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-1a22c6a92e60928931ba21ed2c528dedf8c466c030a8c1af6a02a0252c489083 | A lattice point in an $xy$-coordinate system is any point $(x,y)$ where both $x$ and $y$ are integers. The graph of $y=mx+2$ passes through no lattice point with $0<x \le 100$ for all $m$ such that $\frac{1}{2}<m<a$. What is the maximum possible value of $a$? | [
"\\(\\frac{50}{99}\\)"
] | [
"A lattice point in an $xy$-coordinate system is any point $(x,y)$ where both $x$ and $y$ are integers. The graph of $y=mx+2$ passes through no lattice point with $0<x \\le 100$ for all $m$ such that $\\frac{1}{2}<m<a$. What is the maximum possible value of $a$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 29
} | 77 | 0 | 0 |
omr-ws-dc9abe56e0e7d63a5ede5b71dba9491e5e0c5de4271aa351554b20a2be8a3f2e | A lattice point in the $x,y$-plane is a point both of whose coordinates are integers (not necessarily positive). How many lattice points lie on the graph of the equation $x^2-y^2=47$? | [
"4"
] | [
"A lattice point in the $x,y$-plane is a point both of whose coordinates are integers (not necessarily positive). How many lattice points lie on the graph of the equation $x^2-y^2=47$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fe07a122f2d6ec8716f4a35a2e7fdd21aa3ea0d9f41675fcc882dc03a8ac3c88 | A lattice point in the plane is a point with integer coordinates. Suppose that circles with radius $r$ are drawn using all lattice points as centers. Find the smallest value of $r$ such that any line with slope $2/5$ intersects some of these circles. | [
"\\(\\frac{1}{2\\sqrt{29}}\\)"
] | [
"A lattice point in the plane is a point with integer coordinates. Suppose that circles with radius $r$ are drawn using all lattice points as centers. Find the smallest value of $r$ such that any line with slope $2/5$ intersects some of these circles."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-30cceb5ed43676c3e4616f14aeab24bd3d4d2dfbf8b784510c4c99c62227b86c | A lattice point is a pair $(x, y)$ where $x$ and $y$ are integers. How many lattice points are on the graph of $y = \sqrt{x^2 + 60^4} + 2016$? | [
"350"
] | [
"A lattice point is a pair $(x, y)$ where $x$ and $y$ are integers. How many lattice points are on the graph of $y = \\sqrt{x^2 + 60^4} + 2016$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cbc3798646cb969d80d193888ea5db160169da2290eca3d6577e518b72df0b82 | A lattice point is a point $(a, b)$ on the Cartesian plane where $a$ and $b$ are integers. Compute the number of lattice points in the interior and on the boundary of the triangle with vertices at $(0, 0)$, $(0, 20)$, and $(18, 0)$. | [
"201"
] | [
"A lattice point is a point $(a, b)$ on the Cartesian plane where $a$ and $b$ are integers. Compute the number of lattice points in the interior and on the boundary of the triangle with vertices at $(0, 0)$, $(0, 20)$, and $(18, 0)$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
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"cot": 16,
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} | 33 | 0 | 0 |
omr-ws-c7cd81b7d9a297cb005701eb257a244d4fb314ed3bc76cba3bdc63c200eb6ca7 | A lattice point is a point \((x,y)\) such that \(x\) and \(y\) are both integers. Suppose we color each of the lattice points within the square \(0 \leq x, y \leq 10\) either black or white. Find the number of colorings in which each of the 100 unit squares (bounded by the colored lattice points) has exactly two white ... | [
"6"
] | [
"A lattice point is a point \\((x,y)\\) such that \\(x\\) and \\(y\\) are both integers. Suppose we color each of the lattice points within the square \\(0 \\leq x, y \\leq 10\\) either black or white. Find the number of colorings in which each of the 100 unit squares (bounded by the colored lattice points) has exa... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 5
} | 18 | 0 | 0 |
omr-ws-502a944b1c8e592c4a545bc8f764638bef9b180d67fca5dd9c47668e986d682b | A lattice point is a point in the plane for which both the \( x \) and \( y \) coordinates are integers. Let \( C \) be the number of all the lattice points inside the circle \( x^2 + y^2 = B \). | [
"\\sum_{x=-\\lfloor \\sqrt{B-1} \\rfloor}^{\\lfloor \\sqrt{B-1} \\rfloor} \\left( 2 \\left\\lfloor \\sqrt{B - x^2 - 1} \\right\\rfloor + 1 \\right)"
] | [
"A lattice point is a point in the plane for which both the \\( x \\) and \\( y \\) coordinates are integers. Let \\( C \\) be the number of all the lattice points inside the circle \\( x^2 + y^2 = B \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-0cfaeee2b9dfb824467db0a04ddaf119f6a2299d555e72c89091de0698b90b8b | A lattice point is a point whose coordinates are both integers. How many lattice points are on the boundary or inside the region bounded by $y=|x|$ and $y=-x^2+6$? | [
"19"
] | [
"A lattice point is a point whose coordinates are both integers. How many lattice points are on the boundary or inside the region bounded by $y=|x|$ and $y=-x^2+6$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fc1d27fc22800d15180666bae1625d658c10774317e0a0067e420968e224745c | A lattice point is a point whose coordinates are both integers. Suppose we draw a line segment from (0, 2004) to (a, b) where a and b are integers each randomly selected from 1, 2, 3, ..., 100. What is the probability that the line segment contains an even number of lattice points? | [
"\\dfrac{3}{4}"
] | [
"A lattice point is a point whose coordinates are both integers. Suppose we draw a line segment from (0, 2004) to (a, b) where a and b are integers each randomly selected from 1, 2, 3, ..., 100. What is the probability that the line segment contains an even number of lattice points?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-85b861719b55684b3d43562f0870cea5d5f882b8a474a6a99dd48f6f76cf1b98 | A lattice rectangle has vertices at lattice points and sides parallel to the axes. If the shorter side of the lattice rectangle $R$ is doubled, the new rectangle contains $304$ more lattice points than $R$. Compute the minimum possible area of $R$. | [
"288"
] | [
"A lattice rectangle has vertices at lattice points and sides parallel to the axes. If the shorter side of the lattice rectangle $R$ is doubled, the new rectangle contains $304$ more lattice points than $R$. Compute the minimum possible area of $R$."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-261dd029f69a4fa4f561f26fdff6a25a08bd93d81d7c8d8b9da9b8f8a92e5252 | A lawn sprinkler is constructed such that the angle \( z \) with the ground changes at a constant rate, where \( z \) ranges between 45 degrees and 135 degrees. The horizontal distance the water travels is given by \[ x = \frac{v^2 \sin(2z)}{32} \] where \( v \) is the speed of the water. Find \( \frac{dx}{dt} \) and e... | [
"\\( \\frac{dx}{dt} = \\frac{v^2 C}{32} \\cos(2z) \\) and the lawn receives the most water at \\( z = 45^\\circ \\)."
] | [
"A lawn sprinkler is constructed such that the angle \\( z \\) with the ground changes at a constant rate, where \\( z \\) ranges between 45 degrees and 135 degrees. The horizontal distance the water travels is given by \n\\[ x = \\frac{v^2 \\sin(2z)}{32} \\]\nwhere \\( v \\) is the speed of the water. Find \\( \\f... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 2,
"tir": 2
} | 15 | 0 | 0 |
omr-ws-fa35c1dfc0eb1603d619c4890d73dda43d23e8eeb5c442000db9148a321e2b87 | A lawyer claims to know which of 14 coins are counterfeit (7 of them) and which are genuine. She needs to prove this using only 3 weighings. How can she accomplish this? | [
"\\text{She can prove it by strategically weighing groups to show imbalances that only her known counterfeit set would produce.}"
] | [
"A lawyer claims to know which of 14 coins are counterfeit (7 of them) and which are genuine. She needs to prove this using only 3 weighings. How can she accomplish this?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 5
} | 6 | 0 | 0 |
omr-ws-a1dfccf1aebbecf03ee49140c9ef98a49a7775937efb1b7211190918927a971a | A leaf 30 feet from the ground falls from a branch of a tree at a rate of 2 feet per second. There is a slight wind blowing at a rate of 2 feet per second. Describe the locus of all points at which the leaf can land. | [
"A circle with a radius of 30 feet."
