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-a27e604f056c13cbf5713513a15a82cfe02c17d00655c850151c53255d07098b | A particle moves on the eight vertices of a cube, moving to each of the three adjacent vertices with equal probability at each step. Let \( i \) be the initial vertex occupied by the particle, and let \( o \) be the vertex opposite \( i \). Calculate the expected number of steps until the particle returns to \( i \) fo... | [
"8"
] | [
"A particle moves on the eight vertices of a cube, moving to each of the three adjacent vertices with equal probability at each step. Let \\( i \\) be the initial vertex occupied by the particle, and let \\( o \\) be the vertex opposite \\( i \\). Calculate the expected number of steps until the particle returns to... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a74b5ebf3172a14695c68f2ae6965a6494723ce6fb4ca30f0d9aa02aa24a340d | A particle moves on the eight vertices of a cube, moving to each of the three adjacent vertices with equal probability at each step. Let \( i \) be the initial vertex occupied by the particle, and let \( o \) be the vertex opposite \( i \). Calculate the expected number of visits to \( o \) until the first return to \(... | [
"1"
] | [
"A particle moves on the eight vertices of a cube, moving to each of the three adjacent vertices with equal probability at each step. Let \\( i \\) be the initial vertex occupied by the particle, and let \\( o \\) be the vertex opposite \\( i \\). Calculate the expected number of visits to \\( o \\) until the first... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 2
} | 13 | 0 | 0 |
omr-ws-d53e9bf80c3783dbfb88164fd5b3269c8ce3612ff9d60325e9a13af420991367 | A particle moves rectilinearly under the action of a constant force with the acceleration of 2 m/s$^2$ and with the zero initial velocity. Three seconds after the beginning of motion, the action of the force ceases and the particle begins moving uniformly with the velocity attained. Find the law of motion $S (t)$ of th... | [
"$S(t)= \\left\\{\\begin{array}{lll} t^2 & \\mbox{if} & 0\\le t<3,\\medskip \\\\ 6t+9 & \\mbox{if} &t\\ge 3. \\end{array}\\right.$"
] | [
"A particle moves rectilinearly under the action of a constant force with the acceleration of 2 m/s$^2$ and with the zero initial velocity. Three seconds after the beginning of motion, the action of the force ceases and the particle begins moving uniformly with the velocity attained. Find the law of motion $S (t)$ ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-947f07d02b80d0d81100774d831c8595b566a86dfa6fdb7a82657036884b46f6 | A particle moves rectilinearly with a speed of $v(t)=\sin t\cos t$ m/s. Find the equation of motion of the particle if for $t=\pi /3$ s the distance covered is 17/8 m. | [
"$S(t)=2-0.25\\cos 2t$ (m)"
] | [
"A particle moves rectilinearly with a speed of $v(t)=\\sin t\\cos t$ m/s. Find the equation of motion of the particle if for $t=\\pi /3$ s the distance covered is 17/8 m."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f358e599f2bb68efe52b4d9fc746a1adcd55b8602ef1b4dc45efaa80cc919eb6 | A particle moves so that it is at $(2t + 7, 4t - 13)$ at time $t.$ Find the speed of the particle, measured in unit of distance per unit of time. | [
"2 \\sqrt{5}"
] | [
"A particle moves so that it is at $(2t + 7, 4t - 13)$ at time $t.$ Find the speed of the particle, measured in unit of distance per unit of time."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3783840387aa2144700c97bf857523673aa19dfc42f7059bf22b9611fb53f2e3 | A particle moves through the first quadrant as follows. During the first minute, it moves from the origin to $(1,0)$. Thereafter, it continues to follow the directions indicated, going back and forth between the positive $x$ and $y$ axes, moving one unit of distance parallel to an axis in each minute. At which point wi... | [
"(44,35)"
] | [
"A particle moves through the first quadrant as follows. During the first minute, it moves from the origin to $(1,0)$. Thereafter, it continues to follow the directions indicated, going back and forth between the positive $x$ and $y$ axes, moving one unit of distance parallel to an axis in each minute. At which poi... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d3de7e5f06a4911eee1dddfacef2248f7ff60b72e9e2286bc1a3253a819adc24 | A particle moves through the first quadrant as follows: during the first minute it moves from the origin to (1,0). Then, it continues to follow the pattern, going back and forth between the positive x and y axes, moving one unit of distance parallel to an axis in one minute. Which point will the particle reach after ex... | [
"(23, 11)"
] | [
"A particle moves through the first quadrant as follows: during the first minute it moves from the origin to (1,0). Then, it continues to follow the pattern, going back and forth between the positive x and y axes, moving one unit of distance parallel to an axis in one minute. Which point will the particle reach aft... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fb3b70b07d769c96fda788856def44b621f01b0802cbbf382b9aad5a6bd592b3 | A particle moves through the first quadrant as follows: during the first minute, it moves from the origin to (1, 0). It then moves up 1 unit, left 1 unit, up 1 unit, right 2 units, down 2 units, right 1 unit, up 3 units, left 3 units, up 1 unit, right 4 units, down 4 units, right 1 unit, up 5 units, and so on. At a rat... | [
"(44, 35)"
] | [
"A particle moves through the first quadrant as follows: during the first minute, it moves from the origin to (1, 0). It then moves up 1 unit, left 1 unit, up 1 unit, right 2 units, down 2 units, right 1 unit, up 3 units, left 3 units, up 1 unit, right 4 units, down 4 units, right 1 unit, up 5 units, and so on. At ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 8,
"tir": 6
} | 25 | 0 | 0 |
omr-ws-3c2c23eb14838c19d09ce2bb488fc5919e775858381802981477bda1f8d5d2a0 | A particle moves under a central force inversely proportional to the $k$-th power of the distance. If the particle describes a circle with the central force proceeding from a point on the circumference of the circle, find the value of $k$. | [
"\\( k = 2 \\)"
] | [
"A particle moves under a central force inversely proportional to the $k$-th power of the distance. If the particle describes a circle with the central force proceeding from a point on the circumference of the circle, find the value of $k$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cfaafa31313403a9b08a13390516936f5c5491a020aa313ea780c2f6b5e90355 | A particle moves under the influence of a force that brings it to rest at a distance \( z \). Given that \( z \) can be expressed as \( z = \frac{V}{n} g\left(\frac{kV}{n}\right) \), where \( V \), \( n \), and \( k \) are constants, and \( g(x) \) is a function defined for all real numbers \( x \). If \( V = 10 \), \(... | [
"2e^{-16}"
] | [
"A particle moves under the influence of a force that brings it to rest at a distance \\( z \\). Given that \\( z \\) can be expressed as \\( z = \\frac{V}{n} g\\left(\\frac{kV}{n}\\right) \\), where \\( V \\), \\( n \\), and \\( k \\) are constants, and \\( g(x) \\) is a function defined for all real numbers \\( x... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1c0723206531b87dc5fc190af9c1260802978481f0b3b0009b17f01ed05919cc | A particle moving along a curve in the xy-plane has position \((x(t), y(t))\) at time \(t > 0\) where \[ \frac{dx}{dt} = \left(\frac{6}{t} - 3\right)^{1/3} \quad \text{and} \quad \frac{dy}{dt} = te^{-t} \] At time \(t = 3\), the particle is at the point \((5, 4)\). Is there a time \(t\) at which the particle is farthes... | [
"\\(t = 2\\) and \\(x(2)\\)"
] | [
"A particle moving along a curve in the xy-plane has position \\((x(t), y(t))\\) at time \\(t > 0\\) where\n\n\\[\n\\frac{dx}{dt} = \\left(\\frac{6}{t} - 3\\right)^{1/3} \\quad \\text{and} \\quad \\frac{dy}{dt} = te^{-t}\n\\]\n\nAt time \\(t = 3\\), the particle is at the point \\((5, 4)\\).\n\nIs there a time \\(t... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 8,
"tir": 26
} | 35 | 0 | 0 |
omr-ws-4073e647b0018e0121e79e880e50b6258d25961a88ceb8dbcf667695047fa5ed | A particle moving along a straight line with uniform acceleration travels three equal distances in times \( t_1 \), \( t_2 \), and \( t_3 \). If \( t_1 = 2 \), \( t_2 = 3 \), and \( t_3 = 6 \), calculate the value of \(\frac{1}{t_1} - \frac{1}{t_2} + \frac{1}{t_3}\). | [
"\\dfrac{1}{3}"
] | [
