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-4ddbf5784615cacd2654a83fe3d32d9da6fd4511a740c6e77ff7ddd40e05cf4d | A square with sides parallel to the coordinate axes is inscribed in the region defined by \((x, y) : x, y > 0, y \leq -x^3 + 3x\). If the area of the square is expressed as \(A^{1/3} + B^{1/3}\) square units, where \(A\) and \(B\) are integers and \(A > B\), find \(A\) and \(B\). | [
"A = 216, B = -108"
] | [
"A square with sides parallel to the coordinate axes is inscribed in the region defined by \\((x, y) : x, y > 0, y \\leq -x^3 + 3x\\). If the area of the square is expressed as \\(A^{1/3} + B^{1/3}\\) square units, where \\(A\\) and \\(B\\) are integers and \\(A > B\\), find \\(A\\) and \\(B\\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bc7b5eb2d6a2c4007abb7463458ebedde6731bdc5480cc87fcb1e8864a14d439 | A square with vertices $(0,0)$, $(0,1)$, $(1,0)$, and $(1,1)$ is placed on the Cartesian plane. Inside the square, the walls are all glass mirrors. A light beam is at $\left(\frac{\sqrt{2}}{3},0\right)$ and is aimed at an angle of $\alpha$ with respect to the positive $x$-axis. Let $\mathcal{S}$ be the set of all value... | [
"$\\mathfrak{c}$"
] | [
"A square with vertices $(0,0)$, $(0,1)$, $(1,0)$, and $(1,1)$ is placed on the Cartesian plane. Inside the square, the walls are all glass mirrors. A light beam is at $\\left(\\frac{\\sqrt{2}}{3},0\\right)$ and is aimed at an angle of $\\alpha$ with respect to the positive $x$-axis. Let $\\mathcal{S}$ be the set o... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-bee32f4780c7385c70edfe24c518cb586d7b39a4ac7dd902994a59834fb88613 | A square with vertices \((-100, -100)\), \((-100, 100)\), \((100, 100)\), and \((100, -100)\) is given in the coordinate system. A point \((x, y)\) with integer coordinates is selected inside or on the boundary of the square. Find the probability that the product \(x \times y\) ends with the digit 3. | [
"\\(\\frac{800}{40401}\\)"
] | [
"A square with vertices \\((-100, -100)\\), \\((-100, 100)\\), \\((100, 100)\\), and \\((100, -100)\\) is given in the coordinate system. A point \\((x, y)\\) with integer coordinates is selected inside or on the boundary of the square. Find the probability that the product \\(x \\times y\\) ends with the digit 3."... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-c4ab0c43a861b938fe6c1d3bfb85952f2674d8b567e5b49c8b5879b3e740ae9b | A square “rolls” inside a circle of area $\pi$ in the obvious way. That is, when the square has one corner on the circumference of the circle, it is rotated clockwise around that corner until a new corner touches the circumference, then it is rotated around that corner, and so on. The square goes all the way around the... | [
"2 - \\sqrt{2}"
] | [
"A square “rolls” inside a circle of area $\\pi$ in the obvious way. That is, when the square has one corner on the circumference of the circle, it is rotated clockwise around that corner until a new corner touches the circumference, then it is rotated around that corner, and so on. The square goes all the way arou... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-7dd4b9679dabd171c8d714f7ac235cc2433d3d195be8c884495a4407e6edf6db | A square-based pyramid has a bottom layer with 100 balls, the next level with 81 balls, and continues until the top layer has 1 ball. If each ball has a radius of 1 meter, how tall is the pyramid? Express your answer in simplest radical form. | [
"$2+9\\sqrt{2}$"
] | [
"A square-based pyramid has a bottom layer with 100 balls, the next level with 81 balls, and continues until the top layer has 1 ball. If each ball has a radius of 1 meter, how tall is the pyramid? Express your answer in simplest radical form."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 47,
"genselect": 8,
"tir": 22
} | 77 | 0 | 0 |
omr-ws-24f0ee0b5b6f1e6072c0bc47e89094e8482fb4be13981f636dd87c2804df11bc | A square-based pyramid is made out of rubber balls. There are $100$ balls on the bottom level, 81 on the next level, etc., up to $1$ ball on the top level. How many balls are there in the pyramid? | [
"385"
] | [
"A square-based pyramid is made out of rubber balls. There are $100$ balls on the bottom level, 81 on the next level, etc., up to $1$ ball on the top level. How many balls are there in the pyramid?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-f5e912e51dbdd9df391d5d24ca2dace3a8332e1aca3819d23e417cbe2b49fc26 | A square-shaped field is divided into $n$ rectangular farms whose sides are parallel to the sides of the field. What is the greatest value of $n$, if the sum of the perimeters of the farms is equal to $100$ times the perimeter of the field? | [
"10000"
] | [
"A square-shaped field is divided into $n$ rectangular farms whose sides are parallel to the sides of the field. What is the greatest value of $n$, if the sum of the perimeters of the farms is equal to $100$ times the perimeter of the field?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-c0320ac64bac9b7b88d29c6538510fcc1b395183c0cbea29908bf6de355356e8 | A square-shaped quilt is divided into $16 = 4 \times 4$ equal squares. The quilt is called "UMD certified" if each of these $16$ squares is colored red, yellow, or black, such that (i) all three colors are used at least once and (ii) the quilt looks the same when it is rotated $90, 180,$ or $270$ degrees about its cent... | [
"36"
] | [
"A square-shaped quilt is divided into $16 = 4 \\times 4$ equal squares. The quilt is called \"UMD certified\" if each of these $16$ squares is colored red, yellow, or black, such that (i) all three colors are used at least once and (ii) the quilt looks the same when it is rotated $90, 180,$ or $270$ degrees about ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-cd238fe6722f6b52fd5ae5dc5e3b374df8e14648b99de23cab78cabe10fee147 | A squint man is distance \( d \) away from his car. He sees everything with a deflection of 30 degrees. How long will be the path he follows to his car, given that he walks on a logarithmic spiral? | [
"\\(\\frac{2d}{\\sqrt{3}}\\)"
] | [
"A squint man is distance \\( d \\) away from his car. He sees everything with a deflection of 30 degrees. How long will be the path he follows to his car, given that he walks on a logarithmic spiral?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-4bf0971f316f2da50d00ea69a6da4137357d12a5a90b3ea66a20fb5e486ca44d | A squirrel has infinitely many nuts with masses $1g, 2g, 3g, \ldots$. The squirrel places a finite number of nuts into each of 100 bags and writes the total mass of the nuts inside each bag. What is the maximum number of bags that can have the same mass if the squirrel uses no more than 500 nuts in total? | [
"100"
] | [
"A squirrel has infinitely many nuts with masses $1g, 2g, 3g, \\ldots$. The squirrel places a finite number of nuts into each of 100 bags and writes the total mass of the nuts inside each bag. What is the maximum number of bags that can have the same mass if the squirrel uses no more than 500 nuts in total?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-5a63d7a1d3a7adc7460a021a0ce76ba14341779cdcd2184d759a701ec54c351f | A squirrel runs at a rate of $10$ ft/s and Goldilocks walks at a rate of $5$ ft/s. Whenever the squirrel is $20$ ft behind Goldilocks, it runs up until it catches up with her, then stays in place until Goldilocks is $20$ ft ahead again. If Goldilocks and the squirrel start at the same place, how many seconds will pass ... | [
"8"
] | [
"A squirrel runs at a rate of $10$ ft/s and Goldilocks walks at a rate of $5$ ft/s. Whenever the squirrel is $20$ ft behind Goldilocks, it runs up until it catches up with her, then stays in place until Goldilocks is $20$ ft ahead again. If Goldilocks and the squirrel start at the same place, how many seconds will ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-79638374ebe13b48ae14fdb62ede7bc9f755a0fb2597f2c3d38c3e8e3df3a334 | A squirrel travels at a constant 4 miles per hour. How long does it take for this squirrel to travel 1 mile? Express your answer in minutes. | [
"15"
] | [
