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-0602a2586e62f40799e2c743b5805683a22b500c1ea30b42cfc16fb45d85623a | An object is formed from a solid cube of wood by cutting it according to the following plans: - Semi-circular cross sections are taken parallel to the y-axis along the graph \( y = 2 \sin(x) \) from \( x = \frac{\pi}{4} \) to \( x = \frac{3\pi}{4} \), with the x-axis as its lower bound. - From \( x = \frac{3\pi}{4} \) ... | [
"2.255"
] | [
"An object is formed from a solid cube of wood by cutting it according to the following plans:\n- Semi-circular cross sections are taken parallel to the y-axis along the graph \\( y = 2 \\sin(x) \\) from \\( x = \\frac{\\pi}{4} \\) to \\( x = \\frac{3\\pi}{4} \\), with the x-axis as its lower bound.\n- From \\( x =... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-cf8f82d410dd5ac46021d2c54a842d56c2138d589f661ed59898a9300ca38e69 | An object is moving along the line \( y = 2x + 1 \) such that its projected speed on the \( x \)-axis is 4 m/sec. Find the rate of change of the distance between the object and the point (5,0) when the object's \( x \)-position is 2. | [
"\\(\\frac{14\\sqrt{34}}{17}\\)"
] | [
"An object is moving along the line \\( y = 2x + 1 \\) such that its projected speed on the \\( x \\)-axis is 4 m/sec. Find the rate of change of the distance between the object and the point (5,0) when the object's \\( x \\)-position is 2."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-d86e3c865a788f982ff1215c92ec20168da7d119e093d1cbb663f7853a0f7146 | An object is moving around the unit circle, so its location at time $t$ is $P(t) = (\cos(t), \sin(t))$. Assume $0 < t < \pi/2$. At a given time $t$, the tangent line to the unit circle at the position $P(t)$ will determine a right triangle in the first quadrant (connect the origin with the y-intercept and x-intercept o... | [
"\\dfrac{1}{\\sin 2t}"
] | [
"An object is moving around the unit circle, so its location at time $t$ is $P(t) = (\\cos(t), \\sin(t))$. Assume $0 < t < \\pi/2$. At a given time $t$, the tangent line to the unit circle at the position $P(t)$ will determine a right triangle in the first quadrant (connect the origin with the y-intercept and x-int... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-a042e1bf46880b5cfe31fe89009113fc6d0a3b1c795d3e8ced1d530f29e8b40b | An object is moving away from a light source at a rate of 4 units/s. Find the rate of change of the intensity of illumination when the object is 2 units from the light source. | [
"-1"
] | [
"An object is moving away from a light source at a rate of 4 units/s. Find the rate of change of the intensity of illumination when the object is 2 units from the light source."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-279ac31215a91d20270cc54aa940f94e9f0b6e26e76d3b6429141427e8179b9d | An object is moving towards Earth at a constant speed. When first observed, the object subtends an angle of 2.7 seconds of arc. After 48 minutes, the object subtends an angle of 3.6 seconds of arc. What is the approximate time left before the object collides with Earth? | [
"144"
] | [
"An object is moving towards Earth at a constant speed. When first observed, the object subtends an angle of 2.7 seconds of arc. After 48 minutes, the object subtends an angle of 3.6 seconds of arc. What is the approximate time left before the object collides with Earth?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-abe60ab663a530b9912efb0d8f96d89e96cdf253aebb96601aaa5e4f40ccfd2f | An object is projected onto the plane \( \alpha \) as a square \( P \) with side length \( s \) and onto the plane \( \beta \) as a square \( Q \) with side length \( t \). If one side of \( P \) is parallel to one side of \( Q \), find the value of \( \frac{s^2}{t^2} \). | [
"1"
] | [
"An object is projected onto the plane \\( \\alpha \\) as a square \\( P \\) with side length \\( s \\) and onto the plane \\( \\beta \\) as a square \\( Q \\) with side length \\( t \\). If one side of \\( P \\) is parallel to one side of \\( Q \\), find the value of \\( \\frac{s^2}{t^2} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-701da07b66d8f0ef07a5d4c06367da2e349b9a5c6b46a72b9ff88ce909d951fb | An object is propelled upward with an initial velocity of 20 meters per second. The distance \( s \) (in meters) of the object from the ground after \( t \) seconds is given by \( s = -4.9t^2 + 20t \). What is the minimum height reached by the object? | [
"0"
] | [
"An object is propelled upward with an initial velocity of 20 meters per second. The distance \\( s \\) (in meters) of the object from the ground after \\( t \\) seconds is given by \\( s = -4.9t^2 + 20t \\). What is the minimum height reached by the object?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 5
} | 20 | 0 | 0 |
omr-ws-d41e11924453bcb58344a6d4afa218bddf491264ca1d04e79f3efca2a41dd838 | An object moves in a sequence of unit steps. Each step is N, S, E, or W with equal probability. It starts at the origin. Find the probability that it reaches (2, 2) in less than 7 steps. | [
"\\(\\frac{3}{64}\\)"
] | [
"An object moves in a sequence of unit steps. Each step is N, S, E, or W with equal probability. It starts at the origin. Find the probability that it reaches (2, 2) in less than 7 steps."
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 42,
"genselect": 8,
"tir": 22
} | 72 | 0 | 0 |
omr-ws-8fd4c5ebeadccdde338c1130a3da9b8d5d266b756cd06a921ede685ee4927471 | An object moves one step at a time in any of the four directions (North, South, East, or West) with equal probability. If the object starts at the origin of the plane, what is the probability that it will be at the point (2,2) in less than 7 steps? | [
"\\(\\frac{249}{512}\\)"
] | [
"An object moves one step at a time in any of the four directions (North, South, East, or West) with equal probability. If the object starts at the origin of the plane, what is the probability that it will be at the point (2,2) in less than 7 steps?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9856d72bc491ac736ce994944d4149d97d30ef70e77142eedfe9adb23561b1c7 | An object moves through the 3-D plane using a series of 1 unit steps, starting at (0,0,0) and ending at (50,50,50). It only moves in the positive direction (up, back, and right). What is the probability that it goes through the point (13,14,15)? | [
"2.44\\%"
] | [
"An object moves through the 3-D plane using a series of 1 unit steps, starting at (0,0,0) and ending at (50,50,50). It only moves in the positive direction (up, back, and right). What is the probability that it goes through the point (13,14,15)?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 24
} | 24 | 0 | 0 |
omr-ws-63dfeadf1825176663fe457af5e204045d76c651bc7bf721884384a534829440 | An object moving along a curve in the xy-plane is at position \((x(t), y(t))\) at time \(t\), where \(\frac{dx}{dt} = \tan(e^{-t})\) and \(\frac{dy}{dt} = \sec(e^{-t})\) for \(t \geq 0\). At time \(t = 1\), the object is at position \((2, -3)\). a) Is there a time \(t \geq 0\) at which the object is on the y-axis? Expl... | [
"The object cannot be on the y-axis at any time \\( t \\geq 0 \\)."