"A particle moving along a straight line with uniform acceleration travels three equal distances in times \\( t_1 \\), \\( t_2 \\), and \\( t_3 \\). If \\( t_1 = 2 \\), \\( t_2 = 3 \\), and \\( t_3 = 6 \\), calculate the value of \\(\\frac{1}{t_1} - \\frac{1}{t_2} + \\frac{1}{t_3}\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-632cd2167d0684a155796c9c1f057aaf1e7ba05d627ef5d9f5d7fcdcae33cd67 | A particle moving along the number line has acceleration given by \( a(t) = 2t - 1 \). We also know that \( s(0) = 2 \) and \( v(0) = -2 \). What is the net distance traversed from time \( t = 0 \) to \( t = 3 \)? | [
"\\dfrac{31}{6}"
] | [
"A particle moving along the number line has acceleration given by \\( a(t) = 2t - 1 \\). We also know that \\( s(0) = 2 \\) and \\( v(0) = -2 \\). What is the net distance traversed from time \\( t = 0 \\) to \\( t = 3 \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-d6680773ab0e0066d6758184406f65780c2fa9fb5a0fa05e240acd86232bd842 | A particle moving along the number line has acceleration given by \( a(t) = 2t - 1 \). We also know that \( s(0) = 2 \) and \( v(0) = -2 \). What is the total distance traversed from time \( t = 0 \) to \( t = 3 \)? | [
"\\dfrac{31}{6}"
] | [
"A particle moving along the number line has acceleration given by \\( a(t) = 2t - 1 \\). We also know that \\( s(0) = 2 \\) and \\( v(0) = -2 \\). What is the total distance traversed from time \\( t = 0 \\) to \\( t = 3 \\)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d9e2928ebc2208278cc8dd9c0b5233afd585281ef7217a4648e489095524cea7 | A particle of mass \( m \) is projected vertically upwards with speed \( Ums^{-1} \) and returns to its starting point with speed \( Vms^{-1} \). The resistance to its motion in Newtons has magnitude \( \frac{mv}{10} \) when its speed is \( vms^{-1} \). The acceleration due to gravity is \( 10ms^{-2} \). (i) For the as... | [
"5(U + V)"
] | [
"A particle of mass \\( m \\) is projected vertically upwards with speed \\( Ums^{-1} \\) and returns to its starting point with speed \\( Vms^{-1} \\). The resistance to its motion in Newtons has magnitude \\( \\frac{mv}{10} \\) when its speed is \\( vms^{-1} \\). The acceleration due to gravity is \\( 10ms^{-2} \... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-bcc245017548f0412a123f7794e70520fa856e3e4a49a8d8192010c7037fd277 | A particle of mass \( m \) is released from rest at a distance \( b \) from an origin where it experiences an attractive force \( F(x) = -kx^{-2} \). Calculate the time \( T \) it takes for the particle to reach the origin. Express your answer in terms of \( m \), \( b \), and \( k \). | [
"\\pi \\sqrt{\\dfrac{m b^3}{8k}}"
] | [
"A particle of mass \\( m \\) is released from rest at a distance \\( b \\) from an origin where it experiences an attractive force \\( F(x) = -kx^{-2} \\). Calculate the time \\( T \\) it takes for the particle to reach the origin. Express your answer in terms of \\( m \\), \\( b \\), and \\( k \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-1c0d9a7471aa5e615ceb67c8ee6b9a605df28d2603b4466d7deb60471ac5b4ce | A particle of mass \( m \) moves from point \( \mathbf{X_1} = (x_1, y_1, z_1) \) to point \( \mathbf{X_2} = (x_2, y_2, z_2) \) under the gravitational force field \( \mathbf{F} = -\frac{GMm}{(x^2 + y^2 + z^2)^{3/2}} (x \mathbf{i} + y \mathbf{j} + z \mathbf{k}) \). Given \( G \), \( M \), and \( m \) are constants, calc... | [
"-1"
] | [
"A particle of mass \\( m \\) moves from point \\( \\mathbf{X_1} = (x_1, y_1, z_1) \\) to point \\( \\mathbf{X_2} = (x_2, y_2, z_2) \\) under the gravitational force field \\( \\mathbf{F} = -\\frac{GMm}{(x^2 + y^2 + z^2)^{3/2}} (x \\mathbf{i} + y \\mathbf{j} + z \\mathbf{k}) \\). Given \\( G \\), \\( M \\), and \\(... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-20a46a399eebc9b5aa9efde7945e0fb7b48d74c006b593083d9ffa72369dc2e0 | A particle of mass \( m \) moves in a horizontal straight line under a force equal to \( m(n^2) \) times the displacement from a point \( O \) on the line and directed towards \( O \); in addition, the motion of the particle is resisted by a force equal to \( mk \) times the square of the speed. The particle is project... | [
"0"
] | [
"A particle of mass \\( m \\) moves in a horizontal straight line under a force equal to \\( m(n^2) \\) times the displacement from a point \\( O \\) on the line and directed towards \\( O \\); in addition, the motion of the particle is resisted by a force equal to \\( mk \\) times the square of the speed. The part... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0a6a6857aaf3db56e69499d3fd28600286e9763d43ee3776d4762704a34c9d29 | A particle of mass \( m \) starting from rest at the origin moves under the action of the force \( \mathbf{F} = y^2 \mathbf{i} - \mathbf{j} \). Determine the constant \( C \) in the trajectory equation \( y^3 + Cx = 0 \) of the particle. | [
"15"
] | [
"A particle of mass \\( m \\) starting from rest at the origin moves under the action of the force \\( \\mathbf{F} = y^2 \\mathbf{i} - \\mathbf{j} \\). Determine the constant \\( C \\) in the trajectory equation \\( y^3 + Cx = 0 \\) of the particle."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e5b496ea9e27fef3440e14cbe654f3f93672f618fad488e27ef5a2e76e564637 | A particle of unit mass is projected from level ground with speed \( U\sqrt{2} \) at an elevation of \( \frac{\pi}{4} \) above the horizontal. It experiences a resisting force directly opposing the motion whose magnitude is \( k \) times the square of the particle's speed, where \( \frac{kU^2}{g} \) is a small number. ... | [
"-g - k(U - gt)\\sqrt{U^2 + (U - gt)^2}"
] | [
"A particle of unit mass is projected from level ground with speed \\( U\\sqrt{2} \\) at an elevation of \\( \\frac{\\pi}{4} \\) above the horizontal. It experiences a resisting force directly opposing the motion whose magnitude is \\( k \\) times the square of the particle's speed, where \\( \\frac{kU^2}{g} \\) is... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-389ff36cef89e2f56ff264ee323904fd3138f9b9d9578123e63d2068246e5ca0 | A particle of unit mass moves on a straight line under the action of a force which is a function $f(v)$ of the velocity $v$ of the particle. The distance $x$ covered in time $t$ satisfies the formula $x = at^2 + bt + c$, where $a$, $b$, and $c$ are numerical values determined by observation. Determine the function $f(v... | [
"\\( f(v) = 2a \\)"
] | [
"A particle of unit mass moves on a straight line under the action of a force which is a function $f(v)$ of the velocity $v$ of the particle. The distance $x$ covered in time $t$ satisfies the formula $x = at^2 + bt + c$, where $a$, $b$, and $c$ are numerical values determined by observation. Determine the function... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-91d442c1627247403670638f3584453cfa7b3725d7fc0f550d4025eaa9e057f4 | A particle rests at vertex A of a triangle ABC with side length 1. The particle rockets toward a point D on side BC such that BD:DC = 20:17. The particle continues to bounce around the triangle such that when it collides into a side of the triangle, the angle of reflection is equal to the angle of incidence. This proce... | [
"1029"
] | [
"A particle rests at vertex A of a triangle ABC with side length 1. The particle rockets toward a point D on side BC such that BD:DC = 20:17. The particle continues to bounce around the triangle such that when it collides into a side of the triangle, the angle of reflection is equal to the angle of incidence. This ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-81d89104c8796795240d6ae180d174a6ccd3e5ee3c866ffb7ff3a74855c6fb8f | A particle starts at $(0,0)$ and moves one unit randomly in a direction parallel to the $x$ or $y$ axis every second. After $4092$ seconds, the particle is at $(x,y)$. Find the expected value of $x+y$. | [
"0"
] | [
"A particle starts at $(0,0)$ and moves one unit randomly in a direction parallel to the $x$ or $y$ axis every second. After $4092$ seconds, the particle is at $(x,y)$. Find the expected value of $x+y$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-09ee7f3a972bbb58585613c34276ef7a47fd624c103eb540bd160bab950cc5fa | A particle starts at $(0,0)$ in the Cartesian coordinate plane and moves every second to one of the eight closest lattice points with equal probability $\frac{1}{8}$. The particle will eventually hit a square with vertices at $(2,2)$, $(-2,2)$, $(-2,-2)$, and $(2,-2)$. The probability that it will hit at one of the $4$... | [
"39"
] | [
"A particle starts at $(0,0)$ in the Cartesian coordinate plane and moves every second to one of the eight closest lattice points with equal probability $\\frac{1}{8}$. The particle will eventually hit a square with vertices at $(2,2)$, $(-2,2)$, $(-2,-2)$, and $(2,-2)$. The probability that it will hit at one of t... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 7
