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-c2f9f3b2c7c1788d27d9e57a69f9e9b42ff73b996f5ae102a82d3403952e3a32 | A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0),$ $(2020, 0),$ $(2020, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\tfrac{1}{2}$. (A point $(x, y)$ is a lattice point if $x$ and $y$ are both integers.) What is $d$ to... | [
"0.4"
] | [
"A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0),$ $(2020, 0),$ $(2020, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\\tfrac{1}{2}$. (A point $(x, y)$ is a lattice point if $x$ and $y$ are both integers.) What is ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-f9639ea92172b6c3f8a8f08fd9ce7d36dfe9d96f0f6b2453b1cbd4a750565a7e | A point is chosen at random within the square in the coordinate plane whose vertices are \((0, 0)\), \((2020, 0)\), \((2020, 2020)\), and \((0, 2020)\). The probability that the point is within \(d\) units of a lattice point is \(\frac{1}{2}\). (A point \((x, y)\) is a lattice point if \(x\) and \(y\) are both integers... | [
"0.4"
] | [
"A point is chosen at random within the square in the coordinate plane whose vertices are \\((0, 0)\\), \\((2020, 0)\\), \\((2020, 2020)\\), and \\((0, 2020)\\). The probability that the point is within \\(d\\) units of a lattice point is \\(\\frac{1}{2}\\). (A point \\((x, y)\\) is a lattice point if \\(x\\) and \... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 6,
"tir": 3
} | 24 | 0 | 0 |
omr-ws-0abda017d94af43799fa3baedd081c4d2dc776a2f141d6716c26f57b4e71122a | A point is chosen inside square \(ABCD\) such that its distance from vertex \(A\) is \(a\), its distance from vertex \(B\) is \(b\), its distance from vertex \(C\) is \(c\), and its distance from vertex \(D\) is \(d\). Given that \(a + b + c + d = 168\), \(|a - c| = 47\), and \(|b - d| = 23\), find the area of square \... | [
"4225"
] | [
"A point is chosen inside square \\(ABCD\\) such that its distance from vertex \\(A\\) is \\(a\\), its distance from vertex \\(B\\) is \\(b\\), its distance from vertex \\(C\\) is \\(c\\), and its distance from vertex \\(D\\) is \\(d\\). Given that \\(a + b + c + d = 168\\), \\(|a - c| = 47\\), and \\(|b - d| = 23\... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 8,
"tir": 1
} | 9 | 0 | 0 |
omr-ws-b87ceea11e26e0c584c75a884460c71081b09d7ac24bd2c69a36100104255c3c | A point is chosen on each side of a regular $2009$-gon. Let $S$ be the area of the $2009$-gon with vertices at these points. For each of the chosen points, reflect it across the midpoint of its side to form a new $2009$-gon. Calculate the ratio of the area of the new $2009$-gon to $S$. | [
"1"
] | [
"A point is chosen on each side of a regular $2009$-gon. Let $S$ be the area of the $2009$-gon with vertices at these points. For each of the chosen points, reflect it across the midpoint of its side to form a new $2009$-gon. Calculate the ratio of the area of the new $2009$-gon to $S$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-14359f75c4cc454c784ccddff5607fea9d6eb3f11c37c799b1855fc3348dcb9d | A point is chosen randomly inside a square, and a line is drawn at a random angle through that point. What is the probability that the square is split into 2 quadrilaterals by this line? | [
"\\dfrac{1}{2}"
] | [
"A point is chosen randomly inside a square, and a line is drawn at a random angle through that point. What is the probability that the square is split into 2 quadrilaterals by this line?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 1
} | 10 | 0 | 0 |
omr-ws-1bcbc8b74932420aa45a38482def89b6e82cfa4e34920978dff4a863f18f4f73 | A point is chosen randomly inside the interval [0,1]. A second point is chosen randomly between the first point and 1. What is the probability that the sum of the distances from zero to each point chosen is greater than 1? | [
"\\(\\ln 2\\)"
] | [
"A point is chosen randomly inside the interval [0,1]. A second point is chosen randomly between the first point and 1. What is the probability that the sum of the distances from zero to each point chosen is greater than 1?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-4258d959040d3aed4456ba93e06a0e48430ffc4b9490b10a8d1997490238739f | A point is chosen uniformly at random from the interior of a unit square. Let \( p \) be the probability that any circle centered at the point that intersects a diagonal of the square must also intersect a side of the square. Given that \( p^2 \) can be written as \( m - \sqrt{n} \) for positive integers \( m \) and \(... | [
"11"
] | [
"A point is chosen uniformly at random from the interior of a unit square. Let \\( p \\) be the probability that any circle centered at the point that intersects a diagonal of the square must also intersect a side of the square. Given that \\( p^2 \\) can be written as \\( m - \\sqrt{n} \\) for positive integers \\... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 6,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-aec999caf86a974f57f16fab1ae102051ad64c6d1814abff5db872d5f7034754 | A point is located inside a square. The distance from the point to the nearest corner of the square is $5$ units; to the next nearest, $11$ units, and to the farthest, $17$ units. Find the area of the square. | [
"234"
] | [
"A point is located inside a square. The distance from the point to the nearest corner of the square is $5$ units; to the next nearest, $11$ units, and to the farthest, $17$ units. Find the area of the square."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 8
} | 23 | 0 | 0 |
omr-ws-b15fa8b206bf761f293d3423885bffa75ca4f074134e11ef9ae08ff1024db418 | A point is located on a circle with a known radius. The circle moves horizontally to the right by a distance \( X \) while the point rotates counterclockwise by \( \alpha \) radians. Determine the new coordinates of the point on the circle after these transformations. | [
"\\(( (r \\cos(\\theta) + X) \\cos(\\alpha) - r \\sin(\\theta) \\sin(\\alpha), (r \\cos(\\theta) + X) \\sin(\\alpha) + r \\sin(\\theta) \\cos(\\alpha) )\\)"
] | [
"A point is located on a circle with a known radius. The circle moves horizontally to the right by a distance \\( X \\) while the point rotates counterclockwise by \\( \\alpha \\) radians. Determine the new coordinates of the point on the circle after these transformations."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-2ce00c6da792f56a00a51a544897c5dc91445031a314274d623ef56b4e2701f7 | A point is located outside a square. The distance from the point to the nearest corner of the square is $5$ units; to the next nearest, $11$ units, and to the farthest, $17$ units. Find the area of the square. | [
"80"
] | [
"A point is located outside a square. The distance from the point to the nearest corner of the square is $5$ units; to the next nearest, $11$ units, and to the farthest, $17$ units. Find the area of the square."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 6
} | 21 | 0 | 0 |
omr-ws-43bebdbc05a00b04131d4bf84420835065b9b04580082456056bb20fa739cafb | A point is moving randomly inside an equilateral triangle with a side length of 3 cm. What is the probability that the distance from the point to any corner is greater than 1 cm? | [
"\\(\\frac{9\\sqrt{3} - 2\\pi}{9\\sqrt{3}}\\)"
] | [
"A point is moving randomly inside an equilateral triangle with a side length of 3 cm. What is the probability that the distance from the point to any corner is greater than 1 cm?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 25
} | 38 | 0 | 0 |
omr-ws-8ac15523ca168fafa9d1948ff98b1fa4ed9cab5f5c5631ebaf6a719009bfeecf | A point is picked at random on the graph of \( y = x^2 \) with domain \( (0, n) \). What is the expected value of the point in terms of \( n \)? | [
"\\(\\frac{n^2}{3}\\)"
] | [
"A point is picked at random on the graph of \\( y = x^2 \\) with domain \\( (0, n) \\). What is the expected value of the point in terms of \\( n \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 8,
"tir": 9
} | 17 | 0 | 0 |
omr-ws-83f697fe350093af2ac00c6798513706121ae0e4bff789e6d7a4ddd9ffc84e2c | A point is randomly chosen inside a cube of area 1. That point is the center of a line of length \(a\), where \(1 > a > 0\). A point is randomly chosen on that line. This new point is the center of a circle with diameter \(b\), where \(1 > b > 0\). A point randomly chosen on that circle. This new point is the center of... | [
"(1 - a - b - c)^3"
] | [
