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-14763523122bf9753204c8846a93ed3cd3355b52424c3e1348356931043d5111 | A knight starts at the origin $(0,0)$ on a chessboard and moves by changing its position by $2$ units in one axis-parallel direction and $1$ unit in an orthogonal direction. The number $t$ is updated from $x$ to $ax + b$ when the knight lands at the point $(a, b)$. Find the minimum number of moves required for the knig... | [
"3"
] | [
"A knight starts at the origin $(0,0)$ on a chessboard and moves by changing its position by $2$ units in one axis-parallel direction and $1$ unit in an orthogonal direction. The number $t$ is updated from $x$ to $ax + b$ when the knight lands at the point $(a, b)$. Find the minimum number of moves required for the... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-28d0cf104a2aeba433dc55b1e6f8780d04e2b8648ed9103c24d42b196dcbe22f | A knight starts at the origin on a chessboard. Each step, it moves to one of the 8 possible positions chosen uniformly and at random. What is the expected value of the square of the distance of the knight from the origin after 1000 moves? | [
"5000"
] | [
"A knight starts at the origin on a chessboard. Each step, it moves to one of the 8 possible positions chosen uniformly and at random. What is the expected value of the square of the distance of the knight from the origin after 1000 moves?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-e7676f87f3804fb0a0d790f76e154a7cdc6c5bc6332ac1b4677210e73f3163d3 | A knight starts in the bottom-left corner of an otherwise-empty chessboard. Every second, the knight moves to a square chosen uniformly at random among the legal moves for the knight. What is the expected number of seconds until the knight first lands in the top right corner of the chessboard? | [
"168",
"\\(\\frac{4464173644}{20983073}\\)"
] | [
"A knight starts in the bottom-left corner of an otherwise-empty chessboard. Every second, the knight moves to a square chosen uniformly at random among the legal moves for the knight. What is the expected number of seconds until the knight first lands in the top right corner of the chessboard?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-9262168e549ad9b96b9ff255148b5b142f95f6a5603dfe89d6a8200ae4e29dc3 | A knight starts in the corner of an $8 \times 8$ chessboard and makes a move selected from his valid moves with equal probability. What is the expected number of steps for the knight to return to his starting place? | [
"168"
] | [
"A knight starts in the corner of an $8 \\times 8$ chessboard and makes a move selected from his valid moves with equal probability. What is the expected number of steps for the knight to return to his starting place?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-8b29f3fab726f01275152775b96ebee60a0c7aad9f8a98699e3b058afb197488 | A laboratory has to order a certain number of identical spherical flasks with a total capacity of 100 liters. The cost of one flask consists of the cost of labor, which is proportional to the square of the surface area of the flask, and the cost of the material, which is proportional to the surface area. A flask of cap... | [
"No"
] | [
"A laboratory has to order a certain number of identical spherical flasks with a total capacity of 100 liters. The cost of one flask consists of the cost of labor, which is proportional to the square of the surface area of the flask, and the cost of the material, which is proportional to the surface area. A flask o... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a59deb3dcdbee7311fd213ae9b88c969893f546843da2518096a9c8d278ee44c | A laboratory test for the detection of a certain disease gives a positive result 5 percent of the time for people who do not have the disease. The test gives a negative result 0.3 percent of the time for people who have the disease. Large scale studies have shown that the disease occurs in about 2 percent of the popula... | [
"72.06%"
] | [
"A laboratory test for the detection of a certain disease gives a positive result 5 percent of the time for people who do not have the disease. The test gives a negative result 0.3 percent of the time for people who have the disease. Large scale studies have shown that the disease occurs in about 2 percent of the p... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b969d454180f195e4171c7033cc3c4a5791647dad0743e53abc1afd3fb800454 | A laboratory test for the detection of a certain disease gives a positive result 5 percent of the time for people who do not have the disease. The test gives a negative result 0.3 percent of the time for people who have the disease. Large scale studies have shown that the disease occurs in about 2 percent of the popula... | [
"0.0684"
] | [
"A laboratory test for the detection of a certain disease gives a positive result 5 percent of the time for people who do not have the disease. The test gives a negative result 0.3 percent of the time for people who have the disease. Large scale studies have shown that the disease occurs in about 2 percent of the p... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-0e7783bbac803fa502f7165f597317b098800a130e0c17299736ba7632a90792 | A labyrinth consists of 2024 caves and 2023 non-intersecting bidirectional corridors, each connecting exactly two caves, such that each pair of caves is connected through some sequence of corridors. Initially, Erik is standing in a corridor connecting two caves. In a move, Erik can walk through one of the caves to anot... | [
"1"
] | [
"A labyrinth consists of 2024 caves and 2023 non-intersecting bidirectional corridors, each connecting exactly two caves, such that each pair of caves is connected through some sequence of corridors. Initially, Erik is standing in a corridor connecting two caves. In a move, Erik can walk through one of the caves to... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-ca66bf4895b1e102bf979ad8824fd3ac4df0b0b86bc59cfe759f4ef8031f2d1e | A ladder $AB$, 25 feet long, leans against a vertical wall. The lower end $A$ is 7 feet from the bottom of the wall and is being moved away from the wall at a rate of 2 feet per second. Determine the rate at which the upper end $B$ is moving towards the bottom of the wall in inches per second. | [
"7"
] | [
"A ladder $AB$, 25 feet long, leans against a vertical wall. The lower end $A$ is 7 feet from the bottom of the wall and is being moved away from the wall at a rate of 2 feet per second. Determine the rate at which the upper end $B$ is moving towards the bottom of the wall in inches per second."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 2,
"tir": 7
} | 24 | 0 | 0 |
omr-ws-2ca4c95929fc1db4c345a545e0aa069624e8e680a094660b3bcb49a34d8c7556 | A ladder 10 feet long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a speed of 2 feet per second, how fast is the angle between the top of the ladder and the wall changing when the angle is \(\frac{\pi}{4}\) radians? | [
"\\(\\frac{\\sqrt{2}}{5}\\)"
] | [
"A ladder 10 feet long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a speed of 2 feet per second, how fast is the angle between the top of the ladder and the wall changing when the angle is \\(\\frac{\\pi}{4}\\) radians?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-219437564ca7106a8f90a02827c86752682b3ca34a686536ff31a8fad389df3d | A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1.5 ft/s, how fast is the angle between the ladder and the ground changing when the bottom of the ladder is 8 ft from the wall? | [
"\\(-\\frac{1}{4}\\)"
] | [
"A ladder 10 ft long rests against a vertical wall. If the bottom of the ladder slides away from the wall at a rate of 1.5 ft/s, how fast is the angle between the ladder and the ground changing when the bottom of the ladder is 8 ft from the wall?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-9cc5cfd39f823e9d619d63e1b290fed18d3b43443436b944df5412d4f6d3eda1 | A ladder 10 meters long rests on a flat surface and leans against a vertical wall. A cubical box of side length 3 meters can exactly fit between the surface, the wall, and the ladder. Find to 3 significant figures how far up the wall the ladder reaches. | [
"4.52 or 8.92"
] | [
"A ladder 10 meters long rests on a flat surface and leans against a vertical wall. A cubical box of side length 3 meters can exactly fit between the surface, the wall, and the ladder. Find to 3 significant figures how far up the wall the ladder reaches."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 28,
"genselect": 5,
"tir": 21
} | 54 | 0 | 0 |
omr-ws-3e41071feb4c11a0db548fe16a308aefcc5091c758c7d6d746bab17bdd7b1a5b | A ladder 25 feet long is leaning against the wall of a building. The base of the ladder is pulled away from the wall at a rate of 4 feet per second. At what rate is the angle between the ladder and the wall changing when the base of the ladder is 15 feet from the wall? | [