"A squirrel travels at a constant 4 miles per hour. How long does it take for this squirrel to travel 1 mile? Express your answer in minutes."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9ba51318f0d2d1ba321aa93dcf997a7d5f6adc825928736aca568e914ac781cc | A stack of $2000$ cards is labelled with the integers from $1$ to $2000,$ with different integers on different cards. The cards in the stack are not in numerical order. The top card is removed from the stack and placed on the table, and the next card is moved to the bottom of the stack. The new top card is removed from... | [
"927"
] | [
"A stack of $2000$ cards is labelled with the integers from $1$ to $2000,$ with different integers on different cards. The cards in the stack are not in numerical order. The top card is removed from the stack and placed on the table, and the next card is moved to the bottom of the stack. The new top card is removed... | [
"MATH_training_set"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 11
} | 13 | 0 | 0 |
omr-ws-5d981e3f223defbf067fa7eda298e1b14aa46c09bb0d76707a55c5bf5b384164 | A stack of $n$ books is placed on top of each other. An operation consists of reversing the order of the top $k$ books for $k = 1, 2, \ldots, n$. After performing these operations once, the process is repeated from the beginning. Let $f(n)$ be the minimum number of complete cycles of operations required for the stack o... | [
"6"
] | [
"A stack of $n$ books is placed on top of each other. An operation consists of reversing the order of the top $k$ books for $k = 1, 2, \\ldots, n$. After performing these operations once, the process is repeated from the beginning. Let $f(n)$ be the minimum number of complete cycles of operations required for the s... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 5
} | 8 | 0 | 0 |
omr-ws-2361c8d1346f40ae368a2e48ce164e92451bfdea7d95d2d27b9c36836983fc59 | A stack of 2000 cards is labeled with the integers from 1 to 2000. The cards are not in numerical order. The process of removing the top card and placing it on the table, and then moving the next card to the bottom of the stack, is repeated until all cards are on the table. The cards on the table are in ascending order... | [
"927"
] | [
"A stack of 2000 cards is labeled with the integers from 1 to 2000. The cards are not in numerical order. The process of removing the top card and placing it on the table, and then moving the next card to the bottom of the stack, is repeated until all cards are on the table. The cards on the table are in ascending ... | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 25
} | 25 | 0 | 0 |
omr-ws-fd230d4e93b42d2c41b9d714486aad366d38404ba9ff13b15291634ebb810cca | A stack of 2000 cards labeled from 1 to 2000 is shuffled. Cards are removed from the top and placed on a table, and the next card is moved to the bottom of the stack. This process is repeated until all cards are on the table in ascending order (1, 2, 3, ..., 1999, 2000). How many cards were above the card labeled 1999 ... | [
"1951"
] | [
"A stack of 2000 cards labeled from 1 to 2000 is shuffled. Cards are removed from the top and placed on a table, and the next card is moved to the bottom of the stack. This process is repeated until all cards are on the table in ascending order (1, 2, 3, ..., 1999, 2000). How many cards were above the card labeled ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-d1f34a7eccb7ffa251cd2ae3542c48017bf7e4769f587645dad7e730f3191837 | A stack of 2012 cards labeled from 1 to 2012 is shuffled. Cards are removed from the top and placed on the table, and the next card is moved to the bottom of the stack. This process is repeated until all cards are on the table in ascending order. How many cards were above the card labeled 2011 in the original stack? | [
"1005"
] | [
"A stack of 2012 cards labeled from 1 to 2012 is shuffled. Cards are removed from the top and placed on the table, and the next card is moved to the bottom of the stack. This process is repeated until all cards are on the table in ascending order. How many cards were above the card labeled 2011 in the original stac... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-b134aadacd489a46b89385018b39faeb5f6a0a1116850bea8d8785ed1b0d9b8b | A stack of logs has 12 logs on the bottom row, and one less in each successive row, ending with three logs at the top. How many logs are in the stack? | [
"75"
] | [
"A stack of logs has 12 logs on the bottom row, and one less in each successive row, ending with three logs at the top. How many logs are in the stack?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2026118faff3a7ef2e6f8d93008cc48812f917aafe694514f11eeb0f3243d026 | A stalk of corn is 32 inches tall. It grows by 25% of its height each month for the next three months. What is the height of the stalk, in inches, at the end of the 3-month period? Express your answer as a decimal to the nearest tenth. | [
"62.5"
] | [
"A stalk of corn is 32 inches tall. It grows by 25% of its height each month for the next three months. What is the height of the stalk, in inches, at the end of the 3-month period? Express your answer as a decimal to the nearest tenth."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-ac56921b2e0cf9a58a70f77bd08f8d6f286e0ac9180bf3293011fc47c45ca268 | A stamp collector decides to put all his stamps in a new album. If he puts 20 stamps on one sheet, there will be some left over; if he puts 23 stamps on a sheet, there will be at least one empty sheet left in the album. If the stamp collector is presented with an album of the same kind, each sheet holding 21 stamps, he... | [
"12 sheets"
] | [
"A stamp collector decides to put all his stamps in a new album. If he puts 20 stamps on one sheet, there will be some left over; if he puts 23 stamps on a sheet, there will be at least one empty sheet left in the album. If the stamp collector is presented with an album of the same kind, each sheet holding 21 stamp... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bf2d83091cf36e4320f712e790c280ad38614b41006de7270f5e2a0e0c4c78e3 | A standard 6-sided die is rolled. If it lands on 1 or 2, Eric gets a point. If it lands on 3, 4, 5, or 6, Matthew gets a point. The game ends when the person with the greatest number of points has at least 3 points and at least 2 more points than the other player. What is the probability that Eric wins? | [
"\\(\\frac{23}{135}\\)"
] | [
"A standard 6-sided die is rolled. If it lands on 1 or 2, Eric gets a point. If it lands on 3, 4, 5, or 6, Matthew gets a point. The game ends when the person with the greatest number of points has at least 3 points and at least 2 more points than the other player. What is the probability that Eric wins?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-9f1110d11b18644773ec83c58376ec9a6a4141e5343ecf1b6ff60522c8edd1e0 | A standard 6-sided die is tossed twice. Calculate the following probabilities: a. The probability of rolling doubles. b. The probability of rolling an even number both times. c. The probability of rolling a 4 and a 3, but not necessarily in that order. | [
"a. 1/6, b. 1/4, c. 1/18"
] | [
"A standard 6-sided die is tossed twice. Calculate the following probabilities:\na. The probability of rolling doubles.\nb. The probability of rolling an even number both times.\nc. The probability of rolling a 4 and a 3, but not necessarily in that order."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 26
} | 26 | 0 | 0 |
omr-ws-5e7f745e4f4a4361ebe43042feaac79c79dacc58db23837e3a5d0285c69464a4 | A standard 8 by 8 chessboard is placed on a table, and 64 coins are each placed so that one is on each square of the chessboard. Three players play the game as follows: First, player 1 flips the coins to heads or tails however they want, then player 1 places a key under one of the coins so that the coins on the spaces ... | [
"1"
] | [
