15ad542097017e4ed30ff | Write the equation of the parabola passing through the points (-1, -6), (3, -12), and (2, -4.5). | [
"\\( y = \\frac{-13}{6}x^2 + \\frac{7}{3}x - \\frac{3}{2} \\)"
] | [
"Write the equation of the parabola passing through the points (-1, -6), (3, -12), and (2, -4.5)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7b82445f06fec1cf1cbbf248f0ad73b9586740abc9d3e13177fbbd1cb5c5b61f | Write the equation of the parabola with vertex (5, -3) and passing through the point (3, 9). | [
"\\( y = 3(x - 5)^2 - 3 \\)"
] | [
"Write the equation of the parabola with vertex (5, -3) and passing through the point (3, 9)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-dc5cbe76629df8397efbba9df961b0183899d5e2c70ddfa5ad97f4ea373ec730 | Write the equation of the rational function given the following information: - Domain: $x \neq -0.5, 5, 6$ - $f(0) = 8$ - $f(3) = 0$ - $f(-8) = 0$ - $\lim_{x\to -\infty} f(x)=4$ - $\lim_{x\to 2} f(x)=\frac{1}{2}$ - $f'(x) < 0$ when $x < -0.5$ - $f''(x) > 0$ when $-0.5 < x < 5$, $5 < x < 6$, and $x > 6$ | [
"f(x) = \\frac{(x - 3)(x + 8)(4x - 5)}{(x + 0.5)(x - 5)(x - 6)}"
] | [
"Write the equation of the rational function given the following information:\n- Domain: $x \\neq -0.5, 5, 6$\n- $f(0) = 8$\n- $f(3) = 0$\n- $f(-8) = 0$\n- $\\lim_{x\\to -\\infty} f(x)=4$\n- $\\lim_{x\\to 2} f(x)=\\frac{1}{2}$\n- $f'(x) < 0$ when $x < -0.5$\n- $f''(x) > 0$ when $-0.5 < x < 5$, $5 < x < 6$, and $x >... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-f9384532a64ab59467cfd8fd3fd35c567cb4b95c06ac54f2d5e8d4a796b38e2d | Write the equation of the straight line passing through the point with the coordinates $(1/2, 2)$, and touching the graph $$y(x)=-\frac{x^2}{2}+2$$ and cutting at two points the graph of the function $$y(x)=\sqrt {4-x^2}.$$ | [
"$y=-x+5/2$"
] | [
"Write the equation of the straight line passing through the point with the coordinates $(1/2, 2)$, and touching the graph $$y(x)=-\\frac{x^2}{2}+2$$ and cutting at two points the graph of the function $$y(x)=\\sqrt {4-x^2}.$$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8971eb7413f79fb413f87b0842416daf7dac733f6e4625e2ce2f657877322a29 | Write the equation of the symmetric of the plane $\pi : x-3y+z-1=0$ about the plane $\pi_2 : x+2y-2=0.$ | [
"\\(21x - 8y + 10z - 32 = 0\\)"
] | [
"Write the equation of the symmetric of the plane $\\pi : x-3y+z-1=0$ about the plane $\\pi_2 : x+2y-2=0.$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ef2d13eb815f3167393d5a2a1c96a1cd5c13f9c7c6dbbc365fe65f8373cd3de4 | Write the equation of the tangent line to the curve \( y = \frac{1}{1 + \sqrt{x}} \) at the point \( (4, \frac{1}{3}) \). | [
"\\( y = -\\frac{1}{36}x + \\frac{4}{9} \\)"
] | [
"Write the equation of the tangent line to the curve \\( y = \\frac{1}{1 + \\sqrt{x}} \\) at the point \\( (4, \\frac{1}{3}) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 55
} | 86 | 0 | 0 |
omr-ws-df78faf489109e3731b3bdc633e5eed2e5ac2ff76a4725d42aecf55a0fc71475 | Write the equation of the tangent line to the graph of the function $$y=\cos (2x-\pi /3)+2$$ at the point with abscissa $x_0=\pi /2$. | [
"$y=-\\sqrt 3x+\\frac{\\pi \\sqrt 3+3}{2}$"
] | [
"Write the equation of the tangent line to the graph of the function $$y=\\cos (2x-\\pi /3)+2$$ at the point with abscissa $x_0=\\pi /2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-928984e93a7205722ad101016198020d9ced3edfabfe842d8d7c7c24d6089bd2 | Write the equation of the tangent to the graph of the function $$f(x)=\big|x^2-|x|\big|$$ at the point with abscissa $x=-2$. | [
"$y=-3x-4$"
] | [
"Write the equation of the tangent to the graph of the function $$f(x)=\\big|x^2-|x|\\big|$$ at the point with abscissa $x=-2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7c021fec8b2f5f68f324b2f939c13d7a16c26e49b1a3dfb2d14758add14dedda | Write the equation of the vertical line that is tangent to the curve given by the equation $xy^2 + 5xy = 50$. | [
"\\( x = -8 \\)"
] | [
"Write the equation of the vertical line that is tangent to the curve given by the equation $xy^2 + 5xy = 50$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 5
} | 20 | 0 | 0 |
omr-ws-258d42df677e5791798c9c50d7e91837e9a93fb1ef79eced09cae09cd8aed9de | Write the exact algebraic form of $\sin(40^\circ)$. How many roots does it take? | [