] | [
"An object moving along a curve in the xy-plane is at position \\((x(t), y(t))\\) at time \\(t\\), where \\(\\frac{dx}{dt} = \\tan(e^{-t})\\) and \\(\\frac{dy}{dt} = \\sec(e^{-t})\\) for \\(t \\geq 0\\). At time \\(t = 1\\), the object is at position \\((2, -3)\\).\na) Is there a time \\(t \\geq 0\\) at which the o... | [
"c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-4c450130d309ac18f76cd46477d33927aac5bc7911ac81cd366e0ff14b19190c | An object of mass \( m \) is thrown vertically upward from the surface of the earth with initial velocity \( v_0 \). The gravitational force \( F \) on the object at an altitude \( h \) above the surface of the earth is given by \( F = \frac{mgR^2}{(R+h)^2} \), where \( g \) is the acceleration due to gravity at sea le... | [
"-\\dfrac{g R^2}{(R + h)^2 v}"
] | [
"An object of mass \\( m \\) is thrown vertically upward from the surface of the earth with initial velocity \\( v_0 \\). The gravitational force \\( F \\) on the object at an altitude \\( h \\) above the surface of the earth is given by \\( F = \\frac{mgR^2}{(R+h)^2} \\), where \\( g \\) is the acceleration due to... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-af2caef19351234c52668b7b67fcbba9757b5214924ff1d87bd74df3ab76c01e | An object starts at (0,0) and moves in the coordinate plane via a sequence of steps, each of length one. Each step is left, right, up, or down, all four equally likely. What is the probability that the object reaches (2,2) in six or fewer steps? Express the probability in the form \( \frac{m}{n} \), where \( m \) and \... | [
"67"
] | [
"An object starts at (0,0) and moves in the coordinate plane via a sequence of steps, each of length one. Each step is left, right, up, or down, all four equally likely. What is the probability that the object reaches (2,2) in six or fewer steps? Express the probability in the form \\( \\frac{m}{n} \\), where \\( m... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-79dcf637eaf05c097eb2d305e0cc8e8b7c8f31ef63b7523025721183c180a1d9 | An object starts at the point $(0,0)$ and each unit of time moves either one unit to the right or one unit up. The probability it moves to the right is $r$. If it has moved $m$ units to the right and $n$ units up after $(m + n)$ moves then its coordinates are $(m,n)$ at that time. Find the probability in terms of $r$ t... | [
"(a) \\(\\frac{3}{8}\\), (b) \\(\\frac{3}{32}\\), (c) \\(\\frac{\\binom{m+n}{m}}{2^{m+n}}\\), (d) \\(\\frac{45}{512}\\), (e) \\(\\frac{81}{512}\\), (f) \\(\\frac{29}{128}\\), (g) \"If \\(r\\) is \\(\\frac{1}{2}\\), what is the probability that Team A wins 4 times before it loses 3 times?\""
] | [
"An object starts at the point $(0,0)$ and each unit of time moves either one unit to the right or one unit up. The probability it moves to the right is $r$. If it has moved $m$ units to the right and $n$ units up after $(m + n)$ moves then its coordinates are $(m,n)$ at that time. Find the probability in terms of ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d39ec70bd9bb7333c36692f298828e6ce88872ceaa69e4dd41419001049276ea | An object travels along the x-axis with its acceleration given by $a(t) = \sin(1 + 3^t)$ at any time $t > 0$. If the object's velocity is $1$ at time $t = 2$, then what is its velocity at time $t = 4$? | [
"1 + \\int_{2}^{4} \\sin(1 + 3^t) \\, dt"
] | [
"An object travels along the x-axis with its acceleration given by $a(t) = \\sin(1 + 3^t)$ at any time $t > 0$. If the object's velocity is $1$ at time $t = 2$, then what is its velocity at time $t = 4$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 4
} | 18 | 0 | 0 |
omr-ws-27f30954c011ac0a65d85f02da365357dcadbbb661c59e43101add18e43ee8da | An object travels along the y-axis with its velocity (cm/sec) given by $v(t)=(t^2-2)e^{t^2}$, where $t$ is measured in seconds. What is the total distance traveled by the object from $t=0$ to $t=2$? | [
"e^4 - \\sqrt{2} e^2 + \\frac{5\\sqrt{\\pi}}{4} \\left(2\\,\\text{erf}(\\sqrt{2}) - \\text{erf}(2)\\right)"
] | [
"An object travels along the y-axis with its velocity (cm/sec) given by $v(t)=(t^2-2)e^{t^2}$, where $t$ is measured in seconds. What is the total distance traveled by the object from $t=0$ to $t=2$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-a85e384ec9500df8ef3e29a1de98e0d3ed77a2f3020b9bdf167a6ebb7203e6b7 | An observer at the center of our galaxy measures the apparent magnitude of the Sun, which is located 8.5 kpc away. Given that the absolute magnitude of the Sun is 4.83, calculate the apparent magnitude of the Sun as seen from the center of the galaxy. If the Sun is not visible to the naked eye when its apparent magnitu... | [
"19.48"
] | [
"An observer at the center of our galaxy measures the apparent magnitude of the Sun, which is located 8.5 kpc away. Given that the absolute magnitude of the Sun is 4.83, calculate the apparent magnitude of the Sun as seen from the center of the galaxy. If the Sun is not visible to the naked eye when its apparent ma... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bb3c344e61eb9dba73f0173a4e51e9f952cca8437defb802ec51c0f02e308bb0 | An observer is 1 kilometer away from a launch pad, and the rocket lifts off at \( t = 0 \) straight upwards with altitude \( 10t^2 \) at time \( t \). At \( t = 5 \) seconds, what is \( \frac{dx}{dt} \), if \( x \) is the distance between the rocket and the observer? | [
"\\(\\frac{100\\sqrt{17}}{17}\\)"
] | [
"An observer is 1 kilometer away from a launch pad, and the rocket lifts off at \\( t = 0 \\) straight upwards with altitude \\( 10t^2 \\) at time \\( t \\). At \\( t = 5 \\) seconds, what is \\( \\frac{dx}{dt} \\), if \\( x \\) is the distance between the rocket and the observer?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-437b7c516a714499c71fc37e69456641b1b3f96dfb7ab53c593be02dd1b849fc | An observer with eyes 75 feet above water spots two ships in the distance. The observer, and the two ships are all coplanar. The angle of depression to one ship is 5° and to the other it is 10°. Find the distance between the two ships rounded to the nearest foot. | [
"432 ft"
] | [
"An observer with eyes 75 feet above water spots two ships in the distance. The observer, and the two ships are all coplanar. The angle of depression to one ship is 5° and to the other it is 10°. Find the distance between the two ships rounded to the nearest foot."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-4023ecfec4a2d4fa41bc279a2a15ca79c4b2b9c847ec8a66ea2f42f21ebebac0 | An obtuse triangle is similar to its orthic triangle. Find the angles of the triangle. | [
"\\(\\frac{\\pi}{7}, \\frac{2\\pi}{7}, \\frac{4\\pi}{7}\\)"
] | [
"An obtuse triangle is similar to its orthic triangle. Find the angles of the triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 4
} | 6 | 0 | 0 |
omr-ws-28e76e52179a172cf26c5300628070f42d428e96ce8ff1bd53dc7f59d922886e | An obtuse triangle with acute angles $\alpha $ and $\beta $ and the smaller altitude $h$ rotates about the side opposite the angle $\beta $. Find the surface of the solid thus generated. | [
"$S=\\frac{2\\pi h^2\\sin(\\alpha +\\beta )\\sin\\left(\\alpha +\\frac{\\beta }{2}\\right) \\cos\\frac{\\beta }{2}}{\\sin\\alpha \\sin^2 \\beta }$"
] | [
"An obtuse triangle with acute angles $\\alpha $ and $\\beta $ and the smaller altitude $h$ rotates about the side opposite the angle $\\beta $. Find the surface of the solid thus generated."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-513b39ba86b87e9a3923fbc822ae384a9b1a1464e310b00de8712abd3471be75 | An octagon \( ABCDEFGH \) has all interior angles equal. The lengths of its sides are \( 2, 2\sqrt{2}, 4, 4\sqrt{2}, 6, 7, 7, \) and \( 8 \) in some order. Find the area of the octagon \( ABCDEFGH \). | [
"50 + 56\\sqrt{2}"
] | [