"A standard 8 by 8 chessboard is placed on a table, and 64 coins are each placed so that one is on each square of the chessboard. Three players play the game as follows: First, player 1 flips the coins to heads or tails however they want, then player 1 places a key under one of the coins so that the coins on the sp... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-8de7c3cb83668653ffd06f6dc2c9c1d80c07e8c8342904c011fec42b11d66455 | A standard 8.5" x 11" piece of paper is folded such that one vertex coincides with the interior of the opposite short side. What is the maximum possible length of the line segment from the vertex to the point of intersection? | [
"\\( 11 - \\frac{\\sqrt{195}}{2} \\)"
] | [
"A standard 8.5\" x 11\" piece of paper is folded such that one vertex coincides with the interior of the opposite short side. What is the maximum possible length of the line segment from the vertex to the point of intersection?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e3036e93125e7bb5676423525005111a2e3a4fe287a9482cb613d45dd6adc1f0 | A standard \(8 \times 8\) chessboard is glued onto a torus. Find the maximum number of queens that can be placed on the glued board so that none of them attacks another. | [
"8"
] | [
"A standard \\(8 \\times 8\\) chessboard is glued onto a torus. Find the maximum number of queens that can be placed on the glued board so that none of them attacks another."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 2
} | 13 | 0 | 0 |
omr-ws-c311833c67eb85c4be6f9787975a42637839e38d596a933d6b4861d02909c71d | A standard cubical die is rolled four times, with the results \(a\), \(b\), \(c\), and \(d\) respectively. What is the probability that \(a \leq b \leq c \leq d\)? | [
"\\dfrac{7}{72}"
] | [
"A standard cubical die is rolled four times, with the results \\(a\\), \\(b\\), \\(c\\), and \\(d\\) respectively. What is the probability that \\(a \\leq b \\leq c \\leq d\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-65a589ffff67b35289251c33dd105de313cb4688a0a00d3c3cb0459bc0051b93 | A standard deck of 52 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 4 suits ($\spadesuit$, $\heartsuit$, $\diamondsuit$, and $\clubsuit$), such that there is exactly one card for any given rank and suit. Two of the suits ($\spadesuit$ and $\clubsuit$) are black and the other two suits ($\h... | [
"\\dfrac14"
] | [
"A standard deck of 52 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 4 suits ($\\spadesuit$, $\\heartsuit$, $\\diamondsuit$, and $\\clubsuit$), such that there is exactly one card for any given rank and suit. Two of the suits ($\\spadesuit$ and $\\clubsuit$) are black and the other tw... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d7529e5daaeaf9f8f9187b49ef72dc9476b2a874d28ef49377ec2ccdf751b89f | A standard deck of 52 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 4 suits ($\spadesuit$, $\heartsuit$, $\diamondsuit$, and $\clubsuit$), such that there is exactly one card for any given rank and suit. Two of the suits ($\spadesuit$ and $\clubsuit$) are black and the other two suits ($\h... | [
"\\dfrac{1}{13}"
] | [
"A standard deck of 52 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 4 suits ($\\spadesuit$, $\\heartsuit$, $\\diamondsuit$, and $\\clubsuit$), such that there is exactly one card for any given rank and suit. Two of the suits ($\\spadesuit$ and $\\clubsuit$) are black and the other tw... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7e886ec4820d3e093e2a1f7a66f158b1ec9addc91a08a7cee5cb8cb88b981730 | A standard deck of 52 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 4 suits ($\spadesuit$, $\heartsuit$, $\diamondsuit$, and $\clubsuit$), such that there is exactly one card for any given rank and suit. Two of the suits ($\spadesuit$ and $\clubsuit$) are black and the other two suits ($\h... | [
"\\dfrac{13}{51}"
] | [
"A standard deck of 52 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 4 suits ($\\spadesuit$, $\\heartsuit$, $\\diamondsuit$, and $\\clubsuit$), such that there is exactly one card for any given rank and suit. Two of the suits ($\\spadesuit$ and $\\clubsuit$) are black and the other tw... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2ff8cdab6aa631d951598978a86ea893f7c47af0a2527667b92f30ff0904bba6 | A standard deck of 52 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 4 suits ($\spadesuit$, $\heartsuit$, $\diamondsuit$, and $\clubsuit$), such that there is exactly one card for any given rank and suit. Two of the suits ($\spadesuit$ and $\clubsuit$) are black and the other two suits ($\h... | [
"\\dfrac{1}{52}"
] | [
"A standard deck of 52 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 4 suits ($\\spadesuit$, $\\heartsuit$, $\\diamondsuit$, and $\\clubsuit$), such that there is exactly one card for any given rank and suit. Two of the suits ($\\spadesuit$ and $\\clubsuit$) are black and the other tw... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-92322f09d3b2919379e76121839ad35bdcadf15c07665647860cc2e22db1d55e | A standard deck of 52 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 4 suits ($\spadesuit$, $\heartsuit$, $\diamondsuit$, and $\clubsuit$), such that there is exactly one card for any given rank and suit. Two of the suits ($\spadesuit$ and $\clubsuit$) are black and the other two suits ($\h... | [
"\\dfrac{11}{850}"
] | [
"A standard deck of 52 cards has 13 ranks (Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King) and 4 suits ($\\spadesuit$, $\\heartsuit$, $\\diamondsuit$, and $\\clubsuit$), such that there is exactly one card for any given rank and suit. Two of the suits ($\\spadesuit$ and $\\clubsuit$) are black and the other tw... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-29276774eb89f2f9cc72b9c5dd1c37e096d105ebb282b9017186149b6e9c0a0e | A standard deck of 52 cards is split into two subdecks of 39 and 13 cards. You and a friend take turns selecting one of the subdecks to draw cards from the top one-by-one, aiming to minimize the expected number of cards needed to draw the first Ace (there are 4 Aces in total). If a subdeck has no Aces, the other subdec... | [
"13 cards subdeck"
] | [
"A standard deck of 52 cards is split into two subdecks of 39 and 13 cards. You and a friend take turns selecting one of the subdecks to draw cards from the top one-by-one, aiming to minimize the expected number of cards needed to draw the first Ace (there are 4 Aces in total). If a subdeck has no Aces, the other s... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
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] | {
"cot": 9,
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"tir": 13
} | 25 | 0 | 0 |
omr-ws-5a5d453dc570ebdc3c13cd78c2ecf19690d2c973e90df89136c2184113fc6230 | A standard deck of cards contains 52 cards, half of which are red and half of which are black. You draw 2 cards at a time from a well-shuffled deck. If both cards are red, put them onto a "red pile". If both are black, put them on a "black" pile. If the cards are of different colors, tear them up and throw them in the ... | [
"1"
] | [
"A standard deck of cards contains 52 cards, half of which are red and half of which are black. You draw 2 cards at a time from a well-shuffled deck. If both cards are red, put them onto a \"red pile\". If both are black, put them on a \"black\" pile. If the cards are of different colors, tear them up and throw the... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-a235274a4ddedbd20a5c6026e0ba82621c70bdda4711bbda0ab4282b7c103370 | A standard deck of cards has 52 cards divided into 4 suits, each of which has 13 cards. Two of the suits ($\heartsuit$ and $\diamondsuit$, called 'hearts' and 'diamonds') are red, the other two ($\spadesuit$ and $\clubsuit$, called 'spades' and 'clubs') are black. The cards in the deck are placed in random order (usual... | [
"2652"
] | [
"A standard deck of cards has 52 cards divided into 4 suits, each of which has 13 cards. Two of the suits ($\\heartsuit$ and $\\diamondsuit$, called 'hearts' and 'diamonds') are red, the other two ($\\spadesuit$ and $\\clubsuit$, called 'spades' and 'clubs') are black. The cards in the deck are placed in random o... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b4aaf93f7b3f15a34227ad095e2c39030f74631091addfe2dd4094a9edf0b4c0 | A standard deck of cards has 52 cards divided into 4 suits, each of which has 13 cards. Two of the suits ($\heartsuit$ and $\diamondsuit$, called `hearts' and `diamonds') are red, the other two ($\spadesuit$ and $\clubsuit$, called `spades' and `clubs') are black. The cards in the deck are placed in random order (usual... | [