] | [
"A leaf 30 feet from the ground falls from a branch of a tree at a rate of 2 feet per second. There is a slight wind blowing at a rate of 2 feet per second. Describe the locus of all points at which the leaf can land."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 41,
"genselect": 8,
"tir": 21
} | 70 | 0 | 0 |
omr-ws-d9771b37375a83d306f7d54700b68e65220551ce410e7f753981ac25d6c88afa | A leaf is torn from a book, and the sum of the remaining page numbers is 4851. Find the missing page numbers. | [
"The missing page numbers are either 49 and 50 or 99 and 100."
] | [
"A leaf is torn from a book, and the sum of the remaining page numbers is 4851. Find the missing page numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 6,
"tir": 24
} | 45 | 0 | 0 |
omr-ws-e3aa52fad18768dbc4638742d97786c0377212b15e656bdc76d443087d3dc97b | A leaf is torn from a book. If the sum of the numbers on the remaining pages is 4851, then what are the numbers of the missing pages? | [
"99 and 100"
] | [
"A leaf is torn from a book. If the sum of the numbers on the remaining pages is 4851, then what are the numbers of the missing pages?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 7,
"tir": 27
} | 46 | 0 | 0 |
omr-ws-b3f5a11714aa644da6c2f960448c2c8a964ca0d02c02725ec15ae59b78770d2e | A leaf is torn from a paperback novel. The sum of the numbers on the remaining pages is $15000$. Determine the page numbers on the torn leaf. | [
"\\{112, 113\\}"
] | [
"A leaf is torn from a paperback novel. The sum of the numbers on the remaining pages is $15000$. Determine the page numbers on the torn leaf."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 20,
"genselect": 5,
"tir": 11
} | 36 | 0 | 0 |
omr-ws-9bb3747b269588c9f85d137c1fcfa00beaa54ff6b35c9d6e3a0f30386eac6e01 | A leaf is torn from a paperback novel. The sum of the numbers on the remaining pages is $15000$. What are the page numbers on the torn leaf? | [
"112 and 113",
"25, 26"
] | [
"A leaf is torn from a paperback novel. The sum of the numbers on the remaining pages is $15000$. What are the page numbers on the torn leaf?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 25,
"genselect": 5,
"tir": 37
} | 67 | 0 | 0 |
omr-ws-5fef07198a6482e68141debf9e7ce141c9956e835f24cd755052c8f09d0e37b5 | A leaf is torn from a paperback novel. The sum of the numbers on the remaining pages is 15000. Calculate the page numbers on the torn leaf. | [
"51 and 52"
] | [
"A leaf is torn from a paperback novel. The sum of the numbers on the remaining pages is 15000. Calculate the page numbers on the torn leaf."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6bbdb65960fb65a483aed95be288afec9d8cb12d27b37cf60c8e855f8240196d | A leaf is torn from a paperback novel. The sum of the numbers on the remaining pages is 15000. What are the page numbers on the torn leaf? | [
"(25, 26)"
] | [
"A leaf is torn from a paperback novel. The sum of the numbers on the remaining pages is 15000. What are the page numbers on the torn leaf?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 5,
"tir": 18
} | 23 | 0 | 0 |
omr-ws-f9754628d569b666107f6acafdeadb36ee2ff10971eba29874ea35d9c6836c2d | A leaf is torn from a paperback novel. The sum of the remaining page numbers is 15000. What are the page numbers on the torn leaf? | [
"10 and 11"
] | [
"A leaf is torn from a paperback novel. The sum of the remaining page numbers is 15000. What are the page numbers on the torn leaf?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-92bac09886c4a38e1221ddc741b720cb500a4e98d0082de000a4d3dbcde28079 | A league consists of 2024 players. A balanced round involves splitting the players into two teams of equal size and having every member of one team play with every member of the other team. A tournament consists of several balanced rounds such that any two players have played each other by the end. If a tournament last... | [
"11"
] | [
"A league consists of 2024 players. A balanced round involves splitting the players into two teams of equal size and having every member of one team play with every member of the other team. A tournament consists of several balanced rounds such that any two players have played each other by the end. If a tournament... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6f53cbd5e787a17d3c65c3c8cdbd0a8415d4a00c724148d75a3a6e15f2675d9a | A leaky faucet drips at a steady rate. If it takes 36 seconds for 9 drops to fall, how many seconds would it take for 12 drops to fall? | [
"49.5"
] | [
"A leaky faucet drips at a steady rate. If it takes 36 seconds for 9 drops to fall, how many seconds would it take for 12 drops to fall?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1b2bcd234a155a992a9482438a81fa52b9a43d218778f54a903942cb0ff353ca | A leaky faucet drips at a steady rate. If we measure from the instant the first drop hits the bottom to the instant the last drop falls, it takes 36 seconds for 9 drops to fall. How many seconds would it take for 12 drops to fall? | [
"49.5"
] | [
"A leaky faucet drips at a steady rate. If we measure from the instant the first drop hits the bottom to the instant the last drop falls, it takes 36 seconds for 9 drops to fall. How many seconds would it take for 12 drops to fall?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 13
} | 28 | 0 | 0 |
omr-ws-bfa9e2e75a8fbe2357cd439a00bbdb7a51e238105b561812a45eb7241320bb0a | A lecturer assigns social studies grades randomly to 3 students, with each grade being a real number between $0$ and $4$ (excluding $0$ and $4$). The probability that the difference in grades between any two students is less than $\frac{p-q\sqrt{r}}{s}$, where $p, q, r, s$ are natural numbers with $\gcd(p, q, s) = 1$ a... | [
"8"
] | [
"A lecturer assigns social studies grades randomly to 3 students, with each grade being a real number between $0$ and $4$ (excluding $0$ and $4$). The probability that the difference in grades between any two students is less than $\\frac{p-q\\sqrt{r}}{s}$, where $p, q, r, s$ are natural numbers with $\\gcd(p, q, s... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-54d995341de14893f917aceeb89f99427e7678b2fe531b60a3b4404b8f7a1f59 | A lemming sits at a corner of a square with side length 10 meters. The lemming runs 6.2 meters along a diagonal toward the opposite corner. It stops, makes a 90 degree right turn and runs 2 more meters. A scientist measures the shortest distance between the lemming and each side of the square. What is the average of th... | [
"5"
] | [
"A lemming sits at a corner of a square with side length 10 meters. The lemming runs 6.2 meters along a diagonal toward the opposite corner. It stops, makes a 90 degree right turn and runs 2 more meters. A scientist measures the shortest distance between the lemming and each side of the square. What is the average ... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-12b3ee930ec5987199499b8334186adcdee3314a8b4d81b2051773ade0282dff | A lemonade stand analyzes its earning and operations. For the previous month it had a \$45 dollar budget to divide between production and advertising. If it spent $k$ dollars on production, it could make $2k - 12$ glasses of lemonade. If it spent $k$ dollars on advertising, it could sell each glass at an average price ... | [
"21"
] | [
"A lemonade stand analyzes its earning and operations. For the previous month it had a \\$45 dollar budget to divide between production and advertising. If it spent $k$ dollars on production, it could make $2k - 12$ glasses of lemonade. If it spent $k$ dollars on advertising, it could sell each glass at an average ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-3154651c92fbbcc582cd03a5fea8422a1a2c62d919a9638626f599bb7fb21b20 | A lemonade stand analyzes its earning and operations. For the previous month it had a \$50 dollar budget to divide between production and advertising. If it spent $k$ dollars on production, it could make $2k - 2$ glasses of lemonade. If it spent $k$ dollars on advertising, it could sell each glass at an average price o... | [
"6"
] | [
"A lemonade stand analyzes its earning and operations. For the previous month it had a \\$50 dollar budget to divide between production and advertising. If it spent $k$ dollars on production, it could make $2k - 2$ glasses of lemonade. If it spent $k$ dollars on advertising, it could sell each glass at an average p... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-98afb330c793cfa4dc043c14764cd81576b1a097c27ee7da6876e44fe335847d | A leprechaun makes a journey starting at $(0,0)$ in the coordinate plane. Each move, he may only travel from $(x,y)$ to either $(x,y+1),$ $(x+1,y),$ or $(x-1,y-1).$ After $13$ moves or if he steps into the second or fourth quadrant, he loses. If he manages to arrive at the point $(4,4)$ before either of these events oc... | [
"2254"
] | [
"A leprechaun makes a journey starting at $(0,0)$ in the coordinate plane. Each move, he may only travel from $(x,y)$ to either $(x,y+1),$ $(x+1,y),$ or $(x-1,y-1).$ After $13$ moves or if he steps into the second or fourth quadrant, he loses. If he manages to arrive at the point $(4,4)$ before either of these even... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-062cf44f761692022da7b5770f949576990cb10a0a2dc4e80e231aa368815138 | A letter is known to come from either TATANAGAR or CALCUTTA. On the envelope, only two consecutive letters "TA" are visible. What is the probability that the letter has come from CALCUTTA? | [