"An octagon \\( ABCDEFGH \\) has all interior angles equal. The lengths of its sides are \\( 2, 2\\sqrt{2}, 4, 4\\sqrt{2}, 6, 7, 7, \\) and \\( 8 \\) in some order. Find the area of the octagon \\( ABCDEFGH \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-7455db8e02ceca6f6d9e995bbfc3b7e72cde77d4e6cb39335024ae294f5906d3 | An octagon is inscribed in a circle. One set of alternate vertices forms a square of area $M$ square units. The other set forms a rectangle of area $N$ square units. What is the maximum possible area of the octagon? | [
"\\(\\sqrt{M^2 + MN}\\)"
] | [
"An octagon is inscribed in a circle. One set of alternate vertices forms a square of area $M$ square units. The other set forms a rectangle of area $N$ square units. What is the maximum possible area of the octagon?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 29,
"genselect": 4,
"tir": 7
} | 40 | 0 | 0 |
omr-ws-e7e1f569b8be218676aabbf22e98a11bf3bf4732891b3ebe14f5aae83bbd1ef9 | An octagon is inscribed in a square so that the vertices of the octagon trisect the sides of the square. The perimeter of the square is 108 centimeters. What is the number of square centimeters in the area of the octagon? | [
"567"
] | [
"An octagon is inscribed in a square so that the vertices of the octagon trisect the sides of the square. The perimeter of the square is 108 centimeters. What is the number of square centimeters in the area of the octagon?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e0a9d03716e8b357fccf62ecb6193f295785872ad61ea831f0bba9aa6cc5f0e6 | An octagon is inscribed in circle O with sides of lengths \(a, a, a, a, b, b, b, b\) respectively. What is the area of circle O in terms of \(a\) and \(b\)? | [
"\\(\\pi \\left( \\frac{a^2 + b^2 - \\sqrt{2}ab}{2} \\right)\\)"
] | [
"An octagon is inscribed in circle O with sides of lengths \\(a, a, a, a, b, b, b, b\\) respectively. What is the area of circle O in terms of \\(a\\) and \\(b\\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4f834904b9d83c17121ef3a3fd726aa633fb890cae31f1335f3d87afce337971 | An octagon with side lengths 3, 3, 11, 11, 15, 15, 15, 15 cm is inscribed in a circle. What is the area of the octagon in cm²? | [
"840"
] | [
"An octagon with side lengths 3, 3, 11, 11, 15, 15, 15, 15 cm is inscribed in a circle. What is the area of the octagon in cm²?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 1
} | 8 | 0 | 0 |
omr-ws-c76742f2f4fb0eff48da3ac0891693e82826245fba6c9c34731b4f1e0072f7da | An octahedron has side length of $1$. What is the square of the distance between two opposite vertices? | [
"2"
] | [
"An octahedron has side length of $1$. What is the square of the distance between two opposite vertices?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d9f3dd997420bf6a9856a7093141a9cfd670925ff98c83176b3d4ea453fa970a | An octahedron has vertices numbered from 1 to 26. Each pair of vertices is connected by a path, and each path is colored with one of 666 different colors. For any 20 vertices chosen, all 190 paths connecting these vertices must have distinct colors at least once. What is the minimum number of coloring schemes required ... | [
"1"
] | [
"An octahedron has vertices numbered from 1 to 26. Each pair of vertices is connected by a path, and each path is colored with one of 666 different colors. For any 20 vertices chosen, all 190 paths connecting these vertices must have distinct colors at least once. What is the minimum number of coloring schemes requ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4760a49f81d6faec416bc1194eabb3468e1fc5d951f5e1055b77b15cf3f5b3bb | An octopus told me that his underwater cave is $245_{8}$ years old. How many years is this in base ten? | [
"165"
] | [
"An octopus told me that his underwater cave is $245_{8}$ years old. How many years is this in base ten?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-25a069edc386319fd2796bcc75c196c516cdafce2197e1a36f865377b972d0f8 | An octopus told me that his underwater cave is $567_{8}$ years old. How many years is this in base ten? | [
"375"
] | [
"An octopus told me that his underwater cave is $567_{8}$ years old. How many years is this in base ten?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-78a36fe5ba289d701a4b6682fe65264d035ad7baf5c9f6f752f20d21d412fdef | An odd and periodic function \( f(x) \) with period 4 is defined on the set of real numbers. On the segment \([0, 2]\), the function is defined by \( f(x) = 4x - 2x^2 \). Find all solutions to the equation \( 2f(x) \cdot f(x-8) - 3f(x+12) - 2 = 0 \). | [
"\\(x = 1 + 4k\\), \\(x = 1 + \\frac{\\sqrt{5}}{2} + 4k\\), \\(x = 1 - \\frac{\\sqrt{5}}{2} + 4k\\) for any integer \\(k\\)."
] | [
"An odd and periodic function \\( f(x) \\) with period 4 is defined on the set of real numbers. On the segment \\([0, 2]\\), the function is defined by \\( f(x) = 4x - 2x^2 \\). Find all solutions to the equation \\( 2f(x) \\cdot f(x-8) - 3f(x+12) - 2 = 0 \\)."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-dada9acd013439aec5f3a8a29a05a24860411468f926dd0470352837a16a107f | An odd box is a rectangular parallelepiped box whose three dimensions are integer odd numbers of centimeters. Fill it with as many cubes as possible with the side length of $2$ cm, disposing the edges of the cubes parallel to those of the box. What is the volume of the box, knowing that the cubes occupy $64\%$ of it? (... | [
"26325, 23925, 11625, 8925, 8325, 7425, 525"
] | [
"An odd box is a rectangular parallelepiped box whose three dimensions are integer odd numbers of centimeters. Fill it with as many cubes as possible with the side length of $2$ cm, disposing the edges of the cubes parallel to those of the box. What is the volume of the box, knowing that the cubes occupy $64\\%$ of... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-dd74eee0899d0ccc18de6ddb390b0b1537f5c3bb04bb1f1cb5a78b151a3f8767 | An odd integer \( n > 1 \) is given. If \( n \) is expressible as a sum of three or more consecutive positive integers, find the smallest possible value of \( n \) that is composite. | [
"9"
] | [
"An odd integer \\( n > 1 \\) is given. If \\( n \\) is expressible as a sum of three or more consecutive positive integers, find the smallest possible value of \\( n \\) that is composite."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-793bba0a3b351d48be3a886e5c6eeaa10f85ed9dde1eeebe27a7fcaabfdab071 | An odd number of politicians, specifically in the form $6k+3$, are sitting at a round table. After a break, they come back and sit down again in arbitrary places. Let $m$ be the number of pairs of politicians who have the same number of people sitting between them as before the break. Find the minimum possible value of... | [
"3"
] | [
"An odd number of politicians, specifically in the form $6k+3$, are sitting at a round table. After a break, they come back and sit down again in arbitrary places. Let $m$ be the number of pairs of politicians who have the same number of people sitting between them as before the break. Find the minimum possible val... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-cdb9525828002c3ad7262d58321efa08089d3bb1264aa25d62ab27ce8afbc3bc | An odd number of sticks lie along a straight path, with each pair of consecutive sticks 10 meters apart. The sticks are to be collected at the place where the middle stick lies. A player can pick only one stick at a time and must start from one of the extreme sticks. If the total distance covered by the player is 3 km,... | [
"33"
] | [
"An odd number of sticks lie along a straight path, with each pair of consecutive sticks 10 meters apart. The sticks are to be collected at the place where the middle stick lies. A player can pick only one stick at a time and must start from one of the extreme sticks. If the total distance covered by the player is ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a8ac7f7117f13856841e5fa35c7d4a9e5d44550747c77ce6886152649957c614 | An odd perfect number $N$ is defined as an odd positive integer that is equal to the sum of its proper divisors. Let $S$ be the sum of the reciprocals of all positive divisors of $N$ excluding 3, 5, and 7. If $N$ is an odd perfect number, find the value of $945S$. | [