"\\frac{25}{102}"
] | [
"A standard deck of cards has 52 cards divided into 4 suits, each of which has 13 cards. Two of the suits ($\\heartsuit$ and $\\diamondsuit$, called `hearts' and `diamonds') are red, the other two ($\\spadesuit$ and $\\clubsuit$, called `spades' and `clubs') are black. The cards in the deck are placed in random o... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-25b416ee03a1784757dd4b4189cbdb271bea1907535eaff3a8a692b23131ea0e | A standard deck of cards is shuffled and 3 cards are selected. What is the probability that the 3rd card is either a black face card or a queen, if the queen of hearts and queen of spades are the first two cards selected? | [
"\\(\\frac{3}{25}\\)"
] | [
"A standard deck of cards is shuffled and 3 cards are selected. What is the probability that the 3rd card is either a black face card or a queen, if the queen of hearts and queen of spades are the first two cards selected?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-c441b33b45cc0b02240f3df661ab036bc3d6e1f7fa9b2793cba0725f74caeec3 | A standard deck of playing cards (26 red and 26 black cards) is split into two piles, each with at least one card. In pile A, there are six times as many black cards as red cards. In pile B, the number of black cards is a multiple of the number of red cards. How many red cards are in pile B? | [
"22"
] | [
"A standard deck of playing cards (26 red and 26 black cards) is split into two piles, each with at least one card. In pile A, there are six times as many black cards as red cards. In pile B, the number of black cards is a multiple of the number of red cards. How many red cards are in pile B?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 5
} | 5 | 0 | 0 |
omr-ws-eb5bb88fd52ffea3f82244ce1dbfd8a498b0bcbc7ca50e2442bb110d1b0881cd | A standard deck of playing cards with 26 red and 26 black cards is split into two non-empty piles. In pile A, there are four times as many black cards as red cards. In pile B, the number of red cards is an integer multiple of the number of black cards. How many red cards are in pile B? | [
"20"
] | [
"A standard deck of playing cards with 26 red and 26 black cards is split into two non-empty piles. In pile A, there are four times as many black cards as red cards. In pile B, the number of red cards is an integer multiple of the number of black cards. How many red cards are in pile B?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-c565f9672be20b2499b01d759755d31f0ac1ee9c14004586d0110f78209b8598 | A standard die is rolled six times. What is the probability that the product of all six rolls is odd? Express your answer as a common fraction. | [
"\\frac{1}{64}"
] | [
"A standard die is rolled six times. What is the probability that the product of all six rolls is odd? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d5ffdc3c7db6ec9c22408cc4f37e5e08c2b2d1078230f9774eac58d3908dc199 | A standard die is rolled three times. Find the probability that the product of the outcomes is divisible by $8$. Express your answer as a rational number in lowest terms. | [
"\\(\\frac{1}{3}\\)"
] | [
"A standard die is rolled three times. Find the probability that the product of the outcomes is divisible by $8$. Express your answer as a rational number in lowest terms."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-afb433f4aedb7949a37eb4e112c826fa5e187cd3ffb28979f0cf93a3c8b1dd92 | A standard fair 6-sided die is rolled 10 times. Given that the number 1 appears 4 times, what is the probability that no two 1’s appear on consecutive rolls? | [
"\\(\\frac{1}{6}\\)"
] | [
"A standard fair 6-sided die is rolled 10 times. Given that the number 1 appears 4 times, what is the probability that no two 1’s appear on consecutive rolls?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-3b0eda2d979e8072c4c46cc6ea4504b599f8e409549212db493ee0133f64aeb3 | A standard fair 6-sided die is rolled two times successively. Let \(\alpha\) and \(\beta\) be the numbers appearing on the first and second rolls, respectively. Consider the polynomial equation: \[ (x - \alpha)^2 (x - \beta)^2 = x^4 + ax^3 + bx^2 + cx + d \] Find the expected values of \(a\), \(b\), \(c\), and \(d\). | [
"\\(E(a) = -14\\), \\(E(b) = \\frac{238}{3}\\), \\(E(c) = -\\frac{637}{3}\\), \\(E(d) = \\frac{8281}{36}\\)"
] | [
"A standard fair 6-sided die is rolled two times successively. Let \\(\\alpha\\) and \\(\\beta\\) be the numbers appearing on the first and second rolls, respectively. Consider the polynomial equation:\n\n\\[\n(x - \\alpha)^2 (x - \\beta)^2 = x^4 + ax^3 + bx^2 + cx + d\n\\]\n\nFind the expected values of \\(a\\), \... | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 27
} | 27 | 0 | 0 |
omr-ws-417bb7549c578d5b29f976e5d639bb30c82f3205b08d61f52000e5b0dcc09872 | A standard six-sided die is fashioned out of a cube where the diameter of a dot on the die is $\frac{3}{13}$ the length of the side. The dots are painted on using black paint. Compute the area of the die covered by black paint. | [
"\\dfrac{189}{676} \\pi s^2"
] | [
"A standard six-sided die is fashioned out of a cube where the diameter of a dot on the die is $\\frac{3}{13}$ the length of the side. The dots are painted on using black paint. Compute the area of the die covered by black paint."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bca92572bae1e82e9cb3a930b5c84978b5b3547abea78d149c668f7cdf578868 | A standard six-sided die is fashioned out of a cube where the diameter of a dot on the die is $\frac{3}{13}$ the length of the side. The dots are painted on using black paint. Compute the area of the die that is still white. | [
"\\left(6 - \\dfrac{189\\pi}{676}\\right) s^2"
] | [
"A standard six-sided die is fashioned out of a cube where the diameter of a dot on the die is $\\frac{3}{13}$ the length of the side. The dots are painted on using black paint. Compute the area of the die that is still white."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-30432dd483251f5d3524c32235bde97657954ba2631ff30cda75f50bed84143c | A standard six-sided die is rolled, and $P$ is the product of the five numbers that are visible. What is the largest number that is certain to divide $P$? | [
"12"
] | [
"A standard six-sided die is rolled, and $P$ is the product of the five numbers that are visible. What is the largest number that is certain to divide $P$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c4818bf3b513551370bbb8c7302958b96f1480d7bb9b8d18f87106f9f0bc5dcb | A standard six-sided die is rolled, and \( P \) is the product of the five numbers that are visible. What is the largest number that is certain to divide \( P \)? | [
"12"
] | [
"A standard six-sided die is rolled, and \\( P \\) is the product of the five numbers that are visible. What is the largest number that is certain to divide \\( P \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 22
} | 53 | 0 | 0 |
omr-ws-7ad74e95c03bc0c7f8bc504b8f0fbd24ebec0f1ac118dbe7676e2c50242a803f | A standard six-sided fair die is rolled four times. The probability that the product of all four numbers rolled is a perfect square is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | [
"187"
] | [
"A standard six-sided fair die is rolled four times. The probability that the product of all four numbers rolled is a perfect square is $\\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.",
"A standard six-sided fair die is rolled four times. The probability that the product of ... | [
"MATH_training_set",
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 41,
"genselect": 8,
"tir": 49
} | 98 | 0 | 0 |
omr-ws-5c5cbb645d084dd6064ca275a741dc008fadf729a3d54a749c4d52b08a8610db | A standard, fair 6-sided die is rolled 10 times successively. Find the probability that the number 6 is rolled 6 or more times in a row. | [
"\\(\\frac{5}{6^6} - \\frac{4}{6^7}\\)"
] | [
"A standard, fair 6-sided die is rolled 10 times successively. Find the probability that the number 6 is rolled 6 or more times in a row."