"\\dfrac{1}{5}"
] | [
"A ladder 25 feet long is leaning against the wall of a building. The base of the ladder is pulled away from the wall at a rate of 4 feet per second. At what rate is the angle between the ladder and the wall changing when the base of the ladder is 15 feet from the wall?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-5eafadde55c5332f8c4483d7f3d8f25d72e25c9e99a53ba17cd86ccea4595744 | A ladder 50 feet long is leaning against a wall. If the foot of the ladder is being pulled away from the wall at a rate of 3 ft/min, at what point is the top of the ladder descending at a rate of 4 ft/min? | [
"\\( (x, y) = (40, 30) \\)"
] | [
"A ladder 50 feet long is leaning against a wall. If the foot of the ladder is being pulled away from the wall at a rate of 3 ft/min, at what point is the top of the ladder descending at a rate of 4 ft/min?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 5,
"tir": 11
} | 23 | 0 | 0 |
omr-ws-5545bf57d250d3e61813b328422acc2539731bf4067af911a20379cec580d020 | A ladder is formed by removing some consecutive unit squares of a $10 \times 10$ chessboard such that for each $k$-th row ($k \in \{1, 2, \dots, 10\}$), the leftmost $k-1$ unit squares are removed. How many rectangles formed by composition of unit squares does the ladder have? | [
"715"
] | [
"A ladder is formed by removing some consecutive unit squares of a $10 \\times 10$ chessboard such that for each $k$-th row ($k \\in \\{1, 2, \\dots, 10\\}$), the leftmost $k-1$ unit squares are removed. How many rectangles formed by composition of unit squares does the ladder have?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 16
} | 63 | 0 | 0 |
omr-ws-41e83b10cae974083040aa998596973c6de77762087f5dd5f592fdf017b8e3ca | A ladder is leaning against a house with its lower end 15 feet from the house. When the lower end is pulled 9 feet farther from the house, the upper end slides 13 feet down. How long is the ladder (in feet)? | [
"25",
"25 ft"
] | [
"A ladder is leaning against a house with its lower end 15 feet from the house. When the lower end is pulled 9 feet farther from the house, the upper end slides 13 feet down. How long is the ladder (in feet)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 1
} | 33 | 0 | 0 |
omr-ws-dd3f560ba9296ba5eec633e6ca597b99e2b405fcaf334353b6d22e2426d9c5e8 | A ladder is resting against a wall with its top and base touching the wall and the floor, respectively. If the base is moved 4 feet away from the wall, the top of the ladder moves down the wall by one-fourth the length of the ladder. What is the exact length of the ladder in feet? | [
"\\(\\frac{16 \\sqrt{7}}{7}\\)"
] | [
"A ladder is resting against a wall with its top and base touching the wall and the floor, respectively. If the base is moved 4 feet away from the wall, the top of the ladder moves down the wall by one-fourth the length of the ladder. What is the exact length of the ladder in feet?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-40a7057b03dc6852ba6908511f56f93872c52a7b43706a429a8a929ce5f9225e | A ladder of length $\ell$ rests against a vertical wall. There is a rung on the ladder which has the same distance $d$ from both the wall and the ground. Find the height $h$ from the ground that the ladder reaches up the wall, expressed explicitly in terms of $\ell$ and $d$. | [
"\\(\\frac{d}{2}\\left(1 + \\sqrt{1+\\left(\\frac{\\ell}{d}\\right)^2} \\pm \\sqrt{\\left(\\frac{\\ell}{d}\\right)^2-2-2\\sqrt{1+\\left(\\frac{\\ell}{d}\\right)^2}}\\right)\\)"
] | [
"A ladder of length $\\ell$ rests against a vertical wall. There is a rung on the ladder which has the same distance $d$ from both the wall and the ground. Find the height $h$ from the ground that the ladder reaches up the wall, expressed explicitly in terms of $\\ell$ and $d$."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-c7625ca186605f6773b490164c4a0f907f6fc0c8b8f90bab2e64f0c92aa94a41 | A ladder of length 1 is fixed at both ends, one on the wall and the other on the ground, forming a 45-degree angle. When the end on the wall slides, find the area swept by the ladder. | [
"\\(\\frac{\\pi}{4}\\)"
] | [
"A ladder of length 1 is fixed at both ends, one on the wall and the other on the ground, forming a 45-degree angle. When the end on the wall slides, find the area swept by the ladder."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-477c8cdb136a47c05c98c0f93300f99b47d25c9e4f8b456bb4670d19d15ddcd0 | A ladder of length \(5 \text{ m}\) is leaning against a vertical wall at an angle of \(60^\circ\) to the horizontal. The foot of the ladder moves in by \(50 \text{ cm}\). By how much does the top of the ladder move up the wall? | [
"25.24 cm"
] | [
"A ladder of length \\(5 \\text{ m}\\) is leaning against a vertical wall at an angle of \\(60^\\circ\\) to the horizontal. The foot of the ladder moves in by \\(50 \\text{ cm}\\). By how much does the top of the ladder move up the wall?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 3,
"tir": 27
} | 36 | 0 | 0 |
omr-ws-46deff452380b5a6faf0c9b4696dbb0dc75ae2cd2b2eb14362123aeb805ecd66 | A ladder or pole of finite length is supported by a wall (y-axis) and makes an angle $\alpha$ ($\alpha < \frac{\pi}{2}$) with the ground (x-axis). A cat is sitting at the top of the pole. As the pole falls to the ground, the cat descends such that it reaches the end of the pole exactly when the pole touches the ground.... | [
"x^{2/3} + y^{2/3} = L^{2/3}"
] | [
"A ladder or pole of finite length is supported by a wall (y-axis) and makes an angle $\\alpha$ ($\\alpha < \\frac{\\pi}{2}$) with the ground (x-axis). A cat is sitting at the top of the pole. As the pole falls to the ground, the cat descends such that it reaches the end of the pole exactly when the pole touches th... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-18ffb2cd2583470b161cade75ee2d3ec1f7272be9c1d349a9a606ddc3904c3c0 | A ladder with a length of 13 feet is laid against a wall, perpendicular to the floor. If the tip of the ladder slides down the wall at a rate of B feet per minute, find the rate at which the other tip of the ladder slides away from the wall when it is 5 feet away from the wall. | [
"\\(\\frac{12B}{5}\\)"
] | [
"A ladder with a length of 13 feet is laid against a wall, perpendicular to the floor. If the tip of the ladder slides down the wall at a rate of B feet per minute, find the rate at which the other tip of the ladder slides away from the wall when it is 5 feet away from the wall."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-394ec114676cd5b68952bcb1a72d28ade735c046a889e85d12797c32916002c2 | A lady bought \( x \) flowers for \( y \) dollars, where \( x \) and \( y \) are positive integers. The manager offered to sell her \( x + 10 \) flowers for $2, which would save her $0.80 per dozen flowers bought. Determine the values of \( x \) and \( y \). | [
"\\( x = 50 \\), \\( y = 10 \\)"
] | [
"A lady bought \\( x \\) flowers for \\( y \\) dollars, where \\( x \\) and \\( y \\) are positive integers. The manager offered to sell her \\( x + 10 \\) flowers for $2, which would save her $0.80 per dozen flowers bought. Determine the values of \\( x \\) and \\( y \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bec1cd6be925cbe1f031605a571298c48526213aabfc7ed01c23f1778463d7f3 | A lady has an unlimited supply of quarters, dimes, nickels, and pennies. Considering only the value of the coins involved, find the number of distinct ways the lady can make exactly 50 cents. | [
"49"
] | [
"A lady has an unlimited supply of quarters, dimes, nickels, and pennies. Considering only the value of the coins involved, find the number of distinct ways the lady can make exactly 50 cents."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-a7d0a08d106f83c914786ad5581c11ed4b7854cead8ebcd2302a785233a884f9 | A ladybug is walking at random on a hexagon and starts at vertex A. Every minute, she moves to one of the two adjacent vertices chosen at random. What is the probability that, after 10 minutes, she is back at vertex A? | [
"\\(\\frac{171}{512}\\)"
] | [
"A ladybug is walking at random on a hexagon and starts at vertex A. Every minute, she moves to one of the two adjacent vertices chosen at random. What is the probability that, after 10 minutes, she is back at vertex A?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-cb43441f08bf859faaa52c1823cb042e62021719272228936995aced0c887d25 | A ladybug starts at \( (0,0) \) and moves either one unit right or one unit up every second to reach \( (10,10) \) in 20 seconds. It must avoid points where both coordinates are odd. How many paths can it take to reach \( (10,10) \) without going through an acid puddle? | [