"A point is randomly chosen inside a cube of area 1. That point is the center of a line of length \\(a\\), where \\(1 > a > 0\\). A point is randomly chosen on that line. This new point is the center of a circle with diameter \\(b\\), where \\(1 > b > 0\\). A point randomly chosen on that circle. This new point is ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 7
} | 15 | 0 | 0 |
omr-ws-a8164aeaea78b55c3173972d95e4987eb2f5c6ac61a2e5d74ccbdf793acc1391 | A point is randomly chosen inside a cube of area 1. That point is the center of a line with length \(\frac{1}{2}\). What is the probability that the line is completely inside of the cube? | [
"\\dfrac{1}{8}"
] | [
"A point is randomly chosen inside a cube of area 1. That point is the center of a line with length \\(\\frac{1}{2}\\). What is the probability that the line is completely inside of the cube?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5e8b93fe1191219c99fbacaded6bee35a0412883a50bcd2835c366a56faab80e | A point is randomly chosen inside a cube of area 1. That point is the center of a sphere with diameter \(\frac{1}{2}\). What is the probability that the sphere is completely inside of the cube? | [
"0"
] | [
"A point is randomly chosen inside a cube of area 1. That point is the center of a sphere with diameter \\(\\frac{1}{2}\\). What is the probability that the sphere is completely inside of the cube?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 9
} | 9 | 0 | 0 |
omr-ws-4368fd5d0b8b82b141266a04b5d3fd2920866e4d0fb13830bb87d71b3c1e1c4d | A point is randomly chosen inside a cube of area 1. That point is the center of a sphere with diameter \(n\), where \(1 > n > 0\). What is the probability that the sphere is completely inside of the cube? | [
"\\((1-2n)^3\\)"
] | [
"A point is randomly chosen inside a cube of area 1. That point is the center of a sphere with diameter \\(n\\), where \\(1 > n > 0\\). What is the probability that the sphere is completely inside of the cube?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-92fe2557a63abe2a23cbd37a37e5fad3b5907f65c3e564304d4ce0915c7ecaa7 | A point is randomly chosen inside a square of area 1. That point is the center of a circle with diameter \(n\), where \(1 > n > 0\). What is the probability that the circle is completely inside of the square? Give your answer in terms of \(n\). | [
"\\((1-n)^2\\)"
] | [
"A point is randomly chosen inside a square of area 1. That point is the center of a circle with diameter \\(n\\), where \\(1 > n > 0\\). What is the probability that the circle is completely inside of the square? Give your answer in terms of \\(n\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-b084c9bf066408de8d291bf51d7e402f2c6210459fa12a4d821940739d12b98e | A point is randomly chosen inside a square of area 1. That point is the center of a line with length \(\frac{1}{2}\). What is the probability that the line is completely inside of the square? | [
"\\dfrac{1}{4}"
] | [
"A point is randomly chosen inside a square of area 1. That point is the center of a line with length \\(\\frac{1}{2}\\). What is the probability that the line is completely inside of the square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-2706872a67d10cc1172a5af644a51c297c41b109cf4257ff6c59efcf1cb69d1a | A point is randomly chosen inside a square of area 1. That point is the center of a line with length \(a\), where \(1 > a > 0\). A point is randomly chosen on that line. This new point is the center of a circle with diameter \(b\), where \(1 > b > 0\). What is the probability that the circle is completely inside the sq... | [
"(1 - a - b)^2"
] | [
"A point is randomly chosen inside a square of area 1. That point is the center of a line with length \\(a\\), where \\(1 > a > 0\\). A point is randomly chosen on that line. This new point is the center of a circle with diameter \\(b\\), where \\(1 > b > 0\\). What is the probability that the circle is completely ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-6c3b9cb57c139a1311926e101072c5609a842c014d0171268df3b69988b47867 | A point is randomly chosen inside an \(n\)-dimensional hypercube of area 1. That point is the center of a line of length \(x_1\), where \(1 > x_1 > 0\). A point is randomly chosen on that line. This new point is the center of a circle with diameter \(x_2\), where \(1 > x_2 > 0\). This new point is the center of a spher... | [
"\\prod_{k=1}^{n} (1 - x_k)"
] | [
"A point is randomly chosen inside an \\(n\\)-dimensional hypercube of area 1. That point is the center of a line of length \\(x_1\\), where \\(1 > x_1 > 0\\). A point is randomly chosen on that line. This new point is the center of a circle with diameter \\(x_2\\), where \\(1 > x_2 > 0\\). This new point is the ce... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-f4b18a3efa2898e365e004fcce7bb2d56dd7b1a76ddccf7d1e5a863cad54db58 | A point is randomly chosen inside an equilateral triangle with side length $2$. Find the probability that this point lies within $\sqrt{2}$ units of exactly two of its vertices. | [
"\\(\\sqrt{3}\\pi - 1\\)"
] | [
"A point is randomly chosen inside an equilateral triangle with side length $2$. Find the probability that this point lies within $\\sqrt{2}$ units of exactly two of its vertices."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-25552a468c4fb8badcc79f9c076070b2ce73a2b6e3047c19c4f9100834c04de7 | A point is randomly chosen inside an equilateral triangle with side length $2$. What is the probability that the point is at least one unit away from all three of the triangle's vertices? Give your answer as a decimal rounded to the nearest thousandth. | [
"0.093"
] | [
"A point is randomly chosen inside an equilateral triangle with side length $2$. What is the probability that the point is at least one unit away from all three of the triangle's vertices? Give your answer as a decimal rounded to the nearest thousandth."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-227a938829fc2176411c2d07990c9b74dbea8b87f15680fa5322e6dd859c1d37 | A point is randomly selected inside a square. What is the probability that at least three of the lengths of the four perpendiculars dropped from the point to the sides of the square determine a non-degenerate triangle? | [
"1"
] | [
"A point is randomly selected inside a square. What is the probability that at least three of the lengths of the four perpendiculars dropped from the point to the sides of the square determine a non-degenerate triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 10,
"tir": 4
} | 14 | 0 | 0 |
omr-ws-631013e151cf798cebc6ef761a09f53e9f2c6d2eeabf7e4d0b4a13df5002ce90 | A point is rotated \( x \) degrees counter-clockwise from the point \((1,0)\) to the point \((-0.6, 0.8)\) on the unit circle. What is the value of \( x \)? | [
"127"
] | [
"A point is rotated \\( x \\) degrees counter-clockwise from the point \\((1,0)\\) to the point \\((-0.6, 0.8)\\) on the unit circle. What is the value of \\( x \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-0c8bfe6b4d8ccdfac6866dde69ed5fa9909d4e9c4b23911483b82da8ca71f0fc | A point is selected at random from the interior of a right triangle with legs of length $2\sqrt{3}$ and $4$. Let $p$ be the probability that the distance between the point and the nearest vertex is less than $2$. Then $p$ can be written in the form $a + \sqrt{b}\pi$, where $a$ and $b$ are rational numbers. Compute $(a,... | [
"(1/4, 1/27)"
] | [
"A point is selected at random from the interior of a right triangle with legs of length $2\\sqrt{3}$ and $4$. Let $p$ be the probability that the distance between the point and the nearest vertex is less than $2$. Then $p$ can be written in the form $a + \\sqrt{b}\\pi$, where $a$ and $b$ are rational numbers. Comp... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4a93b06d42989ee8efb03ea5bc36e30ad3898a2ddee8932f5adf9c829a432128 | A point is selected at random inside the triangle defined by {(x, y) : 0 ≤ y ≤ x ≤ 1}. Assume the point is equally likely to fall anywhere in the triangle. Find the joint and marginal probability density functions (pdf) of X and Y. | [
"f_Y(y) = \\begin{cases} 2(1 - y) & \\text{if } 0 \\leq y \\leq 1, \\\\ 0 & \\text{otherwise} \\end{cases}"
] | [
"A point is selected at random inside the triangle defined by {(x, y) : 0 ≤ y ≤ x ≤ 1}. Assume the point is equally likely to fall anywhere in the triangle. Find the joint and marginal probability density functions (pdf) of X and Y."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-db43695536df778e22ecaeeeda1de717c29eb4a3800aa39eec4e48f1d2c9400e | A point is selected within each face of a unit cube. Two selected points are joined by a line segment if and only if they are on adjacent faces. Let $S$ be the sum of the lengths of all such line segments. Find the minimum possible value of $S^2$. | [