"\\(\\frac{4}{11}\\)"
] | [
"A letter is known to come from either TATANAGAR or CALCUTTA. On the envelope, only two consecutive letters \"TA\" are visible. What is the probability that the letter has come from CALCUTTA?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-a42917f44ad7a1368aa64adb781d5aa5a7e1baa0e44b9acb896358d03f75263f | A letter is known to come from either TATANAGAR or CALCUTTA. On the envelope, only two consecutive letters "TA" are visible. What is the probability that the letter has come from TATANAGAR? | [
"\\(\\frac{7}{11}\\)"
] | [
"A letter is known to come from either TATANAGAR or CALCUTTA. On the envelope, only two consecutive letters \"TA\" are visible. What is the probability that the letter has come from TATANAGAR?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-25e5149c8ef2ed9c1acba9f5da9faeed056c1e4f5abab58fdee6c4b6de888bcd | A letter is known to have come from either MAHARASHTRA or MADRAS. On the postmark, only the consecutive letters "RA" can be read clearly. What is the probability that the letter came from MAHARASHTRA? | [
"\\dfrac{1}{2}"
] | [
"A letter is known to have come from either MAHARASHTRA or MADRAS. On the postmark, only the consecutive letters \"RA\" can be read clearly. What is the probability that the letter came from MAHARASHTRA?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 4
} | 13 | 0 | 0 |
omr-ws-73b3709c7ac7d983654fefcc99b6cba38748243b7a758c287d6c6c3b7e976180 | A letter is randomly chosen from the eleven letters of the word MATHEMATICS. What is the probability that this letter has a vertical axis of symmetry? | [
"\\(\\frac{8}{11}\\)"
] | [
"A letter is randomly chosen from the eleven letters of the word MATHEMATICS. What is the probability that this letter has a vertical axis of symmetry?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-a2ad074856c1fb4a1e0671585d604e3c6e5d099dc8ac989e077a506ad34aaccd | A lever of the second order has its fulcrum at $A$ and a load $P$ is suspended from point $B$ $(AB = a)$. The weight per unit length of the lever is $p$. What should the length of the lever be for the load $P$ to be counterbalanced by the least force? (The moment of the balancing force must be equal to the sum of the m... | [
"$\\sqrt {2ap/K}$"
] | [
"A lever of the second order has its fulcrum at $A$ and a load $P$ is suspended from point $B$ $(AB = a)$. The weight per unit length of the lever is $p$. What should the length of the lever be for the load $P$ to be counterbalanced by the least force? (The moment of the balancing force must be equal to the sum of ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2f22f3eec515c7b3a9f83b923f1a4ff39dcca64d5b50255ac21280b71cdb851e | A library charges 100 cents for borrowing a book. For late returns, the fines are as follows: the first 10 days are penalized at 10% per day, the next 10 days at 15% per day, the next 10 days at 20% per day, and so on. Calculate the total fines for a book returned 4 months late, assuming each month consists of 30 days. | [
"4500"
] | [
"A library charges 100 cents for borrowing a book. For late returns, the fines are as follows: the first 10 days are penalized at 10% per day, the next 10 days at 15% per day, the next 10 days at 20% per day, and so on. Calculate the total fines for a book returned 4 months late, assuming each month consists of 30 ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c68df3065f201a27544b2a9b19816302e3d48e3b524b1f8a9408797c3d1add36 | A library has six identical copies of a certain book. At any given time, some of these copies are at the library and some are checked out. How many different ways are there for some of the books to be in the library and the rest to be checked out if at least one book is in the library and at least one is checked out? (... | [
"5"
] | [
"A library has six identical copies of a certain book. At any given time, some of these copies are at the library and some are checked out. How many different ways are there for some of the books to be in the library and the rest to be checked out if at least one book is in the library and at least one is checked o... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a1c62d6935ba4e8a1c92066802ae6979ee61f363d86e26568316a83655fb1382 | A license plate consists of 6 digits (0-9), and the beginning digit may be 0. Each license plate must differ from any other in at least two spots. What is the largest possible number of such license plates? | [
"100000"
] | [
"A license plate consists of 6 digits (0-9), and the beginning digit may be 0. Each license plate must differ from any other in at least two spots. What is the largest possible number of such license plates?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-28761e6d3424d9b0c61b17bb2e3d241a80268628455d46820de3ee2f6ac4a805 | A license plate has 7 places. Four places can be either a letter or a number, two places can only be a letter, and one place can only be a 0, 1, or 2. Repetition is allowed. How many different license plates can be formed? | [
"357657431040"
] | [
"A license plate has 7 places. Four places can be either a letter or a number, two places can only be a letter, and one place can only be a 0, 1, or 2. Repetition is allowed. How many different license plates can be formed?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 6
} | 16 | 0 | 0 |
omr-ws-93713902c1473b224a66fcb13fd76ed6ffb14f36a4a036492f7db0c768a2b5cb | A license plate has seven binary digits ($0$ or $1$), and may have leading zeros. If two plates must always differ in at least three places, what is the largest number of plates that is possible? | [
"16"
] | [
"A license plate has seven binary digits ($0$ or $1$), and may have leading zeros. If two plates must always differ in at least three places, what is the largest number of plates that is possible?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d73d55a6fb047e21a23398f4e5281367cd79a966b16c052a5a1d1db09b797c91 | A license plate has six digits from 0 to 9 and may have leading zeros. If two plates must always differ in at least two places, what is the largest number of plates that is possible? | [
"100000"
] | [
"A license plate has six digits from 0 to 9 and may have leading zeros. If two plates must always differ in at least two places, what is the largest number of plates that is possible?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-52a9a67f8602a37a36d3ad7ab01f68a44d77ef0ea47ace5a96e1c715c86123de | A license plate in a certain state consists of 4 digits, not necessarily distinct, and 2 letters, also not necessarily distinct. These six characters may appear in any order, except that the two letters must appear next to each other. How many distinct license plates are possible? | [
"33,\\!800,\\!000"
] | [
"A license plate in a certain state consists of 4 digits, not necessarily distinct, and 2 letters, also not necessarily distinct. These six characters may appear in any order, except that the two letters must appear next to each other. How many distinct license plates are possible?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4913a6e7d29d68824207b6df8d4313ac99110fe3d88310797d81b553f2f3559a | A license plate number consists of any combination of three letters followed by any combination of three digits. The value of a license plate is determined by adding the value of each character in the license plate, with letters in the first half of the alphabet being worth 4 points, letters in the second half of the a... | [
"239"
] | [
"A license plate number consists of any combination of three letters followed by any combination of three digits. The value of a license plate is determined by adding the value of each character in the license plate, with letters in the first half of the alphabet being worth 4 points, letters in the second half of ... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-1d2520bd4bccdfc65414edd37313049fbcb6c21fe7dd57e4d118346894e5caef | A lie detector is known to be reliable 80 percent of the time when the person is guilty, and 90 percent reliable when the person is innocent. A suspect is chosen at random from a group of suspects of whom only 1 percent have ever committed a crime. If the test indicates that the suspect is guilty, what is the probabili... | [
"\\(\\frac{22}{23}\\)"
] | [
"A lie detector is known to be reliable 80 percent of the time when the person is guilty, and 90 percent reliable when the person is innocent. A suspect is chosen at random from a group of suspects of whom only 1 percent have ever committed a crime. If the test indicates that the suspect is guilty, what is the prob... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f9bfe7abbfc6f797f18b64d87fb4cd2ab9cb512627102e8464e09de594274c12 | A life raft, set adrift from a sinking ship 150 miles offshore, travels directly toward a Coast Guard station at a rate of 5 mph. At the same time, a rescue helicopter is dispatched from the Coast Guard station with an average speed of 90 mph. How long will it take the helicopter to reach the life raft? | [
"\\( \\frac{150}{95} \\) or approximately 1.578947368 hours."