"252"
] | [
"A ladybug starts at \\( (0,0) \\) and moves either one unit right or one unit up every second to reach \\( (10,10) \\) in 20 seconds. It must avoid points where both coordinates are odd. How many paths can it take to reach \\( (10,10) \\) without going through an acid puddle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 7,
"tir": 22
} | 41 | 0 | 0 |
omr-ws-00d34a5cc00503b70ed73b21a29ba2a75d1960ea4b1e13ccd899288a68d0ad3b | A lake containing 140 million gallons of fresh water has a stream flowing through it at a constant rate of 7 million gallons per day. Pollutants are discharged into the stream from 6 a.m. to 6 p.m. each day, with a concentration of 3 mg/gal (3×10^(−6) kg/gal). Assume the lake is well-stirred and the manufacturer operat... | [
"q' = 7c_i - \\frac{1}{20}q, \\quad q(0) = 0"
] | [
"A lake containing 140 million gallons of fresh water has a stream flowing through it at a constant rate of 7 million gallons per day. Pollutants are discharged into the stream from 6 a.m. to 6 p.m. each day, with a concentration of 3 mg/gal (3×10^(−6) kg/gal). Assume the lake is well-stirred and the manufacturer o... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-97067229b72e15d1ca17647a04d05a6f8a250cf36c82a3c09b2e8ae08f73fa25 | A lake is in the shape of trapezoid \(ABCD\) with \(\overline{AB} \parallel \overline{CD}\). Point \(A\) has an equal distance of 5 kilometers from points \(B\), \(C\), and \(D\). Kevin and Jason each take a canoe from point \(A\) on the shortest route to sides \(\overline{BC}\) and \(\overline{CD}\) respectively. If t... | [
"\\(\\frac{94}{5}\\)"
] | [
"A lake is in the shape of trapezoid \\(ABCD\\) with \\(\\overline{AB} \\parallel \\overline{CD}\\). Point \\(A\\) has an equal distance of 5 kilometers from points \\(B\\), \\(C\\), and \\(D\\). Kevin and Jason each take a canoe from point \\(A\\) on the shortest route to sides \\(\\overline{BC}\\) and \\(\\overli... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 11
} | 32 | 0 | 0 |
omr-ws-5d9d503b58df235185a55271cabffb20dc755dd28dedceeefc816115a8b63ae7 | A lame rook lies on a $9 \times 9$ chessboard. It can move one cell horizontally or vertically. The rook made $n$ moves, visited each cell at most once, and did not make two moves consecutively in the same direction. What is the largest possible value of $n$? | [
"67"
] | [
"A lame rook lies on a $9 \\times 9$ chessboard. It can move one cell horizontally or vertically. The rook made $n$ moves, visited each cell at most once, and did not make two moves consecutively in the same direction. What is the largest possible value of $n$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-337feff83031934912b5a8309c26208283e03dbea7ccac29a04602dbde6524e0 | A lame rook moves through an $n \times n$ board ($n \ge 2$), visiting each cell exactly once and returning to the starting cell. This path forms a closed polyline $L$ of length $n^2$. Let $P(n)$ be the minimum length of the longer segment of $L$ when $L$ is divided by the line segment connecting the centers of any two ... | [
"32"
] | [
"A lame rook moves through an $n \\times n$ board ($n \\ge 2$), visiting each cell exactly once and returning to the starting cell. This path forms a closed polyline $L$ of length $n^2$. Let $P(n)$ be the minimum length of the longer segment of $L$ when $L$ is divided by the line segment connecting the centers of a... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fe59027962f3837107c155f62900fff306a158b929ff98e2cecbc4f1506f3bad | A lamina occupies the part of the disk \( x^2 + y^2 \leq 4 \) in the first quadrant, and the density at each point is given by the function \( \rho(x, y) = 3(x^2 + y^2) \). A. What is the total mass? B. What is the moment about the x-axis? C. What is the moment about the y-axis? D. Where is the center of mass? E. What ... | [
"\\( 6\\pi \\) (for the total mass)"
] | [
"A lamina occupies the part of the disk \\( x^2 + y^2 \\leq 4 \\) in the first quadrant, and the density at each point is given by the function \\( \\rho(x, y) = 3(x^2 + y^2) \\).\n\nA. What is the total mass?\n\nB. What is the moment about the x-axis?\n\nC. What is the moment about the y-axis?\n\nD. Where is the c... | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 3
} | 3 | 0 | 0 |
omr-ws-096d1e58cff424d8628294da6737417257c71079ab3b337b69ec98c5ff26d97c | A lamina occupies the part of the disk \(x^2 + y^2 \leq 49\) in the first quadrant. Find its center of mass if the density at any point is proportional to its distance from the x-axis. | [
"\\(\\left( \\frac{21}{8}, \\frac{21\\pi}{16} \\right)\\)"
] | [
"A lamina occupies the part of the disk \\(x^2 + y^2 \\leq 49\\) in the first quadrant. Find its center of mass if the density at any point is proportional to its distance from the x-axis."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-946bb13a17049f11c9a40801a34a1536906fbd81176c976c35bd93d095425cd8 | A lamp is initially turned on at \( t = 0 \) seconds. Then, after one second, the lamp is turned off. After half the previous time it was on, the lamp is turned on again. Then, after half the previous time it was off, the lamp is turned on again. If a random time \( 0 < t < 2 \) is chosen, what is the probability that ... | [
"\\(\\frac{2}{3}\\)"
] | [
"A lamp is initially turned on at \\( t = 0 \\) seconds. Then, after one second, the lamp is turned off. After half the previous time it was on, the lamp is turned on again. Then, after half the previous time it was off, the lamp is turned on again. If a random time \\( 0 < t < 2 \\) is chosen, what is the probabil... | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-236f911e1d0d275fd5450e10439f6015c145885a9f1ce533763cae1de0d5c8f6 | A lamp post stands by the side of a straight road, with the nearest point A on the road being at a distance $a$ from the lamp post. A man of height $b$ is walking along the road. When he is at point B, the length of his shadow is $\frac{3}{2}$ times the length of the shadow when he is at point C, the midpoint of AB. If... | [
"\\( b \\left(1 + \\frac{2\\sqrt{3}}{\\sqrt{7}}\\right) \\)"
] | [
"A lamp post stands by the side of a straight road, with the nearest point A on the road being at a distance $a$ from the lamp post. A man of height $b$ is walking along the road. When he is at point B, the length of his shadow is $\\frac{3}{2}$ times the length of the shadow when he is at point C, the midpoint of ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5dc3304d0d2cc77337d77ea84faee6c532102e00400048794c8f0e7e99e4e63d | A lamp-post, 4.5m high, is 4.2m away from a straight footpath. A boy, 1.5m tall, walks along the path at the rate of 1.8m/s. Find the rate at which his shadow is shortening when he has 14.4 m more to cover in order to arrive at the point on the path nearest to the lamp-post. | [
"-0.9 m/s"
] | [
"A lamp-post, 4.5m high, is 4.2m away from a straight footpath. A boy, 1.5m tall, walks along the path at the rate of 1.8m/s. Find the rate at which his shadow is shortening when he has 14.4 m more to cover in order to arrive at the point on the path nearest to the lamp-post."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 2,
"tir": 8
} | 14 | 0 | 0 |
omr-ws-8690faf399ac7be5554d1885c901438fba91e300de756ece6e14f496e06ec63b | A lamp-post, 4.5m high, is 4.2m away from a straight footpath. A boy, 1.5m tall, walks along the path at the rate of 1.8m/s. Find the rate at which his shadow is shortening when he has 14.4m more to cover in order to arrive at the point on the path nearest to the lamp-post. | [
"\\dfrac{3}{8}"
] | [
"A lamp-post, 4.5m high, is 4.2m away from a straight footpath. A boy, 1.5m tall, walks along the path at the rate of 1.8m/s. Find the rate at which his shadow is shortening when he has 14.4m more to cover in order to arrive at the point on the path nearest to the lamp-post."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-d16f8547685fbdb8c148802ade87c9d6c170116570d880781a8b1b2b47eeab3f | A lampshade is made in the form of the lateral surface of the frustum of a right circular cone. The height of the frustum is \(3\sqrt{3}\) inches, its top diameter is 6 inches, and its bottom diameter is 12 inches. A bug is at the bottom of the lampshade and there is a glob of honey on the top edge of the lampshade at ... | [
"\\(6\\sqrt{3} + \\pi\\)"
] | [
