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-14c76d11033b4de35f872227885448d420628078058674a507b754d903791e3c | A hotel provides four types of rooms with the following capacities, rates, and numbers of rooms: - Type A: 2 persons, Rp. 100,000/day, 10 rooms - Type B: 3 persons, Rp. 150,000/day, 15 rooms - Type C: 4 persons, Rp. 200,000/day, 20 rooms - Type D: 5 persons, Rp. 250,000/day, 25 rooms A group of four families wants to s... | [
"1450000"
] | [
"A hotel provides four types of rooms with the following capacities, rates, and numbers of rooms:\n- Type A: 2 persons, Rp. 100,000/day, 10 rooms\n- Type B: 3 persons, Rp. 150,000/day, 15 rooms\n- Type C: 4 persons, Rp. 200,000/day, 20 rooms\n- Type D: 5 persons, Rp. 250,000/day, 25 rooms\n\nA group of four familie... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-e4139c445af20d1030d540c2a737b099d07a321b6da8b889c0d97e9328bdf993 | A hound chases a fox which flees in a straight line with a constant velocity \( u \). The hound accelerates in the direction of the fox starting with a velocity \( v \), and throughout the chase it maintains a speed of \( |v| \). At time \( t = 0 \), \( u \) is perpendicular to \( v \), and the distance between the fox... | [
"\\dfrac{L v}{v^2 - u^2}"
] | [
"A hound chases a fox which flees in a straight line with a constant velocity \\( u \\). The hound accelerates in the direction of the fox starting with a velocity \\( v \\), and throughout the chase it maintains a speed of \\( |v| \\). At time \\( t = 0 \\), \\( u \\) is perpendicular to \\( v \\), and the distanc... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-d5aa29410a9ab14ee7d99c5afa1ba7b982cbb452f02715be0cfe1d8557b5fbba | A hound chases a fox which flees in a straight line with a constant velocity \( u \). The hound accelerates in the direction of the fox starting with a velocity \( v \), and throughout the chase it maintains a speed of \( |v| \). At time \( t = 0 \), \( u \) is perpendicular to \( v \), and the distance between the fox... | [
"\\( \\frac{uv}{L} \\)"
] | [
"A hound chases a fox which flees in a straight line with a constant velocity \\( u \\). The hound accelerates in the direction of the fox starting with a velocity \\( v \\), and throughout the chase it maintains a speed of \\( |v| \\). At time \\( t = 0 \\), \\( u \\) is perpendicular to \\( v \\), and the distanc... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 6,
"tir": 1
} | 22 | 0 | 0 |
omr-ws-b21e09034747897d3ba0e167d3096d4f452411cdee2dec7c2aca7374ef1dc285 | A house A is located 300 m from the bank of a 200 m wide river. 600 m above and 500 m from the opposite bank is another house B. A bridge has been built over the river to minimize the distance between the two houses. Determine the minimum distance and the location of the bridge on the riverbank. | [
"225"
] | [
"A house A is located 300 m from the bank of a 200 m wide river. 600 m above and 500 m from the opposite bank is another house B. A bridge has been built over the river to minimize the distance between the two houses. Determine the minimum distance and the location of the bridge on the riverbank."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-6dce13acfd2387b4a386136e9696c2ac32f78435484caa508951314d8262f8ed | A house buyer borrows RM50000 from a bank at an annual interest rate of 9%, calculated based on the outstanding amount at the beginning of each year. The loan is to be repaid in monthly installments over 15 years. Calculate the monthly repayment amount. | [
"516.91"
] | [
"A house buyer borrows RM50000 from a bank at an annual interest rate of 9%, calculated based on the outstanding amount at the beginning of each year. The loan is to be repaid in monthly installments over 15 years. Calculate the monthly repayment amount."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-d534c943993035b12976926581359122a4f4e8fc6b31e9854532c0660bf1028a | A house is in the shape of a triangle with perimeter \( P \) meters and area \( A \) square meters. The garden consists of all the land within 5 meters of the house. How much land do the garden and house together occupy? | [
"\\( 5P + A + 25\\pi \\)"
] | [
"A house is in the shape of a triangle with perimeter \\( P \\) meters and area \\( A \\) square meters. The garden consists of all the land within 5 meters of the house. How much land do the garden and house together occupy?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-2ae1015af102eaac435c11e7c2b7143e6c3cce73857c38b6798d5e471573256d | A house number is a 3-digit number with distinct digits. If you permute its digits, you get 5 other numbers. The sum of these 5 numbers is exactly 2017. Determine the original 3-digit house number. | [
"425"
] | [
"A house number is a 3-digit number with distinct digits. If you permute its digits, you get 5 other numbers. The sum of these 5 numbers is exactly 2017. Determine the original 3-digit house number."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-67b33d5177376dcca2c6e827598bcf2fa04b0fc0690a767ff9573f99829ff9aa | A house with a square floor plan is to have each of its four outer walls finished with brick, stone, stucco, or wood. If no two adjacent outer walls are to have the same finish, in how many ways can the house's walls be constructed? | [
"30"
] | [
"A house with a square floor plan is to have each of its four outer walls finished with brick, stone, stucco, or wood. If no two adjacent outer walls are to have the same finish, in how many ways can the house's walls be constructed?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3443a5822b1b3ecc16857f4931561d21898b859fff2a5ebbe634adf2af9bb8f3 | A housewife is preparing a cake for guests, where the number of guests will be $n-1$, $n$, or $n+1$. What is the minimal number $f(n)$ of pieces the cake should be cut into to ensure it can be divided equally among the guests? | [
"\\(2n + O(\\sqrt{n})\\)"
] | [
"A housewife is preparing a cake for guests, where the number of guests will be $n-1$, $n$, or $n+1$. What is the minimal number $f(n)$ of pieces the cake should be cut into to ensure it can be divided equally among the guests?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-502cff5905cc9131ccca49dda0c93e111adb1da790ce85a4fc040cc3b31f70ad | A housewife saved $ \$2.50$ in buying a dress on sale. If she spent $ \$25$ for the dress, what percentage did she save? | [
"9 \\%"
] | [
"A housewife saved $ \\$2.50$ in buying a dress on sale. If she spent $ \\$25$ for the dress, what percentage did she save?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 4,
"tir": 4
} | 9 | 0 | 0 |
omr-ws-ec4d71c38c94aa5c7695422bf9886a2080963b51dc9f4f9ce9c33d7e9cd70459 | A housing developer will build 10 houses consisting of 5 tulip type houses and 5 rose type houses, with the rule that 7 houses must be built on the left of the road and 3 houses built on the right of the road. How many arrangements of the placement of the houses are there if houses of the same type are distinguished? | [
"3628800"
] | [
"A housing developer will build 10 houses consisting of 5 tulip type houses and 5 rose type houses, with the rule that 7 houses must be built on the left of the road and 3 houses built on the right of the road. How many arrangements of the placement of the houses are there if houses of the same type are distinguish... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 8,
"tir": 27
} | 47 | 0 | 0 |
omr-ws-9a2af97cefb1a46e55595bac82a831fcbc7a8898c12c8ac74d71c1e6505ae530 | A housing developer will build 10 houses consisting of 5 tulip type houses and 5 rose type houses, with the rule that 7 houses must be built on the left of the road and 3 houses built on the right of the road. How many arrangements of the placement of the houses are there if houses of the same type are indistinguishabl... | [
"210"
] | [
"A housing developer will build 10 houses consisting of 5 tulip type houses and 5 rose type houses, with the rule that 7 houses must be built on the left of the road and 3 houses built on the right of the road. How many arrangements of the placement of the houses are there if houses of the same type are indistingui... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-de1f21b13ff0d8d8adefbfd8f21529ce792ec130e337e34f238b2456ee8fa556 | A hubcap with a radius of 24 inches rolls down a street that is 200 feet long. How many times will it rotate? | [
"15"
] | [
"A hubcap with a radius of 24 inches rolls down a street that is 200 feet long. How many times will it rotate?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-915acdbb5540602ebcfc2674c2e05d7184488017a907966eb3c3a79b32509356 | A huge room has 1000 doors, each with a lock and numbered from 1 to 1000. A mad-man enters and at the first round, he unlocks all doors. At the second round, he locks only doors that have numbers that are multiples of 2. At the third round, he operates the locks of doors which have numbers that are multiples of 3 (unlo... | [
"969"
] | [
"A huge room has 1000 doors, each with a lock and numbered from 1 to 1000. A mad-man enters and at the first round, he unlocks all doors. At the second round, he locks only doors that have numbers that are multiples of 2. At the third round, he operates the locks of doors which have numbers that are multiples of 3 ... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-1092bd028a362cbceaac5705e01f020ac1956fd02a5eae03831a6882d642a6c1 | A hundred fractions are written, whose numerators are numbers from $1$ to $100$, each used once, and whose denominators are numbers from $1$ to $100$, each used once. Suppose that the sum of these fractions is an irreducible fraction with denominator $2$. Find the number of distinct pairs of fractions that can be excha... | [
"2500"
] | [
"A hundred fractions are written, whose numerators are numbers from $1$ to $100$, each used once, and whose denominators are numbers from $1$ to $100$, each used once. Suppose that the sum of these fractions is an irreducible fraction with denominator $2$. Find the number of distinct pairs of fractions that can be ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-7ce3b8263bca0bdad06cfdb2d2175545e547bee2de437e22cce8f16b80370d65 | A hundred knights, some of whom are at enmity with each other, came to King Arthur. Each of the 2005 pairs of knights who are at enmity refuse to sit at the same table. How many tables does King Arthur need to ensure that he can seat all his guests without any enmity issues? | [
"63"
] | [
"A hundred knights, some of whom are at enmity with each other, came to King Arthur. Each of the 2005 pairs of knights who are at enmity refuse to sit at the same table. How many tables does King Arthur need to ensure that he can seat all his guests without any enmity issues?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 7
} | 15 | 0 | 0 |
omr-ws-be38ab29092968c939985663d04a358799e7814a1a8796f87f07b3ce124021d6 | A hundred people have lined up to board an airplane. The first person has lost his boarding pass and takes a random seat. Every person after him takes his/her own seat if available, and a random unoccupied seat if not. What is the probability that the 100th person takes his own seat? | [
"\\(\\frac{1}{2}\\)"
] | [
"A hundred people have lined up to board an airplane. The first person has lost his boarding pass and takes a random seat. Every person after him takes his/her own seat if available, and a random unoccupied seat if not. What is the probability that the 100th person takes his own seat?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-73b2920327eee341eab4f7e1abaee5dae6c3b710f09d90763bc2f8592a116534 | A hundred people want to take a photo and can stand in any number of rows from 1 to 100. Let $N$ be the number of possible photos they can take. What is the largest integer $k$ such that $2^k \mid N$? | [
"196"
] | [
"A hundred people want to take a photo and can stand in any number of rows from 1 to 100. Let $N$ be the number of possible photos they can take. What is the largest integer $k$ such that $2^k \\mid N$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-374b2d22f1aa73726294b644f27df10b692389967ca3cdd1f5110fcf5f0f805f | A hundred seeds are planted in ten rows of ten seeds per row. Assuming that each seed independently germinates with probability 1/2, find the probability that the row with the maximum number of germinations contains at least 8 seedlings. | [
"0.430"
] | [
"A hundred seeds are planted in ten rows of ten seeds per row. Assuming that each seed independently germinates with probability 1/2, find the probability that the row with the maximum number of germinations contains at least 8 seedlings."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 1
} | 10 | 0 | 0 |
omr-ws-26ba9bff7c56f819976c6b078bdce8f4f34b0c21482af9f2ffec85279da9f845 | A hundred silver coins are laid down in a line. A wizard can convert a silver coin into a golden one in 3 seconds. Each adjacent golden coin reduces the conversion time by 1 second. What is the minimal time required for the wizard to convert all coins to gold? | [
"201 seconds"
] | [
"A hundred silver coins are laid down in a line. A wizard can convert a silver coin into a golden one in 3 seconds. Each adjacent golden coin reduces the conversion time by 1 second. What is the minimal time required for the wizard to convert all coins to gold?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 47,
"genselect": 7
} | 54 | 0 | 0 |
omr-ws-4ba47e5b29c7fb2cda92920acc32c7e75f0023803755628b6404322245f31f07 | A hundred tourists arrive at a hotel with single rooms numbered $1, 2, \ldots, n$. Among these rooms, $k$ rooms are under repair, but the tourists do not know which ones. Each tourist, in turn, checks rooms in any order and takes the first room not under repair and not occupied. The tourists may coordinate their strate... | [
"\\( 50k + 51 \\) if \\( k \\) is odd, \\( 50k + 100 \\) if \\( k \\) is even."
] | [
"A hundred tourists arrive at a hotel with single rooms numbered $1, 2, \\ldots, n$. Among these rooms, $k$ rooms are under repair, but the tourists do not know which ones. Each tourist, in turn, checks rooms in any order and takes the first room not under repair and not occupied. The tourists may coordinate their ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8b51643c53d316bb54b3e79408b2c6117f5fcb2ea51d1f302d0e3c46dc430914 | A hunter and a visible rabbit play a game in the Euclidean plane bounded by an equilateral triangle. The rabbit's starting point, $A_0$, the hunter's starting point, $B_0$, and the centroid of the triangle are the same. After $n-1$ rounds of the game, the rabbit is at point $A_{n-1}$ and the hunter is at point $B_{n-1}... | [
"\\dfrac{1}{4}"
] | [
"A hunter and a visible rabbit play a game in the Euclidean plane bounded by an equilateral triangle. The rabbit's starting point, $A_0$, the hunter's starting point, $B_0$, and the centroid of the triangle are the same. After $n-1$ rounds of the game, the rabbit is at point $A_{n-1}$ and the hunter is at point $B_... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 1
} | 9 | 0 | 0 |
omr-ws-fb380644727617cc36ed16f9b76b8923b3d7a97238a59a597ff282ba0028f1c0 | A hunter and an invisible rabbit play a game in the complex plane. The rabbit's starting point, $A_0$, and the hunter's starting point, $B_0$, are the same. After $n-1$ rounds of the game, the rabbit is at point $A_{n-1}$ and the hunter is at point $B_{n-1}$. In the nth round of the game, three things occur in order: (... | [
"not possible"
] | [
"A hunter and an invisible rabbit play a game in the complex plane. The rabbit's starting point, $A_0$, and the hunter's starting point, $B_0$, are the same. After $n-1$ rounds of the game, the rabbit is at point $A_{n-1}$ and the hunter is at point $B_{n-1}$. In the nth round of the game, three things occur in ord... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-26a4e3668453f2b279812bfcf3b5cb4b8d36d182875b8cc8ef24b9ee5ee2c2ce | A hunter is lost in the forest and knows that the road is exactly 1 unit away from his initial position. Determine the trajectory he should follow to find the road such that the length of this trajectory is minimized. Also, find the length of this trajectory. | [
"\\(\\sqrt{3} + 1 + \\frac{7\\pi}{6}\\)"
] | [
"A hunter is lost in the forest and knows that the road is exactly 1 unit away from his initial position. Determine the trajectory he should follow to find the road such that the length of this trajectory is minimized. Also, find the length of this trajectory."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ff9d029b5505c5d9c0be7bc64e88deb342ec4acaace1b4a3921fadb261e1a987 | A hunter knows that a deer is hidden in one of two nearby bushes, with the probability of it being in bush-I being 4/5. The hunter has a rifle with 10 bullets and decides to fire them all at bush-I or bush-II. Each shot hits one of the two bushes independently with probability 1/2. How many bullets must the hunter fire... | [
"6"
] | [
"A hunter knows that a deer is hidden in one of two nearby bushes, with the probability of it being in bush-I being 4/5. The hunter has a rifle with 10 bullets and decides to fire them all at bush-I or bush-II. Each shot hits one of the two bushes independently with probability 1/2. How many bullets must the hunter... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-3853adccafce84bfbeeeb8a64ab436baeeccd73f8108009531478afd08948fc7 | A hunter shoots at a fox from a distance of 100 yards with a probability of hitting the fox of $1/2$. If he misses, he reloads and shoots again after the fox has run 50 yards further. If he misses again, he shoots a third time after the fox has run another 50 yards. The probability of a hit is inversely proportional to... | [
"\\(\\frac{95}{144}\\)"
] | [
"A hunter shoots at a fox from a distance of 100 yards with a probability of hitting the fox of $1/2$. If he misses, he reloads and shoots again after the fox has run 50 yards further. If he misses again, he shoots a third time after the fox has run another 50 yards. The probability of a hit is inversely proportion... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-1b9f6721cc85659c4e31cb461bfbe8129e5e17a131aef6028c9d4ce20228fc30 | A hunter starts at $(0,0)$ and a rabbit starts at $(8,0)$. Both have the same speed and infinite stamina. The rabbit runs along the parametrization $x = 8 - t$, $y = t$. If the hunter continuously moves towards the rabbit, what is the path (arc) that the hunter follows? | [
"y = x"
] | [
"A hunter starts at $(0,0)$ and a rabbit starts at $(8,0)$. Both have the same speed and infinite stamina. The rabbit runs along the parametrization $x = 8 - t$, $y = t$. If the hunter continuously moves towards the rabbit, what is the path (arc) that the hunter follows?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-124ed39122aca7d281bab26a5bd37eae381ef2ec47c8965d00132ce0f315466d | A hunter with a bow has a probability of $\frac{1}{3}$ of hitting a deer with each shot. What is the expected value and the variance of the number of shots needed to hit the deer at least once? | [
"6"
] | [
"A hunter with a bow has a probability of $\\frac{1}{3}$ of hitting a deer with each shot. What is the expected value and the variance of the number of shots needed to hit the deer at least once?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-a7e62f8e38e48a99cbaab968a2036878fa0c2c7699183052d1d75900e1716d63 | A hut has a semicircular cross-section and a minimum height of 6 meters. The product of its length and width is 500. Find the minimum surface area of the hut, excluding the base surface. | [
"286\\pi"
] | [
"A hut has a semicircular cross-section and a minimum height of 6 meters. The product of its length and width is 500. Find the minimum surface area of the hut, excluding the base surface."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8954b508f4006cd12dc3f3c3e42778e23a70a94ca0736487a0a7abd2ef79e0b4 | A hydrofoil made a round trip of 144 km in 4 hours. Because of head winds, the average speed on the return trip was 15 km/h less than the average speed going out. Find the two speeds. | [
"45 km/h and 30 km/h"
] | [
"A hydrofoil made a round trip of 144 km in 4 hours. Because of head winds, the average speed on the return trip was 15 km/h less than the average speed going out. Find the two speeds."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 28
} | 28 | 0 | 0 |
omr-ws-1306d7ed5e1a52c331c455600d25ac3262f3cfc023f40ee3a22d8c8f59700eb8 | A hydrologist measures the diameter of a well to be 3 feet. Using a flashlight, she creates an angle of depression of 79° to the water on the other side of the well. What is the depth of the well water? | [
"15.43 ft"
] | [
"A hydrologist measures the diameter of a well to be 3 feet. Using a flashlight, she creates an angle of depression of 79° to the water on the other side of the well. What is the depth of the well water?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-f924ee2488bf05bfea9f2eaf1bff3e8f895d8ddbe462a2b8bd9b682843d2e7c3 | A hydrophobic, hungry, and lazy mouse is at \( (0, 0) \), a piece of cheese at \( (26, 26) \), and a circular lake of radius \( 5\sqrt{2} \) is centered at \( (13, 13) \). What is the length of the shortest path that the mouse can take to reach the cheese that also does not pass through the lake? | [
"(16 + 5\\pi)\\sqrt{2}"
] | [
"A hydrophobic, hungry, and lazy mouse is at \\( (0, 0) \\), a piece of cheese at \\( (26, 26) \\), and a circular lake of radius \\( 5\\sqrt{2} \\) is centered at \\( (13, 13) \\). What is the length of the shortest path that the mouse can take to reach the cheese that also does not pass through the lake?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
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"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-dab1a802add15693349dea1bf603c28b7f159416000d058d7f4f7406df3ae3f9 | A hyperbola has its center at (0,0) and the equation of the tangent to it at (1,1) is \(x + y = 2\). The given tangent cuts the conjugate hyperbola at points A and B such that \(AB = 6\sqrt{2}\). Find the equation of the hyperbola and the pair of asymptotes. | [
"\\(\\frac{x^{2}}{\\sqrt{22}+2}-\\frac{y^{2}}{\\sqrt{22}-2}=1\\) and \\(y = \\pm \\frac{\\sqrt{11}-\\sqrt{2}}{3} \\cdot x\\)"
] | [
"A hyperbola has its center at (0,0) and the equation of the tangent to it at (1,1) is \\(x + y = 2\\). The given tangent cuts the conjugate hyperbola at points A and B such that \\(AB = 6\\sqrt{2}\\). Find the equation of the hyperbola and the pair of asymptotes."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0d861ec14706ce7f113e69f8ca99413f154c9ab2c3a65965ad6fcd95d6f104ed | A hyperbola has its two foci at $(5, 0)$ and $(9, 4).$ Find the coordinates of its center. | [
"(7,2)"
] | [
"A hyperbola has its two foci at $(5, 0)$ and $(9, 4).$ Find the coordinates of its center."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2f2375838bd38bc96bb6e5ac08f4b561a36ff32c9da48dc579598a0badb25de5 | A hyperbola has the property that, if F and F' are its foci, then the four points of intersection of this hyperbola with the circle having diameter FF' are the vertices of a square. Find the eccentricity of the hyperbola. | [
"\\(\\frac{\\sqrt2}{\\sqrt{2-\\sqrt2}}\\) or equivalently \\(\\sqrt{2+\\sqrt2}\\)."
] | [
"A hyperbola has the property that, if F and F' are its foci, then the four points of intersection of this hyperbola with the circle having diameter FF' are the vertices of a square. Find the eccentricity of the hyperbola."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
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} | 45 | 0 | 0 |
omr-ws-0338750d4b91f59323ddc8615ae189e48fc5e559f78154c21a950baaaa662493 | A hyperbola is centered at the origin and opens either horizontally or vertically. It passes through the points $(-3, 4),$ $(-2, 0),$ and $(t, 2).$ Find $t^2.$ | [
"\\frac{21}{4}"
] | [
"A hyperbola is centered at the origin and opens either horizontally or vertically. It passes through the points $(-3, 4),$ $(-2, 0),$ and $(t, 2).$ Find $t^2.$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a699551532f41bd640f16427b0609d0bd56c7056ac461b6dd0071d7fc627306b | A hyperbola is tangent to the line \( x + y = 8 \) and has foci at \( (10,0) \) and \( (0,4) \). Find the length of its latus rectum. | [
"\\dfrac{8}{5}"
] | [
"A hyperbola is tangent to the line \\( x + y = 8 \\) and has foci at \\( (10,0) \\) and \\( (0,4) \\). Find the length of its latus rectum."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-e819d4d0980c08d8e4d7c36d736256b694cfa7867c30f57d4ee15ee4a812a784 | A hyperbola is tangent to the x-axis at point M and the y-axis at point N. One focus of the hyperbola is at (6, 2), and the cosine of the angle between its asymptotes is $\frac{13}{37}$. Given that the center of the hyperbola has integer coordinates, find the coordinates of points M and N, and the coordinates of the ot... | [
"M = (8, 0), N = (0, -40), other focus = (-4, -12)"
] | [
"A hyperbola is tangent to the x-axis at point M and the y-axis at point N. One focus of the hyperbola is at (6, 2), and the cosine of the angle between its asymptotes is $\\frac{13}{37}$. Given that the center of the hyperbola has integer coordinates, find the coordinates of points M and N, and the coordinates of ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e8ad146f499fc1475520a43cafca79d2839affc9585ffb5d450ddf3047773253 | A hypercube $H_n$ is an $n$-dimensional analogue of a cube. Its vertices are all the points $(x_1, \ldots, x_n)$ that satisfy $x_i = 0$ or $1$ for all $1 \le i \le n$ and its edges are all segments that connect two adjacent vertices. (Two vertices are adjacent if their coordinates differ at exactly one $x_i$. For examp... | [
"48"
] | [
"A hypercube $H_n$ is an $n$-dimensional analogue of a cube. Its vertices are all the points $(x_1, \\ldots, x_n)$ that satisfy $x_i = 0$ or $1$ for all $1 \\le i \\le n$ and its edges are all segments that connect two adjacent vertices. (Two vertices are adjacent if their coordinates differ at exactly one $x_i$. F... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8d7b5d4320c26c2c3adfc3a05a2e3f8c1fcf3a30e35fe8108d1c2b11c7472c48 | A hypersphere is inscribed in a unit tesseract and circumscribed in a regular hyper-tetrahedron. What is the side length of the hyper-tetrahedron? | [
"$\\frac{\\sqrt{112}}{12}$"
] | [
"A hypersphere is inscribed in a unit tesseract and circumscribed in a regular hyper-tetrahedron. What is the side length of the hyper-tetrahedron?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fb9bb4800b570769f999bb5b882f2918b99f2cea3e347d32f361617c31c27b34 | A is a 3 × 3 matrix with entries from the set {–1, 0, 1}. What is the probability that A is neither symmetric nor skew-symmetric? | [
"\\(\\frac{18927}{19683}\\)"
] | [
"A is a 3 × 3 matrix with entries from the set {–1, 0, 1}. What is the probability that A is neither symmetric nor skew-symmetric?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ffbbd69e04661dabae99b6c70fbe95697ab1b754066e33c7709bb0bbdc64ce18 | A is a three-digit number, and \(A'\) is A with its digits written in reverse order. If \(A - A'\) is a positive three-digit number with the same three digits as A and \(A'\), what is A? | [
"954"
] | [
"A is a three-digit number, and \\(A'\\) is A with its digits written in reverse order. If \\(A - A'\\) is a positive three-digit number with the same three digits as A and \\(A'\\), what is A?"
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-b87aad999b8d6f82a159f3318a4116d00e8d536c1e251afa7303118e7fc90acc | A is engaged to do a piece of work. After working for 2 days, A leaves and B finishes the remaining work in 9 days. If A had worked for 3 days before leaving, B would have finished the remaining work in 6 days. In how many days can each person working alone finish the whole work? | [
"A can finish the work in 5 days, and B can finish the work in 15 days."
] | [
"A is engaged to do a piece of work. After working for 2 days, A leaves and B finishes the remaining work in 9 days. If A had worked for 3 days before leaving, B would have finished the remaining work in 6 days. In how many days can each person working alone finish the whole work?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 28
} | 39 | 0 | 0 |
omr-ws-105a38fcfdef702959d54ddf987088abf576ec7e0acc6b12348a8e83b72ed2a9 | A is one of the points of intersection of two given circles. A variable line through A meets the two circles again at points P and Q. Let M be the midpoint of P and Q. If the two given circles have equations \(x^2 + y^2 = 25\) and \((x-3)^2 + (y-4)^2 = 25\), and A is the point (0,0), find the equation of the circle tha... | [
"\\left(x - \\frac{3}{2}\\right)^2 + \\left(y - 2\\right)^2 = \\left(\\frac{5}{2}\\right)^2"
] | [
"A is one of the points of intersection of two given circles. A variable line through A meets the two circles again at points P and Q. Let M be the midpoint of P and Q. If the two given circles have equations \\(x^2 + y^2 = 25\\) and \\((x-3)^2 + (y-4)^2 = 25\\), and A is the point (0,0), find the equation of the c... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-13d43456ff2ce80c802a4b0decbda0ef6bbc6e7bfaa66c0babc7388a04c61c1d | A is running 3 laps in 10 minutes, and B is running 5 laps in 12 minutes. After how many laps, done by A and B together, will they meet at the start line at the same time? | [
"43"
] | [
"A is running 3 laps in 10 minutes, and B is running 5 laps in 12 minutes. After how many laps, done by A and B together, will they meet at the start line at the same time?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-e57d5b6d9e1adac4deffa0173a99d193498222ed6d51e8bc2d4cf402bf7a34fd | A jar contains $29\frac{5}{7}$ tablespoons of peanut butter. If one serving of peanut butter is 2 tablespoons, how many servings of peanut butter does the jar contain? Express your answer as a mixed number. | [
"14\\frac{6}{7}"
] | [
"A jar contains $29\\frac{5}{7}$ tablespoons of peanut butter. If one serving of peanut butter is 2 tablespoons, how many servings of peanut butter does the jar contain? Express your answer as a mixed number."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a97b512fe3ce60f20010a6434b663b17a7b970f2b6e823a116bd9de2ed83326b | A jar contains 7 blue balls, 9 red balls, and 10 white balls. Balls are drawn at random one by one from the jar until either four balls of the same color or at least two of each color have been drawn. What is the largest number of balls that one may have to draw? | [
"8"
] | [
"A jar contains 7 blue balls, 9 red balls, and 10 white balls. Balls are drawn at random one by one from the jar until either four balls of the same color or at least two of each color have been drawn. What is the largest number of balls that one may have to draw?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-11a0f86b208d31e6e1fc1b0a9f555e2543c1c7a01a84d0153870fa47548d30d2 | A jar contains an acidic solution that is 30% acid and 70% water. An outlet pipe can empty the jar in 1 hour, and an inlet pipe can fill the jar with a 50% acid solution in 2 hours. Both pipes are opened simultaneously. What is the concentration of the acid in the solution after half an hour? | [
"36.7%"
] | [
"A jar contains an acidic solution that is 30% acid and 70% water. An outlet pipe can empty the jar in 1 hour, and an inlet pipe can fill the jar with a 50% acid solution in 2 hours. Both pipes are opened simultaneously. What is the concentration of the acid in the solution after half an hour?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c57842ac91b6157fedc2f0f2e1638973e96142c1badc4d5889929e3d91c12a73 | A jar contains blue, red, and yellow marbles. Yellow marbles make up 20% of the marbles in the jar. After adding 5 blue marbles, blue marbles make up 64% of the marbles. After adding another 5 blue marbles, blue marbles now make up 70% of the marbles. How many red marbles are in the jar? | [
"5"
] | [
"A jar contains blue, red, and yellow marbles. Yellow marbles make up 20% of the marbles in the jar. After adding 5 blue marbles, blue marbles make up 64% of the marbles. After adding another 5 blue marbles, blue marbles now make up 70% of the marbles. How many red marbles are in the jar?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-36a11015a14c028e3802afe37937d8e911e71e543263432d26f81774e3398427 | A jar contains one two-headed coin and three normal coins. A coin is selected at random and tossed three times, resulting in heads each time. What is the probability that the selected coin was the two-headed coin? | [
"8/11"
] | [
"A jar contains one two-headed coin and three normal coins. A coin is selected at random and tossed three times, resulting in heads each time. What is the probability that the selected coin was the two-headed coin?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-896d63166db7705fc3f4e33e6c0aacc6bc64126cfefa17897e8ba6ceafad6055 | A jar contains quarters (worth $\$0.25$ each), nickels (worth $\$0.05$ each) and pennies (worth $\$0.01$ each). The value of the quarters is $\$10.00.$ The value of the nickels is $\$10.00.$ The value of the pennies is $\$10.00.$ If Judith randomly chooses one coin from the jar, what is the probability that it is a qua... | [
"\\dfrac{1}{31}"
] | [
"A jar contains quarters (worth $\\$0.25$ each), nickels (worth $\\$0.05$ each) and pennies (worth $\\$0.01$ each). The value of the quarters is $\\$10.00.$ The value of the nickels is $\\$10.00.$ The value of the pennies is $\\$10.00.$ If Judith randomly chooses one coin from the jar, what is the probability that ... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6c2fe714d2052ced9b13af94330b0aa76134eea8c05322c4fa2dcdeea7dd676a | A jar contains red and blue marbles. If two marbles are picked at random, the probability that both are red is 50%. The total number of marbles is a 2-digit number. Determine the number of red and blue marbles in the jar. | [
"15 red marbles and 6 blue marbles."
] | [
"A jar contains red and blue marbles. If two marbles are picked at random, the probability that both are red is 50%. The total number of marbles is a 2-digit number. Determine the number of red and blue marbles in the jar."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 31
} | 33 | 0 | 0 |
omr-ws-61752c6e9b218c6da615c688f9eddd00d0a6e2a5e01dc98de8f40c0b38630b65 | A jar contains red, yellow, green, and blue marbles. If $1/5$ of the non-yellow marbles are red, $1/3$ of the non-red marbles are blue, and $1/2$ of the non-blue marbles are green, what fraction of the non-green marbles are yellow? | [
"\\(\\frac{5}{16}\\)"
] | [
"A jar contains red, yellow, green, and blue marbles. If $1/5$ of the non-yellow marbles are red, $1/3$ of the non-red marbles are blue, and $1/2$ of the non-blue marbles are green, what fraction of the non-green marbles are yellow?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-4d6ec7c1a499a3f697dd322100185a27878b0258a7dc04ab312e84088779a357 | A jar contains two red marbles, three green marbles, ten white marbles and no other marbles. Two marbles are randomly drawn from this jar without replacement. What is the probability that these two marbles drawn will both be red? Express your answer as a common fraction. | [
"\\frac{1}{105}"
] | [
"A jar contains two red marbles, three green marbles, ten white marbles and no other marbles. Two marbles are randomly drawn from this jar without replacement. What is the probability that these two marbles drawn will both be red? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c03dff93bf208d277a5e57309e5c3740043cde740b059b264d3ed844807ad0dc | A jar has $10$ red candies and $10$ blue candies. Terry picks two candies at random, then Mary picks two of the remaining candies at random. Given that the probability that they get the same color combination, irrespective of order, is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$ | [
"441"
] | [
"A jar has $10$ red candies and $10$ blue candies. Terry picks two candies at random, then Mary picks two of the remaining candies at random. Given that the probability that they get the same color combination, irrespective of order, is $m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m+n.$... | [
"MATH_training_set"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 20
} | 52 | 0 | 0 |
omr-ws-5187f9625004255a8e2a2cdc5689795305981030bed088cbf25ce84dfdf1145a | A jar has 10 red candies and 10 blue candies. Terry picks two candies at random, then Mary picks two of the remaining candies at random. Given that the probability that they get the same color combination, irrespective of order, is $m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$. | [
"441"
] | [
"A jar has 10 red candies and 10 blue candies. Terry picks two candies at random, then Mary picks two of the remaining candies at random. Given that the probability that they get the same color combination, irrespective of order, is $m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-6ce9ba724ef408ddeeda26165a202a7ce0b8a5e2ddd4af73b5d82ede1aa05fd4 | A jar has 10 red candies and 10 blue candies. Terry picks two candies at random, then Mary picks two of the remaining candies at random. Given that the probability that they get the same color combination, irrespective of order, is \( \frac{m}{n} \), where \( m \) and \( n \) are relatively prime positive integers, fin... | [
"441"
] | [
"A jar has 10 red candies and 10 blue candies. Terry picks two candies at random, then Mary picks two of the remaining candies at random. Given that the probability that they get the same color combination, irrespective of order, is \\( \\frac{m}{n} \\), where \\( m \\) and \\( n \\) are relatively prime positive i... | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 30,
"tir": 49
} | 79 | 0 | 0 |
omr-ws-82b5784b3a2287159d6a58c2cf3d9fe985e99b4fc1a1ba0ad37c1a48953234a7 | A jar has 10 red candies and 10 blue candies. Terry picks two candies at random, then Mary picks two of the remaining candies at random. Given that the probability that they get the same color combination, irrespective of order, is \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers, find \(m ... | [
"441"
] | [
"A jar has 10 red candies and 10 blue candies. Terry picks two candies at random, then Mary picks two of the remaining candies at random. Given that the probability that they get the same color combination, irrespective of order, is \\(\\frac{m}{n}\\), where \\(m\\) and \\(n\\) are relatively prime positive integer... | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-85bf810018585dd43c200a2957e099027935dd5a6f87c776e385c0a2e48e2683 | A jar has 4 red marbles and 1 blue marble. You have 3 fair coins, and you choose (without replacement) from the jar a number of marbles equal to the number of heads appearing when you flip the 3 coins. Find the probability that you picked the blue marble. | [
"\\(\\frac{11}{32}\\)"
] | [
"A jar has 4 red marbles and 1 blue marble. You have 3 fair coins, and you choose (without replacement) from the jar a number of marbles equal to the number of heads appearing when you flip the 3 coins. Find the probability that you picked the blue marble."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6080eba4b963964cf1351ede8a03294fbfd7fef6f3f1207e6e5de45db3cd2249 | A jar has 4 red marbles and 1 blue marble. You have 3 fair coins, and you choose (without replacement) from the jar a number of marbles equal to the number of heads appearing when you flip the 3 coins. If you already have the blue marble with you, find the probability that 3 heads must have appeared on the flip of the ... | [
"\\dfrac{1}{4}"
] | [
"A jar has 4 red marbles and 1 blue marble. You have 3 fair coins, and you choose (without replacement) from the jar a number of marbles equal to the number of heads appearing when you flip the 3 coins. If you already have the blue marble with you, find the probability that 3 heads must have appeared on the flip of... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ea51e38fa0a7677471e7d9ffb38e943ce9508e9228f65a6ac62569247c168bec | A jar has blue, red, and yellow marbles. Yellow marbles make up 20% of the marbles in the jar. Once 5 blue marbles are added, blue marbles make up 64% of the marbles. Another 5 blue marbles are added, and now blue marbles make up 70% of the marbles in the jar. How many red marbles are in the jar? | [
"5"
] | [
"A jar has blue, red, and yellow marbles. Yellow marbles make up 20% of the marbles in the jar. Once 5 blue marbles are added, blue marbles make up 64% of the marbles. Another 5 blue marbles are added, and now blue marbles make up 70% of the marbles in the jar. How many red marbles are in the jar?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2509b8e26fe508af430ab5ed85b11fd38abd0db95cd4bb39a055ac23367ca058 | A jar has blue, red, and yellow marbles. Yellow marbles make up 20% of the marbles in this jar. Once 5 blue marbles are added to the jar, blue marbles make up 64% of the marbles. Another 5 blue marbles are added, and now blue marbles make up 70% of the marbles in the jar. How many red marbles are in the jar? | [
"5"
] | [
"A jar has blue, red, and yellow marbles. Yellow marbles make up 20% of the marbles in this jar. Once 5 blue marbles are added to the jar, blue marbles make up 64% of the marbles. Another 5 blue marbles are added, and now blue marbles make up 70% of the marbles in the jar. How many red marbles are in the jar?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ee9cf8e7333d7c5689897357a22f08a19df4138a86a5e1f928f6b299ef092ddc | A jar is filled with red, orange and yellow jelly beans. The probability of randomly selecting a red jelly bean from this jar is $0.2$, and the probability of randomly selecting an orange jelly bean from this jar is $0.5$. What is the probability of randomly selecting a yellow jelly bean from this jar? | [
"0.3"
] | [
"A jar is filled with red, orange and yellow jelly beans. The probability of randomly selecting a red jelly bean from this jar is $0.2$, and the probability of randomly selecting an orange jelly bean from this jar is $0.5$. What is the probability of randomly selecting a yellow jelly bean from this jar?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cd2e2b1f6fc9d9859756762b8a8d7e34aa6cb4116af3c7decb9fd4462dde86ba | A jar of peanut butter which is 3 inches in diameter and 4 inches high sells for $\$$0.60. At the same rate, what would be the price for a jar that is 6 inches in diameter and 6 inches high? | [
"\\$3.60"
] | [
"A jar of peanut butter which is 3 inches in diameter and 4 inches high sells for $\\$$0.60. At the same rate, what would be the price for a jar that is 6 inches in diameter and 6 inches high?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e1e4ed45f8bfd72c1235f0eb485e0e9eefebb8ae23cad6a04f3e1a0fe00a82e0 | A jeweller makes a chain consisting of \( N > 3 \) numbered links. A customer asks the jeweller to rearrange the links in such a way that the number of links the jeweller must open is maximized. What is the maximum number of links the jeweller must open? | [
"\\( N \\)"
] | [
"A jeweller makes a chain consisting of \\( N > 3 \\) numbered links. A customer asks the jeweller to rearrange the links in such a way that the number of links the jeweller must open is maximized. What is the maximum number of links the jeweller must open?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5ef68926471995b9e44ab56925f24074edcfc8da9dfe02ce73d30cc6149bbf88 | A jigsaw puzzle consists of a red, a blue, and a yellow piece that together form a rectangle ABCD. The area of the rectangle is 24, and the area of the red piece ABE is 8. The point M is the midpoint of BE. Determine the areas of the blue and yellow pieces. | [
"10"
] | [
"A jigsaw puzzle consists of a red, a blue, and a yellow piece that together form a rectangle ABCD. The area of the rectangle is 24, and the area of the red piece ABE is 8. The point M is the midpoint of BE. Determine the areas of the blue and yellow pieces."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-7dfe5a9e2b54944ac8bfcdf83cfcfe2548f8f625cae83acb4049c7ccf4e71989 | A jobber buys an article at \$24 less \(12\frac{1}{2}\%\). He then wishes to sell the article at a gain of \(33\frac{1}{3}\%\) of his cost after allowing a \(20\%\) discount on his marked price. At what price, in dollars, should the article be marked? | [
"35"
] | [
"A jobber buys an article at \\$24 less \\(12\\frac{1}{2}\\%\\). He then wishes to sell the article at a gain of \\(33\\frac{1}{3}\\%\\) of his cost after allowing a \\(20\\%\\) discount on his marked price. At what price, in dollars, should the article be marked?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2014fc6fed5a62b85b50ec04eaae089a2d4ca1afa2d369df882202a016118fc0 | A journalist is looking for a person named Z at a meeting of N persons, where Z knows everyone but no one knows Z. The journalist can ask any person if they know another. a) Can the journalist find Z by asking less than N questions? b) What is the minimal number of questions needed to find Z? | [
"N-1"
] | [
"A journalist is looking for a person named Z at a meeting of N persons, where Z knows everyone but no one knows Z. The journalist can ask any person if they know another. \na) Can the journalist find Z by asking less than N questions?\nb) What is the minimal number of questions needed to find Z?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-56781308edcf1bf95b7594c6a00dd92cff626ecb0d32ac55d4944e0fa6ead26e | A judo class has 12 students, and they can only train 4 pairs at a time. The class starts with 10 minutes of exercises with all students together and then students are paired for 2-minute workouts. What is the minimum duration that the class must have so that each student has trained exactly once with each of the colle... | [
"44"
] | [
"A judo class has 12 students, and they can only train 4 pairs at a time. The class starts with 10 minutes of exercises with all students together and then students are paired for 2-minute workouts. What is the minimum duration that the class must have so that each student has trained exactly once with each of the ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-84ef4514f41cae9bb06c53e54c650114d9f150ea99838b3da5b05d1e43e74db7 | A judoka team was weighed before a competition. The three heaviest members weigh 41% of the total weight of the team, and the two lightest members weigh 17% of the total weight. How many judoka are there in the team? | [
"9"
] | [
"A judoka team was weighed before a competition. The three heaviest members weigh 41% of the total weight of the team, and the two lightest members weigh 17% of the total weight. How many judoka are there in the team?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 11
} | 26 | 0 | 0 |
omr-ws-c53e7c0ae14b437baec6bbd3380ae6033723220944dc8f6674a32aab8fac59fe | A juice company makes two kinds of juice: A and B. A gallon of juice A is made by mixing 3 quarts of pumpkin juice and 2 quarts of carrot juice, while one gallon of juice B is made by mixing 5 quarts of pumpkin juice and 3 quarts of carrot juice. A profit of $30 is made on every gallon of juice A sold and $20 is made o... | [
"\\( x = 40 \\) gallons of juice A and \\( y = 0 \\) gallons of juice B."
] | [
"A juice company makes two kinds of juice: A and B. A gallon of juice A is made by mixing 3 quarts of pumpkin juice and 2 quarts of carrot juice, while one gallon of juice B is made by mixing 5 quarts of pumpkin juice and 3 quarts of carrot juice. A profit of $30 is made on every gallon of juice A sold and $20 is m... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8403bb41fb645b4cf6f4bc09e0b1b98eb9d544f730371b82cc771acc78d950f1 | A jury of 9 persons should decide whether a verdict is guilty or not. Each juror votes independently with the probability $1/2$ for each of the two possibilities. Find the probability that a fixed juror will be a part of the majority. | [
"\\(\\frac{1}{2}\\)"
] | [
"A jury of 9 persons should decide whether a verdict is guilty or not. Each juror votes independently with the probability $1/2$ for each of the two possibilities. Find the probability that a fixed juror will be a part of the majority."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6f690d3deea59c52c6241e7997d05ac3efbd41b339bdb1ae372fde1ee7964ac6 | A kangaroo is sitting at the point 1 on the real line and wants to visit the points \(2, 3, \dots, n\) exactly once, and then return to point 1. What is the maximum possible sum of the lengths of his jumps? | [
"\\(\\frac{n(n-1)}{2} + \\left\\lfloor \\frac{n}{2} \\right\\rfloor\\)"
] | [
"A kangaroo is sitting at the point 1 on the real line and wants to visit the points \\(2, 3, \\dots, n\\) exactly once, and then return to point 1. What is the maximum possible sum of the lengths of his jumps?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-eb21ee2e37b1c1bedcd1a2ec1ab52e6e22da27ffb036b7075bb9caacfc66d802 | A kangaroo jumps from lattice point to lattice point in the coordinate plane. It can make only two kinds of jumps: $(A)$ $1$ to the right and $3$ up, and $(B)$ $2$ to the left and $4$ down. Starting from $(0,0)$, find all points $(m,n)$ with $m,n \ge 0$ which the kangaroo can reach. | [
"\\((m, n) \\text{ such that } n \\geq 3m \\text{ and } m \\text{ and } n \\text{ have the same parity}\\)"
] | [
"A kangaroo jumps from lattice point to lattice point in the coordinate plane. It can make only two kinds of jumps: $(A)$ $1$ to the right and $3$ up, and $(B)$ $2$ to the left and $4$ down. Starting from $(0,0)$, find all points $(m,n)$ with $m,n \\ge 0$ which the kangaroo can reach."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 18
} | 39 | 0 | 0 |
omr-ws-375a9c28b16761bbdbc4ca67c7b64fee084c5d905b9fac60b7bb0f1bf36a911e | A kangaroo starts at position 0 on the number axis. On the $n^{\text{th}}$ move, it can jump $2^n + 1$ feet left or right. Let $S$ be the set of all integer positions the kangaroo can reach after any number of moves. Calculate the smallest positive integer $k$ such that $k \notin S$. | [
"4"
] | [
"A kangaroo starts at position 0 on the number axis. On the $n^{\\text{th}}$ move, it can jump $2^n + 1$ feet left or right. Let $S$ be the set of all integer positions the kangaroo can reach after any number of moves. Calculate the smallest positive integer $k$ such that $k \\notin S$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-d055cdcb4d9892aa6942d8c7840b33f1a196a406a2ccbaca597b8b0b416447b5 | A kayak race starts at point P on the shore of a circular lake, with the goal of reaching the diametrically opposite point Q on the lakeshore. The racer can carry the kayak part way around the lake to a point M and then paddle to the finishing point Q. The racer can run with the kayak twice as fast as she can paddle it... | [
"Jog the whole way."
] | [
"A kayak race starts at point P on the shore of a circular lake, with the goal of reaching the diametrically opposite point Q on the lakeshore. The racer can carry the kayak part way around the lake to a point M and then paddle to the finishing point Q. The racer can run with the kayak twice as fast as she can padd... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 3,
"tir": 10
} | 15 | 0 | 0 |
omr-ws-c5915301e5574ad2b3e338945b5e9c210a8fcdc22b8de419639eeea9108ae806 | A kid divided a positive integer \( p \) by another positive integer \( q \) and found the quotient to be \( 0.1982\ldots \). If the quotient were correct, calculate the smallest possible value of \( q \) such that \( p/q = 0.1982\ldots \) could be a valid repeating decimal. | [
"1125"
] | [
"A kid divided a positive integer \\( p \\) by another positive integer \\( q \\) and found the quotient to be \\( 0.1982\\ldots \\). If the quotient were correct, calculate the smallest possible value of \\( q \\) such that \\( p/q = 0.1982\\ldots \\) could be a valid repeating decimal."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-cb0fe550a6fd3dafd10e8ebc80533cb4950d2c1d73a9b5073dedbdb454fcb956 | A king has 100 prisoners standing in a line, each wearing a randomly assigned black or white cap. The 100th prisoner can see the caps of the 99 prisoners in front of him, the 99th prisoner can see the caps of the 98 prisoners in front of him, and so on, with the 1st prisoner seeing no caps. The king will ask each priso... | [
"99 prisoners can be saved. The plan is for the 100th prisoner to count the number of white hats in front of him and announce \"white\" if the number is odd and \"black\" if the number is even. Each subsequent prisoner can then determine the color of their own hat based on the parity of white hats announced by the ... | [
"A king has 100 prisoners standing in a line, each wearing a randomly assigned black or white cap. The 100th prisoner can see the caps of the 99 prisoners in front of him, the 99th prisoner can see the caps of the 98 prisoners in front of him, and so on, with the 1st prisoner seeing no caps. The king will ask each ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 1
} | 49 | 0 | 0 |
omr-ws-2a2fb05f0d9cbe09950083d86845d63b8061525cc78312e1131c24b8fa328948 | A king has 600 barrels of wine, one of which is poisoned. He has 4 enemy spies who can drink the wine to test for poison, but drinking poisoned wine will kill them within 24 hours. The feast starts in exactly 4 days. How can the king find the poisoned barrel using the spies? | [
"Assign each barrel a unique 12-bit code where each bit represents a spy drinking on a specific day. The death pattern of spies reveals the poisoned barrel's code."
] | [
"A king has 600 barrels of wine, one of which is poisoned. He has 4 enemy spies who can drink the wine to test for poison, but drinking poisoned wine will kill them within 24 hours. The feast starts in exactly 4 days. How can the king find the poisoned barrel using the spies?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 8
} | 9 | 0 | 0 |
omr-ws-16049b0b4875db45f45d614c9c78b17eaa3ed438be3f690a31c4c70b6a11cba8 | A king invites \(n\) couples to have dinner with him and seats them around a round table with chairs labeled \(1, 2, \ldots, 2n+1\), with the king occupying chair \(1\). \(n\) parrots pick \(n\) arbitrary cards from \(n\) sets of cards labeled \(1, 2, \ldots, n\). If the \(j\)th parrot picks the card \(k\), then the sp... | [
"45"
] | [
"A king invites \\(n\\) couples to have dinner with him and seats them around a round table with chairs labeled \\(1, 2, \\ldots, 2n+1\\), with the king occupying chair \\(1\\). \\(n\\) parrots pick \\(n\\) arbitrary cards from \\(n\\) sets of cards labeled \\(1, 2, \\ldots, n\\). If the \\(j\\)th parrot picks the ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d30ba38ace0ce7a137e514aa4614bac82e4dce2c329f362deafd97f97cf368ec | A king is located at h8 on an 8x8 chessboard. Each move, it moves to one of the 3, 5, or 8 adjacent squares (including corners) with equal probability. Once it reaches a1, it stops. What is the expected number of moves that the king will make before it stops? | [
"70"
] | [
"A king is located at h8 on an 8x8 chessboard. Each move, it moves to one of the 3, 5, or 8 adjacent squares (including corners) with equal probability. Once it reaches a1, it stops. What is the expected number of moves that the king will make before it stops?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-7876c39392148e0f0927b363c92ec361f66043f56f937de33c917cb400e68582 | A king is located in the lower left corner of a $6 \times 6$ chessboard. In one move, the king can move one cell to the right, one cell up, or one cell diagonally to the right and up. How many different paths can the king take to reach the upper right corner of the board? | [
"8989"
] | [
"A king is located in the lower left corner of a $6 \\times 6$ chessboard. In one move, the king can move one cell to the right, one cell up, or one cell diagonally to the right and up. How many different paths can the king take to reach the upper right corner of the board?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 1
} | 8 | 0 | 0 |
omr-ws-99574958f206f38bd97a288faa385ad4d72174e1ed221c050d712c7782c8c0d8 | A king is placed in the middle of a 15 x 15 chessboard. The king can move one step in any direction (forward, backward, left, right, or diagonally). How many different ways can the king return to the middle point after taking exactly 6 steps? | [
"8840"
] | [
"A king is placed in the middle of a 15 x 15 chessboard. The king can move one step in any direction (forward, backward, left, right, or diagonally). How many different ways can the king return to the middle point after taking exactly 6 steps?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 8,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-e797527618a8534392010987efce061c176086a09745731e1a8eb5a514f95c03 | A king is placed in the middle of a 16x16 chessboard. The king can move one step forward, backward, left, or right in each move. Starting from the middle, the king moves in any direction and then returns to the middle point in six steps. How many different ways can the king return to the starting point? | [
"400"
] | [
"A king is placed in the middle of a 16x16 chessboard. The king can move one step forward, backward, left, or right in each move. Starting from the middle, the king moves in any direction and then returns to the middle point in six steps. How many different ways can the king return to the starting point?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-016cc703feaeef566f8d9e33b651f70052d4764177f86313895133b26877793f | A king moves around an 8x8 chessboard, visiting each square exactly once and returning to the starting square, with the path connecting the centers of the squares and not crossing itself. Let $h$ be the number of horizontal moves and $v$ be the number of vertical moves. Find the minimum possible value of $h + v$. | [
"16"
] | [
"A king moves around an 8x8 chessboard, visiting each square exactly once and returning to the starting square, with the path connecting the centers of the squares and not crossing itself. Let $h$ be the number of horizontal moves and $v$ be the number of vertical moves. Find the minimum possible value of $h + v$."... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-142406fb1de7c1996354ac93724e5ab9473ed4dc55804a93512f1339979448b2 | A king on a 2021×2021 chessboard starts moving on Monday. Every Sunday, the king moves to a diagonally neighboring square, and on other days, the king moves to a horizontally or vertically neighboring square. The king cannot revisit any square. What is the maximum number of moves the king can make? | [
"\\( 2021^2 - 1 \\)"
] | [
"A king on a 2021×2021 chessboard starts moving on Monday. Every Sunday, the king moves to a diagonally neighboring square, and on other days, the king moves to a horizontally or vertically neighboring square. The king cannot revisit any square. What is the maximum number of moves the king can make?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 19
} | 23 | 0 | 0 |
omr-ws-6b66c09c54db81d0ca413232210c0934f1ccbc2441a968e05da384c83b7ccca1 | A king starts in the top left corner of an $8 \times 8$ chessboard. Find the expected number of moves it makes before it revisits its original square. | [
"140"
] | [
"A king starts in the top left corner of an $8 \\times 8$ chessboard. Find the expected number of moves it makes before it revisits its original square."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-1158b60b78eed5ac99873e8924bcd2438731f524ba40a54474f69bbe80689a1b | A king tours every square of an 8 x 8 chessboard, visiting each square exactly once and returning to his starting position. Let $d$ be the number of diagonal moves the king makes during this tour. Find the remainder when $d$ is divided by 2. | [
"0"
] | [
"A king tours every square of an 8 x 8 chessboard, visiting each square exactly once and returning to his starting position. Let $d$ be the number of diagonal moves the king makes during this tour. Find the remainder when $d$ is divided by 2."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dbc67e3deb8c8447a32c418e2aab5a83c72ad954668fefbf19bc5f9c58089dbf | A king went around an 8x8 chessboard, stopping in each square exactly once and ending up in the beginning square. His path was drawn by connecting the centers of the squares he was in, and it did not cross itself. a) Provide an example in which the king made exactly 28 horizontal and vertical moves. | [
"28"
] | [
"A king went around an 8x8 chessboard, stopping in each square exactly once and ending up in the beginning square. His path was drawn by connecting the centers of the squares he was in, and it did not cross itself.\na) Provide an example in which the king made exactly 28 horizontal and vertical moves."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 4
} | 13 | 0 | 0 |
omr-ws-5326b40c970ed38aabe26566a89fc45d7a75a0833b9fed050ed6cc4cd76dd969 | A king went around an 8x8 chessboard, stopping in each square exactly once and ending up in the beginning square. His path was drawn by connecting the centers of the squares he was in, and it did not cross itself. c) Determine the maximum and minimum distances the king could have traveled, if each square has a side of ... | [
"Minimum distance = 64 inches, Maximum distance = \\(28 + 36\\sqrt{2}\\) inches."
] | [
"A king went around an 8x8 chessboard, stopping in each square exactly once and ending up in the beginning square. His path was drawn by connecting the centers of the squares he was in, and it did not cross itself.\nc) Determine the maximum and minimum distances the king could have traveled, if each square has a si... | [
"aops_c5_contests_amp_programs"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-008d393004a4c5f9e4d4974c81d2321a1c3fe9052ccdcafd9385440a8e2f8a2f | A king, his daughter, and his son were locked at the top of a tower. The king weighed 91 kg, the daughter 42 kg, and the son 49 kg. They had a pulley with a rope that reached exactly from the top of the tower to the ground with a basket at each end, and also had a huge rock weighing 35 kg. How did they manage to get of... | [
"They used the daughter and rock to balance the weights step-by-step, ensuring the 7 kg difference was never exceeded."
] | [
"A king, his daughter, and his son were locked at the top of a tower. The king weighed 91 kg, the daughter 42 kg, and the son 49 kg. They had a pulley with a rope that reached exactly from the top of the tower to the ground with a basket at each end, and also had a huge rock weighing 35 kg. How did they manage to g... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-23741910b9575ff7b0bf0edd2a661cf8c8370f7d1a4378935867ff836c47067e | A kite 100 ft above the ground moves horizontally with a rate of 8 ft/s. Find the rate of change of the angle between the string and the horizontal when 200 ft of string has been let out. | [
"\\(-\\frac{1}{50}\\)"
] | [
"A kite 100 ft above the ground moves horizontally with a rate of 8 ft/s. Find the rate of change of the angle between the string and the horizontal when 200 ft of string has been let out."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 21
} | 36 | 0 | 0 |
omr-ws-8509d6f3e054cdeee1a0e7ed1292b7036f2ed2d546fc261bad7a47cd54be5a07 | A kite 75 ft above the ground moves horizontally at a speed of 8 ft/s. At what rate is the angle between the string and the horizontal decreasing when 300 ft of string has been let out? | [
"\\(\\frac{-8}{75} \\text{ rad/sec}\\)"
] | [
"A kite 75 ft above the ground moves horizontally at a speed of 8 ft/s. At what rate is the angle between the string and the horizontal decreasing when 300 ft of string has been let out?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-0d91cff40e090a8240b39d3d49a9240ac8d3b7bce5cabaf6769803b7deb870c7 | A kite \( KITE \) has three vertices on a circle, and its diagonals intersect at the center of the circle. The other vertex \( E \) lies outside the circle. Segment \( EI \) intersects the circle at point \( X \). The area of the kite is 24, and \( EX = 8 \). When the product \( (KI)(KE) \) is written in simplest radic... | [
"21"
] | [
"A kite \\( KITE \\) has three vertices on a circle, and its diagonals intersect at the center of the circle. The other vertex \\( E \\) lies outside the circle. Segment \\( EI \\) intersects the circle at point \\( X \\). The area of the kite is 24, and \\( EX = 8 \\). When the product \\( (KI)(KE) \\) is written ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 6,
"tir": 14
} | 35 | 0 | 0 |
omr-ws-96fec487df88b291eba41524ed09f92329322bbe35672c7a769bc100e1d482ab | A kite with an area of 22 is inscribed in a circle with a diameter of 10. Find the sum of the reciprocals of the side lengths of the kite. | [
"\\dfrac{12}{11}"
] | [
"A kite with an area of 22 is inscribed in a circle with a diameter of 10. Find the sum of the reciprocals of the side lengths of the kite."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-d48ddbcd37e64c0f98a2de1ea9398eddf5fd446258a8c5b9773bb6b0ec9709bc | A knight is on an infinite chessboard. After exactly 100 legal moves, how many different possible squares can it end on? A knight can move to any of the 8 closest squares not on the same row, column, or diagonal. | [
"70401."
] | [
"A knight is on an infinite chessboard. After exactly 100 legal moves, how many different possible squares can it end on? A knight can move to any of the 8 closest squares not on the same row, column, or diagonal."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 3
} | 5 | 0 | 0 |
omr-ws-74b5294cb0579b15513cc3d54ff5d07a392eee14f3a52bb356fb9033274f5f59 | A knight is placed in the bottom-left corner of a $2009 \times 2009$ chessboard. The row and column containing this corner are painted. The knight cannot move into a painted cell, and after each move, the row and column containing the knight's new position become painted. What is the maximum number of unpainted rows an... | [
"0"
] | [
"A knight is placed in the bottom-left corner of a $2009 \\times 2009$ chessboard. The row and column containing this corner are painted. The knight cannot move into a painted cell, and after each move, the row and column containing the knight's new position become painted. What is the maximum number of unpainted r... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-f0d3629a2f8fa9e0f46ffff38f5ffbfa3675024a4f4e45f1d595b3af3b1097e9 | A knight is placed on each square of the first column of a $2017 \times 2017$ board. A move consists of choosing two different knights and moving each of them to a square which is one knight-step away. Find all integers $k$ with $1 \leq k \leq 2017$ such that it is possible for each square in the $k$-th column to conta... | [
"$k$ is odd."
] | [
"A knight is placed on each square of the first column of a $2017 \\times 2017$ board. A move consists of choosing two different knights and moving each of them to a square which is one knight-step away. Find all integers $k$ with $1 \\leq k \\leq 2017$ such that it is possible for each square in the $k$-th column ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 14
} | 36 | 0 | 0 |
omr-ws-8d699c16fc586ad1513ecda848687c43f2d3d7b510cbd25fcc45058026897557 | A knight is standing at the top-right corner of a chessboard (field $h8$). Two players move it sequentially, but they must move it two fields downwards or leftwards (and then one field in any suitable direction). For instance, from $f6$ the knight can be moved only to $g4$, $e4$, $d5$, or $d7$. The player unable to mak... | [
"First player has a winning strategy."
] | [
"A knight is standing at the top-right corner of a chessboard (field $h8$). Two players move it sequentially, but they must move it two fields downwards or leftwards (and then one field in any suitable direction). For instance, from $f6$ the knight can be moved only to $g4$, $e4$, $d5$, or $d7$. The player unable t... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 4
} | 5 | 0 | 0 |
omr-ws-4b07fa9e6c5189c467f2419d4bfbf9cdc1627f7484a1ad36c6d00a55f62cc9ce | A knight on a $4 \times 50$ chessboard starts at the bottom-left corner. Calculate the maximum number of distinct cells the knight can visit without revisiting any cell. | [
"200"
] | [
"A knight on a $4 \\times 50$ chessboard starts at the bottom-left corner. Calculate the maximum number of distinct cells the knight can visit without revisiting any cell."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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