] | [
"A life raft, set adrift from a sinking ship 150 miles offshore, travels directly toward a Coast Guard station at a rate of 5 mph. At the same time, a rescue helicopter is dispatched from the Coast Guard station with an average speed of 90 mph. How long will it take the helicopter to reach the life raft?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-b89be5ddb00e5191afa1650ae1d8223672b20a19e95bfa97941848f180bd7dcb | A lifeguard is at the beach on the sand right next to the water. The swimmer is 75 feet from shore and 195 feet from the lifeguard. The lifeguard can run at 10 feet/second on the sand and swim at 5 feet/second. What is the shortest time the lifeguard needs to reach the drowning swimmer? | [
"\\( 18 + 7.5\\sqrt{3} \\)"
] | [
"A lifeguard is at the beach on the sand right next to the water. The swimmer is 75 feet from shore and 195 feet from the lifeguard. The lifeguard can run at 10 feet/second on the sand and swim at 5 feet/second. What is the shortest time the lifeguard needs to reach the drowning swimmer?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 14
} | 29 | 0 | 0 |
omr-ws-16077c4d2eb54423668864c3ce684b607316b02c3b4351bf564538e2ede1481f | A lifeguard is stationed at a corner of a square pool with a side length of 10 meters. The lifeguard moves at a rate of 1 meter per second in the water and 2 meters per second along the edge of the pool. Calculate the least time required for the lifeguard to reach any point in the pool. Assume the lifeguard can turn fr... | [
"\\( 10\\sqrt{2} \\)"
] | [
"A lifeguard is stationed at a corner of a square pool with a side length of 10 meters. The lifeguard moves at a rate of 1 meter per second in the water and 2 meters per second along the edge of the pool. Calculate the least time required for the lifeguard to reach any point in the pool. Assume the lifeguard can tu... | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-6665d343a40e3672de6ae48f5ff7863ca8666317af489676a3f4edab89335a55 | A lift starts at floor 1 and goes up to floor 6. How many different records are possible of the people leaving the lift? What if the 8 people consisted of 3 men and 5 women, and the computer can distinguish a man from a woman? | [
"4410"
] | [
"A lift starts at floor 1 and goes up to floor 6. How many different records are possible of the people leaving the lift? What if the 8 people consisted of 3 men and 5 women, and the computer can distinguish a man from a woman?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-9ace24ebe17b7c4fc746099f27858425953c5ab8bb8ed4bbd57319602335f555 | A lift with seven people stops at ten floors. From zero to seven people go out at each floor. How many ways are there for the lift to empty? (The ways only differ by the number of people leaving at each floor.) | [
"$\\binom{16}{7}=\\binom{16}{9}$"
] | [
"A lift with seven people stops at ten floors. From zero to seven people go out at each floor. How many ways are there for the lift to empty? (The ways only differ by the number of people leaving at each floor.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8441b515800fcb6966e4cf460c9e89d6e24389c4e35ae274aa8d469d505c0c74 | A light 6 feet above level ground on one edge of a street shines on a $2\text{ ft} \times 2\text{ ft}$ square board propped up vertically along the opposite edge of the street. If the street is 20 feet wide, what is the area, in ft$^2$, of the square's shadow? | [
"25"
] | [
"A light 6 feet above level ground on one edge of a street shines on a $2\\text{ ft} \\times 2\\text{ ft}$ square board propped up vertically along the opposite edge of the street. If the street is 20 feet wide, what is the area, in ft$^2$, of the square's shadow?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0ca5a8d5be17f5f4debecf95736b928db55c48f02cbac2cf3a72128ef4ecc482 | A light beam shines from the origin into the unit square at an angle of $\theta$ to one of the sides such that $\tan \theta = \frac{13}{17}$. The light beam is reflected by the sides of the square. How many times does the light beam hit a side of the square before hitting a vertex of the square? | [
"28"
] | [
"A light beam shines from the origin into the unit square at an angle of $\\theta$ to one of the sides such that $\\tan \\theta = \\frac{13}{17}$. The light beam is reflected by the sides of the square. How many times does the light beam hit a side of the square before hitting a vertex of the square?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1e3058eff420cb25ae1eb9107c176be98eb3f6649fbbc251bbf87e9d868fb4b8 | A light bulb is placed on each vertex of a regular $n$-gon whose sides and diagonals are all drawn. In the beginning, $k$ light bulbs are turned on. In each step, it is allowed to erase a side or a diagonal and change the state of one of the two bulbs at the endpoints of the side/diagonal. Determine all possible values... | [
"k \\equiv \\frac{n(n+1)}{2} \\pmod{2}"
] | [
"A light bulb is placed on each vertex of a regular $n$-gon whose sides and diagonals are all drawn. In the beginning, $k$ light bulbs are turned on. In each step, it is allowed to erase a side or a diagonal and change the state of one of the two bulbs at the endpoints of the side/diagonal. Determine all possible v... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-c969259fa580e9ae6e573be4ad98b7a3b494e9d58abbb3b0695ea05061be6415 | A light elastic string $AB$ of length $l$ is fixed at $A$. If a weight $W$ is attached to $B$, the string stretches to a length $2l$. If a weight $\frac{1}{4}W$ is attached to $B$ and let fall from the level of $A$, calculate the distance through which it falls. | [
"2l"
] | [
"A light elastic string $AB$ of length $l$ is fixed at $A$. If a weight $W$ is attached to $B$, the string stretches to a length $2l$. If a weight $\\frac{1}{4}W$ is attached to $B$ and let fall from the level of $A$, calculate the distance through which it falls."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-9cd3415f49c233e5993935b5b14cbcd01ab6fc02b3c49a13de520e5eb0b9e48d | A light flashes on and off instantaneously at various points in time, with the time between two adjacent flashes distributed exponentially with parameter $L$. If we observe this process for a total time $T$, what is the expected number of flashes that will occur within $\tau$ units of time of another flash? | [
"\\( LT (1 - e^{-L\\tau}) \\)"
] | [
"A light flashes on and off instantaneously at various points in time, with the time between two adjacent flashes distributed exponentially with parameter $L$. If we observe this process for a total time $T$, what is the expected number of flashes that will occur within $\\tau$ units of time of another flash?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-7a12ed60c84d0ead5ef599379e3141e60d7feae1cce674659e2068848b716bd0 | A light in a revolving lighthouse located 1 mile away from a straight shoreline turns at 1 revolution per minute. Find the speed of the spot of light along the shore when it is 2 miles away from the point of shore nearest the light. | [
"\\( 10\\pi \\)"
] | [
"A light in a revolving lighthouse located 1 mile away from a straight shoreline turns at 1 revolution per minute. Find the speed of the spot of light along the shore when it is 2 miles away from the point of shore nearest the light."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 30
} | 44 | 0 | 0 |
omr-ws-556d3b1eb4e687d113db126351d001dca45d7f4435df3c5f8bf37b5e23f26dcc | A light is at the top of a pole 20 meters high. From a point on the ground 8 meters from the base of the pole, a ball is thrown upwards with an initial velocity of 13 meters per second. At what rate is the shadow of the ball moving along the ground 1 second later? | [
"3.62"
] | [