"A lampshade is made in the form of the lateral surface of the frustum of a right circular cone. The height of the frustum is \\(3\\sqrt{3}\\) inches, its top diameter is 6 inches, and its bottom diameter is 12 inches. A bug is at the bottom of the lampshade and there is a glob of honey on the top edge of the lamps... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1405768d4075715eb15ec155fe833403ef9b20712499f7c654d33f1899b154ad | A lampshade is part of the surface of a right circular cone whose axis is vertical. Its upper and lower edges are two horizontal circles. Two points are selected on the upper smaller circle and four points on the lower larger circle. Each of these six points has three of the others that are its nearest neighbors at a d... | [
"\\dfrac{3d^2}{\\theta}"
] | [
"A lampshade is part of the surface of a right circular cone whose axis is vertical. Its upper and lower edges are two horizontal circles. Two points are selected on the upper smaller circle and four points on the lower larger circle. Each of these six points has three of the others that are its nearest neighbors a... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 3
} | 4 | 0 | 0 |
omr-ws-c03436eaa7d0c07b30d34ae60dfe68d9a991ac976785e8f2b16a2c027d9699ba | A landing stage $B$ is $a$ km up the river from $A$. A motor boat makes trips going from $A$ to $B$ and returning to $A$ without stopping in $T$ hours. Find the speed of the boat in still water and the speed of the current if it is known that once, when returning from $B$ to $A$, the motor boat had an accident at a dis... | [
"\\[ v=\\frac{1}{2}\\left[\\frac{2a+b}{T-T_0}+\\frac{a(2a+b)}{T(a+b)+T_0 a}\\right] \\]\n\\[ w=\\frac{1}{2}\\left[\\frac{2a+b}{T-T_0}-\\frac{a(2a+b)}{T(a+b)+T_0 a}\\right] \\]"
] | [
"A landing stage $B$ is $a$ km up the river from $A$. A motor boat makes trips going from $A$ to $B$ and returning to $A$ without stopping in $T$ hours. Find the speed of the boat in still water and the speed of the current if it is known that once, when returning from $B$ to $A$, the motor boat had an accident at ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-556b89fb59d31963dd05fe0aeee31b142193ced0a9d707474fa933e8c53475cc | A landowner has 3000 yards of fencing to enclose a rectangular field bordering a river. The field is to be divided into 2 regions by a fence parallel to the short sides of the field. No fencing will be required along the side by the river. What are the dimensions of the largest field that may be enclosed with the avail... | [
"\\( x = 750 \\) yards, \\( y = 500 \\) yards"
] | [
"A landowner has 3000 yards of fencing to enclose a rectangular field bordering a river. The field is to be divided into 2 regions by a fence parallel to the short sides of the field. No fencing will be required along the side by the river. What are the dimensions of the largest field that may be enclosed with the ... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 15
} | 15 | 0 | 0 |
omr-ws-db1a4a168cdcdfa1d20532767220af8d69bc636c9a79d21b2044cf313fab5cbc | A landscape architect plans to enclose a 3000 square foot rectangular region in a botanical garden. She will use shrubs costing $25 per foot along three sides and fencing costing $10 per foot along the fourth side. Find the minimum cost. | [
"550\\sqrt{210}"
] | [
"A landscape architect plans to enclose a 3000 square foot rectangular region in a botanical garden. She will use shrubs costing $25 per foot along three sides and fencing costing $10 per foot along the fourth side. Find the minimum cost."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-13afffa4fd6c7568f8cb44d63b626e15a14d66ddc33f09d8467dbf346b6ee872 | A landscape gardener uses 36 paving bricks, each measuring 22cm by 11cm, to form a 'circular' flower bed. The gardener wishes to eliminate the V-shape by cutting a portion from the corners of each brick so that the face of each brick is in the shape of an isosceles trapezium ABCD. What is the length of the side AB? | [
"22"
] | [
"A landscape gardener uses 36 paving bricks, each measuring 22cm by 11cm, to form a 'circular' flower bed. The gardener wishes to eliminate the V-shape by cutting a portion from the corners of each brick so that the face of each brick is in the shape of an isosceles trapezium ABCD. What is the length of the side AB... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-b47bb23365e7065bee3f3add040f4f8288bed709497f5c1b9ebd585bf045955b | A landscape gardener uses 36 paving bricks, each measuring 22cm by 11cm, to form a 'circular' flower bed. What is the radius of the circle? | [
"126.21"
] | [
"A landscape gardener uses 36 paving bricks, each measuring 22cm by 11cm, to form a 'circular' flower bed. What is the radius of the circle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 18,
"genselect": 2,
"tir": 21
} | 41 | 0 | 0 |
omr-ws-d8c280a881af1e539a4e591ee04e4da43c8a846cbfcdaf0650cf9b5118cbb74b | A lane runs at a right angle out into a road $18$ feet wide. How many feet wide is the lane if it is just possible to carry a pole $45$ feet long from the road into the lane, keeping it horizontal? | [
"9\\sqrt{21}"
] | [
"A lane runs at a right angle out into a road $18$ feet wide. How many feet wide is the lane if it is just possible to carry a pole $45$ feet long from the road into the lane, keeping it horizontal?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-5379d7d6eb89066bfe62578ff39cec80563e247e080c1cebca26547016f55c2a | A lane runs perpendicular to a 64-feet-wide road. If it is just possible to carry a 125-feet-long pole from the road into the lane, keeping it horizontal, what is the minimum width of the lane (in feet)? | [
"27"
] | [
"A lane runs perpendicular to a 64-feet-wide road. If it is just possible to carry a 125-feet-long pole from the road into the lane, keeping it horizontal, what is the minimum width of the lane (in feet)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-08e4e839768a437ad05ddb7ea6bb0e283f17e61bf0c0b4aaccf4a4821505e1df | A lane runs perpendicular to a road that is 64 feet wide. If it is just possible to carry a 125-foot long pole horizontally from the road into the lane, find the maximum width of the lane. | [
"27"
] | [
"A lane runs perpendicular to a road that is 64 feet wide. If it is just possible to carry a 125-foot long pole horizontally from the road into the lane, find the maximum width of the lane."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-74b89f4a00a3cbfd4196806d116df847b6dda7c33c6feab0cd885eb4052863a3 | A lantern is placed on the ground 100 feet away from a wall. A man 6 feet tall is walking at a speed of 10 feet/sec from the lantern towards the nearest point on the wall. When he is midway between the lantern and the wall, find the rate of change in the length of his shadow. | [
"-2.4"
] | [
"A lantern is placed on the ground 100 feet away from a wall. A man 6 feet tall is walking at a speed of 10 feet/sec from the lantern towards the nearest point on the wall. When he is midway between the lantern and the wall, find the rate of change in the length of his shadow."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-cf4ab09069ea4b1c77f3ed779a10d552c4c465ec6bf2a9b90bf8577475fe7835 | A large 3x3 cube is formed by stacking 27 unit cubes. A plane is perpendicular to one of the internal diagonals of the large cube and bisects that diagonal. What is the number of unit cubes that the plane intersects? | [
"19"
] | [
"A large 3x3 cube is formed by stacking 27 unit cubes. A plane is perpendicular to one of the internal diagonals of the large cube and bisects that diagonal. What is the number of unit cubes that the plane intersects?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 16
} | 39 | 0 | 0 |
omr-ws-aeb51ee5a99da3046508fbf00862a798160c03473c9df8e55dc1f9ad0e16c538 | A large block with dimensions $n$ by $11$ by $10$ is made up of unit cubes and one $2$ by $1$ by $1$ block. There are exactly $2362$ positions in which the $2$ by $1$ by $1$ block can be placed. What is the value of $n$? | [
"8"
] | [
"A large block with dimensions $n$ by $11$ by $10$ is made up of unit cubes and one $2$ by $1$ by $1$ block. There are exactly $2362$ positions in which the $2$ by $1$ by $1$ block can be placed. What is the value of $n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6e7a815b519344eff95ffd7c08e04f3089819e364ea3be6d193503343a83e347 | A large candle is $119$ centimeters tall. It is designed to burn down more quickly when it is first lit and more slowly as it approaches its bottom. Specifically, the candle takes $10$ seconds to burn down the first centimeter from the top, $20$ seconds to burn down the second centimeter, and $10k$ seconds to burn down... | [
"350"
] | [
"A large candle is $119$ centimeters tall. It is designed to burn down more quickly when it is first lit and more slowly as it approaches its bottom. Specifically, the candle takes $10$ seconds to burn down the first centimeter from the top, $20$ seconds to burn down the second centimeter, and $10k$ seconds to bu... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-0d5458f920fc9897c6c5bac5ef91580cd6f016b146fc88bdf4870e52e02ac3d8 | A large candle is $119$ centimeters tall. It is designed to burn down more quickly when it is first lit and more slowly as it approaches its bottom. Specifically, the candle takes $10$ seconds to burn down the first centimeter from the top, $20$ seconds to burn down the second centimeter, and $10k$ seconds to burn down... | [