"1251"
] | [
"An odd perfect number $N$ is defined as an odd positive integer that is equal to the sum of its proper divisors. Let $S$ be the sum of the reciprocals of all positive divisors of $N$ excluding 3, 5, and 7. If $N$ is an odd perfect number, find the value of $945S$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-6fc64c10df928e3cad806480bd896b65f0c6563ef2a00586b576e13745560f6e | An odd positive integer \( n \) is called pretty if there exists at least one permutation \( a_1, a_2, \ldots, a_n \) of \( 1, 2, \ldots, n \) such that all \( n \) sums \( a_1 - a_2 + a_3 - \cdots + a_n \), \( a_2 - a_3 + a_4 - \cdots + a_1 \), \ldots, \( a_n - a_1 + a_2 - \cdots + a_{n-1} \) are positive. Find all pr... | [
"\\( n = 4k + 1 \\) for all positive integers \\( k \\)"
] | [
"An odd positive integer \\( n \\) is called pretty if there exists at least one permutation \\( a_1, a_2, \\ldots, a_n \\) of \\( 1, 2, \\ldots, n \\) such that all \\( n \\) sums \\( a_1 - a_2 + a_3 - \\cdots + a_n \\), \\( a_2 - a_3 + a_4 - \\cdots + a_1 \\), \\ldots, \\( a_n - a_1 + a_2 - \\cdots + a_{n-1} \\) ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 8,
"tir": 23
} | 35 | 0 | 0 |
omr-ws-6b291c4492c96b2c265595c9d0b23e3c8c2d7c75631e136608177bfc56cc512f | An odd triperfect number $N$ is defined as a number for which the sum of its positive divisors is equal to $3N$. If $N$ is an odd triperfect number, calculate the smallest possible number of distinct prime factors that $\sqrt{N}$ can have if it is not square-free. | [
"3"
] | [
"An odd triperfect number $N$ is defined as a number for which the sum of its positive divisors is equal to $3N$. If $N$ is an odd triperfect number, calculate the smallest possible number of distinct prime factors that $\\sqrt{N}$ can have if it is not square-free."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-0e94cabdb4f797bb000cf57969ada101580baeb74252394e422bb80f67d68d3f | An office block has five floors (ground, 1, 2, 3, and 4), all connected by a lift. When it goes up to any floor (except 4), the probability that after it has stopped it will continue to rise is 3/4. When it goes down to any floor (except the ground floor), the probability that after it has stopped it will continue to g... | [
"\\(\\frac{45}{256}\\)"
] | [
"An office block has five floors (ground, 1, 2, 3, and 4), all connected by a lift. When it goes up to any floor (except 4), the probability that after it has stopped it will continue to rise is 3/4. When it goes down to any floor (except the ground floor), the probability that after it has stopped it will continue... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1b93303dbc1e3fdafd341b5fd0d998880490d1a5b7a4f491e862469651bb35cf | An office employs two typists, Annie and Betty. The number of errors Annie and Betty make can be modeled by independent Poisson distributions with means of 0.1 and 0.7, respectively. a.) Betty is given a 200-page document to type. Using a suitable approximation, find the probability that she makes more than 21 errors. ... | [
"0.386"
] | [
"An office employs two typists, Annie and Betty. The number of errors Annie and Betty make can be modeled by independent Poisson distributions with means of 0.1 and 0.7, respectively.\n a.) Betty is given a 200-page document to type. Using a suitable approximation, find the probability that she makes more than 21... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-6fa449991ec7c0efacdd5ac85cf36afca231d0c67c35887498b55f976cddf932 | An office has 8 officers, including a pair of twins. Two teams, Red and Blue, each consisting of 4 officers, are to be formed randomly. Find the probability that the twins will be on the same team (Red). | [
"\\(\\frac{3}{14}\\)"
] | [
"An office has 8 officers, including a pair of twins. Two teams, Red and Blue, each consisting of 4 officers, are to be formed randomly. Find the probability that the twins will be on the same team (Red)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-5647525e92447b0c7d8c569e88469d3ca1b4b616d7298f85292a638577b3f819 | An office has 8 officers, including a pair of twins. Two teams, Red and Blue, each consisting of 4 officers, are to be formed randomly. What is the probability that the twins will be together in the Red team? | [
"$\\frac{3}{14}$"
] | [
"An office has 8 officers, including a pair of twins. Two teams, Red and Blue, each consisting of 4 officers, are to be formed randomly. What is the probability that the twins will be together in the Red team?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-b3ed4a83164a3470db25f72c06cafb0531b1de28a69eb3a6590fc54c2e3a1a9c | An office supply company manufactures desks and chairs. In a given week, the company can assemble 400 units. They must assemble at least 100 desks and 150 chairs to meet their usual demands. Their profit is $ \$30$ per desk and $ \$20$ per chair. How many of each should they sell to maximize their profit? | [
"\\$10500"
] | [
"An office supply company manufactures desks and chairs. In a given week, the company can assemble 400 units. They must assemble at least 100 desks and 150 chairs to meet their usual demands. Their profit is $ \\$30$ per desk and $ \\$20$ per chair. How many of each should they sell to maximize their profit?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 12
} | 12 | 0 | 0 |
omr-ws-ec30031f0c284e65d16054b6e2bfc68e1e2a609168b8eb4866a0d7f2611a7f2e | An officer has to solve a case with 20 suspects, 10 from colony A and 10 from colony B. He has two boxes, P and Q, each containing 20 cards with the names of the suspects. A suspect whose card is drawn from box P always tells the truth, while a suspect whose card is drawn from box Q tells the truth only if the other su... | [
"1024"
] | [
"An officer has to solve a case with 20 suspects, 10 from colony A and 10 from colony B. He has two boxes, P and Q, each containing 20 cards with the names of the suspects. A suspect whose card is drawn from box P always tells the truth, while a suspect whose card is drawn from box Q tells the truth only if the oth... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-cdb2e9b6336bf8f6a9c5a8c8b41c947e3914fd0ea60f02cab8d2b36a7efb9775 | An officer needs to travel through an equilateral triangle room with side length 16 ft to detect any clues lying within 4√3 ft of his path. What is the minimal distance the officer must travel to ensure he detects any clues in the entire room? | [
"24"
] | [
"An officer needs to travel through an equilateral triangle room with side length 16 ft to detect any clues lying within 4√3 ft of his path. What is the minimal distance the officer must travel to ensure he detects any clues in the entire room?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-e3af4046dd8be744a9822ff13f9e7737a162eb07d0e0a100199433ee2affa96f | An offshore wind turbine is 3 km off the shoreline. The power operator is 7 km down the coast to the east of the turbine. The turbine must be connected to distribution lines and then to the network transmission lines located at the power operator. If laying the distribution lines costs twice as much under water as it d... | [
"\\sqrt{3}"
] | [
"An offshore wind turbine is 3 km off the shoreline. The power operator is 7 km down the coast to the east of the turbine. The turbine must be connected to distribution lines and then to the network transmission lines located at the power operator. If laying the distribution lines costs twice as much under water as... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 1
} | 16 | 0 | 0 |
omr-ws-1cda2a87aa12ef13e4f760fb72907fa0b6b3f7681f3d03b957bbe07a486a24d9 | An oil tanker can be emptied by the main pump in 4 hours and by an auxiliary pump in 9 hours. If the main pump is started at 9 AM, when should the auxiliary pump be started so that the tanker is emptied by noon? | [