"A light is at the top of a pole 20 meters high. From a point on the ground 8 meters from the base of the pole, a ball is thrown upwards with an initial velocity of 13 meters per second. At what rate is the shadow of the ball moving along the ground 1 second later?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c0914ce74ff056def0db5ffac0197dec8f8a7a50053811f796d6d2c000242dda | A light is at the top of a tower 80 ft high. A ball is dropped from the same height from a point 20 ft from the light. Assuming that the ball falls according to the law $s=16t^2$, how fast is the shadow of the ball moving toward the base of the tower, along the ground, two seconds after release? | [
"25"
] | [
"A light is at the top of a tower 80 ft high. A ball is dropped from the same height from a point 20 ft from the light. Assuming that the ball falls according to the law $s=16t^2$, how fast is the shadow of the ball moving toward the base of the tower, along the ground, two seconds after release?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-45a9928bada4d71fbd0eab6a99e6ff31910cc816625a78e99bd44ab6abd5929a | A light is atop a 16-foot pole. A boy 5 feet tall walks past the pole at a rate of 4 ft/second. What is the rate at which his shadow is lengthening when he is 18 feet from the pole? | [
"\\dfrac{20}{11}"
] | [
"A light is atop a 16-foot pole. A boy 5 feet tall walks past the pole at a rate of 4 ft/second. What is the rate at which his shadow is lengthening when he is 18 feet from the pole?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d4d9778c960145d09e4b9e11c19381a9e779a7cc2d9cf7b018b8f5ebea91dbd0 | A light is atop a 16-foot pole. A boy 5 feet tall walks past the pole at a rate of 4 ft/second. What is the rate at which the tip of his shadow is moving when he is 18 feet from the pole? | [
"\\dfrac{64}{11}"
] | [
"A light is atop a 16-foot pole. A boy 5 feet tall walks past the pole at a rate of 4 ft/second. What is the rate at which the tip of his shadow is moving when he is 18 feet from the pole?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-728cdfc5510665adc7fb0de7b022d93c4181cebf762f8250e6203c68cea7bf42 | A light is mounted on a pole 15 feet high. A six-foot tall man is walking in a straight line away from the pole at 5 feet per second. Find the speed of the leading edge of his shadow as he passes a point 40 feet from the pole. | [
"\\dfrac{25}{3}"
] | [
"A light is mounted on a pole 15 feet high. A six-foot tall man is walking in a straight line away from the pole at 5 feet per second. Find the speed of the leading edge of his shadow as he passes a point 40 feet from the pole."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8c47212cd003ac4be189d3556a7841c21e235c6f59e31ed60c17daca3aa01f51 | A light is on the ground, 15m from a wall. A woman, 1.6m tall, walks away from the light and towards the wall at 0.5 m/s. Calculate the rate of change of the length of her shadow on the wall when the woman is 10m from the wall. | [
"0.48 m/s",
"\\(-\\frac{12}{25}\\)"
] | [
"A light is on the ground, 15m from a wall. A woman, 1.6m tall, walks away from the light and towards the wall at 0.5 m/s. Calculate the rate of change of the length of her shadow on the wall when the woman is 10m from the wall."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 16,
"genselect": 5,
"tir": 12
} | 33 | 0 | 0 |
omr-ws-f462df5a784c69010880864fa9a5623ec53a9e24c68510b849cfa35c75342733 | A light is placed on the ground at a distance of 30 feet from a house. A man 6 feet tall walks from the light towards the house at a rate of 5 feet per second. Find the rate at which the shadow on the wall is shortening when he is 15 feet from the wall. | [
"-4 feet/second"
] | [
"A light is placed on the ground at a distance of 30 feet from a house. A man 6 feet tall walks from the light towards the house at a rate of 5 feet per second. Find the rate at which the shadow on the wall is shortening when he is 15 feet from the wall."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-38f426d45d8311b3fb5415704c70470b17fa770022a9683b4bb1be802ce5d6ba | A light ray enters a rectangle $ABCD$ at vertex $A$ and exits from the opposite vertex $C$. If the coordinates of $A$ are $(0,0)$ and $C$ are $(a,b)$, find the coordinates of the point where the light ray crosses the center of the rectangle $ABCD$. Express your answer as an ordered pair $(x,y)$. | [
"\\left( \\frac{a}{2}, \\frac{b}{2} \\right)"
] | [
"A light ray enters a rectangle $ABCD$ at vertex $A$ and exits from the opposite vertex $C$. If the coordinates of $A$ are $(0,0)$ and $C$ are $(a,b)$, find the coordinates of the point where the light ray crosses the center of the rectangle $ABCD$. Express your answer as an ordered pair $(x,y)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a013266dbf721360e4a2daf563b556aea739c0efb75f2b39551bdb9769f76df4 | A light ray enters a rectangular box with dimensions 57 cm by 82 cm through corner A at a 45-degree angle with the walls. Through which corner will the light ray come out? | [
"C"
] | [
"A light ray enters a rectangular box with dimensions 57 cm by 82 cm through corner A at a 45-degree angle with the walls. Through which corner will the light ray come out?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-ff88f32da64db03e18412f543a34520e0e33a77ec9357d463988731cc3bea878 | A light ray enters a rectangular box with mirrors on the inner walls through corner A of the base rectangle ABCD, making a 45-degree angle with the walls. The dimensions of the rectangle are AB = CD = 82 cm and BC = DA = 57 cm. Through which corner will the light ray come out? | [
"B"
] | [
"A light ray enters a rectangular box with mirrors on the inner walls through corner A of the base rectangle ABCD, making a 45-degree angle with the walls. The dimensions of the rectangle are AB = CD = 82 cm and BC = DA = 57 cm. Through which corner will the light ray come out?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-7eb6159b9536c9a6a6fa48551a89f7706a691812ec0d9b2e3bccc1435e15ec26 | A light ray is emitted from the center of a square room with mirrors on all four walls. If the ray strikes a corner, it reverses direction. Let $d$ be the distance traveled by the light ray before it returns to the center of the room for the first time, but not from the diametrically opposite direction. Find the smalle... | [
"\\sqrt{5}"
] | [
"A light ray is emitted from the center of a square room with mirrors on all four walls. If the ray strikes a corner, it reverses direction. Let $d$ be the distance traveled by the light ray before it returns to the center of the room for the first time, but not from the diametrically opposite direction. Find the s... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-d035ac8794324ec4f7f638dbda64d8a591dc3aae51e0fd459ca15bd8963c9d01 | A light ray travels along the straight line \( x - 3y + 7 = 0 \) and gets reflected from a spherical mirror whose circular section with the x-y plane is given by \( x^2 + y^2 - 2x - 2y - 3 = 0 \). Determine the equation(s) of the straight line(s) along which the reflected ray travels. | [
"A) y = 2, D) x = -1"
] | [
"A light ray travels along the straight line \\( x - 3y + 7 = 0 \\) and gets reflected from a spherical mirror whose circular section with the x-y plane is given by \\( x^2 + y^2 - 2x - 2y - 3 = 0 \\). Determine the equation(s) of the straight line(s) along which the reflected ray travels."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-86739f4ca6dfd72c82edd24f5dd31acc7efdbdeb3ad0a4e48c2085e0a3df0854 | A light source $C$ moves at a constant velocity of $2 \text{ m/sec}$ along a straight line $(e)$. A circular obstacle with a diameter of $2 \text{ m}$ is centered on a parallel line equidistant from both $(e)$ and another line $(d)$. The obstacle creates a shadow $(AB)$. Find the instantaneous rate of change in the len... | [
"\\( 2 \\text{ m/sec} \\)"
] | [