"35"
] | [
"A large candle is $119$ centimeters tall. It is designed to burn down more quickly when it is first lit and more slowly as it approaches its bottom. Specifically, the candle takes $10$ seconds to burn down the first centimeter from the top, $20$ seconds to burn down the second centimeter, and $10k$ seconds to burn... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9d22b079f6a81e83215230cc983324b6e0dbb2b612dce5c2ee6723bd46e8fe2e | A large circle is just large enough to contain in its interior three non-overlapping, mutually tangent smaller circles, each of which has an area of one square centimeter. It is also just large enough to contain in its interior four non-overlapping smaller circles, each of the same area. What is the combined area of th... | [
"\\dfrac{4(7 + 4\\sqrt{3})}{3(3 + 2\\sqrt{2})}"
] | [
"A large circle is just large enough to contain in its interior three non-overlapping, mutually tangent smaller circles, each of which has an area of one square centimeter. It is also just large enough to contain in its interior four non-overlapping smaller circles, each of the same area. What is the combined area ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-d90cee917d1e991f78a2c004b348411fa5a49885d1bd3e718cbd0301c6312418 | A large circle is tangent to each of the sides of a square with side length 1. A small circle is tangent to the large circle and two sides of the square. Find the radius of the smaller circle. | [
"\\( \\frac{3}{2} - \\sqrt{2} \\)"
] | [
"A large circle is tangent to each of the sides of a square with side length 1. A small circle is tangent to the large circle and two sides of the square. Find the radius of the smaller circle."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-b5d56d502f07f483def5066573257e16a872b62eb924fcd81b9b57abedbd363d | A large circular metal plate has 12 equal smaller holes drilled out along its circumference. A robot arm will randomly place 5 test tubes onto the plate. What is the probability that none of the 5 test tubes will be adjacent to each other? | [
"\\(\\frac{1}{22}\\)"
] | [
"A large circular metal plate has 12 equal smaller holes drilled out along its circumference. A robot arm will randomly place 5 test tubes onto the plate. What is the probability that none of the 5 test tubes will be adjacent to each other?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-1f5de399c491ce7dca66a9a4aae83ca53df47fe649086eead879162042625ac4 | A large cube consisting of $n^3$ small cubes sits on the table. We wish to remove small cubes one by one until no small cubes are left, but we can only remove cubes whose upper face is visible. Find the smallest $n$ such that we can do this in more than $10^{18}$ ways. | [
"3"
] | [
"A large cube consisting of $n^3$ small cubes sits on the table. We wish to remove small cubes one by one until no small cubes are left, but we can only remove cubes whose upper face is visible. Find the smallest $n$ such that we can do this in more than $10^{18}$ ways."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 45,
"genselect": 7,
"tir": 1
} | 53 | 0 | 0 |
omr-ws-15ba8dc81516795ad77b62f68696d5ba9f46e4b6b91f81898ef504cc3f9f8219 | A large cube is dipped in green paint and then divided into 216 smaller, congruent cubes. One of the smaller cubes is then randomly selected. What is the probability, expressed as a common fraction, that the cube selected will be painted on an odd number of sides? | [
"\\(\\frac{13}{27}\\)"
] | [
"A large cube is dipped in green paint and then divided into 216 smaller, congruent cubes. One of the smaller cubes is then randomly selected. What is the probability, expressed as a common fraction, that the cube selected will be painted on an odd number of sides?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-990cb047395a22502e593f34f42d6c5102293f209b4016954819a6fd0bf76c44 | A large cube is dipped into red paint and then divided into 512 smaller congruent cubes. What is the probability that a randomly selected smaller cube will have at least 25% of its area painted red? | [
"\\(\\frac{5}{32}\\)"
] | [
"A large cube is dipped into red paint and then divided into 512 smaller congruent cubes. What is the probability that a randomly selected smaller cube will have at least 25% of its area painted red?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-b87a1e0fad69bca0cfaaddea8ddda35cddb9b6bb27789c05412c3c188d440f25 | A large cube is formed by stacking 125 unit cubes. A plane is perpendicular to one of the internal diagonals of the large cube and bisects that diagonal. What is the number of unit cubes that the plane intersects? | [
"55"
] | [
"A large cube is formed by stacking 125 unit cubes. A plane is perpendicular to one of the internal diagonals of the large cube and bisects that diagonal. What is the number of unit cubes that the plane intersects?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-738a82db7f60b33d506c81f96fc501da144452fb110ecc6962f789aa3b43b619 | A large cube is formed by stacking 27 unit cubes. A plane is perpendicular to one of the internal diagonals of the large cube and bisects that diagonal. How many unit cubes does the plane intersect? | [
"19"
] | [
"A large cube is formed by stacking 27 unit cubes. A plane is perpendicular to one of the internal diagonals of the large cube and bisects that diagonal. How many unit cubes does the plane intersect?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 27,
"genselect": 15,
"tir": 17
} | 59 | 0 | 0 |
omr-ws-e6f3b17b9249a767206edb7e4cb78d3ff90fd6748365cca639998629d2b2d349 | A large cube with dimensions $2011 \times 2011 \times 2011$ is composed of smaller identical cubes. A plane perpendicular to the cube's diagonal and passing through its center intersects the cube. How many of the smaller cubes does this plane intersect? | [
"\\( 2011 \\)"
] | [
"A large cube with dimensions $2011 \\times 2011 \\times 2011$ is composed of smaller identical cubes. A plane perpendicular to the cube's diagonal and passing through its center intersects the cube. How many of the smaller cubes does this plane intersect?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c93146e6fd15e3056c2b73c03f64adc2ebab2ffa5ca94e02608b31b972a51af9 | A large cube with dimensions 2011 x 2011 x 2011 is composed of smaller identical cubes. A plane perpendicular to the cube's diagonal and passing through its center intersects the cube. How many of the smaller cubes does this plane intersect? | [
"9099271"
] | [
"A large cube with dimensions 2011 x 2011 x 2011 is composed of smaller identical cubes. A plane perpendicular to the cube's diagonal and passing through its center intersects the cube. How many of the smaller cubes does this plane intersect?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 7
} | 15 | 0 | 0 |
omr-ws-f65a843819902cc0affd22f636da1c02811302fa7e6d64cf79e7a8ab40f1e632 | A large cube with edge length 5 is formed from smaller identical cubes with edge length 1. All corner cubes and all cubes that contain the center of each face of the large cube are removed. Find the ratio of the surface area of the original large cube to the surface area of the resulting body after the cubes are remove... | [
"\\(\\frac{25}{32}\\)"
] | [
"A large cube with edge length 5 is formed from smaller identical cubes with edge length 1. All corner cubes and all cubes that contain the center of each face of the large cube are removed. Find the ratio of the surface area of the original large cube to the surface area of the resulting body after the cubes are r... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e81834ae0b5e391996e1f61b9d5c780aad8c13ca40d077d905ed573d4a8b12cb | A large mixing tank initially holds 500 liters of water with 20 kg of salt dissolved in it. A brine solution containing 1 kg/liter of salt is pumped into the tank at a rate of 3 liters/min, and the solution is well stirred. The solution is then pumped out of the tank at a rate of 4 liters/min. Determine the amount of s... | [
"S(t) = (500 - t) - \\frac{480(500 - t)^4}{500^4}"
] | [
"A large mixing tank initially holds 500 liters of water with 20 kg of salt dissolved in it. A brine solution containing 1 kg/liter of salt is pumped into the tank at a rate of 3 liters/min, and the solution is well stirred. The solution is then pumped out of the tank at a rate of 4 liters/min. Determine the amount... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f5be9ca058a4ebdf7c5c089f21f7eb86654e0d61e6e99e096434026d6bc68d28 | A large mixing tank initially holds 500 liters of water with 20 kg of salt dissolved in it. A brine solution containing 1 kg/liter of salt is pumped into the tank at a rate of 3 liters/min, and the solution is well stirred. The solution is then pumped out of the tank at a rate of 4 liters/min. Write a differential equa... | [
"\\(\\frac{dA}{dt} = 3 - \\frac{4A(t)}{500 - t}\\)"
] | [
"A large mixing tank initially holds 500 liters of water with 20 kg of salt dissolved in it. A brine solution containing 1 kg/liter of salt is pumped into the tank at a rate of 3 liters/min, and the solution is well stirred. The solution is then pumped out of the tank at a rate of 4 liters/min. Write a differential... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-0e940769ec36c316401ee104a7d9065f8035e4f77683607ff67c5b2c7e4847a4 | A large number of cards are each given a number in the set {1, 2, ..., n} such that the sum of the numbers on all the cards is 5·n! for some positive integer n. Find the minimum value of n for which it is possible to divide these cards into 5 groups such that the sum of the numbers on the cards in each group is n!. | [
"5"
] | [
"A large number of cards are each given a number in the set {1, 2, ..., n} such that the sum of the numbers on all the cards is 5·n! for some positive integer n. Find the minimum value of n for which it is possible to divide these cards into 5 groups such that the sum of the numbers on the cards in each group is n!... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-60539a6c242b6708565b520fa634726c33e64db59163beeec606314bf9a77117 | A large number of line segments are drawn with the following procedure: - A real number \( k \) is chosen with \( 0 < k < 1 \). - The line segment from \( (k,0) \) to \( (0,1-k) \) is drawn. - Repeat until bored. As more and more line segments are drawn in this fashion, they begin to bound a curve in the first quadrant... | [
"\\( y = x + 1 + 2\\sqrt{x} \\)"
] | [
"A large number of line segments are drawn with the following procedure:\n- A real number \\( k \\) is chosen with \\( 0 < k < 1 \\).\n- The line segment from \\( (k,0) \\) to \\( (0,1-k) \\) is drawn.\n- Repeat until bored.\nAs more and more line segments are drawn in this fashion, they begin to bound a curve in t... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f0e58798ca2773b9e297bb8456b44df9108d5cf2d147941b2817671bb0a8c67f | A large open cistern, filled with water, has the shape of a hemisphere with radius 25 ft. The cistern has a circular hole of radius 1 ft. at the bottom. By Torricelli's Law, water will flow out of the hole with the same speed it would attain in falling freely from the level of the water to the hole. How long will it ta... | [
"\\(\\frac{8750}{3\\sqrt{2g}}\\)"
] | [
"A large open cistern, filled with water, has the shape of a hemisphere with radius 25 ft. The cistern has a circular hole of radius 1 ft. at the bottom. By Torricelli's Law, water will flow out of the hole with the same speed it would attain in falling freely from the level of the water to the hole. How long will ... | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-7317e000638b2e73285b1e30dbbf9e8e66c30d0675ea7722833c6b4a32bff61b | A large panel in the shape of a quarter of a circle is divided into four colors, with the segment dividing the sector into two equal parts and the inner arc being a semicircle. Determine which color covers the largest area. | [
"D"
] | [
"A large panel in the shape of a quarter of a circle is divided into four colors, with the segment dividing the sector into two equal parts and the inner arc being a semicircle. Determine which color covers the largest area."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 7
} | 15 | 0 | 0 |
omr-ws-b8ced24a0ccd73a0460cc7ceb7d3a2d4032b890028e37d1207c42d3aeaf883a3 | A large rectangle has side lengths of $(x+7)$ and $(x+5)$. In the large rectangle, there is a rectangular hole with side lengths of $(2x-3)$ and $(x-2)$. What is the area of the large rectangle (not including the area of the hole)? Express your answer as a polynomial in $x$. | [
"-x^2+19x+29"
] | [
"A large rectangle has side lengths of $(x+7)$ and $(x+5)$. In the large rectangle, there is a rectangular hole with side lengths of $(2x-3)$ and $(x-2)$. What is the area of the large rectangle (not including the area of the hole)? Express your answer as a polynomial in $x$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-58e6ec1c7965d1a395da3c67eb20a3b35b35204ce77352a5c0a2750afdde744b | A large rectangle is divided into four identical smaller rectangles by slicing parallel to one of the sides. The perimeter of the large rectangle is $18$ metres more than the perimeter of each of the smaller rectangles. The area of the large rectangle is $18$ m$^{2}$ more than the area of each of the smaller rectangles... | [
"36"
] | [
"A large rectangle is divided into four identical smaller rectangles by slicing parallel to one of the sides. The perimeter of the large rectangle is $18$ metres more than the perimeter of each of the smaller rectangles. The area of the large rectangle is $18$ m$^{2}$ more than the area of each of the smaller recta... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2dfc1cc12dfe71a01e2042650294281b340da9c05faf708491528f1422109601 | A large rectangle is divided into several smaller rectangles, each having at least one side of integer length. If the large rectangle has dimensions $a \times b$, where $a$ and $b$ are real numbers, determine the minimum possible value of $|a - b|$ such that the large rectangle does not have at least one side of intege... | [
"0"
] | [
"A large rectangle is divided into several smaller rectangles, each having at least one side of integer length. If the large rectangle has dimensions $a \\times b$, where $a$ and $b$ are real numbers, determine the minimum possible value of $|a - b|$ such that the large rectangle does not have at least one side of ... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 8
} | 11 | 0 | 0 |
omr-ws-9a0825ece89f3e97256ef58e1c64f51339231686249772bd4db43ea4124ab4ec | A large rectangle is divided into smaller rectangles, each of which has at least one side of integer length. Let the dimensions of the large rectangle be $a \times b$, where $a$ and $b$ are positive real numbers. If $a$ is not an integer, find the minimum possible value of $b$ such that the large rectangle still satisf... | [
"1"
] | [
"A large rectangle is divided into smaller rectangles, each of which has at least one side of integer length. Let the dimensions of the large rectangle be $a \\times b$, where $a$ and $b$ are positive real numbers. If $a$ is not an integer, find the minimum possible value of $b$ such that the large rectangle still ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-21126b24de0510d00a227f713f864c0ba4250b1b71c190b15abb2fb8ed6ca3f4 | A large rectangle is partitioned into four rectangles by two segments parallel to its sides. The areas of three of the resulting rectangles are 6, 14, and 35. What is the area of the fourth rectangle? | [
"15"
] | [
"A large rectangle is partitioned into four rectangles by two segments parallel to its sides. The areas of three of the resulting rectangles are 6, 14, and 35. What is the area of the fourth rectangle?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 1
} | 48 | 0 | 0 |
omr-ws-759c9d16e69f8a06467d52943cfc57bd21627ceb373e87532cf7ab9c8dda8182 | A large regular hexagon is cut out of a triangular grid and tiled with diamonds (pairs of triangles glued together along an edge). Diamonds come in three orientations. If the side length of the hexagon is 10 units, calculate the total number of diamonds used in the tiling. | [
"300"
] | [
"A large regular hexagon is cut out of a triangular grid and tiled with diamonds (pairs of triangles glued together along an edge). Diamonds come in three orientations. If the side length of the hexagon is 10 units, calculate the total number of diamonds used in the tiling."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-1e10b22d7a5dcc6b67e1a2151f0b1fe79856b61c4ab494ab2c1524298a648b15 | A large semicircle has three smaller circles touching it. The first smaller circle is tangent to the diameter and the semicircle, and the other two smaller circles of the same radius are tangent to the first one, the diameter line, and also the large semicircle. If the combined area of the three smaller circles is 120,... | [
"180"
] | [
"A large semicircle has three smaller circles touching it. The first smaller circle is tangent to the diameter and the semicircle, and the other two smaller circles of the same radius are tangent to the first one, the diameter line, and also the large semicircle. If the combined area of the three smaller circles is... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-be3882413d4fa911b9a229d94b39ed158a4200a2b1c337ee6effd074726b85dc | A large sphere has a volume of $288\pi$ cubic units. A smaller sphere has a volume which is $12.5\%$ of the volume of the larger sphere. What is the ratio of the radius of the smaller sphere to the radius of the larger sphere? Express your answer as a common fraction. | [
"\\frac{1}{2}"
] | [
"A large sphere has a volume of $288\\pi$ cubic units. A smaller sphere has a volume which is $12.5\\%$ of the volume of the larger sphere. What is the ratio of the radius of the smaller sphere to the radius of the larger sphere? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-71e97d9cbd1dceed5a8399b754b6a48f654c6d60617d197eb974f05a1cd55ae4 | A large sphere is on a horizontal field on a sunny day. At a certain time the shadow of the sphere reaches out a distance of $10$ m from the point where the sphere touches the ground. At the same instant a meter stick (held vertically with one end on the ground) casts a shadow of length $2$ m. What is the radius of the... | [
"5"
] | [
"A large sphere is on a horizontal field on a sunny day. At a certain time the shadow of the sphere reaches out a distance of $10$ m from the point where the sphere touches the ground. At the same instant a meter stick (held vertically with one end on the ground) casts a shadow of length $2$ m. What is the radius o... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e55ce8569138d7414dbe3a01c489c9d8325f555353e7d8aefc4c139f301d881a | A large sphere is on a horizontal field on a sunny day. At a certain time, the shadow of the sphere reaches out a distance of $10$ meters from the point where the sphere touches the ground. At the same instant, a meter stick (held vertically with one end on the ground) casts a shadow of length $2$ meters. What is the r... | [
"\\(10\\sqrt{5} - 20\\)"
] | [
"A large sphere is on a horizontal field on a sunny day. At a certain time, the shadow of the sphere reaches out a distance of $10$ meters from the point where the sphere touches the ground. At the same instant, a meter stick (held vertically with one end on the ground) casts a shadow of length $2$ meters. What is ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-81d0e41670ef7a066f062b53d46a179f596d32f357539df40a1147102f4e25a0 | A large sphere is on a horizontal field on a sunny day. At a certain time, the shadow of the sphere reaches out a distance of 10 cm from the point where it touches the ground. At the same instant, a meter stick (held vertically with one end on the ground) casts a shadow of length 2 cm. What is the radius of the sphere ... | [
"\\(\\frac{10}{3}\\)"
] | [
"A large sphere is on a horizontal field on a sunny day. At a certain time, the shadow of the sphere reaches out a distance of 10 cm from the point where it touches the ground. At the same instant, a meter stick (held vertically with one end on the ground) casts a shadow of length 2 cm. What is the radius of the sp... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-692b00cf76c34814885129ad01146765b9b736ce8287cd7a60536b0dbd4f9819 | A large sphere is on a horizontal field on a sunny day. At a certain time, the shadow of the sphere reaches out a distance of 10 meters from the point where the sphere touches the ground. At the same instant, a meter stick (held vertically with one end on the ground) casts a shadow of length 2 meters. What is the radiu... | [
"\\( 10\\sqrt{5} - 20 \\)"
] | [
"A large sphere is on a horizontal field on a sunny day. At a certain time, the shadow of the sphere reaches out a distance of 10 meters from the point where the sphere touches the ground. At the same instant, a meter stick (held vertically with one end on the ground) casts a shadow of length 2 meters. What is the ... | [
"aops_c5_contests_amp_programs",
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 93,
"genselect": 8,
"tir": 37
} | 138 | 0 | 0 |
omr-ws-70e770968701136f6fe24deff88405d85f36f9b34af758fcbf2a061b5be60778 | A large sphere with a diameter of 7 touches three walls in a corner. A smaller sphere touches the three walls and the large sphere. What is the diameter of the smaller sphere? | [
"\\( 14 - 7\\sqrt{3} \\)"
] | [
"A large sphere with a diameter of 7 touches three walls in a corner. A smaller sphere touches the three walls and the large sphere. What is the diameter of the smaller sphere?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-f9eefbcf015f65b5f38d2cd58c9913db132817bb219ab7f23c4789bbe454f186 | A large square is divided into a small square surrounded by four congruent rectangles. The perimeter of each of the congruent rectangles is 14. What is the area of the large square? | [
"49"
] | [
"A large square is divided into a small square surrounded by four congruent rectangles. The perimeter of each of the congruent rectangles is 14. What is the area of the large square?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-676f0c02efe980242526bb341fd2d315e13fb70da45a111ad12d266796ec3052 | A large square is divided into unit squares, and the two diagonals of the large square together intersect 21 of the smaller squares. How many unit squares are in the larger square? | [
"121"
] | [
"A large square is divided into unit squares, and the two diagonals of the large square together intersect 21 of the smaller squares. How many unit squares are in the larger square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-1a14c9b100d3b3f01218c362a1e35216f88092554bcebb785f8aab62f0dadc98 | A large square land lot of dimension $100 \times 100$ m is subdivided into $100$ square lots, each measured $10 \times 10$ m. If a particular lot is not dumped with waste and two of its adjacent lots have waste materials, then the lot will be filled with wastes the next day. At least how many lots should be initially f... | [
"10"
] | [
"A large square land lot of dimension $100 \\times 100$ m is subdivided into $100$ square lots, each measured $10 \\times 10$ m. If a particular lot is not dumped with waste and two of its adjacent lots have waste materials, then the lot will be filled with wastes the next day. At least how many lots should be init... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-5f6b9e78bc67954938b1296670180fe73ca4e999815ac730e0ab78fcf624ffd3 | A large square region is paved with \( n^2 \) gray square tiles, each measuring \( s \) inches on a side. A border \( d \) inches wide surrounds each tile. When \( n = 24 \), the 576 gray tiles cover 64% of the area of the large square region. What is the ratio \( \frac{d}{s} \) for this larger value of \( n \)? | [
"\\(\\frac{6}{25}\\)",
"\\dfrac{1}{8}"
] | [
"A large square region is paved with \\( n^2 \\) gray square tiles, each measuring \\( s \\) inches on a side. A border \\( d \\) inches wide surrounds each tile. When \\( n = 24 \\), the 576 gray tiles cover 64% of the area of the large square region. What is the ratio \\( \\frac{d}{s} \\) for this larger value of... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 25,
"genselect": 13,
"tir": 19
} | 57 | 0 | 0 |
omr-ws-4d343a8d903e551f51b8067514df6d2af9ba2297e3b618ac7ec489fbab1cc3a8 | A large tank is filled to capacity with 500 gallons of pure water. Brine containing 2 pounds of salt per gallon is pumped into the tank at a rate of 5 gallons per minute. The well-mixed solution is pumped out at 10 gallons per minute. Find the number \( A(t) \) of pounds of salt in the tank at time \( t \). | [
"A(t) = 10t - \\frac{1}{10}t^2"
] | [
"A large tank is filled to capacity with 500 gallons of pure water. Brine containing 2 pounds of salt per gallon is pumped into the tank at a rate of 5 gallons per minute. The well-mixed solution is pumped out at 10 gallons per minute. Find the number \\( A(t) \\) of pounds of salt in the tank at time \\( t \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-44b7dc4c3a52f2a86f7d24b72be2c93c9cda4ea6009608ba3ec0e874bfc309f8 | A large white cube is painted red, then cut into 27 identical smaller cubes. These smaller cubes are shuffled randomly and reassembled into a large cube by a blind man. What is the probability that the outside of the large cube is completely red? | [
"\\(\\frac{6!8!12!}{27!6^68^812^{12}}\\)"
] | [
"A large white cube is painted red, then cut into 27 identical smaller cubes. These smaller cubes are shuffled randomly and reassembled into a large cube by a blind man. What is the probability that the outside of the large cube is completely red?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 3
} | 4 | 0 | 0 |