"9:45 AM"
] | [
"An oil tanker can be emptied by the main pump in 4 hours and by an auxiliary pump in 9 hours. If the main pump is started at 9 AM, when should the auxiliary pump be started so that the tanker is emptied by noon?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-c8760e45b87b9b20ddd6c43be1463a18281a8a0c395aebeaabf3369007d5175e | An oil-pipe has to connect the oil-well $O$ and the factory $F$, between which there is a river whose banks are parallel. The pipe must cross the river perpendicular to the banks. Let the perpendicular distance between the banks be $d$, the horizontal distance from $O$ to the nearest bank be $a$, and the horizontal dis... | [
"a + b + d"
] | [
"An oil-pipe has to connect the oil-well $O$ and the factory $F$, between which there is a river whose banks are parallel. The pipe must cross the river perpendicular to the banks. Let the perpendicular distance between the banks be $d$, the horizontal distance from $O$ to the nearest bank be $a$, and the horizonta... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-eaac56656a2991bdfa5921cac82254d093723cfca37a33bfde2a37704f482677 | An old circular porthole has two parallel bars, one of 22 cm and the other of 18 cm, spaced 10 cm apart. To meet new safety standards, a third parallel bar is to be added halfway between the two existing bars. What will be the length of this new bar? | [
"\\(6\\sqrt{14}\\)"
] | [
"An old circular porthole has two parallel bars, one of 22 cm and the other of 18 cm, spaced 10 cm apart. To meet new safety standards, a third parallel bar is to be added halfway between the two existing bars. What will be the length of this new bar?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-28d8f14a40d89ccaea71ac380346631c692c1f18d7179f62553ac2868ca62597 | An old lady has a blanket of unit area made up of five, possibly overlapping, patches. Each patch is of area at least 1/2. Calculate the minimum possible area of overlap between any two patches. | [
"\\dfrac{1}{6}"
] | [
"An old lady has a blanket of unit area made up of five, possibly overlapping, patches. Each patch is of area at least 1/2. Calculate the minimum possible area of overlap between any two patches."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-5719a9ec0e09821bfa75126bb59bddeb7d7b93333cea0287fe1a426b957f0ac0 | An old man wants to visit his three grandchildren, Anna, Bob, and Cathy, who live in different cities among 20 cities. He knows the cities where Anna and Bob live but not where Cathy lives. What is the probability that the old man must visit all 20 cities in order to visit all three grandchildren? | [
"\\( \\frac{1}{18} \\)"
] | [
"An old man wants to visit his three grandchildren, Anna, Bob, and Cathy, who live in different cities among 20 cities. He knows the cities where Anna and Bob live but not where Cathy lives. What is the probability that the old man must visit all 20 cities in order to visit all three grandchildren?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-e657b57cca6c0568fe2f9b0feac5979eb7d13332654ce0abf5b303091c47b1c7 | An old receipt reads 88 chickens at a total of $x4.2y, where \( x \) and \( y \) are unreadable digits. How much did each chicken cost? | [
"0.73"
] | [
"An old receipt reads 88 chickens at a total of $x4.2y, where \\( x \\) and \\( y \\) are unreadable digits. How much did each chicken cost?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-f7568ebbaf62a13e31022b6599c2acb830b7938faea486a936f58c7179b66f41 | An online four-sided pyramid $P$ has a right-angled trapezoid $ABB'A'$ as its base with right angles at $A$ and $B$. The lateral surface consists of an equilateral triangle $A'B'C$, two right-angled triangles $A'AC$ and $B'BC$ with right angles at $A$ and $B$, and a triangle $ABC$ with $|AB| = 4$ cm, $|AC| = 2$ cm, and... | [
"\\sqrt{\\dfrac{29 + 2\\sqrt{109}}{3}}"
] | [
"An online four-sided pyramid $P$ has a right-angled trapezoid $ABB'A'$ as its base with right angles at $A$ and $B$. The lateral surface consists of an equilateral triangle $A'B'C$, two right-angled triangles $A'AC$ and $B'BC$ with right angles at $A$ and $B$, and a triangle $ABC$ with $|AB| = 4$ cm, $|AC| = 2$ cm... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 7
} | 10 | 0 | 0 |
omr-ws-e793dc7a18e395d4dd0bff0d10746fc4eb909ff23ad3f86266c166cd313e7c37 | An open box is constructed by gluing 1 cm cubes together to form the bottom and the sides. The outside dimensions of the finished box are 10 cm by 10 cm by 10 cm, and the sides and the bottom are all 1 cm thick. Determine the number of cubes required to construct the box. | [
"424"
] | [
"An open box is constructed by gluing 1 cm cubes together to form the bottom and the sides. The outside dimensions of the finished box are 10 cm by 10 cm by 10 cm, and the sides and the bottom are all 1 cm thick. Determine the number of cubes required to construct the box."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-54b7b8460dedc2060fe320b6d279815ec885d9327766f7523813a0eb4c48d8cb | An open box is to be made from a 6 inches by 6 inches rectangular piece of material by cutting equal squares from each corner and turning up the sides. Find the dimensions of the box that will maximize its volume. | [
"4 inches by 4 inches by 1 inch"
] | [
"An open box is to be made from a 6 inches by 6 inches rectangular piece of material by cutting equal squares from each corner and turning up the sides. Find the dimensions of the box that will maximize its volume."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-fd6867840637eb449cd68fdcbaf35a8d4baa2be773958786a36c83ee3d819d8c | An open box with a rectangular base is to be made from a rectangular piece of cardboard 30 inches by 36 inches by cutting out a square from each corner and turning up the sides. What is the maximum possible volume of the box rounded to the nearest cubic inch? | [
"2613"
] | [
"An open box with a rectangular base is to be made from a rectangular piece of cardboard 30 inches by 36 inches by cutting out a square from each corner and turning up the sides. What is the maximum possible volume of the box rounded to the nearest cubic inch?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0ca6c233d043041c5043c2d5acd78be4e53f92239480db64de45f2d96f743eb8 | An open box with a square base is required to have a volume of 10 cubic feet. How much material is required for a base that is 1 foot by 1 foot? | [
"41 square feet"
] | [
"An open box with a square base is required to have a volume of 10 cubic feet. How much material is required for a base that is 1 foot by 1 foot?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 4,
"tir": 1
} | 12 | 0 | 0 |
omr-ws-4c7348002365e502b7651304b1e27b32c24663e12ab2af3c5c455cfdabea433f | An open rectangular box can be formed by cutting identical squares off the four corners of a rectangular piece of cardboard and folding up the four sections that stick out. For a particular piece of cardboard, the same volume results whether squares of size 1 or squares of size 2 have been cut out. Compute the resultin... | [
"24"
] | [
"An open rectangular box can be formed by cutting identical squares off the four corners of a rectangular piece of cardboard and folding up the four sections that stick out. For a particular piece of cardboard, the same volume results whether squares of size 1 or squares of size 2 have been cut out. Compute the res... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1c291fa5b68199a57639bd172257dfda201d9d608631177d6f8e15975bb58945 | An open-topped glass aquarium with a square base is designed to hold 62.5 cubic feet of water. What is the minimum possible exterior surface area of the aquarium? | [
"75"
] | [