"3"
] | [
"Write the exact algebraic form of $\\sin(40^\\circ)$. How many roots does it take?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 3
} | 17 | 0 | 0 |
omr-ws-82dba389eaa29cbeb64ca85dbb74931f46f997b90143acb289fcc9a7dc186477 | Write the expression $$\frac{a^2b+b^2c+c^2a}{3\sum a^3}+\frac{\sum a^2}{\sum ab}-\frac{3}{4}$$ in SOS (Sum of Squares) form. | [
"\\frac{(a^2 + b^2 + c^2 - ab - bc - ca)^2}{4(ab + bc + ca)^2} + \\frac{(a^2b + b^2c + c^2a - ab^2 - bc^2 - ca^2)^2}{12(a^3 + b^3 + c^3)(ab + bc + ca)}"
] | [
"Write the expression $$\\frac{a^2b+b^2c+c^2a}{3\\sum a^3}+\\frac{\\sum a^2}{\\sum ab}-\\frac{3}{4}$$ in SOS (Sum of Squares) form."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 5
} | 6 | 0 | 0 |
omr-ws-baee72eeb9c06d4a003cdd64501412996e0fde50511541347be30d834f74af36 | Write the expression $1(2m) + 4(2m+1) + 7(2m+2) + \cdots + (3m+1)(3m)$ in terms of $m$. | [
"\\(4m^3 + 7m^2 + 3m\\)"
] | [
"Write the expression $1(2m) + 4(2m+1) + 7(2m+2) + \\cdots + (3m+1)(3m)$ in terms of $m$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 16
} | 31 | 0 | 0 |
omr-ws-e329993d6340743a103e8ff78cd92c923c4653df9901f783da63eab6492cf1b4 | Write the expression $E = a^2 + b^2 + c^2 + d^2 + 2ab + 2bc + 2cd + 2da - 6ac - 6bd$ as a product. | [
"no_valid_answer_found"
] | [
"Write the expression $E = a^2 + b^2 + c^2 + d^2 + 2ab + 2bc + 2cd + 2da - 6ac - 6bd$ as a product."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-318403828225f0fd890e524c1b56da28347d6440691247f103916de53551578f | Write the expression $\frac{4+3c}{7}+2$ as a single fraction. | [
"\\frac{18+3c}{7}"
] | [
"Write the expression $\\frac{4+3c}{7}+2$ as a single fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4aaeae04e54088efb6a77e6ca36d1f64c34d510a853299456efcdd8ae9898b9e | Write the expression $\frac{\tan{2x}-2\tan{x}}{\tan{2x}+2\tan{x}}$ in terms of trigonometric functions of $x$. | [
"E) None of these answers"
] | [
"Write the expression $\\frac{\\tan{2x}-2\\tan{x}}{\\tan{2x}+2\\tan{x}}$ in terms of trigonometric functions of $x$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1a5e9e88612042fac1acbb861a1f80f3990781a5106472ea08f5db1e3eb539d2 | Write the expression $\frac{b^2}{a}+\frac{c^2}{b}+\frac{d^2}{c}+\frac{a^2}{d}-a-b-c-d$ into SOS (Sum of Squares) form, assuming $a, b, c, d > 0$. | [
"\\(\\sum_{cyc} \\frac{(a-b)^2}{b}\\)"
] | [
"Write the expression $\\frac{b^2}{a}+\\frac{c^2}{b}+\\frac{d^2}{c}+\\frac{a^2}{d}-a-b-c-d$ into SOS (Sum of Squares) form, assuming $a, b, c, d > 0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-98f581f61a8ea965fe3b0ec5dcba96153ef17b9b5a099538b561fa2b618b3366 | Write the expression $a+b+2c+|a-b|+|a+b-2c+|a-b||$ in an equivalent form not involving absolute values. | [
"\\( C = \\begin{cases} \n4a & \\text{if } a \\geq b \\text{ and } a \\geq c \\\\\n4c & \\text{if } a \\geq b \\text{ and } a < c \\\\\n4b & \\text{if } a < b \\text{ and } b \\geq c \\\\\n4c & \\text{if } a < b \\text{ and } b < c \n\\end{cases} \\)"
] | [
"Write the expression $a+b+2c+|a-b|+|a+b-2c+|a-b||$ in an equivalent form not involving absolute values."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 17
} | 32 | 0 | 0 |
omr-ws-deba5d012db90c2dafd291bc756fe32e80c180def8f1f5f40f823ae9dd666e86 | Write the expression \(6x^2 - xy + 23x - 2y^2 - 6y + 20\) as a product of two trinomials with integer coefficients. | [
"\\((3x - 2y + 4)(2x + y + 5)\\)",
"\\((4 + 3x - 2y)(5 + 2x + y)\\)"
] | [
"Write the expression \\(6x^2 - xy + 23x - 2y^2 - 6y + 20\\) as a product of two trinomials with integer coefficients."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 46
} | 77 | 0 | 0 |
omr-ws-aac702b6af98e104154e52f7b3b8fcd51418234f9a2d4f3626ce83b99ab21559 | Write the expression \(\cos\left(\frac{1}{2} \arccos(x)\right)\) as an algebraic expression in \(x\) for \(x > 0\). | [
"\\sqrt{\\dfrac{1 + x}{2}}"
] | [
"Write the expression \\(\\cos\\left(\\frac{1}{2} \\arccos(x)\\right)\\) as an algebraic expression in \\(x\\) for \\(x > 0\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-14eee230c5b12eaa7ddc091e2698fc9472ee0e1db3b54fb6ffda5c343ff91508 | Write the expression \(\frac{13}{5 - 12i}\) in the standard form \(a + bi\). | [