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0ea3c5d697b2a66f4954ef2baf8dd6deb05ffde90ae04c361d4eac9ea7803cb6 | A standard, fair, 6-sided die is rolled 8 times. Given that the number 3 appears exactly three times, what is the probability that no two 3's appear on consecutive rolls? | [
"\\(\\frac{5}{14}\\)"
] | [
"A standard, fair, 6-sided die is rolled 8 times. Given that the number 3 appears exactly three times, what is the probability that no two 3's appear on consecutive rolls?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-955c38c92b9ae4baa1f3c9e7650cfc5be4fbb713fc664301a0ccaafc952dea51 | A standing pole has three different flags stuck at varying height levels. The distance between each of the flags increases in geometric progression (GP) as we move up the height of the pole, while the distance between the third flag and the topmost of the pole is the harmonic mean (HM) of the combined three distances. ... | [
"45^\\circ"
] | [
"A standing pole has three different flags stuck at varying height levels. The distance between each of the flags increases in geometric progression (GP) as we move up the height of the pole, while the distance between the third flag and the topmost of the pole is the harmonic mean (HM) of the combined three distan... | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 25,
"genselect": 7
} | 32 | 0 | 0 |
omr-ws-76e1a528d68cb710192ab3371468df9e246e617bc423b56423b51b85b24c0212 | A starship is located in a half-space at a distance $a$ from its boundary. The starship can fly along any trajectory and has a sensor that signals when the boundary plane is reached. Can the starship be guaranteed to reach the boundary with a path no longer than $14a$? | [
"14a"
] | [
"A starship is located in a half-space at a distance $a$ from its boundary. The starship can fly along any trajectory and has a sensor that signals when the boundary plane is reached. Can the starship be guaranteed to reach the boundary with a path no longer than $14a$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 29,
"genselect": 2
} | 31 | 0 | 0 |
omr-ws-6a8cacc6431604bb5dd54231bf8ffea03c8542c3ae4db8714dd3cbb949519b2e | A state requires that all boat licenses consist of the letter A or M followed by any five digits. What is the number of groups of letters and numbers available for boat licenses? | [
"200000"
] | [
"A state requires that all boat licenses consist of the letter A or M followed by any five digits. What is the number of groups of letters and numbers available for boat licenses?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0c98832511185d43d393097946a3440c9069c4140b268bc333b75302388057b1 | A state with ten million cars plans to issue license plates consisting of any four letters followed by an n-digit number. What is the minimum possible value for n to ensure there are enough distinct license plates for all the cars? | [
"2"
] | [
"A state with ten million cars plans to issue license plates consisting of any four letters followed by an n-digit number. What is the minimum possible value for n to ensure there are enough distinct license plates for all the cars?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-309229841efc200ce4624a8220fdfed8b62a4cc93118681a183a0a88ed23517c | A stationery store had a batch of ball pens. On the first day, one half minus 6 pens were sold. On the second day, one half of the remaining pens plus 8 more pens were sold. Finally, 20 ball pens are left. How many ball pens did the stationery store have originally? | [
"76"
] | [
"A stationery store had a batch of ball pens. On the first day, one half minus 6 pens were sold. On the second day, one half of the remaining pens plus 8 more pens were sold. Finally, 20 ball pens are left. How many ball pens did the stationery store have originally?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8fe0b0898451627ab9cb769033db8eff95ced5503d32b8ac9f1258adf89191dc | A stationery store has four sections, each displaying three different items. A number of $n$ people each buy exactly one item from each section. There is at least one section where all items are bought by each group of three people. What is the maximum value of $n$? | [
"9"
] | [
"A stationery store has four sections, each displaying three different items. A number of $n$ people each buy exactly one item from each section. There is at least one section where all items are bought by each group of three people. What is the maximum value of $n$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 2,
"tir": 2
} | 16 | 0 | 0 |
omr-ws-04d53e94cf337cd29f27e0d2b385acfc8f8d6e32e1a9acfe07c897648dda5662 | A statue 4 meters high sits on a column 5.6 meters high. How far from the column must a man, whose eye level is 1.6 meters from the ground, stand in order to see the statue at the greatest angle? | [
"$4\\sqrt 2$ m"
] | [
"A statue 4 meters high sits on a column 5.6 meters high. How far from the column must a man, whose eye level is 1.6 meters from the ground, stand in order to see the statue at the greatest angle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-69fdfa25344598aa31db38f62ed7541356b41c6f473b0fa7731172f54f5aec6a | A statue company uses one roll of wrapping paper to wrap forty 7 inch tall busts of Beethoven. How many rolls are needed to wrap 385 12 inch tall busts? The answer should be an integer. | [
"28"
] | [
"A statue company uses one roll of wrapping paper to wrap forty 7 inch tall busts of Beethoven. How many rolls are needed to wrap 385 12 inch tall busts? The answer should be an integer."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4448d0a10ee4e9a0e90201dc0d2249ca5860db588e865e6b011b13703fa33520 | A steep mountain is inclined at a 74-degree angle to the horizontal and rises to a height of 3400 ft above the surrounding plain. A cable car is to be installed running to the top of the mountain from a point 980 ft out in the plain from the base of the mountain. Find the shortest length of cable needed. Round to the n... | [
"3921.96"
] | [
"A steep mountain is inclined at a 74-degree angle to the horizontal and rises to a height of 3400 ft above the surrounding plain. A cable car is to be installed running to the top of the mountain from a point 980 ft out in the plain from the base of the mountain. Find the shortest length of cable needed. Round to ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 6,
"tir": 10
} | 31 | 0 | 0 |
omr-ws-ad29c2066edd1981b4e07a61979414f8d0f41a77f897374c4e6ede4be4ee81b9 | A stick 5 cm long, a stick 9 cm long, and a third stick $n$ cm long form a triangle. What is the sum of all possible whole number values of $n$? | [
"81"
] | [
"A stick 5 cm long, a stick 9 cm long, and a third stick $n$ cm long form a triangle. What is the sum of all possible whole number values of $n$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9ed8e14f4b41c09d013c320e637054319443af0229ecf4e475a65bcd3941b3e5 | A stick is broken at a point chosen at random along its length. Find the probability that the ratio of the length of the shorter piece to the length of the longer piece is less than \( r \). | [
"\\(\\frac{r}{1+r}\\)"
] | [
"A stick is broken at a point chosen at random along its length. Find the probability that the ratio of the length of the shorter piece to the length of the longer piece is less than \\( r \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-802f87c1c303d52e2009727ea79b7c0d8aa78f861214844df9097ab2d97fc008 | A stick is broken at two random points into three pieces. What is the probability that the three pieces can be used to form a triangle? | [
"\\(\\frac{1}{4}\\)"
] | [
"A stick is broken at two random points into three pieces. What is the probability that the three pieces can be used to form a triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-14046f5f5771490373dc7d4c6098bcb4449d067e1d32cfcc8e369da1d0c1be9d | A stick is randomly cut into three pieces. What is the probability that the pieces can be arranged to form a triangle? | [
"\\(\\frac{1}{4}\\)"
] | [
"A stick is randomly cut into three pieces. What is the probability that the pieces can be arranged to form a triangle?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-383d4794d80e267e82b7233be5d9ceb58e7c567a1bf89801fc30ec75eb5f4b18 | A stick is sawed into two pieces at a randomly chosen point, and each of the two pieces is then also sawed into two at randomly chosen points. What is the probability that the four pieces can be assembled into a quadrilateral? | [
"2 \\ln 2 - 1"
] | [