"A light source $C$ moves at a constant velocity of $2 \\text{ m/sec}$ along a straight line $(e)$. A circular obstacle with a diameter of $2 \\text{ m}$ is centered on a parallel line equidistant from both $(e)$ and another line $(d)$. The obstacle creates a shadow $(AB)$. Find the instantaneous rate of change in ... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4d47bffe86c75ab637b553b078741bddcf604189837bc89bc314afea057fce93 | A light source 6km from the straight coast of a beach completes 4 rotations per minute. With what speed is the focus of light moving along the beach at the instant when the light makes a 30º ($\frac{\pi}{6}$ radians) angle with the coast of the beach? | [
"\\( 64\\pi \\) km/min"
] | [
"A light source 6km from the straight coast of a beach completes 4 rotations per minute. With what speed is the focus of light moving along the beach at the instant when the light makes a 30º ($\\frac{\\pi}{6}$ radians) angle with the coast of the beach?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 9
} | 11 | 0 | 0 |
omr-ws-79a22b04a14843a5ae2845289478a057ce9d6712c93e32e40f2484adf3e4cb86 | A light source at the point $(0, 16)$ in the coordinate plane casts light in all directions. A disc (circle along with its interior) of radius $2$ with center at $(6, 10)$ casts a shadow on the X-axis. The length of the shadow can be written in the form $m\sqrt{n}$ where $m, n$ are positive integers and $n$ is squarefr... | [
"21"
] | [
"A light source at the point $(0, 16)$ in the coordinate plane casts light in all directions. A disc (circle along with its interior) of radius $2$ with center at $(6, 10)$ casts a shadow on the X-axis. The length of the shadow can be written in the form $m\\sqrt{n}$ where $m, n$ are positive integers and $n$ is sq... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-236052c6a702212e549c97bb4a39fd0894c08be1d6efcd14576ab6a9e8338add | A light source at the point (8,0) in the coordinate plane casts light in all directions. A solid circle of radius 1 centered at the point (3,5) casts a shadow on the x-axis. How long is the shadow? | [
"$\\frac{14}{3}$"
] | [
"A light source at the point (8,0) in the coordinate plane casts light in all directions. A solid circle of radius 1 centered at the point (3,5) casts a shadow on the x-axis. How long is the shadow?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-9325e278ffa745790b732d1c9c6cbedbe4fefbd2c800902488bfe5fdffcd8f82 | A light source is located at a point $A$ of a plane $P$. A hemispherical mirror of unit radius is placed above the plane so that its reflecting inner side faces the plane and its axis of symmetry passes through the point $A$ and is perpendicular to the plane $P$. Knowing that the least angle between the rays reflected ... | [
"r=\\frac{3+\\sqrt{3}}{2}, h=\\frac{\\sqrt{3}-1}{2}"
] | [
"A light source is located at a point $A$ of a plane $P$. A hemispherical mirror of unit radius is placed above the plane so that its reflecting inner side faces the plane and its axis of symmetry passes through the point $A$ and is perpendicular to the plane $P$. Knowing that the least angle between the rays refle... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8b67590b8ea3741b159c511e40e4bb03381c03be6d1acde1880230c04b0334d9 | A lighthouse facing East sends out a fan-shaped beam of light extending from S.-E to N.-E. An observer sailing due North continues to see the light for \(10\sqrt{2}\) minutes after meeting the beam. When the ship leaves the fan of light, it is 10 miles from the lighthouse. Find the speed of the ship. | [
"60"
] | [
"A lighthouse facing East sends out a fan-shaped beam of light extending from S.-E to N.-E. An observer sailing due North continues to see the light for \\(10\\sqrt{2}\\) minutes after meeting the beam. When the ship leaves the fan of light, it is 10 miles from the lighthouse. Find the speed of the ship."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 6
} | 21 | 0 | 0 |
omr-ws-1f54920f3d5527438395853b78ba8da1ab6a3bc9ca4b6b9652a4c4d5046849d3 | A lighthouse is on a small island 3 km away from the nearest point P on a straight shoreline and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from P? | [
"\\(\\frac{2\\pi \\sqrt{10}}{15}\\)"
] | [
"A lighthouse is on a small island 3 km away from the nearest point P on a straight shoreline and its light makes four revolutions per minute. How fast is the beam of light moving along the shoreline when it is 1 km from P?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7e1e9f6f3f30f31f14a85d481a703b8f133f57385e6b5869b170fe7e5d1b0fdb | A limping chess knight alternates between a normal move and a short move (to any diagonally neighboring cell) on a $5 \times 6$ chessboard. Starting with a normal move, what is the largest number of moves the knight can make without visiting any cell more than once? | [
"25"
] | [
"A limping chess knight alternates between a normal move and a short move (to any diagonally neighboring cell) on a $5 \\times 6$ chessboard. Starting with a normal move, what is the largest number of moves the knight can make without visiting any cell more than once?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 4,
"tir": 1
} | 14 | 0 | 0 |
omr-ws-fd7100b7b91c9429a5c7a02b84413da5ccec3ff877e8213d5be9fcf84d745a12 | A line $ y=x+2 $ is rotated $ 45^{\circ} $ clockwise about the point $ (0,2) $. A second line $ y=-2x+10 $ is drawn. What is the area of the triangle formed by these three lines? | [
"$\\frac{16}{3}$"
] | [
"A line $ y=x+2 $ is rotated $ 45^{\\circ} $ clockwise about the point $ (0,2) $. A second line $ y=-2x+10 $ is drawn. What is the area of the triangle formed by these three lines?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-6f1276ff5162c7fb9256b0217fe541b4b3c9e68106fb3aaf05bccd9e888fde83 | A line $AB$ is parallel to a plane $\pi $. $CD$ intersects $AB$ at an acute angle $\alpha $ and forms with the plane $\pi $ an angle $\varphi $. Determine the angle between the $\pi $ plane and the plane in which the lines $AB$ and $CD$ lie. | [
"$\\arcsin(\\sin \\varphi {\\rm cosec}\\, \\alpha )$"
] | [
"A line $AB$ is parallel to a plane $\\pi $. $CD$ intersects $AB$ at an acute angle $\\alpha $ and forms with the plane $\\pi $ an angle $\\varphi $. Determine the angle between the $\\pi $ plane and the plane in which the lines $AB$ and $CD$ lie."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-88d589ea0669689e1c23e2d3eda92f8f89b4e0a065a63ea0cd3d787c7e33059d | A line $L$ and two points $A$ and $B$ are given on a plane. How should a point $P$ be chosen on this line so that the largest of the distances $AP$ and $BP$ is minimal? | [
"Point $P$ should be chosen on the line $L$ between points $A$ and $B$."
] | [
"A line $L$ and two points $A$ and $B$ are given on a plane. How should a point $P$ be chosen on this line so that the largest of the distances $AP$ and $BP$ is minimal?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-30aef4532f53a8a3197d1639b30b559101e261435c892423a7b343059a9cde8f | A line $L$ and two points $P$ and $Q$ on the same side of it are given. The distances from $P$ and $Q$ to $L$ are $3$ cm and $5$ cm, respectively. The distance between the feet of the projections of $P$ and $Q$ over $L$ is $4$ cm. There are two circles such that each passes through the points $P$ and $Q$ and is tangent... | [
"40"
] | [
"A line $L$ and two points $P$ and $Q$ on the same side of it are given. The distances from $P$ and $Q$ to $L$ are $3$ cm and $5$ cm, respectively. The distance between the feet of the projections of $P$ and $Q$ over $L$ is $4$ cm. There are two circles such that each passes through the points $P$ and $Q$ and is ta... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-76a57571431485d3a42dcd7401211cf906c22ab64a8bbe02ab7cb6445af5f97b | A line $L$ on the plane $2x + y - 3z + 5 = 0$ is at a distance of 3 units from the point $P(1, 2, 3)$. A spider starts from point $A$ and after moving 4 units along the line $\frac{x-1}{2}=\frac{y-2}{1}=\frac{z-3}{-3}$, it reaches point $P$. From $P$, it jumps to line $L$ along the shortest distance and then moves 12 u... | [