omr-ws-ea7045e2ba41d334e721477b69171627e37fb36fc7025339b38470d0ecb33af9 | A large wooden cube has some of its faces painted red and the rest painted black. When the cube is cut into unit cubes, the number of unit cubes with some red paint is found to be exactly 200 larger than the number of cubes with some black paint. How many unit cubes have no paint at all? | [
"5400"
] | [
"A large wooden cube has some of its faces painted red and the rest painted black. When the cube is cut into unit cubes, the number of unit cubes with some red paint is found to be exactly 200 larger than the number of cubes with some black paint. How many unit cubes have no paint at all?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-63b0b28615275a0e5b583d6ffd64aa89dac0d464bc797ba6ef2122f64e632400 | A larger cube is formed by assembling unit cubes, and some of its faces are painted. After disassembling the larger cube, it is found that 96 unit cubes have no paint on any of their faces. How many faces of the larger cube were painted? | [
"4"
] | [
"A larger cube is formed by assembling unit cubes, and some of its faces are painted. After disassembling the larger cube, it is found that 96 unit cubes have no paint on any of their faces. How many faces of the larger cube were painted?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 4
} | 51 | 0 | 0 |
omr-ws-0b2efda1cf67da73e12d043fa264a2d8e3668d32cf7e4bb9048ab218cc201a2b | A larger cylindrical tube with radius $72$ rolls along a flat surface toward a smaller cylindrical tube with radius $24$. It rolls over the smaller tube and continues rolling until it completes one full revolution and comes to rest on the same point of its circumference as it started. If the smaller tube remains statio... | [
"179"
] | [
"A larger cylindrical tube with radius $72$ rolls along a flat surface toward a smaller cylindrical tube with radius $24$. It rolls over the smaller tube and continues rolling until it completes one full revolution and comes to rest on the same point of its circumference as it started. If the smaller tube remains s... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 1
} | 21 | 0 | 0 |
omr-ws-f71fcd10dd1f303f3588ff6ce2564e6c30478a913b2470460716c5caca91552e | A laser gun is set on a point on a circle of radius 1. How many possible ways can the laser gun shoot the laser such that the laser hits the circle 2014 times before it hits the laser gun for the first time? | [
"936"
] | [
"A laser gun is set on a point on a circle of radius 1. How many possible ways can the laser gun shoot the laser such that the laser hits the circle 2014 times before it hits the laser gun for the first time?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-a95d54825c2c3b155f16615d47f1217a42b08c77b80ec03e7650e23c60b943fb | A laser is placed at the point $(3,5)$. The laser beam travels in a straight line. Larry wants the beam to hit and bounce off the $y$-axis, then hit and bounce off the $x$-axis, then hit the point $(97,5)$. What is the total distance the beam will travel along the path? | [
"10\\sqrt{101}"
] | [
"A laser is placed at the point $(3,5)$. The laser beam travels in a straight line. Larry wants the beam to hit and bounce off the $y$-axis, then hit and bounce off the $x$-axis, then hit the point $(97,5)$. What is the total distance the beam will travel along the path?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-0b26b1cf77c841a82a8fece54e9ca8aa6a2912340a822369231898247692796b | A laser is placed at the point (3,5). The laser beam travels in a straight line, hits and bounces off the $y$-axis, then hits and bounces off the $x$-axis, and finally hits the point $(7,5)$. What is the total distance the beam will travel along this path? | [
"10\\sqrt{2}"
] | [
"A laser is placed at the point (3,5). The laser beam travels in a straight line, hits and bounces off the $y$-axis, then hits and bounces off the $x$-axis, and finally hits the point $(7,5)$. What is the total distance the beam will travel along this path?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-80913ddc30cdb655b45f0970a544b2ac6e9d8822e0b9bee5d38a9a370a7a46c0 | A laser is placed at the point (3,5). The laser beam travels in a straight line. Larry wants the beam to hit and bounce off the \(y\)-axis, then hit and bounce off the \(x\)-axis, then hit the point (7,5). What is the total distance the beam will travel along this path? | [
"10\\sqrt{2}"
] | [
"A laser is placed at the point (3,5). The laser beam travels in a straight line. Larry wants the beam to hit and bounce off the \\(y\\)-axis, then hit and bounce off the \\(x\\)-axis, then hit the point (7,5). What is the total distance the beam will travel along this path?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6a8d1378d146d0903c97af688f3350dac8dadf89beed1cf6490126e1d55f0d9b | A laser mounted at a height of $10$ cm above the $x$-axis moves from the point $(5,10)$ cm to $(4,10)$ cm. There is a mirror in the shape of a half-circle with radius $5$ cm centered at the origin. Calculate the distance that the laser reflection travels along the $x$-axis. | [
"9.28 cm"
] | [
"A laser mounted at a height of $10$ cm above the $x$-axis moves from the point $(5,10)$ cm to $(4,10)$ cm. There is a mirror in the shape of a half-circle with radius $5$ cm centered at the origin. Calculate the distance that the laser reflection travels along the $x$-axis."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-68878a3b037c0898894d98ef6de80a0548369e309dd8eb1264cd1165e46aedd2 | A lateral edge of a regular triangular pyramid is equal to $a$, the angle between the lateral faces is $2\varphi $. Find the length of a side of the base. | [
"$\\frac{a}{\\sin \\varphi }$"
] | [
"A lateral edge of a regular triangular pyramid is equal to $a$, the angle between the lateral faces is $2\\varphi $. Find the length of a side of the base."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6420e9764f1c9ebe49e679ada4dfe42330cbf826c12bc48933b651e9f07c9b12 | A lateral edge of a regular truncated quadrangular pyramid is equal to the side of the smaller base and to $a$. The angle between the lateral edge and the side of the larger base is $\alpha $. Find the area of the diagonal section of the truncated pyramid. | [
"$2a^2\\frac{\\alpha }{2}\\sqrt {-2\\cos 2\\alpha }$"
] | [
"A lateral edge of a regular truncated quadrangular pyramid is equal to the side of the smaller base and to $a$. The angle between the lateral edge and the side of the larger base is $\\alpha $. Find the area of the diagonal section of the truncated pyramid."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-614ae9e27f83baf32b931e0b0dd601a5d2fd44824d0991dc623bf2faa3b56e32 | A lattice point \((x, y)\) is selected at random from the interior of the circle described by the equation \(x^2 + y^2 - 2x + 4y = 22\). Find the probability that the point is within 1.5 units from the origin. | [
"\\dfrac{9}{89}"
] | [
"A lattice point \\((x, y)\\) is selected at random from the interior of the circle described by the equation \\(x^2 + y^2 - 2x + 4y = 22\\). Find the probability that the point is within 1.5 units from the origin."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b0de1ddcf1dbf92fb75826a189a9578af1b3858b2995823edca58654020817bf | A lattice point \((x, y, z) \in \mathbb{Z}^3\) can be seen from the origin if the line from the origin does not contain any other lattice point \((x', y', z')\) with \((x')^2 + (y')^2 + (z')^2 < x^2 + y^2 + z^2\). Let \(p\) be the probability that a randomly selected point on the cubic lattice \(\mathbb{Z}^3\) can be s... | [
"\\( i = 1 \\), \\( k = 1 \\), \\( s = 3 \\)"
] | [
"A lattice point \\((x, y, z) \\in \\mathbb{Z}^3\\) can be seen from the origin if the line from the origin does not contain any other lattice point \\((x', y', z')\\) with \\((x')^2 + (y')^2 + (z')^2 < x^2 + y^2 + z^2\\). Let \\(p\\) be the probability that a randomly selected point on the cubic lattice \\(\\mathb... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 16
} | 16 | 0 | 0 |
omr-ws-60c5b796d244414b79fe627760a4ac4881473ae7780d702ccb4b910c883b5dd5 | A lattice point in an $xy$-coordinate system is any point $(x, y)$ where both $x$ and $y$ are integers. The graph of $y = mx + 2$ passes through no lattice point with $0 < x \leq 100$ for all $m$ such that $\frac{1}{2} < m < a$. What is the maximum possible value of $a$? | [
"\\(\\frac{50}{99}\\)"
] | [
"A lattice point in an $xy$-coordinate system is any point $(x, y)$ where both $x$ and $y$ are integers. The graph of $y = mx + 2$ passes through no lattice point with $0 < x \\leq 100$ for all $m$ such that $\\frac{1}{2} < m < a$. What is the maximum possible value of $a$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 48,
"genselect": 5,
"tir": 29
} | 82 | 0 | 0 |
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