"An open-topped glass aquarium with a square base is designed to hold 62.5 cubic feet of water. What is the minimum possible exterior surface area of the aquarium?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-ff91a0c1131e46cd8d3d3fe1d9a76bdd4cb2097207822b79801f2dc7316d3b0d | An open-topped hemispherical bowl with an inside radius of 10 ft is filled with water. The water flows out of a hole in the bottom at a rate of 5 cubic feet per minute when the water level has dropped 4 ft. What is the rate of change of the height of the water surface at this instant in feet per minute? | [
"\\(\\frac{-5}{84\\pi}\\)"
] | [
"An open-topped hemispherical bowl with an inside radius of 10 ft is filled with water. The water flows out of a hole in the bottom at a rate of 5 cubic feet per minute when the water level has dropped 4 ft. What is the rate of change of the height of the water surface at this instant in feet per minute?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 6,
"tir": 22
} | 42 | 0 | 0 |
omr-ws-8554646b7f5ba704cfbd3e0fd15e98bd876d95bd8c951b81cc05085ad6b48529 | An opera singer is due to perform a series of concerts but has a probability of $\frac{1}{2}$ of pulling out each night. Once she pulls out, she will not sing again until the promoter sends flowers every day until she returns. Flowers costing $x$ thousands of dollars bring about a reconciliation with probability $\sqrt... | [
"0.5"
] | [
"An opera singer is due to perform a series of concerts but has a probability of $\\frac{1}{2}$ of pulling out each night. Once she pulls out, she will not sing again until the promoter sends flowers every day until she returns. Flowers costing $x$ thousands of dollars bring about a reconciliation with probability ... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-dadaaac6310209b47450ba9661b4107e185fabd319d7ba6a0078f5e1179ff51e | An operation on three given non-negative integers consists in increasing two of them by $1$ and decreasing the third by $2$, provided the new numbers are non-negative. The process begins with three non-negative integers that add up to $100$ and are each less than $100$. Find the number of distinct triplets that can be ... | [
"3"
] | [
"An operation on three given non-negative integers consists in increasing two of them by $1$ and decreasing the third by $2$, provided the new numbers are non-negative. The process begins with three non-negative integers that add up to $100$ and are each less than $100$. Find the number of distinct triplets that ca... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-f991e8d1a11e1b44961446e70f24e039fd9014b3518d89ad33174ea7966ad474 | An orange grower has 400 crates of fruit ready for market and will have 20 more crates for each day the grower waits. The present price is 60 dollars per crate and will drop by 2 dollars per day for each day waited. In how many days should the grower ship the crop to maximize his income? | [
"5"
] | [
"An orange grower has 400 crates of fruit ready for market and will have 20 more crates for each day the grower waits. The present price is 60 dollars per crate and will drop by 2 dollars per day for each day waited. In how many days should the grower ship the crop to maximize his income?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1b1b732badc677484396b713e4c34324c5d6d310b2d5f1e15fd81504b2137ac0 | An ordered pair $(b, c)$ of integers, each with an absolute value less than or equal to five, is chosen at random. What is the probability that the quadratic equation $x^2 + bx + c = 0$ will not have distinct positive real roots? | [
"\\(\\frac{108}{121}\\)"
] | [
"An ordered pair $(b, c)$ of integers, each with an absolute value less than or equal to five, is chosen at random. What is the probability that the quadratic equation $x^2 + bx + c = 0$ will not have distinct positive real roots?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-21dfb1c8a503002a86bed8375f68fd59ed0802c703641f7743030635e4535ae3 | An ordered pair $(b,c)$ of integers such that $|b| \le 5$ and $|c| \le 5$ is chosen at random with each such ordered pair having an equal likelihood of being chosen. What is the probability that the equation $x^2 + bx + c = 0$ will not have distinct positive real solutions? | [
"\\(\\frac{111}{121}\\)"
] | [
"An ordered pair $(b,c)$ of integers such that $|b| \\le 5$ and $|c| \\le 5$ is chosen at random with each such ordered pair having an equal likelihood of being chosen. What is the probability that the equation $x^2 + bx + c = 0$ will not have distinct positive real solutions?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-44f7386fc8c8fa00de26f7960f0e1f1c9cb2c486a76d8b18ab820ba9e5a7986a | An ordered pair \((b, c)\) of integers, each with absolute value less than or equal to five, is chosen at random. What is the probability that the quadratic equation \(x^2 + bx + c = 0\) will not have distinct positive real roots? | [
"E (none of these)"
] | [
"An ordered pair \\((b, c)\\) of integers, each with absolute value less than or equal to five, is chosen at random. What is the probability that the quadratic equation \\(x^2 + bx + c = 0\\) will not have distinct positive real roots?"
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-254db4fa3ae0259201f771b0913d6e86b009ec27b65316f83399231416d775e9 | An ordered pair \((n, p)\) is called [i]juicy[/i] if \(n^{2} \equiv 1 \pmod{p^{2}}\) and \(n \equiv -1 \pmod{p}\) for positive integer \(n\) and odd prime \(p\). How many juicy pairs exist such that \(n, p \leq 200\)? | [
"36"
] | [
"An ordered pair \\((n, p)\\) is called [i]juicy[/i] if \\(n^{2} \\equiv 1 \\pmod{p^{2}}\\) and \\(n \\equiv -1 \\pmod{p}\\) for positive integer \\(n\\) and odd prime \\(p\\). How many juicy pairs exist such that \\(n, p \\leq 200\\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 20
} | 35 | 0 | 0 |
omr-ws-a2572e701cb8f45ba705e6a52b7451a2089de4765abb7328d567bc5b47781065 | An ordered pair \((x, y)\) is randomly chosen such that \(0 < x, y < 1\). What is the probability that \(x^2 < y < \sqrt{x}\)? | [
"\\dfrac{1}{3}"
] | [
"An ordered pair \\((x, y)\\) is randomly chosen such that \\(0 < x, y < 1\\). What is the probability that \\(x^2 < y < \\sqrt{x}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-54b19cf58fbc9581bfc01f63fb7fcbbb856a05ed5075fe5218c38623269fe215 | An ordered quadruple $(P_1, P_2, P_3, P_4)$ of corners in a regular $2013$-gon is called crossing if the four corners are all different, and the line segment from $P_1$ to $P_2$ intersects the line segment from $P_3$ to $P_4$. How many crossing quadruples are there in the $2013$-gon? | [
"\\(\\frac{2013 \\cdot 2012 \\cdot 2011 \\cdot 2010}{3}\\)"
] | [
"An ordered quadruple $(P_1, P_2, P_3, P_4)$ of corners in a regular $2013$-gon is called crossing if the four corners are all different, and the line segment from $P_1$ to $P_2$ intersects the line segment from $P_3$ to $P_4$. How many crossing quadruples are there in the $2013$-gon?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 6
} | 21 | 0 | 0 |
omr-ws-c0a795fab3fd360ced8bb9ec23f86433c79aec862033719ca95da63ed03ef09e | An ordered random sample of size $n$ is drawn from a population of $K$ distinct elements with replacement. Let $X$ be the number of distinct elements in the sample. Find the expectation and variance of $X$. | [
"K \\left( \\frac{K-1}{K} \\right)^n + K(K-1) \\left( \\frac{K-2}{K} \\right)^n - K^2 \\left( \\frac{K-1}{K} \\right)^{2n}"
] | [
"An ordered random sample of size $n$ is drawn from a population of $K$ distinct elements with replacement. Let $X$ be the number of distinct elements in the sample. Find the expectation and variance of $X$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-2f45594ad77f546660807f23a976743c5f08bcf3e95d1432512bf59c69330afd | An ordered set of four numbers \((a, b, c, d)\) is said to be "interesting" if \(a^2, b^2, c^2, d^2 + 1\) are, in this order, terms of an increasing arithmetic progression. a) Find an "interesting" set. b) Are there an infinite number of "interesting" sets? | [
"Yes, there are infinitely many \"interesting\" sets."