} | 54 | 0 | 0 |
omr-ws-28aa1a3c2d5b2b3754324c257f969aada24c490d8c0d91ee65350f9746590067 | A particle starts at $(0,0)$ in the Cartesian coordinate plane and moves every second to one of the eight closest lattice points with equal probability. The particle will eventually hit a square with vertices at $(2, 2)$, $(-2, 2)$, $(-2, -2)$, and $(2, -2)$. The probability that the particle will hit at a corner rathe... | [
"39"
] | [
"A particle starts at $(0,0)$ in the Cartesian coordinate plane and moves every second to one of the eight closest lattice points with equal probability. The particle will eventually hit a square with vertices at $(2, 2)$, $(-2, 2)$, $(-2, -2)$, and $(2, -2)$. The probability that the particle will hit at a corner ... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-24842ac47f010d6dbdbf2e756879b9a6e441269450265a02a4fe06f56874fce0 | A particle starts at $(5,0)$ on the coordinate plane. At each move, the particle undergoes a $45^\circ$ counter-clockwise rotation about the origin followed by a translation of $10$ units in the positive $X$-direction. Determine $\mid p \mid + \mid q \mid$ if the position of the particle after $150$ moves is $(p,q)$. | [
"5 + 10\\sqrt{2}"
] | [
"A particle starts at $(5,0)$ on the coordinate plane. At each move, the particle undergoes a $45^\\circ$ counter-clockwise rotation about the origin followed by a translation of $10$ units in the positive $X$-direction. Determine $\\mid p \\mid + \\mid q \\mid$ if the position of the particle after $150$ moves is ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-60b1eeb640b0fdde6567ee09288249da9f68bb1388044d3cf86ff287bd826c82 | A particle starts at $(\frac{1}{2},\frac{1}{2},0)$ inside a cube with vertices $(0,0,0)$, $(0,0,1)$, $(0,1,0)$, $(0,1,1)$, $(1,0,0)$, $(1,0,1)$, $(1,1,0)$, and $(1,1,1)$. After one second, it moves to $(\frac{9}{16},\frac{5}{8},\frac{1}{13})$. The particle travels in straight lines with uniform speed and bounces until ... | [
"4"
] | [
"A particle starts at $(\\frac{1}{2},\\frac{1}{2},0)$ inside a cube with vertices $(0,0,0)$, $(0,0,1)$, $(0,1,0)$, $(0,1,1)$, $(1,0,0)$, $(1,0,1)$, $(1,1,0)$, and $(1,1,1)$. After one second, it moves to $(\\frac{9}{16},\\frac{5}{8},\\frac{1}{13})$. The particle travels in straight lines with uniform speed and boun... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-912cb496ca2fddb85cae117183806276513871181c4c9cb3801692637ccb5ac6 | A particle starts at $0$ on the number line. Each second, it moves $2$ units in the positive direction with probability $\tfrac{1}{3}$ and stays put with probability $\tfrac{2}{3}$. After $2016$ seconds, let $d$ be the distance between the particle and the point $-2$. Compute the expected value of $\tfrac{1}{d}$. | [
"\\dfrac{3}{4034}\\left(1 - \\left(\\dfrac{2}{3}\\right)^{2017}\\right)"
] | [
"A particle starts at $0$ on the number line. Each second, it moves $2$ units in the positive direction with probability $\\tfrac{1}{3}$ and stays put with probability $\\tfrac{2}{3}$. After $2016$ seconds, let $d$ be the distance between the particle and the point $-2$. Compute the expected value of $\\tfrac{1}{d}... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 4
} | 16 | 0 | 0 |
omr-ws-966f54be9e397a8e639d980ce4c96d207a95ffa9cc187ccdbabc544602fd41b8 | A particle starts at 0 on the number line. For positive integers $t$, at time $t$ the particle moves a distance $3^{-t}$ either left or right, chosen at random. What is the expected distance of the final position of the particle (at time $\infty$) from $1/4$? Note that the first move will be $\frac{1}{3}$ units. | [
"**\n - The expected distance of the final position of the particle from \\( \\frac{1}{4} \\) is \\( \\frac{1}{2\\sqrt{\\pi}} \\)."
] | [
"A particle starts at 0 on the number line. For positive integers $t$, at time $t$ the particle moves a distance $3^{-t}$ either left or right, chosen at random. What is the expected distance of the final position of the particle (at time $\\infty$) from $1/4$? Note that the first move will be $\\frac{1}{3}$ units.... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-88bc4a584a2d0e3689f239b0f1ce972a1f234ac0ab81416b3a3cc489f76cd0f7 | A particle starts at \((\frac{1}{2}, \frac{1}{2}, 0)\) and travels to \((\frac{9}{16}, \frac{5}{8}, \frac{1}{13})\) in one second. The particle bounces until it hits an edge or a corner of a cube with vertices \((0,0,0)\), \((0,0,1)\), \((0,1,0)\), \((0,1,1)\), \((1,0,0)\), \((1,0,1)\), \((1,1,0)\), and \((1,1,1)\). Fi... | [
"52"
] | [
"A particle starts at \\((\\frac{1}{2}, \\frac{1}{2}, 0)\\) and travels to \\((\\frac{9}{16}, \\frac{5}{8}, \\frac{1}{13})\\) in one second. The particle bounces until it hits an edge or a corner of a cube with vertices \\((0,0,0)\\), \\((0,0,1)\\), \\((0,1,0)\\), \\((0,1,1)\\), \\((1,0,0)\\), \\((1,0,1)\\), \\((1,... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a55c46c371ff0fc18de8994adc16c5b49c8858294045c425c137bf05abf6a288 | A particle starts at one vertex of a cube. At each step, the particle moves to an adjacent vertex. How many ways can the particle arrive at the opposite vertex of the cube after five steps? (The particle may visit the opposite vertex before the fifth step, as long as it is at the opposite vertex after the fifth step.) | [
"24"
] | [
"A particle starts at one vertex of a cube. At each step, the particle moves to an adjacent vertex. How many ways can the particle arrive at the opposite vertex of the cube after five steps? (The particle may visit the opposite vertex before the fifth step, as long as it is at the opposite vertex after the fifth st... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-558e73a77e8a4fe09bf3eab6e73524fe158cc09df884227741f00b405d06cdd1 | A particle starts at point $(0,0)$ on the coordinate plane. Every move, if the x-coordinate is $0$, the particle can move one unit up, down, or right; otherwise, the particle must move one unit left. Let $P(n)$ be the number of valid sequences of $n$ moves the particle can take. Determine a closed form for $P(n)$. | [
"\\( P(n) = \\frac{(1+\\sqrt{2})^n - (1-\\sqrt{2})^n}{2\\sqrt{2}} \\)"
] | [
"A particle starts at point $(0,0)$ on the coordinate plane. Every move, if the x-coordinate is $0$, the particle can move one unit up, down, or right; otherwise, the particle must move one unit left. Let $P(n)$ be the number of valid sequences of $n$ moves the particle can take. Determine a closed form for $P(n)$.... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d9359092efb8b1f02c0de7afc8183a04e69ac1f19453c51cb2a1459bea5b28ec | A particle starts at point $(21,21,0)$ in a cube defined by endpoints $( \{ 0,42 \}, \; \{ 0,42 \}, \; \{ 0,42 \})$. The particle moves with an x-direction speed of $6$, a y-direction speed of $2$, and a z-direction speed of $3$. When the particle reaches either $0$ or $42$ in any dimension, it reverses direction in th... | [
"\\dfrac{78}{7}"
] | [
"A particle starts at point $(21,21,0)$ in a cube defined by endpoints $( \\{ 0,42 \\}, \\; \\{ 0,42 \\}, \\; \\{ 0,42 \\})$. The particle moves with an x-direction speed of $6$, a y-direction speed of $2$, and a z-direction speed of $3$. When the particle reaches either $0$ or $42$ in any dimension, it reverses di... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-19540cc5464316e488615d09d746257f6a226c95b3536bc230f0362f60a191ea | A particle starts at point P on a horizontal line. Each time a balanced coin is tossed, if heads appears, the particle moves a distance of \(a\) units to the right, and if tails appears, it moves a distance of \(b\) units to the left. After \(n\) coin tosses, let \(L\) be the distance of the particle from point P. Find... | [
"\\frac{1}{2^n} \\sum_{k=0}^n \\left| (a + b)k - bn \\right| \\binom{n}{k}"
] | [
"A particle starts at point P on a horizontal line. Each time a balanced coin is tossed, if heads appears, the particle moves a distance of \\(a\\) units to the right, and if tails appears, it moves a distance of \\(b\\) units to the left. After \\(n\\) coin tosses, let \\(L\\) be the distance of the particle from ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-dd2442faa742b9011541325b750c078943f0cea5d395977dc2080534792a47a0 | A particle starts at point \( A_0 \) on a circle of radius 1". It moves along the circle for 1" and stops at point \( A_1 \). It then continues along the circle for \( \frac{1}{2}" \) to point \( A_2 \), then \( \frac{1}{3}" \) to point \( A_3 \), and so on. The stopping point after moving \( \frac{1}{n}" \) is \( A_n ... | [
"0"
] | [