"13"
] | [
"A line $L$ on the plane $2x + y - 3z + 5 = 0$ is at a distance of 3 units from the point $P(1, 2, 3)$. A spider starts from point $A$ and after moving 4 units along the line $\\frac{x-1}{2}=\\frac{y-2}{1}=\\frac{z-3}{-3}$, it reaches point $P$. From $P$, it jumps to line $L$ along the shortest distance and then mo... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-84558e5d6fd0f8a4f0b98060b3541cb589346343462c8e8b4a90ece56b10d74e | A line $OCD$ runs due East from $O$. Point $A$ is due North from $C$, and point $B$ is due North from $D$. The line $AB$ runs due East. Given that $OA$ is $235$ yards and its direction is $63^{\circ} 20'$ North of East, and the direction of $OB$ is $37^{\circ} 45'$ North of East, find the distances $AC$, $OB$, and $CD$... | [
"\\( AC \\approx 210 \\text{ yards}, OB \\approx 134.8 \\text{ yards}, CD \\approx 106.79 \\text{ yards} \\)"
] | [
"A line $OCD$ runs due East from $O$. Point $A$ is due North from $C$, and point $B$ is due North from $D$. The line $AB$ runs due East. Given that $OA$ is $235$ yards and its direction is $63^{\\circ} 20'$ North of East, and the direction of $OB$ is $37^{\\circ} 45'$ North of East, find the distances $AC$, $OB$, a... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-50362e69da3c6807b5d90ca7c44e04286e92d746a421166e0e5b4f305ab8ab4b | A line $\ell$ divides a circle $\Gamma$ into two regions, $\omega_A$ and $\omega_B$. Let $A$ be the largest circle inscribed inside $\omega_A$, and let $A_1$ and $A_2$ be the two circles tangent to $A$ and inscribed inside $\omega_A$. Let $B$ be the largest circle inscribed inside $\omega_B$, and let $B_1$ and $B_2$ be... | [
"\\dfrac{R}{4}"
] | [
"A line $\\ell$ divides a circle $\\Gamma$ into two regions, $\\omega_A$ and $\\omega_B$. Let $A$ be the largest circle inscribed inside $\\omega_A$, and let $A_1$ and $A_2$ be the two circles tangent to $A$ and inscribed inside $\\omega_A$. Let $B$ be the largest circle inscribed inside $\\omega_B$, and let $B_1$ ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0b8cc309e419b92b5ddd65a2c6a9f415469b2b00dcfc4d5186999bc6702855df | A line $\ell$ is drawn through the vertex $D$ of the square $ABCD$ which does not intersect the sides of the square. Ray $CA$ intersects $\ell$ at point $E$. Point $F$ is selected on ray $ED$ beyond point $D$ such that $\angle DCF = 45^\circ$. It turned out that $AE = CF$. Find the angle $\angle CFD$. | [
"67.5^\\circ"
] | [
"A line $\\ell$ is drawn through the vertex $D$ of the square $ABCD$ which does not intersect the sides of the square. Ray $CA$ intersects $\\ell$ at point $E$. Point $F$ is selected on ray $ED$ beyond point $D$ such that $\\angle DCF = 45^\\circ$. It turned out that $AE = CF$. Find the angle $\\angle CFD$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-c7a7027f7f72813a6440c99edb0a35332a3a7d572757f1df35c6fd697c3cbbdd | A line $\ell$ passes through vertex $C$ of the rhombus $ABCD$ and meets the extensions of $AB$ and $AD$ at points $X$ and $Y$, respectively. Lines $DX$ and $BY$ meet the circumcircle of $\triangle AXY$ for the second time at points $P$ and $Q$. If the circumcircle of $\triangle PCQ$ is tangent to $\ell$ at point $T$, c... | [
"10"
] | [
"A line $\\ell$ passes through vertex $C$ of the rhombus $ABCD$ and meets the extensions of $AB$ and $AD$ at points $X$ and $Y$, respectively. Lines $DX$ and $BY$ meet the circumcircle of $\\triangle AXY$ for the second time at points $P$ and $Q$. If the circumcircle of $\\triangle PCQ$ is tangent to $\\ell$ at poi... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0b85bd998e1c16eca94491dab3e468bd6624e2835150b287dba7e4e7ee167f98 | A line $l$ passes through the right focus $F_2$ of the hyperbola $\frac{x^2}{25} - \frac{y^2}{9} = 1$ and intersects the hyperbola at points $A$ and $B$. If the midpoint of segment $AB$ is in the third quadrant, find the slope $k_l$ of the line $l$. | [
"\\( |k_l| < \\frac{3}{5} \\)"
] | [
"A line $l$ passes through the right focus $F_2$ of the hyperbola $\\frac{x^2}{25} - \\frac{y^2}{9} = 1$ and intersects the hyperbola at points $A$ and $B$. If the midpoint of segment $AB$ is in the third quadrant, find the slope $k_l$ of the line $l$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-04868d6fd19e98cda7e925a41c81d8df00aff2e149a60e3d3de2e5a3064f55b0 | A line $l$ passing through the point $M(2,-2)$ intersects the ellipse $x^2 + 4y^2 = 4$ at two points $A$ and $B$ such that the distance $AB = \frac{4\sqrt{2}}{5}$. Find the equation of the line $l$. | [
"y = -\\frac{1}{2}x - 1"
] | [
"A line $l$ passing through the point $M(2,-2)$ intersects the ellipse $x^2 + 4y^2 = 4$ at two points $A$ and $B$ such that the distance $AB = \\frac{4\\sqrt{2}}{5}$. Find the equation of the line $l$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1d61a94beb432d438ad0b6bafb1ea64838f893f19211e27f8dc0ebd745d9669e | A line \( L \) intersects \( AB \) at \( F \), \( AC \) at \( D \), and \( BC \) at \( E \). The line \( L \) also intersects the internal angle bisector of \( \angle BAC \) at \( P \). Given that \( F \) is on the opposite side of \( A \) with respect to \( BC \), \( CD = CE \), and \( P \) is inside \( \triangle ABC ... | [
"100"
] | [
"A line \\( L \\) intersects \\( AB \\) at \\( F \\), \\( AC \\) at \\( D \\), and \\( BC \\) at \\( E \\). The line \\( L \\) also intersects the internal angle bisector of \\( \\angle BAC \\) at \\( P \\). Given that \\( F \\) is on the opposite side of \\( A \\) with respect to \\( BC \\), \\( CD = CE \\), and \... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-c084b7db91b949de22db040497eacb6f3a31a447e52795532b7d0894e2463839 | A line \( L \) is drawn from point \( P(4, 3) \) to meet the lines \( L_1: 3x + 4y + 5 = 0 \) and \( L_2: 3x + 4y + 15 = 0 \) at points \( A \) and \( B \) respectively. From \( A \), a line perpendicular to \( L \) is drawn meeting \( L_2 \) at \( A_1 \). Similarly, from \( B \), a line perpendicular to \( L \) is dra... | [
"\\( x - 7y + 17 = 0 \\) or \\( 7x + y - 31 = 0 \\)"
] | [
"A line \\( L \\) is drawn from point \\( P(4, 3) \\) to meet the lines \\( L_1: 3x + 4y + 5 = 0 \\) and \\( L_2: 3x + 4y + 15 = 0 \\) at points \\( A \\) and \\( B \\) respectively. From \\( A \\), a line perpendicular to \\( L \\) is drawn meeting \\( L_2 \\) at \\( A_1 \\). Similarly, from \\( B \\), a line perp... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 36,
"tir": 7
} | 43 | 0 | 0 |
omr-ws-cd7b36c5ca90fe4e8bd74c8472f09a950a40d439fe431d1ebaeb42f0ef2b752c | A line \( l \) does not meet a circle \( \omega \) with center \( O \). \( E \) is the point on \( l \) such that \( OE \) is perpendicular to \( l \). \( M \) is any point on \( l \) other than \( E \). The tangents from \( M \) to \( \omega \) touch it at \( A \) and \( B \). \( C \) is the point on \( MA \) such tha... | [
"\\dfrac{5}{8}"
] | [
"A line \\( l \\) does not meet a circle \\( \\omega \\) with center \\( O \\). \\( E \\) is the point on \\( l \\) such that \\( OE \\) is perpendicular to \\( l \\). \\( M \\) is any point on \\( l \\) other than \\( E \\). The tangents from \\( M \\) to \\( \\omega \\) touch it at \\( A \\) and \\( B \\). \\( C ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-6dd154c7eee6f8b1147d3bc8e33bdd3321753e02868c36a9c185533b00cef298 | A line \( l \) in the coordinate plane has the property that if \((x, y)\) is a point on \( l \), then the point \((3x + 2y + 1, x + 4y - 3)\) is also on \( l \). Suppose the equation of \( l \) is \( px + qy = 20 \), where \( p \) and \( q \) are constants. What is the maximum possible value of \( p^2 + q^2 \)? | [
"0"
] | [