"A stick is sawed into two pieces at a randomly chosen point, and each of the two pieces is then also sawed into two at randomly chosen points. What is the probability that the four pieces can be assembled into a quadrilateral?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-baadbae983f8c6c86a281095d24daa1ba219f6a20899bbc2c005f46bb088fad1 | A stick of length 1 is broken into two parts at a random point. The smaller part is discarded, and the process is repeated with the remaining part. Find the probability that the length of the remaining stick is no less than $\frac{1}{2}$. | [
"$1-\\ln(2)$"
] | [
"A stick of length 1 is broken into two parts at a random point. The smaller part is discarded, and the process is repeated with the remaining part. Find the probability that the length of the remaining stick is no less than $\\frac{1}{2}$."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-4d3146e1998f621972928aadeb1cfd85f6cf6468a3552989a51eaad89d45f56e | A stick of length 1 unit is cut at two random points to form three segments. What is the mean length of the longest segment? | [
"\\(\\frac{11}{18}\\)"
] | [
"A stick of length 1 unit is cut at two random points to form three segments. What is the mean length of the longest segment?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 18
} | 27 | 0 | 0 |
omr-ws-b082ee57775b3af40d60684dc83d6d15c9fff9045d035058e199080362fec419 | A stick of length 1 unit is cut at two random points to form three segments. What is the mean length of the middle segment? | [
"\\(\\frac{5}{18}\\)"
] | [
"A stick of length 1 unit is cut at two random points to form three segments. What is the mean length of the middle segment?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 12
} | 21 | 0 | 0 |
omr-ws-936ccfc7e805898dedd021fc1b7f0307c41d73862105852dd1fad69efba22351 | A stick of length 1 unit is cut at two random points to form three segments. What is the mean length of the smallest segment? | [
"\\(\\frac{1}{9}\\)"
] | [
"A stick of length 1 unit is cut at two random points to form three segments. What is the mean length of the smallest segment?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 24
} | 39 | 0 | 0 |
omr-ws-5ac6292a4ee65fe99adcb177ca3a467dff419c5f285cfcdf116d248ddad8d97b | A stick of length 10 is marked with 9 evenly spaced marks (so each is one unit apart). An ant is placed at every mark and at the endpoints, randomly facing right or left. All ants start walking simultaneously at a rate of 1 unit per second. If two ants collide, they immediately reverse direction (assume turning takes n... | [
"\\(\\frac{9899}{1024}\\)"
] | [
"A stick of length 10 is marked with 9 evenly spaced marks (so each is one unit apart). An ant is placed at every mark and at the endpoints, randomly facing right or left. All ants start walking simultaneously at a rate of 1 unit per second. If two ants collide, they immediately reverse direction (assume turning ta... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 13
} | 24 | 0 | 0 |
omr-ws-60311bd674a71c0a5de6c9b4437b777b9dd04138bbab0222d692e7b04bf9c931 | A stick of length \( L \) is broken into 3 pieces by choosing 2 random points \( X \) and \( Y \) on the stick uniformly and independently, and breaking the stick at those 2 points. What is the probability that the 3 pieces can be assembled into a triangle? | [
"\\dfrac{1}{4}"
] | [
"A stick of length \\( L \\) is broken into 3 pieces by choosing 2 random points \\( X \\) and \\( Y \\) on the stick uniformly and independently, and breaking the stick at those 2 points. What is the probability that the 3 pieces can be assembled into a triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a987896d5582fad43c8350f2b8600695313f8daf01341e26f414bfa6356d9314 | A sticker covers one fourth of a clock. For what fraction of the day are both the hands not visible? | [
"\\(\\frac{1}{16}\\)"
] | [
"A sticker covers one fourth of a clock. For what fraction of the day are both the hands not visible?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-e0291dd20eee43642a868f1c8e6c71e0a8615c8d42151d50b80899bba7be6b37 | A stock investment went up $25\%$ in 2006. Starting at this increased value, what percent would it have to go down in 2007 to be back to its original price at the beginning of 2006? | [
"20"
] | [
"A stock investment went up $25\\%$ in 2006. Starting at this increased value, what percent would it have to go down in 2007 to be back to its original price at the beginning of 2006?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3d4bb20aeae4a80237c5b6a704c3e82bc45da60e251f35c8463bd9548147ff8a | A stock market investor owns shares in a stock whose present value is 25. She has decided that she must sell her stock if it either goes down to 10 or up to 40. If each change of price is either up 1 point with probability 0.55 or down 1 point with probability 0.45, and the successive changes are independent, what is t... | [
"0.953"
] | [
"A stock market investor owns shares in a stock whose present value is 25. She has decided that she must sell her stock if it either goes down to 10 or up to 40. If each change of price is either up 1 point with probability 0.55 or down 1 point with probability 0.45, and the successive changes are independent, what... | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 3
} | 14 | 0 | 0 |
omr-ws-09855bf62c52dbceb22b76b3720879fe5d0fa2acafc7beeddbcbf714e3197175 | A stock market investor owns shares in a stock whose present value is 25. She has decided that she must sell her stock if it either goes down to 10 or up to 40. If each change of price is either up 1 point with probability 0.55 or down 1 point with probability 0.45, and the successive changes are independent, what is t... | [
"0.953",
"\\dfrac{11^{15}}{11^{15} + 9^{15}}"
] | [
"A stock market investor owns shares in a stock whose present value is 25. She has decided that she must sell her stock if it either goes down to 10 or up to 40. If each change of price is either up 1 point with probability 0.55 or down 1 point with probability 0.45, and the successive changes are independent, what... | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-73c401cad2647ded76e1559f6c99e80069b01003f703f16576c44130b2e55229 | A stock starts at a price of \$100 and changes by 10% or -10% at the end of each day with equal probability. You will sell the stock if its price is strictly more expensive than \$120 or strictly less expensive than \$80. What is the probability that you will sell the stock when its price is greater than \$120? | [
"\\dfrac{1}{2}"
] | [
"A stock starts at a price of \\$100 and changes by 10% or -10% at the end of each day with equal probability. You will sell the stock if its price is strictly more expensive than \\$120 or strictly less expensive than \\$80. What is the probability that you will sell the stock when its price is greater than \\$120... | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 2
} | 14 | 0 | 0 |
omr-ws-dff181274d04b800763fbd133eae9660b6f030f84a371fcb48f458e3ec127941 | A stock tank that measures 10 feet by 1 yard by 1 yard is being filled by a garden hose, and the water level is rising at a rate of 3 inches per hour. How long will it take to fill the tank? | [
"12"
] | [
"A stock tank that measures 10 feet by 1 yard by 1 yard is being filled by a garden hose, and the water level is rising at a rate of 3 inches per hour. How long will it take to fill the tank?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0b9658d54d977e012689504b06be67ee347c98771e5a21f4346f52d24ae40300 | A stockholder owns 5% of a stock company which has issued 900,000 shares, meaning they own 45,000 shares. The company wants to reward this person by giving them a total of 10% of the shares. How much new stock should be issued for this to happen? | [
"50,000"
] | [
"A stockholder owns 5% of a stock company which has issued 900,000 shares, meaning they own 45,000 shares. The company wants to reward this person by giving them a total of 10% of the shares. How much new stock should be issued for this to happen?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-8d202c7bf783c005954f6f2bc5f6c50d951fe5f41414d64e150bc1a4b3cbcd80 | A stone is placed in a square of a chessboard with $n$ rows and $n$ columns. We can alternately undertake two operations: (a) move the stone to a square that shares a common side with the square in which it stands; (b) move it to a square sharing only one common vertex with the square in which it stands. The first step... | [
"\\( n \\) is even."
] | [
"A stone is placed in a square of a chessboard with $n$ rows and $n$ columns. We can alternately undertake two operations: (a) move the stone to a square that shares a common side with the square in which it stands; (b) move it to a square sharing only one common vertex with the square in which it stands. The first... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 8,
"tir": 10
} | 24 | 0 | 0 |
omr-ws-1d6ab7ece12ab6b4f73942319b2688b5cd4f55db8d55d7282c48368508758a78 | A stone is placed on vertex \( A_0 \) of a regular triangle \( A_0A_1A_2 \). When the stone is on vertex \( A_i \) (\( i = 0, 1, 2 \)), a die is thrown. If the outcome is even, the stone moves to vertex \( A_{i+1} \); if odd, it moves to vertex \( A_{i-1} \). Note that \( A_3 = A_0 \) and \( A_{-1} = A_2 \). After repe... | [
"\\( p_{n+1} = \\frac{1 - p_n}{2} \\)"
] | [
"A stone is placed on vertex \\( A_0 \\) of a regular triangle \\( A_0A_1A_2 \\). When the stone is on vertex \\( A_i \\) (\\( i = 0, 1, 2 \\)), a die is thrown. If the outcome is even, the stone moves to vertex \\( A_{i+1} \\); if odd, it moves to vertex \\( A_{i-1} \\). Note that \\( A_3 = A_0 \\) and \\( A_{-1} ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-49bfdd0d0623387d18492292657c9c072b11687582b68cce093c5ab0465b5c5c | A stone is thrown so that it will hit a bird at the top of a pole. However, at the instant the stone is thrown, the bird flies away in a horizontal straight line at the speed of 10 m/s. The stone reaches a height double that of the pole and, in its descent, touches the bird. Find the horizontal component of the velocit... | [
"10"
] | [
"A stone is thrown so that it will hit a bird at the top of a pole. However, at the instant the stone is thrown, the bird flies away in a horizontal straight line at the speed of 10 m/s. The stone reaches a height double that of the pole and, in its descent, touches the bird. Find the horizontal component of the ve... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-b3ab849c1a952671d6398d3238131b41140681a66df9e48822f8750cb7bcd146 | A stone weighing 0.62 kg is thrown with an initial velocity of 5.5 m/s. At its maximum height, its kinetic energy is 2.3 J. Determine the height of the stone at this point. | [
"1.16 m"
] | [
"A stone weighing 0.62 kg is thrown with an initial velocity of 5.5 m/s. At its maximum height, its kinetic energy is 2.3 J. Determine the height of the stone at this point."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-f82dce5532e5b917e25169574208a6065bb11f5238b528da6b3ecc0a4881f254 | A stopping train left point $A$ for point $B$; three hours later an express followed the stopping train. The express overtook the stopping train halfway between $A$ and $B$. By the time the express reached $B$, the stopping train had covered $\frac{13}{16}$ of the total route. How much time will it have taken the stopp... | [
"16 hours"
] | [
"A stopping train left point $A$ for point $B$; three hours later an express followed the stopping train. The express overtook the stopping train halfway between $A$ and $B$. By the time the express reached $B$, the stopping train had covered $\\frac{13}{16}$ of the total route. How much time will it have taken the... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a56928d5ac8a79dd979882534ae175d6c6fc7f66dfe7466be20eef5eb150ee9c | A storage tank ruptures at time \( t = 0 \) and oil leaks from the tank at a rate of \( r(t) = 100e^{-0.01t} \). How much oil leaks during the first hour? | [
"99.5"
] | [
"A storage tank ruptures at time \\( t = 0 \\) and oil leaks from the tank at a rate of \\( r(t) = 100e^{-0.01t} \\). How much oil leaks during the first hour?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 2,
"tir": 16
} | 25 | 0 | 0 |
omr-ws-78f8129e5b57896eac4a5d87362bfb94d19aa62ab0b19a988fef6ed0e5a5acea | A store has $n$ different products for sale, each with a unique price ranging from $1$ to $n$ dollars. A customer inspects $k$ different products and buys the one with the lowest price among the inspected products. If $n = 2023$, find the average amount in dollars the customer will pay. Express your answer as a simplif... | [
"\\dfrac{2024}{k + 1}"
] | [
"A store has $n$ different products for sale, each with a unique price ranging from $1$ to $n$ dollars. A customer inspects $k$ different products and buys the one with the lowest price among the inspected products. If $n = 2023$, find the average amount in dollars the customer will pay. Express your answer as a si... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b9b9abea7e3e05e70f25805822f2549e9b551dc9f46ccfb9b94fd1e5f19e5eb9 | A store has \( n \) different products for sale, each with a unique price from 1 to \( n \) dollars. A customer inspects \( k \) different products and buys the one with the lowest price among those inspected. If \( n = 2023 \) and \( k = 10 \), calculate the expected price the customer will pay. | [
"184"
] | [
"A store has \\( n \\) different products for sale, each with a unique price from 1 to \\( n \\) dollars. A customer inspects \\( k \\) different products and buys the one with the lowest price among those inspected. If \\( n = 2023 \\) and \\( k = 10 \\), calculate the expected price the customer will pay."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-931a839c8b1e2b54e8b3816b563ff5b22037e6710245890e43a0cb8bf436d23d | A store is selling an item for $1000. You have the option to pay it in 20 parts of $80 each, one part per month. What is the monthly interest rate? | [
"4.96\\%"
] | [
"A store is selling an item for $1000. You have the option to pay it in 20 parts of $80 each, one part per month. What is the monthly interest rate?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-741d0c9fb09c52b352233f98e8b9ba0279c968e5cfa8941d1542728d61eccdc1 | A store normally sells windows at $ \$100$ each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately? | [
"100"
] | [
"A store normally sells windows at $ \\$100$ each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-31dd501d71e34e0afdcf2e8de6a5cca097423b8342e5ac7ec795df3b07ee15c2 | A store normally sells windows at 100 dollars each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately? | [
"100"
] | [
"A store normally sells windows at 100 dollars each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-65ca33e7f808e8beb9591198f3ad9e2fbadd97c66b5c78ed39b1c92c5da308dd | A store normally sells windows at \$100 each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately? | [
"100"
] | [
"A store normally sells windows at \\$100 each. This week the store is offering one free window for each purchase of four. Dave needs seven windows and Doug needs eight windows. How many dollars will they save if they purchase the windows together rather than separately?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fcfe25b303d3c67cc735fdbbc0984d5aac1d7706b047e2c811c62a2bd97aa960 | A store owner plans to sell two types of MP3 players. One type costs the store owner $\$200$ each and yields a profit of $\$25$ each. The other type costs $\$400$ each and yields a profit of $\$30$ each. The total demand for the MP3 players will not exceed $300$ units, and the store owner can invest up to $\$80,000$ on... | [
"8000"
] | [