"A particle starts at point \\( A_0 \\) on a circle of radius 1\". It moves along the circle for 1\" and stops at point \\( A_1 \\). It then continues along the circle for \\( \\frac{1}{2}\" \\) to point \\( A_2 \\), then \\( \\frac{1}{3}\" \\) to point \\( A_3 \\), and so on. The stopping point after moving \\( \\... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-895ac6d440fb5289b954a5e5a1556adfca4f9880ac31333c4302c872b46002b5 | A particle starts at the center of a 4x4 chessboard. The particle has a 1/8 probability to move to any of the 8 grid points surrounding its current position. Find the probability that the particle finally ends at one of the 4 vertices of the chessboard without landing on any of the other 12 grid points on the boundary ... | [
"16/141"
] | [
"A particle starts at the center of a 4x4 chessboard. The particle has a 1/8 probability to move to any of the 8 grid points surrounding its current position. Find the probability that the particle finally ends at one of the 4 vertices of the chessboard without landing on any of the other 12 grid points on the boun... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ead6dc592ce59f8f427b63de351121b7d648685940ab3553b750bbde0b402962 | A particle starts at the origin $O$ in three-dimensional space within a cube that is 2 units on each side. At each move, the particle moves to one of the eight corners of the cube with equal probability. Each move changes the particle's position by one unit in one of the three dimensions (up/down, east/west, north/sout... | [
"0"
] | [
"A particle starts at the origin $O$ in three-dimensional space within a cube that is 2 units on each side. At each move, the particle moves to one of the eight corners of the cube with equal probability. Each move changes the particle's position by one unit in one of the three dimensions (up/down, east/west, north... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-18b45c573fd0b51ab25ad461a5018e14bef9314c6e95faa979aedc0e84d161ac | A particle starts at the origin O in three-dimensional space. The origin is centered in a cube that is 2 units on each side. In each move, the particle moves to one of the eight corners of the cube with equal probability. Each move changes the particle's position by one unit in one of the three dimensions (up/down, eas... | [
"0.3405"
] | [
"A particle starts at the origin O in three-dimensional space. The origin is centered in a cube that is 2 units on each side. In each move, the particle moves to one of the eight corners of the cube with equal probability. Each move changes the particle's position by one unit in one of the three dimensions (up/down... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 4
} | 9 | 0 | 0 |
omr-ws-13d525e9abca8a3d65e07ac4cafbd604930ada251c49494208edf9337f2cb823 | A particle starts at the origin and moves each second with a $1/4$ chance of moving one unit in the negative $x$ direction and a $1/2$ chance of moving one unit in the positive $x$ direction. If the particle reaches $(-3,0)$, it ignites a fission that will consume the earth. If it reaches $(7, 0)$, it is harmlessly dif... | [
"919"
] | [
"A particle starts at the origin and moves each second with a $1/4$ chance of moving one unit in the negative $x$ direction and a $1/2$ chance of moving one unit in the positive $x$ direction. If the particle reaches $(-3,0)$, it ignites a fission that will consume the earth. If it reaches $(7, 0)$, it is harmlessl... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-9fffdb77abebb2050c7944a8e9ea41781d23b7f4394cc9314d29bc229d2ece44 | A particle starts at the origin and moves each second with a 1/4 chance of moving one unit in the negative \( x \) direction and a 1/2 chance of moving one unit in the positive \( x \) direction. If the particle reaches \((-3, 0)\), it ignites a fission that will consume the earth. If it reaches \((7, 0)\), it is harml... | [
"919"
] | [
"A particle starts at the origin and moves each second with a 1/4 chance of moving one unit in the negative \\( x \\) direction and a 1/2 chance of moving one unit in the positive \\( x \\) direction. If the particle reaches \\((-3, 0)\\), it ignites a fission that will consume the earth. If it reaches \\((7, 0)\\)... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2e4404926948eaf2a871c020bb2616c17b09311cf3e55501e185b0ec84a4fd02 | A particle starts at the origin and takes a series of steps in the following manner: If the particle is at the point $(x,y)$, it then moves randomly to one of $(x-1,y)$, $(x+1,y)$, $(x-1,y+1)$, $(x,y+1)$, or $(x+1,y+1)$. What is the expected number of steps the particle takes to reach the line $y=3$? | [
"5"
] | [
"A particle starts at the origin and takes a series of steps in the following manner: If the particle is at the point $(x,y)$, it then moves randomly to one of $(x-1,y)$, $(x+1,y)$, $(x-1,y+1)$, $(x,y+1)$, or $(x+1,y+1)$. What is the expected number of steps the particle takes to reach the line $y=3$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a8a9a0ed05c6ce608b99a02c1c4a98dc23cb3bea9bb509fd160c39eaca65cc4e | A particle starts at the origin in the $xy$-coordinate plane. A move is defined as going from $(x,y)$ to any one of $(x,y+1)$, $(x+1,y+1)$, $(x+1,y)$, $(x+1,y-1)$, $(x,y-1)$, $(x-1,y-1)$, $(x-1,y)$, or $(x-1,y+1)$ with the additional rule that the $x$- and $y$-coordinates of the particle must always stay between 0 and ... | [
"111"
] | [
"A particle starts at the origin in the $xy$-coordinate plane. A move is defined as going from $(x,y)$ to any one of $(x,y+1)$, $(x+1,y+1)$, $(x+1,y)$, $(x+1,y-1)$, $(x,y-1)$, $(x-1,y-1)$, $(x-1,y)$, or $(x-1,y+1)$ with the additional rule that the $x$- and $y$-coordinates of the particle must always stay between 0... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-6760c1638a57edb592bcf01dfe0a7751122ebb004f816f75f4c0a228fcc40597 | A particle starts at the origin of a 3-dimensional coordinate grid and moves with equal likelihood to one of the eight corners of the surrounding cube, with a 50-50 chance of moving one unit up or down, one unit east or west, and one unit north or south. What is the probability that the particle returns to the origin? ... | [
"\\frac{1}{4}"
] | [
"A particle starts at the origin of a 3-dimensional coordinate grid and moves with equal likelihood to one of the eight corners of the surrounding cube, with a 50-50 chance of moving one unit up or down, one unit east or west, and one unit north or south. What is the probability that the particle returns to the ori... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-56cf827f7c3c3b79bf3db9d6a134555daf287aaf1be4015f5976c63d2850e16b | A particle starts at the origin of an infinite coordinate grid and moves diagonally to another lattice point with a 50-50 chance of moving north or south and a 50-50 chance of moving east or west. What is the probability that the particle will be back at the origin after n moves? What is the probability that it will ev... | [
"Probability of being back at the origin after n moves: \\(\\frac{(2nCn)^2}{4^{2n}}\\), Probability of ever returning to the origin: 1"
] | [
"A particle starts at the origin of an infinite coordinate grid and moves diagonally to another lattice point with a 50-50 chance of moving north or south and a 50-50 chance of moving east or west. What is the probability that the particle will be back at the origin after n moves? What is the probability that it wi... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 10
} | 10 | 0 | 0 |
omr-ws-12de695c616645f8cc4af86865a33ac507b8b03684518cca61918f7130f435f7 | A particle starts at the origin of the Cartesian plane. At the end of every second, it jumps one unit either to the East, West, North, or South with equal probability. What is the probability that the particle is back at the origin after six seconds? | [
"\\(\\frac{5}{1024}\\)"
] | [
"A particle starts at the origin of the Cartesian plane. At the end of every second, it jumps one unit either to the East, West, North, or South with equal probability. What is the probability that the particle is back at the origin after six seconds?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-71d59526191e07c493f004dcb5587d3e294f21f8974a2b2e8c6b3e7ad42a6375 | A particle starts at the origin on an infinite number line and makes a series of moves, each with a 50-50 chance of going left or right. What is the probability that the particle will be back at the origin after n moves? What is the probability that it will ever return to the origin? | [
"\\(\\frac{\\binom{n}{n/2}}{2^n}\\) for the probability after n moves, and 1 for the probability that it will ever return to the origin."