] | [
"An ordered set of four numbers \\((a, b, c, d)\\) is said to be \"interesting\" if \\(a^2, b^2, c^2, d^2 + 1\\) are, in this order, terms of an increasing arithmetic progression.\na) Find an \"interesting\" set.\nb) Are there an infinite number of \"interesting\" sets?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-64d9c373a3d38a4bcde240f75f1aa107932c04a41e0f61053807d3e454ede24e | An ordered triple \((p, q, r)\) of prime numbers is called "parcera" if \(p\) divides \(q^2-4\), \(q\) divides \(r^2-4\), and \(r\) divides \(p^2-4\). Find all parcera triples. | [
"\\((2, 2, 2), (5, 3, 7), (7, 5, 3), (3, 7, 5)\\)"
] | [
"An ordered triple \\((p, q, r)\\) of prime numbers is called \"parcera\" if \\(p\\) divides \\(q^2-4\\), \\(q\\) divides \\(r^2-4\\), and \\(r\\) divides \\(p^2-4\\). Find all parcera triples."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 22
} | 22 | 0 | 0 |
omr-ws-74969083bd1eae9543b0b34cce1100d86ba590f60ec2cd9528cdeb83ed73b41a | An ordered triple of integers \((a, b, c)\) is called [i]zesty[/i] if there exists a monic quadratic polynomial \(P\) with integer coefficients such that \(P(a) = 2(a - b)\), \(P(b) = 2(b - c)\), \(P(c) = 2(c - a)\), and \(0 < |P(0)| < 2019\). Let \(Z\) be the number of zesty triples. Find the remainder when \(Z\) is d... | [
"576"
] | [
"An ordered triple of integers \\((a, b, c)\\) is called [i]zesty[/i] if there exists a monic quadratic polynomial \\(P\\) with integer coefficients such that \\(P(a) = 2(a - b)\\), \\(P(b) = 2(b - c)\\), \\(P(c) = 2(c - a)\\), and \\(0 < |P(0)| < 2019\\). Let \\(Z\\) be the number of zesty triples. Find the remain... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-638cf9a4aadbbe4817115e24e3cb79acad4b5786f0fefb7d9030e6a008882544 | An ordinary $6$-sided die has a number on each face from $1$ to $6$ (each number appears on one face). How many ways can I paint two faces of a die red, so that the numbers on the red faces don't add up to $7$? | [
"12"
] | [
"An ordinary $6$-sided die has a number on each face from $1$ to $6$ (each number appears on one face). How many ways can I paint two faces of a die red, so that the numbers on the red faces don't add up to $7$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-710c594b3963f5fa90fb34f748a023327c6a9fc3f0720014b24bce67d411fb5f | An ordinary clock in a factory is running slow so that the minute hand passes the hour hand at the usual dial position (12 o'clock, etc.) but only every 69 minutes. At time and one-half for overtime, the extra pay to which a $4.00 per hour worker should be entitled after working a normal 8-hour day by that slow running... | [
"\\dfrac{b p^2 - a q^2}{2(a q - b p)}"
] | [
"An ordinary clock in a factory is running slow so that the minute hand passes the hour hand at the usual dial position (12 o'clock, etc.) but only every 69 minutes. At time and one-half for overtime, the extra pay to which a $4.00 per hour worker should be entitled after working a normal 8-hour day by that slow ru... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-299a4306a7d944fedc656746add661da5e311e8a025961335039ae90b1c2f4d2 | An ordinary cube has 4 blank faces, one face marked with a 2, and one face marked with a 3. What is the probability of obtaining a sum of 12 in 5 throws? | [
"\\(\\frac{5}{1296}\\)"
] | [
"An ordinary cube has 4 blank faces, one face marked with a 2, and one face marked with a 3. What is the probability of obtaining a sum of 12 in 5 throws?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-ca9d4a0409e476fe60f7483be4a67d8acae0dc03448fcefa1c05690fd723cfb3 | An ordinary deck of 52 cards with 4 aces is shuffled, and cards are drawn one by one until the first ace appears. On average, how many cards are drawn? | [
"10"
] | [
"An ordinary deck of 52 cards with 4 aces is shuffled, and cards are drawn one by one until the first ace appears. On average, how many cards are drawn?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d67f15e144a7ef24d92f397e2e90c02c55e05a9d7c2c5da52f5e2d28af86214b | An ordinary deck of 52 cards with 4 aces is shuffled, and then the cards are drawn one by one until the first ace appears. On average, how many cards are drawn? | [
"\\(\\frac{53}{5}\\)"
] | [
"An ordinary deck of 52 cards with 4 aces is shuffled, and then the cards are drawn one by one until the first ace appears. On average, how many cards are drawn?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 9
} | 57 | 0 | 0 |
omr-ws-9ff1b53a211229ced66c0d8258dafa19197feb9d83adecad68c3e8e32951f338 | An ordinary fair die is tossed repeatedly until the face with six dots appears on top. On average, what is the sum of the numbers that appear on top before the six? (Do not include the 6 in the sum.) | [
"15"
] | [
"An ordinary fair die is tossed repeatedly until the face with six dots appears on top. On average, what is the sum of the numbers that appear on top before the six? (Do not include the 6 in the sum.)"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5f473112f74ce6136b6f445e7c2bb2fc98350bbfe7aaede9d4a964eb8f2e9e18 | An ordinary fair die is tossed repeatedly until the face with six dots appears on top. On average, what is the sum of the numbers that appear on top before the six? Do not include the $6$ in the sum. | [
"15"
] | [
"An ordinary fair die is tossed repeatedly until the face with six dots appears on top. On average, what is the sum of the numbers that appear on top before the six? Do not include the $6$ in the sum."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-aeda6a98b99027aa1337ab54fd3aea50268cf54fb5c12ad205442f3c9f2b2d56 | An ordinary irreducible fraction was written on the board, the numerator and denominator of which were positive integers. The numerator was added to its denominator and a new fraction was obtained. The denominator was added to the numerator of the new fraction to form a third fraction. When the numerator was added to t... | [
"\\dfrac{3}{7}"
] | [
"An ordinary irreducible fraction was written on the board, the numerator and denominator of which were positive integers. The numerator was added to its denominator and a new fraction was obtained. The denominator was added to the numerator of the new fraction to form a third fraction. When the numerator was added... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-af18a62989ef26f9d94611ad036acc526ee85b8f0767b3ae65713cac3158a32e | An ordinary six-sided die is rolled repeatedly until a 6 appears. What is the probability that each of the numbers 1 through 5 is rolled at least once before the first 6 appears? | [
"\\(\\frac{1}{6}\\)"
] | [
"An ordinary six-sided die is rolled repeatedly until a 6 appears. What is the probability that each of the numbers 1 through 5 is rolled at least once before the first 6 appears?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 3,
"tir": 27
} | 44 | 0 | 0 |
omr-ws-f141740e2b50bb2e9056dba720709468fdbdad9223e0d9f039d716e3678e4554 | An ordinary soda can has a volume of 355 cm³. Find the dimensions (radius \( r \), height \( h \)) that will minimize the surface area of the can. | [
"r \\approx 3.84 \\, \\text{cm}, \\, h \\approx 7.67 \\, \\text{cm}"
] | [
"An ordinary soda can has a volume of 355 cm³. Find the dimensions (radius \\( r \\), height \\( h \\)) that will minimize the surface area of the can."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-e8222dea025df3e4d0aca988e189a0dc83f2ae5ce8cd75479cf636fa19eb3ec6 | An ordinary soda can has a volume of 355 cm³. Find the dimensions (radius \( r \), height \( h \)) that will minimize the surface area of the can. Describe this can. | [
"r \\approx 3.84 \\, \\text{cm}, \\, h \\approx 7.68 \\, \\text{cm}"
] | [
"An ordinary soda can has a volume of 355 cm³. Find the dimensions (radius \\( r \\), height \\( h \\)) that will minimize the surface area of the can. Describe this can."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-beef473cc02fd0b4f69faec818df4b819b88d20e239069b092bf54fc835c636a | An ore contains 40\% mass impurity, while the metal smelted from it contains 4\% impurity. How much metal will 24 tons of the ore yield? | [
"15 tonnes"
] | [
"An ore contains 40\\% mass impurity, while the metal smelted from it contains 4\\% impurity. How much metal will 24 tons of the ore yield?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-339ddb842b4bcb20be3114ab1b7137fdb8d5bd1b38c16f2db8c08cd5afba00e1 | An oreo shop sells $5$ different flavors of oreos and $3$ different flavors of milk. Alpha and Beta decide to purhcase some oreos. Since Alpha is picky, he will not order more than 1 of the same flavor. To be just as weird, Beta will only order oreos, but she will be willing to have repeats of flavors. How many ways co... | [
"351"
] | [
"An oreo shop sells $5$ different flavors of oreos and $3$ different flavors of milk. Alpha and Beta decide to purhcase some oreos. Since Alpha is picky, he will not order more than 1 of the same flavor. To be just as weird, Beta will only order oreos, but she will be willing to have repeats of flavors. How many wa... | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-d78303bb94216c23faba4575d5f37b876d9ebdc0fa24f0ab780c13affb75041a | An oreo shop sells 5 different flavors of oreos and 3 different flavors of milk. Alpha and Beta decide to purchase some oreos. Alpha will not order more than 1 of the same flavor, while Beta will order oreos but is willing to have repeats of flavors. How many ways could they have left the store with 3 products collecti... | [
"351"
] | [