"72"
] | [
"A point is selected within each face of a unit cube. Two selected points are joined by a line segment if and only if they are on adjacent faces. Let $S$ be the sum of the lengths of all such line segments. Find the minimum possible value of $S^2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4a812506ae6d0465d5bb79bbc014b442e44581ce8af21091c6cf9053694a9f15 | A point is situated inside an angle of $60^\circ$ at distances $a$ and $b$ from its sides. Find the distance of this point from the vertex of the given angle. | [
"$OM=\\frac{2}{\\sqrt 3}\\sqrt {a^2+ab+b^2}$"
] | [
"A point is situated inside an angle of $60^\\circ$ at distances $a$ and $b$ from its sides. Find the distance of this point from the vertex of the given angle."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-78df03d90b5a93fc802a6604d7a8dac57820c7dda33622982974324862c0f58d | A point is situated inside an angle of measure 60 degrees, at a distance 'a' and 'b' from its sides. Find the distance of the point from the vertex of the given angle. | [
"\\dfrac{2\\sqrt{3(a^2 + ab + b^2)}}{3}"
] | [
"A point is situated inside an angle of measure 60 degrees, at a distance 'a' and 'b' from its sides. Find the distance of the point from the vertex of the given angle."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3dfb0e90db1036837b8b7a3a7df89510dd0416ef82d1cbe8d5a5ee278a56daae | A point is taken on each side of a parallelogram with area \(S\). The area of the quadrilateral with vertices at these points is \(S/2\). Let \(d_1\) and \(d_2\) be the lengths of the diagonals of this quadrilateral. If one of the diagonals is parallel to one of the sides of the parallelogram, find the ratio of the len... | [
"\\sqrt{2}"
] | [
"A point is taken on each side of a parallelogram with area \\(S\\). The area of the quadrilateral with vertices at these points is \\(S/2\\). Let \\(d_1\\) and \\(d_2\\) be the lengths of the diagonals of this quadrilateral. If one of the diagonals is parallel to one of the sides of the parallelogram, find the rat... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-0468e3edfa7a417e359af7b1a88dfe19b6bd80dae65ce78679e32beabd1618ec | A point is taken on each side of an equilateral triangle. The sides of the triangle with vertices at these points are respectively perpendicular to the sides of the original triangle. Determine the type of triangle with vertices at the specified points. | [
"Equilateral triangle"
] | [
"A point is taken on each side of an equilateral triangle. The sides of the triangle with vertices at these points are respectively perpendicular to the sides of the original triangle. Determine the type of triangle with vertices at the specified points."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-f2a36b9cc31d1670c4864653747863282ac7dd297afc2dc3d611fcc4d7695877 | A point lies on the diameter of a circle, splitting the diameter into two lengths of 3 and 1. What is the maximum area that two circles drawn inside the large circle but not containing the point in their interior can cover? | [
"\\(\\frac{5}{4} \\pi\\)"
] | [
"A point lies on the diameter of a circle, splitting the diameter into two lengths of 3 and 1. What is the maximum area that two circles drawn inside the large circle but not containing the point in their interior can cover?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ae5b3b2435de6706132b606abd409b803a7d8566c0552aff7d75527998150e0d | A point moves along a line such that the average speed in every time interval is equal to the arithmetic mean of the speeds at the ends of the interval. If the point's speed at time $t=0$ is $v_0$ and at time $t=T$ is $v_T$, find the acceleration $a$ of the point in terms of $v_0$, $v_T$, and $T$. | [
"\\dfrac{v_T - v_0}{T}"
] | [
"A point moves along a line such that the average speed in every time interval is equal to the arithmetic mean of the speeds at the ends of the interval. If the point's speed at time $t=0$ is $v_0$ and at time $t=T$ is $v_T$, find the acceleration $a$ of the point in terms of $v_0$, $v_T$, and $T$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-89136773c3952a30576571588ac04c268bc31147394a4647aa9a397e170e8a9f | A point moves along a polar curve \( r = f(\theta) \) at a constant speed and is also moving away from the origin at a constant speed. If the curve is of the form \( r = ae^{b\theta} \), find the value of \( b \) given that the point is moving away from the origin at a speed of 2 units per second and the angular speed ... | [
"1"
] | [
"A point moves along a polar curve \\( r = f(\\theta) \\) at a constant speed and is also moving away from the origin at a constant speed. If the curve is of the form \\( r = ae^{b\\theta} \\), find the value of \\( b \\) given that the point is moving away from the origin at a speed of 2 units per second and the a... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-09ad9c668b1c93dacca6d59e9a40e32a30d3fd0d93ee90275136b79d2661b5c6 | A point moves along the curve \( y = \sqrt{x} \) in such a way that the y-value is increasing at a rate of 2 units per second. At what rate is \( x \) changing when \( x = 4 \)? | [
"8"
] | [
"A point moves along the curve \\( y = \\sqrt{x} \\) in such a way that the y-value is increasing at a rate of 2 units per second. At what rate is \\( x \\) changing when \\( x = 4 \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 1
} | 15 | 0 | 0 |
omr-ws-c371ab730caebcb68e8087db809d911d8809052de5500eb08e094d0e089cf230 | A point moves along the curve \( y = \sqrt{x} \) such that the y-value is increasing at a rate of 2 units per second. Find the rate of change of the distance between the origin and a point \((x, y)\) on the graph when \( x = 4 \). | [
"\\(\\frac{18\\sqrt{5}}{5}\\)"
] | [
"A point moves along the curve \\( y = \\sqrt{x} \\) such that the y-value is increasing at a rate of 2 units per second. Find the rate of change of the distance between the origin and a point \\((x, y)\\) on the graph when \\( x = 4 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-6ded3ed9460ac315fd3a383fb9191887af7f80643ecda4c810bcb677fead574f | A point moves along the curve \( y = x^2 + 1 \) so that the x-coordinate is increasing at the constant rate of \( \frac{3}{2} \) units per second. What is the rate, in units per second, at which the distance from the origin is changing when the point has coordinates \( (1,2) \)? | [
"\\dfrac{3\\sqrt{5}}{2}"
] | [
"A point moves along the curve \\( y = x^2 + 1 \\) so that the x-coordinate is increasing at the constant rate of \\( \\frac{3}{2} \\) units per second. What is the rate, in units per second, at which the distance from the origin is changing when the point has coordinates \\( (1,2) \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fb79666d5394f68ee479b02e838b129f8957ace2545f8f30c086bd9f475705b6 | A point moves along the parabola \( y = x^2 \). Its \( x \)-coordinate is increasing uniformly at 2 units per minute. At what rate is its distance from the point \( (1, -1) \) changing when it passes through \( (2, 4) \)? | [
"\\(\\frac{21 \\sqrt{26}}{13}\\)"
] | [
"A point moves along the parabola \\( y = x^2 \\). Its \\( x \\)-coordinate is increasing uniformly at 2 units per minute. At what rate is its distance from the point \\( (1, -1) \\) changing when it passes through \\( (2, 4) \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-d4507479c32066952c201e417b3840273302d94a2790f9749d907ad0afab88ab | A point moves on the hyperbola \(3x^2 - y^2 = 23\) so that its y-coordinate is increasing at a constant rate of 4 units per second. How fast is the x-coordinate changing when \(x = 4\)? | [
"\\( \\pm \\frac{5}{3} \\)"
] | [
"A point moves on the hyperbola \\(3x^2 - y^2 = 23\\) so that its y-coordinate is increasing at a constant rate of 4 units per second. How fast is the x-coordinate changing when \\(x = 4\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 8,
"tir": 17
} | 37 | 0 | 0 |
omr-ws-5af9da3cccf74334297a1c6458db8ec2bbfab25148734bab30bd139efc0ce9de | A point moves so that the sum of the squares of its distances from the six faces of a cube is constant. Find the locus of the point. | [
"The locus of the point is a sphere centered at the origin with radius \\(\\sqrt{\\frac{k - 6}{2}}\\)."