] | [
"A particle starts at the origin on an infinite number line and makes a series of moves, each with a 50-50 chance of going left or right. What is the probability that the particle will be back at the origin after n moves? What is the probability that it will ever return to the origin?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 13
} | 13 | 0 | 0 |
omr-ws-e6b32ab84218a1b37a0806af28d9cac0972bc91ea891ae85c8ab4a68595c9749 | A particle starts at the origin on the real line. In each second, it jumps one unit to the left or right with equal probability $\frac{1}{2}$. What is the probability that the particle reaches the point $+1$? | [
"1"
] | [
"A particle starts at the origin on the real line. In each second, it jumps one unit to the left or right with equal probability $\\frac{1}{2}$. What is the probability that the particle reaches the point $+1$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-83d21c5e91f2a4778e9ca117745b5089cbeb5213bdad812188a3f6de08afe184 | A particle starts at the point \((4,4)\) and moves to one of the points \((a-1,b)\), \((a,b-1)\), or \((a-1,b-1)\) with equal probability \(\frac{1}{3}\) each time. The particle stops when it hits one of the coordinate axes. What is the probability that the particle will hit the coordinate axes at \((0,0)\)? Express th... | [
"252"
] | [
"A particle starts at the point \\((4,4)\\) and moves to one of the points \\((a-1,b)\\), \\((a,b-1)\\), or \\((a-1,b-1)\\) with equal probability \\(\\frac{1}{3}\\) each time. The particle stops when it hits one of the coordinate axes. What is the probability that the particle will hit the coordinate axes at \\((0... | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-f1205ba24ab32be9bced677c8973471caa031f1ee99af722ed6692bf0de80724 | A particle starts at the root of a binary tree that ends on the 4th row (with 16 final nodes). The particle moves to the left or right with equal probability initially. If it lands on an even node, it loses. If it lands on an odd node other than the first node (node 1), it is transported back to the start. Each time it... | [
"\\(0.27687718\\)"
] | [
"A particle starts at the root of a binary tree that ends on the 4th row (with 16 final nodes). The particle moves to the left or right with equal probability initially. If it lands on an even node, it loses. If it lands on an odd node other than the first node (node 1), it is transported back to the start. Each ti... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f4e322b0e8dbd2eb083775e1ca4cf8dce1b0b3716e4bcbc35bcf860c00b205e9 | A particle undergoes a random walk on the real line, starting at the origin. Its steps $(X_n)_{n=1}^{\infty}$ are mutually independent and identically distributed, so that at the $n$-th stage the particle moves $a$ units to the right with probability $p$, and $b$ units to the left with probability $1-p$, where $a, b > ... | [
"a p - b(1 - p)"
] | [
"A particle undergoes a random walk on the real line, starting at the origin. Its steps $(X_n)_{n=1}^{\\infty}$ are mutually independent and identically distributed, so that at the $n$-th stage the particle moves $a$ units to the right with probability $p$, and $b$ units to the left with probability $1-p$, where $a... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a6461976924f19a166aa256562259e5ee0d103b55fe153df90185d2b90d9c8f6 | A particle undergoes simple harmonic motion (S.H.M.) along a straight line with a period of $T$ seconds. If the maximum displacement from the equilibrium position is $\frac{3}{4}l$, calculate the total distance traveled by the particle in $2T$ seconds. | [
"6l"
] | [
"A particle undergoes simple harmonic motion (S.H.M.) along a straight line with a period of $T$ seconds. If the maximum displacement from the equilibrium position is $\\frac{3}{4}l$, calculate the total distance traveled by the particle in $2T$ seconds."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8aba186e791ab958ed725836fa2c1d0298c74daf10fb889c9ee7d3fa8dc91138 | A particle's velocity is given by \( v = \frac{1}{d} \), where \( d \) is the distance it has traveled so far. Find an expression for the particle's distance \( d \) as a function of time \( t \). | [
"\\( \\sqrt{2t} \\)"
] | [
"A particle's velocity is given by \\( v = \\frac{1}{d} \\), where \\( d \\) is the distance it has traveled so far. Find an expression for the particle's distance \\( d \\) as a function of time \\( t \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 20
} | 21 | 0 | 0 |
omr-ws-d2be29312f2c1a46a714f70d8e5a17b2c6d65f53dc394a6ead0237643a55143c | A particle, initially located at the origin, moves one unit to the right with probability \( p \) and one unit to the left with probability \( q = 1 - p \). What is the probability that after \( 2k \) moves (\( k \ge 5 \)), the particle is 10 units away from the origin? | [
"\\(\\binom{2k}{k+5} p^{k+5} q^{k-5}\\)"
] | [
"A particle, initially located at the origin, moves one unit to the right with probability \\( p \\) and one unit to the left with probability \\( q = 1 - p \\). What is the probability that after \\( 2k \\) moves (\\( k \\ge 5 \\)), the particle is 10 units away from the origin?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d73925d2c496ad57e4f57a2229b8fa4288ae0b9ed7f799a1a919e32f06ed0d00 | A particular 12-hour digital clock displays the hour and minute of a day. Whenever it is supposed to display a 1, it mistakenly displays a 9. For example, when it is 1:16 PM, the clock incorrectly shows 9:96 PM. What fraction of the day will the clock show the correct time? | [
"\\(\\frac{1}{2}\\)"
] | [
"A particular 12-hour digital clock displays the hour and minute of a day. Whenever it is supposed to display a 1, it mistakenly displays a 9. For example, when it is 1:16 PM, the clock incorrectly shows 9:96 PM. What fraction of the day will the clock show the correct time?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 26
} | 74 | 0 | 0 |
omr-ws-ee7d7739e3ea9192f64fad0373af0b92f17f0960b4235fed828dfc4f10c5a6b1 | A particular convex polygon with seven sides has exactly one right angle. How many diagonals does this seven-sided polygon have? | [
"14"
] | [
"A particular convex polygon with seven sides has exactly one right angle. How many diagonals does this seven-sided polygon have?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-85f28c342ab3c53881e059616fd659bc34082ae0853f95de9a71b57a4c45889d | A particular fortune cookie lists four two-digit positive integers as your lucky numbers. The first three are 57, 13, and 72, but you got sauce on the last one and can't read it. If the sum of the digits of all four numbers equals $\frac{1}{5}$ of the sum of all four numbers, what is the smallest possibility for the fo... | [
"38"
] | [
"A particular fortune cookie lists four two-digit positive integers as your lucky numbers. The first three are 57, 13, and 72, but you got sauce on the last one and can't read it. If the sum of the digits of all four numbers equals $\\frac{1}{5}$ of the sum of all four numbers, what is the smallest possibility for ... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f09ad209bc04f3678f38cd08feb3aae4586eb8fadf649946311e03fc85149b3a | A particular novel contains $40,\!000$ words. If the author completed the novel in $80$ hours, on average how many words per hour did she write? | [
"500"
] | [
"A particular novel contains $40,\\!000$ words. If the author completed the novel in $80$ hours, on average how many words per hour did she write?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-48adea316d0a7cc8f582a64c647a77597fca3129e1972727422729f592cfc11f | A particular right prism has a rectangular octagonal base with each base edge measuring 4 cm and a height of 3 cm. Its volume can be expressed in the form \(a(b + \sqrt{c})\) cubic cm, where \(c\) is not divisible by the square of a prime. What is the value of \(a + b + c\)? | [
"99"
] | [
"A particular right prism has a rectangular octagonal base with each base edge measuring 4 cm and a height of 3 cm. Its volume can be expressed in the form \\(a(b + \\sqrt{c})\\) cubic cm, where \\(c\\) is not divisible by the square of a prime. What is the value of \\(a + b + c\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-0b281065a652523e5f8fc965b21ecb475180548deed81026a129fc20c7908621 | A particular triangle has sides of length 14 cm, 8 cm and 9 cm. In centimeters, what is the perimeter of the triangle? | [
"31"
] | [