"An oreo shop sells 5 different flavors of oreos and 3 different flavors of milk. Alpha and Beta decide to purchase some oreos. Alpha will not order more than 1 of the same flavor, while Beta will order oreos but is willing to have repeats of flavors. How many ways could they have left the store with 3 products col... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 8,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-1a81fe14417e272a05bcbc111006349c09f755c015de257a541f9fafe1fb9a09 | An organisation consisting of $n > 5$ members has $n+1$ three-member committees, no two of which have identical membership. Find the maximum number of pairs of committees that have exactly one member in common, and denote this number as $M(n)$. Calculate $M(10)$. | [
"39"
] | [
"An organisation consisting of $n > 5$ members has $n+1$ three-member committees, no two of which have identical membership. Find the maximum number of pairs of committees that have exactly one member in common, and denote this number as $M(n)$. Calculate $M(10)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-47cc411d479a58a265beb82ce3094051c8ce09851545e4f3a6c100941e715fa1 | An organisation consisting of $n$ members $(n>5)$ has $(n+1)$ three-member committees, no two of which have identical membership. Let $k$ be the number of pairs of committees that have exactly one member in common. Find the minimum possible value of $k$. | [
"3(n-3)"
] | [
"An organisation consisting of $n$ members $(n>5)$ has $(n+1)$ three-member committees, no two of which have identical membership. Let $k$ be the number of pairs of committees that have exactly one member in common. Find the minimum possible value of $k$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-b9ed03e21bec3d17d796231c293cd46f4c28ec015941312f8c44f21d7e2ae4c4 | An organism is infected with Protozoa HB-CSE069. The Protozoa undergo the following operations: - Every 5 minutes, the number of Protozoa quadruples. - Every 1 minute, the number of Protozoa increases by $[\sqrt{x}]^2$, where $x$ is the current number of Protozoa. - The organism dies if the number of Protozoa exceeds 1... | [
"28"
] | [
"An organism is infected with Protozoa HB-CSE069. The Protozoa undergo the following operations:\n- Every 5 minutes, the number of Protozoa quadruples.\n- Every 1 minute, the number of Protozoa increases by $[\\sqrt{x}]^2$, where $x$ is the current number of Protozoa.\n- The organism dies if the number of Protozoa ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 6
} | 17 | 0 | 0 |
omr-ws-a684d7d49524117dc52ca4c71e76e7847f4a8d5db68fde72273b7f946c8e2c43 | An organization called [i]sigmaboys[/i] has $100$ members, where every member is friends with exactly $56$ other members (friendship is mutual). It is also known that there are $50$ members who are friends with each other. Determine the number of ways to divide all $100$ members into two groups such that within each gr... | [
"1"
] | [
"An organization called [i]sigmaboys[/i] has $100$ members, where every member is friends with exactly $56$ other members (friendship is mutual). It is also known that there are $50$ members who are friends with each other. Determine the number of ways to divide all $100$ members into two groups such that within ea... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 8
} | 14 | 0 | 0 |
omr-ws-0c11113439f59f3ff7d4b6dd9233cbc4ce70c4d4f856c03381632a812b6d3790 | An organization decides to raise funds by holding a $60 a plate dinner. They get prices from two caterers. The first caterer charges $50 a plate. The second caterer charges according to the following schedule: $500 set-up fee plus $40 a plate for up to and including 61 plates, and $2500 \log(p/4) for \( p > 61 \) plate... | [
"$287.25"
] | [
"An organization decides to raise funds by holding a $60 a plate dinner. They get prices from two caterers. The first caterer charges $50 a plate. The second caterer charges according to the following schedule: $500 set-up fee plus $40 a plate for up to and including 61 plates, and $2500 \\log(p/4) for \\( p > 61 \... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-12d6ca7a4e79e3ce90cf2d68da5794eee1bc3fa08b329e9e863b4ccba0e8e53c | An organization decides to raise funds by holding a $\$60$ a plate dinner. They get prices from two caterers. The first caterer charges $\$50$ a plate. The second caterer charges according to the following schedule: $\$500$ set-up fee plus $\$40$ a plate for up to and including $61$ plates, and $\$2500$ $\log_{10}\left... | [
"19000"
] | [
"An organization decides to raise funds by holding a $\\$60$ a plate dinner. They get prices from two caterers. The first caterer charges $\\$50$ a plate. The second caterer charges according to the following schedule: $\\$500$ set-up fee plus $\\$40$ a plate for up to and including $61$ plates, and $\\$2500$ $\\lo... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-43913739afa20c55f913287c3de8221750fd2095ba503e0600f8aef3818ae738 | An organization has $30$ employees, $20$ of whom have a brand A computer while the other $10$ have a brand B computer. For security, the computers can only be connected to each other and only by cables. The cables can only connect a brand A computer to a brand B computer. Employees can communicate with each other if th... | [
"20"
] | [
"An organization has $30$ employees, $20$ of whom have a brand A computer while the other $10$ have a brand B computer. For security, the computers can only be connected to each other and only by cables. The cables can only connect a brand A computer to a brand B computer. Employees can communicate with each other ... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c0bf783c1d7752ab060de677b488451e33cbfcf5da1fdb898f1ce3657bb46aa2 | An organization has $n$ members and $n + 1$ three-member committees, no two of which have identical membership. Let $f(n)$ be the minimum number of pairs of committees that share exactly one member. Find $f(10)$. | [
"9"
] | [
"An organization has $n$ members and $n + 1$ three-member committees, no two of which have identical membership. Let $f(n)$ be the minimum number of pairs of committees that share exactly one member. Find $f(10)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a34f15c3cdb62538caa7bd234cda2a57040d8b7540e229fba590e917372d3c5a | An organization has $n$ members and $n+1$ three-member committees, with no two committees having identical membership. Let $k$ be the number of pairs of committees that share exactly one member. Find the minimum possible value of $k$ for $n = 10$. | [
"no_valid_answer_found"
] | [
"An organization has $n$ members and $n+1$ three-member committees, with no two committees having identical membership. Let $k$ be the number of pairs of committees that share exactly one member. Find the minimum possible value of $k$ for $n = 10$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-03f3a5e3e2669374b909e94693c9d563e2f64333bfb55a30acd2814651d19c7d | An organization has 30 employees, 20 of whom have a brand A computer and the other 10 have a brand B computer. Computers can only be connected by cables, and each cable connects one brand A computer to one brand B computer. The technician installs cables between one computer of each brand at a time, ensuring no duplica... | [
"191"
] | [
"An organization has 30 employees, 20 of whom have a brand A computer and the other 10 have a brand B computer. Computers can only be connected by cables, and each cable connects one brand A computer to one brand B computer. The technician installs cables between one computer of each brand at a time, ensuring no du... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 38,
"genselect": 3,
"tir": 2
} | 43 | 0 | 0 |
omr-ws-7c10f0611cf0e3a9bf1f87cd27966f627c4f348cfb7f98f833453f3860e03e1c | An organization has 30 employees, 20 of whom have a brand A computer and the other 10 have a brand B computer. Computers can only be connected by cables, and each cable connects one brand A computer to one brand B computer. The technician installs cables between one computer of each brand at a time, ensuring no duplica... | [
"191"
] | [
"An organization has 30 employees, 20 of whom have a brand A computer and the other 10 have a brand B computer. Computers can only be connected by cables, and each cable connects one brand A computer to one brand B computer. The technician installs cables between one computer of each brand at a time, ensuring no du... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 42,
"genselect": 6,
"tir": 1
} | 49 | 0 | 0 |
omr-ws-12dff502f1a8746da9606e6178086fdf9de50b289836414de8e01aa1b9ef8316 | An organization has \( n \) members and \( n + 1 \) three-member committees, no two of which have identical membership. Let \( S \) be the sum of the number of committees that share exactly one member with each committee. Find the minimum possible value of \( S \). | [
""
] | [
"An organization has \\( n \\) members and \\( n + 1 \\) three-member committees, no two of which have identical membership. Let \\( S \\) be the sum of the number of committees that share exactly one member with each committee. Find the minimum possible value of \\( S \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 1 |
omr-ws-33198ef4f56697683c59f8100567f55351c92349fc5f71f75da58536a244a119 | An organization of 100 people set up a telephone call system so that the initial contact person calls three people, each of whom calls three others, and so on, until all have been contacted. What is the maximum number of people who do not need to make a call? | [
"67"
] | [
"An organization of 100 people set up a telephone call system so that the initial contact person calls three people, each of whom calls three others, and so on, until all have been contacted. What is the maximum number of people who do not need to make a call?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 6,
"tir": 6
} | 25 | 0 | 0 |
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