"\\(\\frac{5}{13} - \\frac{12}{13}i\\)"
] | [
"Write the expression \\(\\frac{13}{5 - 12i}\\) in the standard form \\(a + bi\\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3b9037b485ae93e0311452760fb1bc7bc5a19ace688a680d6348be859b0619df | Write the expression \(\sum b^2c^2(a-b)(a-c)(a+b-kc)(a+c-kb)\) in the Sum of Squares (SOS) form, where \(a, b, c, k\) are non-negative numbers. | [
"\\sum a^2b^2(a - b)^2(a + b - kc)^2"
] | [
"Write the expression \\(\\sum b^2c^2(a-b)(a-c)(a+b-kc)(a+c-kb)\\) in the Sum of Squares (SOS) form, where \\(a, b, c, k\\) are non-negative numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-ff9735e70c065cfa32b696a7b78a161d1f00a48e102265f4de4905b3fb851402 | Write the expression \(\tan(2 \cos^{-1}(1/x))\) as an algebraic expression in \(x\) for \(x > 0\). | [
"\\dfrac{2\\sqrt{x^2 - 1}}{2 - x^2}"
] | [
"Write the expression \\(\\tan(2 \\cos^{-1}(1/x))\\) as an algebraic expression in \\(x\\) for \\(x > 0\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b2bb77ec131652a6415ace163931298323575e371b5f73cd45c3fd50756226a4 | Write the expression of the trapezoidal rule of integration for approximating \( I = \int_a^b f(x) \, dx \). | [
"\\( (b-a)\\frac{f(a) + f(b)}{2} \\)"
] | [
"Write the expression of the trapezoidal rule of integration for approximating \\( I = \\int_a^b f(x) \\, dx \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-d19911f5c3291baf83b39a748d5c4251b03a000138b4855dcb0ddf33f58aa3ca | Write the factorization of $x^n + 1$ using the polar form of $-1$ and the $n$th roots of unity. | [
"\\( x^n + 1 = \\prod_{j = 1}^{n} \\left( x - e^{2\\pi i \\frac{(2j-1)}{2n}} \\right) \\)"
] | [
"Write the factorization of $x^n + 1$ using the polar form of $-1$ and the $n$th roots of unity."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 16
} | 31 | 0 | 0 |
omr-ws-21d0309e08a005e1f8f916bee5341c930b6a1ee05233f1c7ab03d3cb68d699d1 | Write the first $2n$ integers where $n$ is an odd integer. Select two numbers randomly, $a$ and $b$, erase them, and replace them with $|a - b|$. Keep doing this until one number is left. If the process is repeated for all possible initial selections of $a$ and $b$, calculate the sum of all possible last remaining numb... | [
"25"
] | [
"Write the first $2n$ integers where $n$ is an odd integer. Select two numbers randomly, $a$ and $b$, erase them, and replace them with $|a - b|$. Keep doing this until one number is left. If the process is repeated for all possible initial selections of $a$ and $b$, calculate the sum of all possible last remaining... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-34bc62885f90e21b9daf5c3f232b28fd690fa16b9b4e2f12ad1d0170b233753c | Write the first few nonzero terms and determine a general form for the series representing \(\int_0^2 f(x) \, dx\) where \(f(x) = \sum_{k=2}^{\infty} \frac{k}{4^k} \cdot x^{k-2}\). | [
"\\(\\frac{1 + \\ln 2}{4}\\)"
] | [
"Write the first few nonzero terms and determine a general form for the series representing \\(\\int_0^2 f(x) \\, dx\\) where \\(f(x) = \\sum_{k=2}^{\\infty} \\frac{k}{4^k} \\cdot x^{k-2}\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 1,
"tir": 18
} | 28 | 0 | 0 |
omr-ws-12ebea85f8fc8eb134084496a0f7016ab4a9bdb12c1916d30340d65a27d11487 | Write the first three nonzero terms and the general term for an infinite series that represents \( \int_0^1 f(x) \, dx \) for the function \( f(x) = \frac{1}{3} + \frac{2}{3^2}x + \cdots + \frac{n + 1}{3^{n + 1}}x^n \). | [
"\\(\\frac{1}{3}, \\frac{1}{3^2}, \\frac{1}{3^3}, \\frac{1}{3^{n + 1}}\\)"
] | [
"Write the first three nonzero terms and the general term for an infinite series that represents \\( \\int_0^1 f(x) \\, dx \\) for the function \\( f(x) = \\frac{1}{3} + \\frac{2}{3^2}x + \\cdots + \\frac{n + 1}{3^{n + 1}}x^n \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-2a8371dbfb6472344434303ca330ee523c88b97fff3b1e83df9c676ae107aacc | Write the first three nonzero terms and the general term for the series representing $\int_0^2 f(x)\,dx$ for the function $f(x)=\sum_{k=2}^{\infty}\frac{k}{4^k}x^{k-2}$. | [
"First three nonzero terms: \\(\\frac{1}{4}, \\frac{3}{32}, \\frac{1}{24}\\). General term: \\(\\frac{k}{2^{k+1} (k-1)}\\)."