"A particular triangle has sides of length 14 cm, 8 cm and 9 cm. In centimeters, what is the perimeter of the triangle?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b9203e83f481996afd2fcee132366ded834086843f54bb9ec15fb465a839ca08 | A partition of \(n\) is called "distinct" if all the summands are distinct. Let \(A_n\) be the set of all distinct partitions of \(n\) with an even number of summands, and \(B_n\) be the set of all distinct partitions of \(n\) with an odd number of summands. Find an explicit formula for \(|A_n| - |B_n|\). | [
"(-1)^k \\text{ if } n = \\frac{k(3k \\pm 1)}{2} \\text{ for some integer } k, \\text{ otherwise } 0"
] | [
"A partition of \\(n\\) is called \"distinct\" if all the summands are distinct. Let \\(A_n\\) be the set of all distinct partitions of \\(n\\) with an even number of summands, and \\(B_n\\) be the set of all distinct partitions of \\(n\\) with an odd number of summands. Find an explicit formula for \\(|A_n| - |B_n... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7788d8220c68444b23fed706945cc76c51454600c78a75051949f28fdb4fc598 | A party consisting of 5 men and 4 women arrives at a tennis club. They decide to arrange a mixed doubles match using only players from their party. How many different mixed doubles matches can be arranged? | [
"120"
] | [
"A party consisting of 5 men and 4 women arrives at a tennis club. They decide to arrange a mixed doubles match using only players from their party. How many different mixed doubles matches can be arranged?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4441a66567c9a24d2f4f1da60aac86216cc2a256bf7a767bd029ebfbf1b3ad92 | A party of 18 people went to a restaurant. They each chose a $21 meal, but 4 of them forgot to bring their money. In order to settle the total bill, how much extra did those who brought their money have to pay? | [
"6"
] | [
"A party of 18 people went to a restaurant. They each chose a $21 meal, but 4 of them forgot to bring their money. In order to settle the total bill, how much extra did those who brought their money have to pay?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 14
} | 37 | 0 | 0 |
omr-ws-d2f04e39d81a6c6048e184376ccd7166b45476b2075de5f5987f6e4736f8dd27 | A party was held at the house of the Schlobodkins. There were four other couples present, and many, but not all, pairs of people shook hands. Nobody shook hands with anyone twice, and nobody shook hands with their spouse. Both the host and the hostess shook some hands. At the end of the party, Mr. Schlobodkins polls ev... | [
"4"
] | [
"A party was held at the house of the Schlobodkins. There were four other couples present, and many, but not all, pairs of people shook hands. Nobody shook hands with anyone twice, and nobody shook hands with their spouse. Both the host and the hostess shook some hands. At the end of the party, Mr. Schlobodkins pol... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8faf0b340e4bdd017f5e6e29b7e8c3eafc08fa1afc942c00ab07d5e113dc0c40 | A passcode on a lock has 10 dials, and each number is either a 1 or a 0. What are the odds that the password converted to base 10 is divisible by 6? | [
"\\(\\frac{171}{1024}\\)"
] | [
"A passcode on a lock has 10 dials, and each number is either a 1 or a 0. What are the odds that the password converted to base 10 is divisible by 6?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 8,
"tir": 25
} | 44 | 0 | 0 |
omr-ws-0f88de9c966330e3cd290d07dba20cb575f991368b8fb6569a8920c9ada8238e | A passenger airliner is flying a round trip from New York City to San Francisco (2,500 miles one-way). The round-trip flight time is 10 hours and 25 minutes. Given that the speed of the wind during the round-trip is constant and the speed of the aircraft is 500 mph, calculate the speed of the wind. | [
"100 mph"
] | [
"A passenger airliner is flying a round trip from New York City to San Francisco (2,500 miles one-way). The round-trip flight time is 10 hours and 25 minutes. Given that the speed of the wind during the round-trip is constant and the speed of the aircraft is 500 mph, calculate the speed of the wind."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 20
} | 35 | 0 | 0 |
omr-ws-8754d14d05824d7eaa1493d329e40ac979834e415907e477f1c96f1dbddb7888 | A passenger in a train which moves with a speed of 40 km/h noted that another train passed his window in the opposite direction for 3 seconds. What is the speed of the second train if its length is 75 meters? | [
"50 km/h"
] | [
"A passenger in a train which moves with a speed of 40 km/h noted that another train passed his window in the opposite direction for 3 seconds. What is the speed of the second train if its length is 75 meters?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-185d7548448a2b0c2e69ff4b672b3bf77e152a13ec9c798d62eedfe15bf56b73 | A passenger train and a fast train left simultaneously two points which are 2400 km apart to meet each other. Each of them moves with a constant speed, and at a certain instant they meet. If both trains had moved with the speed of the fast train, then they would have met three hours earlier. If both trains had moved wi... | [
"60 km/h and 100 km/h"
] | [
"A passenger train and a fast train left simultaneously two points which are 2400 km apart to meet each other. Each of them moves with a constant speed, and at a certain instant they meet. If both trains had moved with the speed of the fast train, then they would have met three hours earlier. If both trains had mov... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ad3a97b39c41851c1a570ebb3bb0b19ecca15996c0b8b41b1ef4720def15c61b | A passenger train covers the distance between towns $A$ and $B$ 4 hours quicker than a goods train. If it took each train the same time it had taken the other train to cover the distance from $A$ to $B$, then the passenger train would cover 280 km more than the goods train. Now if each train increased its speed by 10 k... | [
"480 km"
] | [
"A passenger train covers the distance between towns $A$ and $B$ 4 hours quicker than a goods train. If it took each train the same time it had taken the other train to cover the distance from $A$ to $B$, then the passenger train would cover 280 km more than the goods train. Now if each train increased its speed ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9a02474792e37fd581411ea1988f27b6cbe0d7fca919842981d7c73852dfbd8b | A passenger train set out from town $A$ for town $B$ and at the same time a goods train started from $B$ to $A$. The speed of each train is constant throughout the way. Two hours after they met, the distance between them was 280 km. The passenger train arrived at the point of destination 9 h and the goods train 16 h af... | [
"The passenger train, 21 h, the goods train, 28 h"
] | [
"A passenger train set out from town $A$ for town $B$ and at the same time a goods train started from $B$ to $A$. The speed of each train is constant throughout the way. Two hours after they met, the distance between them was 280 km. The passenger train arrived at the point of destination 9 h and the goods train 1... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-65beb81230c16cbcf1243682909d3b89081cd098e49a034eaefb548010cb22a0 | A password consists of seven characters and must contain at least one digit and at least one letter. Given that there are 26 letters and 10 digits (0-9), how many different passwords can be formed? | [
"70322353920",
"\\(36^7 - 26^7 - 10^7\\)"
] | [
"A password consists of seven characters and must contain at least one digit and at least one letter. Given that there are 26 letters and 10 digits (0-9), how many different passwords can be formed?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 31,
"genselect": 8,
"tir": 28
} | 67 | 0 | 0 |
omr-ws-f9e9a48b6c18c39a0cce71604d92b9d13dd050d6b900fe0195e634f7b9d4f924 | A password is considered "cute" if it follows these rules: - It contains only digits from 0 to 9. - It is no longer than 6 digits. - It contains the subsequence 926. - It does not contain any prefix zeros. Let \( m \) be the number of "cute" passwords. What is the sum of the digits of \( m \)? | [
"2"
] | [
"A password is considered \"cute\" if it follows these rules:\n- It contains only digits from 0 to 9.\n- It is no longer than 6 digits.\n- It contains the subsequence 926.\n- It does not contain any prefix zeros.\nLet \\( m \\) be the number of \"cute\" passwords. What is the sum of the digits of \\( m \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 2
} | 8 | 0 | 0 |