] | [
"A point moves so that the sum of the squares of its distances from the six faces of a cube is constant. Find the locus of the point."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 46,
"genselect": 4,
"tir": 4
} | 54 | 0 | 0 |
omr-ws-c42e2522cfe0f0726bbacd84afc6ed66ba86373efd1668be98dc072d1b9ebd8d | A point on a chord of a circle is 8 units from one endpoint of the chord and 7 units from the center of the circle. If the radius of the circle is 13 units, what is the length of the chord? | [
"23"
] | [
"A point on a chord of a circle is 8 units from one endpoint of the chord and 7 units from the center of the circle. If the radius of the circle is 13 units, what is the length of the chord?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-44b2c4215f4bae7777c0f93dbb66576a048433a1596d161cc4d49d635bcf2b22 | A point on a circle inscribed in a square is 1 unit and 2 units from the two closest sides of the square. If the area of the square is \( A \), find \( \frac{A}{5} \). | [
"20"
] | [
"A point on a circle inscribed in a square is 1 unit and 2 units from the two closest sides of the square. If the area of the square is \\( A \\), find \\( \\frac{A}{5} \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-67d3b93d80473c64bf2e78a0166c639efe6729a46b26b1b39cef241cdebff92d | A point on a plane is called an integer if both its coordinates on the plane are integers. Consider all triangles that satisfy the following conditions: all three vertices of these triangles are integer points, there are no other integer points on the edges of the triangle, and exactly 4 integer points are located insi... | [
"The four interior points are collinear if and only if the line containing them is parallel to one of the triangle's edges."
] | [
"A point on a plane is called an integer if both its coordinates on the plane are integers. Consider all triangles that satisfy the following conditions: all three vertices of these triangles are integer points, there are no other integer points on the edges of the triangle, and exactly 4 integer points are located... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-914453fc1bb6cc19b4f1410cc5f13bb091d2f8e7e469b57b227b950a1219beeb | A point on the circumference of a wheel, which remains in a vertical plane and rolls along a horizontal path, describes a curve. If the wheel has a diameter of \(d\), calculate the length of the curve described by the point after one complete revolution of the wheel, expressed in terms of \(d\). | [
"4d"
] | [
"A point on the circumference of a wheel, which remains in a vertical plane and rolls along a horizontal path, describes a curve. If the wheel has a diameter of \\(d\\), calculate the length of the curve described by the point after one complete revolution of the wheel, expressed in terms of \\(d\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ee085c1f09797da9e324e2e27ff781d60b73903b1222799161bc155bbc23705c | A point on the surface of a lake exhibits damped harmonic motion after a stone is dropped. After 30 seconds, the amplitude has been reduced by 60%. Find the damping constant to the nearest tenth, given the model \( y = ke^{-ct} \). | [
"0.0"
] | [
"A point on the surface of a lake exhibits damped harmonic motion after a stone is dropped. After 30 seconds, the amplitude has been reduced by 60%. Find the damping constant to the nearest tenth, given the model \\( y = ke^{-ct} \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2472b64dd54da55b4f28e8a5b221e3847bee4561e606fa1f826b76d9a4002fc9 | A point starts at (0,0) in the Euclidean plane and makes three jumps. Each jump is a vector of length 1 in a uniformly random direction. What is the probability that the point is within the unit disk {w ∈ R² | |w| ≤ 1} after three jumps? | [
"\\(\\frac{1}{4}\\)"
] | [
"A point starts at (0,0) in the Euclidean plane and makes three jumps. Each jump is a vector of length 1 in a uniformly random direction. What is the probability that the point is within the unit disk {w ∈ R² | |w| ≤ 1} after three jumps?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 21
} | 34 | 0 | 0 |
omr-ws-ce5523236b01fcab51202d6bfc06582f6b5190e6057b72402879ca37dda80424 | A point starts at the origin of a plane. At each turn, it moves 1 unit in a random direction. What is the expected distance to the origin after n turns? | [
"\\sqrt{n}"
] | [
"A point starts at the origin of a plane. At each turn, it moves 1 unit in a random direction. What is the expected distance to the origin after n turns?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-b5ab144666ba5f293095299b52d7fc0866cc641f0774386ea88cf1c2013b1823 | A point starts at the origin of the coordinate plane. Every minute, it either moves one unit in the $x$-direction or is rotated $45^\circ$ counterclockwise about the origin. For a given point $L = (a, b)$ in the coordinate plane and a positive number $\varepsilon = 0.1$, determine the minimum number of moves required t... | [
"22"
] | [
"A point starts at the origin of the coordinate plane. Every minute, it either moves one unit in the $x$-direction or is rotated $45^\\circ$ counterclockwise about the origin. For a given point $L = (a, b)$ in the coordinate plane and a positive number $\\varepsilon = 0.1$, determine the minimum number of moves req... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 7
} | 9 | 0 | 0 |
omr-ws-ab46620dbd125cebc5716cd395e9babe5a8377657880eabec1ea5a17cdcd89fe | A point starts at the origin of the coordinate plane. Every minute, it either moves one unit in the $x$-direction or is rotated $90^\circ$ counterclockwise about the origin. Determine all possible locations where the point could end up. | [
"All points with integer coordinates $(m, n)$ where $m, n \\in \\mathbb{Z}$."