"A store owner plans to sell two types of MP3 players. One type costs the store owner $\\$200$ each and yields a profit of $\\$25$ each. The other type costs $\\$400$ each and yields a profit of $\\$30$ each. The total demand for the MP3 players will not exceed $300$ units, and the store owner can invest up to $\\$... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-7931b6cab5cfeedad5d4453aeba687ca478386397d0467b0dd90c0d56721bc9b | A store owner reduces the price of a $10$ dollar scarf by $20\%$. Later, the price is lowered again by one-half of the reduced price. What is the final price of the scarf? | [
"4.00 dollars"
] | [
"A store owner reduces the price of a $10$ dollar scarf by $20\\%$. Later, the price is lowered again by one-half of the reduced price. What is the final price of the scarf?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-3aa71aa6155e485701616143d6b3ba7169213544da9a691e24358cd99550d6a9 | A store prices an item in dollars and cents so that when 4% sales tax is added, no rounding is necessary because the result is exactly $n$ dollars where $n$ is a positive integer. What is the smallest value of $n$? | [
"13"
] | [
"A store prices an item in dollars and cents so that when 4% sales tax is added, no rounding is necessary because the result is exactly $n$ dollars where $n$ is a positive integer. What is the smallest value of $n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-78e545cb528031580b7435f17553fde4d4ff9fa3be6d841f3df52dacb24eddb4 | A store sells i-Phones and has six customers per day, with each customer having a 50% probability of purchasing a phone. At the beginning of each day, the store restocks to the expected number of daily sales. What is the probability that the store sells at least 15 phones over a 5-day work week? | [
"\\(\\left(\\frac{21}{32}\\right)^5\\)"
] | [
"A store sells i-Phones and has six customers per day, with each customer having a 50% probability of purchasing a phone. At the beginning of each day, the store restocks to the expected number of daily sales. What is the probability that the store sells at least 15 phones over a 5-day work week?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-664de3059fe03e459dfbe3d5f570e621702a051f40d0a1018f7fe266a9189399 | A store sold 72 decks of cards for \$a67.9b. Find the value of \(a + b\). | [
"5"
] | [
"A store sold 72 decks of cards for \\$a67.9b. Find the value of \\(a + b\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 7
} | 22 | 0 | 0 |
omr-ws-d4f0d5472babd7285554f0fdb31cf6f7c5a6ad5875152621652968035246fa36 | A storm in Sydney, Australia, caused $\$$30 million in damage. That estimate was in Australian dollars. At that time, 1.5 Australian dollars were worth 1 American dollar. Determine the number of American dollars of damage the storm caused. | [
"20,\\!000,\\!000"
] | [
"A storm in Sydney, Australia, caused $\\$$30 million in damage. That estimate was in Australian dollars. At that time, 1.5 Australian dollars were worth 1 American dollar. Determine the number of American dollars of damage the storm caused."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d0b0375785723023c0a2aa90c62ca739c8a686f831ea692d8c469193ed2130d5 | A straight circular cone and a straight circular cylinder have bases of the same radius. The volume of the cone is half the volume of the cylinder, and the lateral area of the cone is equal to the lateral area of the cylinder. Determine, as a function of the height \( h \) of the cone, the radius of the sphere inscribe... | [
"\\(\\frac{h (3\\sqrt{7} - 7)}{2}\\)"
] | [
"A straight circular cone and a straight circular cylinder have bases of the same radius. The volume of the cone is half the volume of the cylinder, and the lateral area of the cone is equal to the lateral area of the cylinder. Determine, as a function of the height \\( h \\) of the cone, the radius of the sphere i... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d6928c6315f80efaa927f8d01587b4a8a6c2abd6f88d27960d71896d3ea503fa | A straight circular cone is inscribed in a cylinder such that they share a common base area and the tip of the cone is the center of the top surface of the cylinder. Six equally sized balls touch the cone from the outside, the outer surface of the cylinder, the top surface of the cylinder, and exactly two of the other ... | [
"\\dfrac{R}{2}"
] | [
"A straight circular cone is inscribed in a cylinder such that they share a common base area and the tip of the cone is the center of the top surface of the cylinder. Six equally sized balls touch the cone from the outside, the outer surface of the cylinder, the top surface of the cylinder, and exactly two of the o... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 4
} | 15 | 0 | 0 |
omr-ws-6d6aaa0c2b7538a10eaabb85d86901245a237a5ac9201605fba8a7e7cf673470 | A straight ladder starts upright against a vertical wall and slides down until it is horizontal, with the top always along the wall and the bottom on the floor. What shape does the midpoint of the ladder trace out? | [
"Quarter Circle"
] | [
"A straight ladder starts upright against a vertical wall and slides down until it is horizontal, with the top always along the wall and the bottom on the floor. What shape does the midpoint of the ladder trace out?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 8,
"tir": 10
} | 21 | 0 | 0 |
omr-ws-1005ef5f34fdc8613736207156cd17c8c0b7601d125670a605144f0e5448f0d6 | A straight line $AD$ drawn in a right triangle with legs $BC$ and $BA$ ($| BC | = 4$, $| BA | = 3$) divides the side $BC$ in the ratio $| BD | : | DC | = 3 : 5$. Calculate the angle between the vectors $\overrightarrow{AD}$ and $\overrightarrow{BC}$. | [
"$\\arccos \\frac{1}{\\sqrt 5}$"
] | [
"A straight line $AD$ drawn in a right triangle with legs $BC$ and $BA$ ($| BC | = 4$, $| BA | = 3$) divides the side $BC$ in the ratio $| BD | : | DC | = 3 : 5$. Calculate the angle between the vectors $\\overrightarrow{AD}$ and $\\overrightarrow{BC}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4446315964100662649525af38a960246d1f79d149c2f2bab6b2e476b1d8df9f | A straight line $l$ passes through the points $(3, 0)$ and $(0, 4)$. The point $A$ lies on the parabola $y=2x-x^2$. Find the distance $p$ from point $A$ to the straight line when $A$ coincides with the origin, and indicate the coordinates of the point $A(x_0,y_0)$ on the parabola for which the distance from the parabol... | [
"$\\rho =2.4$ ($x_0=5/3,\\ y_0=5/9$)"
] | [
"A straight line $l$ passes through the points $(3, 0)$ and $(0, 4)$. The point $A$ lies on the parabola $y=2x-x^2$. Find the distance $p$ from point $A$ to the straight line when $A$ coincides with the origin, and indicate the coordinates of the point $A(x_0,y_0)$ on the parabola for which the distance from the pa... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cdfc2142a047ecfa76e478fc9b69fece523b2fed725b9567dd3dcc21af7fbab3 | A straight line \( L \) with a negative slope passes through the point \( (9, 24) \) and cuts the positive coordinate axes at points \( P \) and \( Q \) respectively. Find the minimum value of the distance \( PQ \) as \( L \) varies. | [
"45"
] | [
"A straight line \\( L \\) with a negative slope passes through the point \\( (9, 24) \\) and cuts the positive coordinate axes at points \\( P \\) and \\( Q \\) respectively. Find the minimum value of the distance \\( PQ \\) as \\( L \\) varies."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
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