omr-ws-fa6a70604448fbb7ed26baa5fd9cd379d1d6e912f4ae59df239cb1b01189257a | A password must be exactly 6 characters long and consist of digits (0-9), upper and lower case letters (a-z and A-Z), and five special characters ($, £, *, &, %). The password must contain at least one letter or special character, and any letter or special character must be followed by a digit (so it must end in a digi... | [
"\\(\\frac{\\text{Number of valid passwords}}{67 \\times 66 \\times 65 \\times 64 \\times 63 \\times 62}\\)"
] | [
"A password must be exactly 6 characters long and consist of digits (0-9), upper and lower case letters (a-z and A-Z), and five special characters ($, £, *, &, %). The password must contain at least one letter or special character, and any letter or special character must be followed by a digit (so it must end in a... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-353340e623bbc5b45c368e0341d3ccecc0c1483a2ddc999debd2d38f0cb6c6d8 | A path from \( (0,0) \) to \( (n,n) \) on the lattice is made up of unit moves upward or rightward. It is balanced if the sum of the x-coordinates of its \( 2n+1 \) vertices equals the sum of their y-coordinates. Let \( n = 10 \). Find the number of balanced paths that divide the square with vertices \( (0,0) \), \( (1... | [
"no_valid_answer_found"
] | [
"A path from \\( (0,0) \\) to \\( (n,n) \\) on the lattice is made up of unit moves upward or rightward. It is balanced if the sum of the x-coordinates of its \\( 2n+1 \\) vertices equals the sum of their y-coordinates. Let \\( n = 10 \\). Find the number of balanced paths that divide the square with vertices \\( (... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-093d20d889cce73a9c9625117b6f9fbc693fce28db4ea939381c47485066397c | A path of uniform width runs all around the inside of a 116 metres by 68 metres rectangle and occupies 720 square metres. Find the width of the path. | [
"28"
] | [
"A path of uniform width runs all around the inside of a 116 metres by 68 metres rectangle and occupies 720 square metres. Find the width of the path."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-18666dacb86557bd7dba496b37dc1857a42b7f981fa5772d4fe7ee27c5553f52 | A path zig-zags from $(1, 0)$ to $(0, 0)$ along line segments $P_nP_{n+1}$, where $P_0$ is $(1, 0)$ and $P_n$ is $(2^{-n}, (-2)^{-n})$ for $n > 0$. Find the length of the path. | [
"\\(\\frac{\\sqrt{5}+1}{\\sqrt{2}}\\)"
] | [
"A path zig-zags from $(1, 0)$ to $(0, 0)$ along line segments $P_nP_{n+1}$, where $P_0$ is $(1, 0)$ and $P_n$ is $(2^{-n}, (-2)^{-n})$ for $n > 0$. Find the length of the path."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-f48c145141a9ce5e99d684f3a4d2eb823967fe2e4c565674eb9ce752a1b4ac90 | A path zig-zags from (1,0) to (0,0) along line segments (P_n)(P_(n+1)), where P_0 is (1,0) and P_n is (2^−n, (−2)^−n), for n > 0. Find the length of the path. | [
"\\(\\frac{\\sqrt{10} + \\sqrt{2}}{2}\\)"
] | [
"A path zig-zags from (1,0) to (0,0) along line segments (P_n)(P_(n+1)), where P_0 is (1,0) and P_n is (2^−n, (−2)^−n), for n > 0. Find the length of the path."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 43,
"genselect": 8,
"tir": 26
} | 77 | 0 | 0 |
omr-ws-6db0a30fc2946f82b9c4f51eba7966b60e6a10678a076cc1e39f7662757d16e6 | A patient can contract diseases A, B, or C with probabilities 0.3, 0.4, and 0.3, respectively. Four doctors independently examine the patient and make the following guesses: the first doctor guesses disease A, the second doctor guesses disease B, the third doctor guesses disease C, and the fourth doctor guesses disease... | [
"\\( P(A) = 0.5625, P(B) = 0.25, P(C) = 0.1875 \\)"
] | [
"A patient can contract diseases A, B, or C with probabilities 0.3, 0.4, and 0.3, respectively. Four doctors independently examine the patient and make the following guesses: the first doctor guesses disease A, the second doctor guesses disease B, the third doctor guesses disease C, and the fourth doctor guesses di... | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 12,
"tir": 32
} | 44 | 0 | 0 |
omr-ws-bcc5140e12893ff3797371863eb1ac6b2dde1436b2b5fe3d910eba49aaef7fec | A pawn is on the middle square of a 9x9 blackboard. In each move, the pawn moves to an adjacent square with probability 1/4. What is the probability that after 4 moves, the pawn will return to the starting point? | [
"\\dfrac{9}{64}"
] | [
"A pawn is on the middle square of a 9x9 blackboard. In each move, the pawn moves to an adjacent square with probability 1/4. What is the probability that after 4 moves, the pawn will return to the starting point?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-129995f0401a34a67c7071f1424277cbdc8d5c02e50643aaae406ca2502613e6 | A pawn is placed at the origin of the Cartesian plane. Each turn, the pawn moves to one of four possible locations (moving up and to the left or right, or moving down and to the left or right). Let \( E \) be the expected value of the square of the distance of the pawn from the origin after 2018 such turns. What is the... | [
"36"
] | [
"A pawn is placed at the origin of the Cartesian plane. Each turn, the pawn moves to one of four possible locations (moving up and to the left or right, or moving down and to the left or right). Let \\( E \\) be the expected value of the square of the distance of the pawn from the origin after 2018 such turns. What... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b0dade3b30d55c3ebbe488b832defba71c2321330c87d7937fac7369e2610e32 | A pawn is placed in the central square of an $11 \times 11$ chessboard. Two players move the pawn in succession to any other square, but each move (beginning with the second) must be longer than the previous one. The player who cannot make such a move loses. Who wins in an errorless game? | [
"\\text{Second player}"
] | [
"A pawn is placed in the central square of an $11 \\times 11$ chessboard. Two players move the pawn in succession to any other square, but each move (beginning with the second) must be longer than the previous one. The player who cannot make such a move loses. Who wins in an errorless game?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-ffe296636d856a9c08e8ecd43f24c3fbee0535c469f4ba44085dd8c8f5b07217 | A pawn starts from square \( n \) on an infinite strip of unit squares numbered 1, 2, 3, ... The pawn can move from square \( n \) to square \( 2n \) or \( 3n + 1 \), and vice-versa. Determine the smallest positive integer \( n \) such that the pawn can reach square 1 after exactly 5 moves. | [
"9"
] | [
"A pawn starts from square \\( n \\) on an infinite strip of unit squares numbered 1, 2, 3, ... The pawn can move from square \\( n \\) to square \\( 2n \\) or \\( 3n + 1 \\), and vice-versa. Determine the smallest positive integer \\( n \\) such that the pawn can reach square 1 after exactly 5 moves."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-3b44a39e42bc7902823e7321a1dda29c8c4360eca497ae65d31f1e8a6f5810ca | A peacock is sitting on top of a 9m high pillar. From a point 27m away from the pillar, a snake is moving towards the base of the pillar. If the peacock and the snake have equal speeds, at what distance from the base of the pillar is the snake caught? | [
"12 m."
] | [
"A peacock is sitting on top of a 9m high pillar. From a point 27m away from the pillar, a snake is moving towards the base of the pillar. If the peacock and the snake have equal speeds, at what distance from the base of the pillar is the snake caught?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-169c9c1f1e153f980b0ef23b263c47a8f7e57a220733e42d5041329a9a82a5d7 | A pearl diver starts with 98 red, 100 green, and 102 blue pearls. He can trade in 2 pearls of the same color for 1 pearl of each of the other two colors. After trading, he ends up with \(a\) red, \(b\) green, and \(c\) blue pearls. Find the minimum possible value of \(T = |a-b| + |b-c| + |c-a|\). | [
"4"
] | [
"A pearl diver starts with 98 red, 100 green, and 102 blue pearls. He can trade in 2 pearls of the same color for 1 pearl of each of the other two colors. After trading, he ends up with \\(a\\) red, \\(b\\) green, and \\(c\\) blue pearls. Find the minimum possible value of \\(T = |a-b| + |b-c| + |c-a|\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-3c750dab695cf6a9eba9567817ed08be08a935925bd6f8d32a49648c65a7a7f0 | A pebble is shaped as the intersection of a cube with side length 1 with a solid sphere tangent to all of the cube's edges. What is the surface area of this pebble? | [
"(3\\sqrt{2} - 4)\\pi"
] | [
"A pebble is shaped as the intersection of a cube with side length 1 with a solid sphere tangent to all of the cube's edges. What is the surface area of this pebble?"