] | [
"A point starts at the origin of the coordinate plane. Every minute, it either moves one unit in the $x$-direction or is rotated $90^\\circ$ counterclockwise about the origin. Determine all possible locations where the point could end up."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 16
} | 29 | 0 | 0 |
omr-ws-55572ec7c61c9c4a510b220c000594f16192c6c85a41a05e36f73dba7cfcb124 | A point starts at the origin of the coordinate plane. Every minute, it either moves one unit in the $x$-direction or is rotated $\theta$ degrees counterclockwise about the origin. Determine all rational numbers $\theta$ such that for every location $L$ in the coordinate plane and every positive number $\varepsilon$, th... | [
"\\text{There are no such rational numbers}"
] | [
"A point starts at the origin of the coordinate plane. Every minute, it either moves one unit in the $x$-direction or is rotated $\\theta$ degrees counterclockwise about the origin. Determine all rational numbers $\\theta$ such that for every location $L$ in the coordinate plane and every positive number $\\varepsi... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-938746fa078cc2876bb8d2f7b6f35e5aeb6912295ae0024b58ae9f0b3a16f701 | A point starts at the origin of the coordinate plane. Every minute, it either moves one unit in the $x$-direction or is rotated $\theta$ degrees counterclockwise about the origin. Let $\theta$ be an irrational number. For a given location $L = (3, 4)$ in the coordinate plane and $\varepsilon = 0.1$, find the minimum nu... | [
"5"
] | [
"A point starts at the origin of the coordinate plane. Every minute, it either moves one unit in the $x$-direction or is rotated $\\theta$ degrees counterclockwise about the origin. Let $\\theta$ be an irrational number. For a given location $L = (3, 4)$ in the coordinate plane and $\\varepsilon = 0.1$, find the mi... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-5521a08666f7be3eda04ac4ef27ecf4cd0efb2308d3557e3bafbfb937ddb119b | A point starts on the circumference of a circle and rotates at a constant rate around the center while simultaneously moving towards the center at a constant rate. By the time it completes one full rotation, it reaches the center. What is the length of the path that the point follows? | [
"\\( 3.383 R \\)"
] | [
"A point starts on the circumference of a circle and rotates at a constant rate around the center while simultaneously moving towards the center at a constant rate. By the time it completes one full rotation, it reaches the center. What is the length of the path that the point follows?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-4244d041b2f95f99b1bfaab8145992f5cbbe9ffe2040181f5b04b7c921d683f2 | A point taken on each median of the triangle divides the median in the ratio $1 :3 $, reckoning from the vertex. Find the ratio of the area of the triangle with vertices at those points to that of the original triangle. | [
"$\\frac{25}{64}$"
] | [
"A point taken on each median of the triangle divides the median in the ratio $1 :3 $, reckoning from the vertex. Find the ratio of the area of the triangle with vertices at those points to that of the original triangle."
] | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 23
} | 71 | 0 | 0 |
omr-ws-c393dccc5d364e808e245c95fd06703b643d3f6d5dac1f129ad8c58367288e72 | A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola $x^2 - y^2 = 2000^2$? | [
"98"
] | [
"A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola $x^2 - y^2 = 2000^2$?\n"
] | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 5
} | 20 | 0 | 0 |
omr-ws-50fd44a9be31fb3f400dfaea0f10d8d3fb4be360fa098478df454ff686318f9b | A point within an equilateral triangle is at distances of 5, 7, and 8 from each of the vertices. What is the length of a side of the triangle? | [
"\\(\\sqrt{129}\\)"
] | [
"A point within an equilateral triangle is at distances of 5, 7, and 8 from each of the vertices. What is the length of a side of the triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-b6a01e12461fe82e9e484f813bcd78b82147fcabdd86020cca0a34668068c0fb | A point within an equilateral triangle with a perimeter of 30 meters is 2 meters from one side and 3 meters from another side. Find its distance from the third side. | [
"\\( 5\\sqrt{3} - 5 \\)"
] | [
"A point within an equilateral triangle with a perimeter of 30 meters is 2 meters from one side and 3 meters from another side. Find its distance from the third side."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-6528343e2f2d512e219134d6096d6396f345785615ff905d1916949220b8cf00 | A point-sized ant is located at the left end of a rubber band $e$ meters long at time $t = 0$. The ant walks to the right at a rate of $\frac{1}{\ln(t + e)}$ centimeters per second, while the child uniformly stretches the rubber band at a rate of one meter per second. The rubber band is infinitely stretchable and the a... | [
"\\( e^{e^{100}} - e \\)"
] | [
"A point-sized ant is located at the left end of a rubber band $e$ meters long at time $t = 0$. The ant walks to the right at a rate of $\\frac{1}{\\ln(t + e)}$ centimeters per second, while the child uniformly stretches the rubber band at a rate of one meter per second. The rubber band is infinitely stretchable an... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-eb523e596b0ae249e9dece3d105b782b10167624d0115e6b4e39f3b8f62c7d25 | A poker hand consists of 5 cards. If the cards have distinct consecutive values and are not all of the same suit, we say that the hand is a straight. What is the probability of being dealt a straight? | [
"\\(\\frac{9 \\times 4^5}{{52 \\choose 5}}\\) or \\(\\frac{10 \\times 4^5}{{52 \\choose 5}}\\)",
"\\dfrac{5}{1274}"
] | [
"A poker hand consists of 5 cards. If the cards have distinct consecutive values and are not all of the same suit, we say that the hand is a straight. What is the probability of being dealt a straight?"
] | [
"aops_c4_high_school_math",
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 26
} | 58 | 0 | 0 |
omr-ws-dfc653be2fef35054726381f05f7d3f28c72662a9b7a485f204a1a2fb3ce3e5c | A polar bear has 3000 pounds of fish and plans to travel 1000 miles across the Arctic. He can pull a sled that holds up to 1000 pounds of fish, but he must eat 1 pound of fish at every mile to keep his energy up. What is the maximum amount of fish (in pounds) the polar bear can transport across the Arctic? | [
"533"
] | [
"A polar bear has 3000 pounds of fish and plans to travel 1000 miles across the Arctic. He can pull a sled that holds up to 1000 pounds of fish, but he must eat 1 pound of fish at every mile to keep his energy up. What is the maximum amount of fish (in pounds) the polar bear can transport across the Arctic?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 7,
"tir": 2
} | 15 | 0 | 0 |
omr-ws-63ea84f1aeadc18605e4c30c5c036ff841b24b39c68461c4d163f9a7ea899c72 | A pole is 28 feet high. Point A is on the ground at a 60-degree angle from the base of the pole. Point B is at the top of the pole, and point C is at the bottom of the pole. What is the length from point A to point B? | [
"56"
] | [
"A pole is 28 feet high. Point A is on the ground at a 60-degree angle from the base of the pole. Point B is at the top of the pole, and point C is at the bottom of the pole. What is the length from point A to point B?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-32ff00059f846e6e2ca1c5d6757b361b457f65e301704f792fc00682aac8092a | A pole of radius 1 inch and height \( h \) inches is perpendicular to the ground. A snake is coiled around the pole in a spiral pattern, making a 60-degree angle with the ground. If the snake is \( 9\pi \) inches long and reaches from the ground to the top of the pole, then \( h \) can be written in the form \( a\pi\sq... | [
"\\(\\frac{15}{2}\\)"
] | [
"A pole of radius 1 inch and height \\( h \\) inches is perpendicular to the ground. A snake is coiled around the pole in a spiral pattern, making a 60-degree angle with the ground. If the snake is \\( 9\\pi \\) inches long and reaches from the ground to the top of the pole, then \\( h \\) can be written in the for... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-6a3e84e726c879564833de42bffb5dda29ddca678c2acffab1cd60f4b3cf320a | A police helicopter is flying due north at 100 mi/h and at a constant altitude of 0.5 mi. Below, a car is traveling west on a highway at 75 mi/h. At the moment the helicopter crosses over the highway, the car is 2 miles east of the helicopter. How fast is the distance between the car and helicopter changing at the mome... | [
"\\(-\\frac{300 \\sqrt{17}}{17}\\)"
] | [
"A police helicopter is flying due north at 100 mi/h and at a constant altitude of 0.5 mi. Below, a car is traveling west on a highway at 75 mi/h. At the moment the helicopter crosses over the highway, the car is 2 miles east of the helicopter. How fast is the distance between the car and helicopter changing at the... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-a06d0c9d6b54ae6b0a665387eaf528a57d49588cde1c4c552ffe63ce110e91e1 | A police officer invests a total of $6000 in two accounts: one earning 8% annual simple interest and the other earning 11% annual simple interest. How much should be invested in each account so that the amount earned on the 8% account is twice the amount earned on the 11% account? | [
"$4400 in the 8% account and $1600 in the 11% account."