"A line \\( l \\) in the coordinate plane has the property that if \\((x, y)\\) is a point on \\( l \\), then the point \\((3x + 2y + 1, x + 4y - 3)\\) is also on \\( l \\). Suppose the equation of \\( l \\) is \\( px + qy = 20 \\), where \\( p \\) and \\( q \\) are constants. What is the maximum possible value of ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5d82f1c342a79f4536815456290a72f2e43b5608407d5e311cfc3f4ccac2317c | A line \( l \) is in the same plane as triangle \( ABC \) and does not pass through any of the vertices. How many sides of the triangle could the line \( l \) pass through? | [
"2"
] | [
"A line \\( l \\) is in the same plane as triangle \\( ABC \\) and does not pass through any of the vertices. How many sides of the triangle could the line \\( l \\) pass through?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-8e87a8da7d2adad8b6b19e4c0723a5601215c18551e0a4b097c7f6612e0752ae | A line \( l \) is tangent to a circle \( S \) at point \( A \). For any points \( B \) and \( C \) on \( l \) on opposite sides of \( A \), let the other tangents from \( B \) and \( C \) to \( S \) intersect at a point \( P \). If \( B \) and \( C \) vary on \( l \) such that the product \( AB \cdot AC \) is constant,... | [
"\\text{A line parallel to } l"
] | [
"A line \\( l \\) is tangent to a circle \\( S \\) at point \\( A \\). For any points \\( B \\) and \\( C \\) on \\( l \\) on opposite sides of \\( A \\), let the other tangents from \\( B \\) and \\( C \\) to \\( S \\) intersect at a point \\( P \\). If \\( B \\) and \\( C \\) vary on \\( l \\) such that the produ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 5
} | 17 | 0 | 0 |
omr-ws-7f42cbd9188d32ad5f819fecb5d66fcf80a451ad032665991e2e37e4b74c5239 | A line \( l \) is tangent to a circle of radius \( r \) at a point \( P \). From a variable point \( R \) on the circle, a perpendicular \( RQ \) is drawn to \( l \) with \( Q \in l \). Determine the maximum area of triangle \( PQR \). | [
"\\(\\frac{3\\sqrt{3}}{8}r^2\\)"
] | [
"A line \\( l \\) is tangent to a circle of radius \\( r \\) at a point \\( P \\). From a variable point \\( R \\) on the circle, a perpendicular \\( RQ \\) is drawn to \\( l \\) with \\( Q \\in l \\). Determine the maximum area of triangle \\( PQR \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-4a581a116262dc636fa7622564531859ffd71113f040f535ad199ab44764de7c | A line \( l \) passes through point \( A \) of a given \( \triangle ABC \) without intersecting segment \( BC \). Circle with center \( O_1 \) is tangent to segment \( AC \), line \( l \), and line \( BC \) (located outside \( \triangle ABC \)). Circle with center \( O_2 \) is tangent to segment \( AB \), line \( l \),... | [
"\\( BC \\)"
] | [
"A line \\( l \\) passes through point \\( A \\) of a given \\( \\triangle ABC \\) without intersecting segment \\( BC \\). Circle with center \\( O_1 \\) is tangent to segment \\( AC \\), line \\( l \\), and line \\( BC \\) (located outside \\( \\triangle ABC \\)). Circle with center \\( O_2 \\) is tangent to segm... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 2
} | 4 | 0 | 0 |
omr-ws-eb9be6eef0888406e8333e25b87f12d3b47bb6898ba386eb9d7ad9b59a58dcb6 | A line \( l \) passes through the center \( O \) of an equilateral triangle \( \Delta ABC \), intersecting \( CA \) at \( N \) and \( BC \) at \( M \). Let \( AM = a \), \( BN = b \), and \( MN = c \). If a triangle can be constructed with sides \( a \), \( b \), and \( c \), find the length of the altitude to side \( ... | [
"2\\sqrt{6}"
] | [
"A line \\( l \\) passes through the center \\( O \\) of an equilateral triangle \\( \\Delta ABC \\), intersecting \\( CA \\) at \\( N \\) and \\( BC \\) at \\( M \\). Let \\( AM = a \\), \\( BN = b \\), and \\( MN = c \\). If a triangle can be constructed with sides \\( a \\), \\( b \\), and \\( c \\), find the le... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ca7896ac17c17f920de1b6a91d4601970eb142c39a50803753f9eeca198b2991 | A line \( l \) passes through the circumcenter \( O \) of an equilateral triangle \( \Delta ABC \) and intersects \( AB \) and \( AC \) at points \( M \) and \( N \), respectively. The circumcircle of \( \Delta AMN \) with center \( K \) intersects the circumcircle \( (O) \) at point \( P \). Let \( OK \) meet \( AB \)... | [
"2"
] | [
"A line \\( l \\) passes through the circumcenter \\( O \\) of an equilateral triangle \\( \\Delta ABC \\) and intersects \\( AB \\) and \\( AC \\) at points \\( M \\) and \\( N \\), respectively. The circumcircle of \\( \\Delta AMN \\) with center \\( K \\) intersects the circumcircle \\( (O) \\) at point \\( P \\... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-c0b79b4fc43dc96f8c84e00befcd18279bbb103a6616f0934ee12129dece2f8e | A line \( l \) passing through the orthocenter \( H \) of \( \Delta ABC \) intersects the circumcircles \( (BHC) \), \( (CHA) \), and \( (AHB) \) at points \( D \), \( E \), and \( F \) respectively. Points \( X \in (BHC) \), \( Y \in (CHA) \), and \( Z \in (AHB) \) are such that \( DX \parallel BC \), \( EY \parallel ... | [
"\\dfrac{1}{2}"
] | [
"A line \\( l \\) passing through the orthocenter \\( H \\) of \\( \\Delta ABC \\) intersects the circumcircles \\( (BHC) \\), \\( (CHA) \\), and \\( (AHB) \\) at points \\( D \\), \\( E \\), and \\( F \\) respectively. Points \\( X \\in (BHC) \\), \\( Y \\in (CHA) \\), and \\( Z \\in (AHB) \\) are such that \\( DX... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-b3f16523f28c593a641f97e62cf54bdbfad56c4a1ce324bfb0eed9938db61ae9 | A line \( r \) contains the points \( A \), \( B \), \( C \), \( D \) in that order. Let \( P \) be a point not on \( r \) such that \( \angle APB = \angle CPD \). The angle bisector of \( \angle APD \) intersects the line \( r \) at a point \( G \). If \( GA = 3 \), \( GB = 4 \), \( GC = 5 \), and \( GD = 6 \), find t... | [
"\\dfrac{7}{60}"
] | [
"A line \\( r \\) contains the points \\( A \\), \\( B \\), \\( C \\), \\( D \\) in that order. Let \\( P \\) be a point not on \\( r \\) such that \\( \\angle APB = \\angle CPD \\). The angle bisector of \\( \\angle APD \\) intersects the line \\( r \\) at a point \\( G \\). If \\( GA = 3 \\), \\( GB = 4 \\), \\( ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8dce137188eda013e8ccd8ccc5bc9c335c05883a813c748b6823afd85f993d25 | A line \( x = k \) intersects the graph of \( y = \log_5 k \) and the graph of \( y = \log_5 (x + 4) \). The distance between the points of intersection is 0.5. Given that \( k = a + \sqrt{b} \), where \( a \) and \( b \) are integers, find \( a + b \). | [
"6"
] | [
"A line \\( x = k \\) intersects the graph of \\( y = \\log_5 k \\) and the graph of \\( y = \\log_5 (x + 4) \\). The distance between the points of intersection is 0.5. Given that \\( k = a + \\sqrt{b} \\), where \\( a \\) and \\( b \\) are integers, find \\( a + b \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-11113d5d4e6f7bdfda14d39578f12057b64da688213ec367ab1ccb1cc248975b | A line containing the points $(2, 16)$ and $(-8, -4)$ intersects the $y$-axis at what point? | [
"(0, 12)"
] | [
"A line containing the points $(2, 16)$ and $(-8, -4)$ intersects the $y$-axis at what point?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ed20dad80ad3ee9dfc2dd36bdc1b713fa5b199f5a9afccfb243dd4fda754da3e | A line containing the points $(9, 1)$ and $(5, 5)$ intersects the $x$ -axis at what point? | [
"(10, 0)"
] | [
"A line containing the points $(9, 1)$ and $(5, 5)$ intersects the $x$ -axis at what point?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e92d0c1116f1cdcd7bc8da5ffc29abaa147414a09a46ae93748e7328dfcf48b8 | A line contains the points $(-1, 6)$, $(6, k)$ and $(20, 3)$. What is the value of $k$? | [
"5"
] | [
"A line contains the points $(-1, 6)$, $(6, k)$ and $(20, 3)$. What is the value of $k$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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