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 3
} | 3 | 0 | 0 |
omr-ws-f349645bc881620923ee403b40ca46d5e0313511ff487424e523a963b4e7690a | A peculiar frog performs a series of hops on the positive integer number line starting at the number $0.$ Every second, he either hops one number to its right with probability $\frac{3}{8},$ hops two numbers to its right with probability $\frac{3}{8},$ hops three numbers to its right with probability $\frac{1}{8},$ or ... | [
"5"
] | [
"A peculiar frog performs a series of hops on the positive integer number line starting at the number $0.$ Every second, he either hops one number to its right with probability $\\frac{3}{8},$ hops two numbers to its right with probability $\\frac{3}{8},$ hops three numbers to its right with probability $\\frac{1}{... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-2da3fa1cb2ae12050ba40bdfd13de32d58a048e25d9fb8ba8e04950e1e1c2bbc | A pedestal in the form of a rectangular parallelepiped is to be cut out of granit. The altitude must be equal to the diagonal of the base, and the area of the base must be 4 square meters. For which sides of the base will the total surface area of the pedestal be a minimum? | [
"2 meters and 2 meters"
] | [
"A pedestal in the form of a rectangular parallelepiped is to be cut out of granit. The altitude must be equal to the diagonal of the base, and the area of the base must be 4 square meters. For which sides of the base will the total surface area of the pedestal be a minimum?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-52fbdb701d9f002b333941aa2a917c1e1659c106558b2574eff9e00e222859dd | A pedestrian and a cyclist simultaneously left point $A$ for point $B$. Having reached $B$, the cyclist turned and an hour after starting met the pedestrian. The pedestrian continued walking toward $B$, while the cyclist turned once more and also rode toward $B$. Upon reaching $B$, the cyclist turned and rode back to $... | [
"3 hours"
] | [
"A pedestrian and a cyclist simultaneously left point $A$ for point $B$. Having reached $B$, the cyclist turned and an hour after starting met the pedestrian. The pedestrian continued walking toward $B$, while the cyclist turned once more and also rode toward $B$. Upon reaching $B$, the cyclist turned and rode back... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-711431bab8fedb20ba424a90b2bc8d95a4bb16ff106d1ec70f0c854c8fb331cb | A pedestrian and a cyclist start out from points $A$ and $B$ towards one another. After they meet the pedestrian continues to go in the direction from $A$ to $B$ while the cyclist turns and also goes towards $B$. The pedestrian reaches $B$ $t$ hours later than the cyclist. Find the time period between the start and mee... | [
"\\frac{t}{k-1}"
] | [
"A pedestrian and a cyclist start out from points $A$ and $B$ towards one another. After they meet the pedestrian continues to go in the direction from $A$ to $B$ while the cyclist turns and also goes towards $B$. The pedestrian reaches $B$ $t$ hours later than the cyclist. Find the time period between the start an... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-251a23336b23e27896ff31a48118268888325523a2c5a0453d89f180a1ee741f | A pedestrian left $A$ for $B$ and $\frac{3}{4}$ hour later a cyclist followed him. When the cyclist reached point $B$, the pedestrian had $\frac{3}{8}$ of the entire path to walk. How much time would it take the pedestrian to walk from $A$ to $B$ if the cyclist caught up with the pedestrian halfway between $A$ and $B$? | [
"2 hours"
] | [
"A pedestrian left $A$ for $B$ and $\\frac{3}{4}$ hour later a cyclist followed him. When the cyclist reached point $B$, the pedestrian had $\\frac{3}{8}$ of the entire path to walk. How much time would it take the pedestrian to walk from $A$ to $B$ if the cyclist caught up with the pedestrian halfway between $A$ a... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d02452a14e6e0a99b0692fedb1286206332e5922f25933ccf510b1fea8467bb1 | A pedestrian left point $A$ for point $B$. At the same time a motor-cyclist started from $B$ toward $A$ to meet the pedestrian. After meeting the pedestrian the motor-cyclist took the pedestrian to $B$ and returned to $A$ at once. As a result, the pedestrian got to $B$ four times faster than he had planned. How many ti... | [
"2.75"
] | [
"A pedestrian left point $A$ for point $B$. At the same time a motor-cyclist started from $B$ toward $A$ to meet the pedestrian. After meeting the pedestrian the motor-cyclist took the pedestrian to $B$ and returned to $A$ at once. As a result, the pedestrian got to $B$ four times faster than he had planned. How ma... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0bf0ce45c9c6ba80d83823f46232d53960488ce8d96951e6a98b83dd78d78b3b | A pedestrian started walking from A to B. At the same time, a biker started cycling from B to A. When the biker met the pedestrian, he immediately picked him up, turned around, and took him to B. Then, without any delay, the biker went to A. As a result, the biker spent two and a half times longer cycling than if he ha... | [
"2"
] | [
"A pedestrian started walking from A to B. At the same time, a biker started cycling from B to A. When the biker met the pedestrian, he immediately picked him up, turned around, and took him to B. Then, without any delay, the biker went to A. As a result, the biker spent two and a half times longer cycling than if ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-f2fc119f7dc69d1258192a8f96994ca584b3878f1c2caae13af2411b8ebb3582 | A pedestrian walks along the highway at a speed of 5 km/h. Along this highway in both directions at the same speed, buses run, meeting every 5 minutes. At 12 o'clock, the pedestrian noticed that the buses met near him and, continuing to walk, he began to count those oncoming and overtaking buses. At 2 p.m., buses met n... | [
"30"
] | [
"A pedestrian walks along the highway at a speed of 5 km/h. Along this highway in both directions at the same speed, buses run, meeting every 5 minutes. At 12 o'clock, the pedestrian noticed that the buses met near him and, continuing to walk, he began to count those oncoming and overtaking buses. At 2 p.m., buses ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-765391138324601e8e20deb6978cff48c32880d693266c378c0b8e0444800099 | A pen and its ink refill together cost $\;\$1.10$. The pen costs $\;\$1$ more than the ink refill. What is the cost of the pen in dollars? | [
"1.05"
] | [
"A pen and its ink refill together cost $\\;\\$1.10$. The pen costs $\\;\\$1$ more than the ink refill. What is the cost of the pen in dollars?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-532048b0c13e81019e13f01ef2601fe87e9c379f36199ce40ddf01970fcfa32c | A pen costs $11 and a notebook costs $13. Find the number of ways a person can spend exactly $1000 to buy pens and notebooks. | [
"7"
] | [
"A pen costs $11 and a notebook costs $13. Find the number of ways a person can spend exactly $1000 to buy pens and notebooks."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-f9eb9e78158deaa60777bae04a0e2c89b3d6a2fa0b331d836dd6f810df3b9599 | A pen costs $11$ units and a notebook costs $13$ units. Find the number of ways in which a person can spend exactly $1000$ units of money to buy pens and notebooks. | [
"7"
] | [
"A pen costs $11$ units and a notebook costs $13$ units. Find the number of ways in which a person can spend exactly $1000$ units of money to buy pens and notebooks."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b8a516ccc120b35da1ef5ee3c60c70372fad27e91671b6146d258fc217daef9f | A pen costs $\mathrm{Rs.}\, 13$ and a notebook costs $\mathrm{Rs.}\, 17$. A school spends exactly $\mathrm{Rs.}\, 10000$ in the year $2017-18$ to buy $x$ pens and $y$ notebooks such that $x$ and $y$ are as close as possible (i.e., $|x-y|$ is minimum). Next year, in $2018-19$, the school spends a little more than $\math... | [
"40"
] | [
"A pen costs $\\mathrm{Rs.}\\, 13$ and a notebook costs $\\mathrm{Rs.}\\, 17$. A school spends exactly $\\mathrm{Rs.}\\, 10000$ in the year $2017-18$ to buy $x$ pens and $y$ notebooks such that $x$ and $y$ are as close as possible (i.e., $|x-y|$ is minimum). Next year, in $2018-19$, the school spends a little more ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-6d9ce1d19e1200c161f9ac7c4e690614b128e95480d9d594548f85adb1dc1ace | A pendulum has a length of 8 inches and swings through a 60-degree angle from left to right. Use right triangle trigonometry to find the straight-line distance between the left and right maximum swing positions. | [
"8"
] | [
"A pendulum has a length of 8 inches and swings through a 60-degree angle from left to right. Use right triangle trigonometry to find the straight-line distance between the left and right maximum swing positions."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 17
} | 40 | 0 | 0 |
omr-ws-bbb5cb63312809173084ab421d70eda2361757f575a7b399f8410ebc21253b37 | A pendulum is 100 cm long. From one end of the swing to the other, the straight-line distance is 20 cm. Find the angle through which the pendulum swings. | [
"11.5^\\circ"
] | [
"A pendulum is 100 cm long. From one end of the swing to the other, the straight-line distance is 20 cm. Find the angle through which the pendulum swings."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3469de1ab7e4c6788984ce892fd9500fb1e04b0e39abb6f9f9c4d6ddb64d7d03 | A pendulum with length $l$ oscillates under the influence of gravity $g$. The period of oscillation of this pendulum is given by the expression: $$T = \sqrt{\frac{l}{4g}}\left(4\sqrt{2}+\pi+2\arcsin \frac{1}{3}\right)$$ Calculate the value of $T$ when $l = 1$ meter and $g = 9.8 \text{ m/s}^2$. | [
"1.514"
] | [
"A pendulum with length $l$ oscillates under the influence of gravity $g$. The period of oscillation of this pendulum is given by the expression:\n$$T = \\sqrt{\\frac{l}{4g}}\\left(4\\sqrt{2}+\\pi+2\\arcsin \\frac{1}{3}\\right)$$\nCalculate the value of $T$ when $l = 1$ meter and $g = 9.8 \\text{ m/s}^2$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-559981cd1066319c8527717eb79c0371014dfba84795dd07d2daf57f758a7bab | A penny is thrown straight down off the top of the Empire State building at 30 feet/sec. If the Empire State building is 1440 feet high, and the equation for the height of a falling projectile at time \( t \) is \( h = -16t^2 + vt + d \), where \( v \) is the initial velocity and \( d \) is the initial height, how long... | [
"8.6"
] | [
"A penny is thrown straight down off the top of the Empire State building at 30 feet/sec. If the Empire State building is 1440 feet high, and the equation for the height of a falling projectile at time \\( t \\) is \\( h = -16t^2 + vt + d \\), where \\( v \\) is the initial velocity and \\( d \\) is the initial hei... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 19
} | 41 | 0 | 0 |
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