] | [
"A police officer invests a total of $6000 in two accounts: one earning 8% annual simple interest and the other earning 11% annual simple interest. How much should be invested in each account so that the amount earned on the 8% account is twice the amount earned on the 11% account?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 25
} | 27 | 0 | 0 |
omr-ws-5cb02bd120bc882118b978869de9c8cd357951a0e979a92eb3bf3eb57f197ace | A policeman decides to let the first two gangsters leave and then follow the first one that is taller than those two. Given that the tallest gangster is the clan leader, what is the probability that the policeman catches the clan leader using this strategy? | [
"0.4333"
] | [
"A policeman decides to let the first two gangsters leave and then follow the first one that is taller than those two. Given that the tallest gangster is the clan leader, what is the probability that the policeman catches the clan leader using this strategy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0d47488adcb77f34593e208ba8a0eaf5ffdc4c4adf1409b7372b7736a8a1ca32 | A policeman wants to visit each intersection of a grid starting from the upper right corner and return to the starting point, visiting each intersection exactly once. The grid consists of 3 horizontal and 10 vertical streets. In how many ways is this possible? | [
"32"
] | [
"A policeman wants to visit each intersection of a grid starting from the upper right corner and return to the starting point, visiting each intersection exactly once. The grid consists of 3 horizontal and 10 vertical streets. In how many ways is this possible?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 2,
"tir": 1
} | 10 | 0 | 0 |
omr-ws-d1f18770a27e34a7e74a72c1047787fe5c29c9bf4e1b0efb43ada09b7bd6e2de | A polygon \( P \) with integer coordinates in a coordinate system is covered with \( S \) and \( Z \) tetrominoes. If the polygon \( P \) can be covered using only \( S \) tetrominoes, find the remainder when the number of possible coverings of \( P \) using an even number of \( Z \) tetrominoes is divided by 1000. | [
"500"
] | [
"A polygon \\( P \\) with integer coordinates in a coordinate system is covered with \\( S \\) and \\( Z \\) tetrominoes. If the polygon \\( P \\) can be covered using only \\( S \\) tetrominoes, find the remainder when the number of possible coverings of \\( P \\) using an even number of \\( Z \\) tetrominoes is d... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-2f58266806cc26e06495d9f6b53d84af2286455397d32db66010be334264dd5b | A polygon \(\prod\) is given in the \(OXY\) plane with an area exceeding \(n\). Consider \(n+1\) points \(P_{1}(x_1, y_1), P_{2}(x_2, y_2), \ldots, P_{n+1}(x_{n+1}, y_{n+1})\) in \(\prod\) such that for all \(i, j \in \{1, 2, \ldots, n+1\}\), \(x_j - x_i\) and \(y_j - y_i\) are integers. Calculate the minimum possible ... | [
"n"
] | [
"A polygon \\(\\prod\\) is given in the \\(OXY\\) plane with an area exceeding \\(n\\). Consider \\(n+1\\) points \\(P_{1}(x_1, y_1), P_{2}(x_2, y_2), \\ldots, P_{n+1}(x_{n+1}, y_{n+1})\\) in \\(\\prod\\) such that for all \\(i, j \\in \\{1, 2, \\ldots, n+1\\}\\), \\(x_j - x_i\\) and \\(y_j - y_i\\) are integers. C... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b217039071b12b735f3a3d77f2e58338f3d85ab681ae01e7d9a684713fa0f849 | A polygon consists of 4 congruent squares joined edge to edge. One more congruent square is attached to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing? | [
"6"
] | [
"A polygon consists of 4 congruent squares joined edge to edge. One more congruent square is attached to an edge at one of the nine positions indicated. How many of the nine resulting polygons can be folded to form a cube with one face missing?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-dcb90dcc8263de520bdf61eed9634009a8fa27633183bcc849c46e5fb9d095fb | A polygon is composed of only angles of degrees $138$ and $150$, with at least one angle of each degree. How many sides does the polygon have? | [
"10"
] | [
"A polygon is composed of only angles of degrees $138$ and $150$, with at least one angle of each degree. How many sides does the polygon have?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-00e3fc580f4ed8734d587d2f2662679fbfe677465063131c2e8538cbff17ec32 | A polygon is deformed in such a way that its sides remain respectively parallel to given directions whereas all its vertices but one slide along given straight lines. Find the locus of positions of that vertex. | [
"The straight line through $E_1$ and $E_2$."
] | [
"A polygon is deformed in such a way that its sides remain respectively parallel to given directions whereas all its vertices but one slide along given straight lines. Find the locus of positions of that vertex."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1d36448f058c10afe98f36069d4255e15c578f788ba8085555cf61d31fbee195 | A polygon on a coordinate grid is built using 2005 dominoes of size $1 \times 2$. What is the smallest number of sides of an even length that such a polygon can have? | [
"2"
] | [
"A polygon on a coordinate grid is built using 2005 dominoes of size $1 \\times 2$. What is the smallest number of sides of an even length that such a polygon can have?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-3769f6d754f250a59bd34dcfe23eb3b4f9f51a816cd614c8e9f02bd3e426c6a9 | A polygon on a coordinate grid is built using 2005 dominoes of size $1 \times 2$. What is the smallest number of sides of even length that the polygon can have? | [
"2"
] | [
"A polygon on a coordinate grid is built using 2005 dominoes of size $1 \\times 2$. What is the smallest number of sides of even length that the polygon can have?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 5
} | 18 | 0 | 0 |
omr-ws-686ca0630845fbcf8225a3a339b4bf0706dae33bd77548fd88fdcdaf914cd186 | A polygon which does not cross itself, and all of whose vertices lie on lattice points is called a simple lattice polygon. (Lattice points are the points whose coordinates are integers.) A theorem of Pick asserts that there are coefficients $a$, $b$, and $c$ with the property that if $H$ is a simple lattice polygon wit... | [
"-1"
] | [
"A polygon which does not cross itself, and all of whose vertices lie on lattice points is called a simple lattice polygon. (Lattice points are the points whose coordinates are integers.) A theorem of Pick asserts that there are coefficients $a$, $b$, and $c$ with the property that if $H$ is a simple lattice polygo... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-b1060da66f4cb664c211bd327df963cc09c52972d42f66276662f7ebf01546c6 | A polygon with \( n \) sides has a certain number of diagonals. Calculate the total number of diagonals for a polygon with 12 sides. | [
"54"
] | [
"A polygon with \\( n \\) sides has a certain number of diagonals. Calculate the total number of diagonals for a polygon with 12 sides."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5a669d08ef9ce6dfd02c7e018f7c5946d75c8847c1ebee181b177b052f21d6b6 | A polygon with all angles equal to \( 90^\circ \) or \( 270^\circ \) has the property that its sides have distinct prime lengths. Find the minimum possible perimeter. | [
"100"
] | [
"A polygon with all angles equal to \\( 90^\\circ \\) or \\( 270^\\circ \\) has the property that its sides have distinct prime lengths. Find the minimum possible perimeter."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 2
} | 11 | 0 | 0 |
omr-ws-fdfa9e7d608068354ab87846a32a8d84731ef35bd61760f4f5a6d066b5f437f1 | A polygon with area $S$ is contained inside a square with side length $a$. Let $d$ be the maximum distance between any two points of the polygon. If $S = 100$ and $a = 10$, find the minimum possible value of $d$. | [
"10\\sqrt{2}"
] | [
"A polygon with area $S$ is contained inside a square with side length $a$. Let $d$ be the maximum distance between any two points of the polygon. If $S = 100$ and $a = 10$, find the minimum possible value of $d$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5c02e79c63b59d570b984cf98298c240e939bfdfa1b9c535adacbc124bc81a6f | A polygon with twelve sides, whose vertices belong to a circle $C$, has six sides of length $2$ and six sides of length $\sqrt{3}$. Calculate the radius of the circle $C$. | [
"\\(\\sqrt{13}\\)"
] | [
"A polygon with twelve sides, whose vertices belong to a circle $C$, has six sides of length $2$ and six sides of length $\\sqrt{3}$. Calculate the radius of the circle $C$."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-8bd6c9a82f877a87bf6a73a2f1b1ba09acc785726bd476540d6cb988bbcd3e8f | A polygonal line is drawn in a rectangle, with adjacent links perpendicular and equal to the smaller side of the rectangle. Find the ratio of the sides of the rectangle. | [
"2"
] | [
"A polygonal line is drawn in a rectangle, with adjacent links perpendicular and equal to the smaller side of the rectangle. Find the ratio of the sides of the rectangle."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-52390a995dac8d815a09a0020fd2871b5497bb1dbfb6b57ae52fdfd261d2c57a | A polyhedron $ABCA_1B_1C_1$ has six vertices. Faces $ABC$ and $A_1B_1C_1$ are triangles. The edges $AA_1$, $BB_1$, and $CC_1$ are parallel. Faces $AA_1B_1B$, $BB_1C_1C$, and $CC_1A_1A$ are trapeziums in which the diagonals intersect at points $P$, $Q$, and $R$, respectively. Calculate the ratio of the volume of polyhed... | [
"1"
] | [
"A polyhedron $ABCA_1B_1C_1$ has six vertices. Faces $ABC$ and $A_1B_1C_1$ are triangles. The edges $AA_1$, $BB_1$, and $CC_1$ are parallel. Faces $AA_1B_1B$, $BB_1C_1C$, and $CC_1A_1A$ are trapeziums in which the diagonals intersect at points $P$, $Q$, and $R$, respectively. Calculate the ratio of the volume of po... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-fb09d4a9b178578c20b44e89d31ba106a68fdc9c74bab2343fb66a74542abd6f | A polyhedron has 12 faces, all of which are isosceles triangles. All edges have lengths either $x$ or $y$, and at each vertex, either 3 or 6 edges meet. Additionally, all dihedral angles are equal. Find the ratio $x/y$. | [
"\\dfrac{5}{3}"
] | [
"A polyhedron has 12 faces, all of which are isosceles triangles. All edges have lengths either $x$ or $y$, and at each vertex, either 3 or 6 edges meet. Additionally, all dihedral angles are equal. Find the ratio $x/y$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-23086823cdc329e6213f16fcf2d73d6d95ae365b719b97f855f13728869bb12b | A polyhedron has 32 faces, each of which has 3 or 5 sides. At each of its \( V \) vertices, it has \( T \) triangles and \( P \) pentagons. What is the value of \( 100P + 10T + V \)? | [
"2158"
] | [
"A polyhedron has 32 faces, each of which has 3 or 5 sides. At each of its \\( V \\) vertices, it has \\( T \\) triangles and \\( P \\) pentagons. What is the value of \\( 100P + 10T + V \\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9e07a22054371b77370110ed7e65893493ebf0239f71abe983410a0ca5451bad | A polyhedron has all faces as congruent polygons, 2.5 times as many edges as faces, and 3 faces meet at each vertex. Determine the number of vertices of this polyhedron. | [
"20"
] | [
"A polyhedron has all faces as congruent polygons, 2.5 times as many edges as faces, and 3 faces meet at each vertex. Determine the number of vertices of this polyhedron."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-7c42d6a475e2253422237277bcbec0b28f0e1066ab787eddd4af08836da92137 | A polyhedron has an equal number of quadrilateral and pentagon faces. Exactly three faces meet at each vertex. How many faces does this polyhedron have? | [
"8"
] | [
"A polyhedron has an equal number of quadrilateral and pentagon faces. Exactly three faces meet at each vertex. How many faces does this polyhedron have?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-07cfb25bb71ac55e78d2e9d06c2bb036aa15b0469dfd8a090e53edbf9e2f548e | A polyhedron is circumscribed around a sphere. A face is called "large" if the projection of the sphere onto the plane of the face falls entirely within the face. If the polyhedron has exactly 6 large faces, find the sum of all possible values of the total number of faces of the polyhedron. | [
"14"
] | [
"A polyhedron is circumscribed around a sphere. A face is called \"large\" if the projection of the sphere onto the plane of the face falls entirely within the face. If the polyhedron has exactly 6 large faces, find the sum of all possible values of the total number of faces of the polyhedron."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f80a9a662bd75d2b62f3de67e312afbe707dcaf3625288a4f921f830efc49dd2 | A polyhedron is constructed using only pentagons and hexagons. Let $P$ be the number of pentagons and $H$ be the number of hexagons in this polyhedron. Determine the value of $P$ if the polyhedron has a total of 12 vertices. | [
"12"
] | [
"A polyhedron is constructed using only pentagons and hexagons. Let $P$ be the number of pentagons and $H$ be the number of hexagons in this polyhedron. Determine the value of $P$ if the polyhedron has a total of 12 vertices."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-e3fa3c8a8b380016dcadf341d30747fb4c134a8b4058f465fbf80bf1e6fcf256 | A polyhedron is formed by 12 identical rhombi as its faces. Some vertices have 3 shared faces, and some have 4 shared faces. How many vertices have 4 shared faces? | [
"8"
] | [
"A polyhedron is formed by 12 identical rhombi as its faces. Some vertices have 3 shared faces, and some have 4 shared faces. How many vertices have 4 shared faces?"
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e8a6a8e5de0533f57e8af019a9a7af380fb5cf9842fda890adc6a094896a0f6f | A polyhedron is inscribed in a sphere of radius $K$, and its volume is numerically equal to its surface area. Calculate the minimum possible value of $K^2$. | [
"135 - 54\\sqrt{5}"
] | [
"A polyhedron is inscribed in a sphere of radius $K$, and its volume is numerically equal to its surface area. Calculate the minimum possible value of $K^2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-fdfdd0fb1c545217461b2d1e9c41fe9a7576b68174ea0f5a0c5b332268bb1e1b | A polyhedron is inscribed in a sphere of radius $K$, and its volume is numerically equal to its surface area. Determine if $K$ could be greater than $1000$. | [
"Yes"
] | [
"A polyhedron is inscribed in a sphere of radius $K$, and its volume is numerically equal to its surface area. Determine if $K$ could be greater than $1000$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4273a1607f34ba653a849e9452ef249d0299a808656851919becba39f43f3c27 | A polyhedron with 2021 faces, each labeled with a unique number from 1 to 2021, is flipped repeatedly with uniform distribution and independence. The process stops when each number from 1 to 2021 has appeared at least 2021 times. What is the expected value of the number of flips required to stop the process? | [
"2021^2 \\cdot H_{2021}"
] | [
"A polyhedron with 2021 faces, each labeled with a unique number from 1 to 2021, is flipped repeatedly with uniform distribution and independence. The process stops when each number from 1 to 2021 has appeared at least 2021 times. What is the expected value of the number of flips required to stop the process?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-109b3dad33576762892b54781111f06c023abd3da347ed508f8acdf82bde4077 | A polyline $AMB$ is inscribed in the arc $AB$, consisting of two segments, with $AM > MB$. Let $K$ be the midpoint of the arc $AB$. If $H$ is the foot of the perpendicular from $K$ onto $AM$, and $AH = HM + MB$, find the value of $\frac{AM}{MB}$ when $AH = 10$ and $MB = 2$. | [
"9"
] | [
"A polyline $AMB$ is inscribed in the arc $AB$, consisting of two segments, with $AM > MB$. Let $K$ be the midpoint of the arc $AB$. If $H$ is the foot of the perpendicular from $K$ onto $AM$, and $AH = HM + MB$, find the value of $\\frac{AM}{MB}$ when $AH = 10$ and $MB = 2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1da14dfb250d4008cd029274befb6777c51a521d2d4995cab6a6bb5bef97d39d | A polyline is drawn inside a $50 \times 50$ square such that the distance from any point inside the square to the polyline is not more than $1$. Calculate the minimum possible length of the polyline. | [
"50\\sqrt{626}"
] | [
"A polyline is drawn inside a $50 \\times 50$ square such that the distance from any point inside the square to the polyline is not more than $1$. Calculate the minimum possible length of the polyline."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1b86d88cee71d7308b8997fc5d74acd4b0bc2ab31a536807586b2a3d3f62560b | A polyline is placed inside a cube such that the sum of the lengths of the edges of the cube is half the length of the polyline. What is the smallest number of segments (links) the polyline can have? | [
"14"
] | [
"A polyline is placed inside a cube such that the sum of the lengths of the edges of the cube is half the length of the polyline. What is the smallest number of segments (links) the polyline can have?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-323797f722b87e63d1341b39c288980b9532118c2aba91cf5e64a1152ccaa5ba | A polynomial $P$ has integer coefficients and satisfies $P(3)=4$ and $P(4)=3$. How many values of $x$ satisfy $P(x)=x$? | [
"0"
] | [
"A polynomial $P$ has integer coefficients and satisfies $P(3)=4$ and $P(4)=3$. How many values of $x$ satisfy $P(x)=x$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-0510bb53c5f88074143c53d76361b07d941cf24b0b28f4c069c5738147396a64 | A polynomial $P$ with integer coefficients satisfies $P(1) = 1, P(2) = 2, P(3) = 3, P(4) = 4,$ and $P(5) \neq 5.$ Determine the minimum value of $| P(5) |.$ | [
"19"
] | [
"A polynomial $P$ with integer coefficients satisfies $P(1) = 1, P(2) = 2, P(3) = 3, P(4) = 4,$ and $P(5) \\neq 5.$ Determine the minimum value of $| P(5) |.$"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-11ba55926e29bf733ed411ef8571bf9c4b7d3749388e15bff05c1bcd8f872b9f | A polynomial $P(x)$ is called $1$-good if $P(1) = 1$. A polynomial $Q(x)$ is called a primordial multiple of $P(x)$ if $Q(x)$ is a multiple of $P(x)$ and the coefficients of $Q(x)$ are all prime numbers or zero. Let $P(x) = x^3 - 2x + 1$. Find the smallest degree of a primordial multiple of $P(x)$. | [
"6"
] | [
"A polynomial $P(x)$ is called $1$-good if $P(1) = 1$. A polynomial $Q(x)$ is called a primordial multiple of $P(x)$ if $Q(x)$ is a multiple of $P(x)$ and the coefficients of $Q(x)$ are all prime numbers or zero. Let $P(x) = x^3 - 2x + 1$. Find the smallest degree of a primordial multiple of $P(x)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-a6a246747631a05e75333fb93d8b23d440d5670d85ae5102cc93ed90155eda38 | A polynomial $P(x)$ is called faithful if there exists an integer $n$ such that $P(n)$ is not divisible by any prime less than or equal to $10$. Let $F(x) = ax^2 + bx + c$ be a faithful polynomial with $a, b, c \in \mathbb{Q}$ and at least one of $a, b, c$ not an integer. Find the smallest possible value of $|a| + |b| ... | [
"1"
] | [
"A polynomial $P(x)$ is called faithful if there exists an integer $n$ such that $P(n)$ is not divisible by any prime less than or equal to $10$. Let $F(x) = ax^2 + bx + c$ be a faithful polynomial with $a, b, c \\in \\mathbb{Q}$ and at least one of $a, b, c$ not an integer. Find the smallest possible value of $|a|... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-d79013d8b1e2caeef81e97f2b69e1a3affc4e083a28e0d2575d32d5c631f5c9f | A polynomial $P(x)$ of degree $2013$ with real coefficients satisfies $P(n) = 3^n$ for $n = 0, 1, \ldots, 2013$. Evaluate $P(2014)$. | [
"\\( 3^{2014} - 2^{2014} \\)"
] | [
"A polynomial $P(x)$ of degree $2013$ with real coefficients satisfies $P(n) = 3^n$ for $n = 0, 1, \\ldots, 2013$. Evaluate $P(2014)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 10
} | 57 | 0 | 0 |
omr-ws-222eaaa7da892ad0b560e2d01f16d5e6bad8ef856c17e09bfa57433aad3b7d4c | A polynomial $P(x)$ of minimal degree has roots $73$, $-47$, and $13$. If the sum of the coefficients of $P(x)$ can be expressed as $a^2-n$, where $n$ is a minimal natural number and $a$ is a natural number, find $an$. Problem 6a: Bob draws an isosceles trapezoid $\text{ABCD}$ such that $AB \parallel CD$ and $\overline... | [
"29376"
] | [
"A polynomial $P(x)$ of minimal degree has roots $73$, $-47$, and $13$. If the sum of the coefficients of $P(x)$ can be expressed as $a^2-n$, where $n$ is a minimal natural number and $a$ is a natural number, find $an$.\n\nProblem 6a: Bob draws an isosceles trapezoid $\\text{ABCD}$ such that $AB \\parallel CD$ and ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 2,
"tir": 9
} | 12 | 0 | 0 |
omr-ws-01f2360dbaaf06b776cdaefe8e8d31ea832336c55613ee0897d27a179345c1ef | A polynomial $P(x)$ with real coefficients and of degree $n \ge 3$ has $n$ real roots $x_1 < x_2 < \cdots < x_n$ such that \[x_2 - x_1 < x_3 - x_2 < \cdots < x_n - x_{n-1}.\] Let $M$ be the maximum value of $|P(x)|$ on the interval $[x_1, x_n]$. Calculate the value of $M$ if $P(x) = (x-x_1)(x-x_2)\cdots(x-x_n)$ and $x_... | [
"\\dfrac{20 + 14\\sqrt{7}}{27}"
] | [
"A polynomial $P(x)$ with real coefficients and of degree $n \\ge 3$ has $n$ real roots $x_1 < x_2 < \\cdots < x_n$ such that\n\\[x_2 - x_1 < x_3 - x_2 < \\cdots < x_n - x_{n-1}.\\]\nLet $M$ be the maximum value of $|P(x)|$ on the interval $[x_1, x_n]$. Calculate the value of $M$ if $P(x) = (x-x_1)(x-x_2)\\cdots(x-... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-6153588927e2cfeea31abeb4bf09270f94075cb21f752dd0e61a772cb9cb53b2 | A polynomial $f$ with integer coefficients satisfies the following conditions: - $f(7) = 77$ - $f(b) = 85$, where $b$ is Beth's age - $f(c) = 0$, where $c$ is Charles' age Determine the ages of Beth and Charles. | [
"Beth is 9 years old and Charles is 14 years old."
] | [
"A polynomial $f$ with integer coefficients satisfies the following conditions:\n- $f(7) = 77$\n- $f(b) = 85$, where $b$ is Beth's age\n- $f(c) = 0$, where $c$ is Charles' age\n\nDetermine the ages of Beth and Charles."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
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