] | [
"Write the first three nonzero terms and the general term for the series representing $\\int_0^2 f(x)\\,dx$ for the function $f(x)=\\sum_{k=2}^{\\infty}\\frac{k}{4^k}x^{k-2}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 24
} | 37 | 0 | 0 |
omr-ws-60b5a61b57616ecc52d8c0721f3c1dc9db8edaf44bbc974a0ef184278589af16 | Write the first three terms of the expansion of \((a - b)^{10}\). | [
"a^{10} - 10a^{9}b + 45a^{8}b^{2}"
] | [
"Write the first three terms of the expansion of \\((a - b)^{10}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-275cea4aaae51a88374172de204cb451f19d487f2dcce242e500c857a55f6774 | Write the following as definite integrals: \[ \lim_{n\to\infty} \frac{1}{\sqrt{2}n^{3/2}}\sum_{i=1}^n \sqrt{2i-1+2n}. \] | [
"\\(\\int_0^1 \\sqrt{x+1} \\, dx\\)"
] | [
"Write the following as definite integrals:\n\\[ \\lim_{n\\to\\infty} \\frac{1}{\\sqrt{2}n^{3/2}}\\sum_{i=1}^n \\sqrt{2i-1+2n}. \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 6,
"tir": 26
} | 45 | 0 | 0 |
omr-ws-295d93eb1ad4ca865ad19d63c4319fd7f1382d29aed6f4cd4e9ba972490b3442 | Write the following as definite integrals: \[ \lim_{n\to\infty} \frac{1}{n^{3/2}}\sum_{i=1}^{n} \sqrt{i+n}. \] | [
"\\(\\int_0^1 \\sqrt{x+1} \\, dx\\)"
] | [
"Write the following as definite integrals:\n\\[ \\lim_{n\\to\\infty} \\frac{1}{n^{3/2}}\\sum_{i=1}^{n} \\sqrt{i+n}. \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 12
} | 34 | 0 | 0 |
omr-ws-1e8574b3bcc4a41e114315e958d50cb651065f9deaf118f5769b4b6884b39bdb | Write the following as definite integrals: \[ \lim_{n\to\infty} \frac{1}{n^{3/2}}\sum_{i=1}^{n}\sqrt{i-1+n}. \] | [
"\\(\\int_0^1 \\sqrt{x + 1} \\, dx\\)"
] | [
"Write the following as definite integrals:\n\\[ \\lim_{n\\to\\infty} \\frac{1}{n^{3/2}}\\sum_{i=1}^{n}\\sqrt{i-1+n}. \\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-7f35e48132a15b5d135fe79308d1b99d6da4e54d18f6ce8d8899c96a87b4e140 | Write the following expression as a polynomial: $$(2x^2+3x+7)(x+1)-(x+1)(x^2+4x-63)+(3x-14)(x+1)(x+5).$$ | [
"4x^3+4x^2"
] | [
"Write the following expression as a polynomial: $$(2x^2+3x+7)(x+1)-(x+1)(x^2+4x-63)+(3x-14)(x+1)(x+5).$$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-688ef8d4a67ccfbb671d66976a0341dff03b1dc9ab97e8f1b46b80fd44dfc381 | Write the following expression in simplest radical form with the smallest possible index for the radical: \(\frac{(12^{1/3})(72^{1/6})}{((108^{1/3})^{1/4})((3^{1/3})^{1/2})}\). | [
"\\(2 \\cdot \\sqrt[4]{3}\\)"
] | [
"Write the following expression in simplest radical form with the smallest possible index for the radical: \\(\\frac{(12^{1/3})(72^{1/6})}{((108^{1/3})^{1/4})((3^{1/3})^{1/2})}\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-bd170ab091e5c3a70498b4b2da5e378bedd2333219baf3fcd27b7b5ffac2fb6f | Write the following product as a common fraction in its lowest terms: \[ \frac{1}{1 \cdot 2} \cdot \frac{1}{2 \cdot 3} \cdots \frac{1}{99 \cdot 100} \] | [
"\\dfrac{1}{99! \\cdot 100!}"
] | [
"Write the following product as a common fraction in its lowest terms:\n\n\\[\n\\frac{1}{1 \\cdot 2} \\cdot \\frac{1}{2 \\cdot 3} \\cdots \\frac{1}{99 \\cdot 100}\n\\]"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-711dff1757c9eb5a52706462661b213c9854b0cf28163a80ca1f1250123e83ca | Write the following sequence in summation notation: Sequence: $\frac{4}{25}+\frac{4}{125}-\frac{4}{625}-\frac{4}{3125}+\frac{4}{15625}+\frac{4}{78125}-\frac{4}{390625}...$ | [
"\\(\\sum_{n = 2}^{\\infty}\\frac{4}{5^n}(-1)^{\\lfloor\\frac{n}{2}\\rfloor+1}\\)"
] | [
"Write the following sequence in summation notation:\nSequence: $\\frac{4}{25}+\\frac{4}{125}-\\frac{4}{625}-\\frac{4}{3125}+\\frac{4}{15625}+\\frac{4}{78125}-\\frac{4}{390625}...$"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-40386fa8b5c4ba09dc29971df2bd354fb9f041b3ce7792aa4cea4d775f9fa6d2 | Write the following sum in summation notation: \[32 - 16 + 8 - 4 + 2 - 1\] | [
"\\(\\sum_{n=0}^{5} (-1)^{n+1}(2)^{n}\\)"
] | [
"Write the following sum in summation notation:\n\\[32 - 16 + 8 - 4 + 2 - 1\\]"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 8
} | 10 | 0 | 0 |
omr-ws-1a8c431bfccf0600d018d93d34aeab5f3cb1387c5826723a3a4c6ecb3541bfd9 | Write the formula \( z + \frac{1}{1!}(n-1)a_1 + \frac{1}{2!}(n-1)(n-2)a_2 + \cdots + \frac{1}{t!}(n-1)\cdots(n-t)a_t \) in binomial form and simplify it. | [
"\\( z + \\sum_{i=1}^{t} \\binom{n-1}{i} a_i \\)"
] | [
"Write the formula \\( z + \\frac{1}{1!}(n-1)a_1 + \\frac{1}{2!}(n-1)(n-2)a_2 + \\cdots + \\frac{1}{t!}(n-1)\\cdots(n-t)a_t \\) in binomial form and simplify it."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 1,
"tir": 6
} | 17 | 0 | 0 |
omr-ws-1627f4c4c054f68dbec2235c8cf64e07a963ed80da2ac46f29b7630ce7db596b | Write the formula for rotating a point in Cartesian coordinates around the vector \((1, 2, 3)\) by an angle of \(\pi / 3\) in \(\mathbb{R}^3\). | [
"\\(\\mathbf{v}' = \\frac{1}{2} (x, y, z) + \\frac{\\sqrt{3}}{2\\sqrt{14}} (2z - 3y, 3x - z, y - 2x) + \\frac{1}{28} (x + 2y + 3z) (1, 2, 3)\\)"
] | [
"Write the formula for rotating a point in Cartesian coordinates around the vector \\((1, 2, 3)\\) by an angle of \\(\\pi / 3\\) in \\(\\mathbb{R}^3\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 2,
"tir": 5
} | 13 | 0 | 0 |
omr-ws-ecd33fc5374a54c21a11df569b7f66b9d7d02bbece3b6474dfbeb8059938132e | Write the formula for the composite application of the trapezoidal rule of integration using \( n \) sub-intervals of integration. | [
"\\(\\int_a^b f(x)\\,dx \\approx \\frac{b-a}{n} \\left[ {f(a) + f(b) \\over 2} + \\sum_{k=1}^{n-1} f \\left( a+k \\frac{b-a}{n} \\right) \\right]\\)"
] | [
"Write the formula for the composite application of the trapezoidal rule of integration using \\( n \\) sub-intervals of integration."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-0a45f246b49608060f0c484ae02e97c67a9c1bd4978ec7d2cefe0346726550b2 | Write the fourth-degree Taylor polynomial for \( g(x) = \int_4^x f(t) \, dt \). | [
"\\( 7(x-4) - \\frac{3}{2}(x-4)^2 + \\frac{5}{3}(x-4)^3 - \\frac{1}{2}(x-4)^4 \\)"
] | [
"Write the fourth-degree Taylor polynomial for \\( g(x) = \\int_4^x f(t) \\, dt \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c6b91874b2195f924f283eb8b64f8ac047095ac39db13ec74350486b240cbe80 | Write the function \( f(t) = \begin{cases} 0 & \text{if } 0 \leq t < 1 \\ t^2 & \text{if } t \geq 1 \end{cases} \) in terms of the unit step function and find its Laplace transform. | [
"\\(\\left(\\frac{2}{s^3} + \\frac{2}{s^2} + \\frac{1}{s}\\right)e^{-s}\\)"
] | [
"Write the function \\( f(t) = \\begin{cases} \n0 & \\text{if } 0 \\leq t < 1 \\\\ \nt^2 & \\text{if } t \\geq 1 \n\\end{cases} \\) in terms of the unit step function and find its Laplace transform."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-f829cd8c1b73d72a24e9607e3e7813ef34a7d5b6b1a48b349775f42119d4e942 | Write the function \( f(x) = \sqrt{1-x} \ln(1+2x) \) as a series. Consider using the series expansions for \( \sqrt{1-x} \) and \( \ln(1+2x) \) and then applying the Cauchy product for series. | [
"\\sum_{n=1}^{\\infty} \\left( (-1)^{n+1} 2^n \\sum_{k=0}^{n-1} \\frac{\\binom{\\frac{1}{2}}{k}}{2^k (n - k)} \\right) x^n"
] | [
"Write the function \\( f(x) = \\sqrt{1-x} \\ln(1+2x) \\) as a series. Consider using the series expansions for \\( \\sqrt{1-x} \\) and \\( \\ln(1+2x) \\) and then applying the Cauchy product for series."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-9f8576c2047e5ad12a3864596fd6fd79d7d3ace39ac7eb4b77a2b3399ca6e7d3 | Write the function \( f(x) = x^6 \left( \frac{1}{x^2} + \sum_{n=1}^{\infty} \frac{2^{2n}(2n-1)|B_{2n}|}{(2n)!} x^{2n-2} \right) \) as an infinite sum, where \( B_n \) is the Bernoulli number. | [
"\\( x^4 + \\sum_{n=1}^{\\infty} \\frac{2^{2n}(2n-1)|B_{2n}|}{(2n)!} x^{2n+4} \\)"
] | [
"Write the function \\( f(x) = x^6 \\left( \\frac{1}{x^2} + \\sum_{n=1}^{\\infty} \\frac{2^{2n}(2n-1)|B_{2n}|}{(2n)!} x^{2n-2} \\right) \\) as an infinite sum, where \\( B_n \\) is the Bernoulli number."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 6,
"tir": 20
} | 41 | 0 | 0 |
omr-ws-6289f00f82463fe06e03a524cb4a141658bf1ddfff3376ee87b29f22f49765d5 | Write the function \( f(x) = x^x \) for \( x > 0 \) as a composition of two other functions. | [
"\\( g(x) = e^x \\) and \\( h(x) = x \\ln x \\)"
] | [
"Write the function \\( f(x) = x^x \\) for \\( x > 0 \\) as a composition of two other functions."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 40,
"genselect": 8,
"tir": 27
} | 75 | 0 | 0 |
omr-ws-5c7e2901b1f0870e4e836f85d0739ed0fb4a22bf71e5598adcec87ad39b4031b | Write the function \( f^{-1}(x) = \frac{5^x}{5^x + 2} \) in logarithmic form. | [
"\\( y = \\log_5 \\left(\\frac{2x}{1-x}\\right) \\)"
] | [
"Write the function \\( f^{-1}(x) = \\frac{5^x}{5^x + 2} \\) in logarithmic form."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-4620d2202a9b0f99b07841bb1966fa9e004cec65aeac61426f4ff90d6745fb82 | Write the general solution to the following linear system in the form \( x = x^* + z \). Clearly identify the particular solution \( x^* \) and the element \( z \) of the kernel. \[ \begin{pmatrix} 1 & -2 & 0 \\ 2 & 3 & 1 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 3 \\ -1 \end{pmatrix} \] | [
"\\( X = \\begin{pmatrix} 1 \\\\ -1 \\\\ 0 \\end{pmatrix} + \\lambda \\begin{pmatrix} 2 \\\\ 1 \\\\ -7 \\end{pmatrix} \\)"
] | [
"Write the general solution to the following linear system in the form \\( x = x^* + z \\). Clearly identify the particular solution \\( x^* \\) and the element \\( z \\) of the kernel.\n\n\\[\n\\begin{pmatrix}\n1 & -2 & 0 \\\\\n2 & 3 & 1\n\\end{pmatrix}\n\\begin{pmatrix}\nx \\\\\ny \\\\\nz\n\\end{pmatrix}\n=\n\\be... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 6,
"tir": 25
} | 44 | 0 | 0 |
omr-ws-42484202cf25c3d70b2b71e121beab4687cbb0bfed0239a646d0a1f68fd19f2a | Write the given $\widehat{h}(\omega)$ as a product of two familiar Fourier transforms $\widehat{f}(\omega)$ and $\widehat{g}(\omega)$. Identify the functions $f(x)$ and $g(x)$ and express $h(x)$ as a convolution. \[ \widehat{h}(\omega)=\left\{\begin{array}{ll} \frac{1}{1+\omega^{2}} & \text { if }|\omega|<1 \\ 0 & \tex... | [
"\\(f(x) = \\frac{\\sin(x)}{x}\\), \\(g(x) = e^{-|x|}\\), \\(h(x) = (f * g)(x)\\)"
] | [
"Write the given $\\widehat{h}(\\omega)$ as a product of two familiar Fourier transforms $\\widehat{f}(\\omega)$ and $\\widehat{g}(\\omega)$. Identify the functions $f(x)$ and $g(x)$ and express $h(x)$ as a convolution.\n\\[\n\\widehat{h}(\\omega)=\\left\\{\\begin{array}{ll}\n\\frac{1}{1+\\omega^{2}} & \\text { if ... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 15
} | 29 | 0 | 0 |
omr-ws-6dcfc2dd7bc88b960bc1333a9e0edb40e858394cb50b4a0340ee03cfdedecad4 | Write the inequalities $z^2 \ge x^2 + y^2$, $x^2 + y^2 + z^2 \le 1$, and $z \ge 0$ in the form $a \le z \le b$, $f(y, z) \le x \le g(y, z)$, and $h(x, z) \le y \le k(x, z)$. | [
"0 \\le z \\le \\frac{\\sqrt{2}}{2}, \\quad -\\sqrt{z^2 - y^2} \\le x \\le \\sqrt{z^2 - y^2}, \\quad -\\sqrt{z^2 - x^2} \\le y \\le \\sqrt{z^2 - x^2}"
] | [
"Write the inequalities $z^2 \\ge x^2 + y^2$, $x^2 + y^2 + z^2 \\le 1$, and $z \\ge 0$ in the form $a \\le z \\le b$, $f(y, z) \\le x \\le g(y, z)$, and $h(x, z) \\le y \\le k(x, z)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 1
} | 11 | 0 | 0 |
omr-ws-61feb5e5213efee7df213b24b458bbeb65f1ac1759fc6372aac5d0a622b58fa9 | Write the inequality \(\sum c^4(a-b)^2(a^2+b^2-c^2) \geq 0\) in the SOS (Sum of Squares) form where \(S_a, S_b, S_c \geq 0\). | [
"\\sum c^4(a - b)^2(a^2 + b^2 - c^2) \\geq 0"
] | [
"Write the inequality \\(\\sum c^4(a-b)^2(a^2+b^2-c^2) \\geq 0\\) in the SOS (Sum of Squares) form where \\(S_a, S_b, S_c \\geq 0\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-681aea38e208f4b56cd4d8bf9761cf3a7619fc02e6ea6f0d1d4f2933f76493b3 | Write the integral \( \int \int \int_{A} f(x,y,z) \, dxdydz \) in the form: \[ \int \int_{E} \left( \int_{g^1(x,y)}^{g^2(x,y)} f(x,y,z) \, dz \right) dxdy \] where \( A = \{ (x,y,z) \mid (x,y) \in E, g^1(x,y) \le z \le g^2(x,y) \} \), and \( A \) is the region in space bounded by the spheres \( x^2 + y^2 + z^2 = 1 \) a... | [
"\\( g^1(x,y) = \\sqrt{x^2 + y^2} \\) and \\( g^2(x,y) = \\sqrt{4 - x^2 - y^2} \\)"
] | [
"Write the integral \\( \\int \\int \\int_{A} f(x,y,z) \\, dxdydz \\) in the form:\n\\[ \\int \\int_{E} \\left( \\int_{g^1(x,y)}^{g^2(x,y)} f(x,y,z) \\, dz \\right) dxdy \\]\nwhere \\( A = \\{ (x,y,z) \\mid (x,y) \\in E, g^1(x,y) \\le z \\le g^2(x,y) \\} \\), and \\( A \\) is the region in space bounded by the sphe... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 3,
"tir": 4
} | 18 | 0 | 0 |
omr-ws-b699acd4a28911040136c383a88b5b99e8b3c9305676612a8c9338fea17ff80f | Write the integral of $f(x, y) = x^2 + y^2$ over the region $D$ defined by the cone $z = \sqrt{x^2 + y^2}$ and the sphere with radius one centered at the origin, using cylindrical coordinates. | [
"\\int_{0}^{2\\pi} \\int_{0}^{\\frac{1}{\\sqrt{2}}} \\int_{r}^{\\sqrt{1 - r^2}} r^3 \\, dz \\, dr \\, d\\theta"
] | [
"Write the integral of $f(x, y) = x^2 + y^2$ over the region $D$ defined by the cone $z = \\sqrt{x^2 + y^2}$ and the sphere with radius one centered at the origin, using cylindrical coordinates."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 2
} | 12 | 0 | 0 |
omr-ws-88d751553b952fab89de889f5ab1113175913d83abe0e95755dd5d356ee39cca | Write the integral of $f(x, y) = x^2 + y^2$ over the region $D$ defined by the cone $z = \sqrt{x^2 + y^2}$ and the sphere with radius one centered at the origin, using spherical coordinates. | [
"\\int_{0}^{2\\pi} \\int_{0}^{\\pi/4} \\int_{0}^{1} \\rho^4 \\sin^3 \\phi \\, d\\rho \\, d\\phi \\, d\\theta"
] | [
"Write the integral of $f(x, y) = x^2 + y^2$ over the region $D$ defined by the cone $z = \\sqrt{x^2 + y^2}$ and the sphere with radius one centered at the origin, using spherical coordinates."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 2
} | 9 | 0 | 0 |
omr-ws-623f03f8e93168c171054864fe7e70c79e8f2f79fe7c6f5bab6e52e70a4bd66d | Write the inverse of the function \( h:[1,+\infty) \to [1,+\infty) \) defined by \( h(x) = x^{x^a} \), where \( a \) is a positive number, in terms of \( \phi_2 \) and \( \phi_3 \). | [
"\\(\\phi_2(x^a)^{\\frac{1}{a}}\\)"
] | [
"Write the inverse of the function \\( h:[1,+\\infty) \\to [1,+\\infty) \\) defined by \\( h(x) = x^{x^a} \\), where \\( a \\) is a positive number, in terms of \\( \\phi_2 \\) and \\( \\phi_3 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 8
} | 14 | 0 | 0 |
omr-ws-2f7dd85623012a9823b6f4b485d1bcdeccaf45a470c4ad2209913ae607902a74 | Write the inverse of the function \( m:[1,+\infty) \to [1,+\infty) \) defined by \( m(x) = x^{x^{x^a}} \), where \( a \) is a positive number, in terms of \( \phi_2 \) and \( \phi_3 \). | [
"\\left( \\phi_2(\\phi_3(y)) \\right)^{1/a}"
] | [
"Write the inverse of the function \\( m:[1,+\\infty) \\to [1,+\\infty) \\) defined by \\( m(x) = x^{x^{x^a}} \\), where \\( a \\) is a positive number, in terms of \\( \\phi_2 \\) and \\( \\phi_3 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-e944186cba4ce38cc64d8e79b9eced8f00bca542d69829f34c0d809d47b97e70 | Write the last three terms of the expansion of \((a - b)^7\). | [
"-21a^2b^5 + 7ab^6 - b^7"
] | [
"Write the last three terms of the expansion of \\((a - b)^7\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-c3525b1c0f2f317eb961761039f0ac4f5d414235908bf8625f1de3f3a2a1dded | Write the linear programming problem in its canonical form to maximize the profit margin from the production of soursop and cashew juices, given the constraints on processing times and costs. | [
"\\(\\text{Maximize } P = 7.50x + 4.50y \\text{ subject to } 8x + 4y \\leq 480, 6x + 2y \\leq 200, x \\geq 0, y \\geq 0\\)"
] | [
"Write the linear programming problem in its canonical form to maximize the profit margin from the production of soursop and cashew juices, given the constraints on processing times and costs."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2dbacf2f28b824daeb7355498138388c90f079adc0448c421d40b2b77bb18322 | Write the list of the first 2005 integers: 1, 2, 3, ..., 2005. Delete the first two and write down their sum at the end of the list: 3, 4, ..., 2005, 3. Continue this process, canceling the first two numbers and writing their sum at the end of the list, until only one number remains. What is the sum of all the numbers ... | [
"24046868"
] | [
"Write the list of the first 2005 integers: 1, 2, 3, ..., 2005. Delete the first two and write down their sum at the end of the list: 3, 4, ..., 2005, 3. Continue this process, canceling the first two numbers and writing their sum at the end of the list, until only one number remains. What is the sum of all the num... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 5
} | 6 | 0 | 0 |
omr-ws-2bb7154e020d41253137a26a4d9eb1629e6e3b76616e192e67d79835dc864ec9 | Write the matrix \[ A = \left( \begin{array}{ccc} 4 & -1 & -1 \\ -1 & 4 & -1 \\ -1 & -1 & 4 \end{array} \right) \] in the form \(LL^T\), where \(L\) is a lower triangular matrix and \(L^T\) is its transpose. | [
"\\[ L = \\left( \\begin{array}{ccc}\n2 & 0 & 0 \\\\\n-\\frac{1}{2} & \\frac{\\sqrt{15}}{2} & 0 \\\\\n-\\frac{1}{2} & -\\frac{\\sqrt{15}}{6} & \\frac{\\sqrt{30}}{3} \\end{array} \\right) \\]"
] | [
"Write the matrix \n\\[ A = \\left( \\begin{array}{ccc}\n4 & -1 & -1 \\\\\n-1 & 4 & -1 \\\\\n-1 & -1 & 4 \\end{array} \\right) \\]\nin the form \\(LL^T\\), where \\(L\\) is a lower triangular matrix and \\(L^T\\) is its transpose."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-9721fea319c227ee623b20b4ecfe6fe11b6ae1f11fd552ecec07c0ed3548eac5 | Write the matrix in row-echelon form: \[ \left(\begin{array}{ccc} 0 & 1 & 1\\ 1 & 2 & 3\\ 2 & 2 & 2 \end{array}\right) \] | [
"\\left(\\begin{array}{ccc}\n1 & 2 & 3\\\\ \n0 & 1 & 1\\\\ \n0 & 0 & 1\n\\end{array}\\right)"
] | [
"Write the matrix in row-echelon form: \\[\n\\left(\\begin{array}{ccc}\n0 & 1 & 1\\\\ \n1 & 2 & 3\\\\ \n2 & 2 & 2\n\\end{array}\\right)\n\\]"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 1
} | 15 | 0 | 0 |
omr-ws-768de994051379242804508086db54a7846f07203f5d113d02885a881ada19a1 | Write the number $1105 = 5 \times 13 \times 17$ as a product of three distinct prime numbers in two different ways. | [
"13 \\times 5 \\times 17"
] | [
"Write the number $1105 = 5 \\times 13 \\times 17$ as a product of three distinct prime numbers in two different ways."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8cf47944b0a05987d5a94b3dac75cf17fa2679b91cb0de7c6f6abe03894c42d2 | Write the number $\frac{1997}{1998}$ as a sum of distinct reciprocals of natural numbers. | [
"\\(\\frac{1}{2} + \\sum_{k=3}^{1998} \\frac{1}{k(k-1)}\\)"
] | [
"Write the number $\\frac{1997}{1998}$ as a sum of distinct reciprocals of natural numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-64b2d0a66115934ca0227e2f1ab90e7cd57b26b2aff0dc656b6f6ca0ff23dc0d | Write the number 1 once, the number 2 twice, the number 3 three times, and so forth until you have written the number 99 ninety-nine times. What is the 2005th digit that you wrote? | [
"63"
] | [
"Write the number 1 once, the number 2 twice, the number 3 three times, and so forth until you have written the number 99 ninety-nine times. What is the 2005th digit that you wrote?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-31791de68f1d6fe7d7dc1f45cfb8ccb505ac7914b5699bc4431558297cf18baf | Write the number 1997 as the sum of positive integers such that the product of these integers is maximal. Let $P$ be the maximum product. Find the remainder when $P$ is divided by 1000. | [
"686"
] | [
"Write the number 1997 as the sum of positive integers such that the product of these integers is maximal. Let $P$ be the maximum product. Find the remainder when $P$ is divided by 1000."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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