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-4be5f10fc30c02dfc40e4da5bb0194aec0ff559ad16ab50ad7e7d51f737226fa | A board with $(n + 1)(n - 1)$ squares is painted using 3 colors such that for any two different columns and any two different rows, the four squares at their intersections are not all the same color. Find the maximum value of $n$. | [
"4"
] | [
"A board with $(n + 1)(n - 1)$ squares is painted using 3 colors such that for any two different columns and any two different rows, the four squares at their intersections are not all the same color. Find the maximum value of $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-8d93b157506ab761563718ebb03c61ee9f1cc90dd62b100968282dcabe7a8254 | A board with $6$ columns and $n \geq 2$ rows is filled with $0$s and $1$s such that all rows are distinct. Additionally, for every two rows $(a_{1},a_{2},...,a_{6})$ and $(b_{1},b_{2},...,b_{6})$, there exists a row $(a_{1}b_{1},a_{2}b_{2},...,a_{6}b_{6})$. Let $f(n)$ be the minimum number of $0$s in any column for a g... | [
"0"
] | [
"A board with $6$ columns and $n \\geq 2$ rows is filled with $0$s and $1$s such that all rows are distinct. Additionally, for every two rows $(a_{1},a_{2},...,a_{6})$ and $(b_{1},b_{2},...,b_{6})$, there exists a row $(a_{1}b_{1},a_{2}b_{2},...,a_{6}b_{6})$. Let $f(n)$ be the minimum number of $0$s in any column f... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 8
} | 12 | 0 | 0 |
omr-ws-5d821a9af89794942c9f527bc8dde4f110b5f504a76c2cf51905582757c4873e | A boat and a raft started simultaneously moving downstream from a landing-stage $A$ on the bank of a river. The boat reached another landing-stage $B$, 324 km away from $A$, and after 18 hours left $B$ to return to $A$. When the boat was 180 km away from landing-stage $A$ a second boat having left $A$ 40 hours later th... | [
"The speeds of the steam-launches are equal to 15 km/h, the rate of flow of the river is equal to 3 km/h"
] | [
"A boat and a raft started simultaneously moving downstream from a landing-stage $A$ on the bank of a river. The boat reached another landing-stage $B$, 324 km away from $A$, and after 18 hours left $B$ to return to $A$. When the boat was 180 km away from landing-stage $A$ a second boat having left $A$ 40 hours lat... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3de1cc3e46ddec6b745b34d369859f54f3dc3c1d0e7148150b338764808aab76 | A boat follows the curve given by \( x = -2 + 3h \) and \( y = 9 - 4h \). Lengths are in meters and time in seconds. A) Draw the curve for \( h \) between 0 and 3. B) Determine the speed vector. | [
"(3, -4)"
] | [
"A boat follows the curve given by \\( x = -2 + 3h \\) and \\( y = 9 - 4h \\). Lengths are in meters and time in seconds.\nA) Draw the curve for \\( h \\) between 0 and 3.\nB) Determine the speed vector."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6015d8d2327ed1ea4a5fa3fe627adf616ed000fc079b47552afab16ba799f69a | A boat heads in the direction N 40 degrees W at a speed of 24 knots. The current runs at 6 knots in the direction N 25 degrees E. Find the resultant speed and direction of the boat. | [
"Speed = 27.087 knots, Direction = N28.42W"
] | [
"A boat heads in the direction N 40 degrees W at a speed of 24 knots. The current runs at 6 knots in the direction N 25 degrees E. Find the resultant speed and direction of the boat."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 10,
"tir": 26
} | 36 | 0 | 0 |
omr-ws-d597daa6006a1244e728e028d9165dbb3047504e44c190f814d81739db9bf44f | A boat heads in the direction N40 degrees W at a speed of 24 knots. The current runs at 6 knots in the direction N25 degrees E. Find the resultant speed and direction of the boat. | [
"27.1 \\text{ knots at } N28.4^\\circ W"
] | [
"A boat heads in the direction N40 degrees W at a speed of 24 knots. The current runs at 6 knots in the direction N25 degrees E. Find the resultant speed and direction of the boat."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-0fe512223669aa0ef1006b50cb2a76cded405f651b7b8c327e40aa1d7aa150b2 | A boat is crossing a river 260 m wide. In still water, the boat travels at 2.2 m/s. When it reaches the other shore, it is 110 m upstream from the starting point. The boat must sail at an angle of 45 degrees away from a straight line perpendicular to the shore it started from. What is the rate of the current? | [
"0.8974 m/s"
] | [
"A boat is crossing a river 260 m wide. In still water, the boat travels at 2.2 m/s. When it reaches the other shore, it is 110 m upstream from the starting point. The boat must sail at an angle of 45 degrees away from a straight line perpendicular to the shore it started from. What is the rate of the current?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 3,
"tir": 32
} | 49 | 0 | 0 |
omr-ws-b153963b31013b12886d7967ecdc70a1c96cd0144e8fe147ffdea0d9e419a893 | A boat is manned by 8 men, of whom 2 can only row on the bow side and 1 can only row on the stroke side. In how many ways can the crew be arranged? | [
"5760"
] | [
"A boat is manned by 8 men, of whom 2 can only row on the bow side and 1 can only row on the stroke side. In how many ways can the crew be arranged?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fae58e76b7e80c62544200099871969f5670950f644e567b9c68e3f10e5b6e05 | A boat is manned by 8 men, where 2 can only row on the bow side and 3 can only row on the stroke side. How many ways can the crew be arranged? | [
"1728"
] | [
"A boat is manned by 8 men, where 2 can only row on the bow side and 3 can only row on the stroke side. How many ways can the crew be arranged?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 1
} | 48 | 0 | 0 |
omr-ws-133af7c7676a37a203be8e7c239e791d452c02b9b4d3f6a9cfd9c161995d3056 | A boat is traveling upstream at 5 mph relative to the current flowing against it at 1 mph. If a tree branch 10 miles upstream from the boat falls into the current of the river, how many hours does it take for the branch to reach the boat? | [
"2"
] | [
"A boat is traveling upstream at 5 mph relative to the current flowing against it at 1 mph. If a tree branch 10 miles upstream from the boat falls into the current of the river, how many hours does it take for the branch to reach the boat?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-e9d167d82b1ba246af8634d61f806ab183b3bf5291b5bdf92c3d068269cdef06 | A boat is travelling at a speed of 30 knots. A yacht is sailing northwards at a speed of 10 knots. At 1300 hours, the boat is 14 nautical miles to the north-east of the yacht. At what time does the interception occur? | [
"13:23:11."
] | [
"A boat is travelling at a speed of 30 knots. A yacht is sailing northwards at a speed of 10 knots. At 1300 hours, the boat is 14 nautical miles to the north-east of the yacht. At what time does the interception occur?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 9
} | 30 | 0 | 0 |
omr-ws-61f6b7f0673805ff8ef168dc277918608b3e0c1321f573942d3bf00e716ef16d | A boat is travelling at a speed of 30 knots. A yacht is sailing northwards at a speed of 10 knots. At 1300 hours, the boat is 14 nautical miles to the north-east of the yacht. Determine the direction in which the boat should be travelling in order to intercept the yacht. | [
"\\( S56.63^\\circ W \\)"
] | [
"A boat is travelling at a speed of 30 knots. A yacht is sailing northwards at a speed of 10 knots. At 1300 hours, the boat is 14 nautical miles to the north-east of the yacht. Determine the direction in which the boat should be travelling in order to intercept the yacht."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 15
} | 16 | 0 | 0 |
omr-ws-5924daa541991b5497a7481ffb9887ea43587bd65f4a15f460c63c267d67dd28 | A boat made of wood 1.5 times heavier than water has an internal volume 2/3 of its external volume. When filled with oil to the brim, it barely floats in water. Find the density of the oil. | [
"\\(\\frac{3}{4}\\)"
] | [
"A boat made of wood 1.5 times heavier than water has an internal volume 2/3 of its external volume. When filled with oil to the brim, it barely floats in water. Find the density of the oil."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 5,
"tir": 12
} | 31 | 0 | 0 |
omr-ws-9b579b3348c5f0d59566ef0f4b2ef55b440e154a2077a199191cd94cf3862aa2 | A boat moves 10 km downstream and then 6 km upstream. The river current is 1 km/h. Within what limits must the relative speed of the boat lie for the entire trip to take from 3 to 4 hours? | [
"4\\le v\\le \\frac{8+\\sqrt {61}}{3}"
] | [
"A boat moves 10 km downstream and then 6 km upstream. The river current is 1 km/h. Within what limits must the relative speed of the boat lie for the entire trip to take from 3 to 4 hours?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-16f535206a039d9ff4ef448b8c132c99432926e679fb626a635111fd6c40dcf6 | A boat moves in such a way that its path is described by the function \( y = f(x) \). Given that the relationship between the derivative of \( y \) with respect to \( x \) and \( x \) is given by \( f'(x) = \frac{dy}{dx} \), calculate \( f'(x) \) if the path of the boat satisfies the condition \( x^2 + y^2 = L^2 \) for... | [
"-\\dfrac{x}{\\sqrt{L^2 - x^2}}"
] | [
"A boat moves in such a way that its path is described by the function \\( y = f(x) \\). Given that the relationship between the derivative of \\( y \\) with respect to \\( x \\) and \\( x \\) is given by \\( f'(x) = \\frac{dy}{dx} \\), calculate \\( f'(x) \\) if the path of the boat satisfies the condition \\( x^2... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-818c47be620dadba81c5698a0989407cb0a28b4653f75328ab84d2a62505fa46 | A boat sails 10 km down the river and then 6 km up the river. The speed of the river flow is 1 km/h. What must be the range of the inherent speed of the boat for the whole trip to take from 3 to 4 hours? | [
"$4\\le v\\le \\frac{8+\\sqrt {61}}{3}$"
] | [
"A boat sails 10 km down the river and then 6 km up the river. The speed of the river flow is 1 km/h. What must be the range of the inherent speed of the boat for the whole trip to take from 3 to 4 hours?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3d7f74feb3b9c1c379201cb9d40c1a7d8801d94c89b2cfa76669eaf8c84e0493 | A boat sails up the river, the speed of the river flow being $v$ km/h. Having sailed $l$ km, it "gets into a lake with dead water". What should the inherent speed of the boat be for the total distance of $S$ km to be covered in not more than $t$ h? | [
"With the speed higher than $\\frac{S+vt+\\sqrt {(S-vt)^2+4tvl}}{2t}$"
] | [
"A boat sails up the river, the speed of the river flow being $v$ km/h. Having sailed $l$ km, it \"gets into a lake with dead water\". What should the inherent speed of the boat be for the total distance of $S$ km to be covered in not more than $t$ h?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e471d072172601224049a0dd4c73666752b0465ea0ef3662bbc74c0c45db3e42 | A boat sinks in the ocean and the 5 survivors each jump in a different lifeboat. They meet on a desert island. The next morning, the first survivor finds a huge pile of oranges and separates them into 5 equal parts, throwing away the remaining one. Each subsequent survivor does the same. What is the minimum number of o... | [
"3121"
] | [
"A boat sinks in the ocean and the 5 survivors each jump in a different lifeboat. They meet on a desert island. The next morning, the first survivor finds a huge pile of oranges and separates them into 5 equal parts, throwing away the remaining one. Each subsequent survivor does the same. What is the minimum number... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 6,
"tir": 1
} | 21 | 0 | 0 |
omr-ws-a00ba1d2692ead50ff3d60557393f325c75fa77ede433f8b7753f77b83690387 | A boat springs a leak at time $t=0$ with water coming in at a constant rate. At time $t=\rho>0$ someone notices the leak and starts to record the distance the boat travels. The boat's speed is inversely related to the amount of water in the boat. If the boat travels twice as far in the first hour as the second, compute... | [
"\\dfrac{\\sqrt{5} - 1}{2}"
] | [
"A boat springs a leak at time $t=0$ with water coming in at a constant rate. At time $t=\\rho>0$ someone notices the leak and starts to record the distance the boat travels. The boat's speed is inversely related to the amount of water in the boat. If the boat travels twice as far in the first hour as the second, c... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-5c366a441dbcb5a481ce00242579de5f353a44e0bd5f2e06eb22b126afce7bb3 | A boat travels a distance $x$ to the west, then turns $30^\circ$ relative to the west and goes $42$ meters, and is now $62$ meters away from the starting point. What is $x$? | [
"\\( 46.25 \\)"
] | [
"A boat travels a distance $x$ to the west, then turns $30^\\circ$ relative to the west and goes $42$ meters, and is now $62$ meters away from the starting point. What is $x$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-38bd92be6bed649e04090415eabd0d71e458adce0aa520ee3c1a92e3be9a1286 | A boat travels against the current from point X for one mile, passing a log floating with the current. The boat continues against the current for another hour and then returns to point X with the current. The boat and the log reach point X at the same time. Determine the rate of the current. Let \( x \) be the rate of ... | [
"\\( \\frac{1}{2} \\) mph"
] | [
"A boat travels against the current from point X for one mile, passing a log floating with the current. The boat continues against the current for another hour and then returns to point X with the current. The boat and the log reach point X at the same time. Determine the rate of the current.\n\nLet \\( x \\) be th... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-5591be6718604a7e4f073794f196b1a37648e03af86c013aae28abb811941e7e | A boat travels between two banks of a river at a speed of 10 km/h relative to the water. The river itself flows at a rate of 10 km/h. What is the velocity of the boat relative to its starting point on shore? (Remember to name the velocity, not the speed!) | [
"\\( 10 \\sqrt{2} \\) km/h at an angle of \\( \\frac{\\pi}{4} \\) radians to the river."
] | [
"A boat travels between two banks of a river at a speed of 10 km/h relative to the water. The river itself flows at a rate of 10 km/h. What is the velocity of the boat relative to its starting point on shore? (Remember to name the velocity, not the speed!)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-6a3f93065b4a0758229a26018c0f40a3e65c0dbf0632a26f0d08ee9b3334fa10 | A bocce ball is accidentally dropped from a building 98 meters high. How long does it take for the bocce ball to hit the ground, given that the acceleration due to gravity is \( 9.8 \) meters per second per second? | [
"\\( 2\\sqrt{5} \\)"
] | [
"A bocce ball is accidentally dropped from a building 98 meters high. How long does it take for the bocce ball to hit the ground, given that the acceleration due to gravity is \\( 9.8 \\) meters per second per second?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 2,
"tir": 9
} | 16 | 0 | 0 |
omr-ws-f22f3821f2f72dfb1e6d171a3a1534a6ba7f6c8274c9918cbb9785a32c04af4d | A body falling under gravity experiences a resisting force of air proportional to the square of the velocity of the body. Write down the differential equation governing the motion satisfied by the distance $y(t)$ traveled by the body in time $t$. | [
"\\( M \\frac{d^2y}{dt^2} + k \\left( \\frac{dy}{dt} \\right)^2 = Mg \\)"
] | [
"A body falling under gravity experiences a resisting force of air proportional to the square of the velocity of the body. Write down the differential equation governing the motion satisfied by the distance $y(t)$ traveled by the body in time $t$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-6814055e213bee6103c48eff3b6d425a8084f35744467a7bf25fd317348aae3f | A body falls from a height of 20 meters. Find the rate of change of the body's position after one second. | [
"-9.8 \\text{ m/s}"
] | [
"A body falls from a height of 20 meters. Find the rate of change of the body's position after one second."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 29
} | 50 | 0 | 0 |
omr-ws-fbb28d9e1e5e8c2e771dc9e52eb1e2d14a79ba9eb726114af31887eb2b75f493 | A body moves rectilinearly with the speed $v=12t-t^2$ m/s. Find the length of the path traversed by the body from the beginning of motion to its stop. | [
"288"
] | [
"A body moves rectilinearly with the speed $v=12t-t^2$ m/s. Find the length of the path traversed by the body from the beginning of motion to its stop."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-59617cc47aa7e82c5c935837339e7c8161c45e28f6a06c2a975f9df3672ad170 | A body of weight $P$ is lying on an inclined plane. A thread runs from the body over a pulley at the top of the inclined plane to a weight $p$ $(p < P)$. At what angle $\alpha $ will the body be kept on the inclined plane by the least weight if the coefficient of friction is $k$ and $\alpha \in [\arctan k,\pi /2]$? | [
"$\\alpha =\\beta $"
] | [
"A body of weight $P$ is lying on an inclined plane. A thread runs from the body over a pulley at the top of the inclined plane to a weight $p$ $(p < P)$. At what angle $\\alpha $ will the body be kept on the inclined plane by the least weight if the coefficient of friction is $k$ and $\\alpha \\in [\\arctan k,\\pi... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2f509265e69ad8710db32bb7a2f57b33d8a49a7fdde225dd62a5c91f279276b1 | A bomb is dropped in city 1 of a network of cities numbered 1 to 4, connected as follows: 1 <--> 2 <--> 3 <--> 4. At each city, the bomb has a 50% chance of exploding. If it doesn't explode, it randomly chooses a neighboring city with equal probability and travels to that city. What is the probability that the bomb exp... | [
"\\(\\left(\\frac{26}{45}, \\frac{14}{45}, \\frac{4}{45}, \\frac{1}{45}\\right)\\)"
] | [
"A bomb is dropped in city 1 of a network of cities numbered 1 to 4, connected as follows: 1 <--> 2 <--> 3 <--> 4. At each city, the bomb has a 50% chance of exploding. If it doesn't explode, it randomly chooses a neighboring city with equal probability and travels to that city. What is the probability that the bom... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 14
} | 14 | 0 | 0 |
omr-ws-d663f2606905a871cff8d67042f309a95e369efda7e1e1ec0699d318a0df012f | A bomb is planted at the point (15,4) and will explode if anything comes within $3\sqrt{2}$ units of it. A tennis ball lands at a random point within the rectangle with corners at (1,1), (1,7), (18,1), and (18,7). What is the probability that the bomb does not explode as a result? | [
"\\dfrac{50 - 3\\pi}{68}"
] | [
"A bomb is planted at the point (15,4) and will explode if anything comes within $3\\sqrt{2}$ units of it. A tennis ball lands at a random point within the rectangle with corners at (1,1), (1,7), (18,1), and (18,7). What is the probability that the bomb does not explode as a result?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-c7f72888b4121c0f6ac1a3054465527d99e0c58debad512f7d8584a5823666b8 | A book contains 30 stories, each starting on a new page. The lengths of these stories are 1, 2, 3,..., 30 pages, respectively. The first story starts on the first page. What is the largest number of stories that can start on an odd-numbered page? | [
"23"
] | [
"A book contains 30 stories, each starting on a new page. The lengths of these stories are 1, 2, 3,..., 30 pages, respectively. The first story starts on the first page. What is the largest number of stories that can start on an odd-numbered page?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 3
} | 25 | 0 | 0 |
omr-ws-3c29a01413acec90cd201e665e4c276cbb0f5bdc23b2704ebb09434b3e4ef251 | A book contains 30 stories, each with a different number of pages under 31. The first story starts on page 1 and each story starts on a new page. What is the largest possible number of stories that can begin on odd page numbers? | [
"15"
] | [
"A book contains 30 stories, each with a different number of pages under 31. The first story starts on page 1 and each story starts on a new page. What is the largest possible number of stories that can begin on odd page numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fb6a69fcd6a8ba5b28a6b8020922e33008de8cadfdeab17f323c786a24c7106f | A book has 136 pages. Each page has the same number of words, and each page has no more than 100 words on it. The number of words in the book is congruent to 184, modulo 203. How many words are on each page? | [
"73"
] | [
"A book has 136 pages. Each page has the same number of words, and each page has no more than 100 words on it. The number of words in the book is congruent to 184, modulo 203. How many words are on each page?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-56cc1bd125d17210b9ffe01d89909e29f6fc256298b598fca0b46baea71b0bd1 | A book has pages numbered from $2$ to $400$. The reading order is as follows: take the last unread page, then read all pages which are not relatively prime to it and which have not been read before. Repeat until all pages are read. What is the last page to be read? | [
"$37$"
] | [
"A book has pages numbered from $2$ to $400$. The reading order is as follows: take the last unread page, then read all pages which are not relatively prime to it and which have not been read before. Repeat until all pages are read. What is the last page to be read?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 11
} | 11 | 0 | 0 |
omr-ws-3c9d98685415bee8701c132db3890ad75e366f8135d81dbce8556c76a897dbde | A book is published in three volumes with continuous page numbering. The second volume has 50 more pages than the first volume, and the third volume has one and a half times as many pages as the second volume. The sum of the first page numbers of the three volumes is 1709. If \( n \) is the last page number, what is th... | [
"17"
] | [
"A book is published in three volumes with continuous page numbering. The second volume has 50 more pages than the first volume, and the third volume has one and a half times as many pages as the second volume. The sum of the first page numbers of the three volumes is 1709. If \\( n \\) is the last page number, wha... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-2118004f0dd9c7d1edf8c11b650497d0713961df2c3922ef5f769891727ddaad | A book that originally cost \$B is discounted by \(A\%\), and now costs \$A. If \(A\) and \(B\) are positive integers less than 200, how many possibilities are there for the ordered pair \((A, B)\)? | [
"3"
] | [
"A book that originally cost \\$B is discounted by \\(A\\%\\), and now costs \\$A. If \\(A\\) and \\(B\\) are positive integers less than 200, how many possibilities are there for the ordered pair \\((A, B)\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c5f01f4d5556fc3ec6183fd7353ccffbf91f1e6797d7216a902a7e2c2614e05f | A book that takes 412 minutes to read aloud is to be recorded onto compact discs, with each disc holding up to 56 minutes of reading. Using the smallest possible number of discs and ensuring each disc contains the same length of reading, how many minutes of reading will each disc contain? | [
"51.5"
] | [
"A book that takes 412 minutes to read aloud is to be recorded onto compact discs, with each disc holding up to 56 minutes of reading. Using the smallest possible number of discs and ensuring each disc contains the same length of reading, how many minutes of reading will each disc contain?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 5,
"tir": 13
} | 26 | 0 | 0 |
omr-ws-62f0abe59b167f81ced04fe062c6c12c52cd9c301aefac2602b19b7566129c53 | A book with 53 pages numbered 1 to 53 has its pages renumbered in reverse, from 53 to 1. For how many pages do the new page number and old page number share the same units digit? | [
"11"
] | [
"A book with 53 pages numbered 1 to 53 has its pages renumbered in reverse, from 53 to 1. For how many pages do the new page number and old page number share the same units digit?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-710fcb5dc7796b255cb90751dc174e453e05e4c86cc0cfc8b1b96e9da78b59a2 | A booking office sells tickets to 200 destinations and issues 3800 tickets on a particular day. Let $N$ be the minimum number of destinations with the same number of passenger arrivals. Calculate the value of $N$. | [
"20"
] | [
"A booking office sells tickets to 200 destinations and issues 3800 tickets on a particular day. Let $N$ be the minimum number of destinations with the same number of passenger arrivals. Calculate the value of $N$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 1
} | 4 | 0 | 0 |
omr-ws-4557a1d212b577002bfda2811d735a4cbe1aad3bc0f838cab0afe61cf2cb0a05 | A bookshelf contains \( n \) volumes, labeled 1 to \( n \), in some order. The librarian selects a volume that is too far to the right, say the volume with label \( k \), takes it out, and inserts it in the \( k \)-th position. Determine the largest number of steps that this process can take to arrange all volumes in t... | [
"\\(\\frac{n(n-1)}{2}\\)"
] | [
"A bookshelf contains \\( n \\) volumes, labeled 1 to \\( n \\), in some order. The librarian selects a volume that is too far to the right, say the volume with label \\( k \\), takes it out, and inserts it in the \\( k \\)-th position. Determine the largest number of steps that this process can take to arrange all... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 3
} | 12 | 0 | 0 |
omr-ws-ea61c3fc62044b887550f35042d118e9e1f529ee8347e304dfb1caf85d0290cf | A bookshelf contains \( n \) volumes, labeled 1 to \( n \), in some order. The librarian selects a volume that is too far to the right, say the volume with label \( k \), takes it out, and inserts it in the \( k \)-th position. Let \( S(n) \) be the maximum number of moves required to sort the volumes into the correct ... | [
"9"
] | [
"A bookshelf contains \\( n \\) volumes, labeled 1 to \\( n \\), in some order. The librarian selects a volume that is too far to the right, say the volume with label \\( k \\), takes it out, and inserts it in the \\( k \\)-th position. Let \\( S(n) \\) be the maximum number of moves required to sort the volumes in... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8357a82716d4ffb4b0d52d97796245c2692bfa62c99f55a064410c5a14da41a7 | A bookstore has a sale on days of the month that are multiples of 5 (such as June 5, June 10...). A shoe store has a sale every 6 days. The shoe store has a sale on July 3. How many times in July do the two stores have sales on the same date? | [
"1"
] | [
"A bookstore has a sale on days of the month that are multiples of 5 (such as June 5, June 10...). A shoe store has a sale every 6 days. The shoe store has a sale on July 3. How many times in July do the two stores have sales on the same date?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2062463740561ec098b00631ad6896d716e262b3f38d2abcfda8a825c4c770ce | A bookstore is deciding what price it should charge for a certain book. After research, the store finds that if the book's price is $p$ dollars (where $p \le 26$), then the number of books sold per month is $130-5p$. What price should the store charge to maximize its revenue? | [
"13"
] | [
"A bookstore is deciding what price it should charge for a certain book. After research, the store finds that if the book's price is $p$ dollars (where $p \\le 26$), then the number of books sold per month is $130-5p$. What price should the store charge to maximize its revenue?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-87327f938f1977ca5a1c123b5de3862257179b57971131c36445705de3b8391e | A bored and hungry Goldilocks finds an infinite number of raisins in her pocket. On the kitchen table lies an infinite number of bowls of porridge. She labels the bowls, numbering the leftmost bowl $1$, the second leftmost bowl $2$, and so on. She then distributes her raisins, starting by adding $1$ raisin to Bowl $1$.... | [
"2037171"
] | [
"A bored and hungry Goldilocks finds an infinite number of raisins in her pocket. On the kitchen table lies an infinite number of bowls of porridge. She labels the bowls, numbering the leftmost bowl $1$, the second leftmost bowl $2$, and so on. She then distributes her raisins, starting by adding $1$ raisin to Bowl... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-88209518dc72c920d5a4f0a8d8619ee34e7e8e1e831eca08e47bd7f7e0c101d8 | A bored mathematician notices that the increase of water droplets in his shower every minute can be expressed as $WI = \frac{3 WO}{5} \cdot \frac{3T}{4}$, where $WI$ is the increase in droplets, $WO$ is the original amount, and $T$ is the temperature. If his shower starts with $300,000$ droplets, and he leaves the show... | [
"183000000"
] | [
"A bored mathematician notices that the increase of water droplets in his shower every minute can be expressed as $WI = \\frac{3 WO}{5} \\cdot \\frac{3T}{4}$, where $WI$ is the increase in droplets, $WO$ is the original amount, and $T$ is the temperature. If his shower starts with $300,000$ droplets, and he leaves ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-521b3357f7f7de3a525276b5972a7a5c608eda2475a754a97845c33d0ff9adad | A bored student named Abraham is writing $n$ numbers $a_1, a_2, \ldots, a_n$. The value of each number is either $1, 2$, or $3$; that is, $a_i$ is $1, 2$ or $3$ for $1 \le i \le n$. Abraham notices that the ordered triples \[ (a_1, a_2, a_3), (a_2, a_3, a_4), \ldots, (a_{n-2}, a_{n-1}, a_n), (a_{n-1}, a_n, a_1), (a_n, ... | [
"(27, 111222333121131332123221323)"
] | [
"A bored student named Abraham is writing $n$ numbers $a_1, a_2, \\ldots, a_n$. The value of each number is either $1, 2$, or $3$; that is, $a_i$ is $1, 2$ or $3$ for $1 \\le i \\le n$. Abraham notices that the ordered triples\n\\[ (a_1, a_2, a_3), (a_2, a_3, a_4), \\ldots, (a_{n-2}, a_{n-1}, a_n), (a_{n-1}, a_n, a... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 16
} | 36 | 0 | 0 |
omr-ws-6fee7eac957268cf0e743a080101120679660010592c7fc5be6d897029d5e25a | A bored student walks down a hall that contains a row of closed lockers, numbered $1$ to $1024$. He opens the locker numbered 1, and then alternates between skipping and opening each locker thereafter. When he reaches the end of the hall, the student turns around and starts back. He opens the first closed locker he enc... | [
"342"
] | [
"A bored student walks down a hall that contains a row of closed lockers, numbered $1$ to $1024$. He opens the locker numbered 1, and then alternates between skipping and opening each locker thereafter. When he reaches the end of the hall, the student turns around and starts back. He opens the first closed locker h... | [
"MATH_training_set"
] | [
"additional_problems",
"cot",
"genselect",
"tir"
] | {
"additional_problems": 1,
"cot": 1,
"genselect": 8,
"tir": 4
} | 14 | 0 | 0 |
omr-ws-0f41f1ff476803385dde7a493d95ca7d744cb3b375d9b36c1c0924686e44289f | A bored warden decides to play a game. He puts 10 prisoners in a single-file line, and then places either a white or black hat on every prisoner's head. The number of white and black hats is not known by the prisoners. Each prisoner is only able to see the colors of the hats in front of them. Starting from the back of ... | [
"9"
] | [
"A bored warden decides to play a game. He puts 10 prisoners in a single-file line, and then places either a white or black hat on every prisoner's head. The number of white and black hats is not known by the prisoners. Each prisoner is only able to see the colors of the hats in front of them. Starting from the bac... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1041102d0da0fc965199037f2a8cbaf57142669ca3cd70b32cbaa1373c9c91c3 | A bored warden decides to play a game. He puts 100 prisoners in a single-file line, and then on each of their heads he places one of his 101 different-colored hats. The prisoners are aware of the 101 different colors, and that every prisoner gets a different hat color. Each prisoner is only able to see the colors of th... | [
"99"
] | [
"A bored warden decides to play a game. He puts 100 prisoners in a single-file line, and then on each of their heads he places one of his 101 different-colored hats. The prisoners are aware of the 101 different colors, and that every prisoner gets a different hat color. Each prisoner is only able to see the colors ... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 5
} | 18 | 0 | 0 |
omr-ws-c27aea83475e5333029e2bc03370ed606a86f8a23cb44cdd1f1ab866d0a4eb87 | A boss plans a business meeting at Starbucks with the two engineers below him. However, he fails to set a time, and all three arrive at Starbucks at a random time between 2:00 and 4:00 p.m. When the boss shows up, if both engineers are not already there, he storms out and cancels the meeting. Each engineer is willing t... | [
"\\frac{7}{24}"
] | [
"A boss plans a business meeting at Starbucks with the two engineers below him. However, he fails to set a time, and all three arrive at Starbucks at a random time between 2:00 and 4:00 p.m. When the boss shows up, if both engineers are not already there, he storms out and cancels the meeting. Each engineer is will... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-6efafe6b3cf9015742e9aae98d901ff5c135732ac67d36280b49b26ea945fba5 | A bottle containing c ounces of juice can be emptied to fill g identical glasses. How many such bottles would be needed to fill n of these glasses with juice? | [
"\\(\\frac{n}{g}\\)"
] | [
"A bottle containing c ounces of juice can be emptied to fill g identical glasses. How many such bottles would be needed to fill n of these glasses with juice?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c8b5fb525a5703f970e13d8ecb3ce0c4f97152fae885e68dd66763dcd6279333 | A bottle is in the shape of a pentagonal pyramid with a regular pentagon base of 15 cm per side and a height of 32 cm. The flow of potion into the bottle starts at zero and increases at a rate of 1 milliliter per second per second. How many seconds will it take to completely fill the bottle? Round to the nearest whole ... | [
"28"
] | [
"A bottle is in the shape of a pentagonal pyramid with a regular pentagon base of 15 cm per side and a height of 32 cm. The flow of potion into the bottle starts at zero and increases at a rate of 1 milliliter per second per second. How many seconds will it take to completely fill the bottle? Round to the nearest w... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e991791b33c7687e354a87be4405d4829d5feea38e692ab73de4d6d450561e83 | A bottle is in the shape of a pentagonal pyramid with a regular pentagon base of 15 cm per side and a height of 32 cm. The flow of potion into the bottle starts at zero and increases at a rate of 1 millilitre per second per second. How many seconds will it take to completely fill the bottle? Round to the nearest whole ... | [
"95"
] | [
"A bottle is in the shape of a pentagonal pyramid with a regular pentagon base of 15 cm per side and a height of 32 cm. The flow of potion into the bottle starts at zero and increases at a rate of 1 millilitre per second per second. How many seconds will it take to completely fill the bottle? Round to the nearest w... | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 1,
"tir": 2
} | 6 | 0 | 0 |
omr-ws-18eaefca4bf3c8c27acd692ce6105db105195b749a444384b286a92eaf2a4a55 | A bottle of cola costs 2.60 €, but a bottle of cola without a plug costs 2.40 €. How much does the plug cost? | [
"0.20"
] | [
"A bottle of cola costs 2.60 €, but a bottle of cola without a plug costs 2.40 €. How much does the plug cost?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c91964dbf2217376d01df1672424a7ec2ec3418842d42d6227ba9a6a2a04c0aa | A bottle of cola costs \$1, and you have \$10. You can trade 3 empty bottles or 4 bottlecaps for a new bottle of cola (for free!). Using trade-ins, how many bottles of cola can you have? | [
"17"
] | [
"A bottle of cola costs \\$1, and you have \\$10. You can trade 3 empty bottles or 4 bottlecaps for a new bottle of cola (for free!). Using trade-ins, how many bottles of cola can you have?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dab7cf59f54bdbc1e33ccd49fcef7586e30e2ef899629d0559cf348f32c1ba6d | A bottle of syrup can be used to make $60$ glasses of drink if dissolved in water with a ratio of $1$ part syrup to $4$ parts water. How many glasses of drink are obtained from a bottle of syrup if the ratio of the solution is $1$ part syrup to $5$ parts water? | [
"72"
] | [
"A bottle of syrup can be used to make $60$ glasses of drink if dissolved in water with a ratio of $1$ part syrup to $4$ parts water. How many glasses of drink are obtained from a bottle of syrup if the ratio of the solution is $1$ part syrup to $5$ parts water?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-21db260425997a70d74a21a9d8ad1c6f4b008dbcfbc09037254721dba232c60f | A bottled water company has designed a new cup for its dispenser. The cup will be a right circular cone with a three-inch radius. How tall will the cup need to be to hold 93 cubic inches of water? Express your answer to the nearest whole number. | [
"10"
] | [
"A bottled water company has designed a new cup for its dispenser. The cup will be a right circular cone with a three-inch radius. How tall will the cup need to be to hold 93 cubic inches of water? Express your answer to the nearest whole number."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-33783f9030c6e53cfc871a8ff03dd51a4dba473b37e3e4ef4434cf799d798efe | A bowl contains 10 jellybeans (four red, one blue and five white). If you pick three jellybeans from the bowl at random and without replacement, what is the probability that exactly two will be red? Express your answer as a common fraction. | [
"\\frac{3}{10}"
] | [
"A bowl contains 10 jellybeans (four red, one blue and five white). If you pick three jellybeans from the bowl at random and without replacement, what is the probability that exactly two will be red? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-94dd06144cea074c0f2340acc2bc715e4b0c706c143c73581cf2bb0a1b526847 | A bowl contains 13 black, 20 green, and 18 purple beads. After drawing a bead 11,543 times with replacement, what is the probability of drawing a green bead on the next draw? | [
"\\(\\frac{20}{51}\\)"
] | [
"A bowl contains 13 black, 20 green, and 18 purple beads. After drawing a bead 11,543 times with replacement, what is the probability of drawing a green bead on the next draw?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-7fb6b903c8c55754f3067c442ce01eb0111d69c1fbb38ef313b0455917e5254d | A bowl contains only red marbles, blue marbles, and green marbles. The probability of selecting a red marble from the bowl is $\frac{3}{13}$. The probability of selecting a blue marble is $\frac{2}{5}$. There are fewer than 100 marbles in the bowl. What is the probability of selecting, at random and without replacement... | [
"\\(\\frac{9}{104}\\)"
] | [
"A bowl contains only red marbles, blue marbles, and green marbles. The probability of selecting a red marble from the bowl is $\\frac{3}{13}$. The probability of selecting a blue marble is $\\frac{2}{5}$. There are fewer than 100 marbles in the bowl. What is the probability of selecting, at random and without repl... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-29663c8bc31c2e202bb5475b458ca814071065e7170330f2bcc4f3c5323abbb7 | A bowl has the shape of the graph of \( y = x^4 \) between the points \( (1,1) \) and \( (-1,1) \) rotated about the y-axis. When the bowl contains water to a depth of \( h \) units, it flows out through a hole in the bottom at a rate proportional to \( \sqrt{h} \), with a constant of proportionality 6. If the bowl is ... | [
"\\dfrac{\\pi}{12}"
] | [
"A bowl has the shape of the graph of \\( y = x^4 \\) between the points \\( (1,1) \\) and \\( (-1,1) \\) rotated about the y-axis. When the bowl contains water to a depth of \\( h \\) units, it flows out through a hole in the bottom at a rate proportional to \\( \\sqrt{h} \\), with a constant of proportionality 6.... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-986eb8afa7e9a731fa8cadd7d3d60c3dda99cef0d00be6a00f6d59f094dbfd06 | A bowl in the shape of a conical frustum is placed out in the rain. At the end of the downpour, the water level $r$ is equal to (i) half the height of the bowl, (ii) the radius of the base of the bowl, and (iii) half the radius of the top of the bowl. What was the reported rainfall over the catchment area? | [
"\\dfrac{19}{48}r"
] | [
"A bowl in the shape of a conical frustum is placed out in the rain. At the end of the downpour, the water level $r$ is equal to (i) half the height of the bowl, (ii) the radius of the base of the bowl, and (iii) half the radius of the top of the bowl. What was the reported rainfall over the catchment area?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 4
} | 11 | 0 | 0 |
omr-ws-e9c5087b63b005cac6efde05de50679ea3b43f1737f5ad276f2d9841b2ef0930 | A bowl is being filled with water, and its cross-sectional area at height \( x \) is given by \( A(x) = \pi \left( 15x - x^2 \right) \). The bowl is half full when \( x \) equals \( 5 \sqrt{3} \sin\left(\frac{\pi}{9}\right) - 5 \cos\left(\frac{\pi}{9}\right) + 5 \). Calculate the value of \( x \) when the bowl is half ... | [
"17"
] | [
"A bowl is being filled with water, and its cross-sectional area at height \\( x \\) is given by \\( A(x) = \\pi \\left( 15x - x^2 \\right) \\). The bowl is half full when \\( x \\) equals \\( 5 \\sqrt{3} \\sin\\left(\\frac{\\pi}{9}\\right) - 5 \\cos\\left(\\frac{\\pi}{9}\\right) + 5 \\). Calculate the value of \\(... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-768175838e690cd24a15717761c0ef8b48600c07710b80b8f61db37a0eb28a4b | A bowl is created by rotating \(y = f(x)\) about the y-axis and is filled with water to a height \(h_0\). At its bottom \((x = y = 0)\), a small hole of radius \(r\) allows the fluid to drain. The speed of the exiting fluid is given by Torricelli’s law: \(\sqrt{2gh}\), where \(h\) is the instantaneous height of the liq... | [
"\\frac{dV}{dt} = -\\pi r^2 \\sqrt{2gh}"
] | [
"A bowl is created by rotating \\(y = f(x)\\) about the y-axis and is filled with water to a height \\(h_0\\). At its bottom \\((x = y = 0)\\), a small hole of radius \\(r\\) allows the fluid to drain. The speed of the exiting fluid is given by Torricelli’s law: \\(\\sqrt{2gh}\\), where \\(h\\) is the instantaneous... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-82668196faa9141445442803d4c4442e51d4cf4a7d9aec64f4943bca36db200a | A bowl is formed by attaching four regular hexagons of side $1$ to a square of side $1$. The edges of the adjacent hexagons coincide. What is the area of the octagon obtained by joining the top eight vertices of the four hexagons, situated on the rim of the bowl? | [
"7"
] | [
"A bowl is formed by attaching four regular hexagons of side $1$ to a square of side $1$. The edges of the adjacent hexagons coincide. What is the area of the octagon obtained by joining the top eight vertices of the four hexagons, situated on the rim of the bowl?"
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0489f5a2738a490bd2c20600e64d5ee7e46c0cdd846ca82cfe0828726b9c4448 | A bowl is formed by attaching four regular hexagons of side 1 to a square of side 1. The edges of adjacent hexagons coincide. What is the area of the octagon obtained by joining the top eight vertices of the four hexagons, situated on the rim of the bowl? | [
"7 + 4\\sqrt{3}"
] | [
"A bowl is formed by attaching four regular hexagons of side 1 to a square of side 1. The edges of adjacent hexagons coincide. What is the area of the octagon obtained by joining the top eight vertices of the four hexagons, situated on the rim of the bowl?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-760c62a3deba2bf08e4050336345433ef97f841eb6c31827160900e6aca1b4c8 | A bowl is formed by rotating the arc of the curve \( ay = x^2 \) from \( x = 0 \) to \( x = a \) about the \( y \)-axis. Initially, the bowl is full of water. The water evaporates from the surface so that, at any instant, the rate of decrease in its volume is proportional to the surface area. If \( T \) is the time for... | [
"\\dfrac{a}{T}"
] | [
"A bowl is formed by rotating the arc of the curve \\( ay = x^2 \\) from \\( x = 0 \\) to \\( x = a \\) about the \\( y \\)-axis. Initially, the bowl is full of water. The water evaporates from the surface so that, at any instant, the rate of decrease in its volume is proportional to the surface area. If \\( T \\) ... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d9f8a5c512ceb0d16f52f91b014fe10f663f03ad768ec51cd55d85957dceab84 | A bowling ball cannot weigh more than 16 pounds and must have a diameter of $8 \frac{1}{2}$ inches. How many square inches are in the surface area of a bowling ball before the finger holes are drilled? Express your answer as a common fraction in terms of $\pi$. | [
"\\frac{289\\pi}{4}"
] | [
"A bowling ball cannot weigh more than 16 pounds and must have a diameter of $8 \\frac{1}{2}$ inches. How many square inches are in the surface area of a bowling ball before the finger holes are drilled? Express your answer as a common fraction in terms of $\\pi$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0cfb39d19274b8b7da9150fc0169cfcd1340551e8d3e8f62ca4a934f76fd97ec | A bowling ball is a solid ball with a spherical surface and diameter 30 cm. To custom fit a bowling ball for each bowler, three holes are drilled in the ball. Bowler Kris has holes drilled that are 8 cm deep and have diameters of 2 cm, 2 cm, and 3 cm. Assuming the three holes are right circular cylinders, find the numb... | [
"4466\\pi"
] | [
"A bowling ball is a solid ball with a spherical surface and diameter 30 cm. To custom fit a bowling ball for each bowler, three holes are drilled in the ball. Bowler Kris has holes drilled that are 8 cm deep and have diameters of 2 cm, 2 cm, and 3 cm. Assuming the three holes are right circular cylinders, find the... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6d985dec98712670e3f7894820dca514e479034e511c4c0e498f5a12dbbd57e8 | A box $3 \times 5 \times 7$ is divided into unit cube cells, each containing a cockchafer. At a signal, every cockchafer moves through a face of its cell to a neighboring cell. What is the minimum number of empty cells after the signal? | [
"1"
] | [
"A box $3 \\times 5 \\times 7$ is divided into unit cube cells, each containing a cockchafer. At a signal, every cockchafer moves through a face of its cell to a neighboring cell. What is the minimum number of empty cells after the signal?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-43cdeaf58a24d4d7cccea0971f824ca96c820f579b5af995187a56ef72a88d06 | A box $3 \times 5 \times 7$ is divided into unit cube cells, each containing a cockchafer. At a signal, every cockchafer moves to a diagonally adjacent cell (sharing exactly one vertex). What is the minimum number of empty cells after the signal? | [
"35"
] | [
"A box $3 \\times 5 \\times 7$ is divided into unit cube cells, each containing a cockchafer. At a signal, every cockchafer moves to a diagonally adjacent cell (sharing exactly one vertex). What is the minimum number of empty cells after the signal?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-c8eaad06038a3d78ec4de8dc7e785924b45818f2a4e6d12b49cc0ae7eb81007b | A box B1 contains 1 white ball, 3 red balls, and 2 black balls. Another box B2 contains 2 white balls, 3 red balls, and 4 black balls. A third box B3 contains 3 white balls, 4 red balls, and 5 black balls. If 2 balls are drawn (without replacement) from a randomly selected box and one of the balls is white and the othe... | [
"\\(\\frac{55}{181}\\)"
] | [
"A box B1 contains 1 white ball, 3 red balls, and 2 black balls. Another box B2 contains 2 white balls, 3 red balls, and 4 black balls. A third box B3 contains 3 white balls, 4 red balls, and 5 black balls. If 2 balls are drawn (without replacement) from a randomly selected box and one of the balls is white and the... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-ea19593d393e8e016a6fa137a61cb9f177b30ce9ec9427c7504f104990461673 | A box contains $300$ matches. Players take turns removing no more than half the matches from the box. The player after whose move only one match remains in the box wins. Which of the two players has the winning strategy, and what is it? | [
"\\text{The first player has a winning strategy.}"
] | [
"A box contains $300$ matches. Players take turns removing no more than half the matches from the box. The player after whose move only one match remains in the box wins. Which of the two players has the winning strategy, and what is it?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2fa71ebd7b741ec311896af4a1c5edd67c08600b682524f01510da05d8e5ecc9 | A box contains $7.15 in nickels, dimes, and quarters. There are 42 coins in total, and the sum of the number of nickels and dimes is 2 less than the number of quarters. How many coins of each kind are there? | [
"\\( N = 7 \\), \\( D = 13 \\), \\( Q = 22 \\)"
] | [
"A box contains $7.15 in nickels, dimes, and quarters. There are 42 coins in total, and the sum of the number of nickels and dimes is 2 less than the number of quarters. How many coins of each kind are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 11,
"tir": 29
} | 40 | 0 | 0 |
omr-ws-408e868cdf76aa5c0caeade3e4e3454c0a8177c700743d50b51a4c7e15e0d435 | A box contains $N > 1$ white and black balls, of which $N_B (0 < N_B \leq N)$ are black. Balls are taken randomly from the box without replacement. Find the probability $P(B_n)$ of taking a black ball at the $n$th withdrawal for $1 \leq n \leq N$. | [
"\\(\\frac{N_B}{N}\\)"
] | [
"A box contains $N > 1$ white and black balls, of which $N_B (0 < N_B \\leq N)$ are black. Balls are taken randomly from the box without replacement. Find the probability $P(B_n)$ of taking a black ball at the $n$th withdrawal for $1 \\leq n \\leq N$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-392cd74cec46674a99f91fb08ea4f7e1cd692057068f7989b4e446e752c77e46 | A box contains $a$ white balls and $b$ black balls. Balls are drawn with replacement. What is the probability that the first person to draw a ball gets a white ball? | [
"\\(\\frac{a+b}{a+2b}\\)"
] | [
"A box contains $a$ white balls and $b$ black balls. