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-ca34f26731b62900428b98778dcfcc8e0889614ba260e0bd61621ec0401b4e85 | A bee starts flying from point $P_0$. She flies $1$ inch due east to point $P_1$. For $j \ge 1$, once the bee reaches point $P_j$, she turns $30^{\circ}$ counterclockwise and then flies $j+1$ inches straight to point $P_{j+1}$. When the bee reaches $P_{2016}$, she is exactly $a \sqrt{b} + c \sqrt{d}$ inches away from $... | [
"2024"
] | [
"A bee starts flying from point $P_0$. She flies $1$ inch due east to point $P_1$. For $j \\ge 1$, once the bee reaches point $P_j$, she turns $30^{\\circ}$ counterclockwise and then flies $j+1$ inches straight to point $P_{j+1}$. When the bee reaches $P_{2016}$, she is exactly $a \\sqrt{b} + c \\sqrt{d}$ inches aw... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-731a048b2d41c5e940536dccb21c3cf2a971595fb2980c4f1476ee33642b572f | A bee starts flying from point \( P_0 \). She flies 1 inch due east to point \( P_1 \). For \( j \ge 1 \), once the bee reaches point \( P_j \), she turns \( 30^\circ \) counterclockwise and then flies \( j+1 \) inches straight to point \( P_{j+1} \). When the bee reaches \( P_{2015} \), she is exactly \( a\sqrt{b} + c... | [
"2024"
] | [
"A bee starts flying from point \\( P_0 \\). She flies 1 inch due east to point \\( P_1 \\). For \\( j \\ge 1 \\), once the bee reaches point \\( P_j \\), she turns \\( 30^\\circ \\) counterclockwise and then flies \\( j+1 \\) inches straight to point \\( P_{j+1} \\). When the bee reaches \\( P_{2015} \\), she is e... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 36,
"tir": 7
} | 43 | 0 | 0 |
omr-ws-ca3e426985d59a84d08d9bd9fd93d9e18b199b3db19b2e21a2766c428d853383 | A bee travels in a series of steps of length $1$: north, west, north, west, up, south, east, south, east, down. There exists a plane $P$ that passes through the midpoints of each step. Suppose we orthogonally project the bee’s path onto the plane $P$, and let $A$ be the area of the resulting figure. What is $A^2$? | [
"\\( 1 \\)"
] | [
"A bee travels in a series of steps of length $1$: north, west, north, west, up, south, east, south, east, down. There exists a plane $P$ that passes through the midpoints of each step. Suppose we orthogonally project the bee’s path onto the plane $P$, and let $A$ be the area of the resulting figure. What is $A^2$?... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-97c5c5077e6bc22cb689f7a9f0d875e29ad986bfe07b8ce0ea9296afe5ffa543 | A bee, which is a perfect circle with a radius of 1/6 inch, randomly flies to a point within an 8 inch by 8 inch square. The square has strings that are 1/2 inch apart. What is the probability that the bee gets hit by one of the strings? | [
"\\( \\frac{1}{4} - \\frac{\\pi}{36} \\)"
] | [
"A bee, which is a perfect circle with a radius of 1/6 inch, randomly flies to a point within an 8 inch by 8 inch square. The square has strings that are 1/2 inch apart. What is the probability that the bee gets hit by one of the strings?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-450ad50a920a3943456003b277c5f34904c8b9498c8d7cbf4e2a8e6a909e63e3 | A beehive cell is modeled as a right regular 6-angled prism with a regular 3-sided pyramidical mantle. If the side length of the hexagonal base is $s$ and the height of the prism is $h$, find the total surface area of the beehive cell in terms of $s$ and $h$. Express your answer in the form $A(s, h) = as^2 + bsh + ch^2... | [
"9"
] | [
"A beehive cell is modeled as a right regular 6-angled prism with a regular 3-sided pyramidical mantle. If the side length of the hexagonal base is $s$ and the height of the prism is $h$, find the total surface area of the beehive cell in terms of $s$ and $h$. Express your answer in the form $A(s, h) = as^2 + bsh +... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-9da18db500a3afcf46fc745b52b22fb99faf8e990f3124511d4d3c99c6dccee0 | A beetle appears at the origin with velocity \( v(t) = 5\sin^2 t \). Find the maximum distance between the ant (with velocity \( v(t) = 10\sin^2 t \)) and the beetle. | [
"\\infty"
] | [
"A beetle appears at the origin with velocity \\( v(t) = 5\\sin^2 t \\). Find the maximum distance between the ant (with velocity \\( v(t) = 10\\sin^2 t \\)) and the beetle."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-fa328dd019b174abd5757e421750f6ff17805a610f441d3cd85147162363bbff | A beetle crawls along the edges of an $n$-lateral pyramid, starting and ending at the midpoint $A$ of a base edge and passing through each point at most once. How many ways are there for the beetle to do this (two ways are said to be equal if they go through the same vertices)? | [
"2^{n-1}"
] | [
"A beetle crawls along the edges of an $n$-lateral pyramid, starting and ending at the midpoint $A$ of a base edge and passing through each point at most once. How many ways are there for the beetle to do this (two ways are said to be equal if they go through the same vertices)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-4220b6ce18253cce8caa180ac255fc4facd91fd18baec7c702bccb0c94cb2131 | A beetle crawls along the edges of an $n$-sided polygon, starting from vertex $1$ and visiting each vertex exactly once before returning to vertex $1$. Let $S(n)$ be the sum of the numbers of passed vertices over all possible paths the beetle can take. Calculate $S(10)$. | [
"112"
] | [
"A beetle crawls along the edges of an $n$-sided polygon, starting from vertex $1$ and visiting each vertex exactly once before returning to vertex $1$. Let $S(n)$ be the sum of the numbers of passed vertices over all possible paths the beetle can take. Calculate $S(10)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-34328682b073d831fc965f5388d157bb459b40ee79d758baf7de6e5ecdbb5715 | A beetle crawls on the surface of a Rubik's cube ($3 \times 3 \times 3$) and can move to an adjacent cell on the same face or to an edge-adjacent cell on an adjacent face. The beetle visits each cell exactly once and returns to the starting cell. Let $T$ be the total number of transitions the beetle makes to an adjacen... | [
"0"
] | [
"A beetle crawls on the surface of a Rubik's cube ($3 \\times 3 \\times 3$) and can move to an adjacent cell on the same face or to an edge-adjacent cell on an adjacent face. The beetle visits each cell exactly once and returns to the starting cell. Let $T$ be the total number of transitions the beetle makes to an ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 4
} | 11 | 0 | 0 |
omr-ws-a2b0ba32c123743d497d1a9aaa7591874d1478d2b93a434ff074cb29f5683c04 | A beetle sits on each square of a $9 \times 9$ chessboard. At a signal, each beetle crawls diagonally onto a neighboring square. Find the maximal possible number of free squares after all beetles have moved. | [
"40"
] | [
"A beetle sits on each square of a $9 \\times 9$ chessboard. At a signal, each beetle crawls diagonally onto a neighboring square. Find the maximal possible number of free squares after all beetles have moved."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-d6f94720edc61284c68b1bed6528863bacc58364ad16230c408cff8573fed426 | A beetle sits on each square of a chessboard. At a signal, each beetle crawls diagonally onto a neighboring square. Determine the minimum possible number of free squares after the beetles move. | [
"0"
] | [
"A beetle sits on each square of a chessboard. At a signal, each beetle crawls diagonally onto a neighboring square. Determine the minimum possible number of free squares after the beetles move."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-599102cdb7f2938e6913ca79a96c3c7d307095126a7cc9e91caaaef7576f9cc4 | A beetle starts at vertex $1$ of a regular $1000$-sided polygon with each side of length $1$. The vertices are numbered $1, 2, \ldots, 1000$ in some order. The beetle moves clockwise from vertex $1$ to vertex $2$, then from vertex $2$ to vertex $3$, and so on, until it reaches vertex $1000$. Determine the number of way... | [
"\\(3^{16} \\times 2^{82}\\)"
] | [
"A beetle starts at vertex $1$ of a regular $1000$-sided polygon with each side of length $1$. The vertices are numbered $1, 2, \\ldots, 1000$ in some order. The beetle moves clockwise from vertex $1$ to vertex $2$, then from vertex $2$ to vertex $3$, and so on, until it reaches vertex $1000$. Determine the number ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-aa16a5de4539f2ee9baf325576905d858beef248dc78804ef59f7aa97d080cf7 | A beetle starts at vertex 1 of a regular 1000-sided polygon with each side of length 1. The beetle moves clockwise from vertex 1 to vertex 2, then from vertex 2 to vertex 3, and so on, until it reaches vertex 1000. Determine the number of ways to assign the numbers 1 through 1000 to the vertices such that the total len... | [
"0"
] | [
"A beetle starts at vertex 1 of a regular 1000-sided polygon with each side of length 1. The beetle moves clockwise from vertex 1 to vertex 2, then from vertex 2 to vertex 3, and so on, until it reaches vertex 1000. Determine the number of ways to assign the numbers 1 through 1000 to the vertices such that the tota... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-1810287f76b88f345c1826ac2f77a14e6d556ea6d915772a1616f35709b5a18d | A belt fits tightly around two pulleys with diameters of 24 units and 6 units. If the centers of the pulleys are 18 units apart, how long is the belt? | [
"\\( 18\\sqrt{3} + 18\\pi \\)"
] | [
"A belt fits tightly around two pulleys with diameters of 24 units and 6 units. If the centers of the pulleys are 18 units apart, how long is the belt?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 3,
"tir": 7
} | 18 | 0 | 0 |
omr-ws-d9ecb25db5316111af4772a75c92fd3b8258573c323d8ea8292a2c1e83d2b852 | A belt is drawn tight around 3 circles of radius 10 cm. How long is the belt? | [
"\\(60 + 20\\pi\\)"
] | [
"A belt is drawn tight around 3 circles of radius 10 cm. How long is the belt?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 5,
"tir": 2
} | 14 | 0 | 0 |
omr-ws-444529488b2dd7f369b947335ee4f590b7473a54bd1ee488985105f1b4051e7b | A belt is wrapped tightly around a circle with radius 6 and forms a right angle at a point P outside the circle. Find the length of the belt. | [
"\\( 12 + 9\\pi \\)"
] | [
"A belt is wrapped tightly around a circle with radius 6 and forms a right angle at a point P outside the circle. Find the length of the belt."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-fc8eaeecf29ec65433ac78b8e79fa2ab8a88c02266c64ad66e517b43618a7b89 | A besieged fortress is held by 5700 men, who have provisions for 66 days. If the garrison loses 20 men each day, for how many days can the provisions hold out? | [
"76"
] | [
"A besieged fortress is held by 5700 men, who have provisions for 66 days. If the garrison loses 20 men each day, for how many days can the provisions hold out?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 20
} | 35 | 0 | 0 |
omr-ws-e24f8cab6c168febe05db19d0931bc40465ef5237b8ef51878584c661e5b5a2d | A bettor follows a strategy where he bets $1 on heads at each coin toss. If he wins, he takes his profit and continues betting $1. If he loses, he bets his current winnings until he reaches a $10 profit or more. What is the minimum amount of money he needs to ensure he reaches his goal of a $10 profit in 98% of the cas... | [
"63"
] | [
"A bettor follows a strategy where he bets $1 on heads at each coin toss. If he wins, he takes his profit and continues betting $1. If he loses, he bets his current winnings until he reaches a $10 profit or more. What is the minimum amount of money he needs to ensure he reaches his goal of a $10 profit in 98% of th... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-f72970d59dbd6052f1cad546d4b0c2f1ac22e15d7a63f62683c8cd3eb9fcd969 | A biased coin has a $ \dfrac{6 + 2\sqrt{3}}{12} $ chance of landing heads and a $ \dfrac{6 - 2\sqrt{3}}{12} $ chance of landing tails. What is the probability that the number of times the coin lands heads after being flipped 100 times is a multiple of 4? The answer can be expressed as $ \dfrac{1}{4} + \dfrac{1 + a^b}{c... | [
"67"
] | [
"A biased coin has a $ \\dfrac{6 + 2\\sqrt{3}}{12} $ chance of landing heads and a $ \\dfrac{6 - 2\\sqrt{3}}{12} $ chance of landing tails. What is the probability that the number of times the coin lands heads after being flipped 100 times is a multiple of 4? The answer can be expressed as $ \\dfrac{1}{4} + \\dfrac... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 6,
"tir": 1
} | 11 | 0 | 0 |
omr-ws-954e7889facd86b7ec052e406574337904e4232530ab79d06529b1f89edc729f | A biased coin has a probability of $\frac{2}{3}$ for heads and $\frac{1}{3}$ for tails. In Game A, a player tosses the coin three times and wins if all three outcomes are the same. In Game B, a player tosses the coin four times and wins if the outcomes of the first and second tosses are the same and the outcomes of the... | [
"\\(\\frac{2}{81}\\)"
] | [
"A biased coin has a probability of $\\frac{2}{3}$ for heads and $\\frac{1}{3}$ for tails. In Game A, a player tosses the coin three times and wins if all three outcomes are the same. In Game B, a player tosses the coin four times and wins if the outcomes of the first and second tosses are the same and the outcomes... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-81e95b680ae7d631b5593ba6376ba49472119ec4ef4985f1148ff65566dcc036 | A biased coin has a probability of \(\frac{1}{4}\) of landing on a digit. If the coin is tossed \(n\) times, the probability of getting exactly 2 digits is equal to the probability of getting exactly 3 digits. What is the value of \(n\)? | [
"11"
] | [
"A biased coin has a probability of \\(\\frac{1}{4}\\) of landing on a digit. If the coin is tossed \\(n\\) times, the probability of getting exactly 2 digits is equal to the probability of getting exactly 3 digits. What is the value of \\(n\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-ca48ff3b66e757d458c740e3c2f6d62f945ad4ce5e352f524389d44aab330015 | A biased coin is flipped $n$ times. Assuming that the flips are independent, with each one coming up heads with probability $p$, what is the probability that there is a string of $k$ consecutive heads? | [
"1 - Q(n, k)"
] | [
"A biased coin is flipped $n$ times. Assuming that the flips are independent, with each one coming up heads with probability $p$, what is the probability that there is a string of $k$ consecutive heads?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 8
} | 9 | 0 | 0 |
omr-ws-8fbeab6d7fa92a07d7cfd16c4fe15d16f75664ac01b739fbc80203708cbadc8e | A bicycle has a pedal gear with 24 teeth rotating at 45 revolutions per minute and a rear gear with 16 teeth. If the diameter of the tire is 20 inches, what is the speed of the bicycle in miles per hour? | [
"4.02"
] | [
"A bicycle has a pedal gear with 24 teeth rotating at 45 revolutions per minute and a rear gear with 16 teeth. If the diameter of the tire is 20 inches, what is the speed of the bicycle in miles per hour?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-66956ce8ab615215df54f3688c7e44f9042c93acf2fc5b368226ac49506782f2 | A bicycle leaves point A for point B at the same time a motorcycle leaves point B for point A. When the motorcycle has traveled 1/3 of the distance, the bicycle still has 26 km to go. When the bicycle has traveled 1/3 of the distance, the motorcycle has 5 km left to travel. Find the distance between points A and B. | [
"30"
] | [
"A bicycle leaves point A for point B at the same time a motorcycle leaves point B for point A. When the motorcycle has traveled 1/3 of the distance, the bicycle still has 26 km to go. When the bicycle has traveled 1/3 of the distance, the motorcycle has 5 km left to travel. Find the distance between points A and B... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0f1f7c8d5ef7ed1b4d40bd16db60fc4203f9342f5bfcb8e12cb696119460646f | A bicycle shop has a display of bicycles, tricycles, and unicycles. There are a total of 200 wheels, 95 seats, and 144 handle grips (one on each side). Unicycles do not have handle grips. Determine the number of bicycles, tricycles, and unicycles on display. | [
"39 bicycles, 33 tricycles, 23 unicycles"
] | [
"A bicycle shop has a display of bicycles, tricycles, and unicycles. There are a total of 200 wheels, 95 seats, and 144 handle grips (one on each side). Unicycles do not have handle grips. Determine the number of bicycles, tricycles, and unicycles on display."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 29
} | 34 | 0 | 0 |
omr-ws-234b675b6869f5ef04a527344fa62dae5df275f9e39542d5289ff6fb6fdec22a | A bicycle tire has a 30-inch diameter. If the tire spins at a rate of 80 revolutions per minute, what distance is traveled in one minute? | [
"\\(2400\\pi\\) inches per minute."
] | [
"A bicycle tire has a 30-inch diameter. If the tire spins at a rate of 80 revolutions per minute, what distance is traveled in one minute?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-c3f6b342619a48e3419e3b5e4bbd18b391397aebb600d5bda32dec69d731225c | A bicycle wheel with a radius of 13 inches makes 6 revolutions per second. What is the speed of the bicycle in inches per second? | [
"\\( 156\\pi \\)"
] | [
"A bicycle wheel with a radius of 13 inches makes 6 revolutions per second. What is the speed of the bicycle in inches per second?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-311b3c03ed858bccb0111f8079ccb5335ddb1dc67f0b5a9f86157c25d7f7e7fe | A big box contains 10 small boxes. Each of these small boxes is either empty or contains 10 even smaller boxes, and so on. After opening all the boxes, it was found that exactly 2008 boxes were nonempty. How many boxes were empty? | [
"18073"
] | [
"A big box contains 10 small boxes. Each of these small boxes is either empty or contains 10 even smaller boxes, and so on. After opening all the boxes, it was found that exactly 2008 boxes were nonempty. How many boxes were empty?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6e897338362ca08f632e60fffba189201027b69de56e96c0351bc4c2a7412213 | A big square is dissected by $10$ horizontal and $10$ vertical lines into $100$ rectangular pieces. If exactly $9$ of these pieces are squares, find the minimum number of congruent square pieces among these $9$ squares. | [
"3"
] | [
"A big square is dissected by $10$ horizontal and $10$ vertical lines into $100$ rectangular pieces. If exactly $9$ of these pieces are squares, find the minimum number of congruent square pieces among these $9$ squares."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 1
} | 5 | 0 | 0 |
omr-ws-2621eac16925f32aa1be1fcef74059208594116502f840b3ec514ec97ba5dedc | A bill for $376.61 was paid by a husband and wife. The husband mistakenly paid $220, and the wife paid the remaining $156.61. To make the payment half and half, who owes who money and how much is owed? | [
"36.995"
] | [
"A bill for $376.61 was paid by a husband and wife. The husband mistakenly paid $220, and the wife paid the remaining $156.61. To make the payment half and half, who owes who money and how much is owed?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2deee2f838e562cf45c981498a37bde748340fba9e4b43d654b4c9005c9fb937 | A bill of exchange in the amount of $8,000,000 with a maturity of 3 years must be replaced by two bills of the same nominal value each, but with maturities of 2 years and 5 years, respectively. Calculate the face value of the new bills, given that the compound interest rate is 8% per semester and the discount compound ... | [
"3766859"
] | [
"A bill of exchange in the amount of $8,000,000 with a maturity of 3 years must be replaced by two bills of the same nominal value each, but with maturities of 2 years and 5 years, respectively. Calculate the face value of the new bills, given that the compound interest rate is 8% per semester and the discount comp... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ff906be30a9332ce515d26fec0e6bbd53165fd5c0af09998f2069c35bfa42673 | A billboard is to be made with 100 m^2 of printed area, with margins of 2 m at the top and bottom and 4 m on each side. Find the outside dimensions of the billboard if its total area is to be a minimum. | [
"\\(20\\sqrt{2} + 8 \\text{ meters by } 10\\sqrt{2} + 4 \\text{ meters}\\)"
] | [
"A billboard is to be made with 100 m^2 of printed area, with margins of 2 m at the top and bottom and 4 m on each side. Find the outside dimensions of the billboard if its total area is to be a minimum."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 1,
"tir": 19
} | 33 | 0 | 0 |
omr-ws-fa63bd762b96c4b8d6cc405ec0c5e3bfa2cc845bea9ccf50c5393fab890d38d8 | A billiard ball (of infinitesimal diameter) strikes ray $\overrightarrow{BC}$ at point $C$, with angle of incidence $\alpha = 2.5^\circ$. The billiard ball continues its path, bouncing off line segments $AB$ and $BC$, which are making an angle $\beta = 17^\circ$, according to the rule "angle of incidence equals angle o... | [
"11"
] | [
"A billiard ball (of infinitesimal diameter) strikes ray $\\overrightarrow{BC}$ at point $C$, with angle of incidence $\\alpha = 2.5^\\circ$. The billiard ball continues its path, bouncing off line segments $AB$ and $BC$, which are making an angle $\\beta = 17^\\circ$, according to the rule \"angle of incidence equ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 6,
"tir": 1
} | 11 | 0 | 0 |
omr-ws-5a081843f684c68ebe1183b14a0086140687170a7ff0ae19ca55a4f0ec7a0d81 | A billiard ball is released from the top $A$ of the square table $ABCD$. It is reflected from the sides according to the law “the angle of incidence is equal to the angle of reflection”. It is known that no two consecutive blows fell on opposite sides, with the first blow coming from the side $BC$. After $2010$ shots f... | [
"$C$"
] | [
"A billiard ball is released from the top $A$ of the square table $ABCD$. It is reflected from the sides according to the law “the angle of incidence is equal to the angle of reflection”. It is known that no two consecutive blows fell on opposite sides, with the first blow coming from the side $BC$. After $2010$ sh... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-121ca5d762fc27b255d15bd90fe4161993f9b9ada0353e1737aec72c8c41fd8c | A billiard ball is shot from one of the corners of a billiards table such that its trajectory forms an angle $\alpha$ with the side of the billiard table, where $\tan \alpha \in \mathbb{Q}$. When the ball hits the border of the billiard table, it reflects elastically. Prove that the ball will necessarily fall into a ho... | [
"\\text{The ball will eventually fall into a hole.}"
] | [
"A billiard ball is shot from one of the corners of a billiards table such that its trajectory forms an angle $\\alpha$ with the side of the billiard table, where $\\tan \\alpha \\in \\mathbb{Q}$. When the ball hits the border of the billiard table, it reflects elastically. Prove that the ball will necessarily fall... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-e9b8da092bfb64eb790d01f2153030a3e2ea755cceb4bcc94b8665510315cecc | A billiard ball strikes ray \( BC \) at point \( C \) with an angle of incidence \( \alpha \). The ball continues its path, bouncing off line segments \( AB \) and \( BC \) according to the rule: angle of incidence equals angle of reflection. If \( AB = BC \), determine the number of times the ball will bounce off the ... | [
"\\( \\left\\lceil \\frac{180 - \\alpha}{\\beta} \\right\\rceil \\)"
] | [
"A billiard ball strikes ray \\( BC \\) at point \\( C \\) with an angle of incidence \\( \\alpha \\). The ball continues its path, bouncing off line segments \\( AB \\) and \\( BC \\) according to the rule: angle of incidence equals angle of reflection. If \\( AB = BC \\), determine the number of times the ball wi... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-03e0a2840a904f582162e0599da3c264dcc0cfbbc8f5e097ee9f72f6a1711ed6 | A bin contains 100 Style A books, 100 Style B books, and 100 Style C books. John will select 3 notebooks from the bin. How many different selections of 3 notebook styles are possible? (Note: the order in which the notebooks are selected does not matter.) | [
"10"
] | [
"A bin contains 100 Style A books, 100 Style B books, and 100 Style C books. John will select 3 notebooks from the bin. How many different selections of 3 notebook styles are possible? (Note: the order in which the notebooks are selected does not matter.)"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 32
} | 32 | 0 | 0 |
omr-ws-c7fd97b8ab7983949667f64efd3efb02071b62602785107f67e0f8d75a753ef7 | A bin contains 100 balls: 30 green, 45 yellow, and 25 blue. If two balls are selected from the bin without replacement, what is the probability that one will be green and one will be blue? | [
"\\(\\frac{5}{33}\\)"
] | [
"A bin contains 100 balls: 30 green, 45 yellow, and 25 blue. If two balls are selected from the bin without replacement, what is the probability that one will be green and one will be blue?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-8a529a10062d233d1d1ec6f0240fd81dde3016e38b4d0219fc0f485879b697c9 | A bin contains 100 balls: 30 green, 45 yellow, and 25 blue. Two balls are drawn in succession without replacement. What is the probability that the first drawn is blue given that the second is green? | [
"\\(\\frac{25}{99}\\)"
] | [
"A bin contains 100 balls: 30 green, 45 yellow, and 25 blue. Two balls are drawn in succession without replacement. What is the probability that the first drawn is blue given that the second is green?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-86aaf4ab481b91b6499be85104501a7b561dfb462855d6684394ecd29cb89037 | A bin contains 16 balls of two colors, black and white. When four balls are drawn from the bin without replacement, the probability of getting exactly two white balls and two black balls is 1/20. Given that the bin contains more white balls than black balls, how many black balls were originally in the bin? | [
"2"
] | [
"A bin contains 16 balls of two colors, black and white. When four balls are drawn from the bin without replacement, the probability of getting exactly two white balls and two black balls is 1/20. Given that the bin contains more white balls than black balls, how many black balls were originally in the bin?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d60185e2da51bee1be48d87616f0839af0d9675a75d79bf5df7b9dd1aa4fbf72 | A bin contains 20 red balls, 30 green balls, and 60 blue balls. The balls are drawn randomly from the bin, one at a time, without replacement. What is the probability that the color of the first ball that is exhausted is red? | [
"\\dfrac{117}{220}"
] | [
"A bin contains 20 red balls, 30 green balls, and 60 blue balls. The balls are drawn randomly from the bin, one at a time, without replacement. What is the probability that the color of the first ball that is exhausted is red?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-50c99b8a4c08955d7a6713fad42428a1d1092f8b5700c0bb7c900f6297dbab26 | A bin has 10 red balls and 8 blue balls. We randomly draw out 6 balls, one at a time without replacement. What is the probability that at some point we choose two consecutive balls that are red? | [
"78.28 \\%"
] | [
"A bin has 10 red balls and 8 blue balls. We randomly draw out 6 balls, one at a time without replacement. What is the probability that at some point we choose two consecutive balls that are red?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-8dc924ac6f85adaa47b20875ada1b9a15fc12277155005f25cf422151ff66590 | A bin has 10 red balls and 8 blue balls. We randomly draw out 6 balls, one at a time, without replacement. What is the probability that, at some point, we choose two consecutive balls that are red? | [
"\\(\\frac{181}{221}\\)"
] | [
"A bin has 10 red balls and 8 blue balls. We randomly draw out 6 balls, one at a time, without replacement. What is the probability that, at some point, we choose two consecutive balls that are red?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1e91bc023f7768574084fd03708a70860b07a8c7ed0cb20aeb9f4f34340ad8d8 | A bin has 5 green balls and $k$ purple balls in it, where $k$ is an unknown positive integer. A ball is drawn at random from the bin. If a green ball is drawn, the player wins 2 dollars, but if a purple ball is drawn, the player loses 2 dollars. If the expected amount won for playing the game is 50 cents, then what is ... | [
"k=3"
] | [
"A bin has 5 green balls and $k$ purple balls in it, where $k$ is an unknown positive integer. A ball is drawn at random from the bin. If a green ball is drawn, the player wins 2 dollars, but if a purple ball is drawn, the player loses 2 dollars. If the expected amount won for playing the game is 50 cents, then ... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e1a3a4959375197f9c0dc5e35749f57b0246dc57c70f0c484f78ce38e8fb9c69 | A bin has 8 black balls and 7 white balls. 3 of the balls are drawn at random. What is the probability of drawing 2 of one color and 1 of the other color? | [
"\\frac{4}{5}"
] | [
"A bin has 8 black balls and 7 white balls. 3 of the balls are drawn at random. What is the probability of drawing 2 of one color and 1 of the other color?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b6beae87d95ef72bca918cbd5a362a0e3af4fd2687c4091a7d0c32d1f8f6ff2c | A bin of one hundred marbles contains two golden marbles. Shen, Li, and Park take turns, in that order, removing marbles from the bin. If a player draws a golden marble, the game ends and the player wins. What is the probability that Shen wins? | [
"\\dfrac{17}{50}"
] | [
"A bin of one hundred marbles contains two golden marbles. Shen, Li, and Park take turns, in that order, removing marbles from the bin. If a player draws a golden marble, the game ends and the player wins. What is the probability that Shen wins?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-90a1c9cff9c4402b2040fe3599d023fbb1a45d9b31ba58cb7e29fb49117f6c70 | A binary operation $\diamondsuit$ has the properties that $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ and that $a\,\diamondsuit \,a=1$ for all nonzero real numbers $a, b,$ and $c$. (Here $\cdot$ represents multiplication). Find the solution to the equation $2016 \,\diamondsuit\, (6\,\diamon... | [
"\\frac{25}{84}"
] | [
"A binary operation $\\diamondsuit$ has the properties that $a\\,\\diamondsuit\\, (b\\,\\diamondsuit \\,c) = (a\\,\\diamondsuit \\,b)\\cdot c$ and that $a\\,\\diamondsuit \\,a=1$ for all nonzero real numbers $a, b,$ and $c$. (Here $\\cdot$ represents multiplication). Find the solution to the equation $2016 \\,\\dia... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9ff5ecbebab128209b455df53f5b34245e004b6efb7b0e8b77afc893695fa565 | A binary operation $\diamondsuit$ has the properties that $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ and that $a\,\diamondsuit \,a=1$ for all nonzero real numbers $a, b,$ and $c$. (Here $\cdot$ represents multiplication). The solution to the equation $2016 \,\diamondsuit\, (6\,\diamondsuit... | [
"109"
] | [
"A binary operation $\\diamondsuit$ has the properties that $a\\,\\diamondsuit\\, (b\\,\\diamondsuit \\,c) = (a\\,\\diamondsuit \\,b)\\cdot c$ and that $a\\,\\diamondsuit \\,a=1$ for all nonzero real numbers $a, b,$ and $c$. (Here $\\cdot$ represents multiplication). The solution to the equation $2016 \\,\\diamonds... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-4e594f2e3df47a7d2d9327f22612f37c4a0dab97753fe8610b5e682281207a8d | A binary operation $\diamondsuit$ has the properties that $a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c$ and that $a\,\diamondsuit \,a=1$ for all nonzero real numbers $a, b,$ and $c$. (Here $\cdot$ represents multiplication). The solution to the equation $2016 \,\diamondsuit\, (6\,\diamondsuit... | [
"109"
] | [
"A binary operation $\\diamondsuit$ has the properties that $a\\,\\diamondsuit\\, (b\\,\\diamondsuit \\,c) = (a\\,\\diamondsuit \\,b)\\cdot c$ and that $a\\,\\diamondsuit \\,a=1$ for all nonzero real numbers $a, b,$ and $c$. (Here $\\cdot$ represents multiplication). The solution to the equation $2016 \\,\\diamonds... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 40
} | 72 | 0 | 0 |
omr-ws-46b60bc678c4f786be1abb04cd3451aee4e23436e703e4ea4afd55a3115f0f45 | A binary operation $\diamondsuit$ has the properties that $a\diamondsuit(b\diamondsuit c)=(a\diamondsuit b)\cdot c$ and that $a\diamondsuit a=1$ for all nonzero real numbers $a$, $b$, and $c$. (Here the dot $\cdot$ represents the usual multiplication operation.) The solution to the equation $2016\diamondsuit(6\diamonds... | [
"109"
] | [
"A binary operation $\\diamondsuit$ has the properties that $a\\diamondsuit(b\\diamondsuit c)=(a\\diamondsuit b)\\cdot c$ and that $a\\diamondsuit a=1$ for all nonzero real numbers $a$, $b$, and $c$. (Here the dot $\\cdot$ represents the usual multiplication operation.) The solution to the equation $2016\\diamondsu... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-43817d64ecfbfa826e00ec4cb7f53e0ee6bf4667ab9de37aa2b9d48b4260c218 | A binary operation, *, is defined on all x, y satisfying: - \( x * x = x + 2 \) - \( x * y = y * x \) - \( \frac{x * (x + y)}{x * y} = \frac{x + y}{y} \) Find \( 11 * 8 \). | [
"264"
] | [
"A binary operation, *, is defined on all x, y satisfying:\n- \\( x * x = x + 2 \\)\n- \\( x * y = y * x \\)\n- \\( \\frac{x * (x + y)}{x * y} = \\frac{x + y}{y} \\)\nFind \\( 11 * 8 \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-6d7e8d94eb39e602fc80de29b3facfd7d9be9fa345f08fcd7efb4b8375137cd1 | A binary sequence \( x = x_1, x_2, \dots, x_{2015} \) (where \( x_i \in \{0,1\} \) for \( i = 1, 2, \dots, 2015 \)) is called good if there exists a unique binary sequence \( y = y_1, y_2, \dots, y_{2015} \), not equal to \( x \), such that every sequence obtained from \( x \) by deleting one of its elements can also b... | [
"4028"
] | [
"A binary sequence \\( x = x_1, x_2, \\dots, x_{2015} \\) (where \\( x_i \\in \\{0,1\\} \\) for \\( i = 1, 2, \\dots, 2015 \\)) is called good if there exists a unique binary sequence \\( y = y_1, y_2, \\dots, y_{2015} \\), not equal to \\( x \\), such that every sequence obtained from \\( x \\) by deleting one of ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-3825b1708aee7ec60624301059a01269dd2abecae82e445ee1174aba965aa481 | A binary sequence is called $friendly$ if each term is adjacent to at least one term that is equal to $1$. Let $F_{n}$ denote the number of $friendly$ binary sequences with $n$ terms. Find the smallest positive integer $n \geq 2$ such that $F_{n} > 100$. | [
"10"
] | [
"A binary sequence is called $friendly$ if each term is adjacent to at least one term that is equal to $1$. Let $F_{n}$ denote the number of $friendly$ binary sequences with $n$ terms. Find the smallest positive integer $n \\geq 2$ such that $F_{n} > 100$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-87550771f931656b928fd81f18363ec1e965ffb0ff9d575285849a3f3d2b5339 | A biologist catches a random sample of 60 fish from a lake, tags them, and releases them. Six months later, she catches a random sample of 70 fish and finds that 3 are tagged. She assumes that 25% of the fish in the lake on the earlier date have died or moved away and that 40% of the fish on the later date have arrived... | [
"840"
] | [
"A biologist catches a random sample of 60 fish from a lake, tags them, and releases them. Six months later, she catches a random sample of 70 fish and finds that 3 are tagged. She assumes that 25% of the fish in the lake on the earlier date have died or moved away and that 40% of the fish on the later date have ar... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 16
} | 31 | 0 | 0 |
omr-ws-1593b062b528d8de8e6d2d27f40fc672e9e12a5ba84fccb97100b2ffb3e23824 | A biologist catches and tags 60 fish in a lake on May 1 and releases them. On September 1, she catches a random sample of 70 fish and finds that 3 of them are tagged. Assuming that $25\%$ of the fish from May 1 are no longer in the lake on September 1, and $40\%$ of the fish in the September 1 sample were not in the la... | [
"840"
] | [
"A biologist catches and tags 60 fish in a lake on May 1 and releases them. On September 1, she catches a random sample of 70 fish and finds that 3 of them are tagged. Assuming that $25\\%$ of the fish from May 1 are no longer in the lake on September 1, and $40\\%$ of the fish in the September 1 sample were not in... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-40b5b83572d035852c400edcdf663d56e00efc5c6c0735558fcc840037e5d258 | A biologist catches and tags 60 fish in a lake on May 1 and releases them. On September 1, she catches a random sample of 70 fish and finds that 3 of them are tagged. Assuming that 25% of the fish from May 1 are no longer in the lake on September 1, and 40% of the fish in the lake on September 1 were not in the lake on... | [
"840"
] | [
"A biologist catches and tags 60 fish in a lake on May 1 and releases them. On September 1, she catches a random sample of 70 fish and finds that 3 of them are tagged. Assuming that 25% of the fish from May 1 are no longer in the lake on September 1, and 40% of the fish in the lake on September 1 were not in the la... | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 37
} | 69 | 0 | 0 |
omr-ws-402dbdcbb9c4f4ad44a55f0f5dfa68171f9bf34290a77745f49c3c1a79bdd647 | A biologist catches and tags 60 fish in a lake on May 1st and releases them. On September 1st, she catches a random sample of 70 fish and finds that 3 of them are tagged. Assuming that 25% of the fish from May 1st are no longer in the lake on September 1st, and 40% of the fish in the lake on September 1st were not in t... | [
"840"
] | [
"A biologist catches and tags 60 fish in a lake on May 1st and releases them. On September 1st, she catches a random sample of 70 fish and finds that 3 of them are tagged. Assuming that 25% of the fish from May 1st are no longer in the lake on September 1st, and 40% of the fish in the lake on September 1st were not... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-595a4ba3db763353d5487d8fbe68a5796e6484e35ce8c40ff9b381e31805d61b | A biologist found a pond with frogs. The 50 lightest frogs represented 30% of the total mass of all the frogs in the pond, while the 44 heaviest frogs represented 27% of the total mass. How many frogs were in the pond? | [
"165"
] | [
"A biologist found a pond with frogs. The 50 lightest frogs represented 30% of the total mass of all the frogs in the pond, while the 44 heaviest frogs represented 27% of the total mass. How many frogs were in the pond?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-e2e16ce61e6263d62424d4055c67d9fad4335b12584c72c532cd7a0355084f31 | A biologist has $n \ge 3$ Petri dishes arranged in a line. Initially, some of them contain one bacterium each, and others are empty. The biologist performs the following experiments repeatedly: 1. Each bacterium divides into 2, with one bacterium remaining in its Petri dish and the other moving to the next one. 2. If a... | [
"\\( 2^{n-2} \\)"
] | [
"A biologist has $n \\ge 3$ Petri dishes arranged in a line. Initially, some of them contain one bacterium each, and others are empty. The biologist performs the following experiments repeatedly:\n\n1. Each bacterium divides into 2, with one bacterium remaining in its Petri dish and the other moving to the next one... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 8,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-236ec82886e9745d1926b69383029e58745d9d642e8eb487fa6531bc149bc49f | A biologist wants to calculate the number of fish in a lake. On May 1 she catches a random sample of 60 fish, tags them, and releases them. On September 1 she catches a random sample of 70 fish and finds that 3 of them are tagged. To calculate the number of fish in the lake on May 1, she assumes that 25% of these fish ... | [
"840"
] | [
"A biologist wants to calculate the number of fish in a lake. On May 1 she catches a random sample of 60 fish, tags them, and releases them. On September 1 she catches a random sample of 70 fish and finds that 3 of them are tagged. To calculate the number of fish in the lake on May 1, she assumes that 25% of these ... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 25
} | 57 | 0 | 0 |
omr-ws-61c663cfc0a87151ad1ae414ceebdbc79b38896cbd0116ad4dcde2529fead178 | A biopoint is placed at a point on a line. After one second, it divides into 2 biopoints, with one moving 1 centimeter to the left and the other 1 centimeter to the right. This process repeats every second. When two biopoints end up in the same location, they both disappear. Let \( n \) be a positive integer with \( k ... | [
"2^k"
] | [
"A biopoint is placed at a point on a line. After one second, it divides into 2 biopoints, with one moving 1 centimeter to the left and the other 1 centimeter to the right. This process repeats every second. When two biopoints end up in the same location, they both disappear. Let \\( n \\) be a positive integer wit... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0c63b1863f83d7a522a2d32940c778874300f5b6918d6750ce9d0f8dd760f660 | A bird flies from an island \( P \) located near a circular beach centered at \( O \) with radius \( R \). The bird flies to a point \( T \) on the coastline, then along the coastline to its nest at \( A \). The island \( P \) is located at the midpoint of the line connecting \( O \) and the nearest point \( B \) on th... | [
"\\dfrac{\\pi}{3}"
] | [
"A bird flies from an island \\( P \\) located near a circular beach centered at \\( O \\) with radius \\( R \\). The bird flies to a point \\( T \\) on the coastline, then along the coastline to its nest at \\( A \\). The island \\( P \\) is located at the midpoint of the line connecting \\( O \\) and the nearest ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-1933e4b97c17b3e3232929e33f4221065b47b5734cf61cfc0193a6c3475088ab | A bird is at point (0,0) and can travel one unit vertically or horizontally with equal probability, restricted to the space where \(0 \leq x \leq 10\) and \(0 \leq y \leq 10\). What is the expected number of moves it will take for the bird to reach point (5,5)? | [
"100"
] | [
"A bird is at point (0,0) and can travel one unit vertically or horizontally with equal probability, restricted to the space where \\(0 \\leq x \\leq 10\\) and \\(0 \\leq y \\leq 10\\). What is the expected number of moves it will take for the bird to reach point (5,5)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 3
} | 13 | 0 | 0 |
omr-ws-668704f2c441542a9b6f9ee8f45db0aa2c34e2f6c29b91d4d3491b11a99111bc | A bird named Rainbow has $n$ colors and its colors in two consecutive days are not equal. There doesn't exist four days $a, b, c, d$ such that $a < b < c < d$ and the bird has the same color on days $a$ and $c$, and the same color on days $b$ and $d$, different from the colors on days $a$ and $c$. What is the maximum n... | [
"2n - 1"
] | [
"A bird named Rainbow has $n$ colors and its colors in two consecutive days are not equal. There doesn't exist four days $a, b, c, d$ such that $a < b < c < d$ and the bird has the same color on days $a$ and $c$, and the same color on days $b$ and $d$, different from the colors on days $a$ and $c$. What is the maxi... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ff65bd7cfb21509e20dc256ca1dfedbb277ad8486357257a419cc8006bf1b72e | A bird named Rainbow has $n$ colors and its colors in two consecutive days are not the same. Additionally, there do not exist four days $i, j, k, l$ such that $i < j < k < l$ and the bird has the same color on days $i$ and $k$, and the same color on days $j$ and $l$, different from the colors on days $i$ and $k$. What ... | [
"\\( 2n-1 \\)"
] | [
"A bird named Rainbow has $n$ colors and its colors in two consecutive days are not the same. Additionally, there do not exist four days $i, j, k, l$ such that $i < j < k < l$ and the bird has the same color on days $i$ and $k$, and the same color on days $j$ and $l$, different from the colors on days $i$ and $k$. ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-edcce7cade00c7064931a05fa4bca44b778f677fea92fabb1fb9bbdef34ea822 | A bird named Rainbow has \( n \) colors and its colors on two consecutive days are not the same. There do not exist 5 days \( i, j, k, l, m \) such that \( i < j < k < l < m \) and the bird has the same color on days \( i, k, \) and \( m \), and the same color on days \( j \) and \( l \), where these colors are differe... | [
"14"
] | [
"A bird named Rainbow has \\( n \\) colors and its colors on two consecutive days are not the same. There do not exist 5 days \\( i, j, k, l, m \\) such that \\( i < j < k < l < m \\) and the bird has the same color on days \\( i, k, \\) and \\( m \\), and the same color on days \\( j \\) and \\( l \\), where these... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-d08b05454ddbd37993a7862cc4b38d1583a242efb501bcd2556054489cf728b9 | A bird sits on each of the 8 vertices of a cube. Whenever a gunshot is fired, each bird flies to the diagonally opposite vertex within the same face. How many ways are there for the birds to fly such that each vertex still has one bird on it after the gunshot? | [
"81"
] | [
"A bird sits on each of the 8 vertices of a cube. Whenever a gunshot is fired, each bird flies to the diagonally opposite vertex within the same face. How many ways are there for the birds to fly such that each vertex still has one bird on it after the gunshot?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 27,
"genselect": 6,
"tir": 5
} | 38 | 0 | 0 |
omr-ws-90882c2e4fbd292e0611469f76ed8599a193a7dcfb35359eefc8e93b52a99092 | A bird starts with 300 ml of blood at 100 degrees in its body and 50 ml of blood at 0 degrees in its feet. Every minute, 50 ml of blood flows from the body to the feet, and 50 ml of blood at 40% of the body temperature flows from the feet to the body. The bird feels cold once its internal body temperature (not includin... | [
"5"
] | [
"A bird starts with 300 ml of blood at 100 degrees in its body and 50 ml of blood at 0 degrees in its feet. Every minute, 50 ml of blood flows from the body to the feet, and 50 ml of blood at 40% of the body temperature flows from the feet to the body. The bird feels cold once its internal body temperature (not inc... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d17454eafaeb3338f6f139862b6eab565da9407f5dcd2b7b34cc45128d515925 | A blackboard contains 68 pairs of nonzero integers. For each positive integer $k$, at most one of the pairs $(k, k)$ and $(-k, -k)$ is written on the blackboard. A student erases some of the 136 integers, subject to the condition that no two erased integers may add to 0. The student scores one point for each of the 68 ... | [
"43"
] | [
"A blackboard contains 68 pairs of nonzero integers. For each positive integer $k$, at most one of the pairs $(k, k)$ and $(-k, -k)$ is written on the blackboard. A student erases some of the 136 integers, subject to the condition that no two erased integers may add to 0. The student scores one point for each of th... | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-42bd31abe752b938248bd041629e8f42c59725855768ff135c843315d7109934 | A blackboard contains 68 pairs of nonzero integers. Suppose that for each positive integer \( k \), at most one of the pairs \((k, k)\) and \((-k, -k)\) is written on the blackboard. A student erases some of the 136 integers, subject to the condition that no two erased integers may add to 0. The student then scores one... | [
"68"
] | [
"A blackboard contains 68 pairs of nonzero integers. Suppose that for each positive integer \\( k \\), at most one of the pairs \\((k, k)\\) and \\((-k, -k)\\) is written on the blackboard. A student erases some of the 136 integers, subject to the condition that no two erased integers may add to 0. The student then... | [
"aops_c6_high_school_olympiads"
] | [
"genselect"
] | {
"genselect": 4
} | 4 | 0 | 0 |
omr-ws-ac3203c2e11fb77095e909bad423666f49c4a6ade9735f2f258fda46276be45a | A blacksmith wants to connect 5 chains, each containing 3 links, into one long chain by opening several links and then closing them. What is the least number of links that need to be opened? | [
"4"
] | [
"A blacksmith wants to connect 5 chains, each containing 3 links, into one long chain by opening several links and then closing them. What is the least number of links that need to be opened?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-b1082001d71734d3f8b32c1d8d25ef93767bca780ba82b86f7709fe0d0e57afd | A blimp is suspended at a height of 600 feet directly over the line from Soldier Field to the Adler Planetarium. The angle of depression from the blimp to Soldier Field is 42 degrees, and the angle of depression from the blimp to the Adler Planetarium is 33 degrees. Find the distance between Soldier Field and the Adler... | [
"\\( 600(\\tan{42} + \\tan{33}) \\)"
] | [
"A blimp is suspended at a height of 600 feet directly over the line from Soldier Field to the Adler Planetarium. The angle of depression from the blimp to Soldier Field is 42 degrees, and the angle of depression from the blimp to the Adler Planetarium is 33 degrees. Find the distance between Soldier Field and the ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2ee4241cdbd1ed7148dfe40d3178aa39955335404162d8c9e4eded1e7d57969b | A blind man rolls a fair die marked with numbers 1 through 6. After each roll, he erases the number on the face that turns up and marks it with the number corresponding to the roll count (1 for the first roll, 2 for the second roll, etc.). This process is repeated six times. What is the probability that the face initia... | [
"\\(\\frac{15625}{46656}\\)"
] | [
"A blind man rolls a fair die marked with numbers 1 through 6. After each roll, he erases the number on the face that turns up and marks it with the number corresponding to the roll count (1 for the first roll, 2 for the second roll, etc.). This process is repeated six times. What is the probability that the face i... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 16
} | 31 | 0 | 0 |
omr-ws-bf5a4111b0360e770de22d837c9f11dc3faa9ffb2db9cc0e93f4cef99e875edd | A block $Z$ is formed by gluing one face of a solid cube with side length 6 onto one of the circular faces of a right circular cylinder with radius 10 and height 3 so that the centers of the square and circle coincide. If $V$ is the smallest convex region that contains $Z$, calculate $\lfloor \operatorname{vol} V \rflo... | [
"2054"
] | [
"A block $Z$ is formed by gluing one face of a solid cube with side length 6 onto one of the circular faces of a right circular cylinder with radius 10 and height 3 so that the centers of the square and circle coincide. If $V$ is the smallest convex region that contains $Z$, calculate $\\lfloor \\operatorname{vol} ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-360fa44b4fbb235ab39c3b2be5b332a313f55df207e2b0540277cd70bebea791 | A block attached to a spring bounces up and down, with the distance from the block to the ceiling varying sinusoidally between 1.2 meters and 1.8 meters. It takes 1 second for the block to go from the high point to the low point. How far from the ceiling is the block 2.7 seconds after it is at a high point? | [
"1.35"
] | [
"A block attached to a spring bounces up and down, with the distance from the block to the ceiling varying sinusoidally between 1.2 meters and 1.8 meters. It takes 1 second for the block to go from the high point to the low point. How far from the ceiling is the block 2.7 seconds after it is at a high point?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-181b608fd4ec7f85e7888cd30eb75101d7c0eb15acf9e8b2478667c979c7c943 | A block is a rectangular parallelepiped with integer sides \(a, b, c\) which is not a cube. \(N\) blocks are used to form a \(10 \times 10 \times 10\) cube. The blocks may be different sizes. Let \(S\) be the number of distinct sets of dimensions \((a, b, c)\) used among the \(N\) blocks, where \(N \geq 100\). If all b... | [
"1"
] | [
"A block is a rectangular parallelepiped with integer sides \\(a, b, c\\) which is not a cube. \\(N\\) blocks are used to form a \\(10 \\times 10 \\times 10\\) cube. The blocks may be different sizes. Let \\(S\\) be the number of distinct sets of dimensions \\((a, b, c)\\) used among the \\(N\\) blocks, where \\(N ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-27a1e4c775fe374bfd61457c9ee5f7fdc690bb8e2e533bffd922b0ff444a34a8 | A block is called an Euler block if the sum of the squares of any two of its dimensions is a perfect square. For a block with dimensions 44, 117, and 240, calculate the sum of all distinct values of $k$ such that $k^2$ is the sum of the squares of any two dimensions of the block. | [
"636"
] | [
"A block is called an Euler block if the sum of the squares of any two of its dimensions is a perfect square. For a block with dimensions 44, 117, and 240, calculate the sum of all distinct values of $k$ such that $k^2$ is the sum of the squares of any two dimensions of the block."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-376703157bdcd74f4c8b62d5307d00da1dcf22d328957da9b4b5f4a16f9268a0 | A block measuring $6$ cm by $8$ cm by $10$ cm is built using $480$ unit cubes, each $1$ cm on a side. An ant chews a straight line from one vertex of the block to the furthest vertex. How many unit cubes does the ant pass through? The ant does not pass through cubes if he is merely passing through where their edges or ... | [
"20"
] | [
"A block measuring $6$ cm by $8$ cm by $10$ cm is built using $480$ unit cubes, each $1$ cm on a side. An ant chews a straight line from one vertex of the block to the furthest vertex. How many unit cubes does the ant pass through? The ant does not pass through cubes if he is merely passing through where their edge... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 6
} | 21 | 0 | 0 |
omr-ws-a9f02a85a027e238db782d552f63a9bd4d32ab2ddb41989380b7edd569c0f604 | A block of cheese is 16 inches long, 8 inches wide, and 7.5 inches deep. What is the largest number of whole pieces measuring 5 × 3 × 0.5 inches which can be cut from it? | [
"114"
] | [
"A block of cheese is 16 inches long, 8 inches wide, and 7.5 inches deep. What is the largest number of whole pieces measuring 5 × 3 × 0.5 inches which can be cut from it?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 3
} | 11 | 0 | 0 |
omr-ws-3aba097640216c86af9b5ca5a861e9f7655c7c4c6ce25c74d840d18920f4a88e | A block of size \( a \times b \times c \) is composed of \( 1 \times 1 \times 2 \) domino blocks. Assuming that each of the three possible directions of domino blocks occurs equally many times, what are the possible values of \( a \), \( b \), and \( c \)? | [
"\\( a \\), \\( b \\), and \\( c \\) must all be even numbers."
] | [
"A block of size \\( a \\times b \\times c \\) is composed of \\( 1 \\times 1 \\times 2 \\) domino blocks. Assuming that each of the three possible directions of domino blocks occurs equally many times, what are the possible values of \\( a \\), \\( b \\), and \\( c \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-03dc130ba61be26995130642e06299544dfc3c1481397e43c84abf8ebb10a441 | A block of wood has the shape of a right circular cylinder with radius $6$ and height $8$, and its entire surface has been painted blue. Points $A$ and $B$ are chosen on the edge of one of the circular faces of the cylinder so that $\overarc{AB}$ on that face measures $120^\circ$. The block is then sliced in half along... | [
"053"
] | [
"A block of wood has the shape of a right circular cylinder with radius $6$ and height $8$, and its entire surface has been painted blue. Points $A$ and $B$ are chosen on the edge of one of the circular faces of the cylinder so that $\\overarc{AB}$ on that face measures $120^\\circ$. The block is then sliced in h... | [
"MATH_training_set",
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 6
} | 7 | 0 | 0 |
omr-ws-631445a468b7abba6e66369c8d1e4a0865dc99e245ebd596b1da79148468432f | A block of wood has the shape of a right circular cylinder with radius $6$ and height $8$, and its entire surface has been painted blue. Points $A$ and $B$ are chosen on the edge of one of the circular faces of the cylinder so that $\overarc{AB}$ on that face measures $120^\text{o}$. The block is then sliced in half al... | [
"53"
] | [
"A block of wood has the shape of a right circular cylinder with radius $6$ and height $8$, and its entire surface has been painted blue. Points $A$ and $B$ are chosen on the edge of one of the circular faces of the cylinder so that $\\overarc{AB}$ on that face measures $120^\\text{o}$. The block is then sliced in ... | [
"MATH_training_set"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1e6210ea7378d9de99a7550482ace6ca4ec88387871372c2ae76b486bf63b92e | A block of wood has the shape of a right circular cylinder with radius 6 and height 8. Points \( A \) and \( B \) are chosen on the edge of one of the circular faces such that the arc \(\overarc{AB}\) measures \(120^\circ\). The block is sliced in half along the plane passing through points \( A \), \( B \), and the ce... | [
"53"
] | [
"A block of wood has the shape of a right circular cylinder with radius 6 and height 8. Points \\( A \\) and \\( B \\) are chosen on the edge of one of the circular faces such that the arc \\(\\overarc{AB}\\) measures \\(120^\\circ\\). The block is sliced in half along the plane passing through points \\( A \\), \\... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 5
} | 7 | 0 | 0 |
omr-ws-df7695f2d7f358295ac822ef2849e4f5faed556fab4675a11a0affb72adada76 | A block wall 100 feet long and 7 feet high will be constructed using blocks that are 1 foot high and either 2 feet long or 1 foot long (no blocks may be cut). The vertical joins in the blocks must be staggered, and the wall must be even on the ends. What is the smallest number of blocks needed to build this wall? | [
"353"
] | [
"A block wall 100 feet long and 7 feet high will be constructed using blocks that are 1 foot high and either 2 feet long or 1 foot long (no blocks may be cut). The vertical joins in the blocks must be staggered, and the wall must be even on the ends. What is the smallest number of blocks needed to build this wall?"... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-c9e8e60c330156d3a2bd84871791b130c982d5287dfe53c676e69af748959718 | A bluebird can eat all the feed in a birdfeeder in 6 hours. At 9:00 am, the bluebird leaves and a finch arrives, which can eat all the remaining feed in 7 hours. How much time would it take for both birds to eat all the feed if they started eating at the same time? | [
"3.5 hours"
] | [
"A bluebird can eat all the feed in a birdfeeder in 6 hours. At 9:00 am, the bluebird leaves and a finch arrives, which can eat all the remaining feed in 7 hours. How much time would it take for both birds to eat all the feed if they started eating at the same time?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-b5f275c7e7cd16f647358ad253d2af12c1e222bbc0365e9938e81796092d45a9 | A board $1 \times k$ is called "guayaco" if: - Each unit square is painted with exactly one of $k$ available colors. - If $\gcd(i, k) > 1$, the $i$th unit square is painted with the same color as the $(i-1)$th unit square. - If $\gcd(i, k) = 1$, the $i$th unit square is painted with the same color as the $(k-i)$th unit... | [
"(a, a) \\text{ for all positive integers } a"
] | [
"A board $1 \\times k$ is called \"guayaco\" if:\n- Each unit square is painted with exactly one of $k$ available colors.\n- If $\\gcd(i, k) > 1$, the $i$th unit square is painted with the same color as the $(i-1)$th unit square.\n- If $\\gcd(i, k) = 1$, the $i$th unit square is painted with the same color as the $... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-c6215ab2d254acc3ffe9f182474b1a9d94f2b33488d601d4a91b3e76ae29b67f | A board $n \times n$ ($n \ge 3$) is divided into $n^2$ unit squares. Integers from $0$ to $n$ are written in each unit square such that the sums of integers in each $2 \times 2$ square are different. Find all values of $n$ for which such boards exist. | [
"\\( n = 3, 4, 5, 6 \\)"
] | [
"A board $n \\times n$ ($n \\ge 3$) is divided into $n^2$ unit squares. Integers from $0$ to $n$ are written in each unit square such that the sums of integers in each $2 \\times 2$ square are different. Find all values of $n$ for which such boards exist."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 4
} | 11 | 0 | 0 |
omr-ws-285299aded476b5505a08196b94d0777100e938d816bb1e2b4ec503e10fffc7b | A board $n \times n$ is colored black and white like a chessboard. The following steps are permitted: Choose a rectangle inside the board (consisting of entire cells) whose side lengths are both odd or both even, but not both equal to $1$, and invert the colors of all cells inside the rectangle. Determine the values of... | [
"\\( n > 2 \\)"
] | [
"A board $n \\times n$ is colored black and white like a chessboard. The following steps are permitted: Choose a rectangle inside the board (consisting of entire cells) whose side lengths are both odd or both even, but not both equal to $1$, and invert the colors of all cells inside the rectangle. Determine the val... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-15494e39c853ef7bb4b99583bdf6cb406c229b7f11e11e949bf25652a3503991 | A board contains the integers 1, 2, ..., 10. Anna repeatedly erases two numbers \( a \) and \( b \) and replaces it with \( a + b \), gaining \( ab(a + b) \) lollipops in the process. She stops when there is only one number left in the board. Assuming Anna uses the best strategy to get the maximum number of lollipops, ... | [
"54450"
] | [
"A board contains the integers 1, 2, ..., 10. Anna repeatedly erases two numbers \\( a \\) and \\( b \\) and replaces it with \\( a + b \\), gaining \\( ab(a + b) \\) lollipops in the process. She stops when there is only one number left in the board. Assuming Anna uses the best strategy to get the maximum number o... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-15bf96ecfa0255d494aa64eba75bfbf8a532ce507baa2a2adf1b454e3a56cc22 | A board game spinner is divided into four regions labeled $A$, $B$, $C$, and $D$. The probability of the arrow stopping on region $A$ is $\frac{3}{8}$, the probability of it stopping in $B$ is $\frac{1}{4}$, and the probability of it stopping in region $C$ is equal to the probability of it stopping in region $D$. What ... | [
"\\frac{3}{16}"
] | [
"A board game spinner is divided into four regions labeled $A$, $B$, $C$, and $D$. The probability of the arrow stopping on region $A$ is $\\frac{3}{8}$, the probability of it stopping in $B$ is $\\frac{1}{4}$, and the probability of it stopping in region $C$ is equal to the probability of it stopping in region $D... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c19e6acdd0440423eda2269dbf41833bd07cb5bee31ff2e0a8f66cc5e876bf3a | A board game spinner is divided into three parts labeled $A$, $B$ and $C$. The probability of the spinner landing on $A$ is $\frac{1}{3}$ and the probability of the spinner landing on $B$ is $\frac{5}{12}$. What is the probability of the spinner landing on $C$? Express your answer as a common fraction. | [
"\\frac{1}{4}"
] | [
"A board game spinner is divided into three parts labeled $A$, $B$ and $C$. The probability of the spinner landing on $A$ is $\\frac{1}{3}$ and the probability of the spinner landing on $B$ is $\\frac{5}{12}$. What is the probability of the spinner landing on $C$? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2f5072fd9c480d18dfffdca63caeff3ed5e95616fbb8d1248b159bc797279277 | A board game spinner is divided into three regions labeled $A$, $B$ and $C$. The probability of the arrow stopping on region $A$ is $\frac{1}{2}$ and on region $B$ is $\frac{1}{5}$. What is the probability of the arrow stopping on region $C$? Express your answer as a common fraction. | [
"\\frac{3}{10}"
] | [
"A board game spinner is divided into three regions labeled $A$, $B$ and $C$. The probability of the arrow stopping on region $A$ is $\\frac{1}{2}$ and on region $B$ is $\\frac{1}{5}$. What is the probability of the arrow stopping on region $C$? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4b01e6f232aea5406f6d60dd30396ebe5f1e91853048aaffbdd58fce11052d02 | A board game spinner is divided into three regions labeled $A$, $B$ and $C$. The probability of the arrow stopping on region $A$ is $\frac{1}{3}$ and on region $B$ is $\frac{1}{2}$. What is the probability of the arrow stopping on region $C$? Express your answer as a common fraction. | [
"\\frac{1}{6}"
] | [
"A board game spinner is divided into three regions labeled $A$, $B$ and $C$. The probability of the arrow stopping on region $A$ is $\\frac{1}{3}$ and on region $B$ is $\\frac{1}{2}$. What is the probability of the arrow stopping on region $C$? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8a63a9c3de62a8dc5ae733af6c67795ea21695ce41ffdd2120c9dd585dc60de3 | A board has 4 rows and 4 columns. A piece starts at the bottom left square $(1, 1)$ and must move to the upper right square $(4, 4)$. The piece can only move to the square immediately above or immediately to the right. How many different paths can the piece take to complete this route? | [
"20"
] | [
"A board has 4 rows and 4 columns. A piece starts at the bottom left square $(1, 1)$ and must move to the upper right square $(4, 4)$. The piece can only move to the square immediately above or immediately to the right. How many different paths can the piece take to complete this route?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-8643fd3ecf0f98f1535544d12dea85071205552bae4be55ab85505c6e675be1c | A board has \( 2n+1 \) holes in a line, with \( n \) red pegs starting in the first \( n \) holes, followed by a gap, then \( n \) blue pegs in the remaining \( n \) holes. The pegs may be moved either one hole forward into an empty hole or a peg of one color may jump over a peg of the other color into an empty hole on... | [
"n(n + 2)"
] | [
"A board has \\( 2n+1 \\) holes in a line, with \\( n \\) red pegs starting in the first \\( n \\) holes, followed by a gap, then \\( n \\) blue pegs in the remaining \\( n \\) holes. The pegs may be moved either one hole forward into an empty hole or a peg of one color may jump over a peg of the other color into a... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-2c1e998ad95061986239fea297b8a61eeeba19173e60cff1151b116d91f8855b | A board has \(2n + 1\) holes in a line, with \(n\) red pegs starting in the first \(n\) holes, followed by a gap, then \(n\) blue pegs in the remaining \(n\) holes. The pegs may be moved either one hole forward into an empty hole or a peg of one color may jump over a peg of the other color into an empty hole on the oth... | [
"\\(3n^2 + n\\)"
] | [
"A board has \\(2n + 1\\) holes in a line, with \\(n\\) red pegs starting in the first \\(n\\) holes, followed by a gap, then \\(n\\) blue pegs in the remaining \\(n\\) holes. The pegs may be moved either one hole forward into an empty hole or a peg of one color may jump over a peg of the other color into an empty ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b5e13a4f7f43e900660888a6c69d6dd0e38ad2779bab8472f564a486c53e1d78 | A board is a set of $2020 \times 2021 - 20 \times 21$ cells, arranged in a $2020 \times 2021$ grid, missing a $20 \times 21$ rectangle of cells in the center, so that each board has rotational symmetry. Alice has $n$ boards, and tiles each with $1 \times 2$ and $2 \times 1$ dominoes. Bob can then make moves as follows:... | [
"2"
] | [
"A board is a set of $2020 \\times 2021 - 20 \\times 21$ cells, arranged in a $2020 \\times 2021$ grid, missing a $20 \\times 21$ rectangle of cells in the center, so that each board has rotational symmetry. Alice has $n$ boards, and tiles each with $1 \\times 2$ and $2 \\times 1$ dominoes. Bob can then make moves ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 5
} | 13 | 0 | 0 |
omr-ws-8aab661362937a2d5257476aa8bdbc7383f56b57a0c1cbce6d2f77a19226b309 | A board is divided into two parts. Alice and Bob take turns writing a natural number on the board such that the sum of all numbers on the board does not exceed 2020. The game ends when the sum of all numbers on the board equals 2020. The player with the smaller sum of the digits of the numbers they wrote wins. Can eith... | [
"Bob"
] | [
"A board is divided into two parts. Alice and Bob take turns writing a natural number on the board such that the sum of all numbers on the board does not exceed 2020. The game ends when the sum of all numbers on the board equals 2020. The player with the smaller sum of the digits of the numbers they wrote wins. Can... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-0a967248385ad5f8c193b749408f9e3d5a42db77ff70d4d0396a6c7fb2bb9d8d | A board of size $3 \times N$ initially has all of its cells painted white. Let $a(N)$ be the maximum number of cells that can be painted black in such a way that no three consecutive cells (either horizontally, vertically, or diagonally) are painted black. Find the value of $\frac{a(100)}{100}$. | [
"2"
] | [
"A board of size $3 \\times N$ initially has all of its cells painted white. Let $a(N)$ be the maximum number of cells that can be painted black in such a way that no three consecutive cells (either horizontally, vertically, or diagonally) are painted black. Find the value of $\\frac{a(100)}{100}$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-08b6f4b7388b1dee503c00cdcf863cd309185c1377ef0a9ac81e8564d1d8f500 | A board of size \( n \times n \) is divided into \( n^2 \) unit squares. Integers from 0 to \( n \) (inclusive) are written in each unit square such that the sums of integers in each \( 2 \times 2 \) square are different. Find all values of \( n \) for which such a board exists. | [
"\\( n = 1, 2, 3, 4, 5 \\)"
] | [
"A board of size \\( n \\times n \\) is divided into \\( n^2 \\) unit squares. Integers from 0 to \\( n \\) (inclusive) are written in each unit square such that the sums of integers in each \\( 2 \\times 2 \\) square are different. Find all values of \\( n \\) for which such a board exists."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
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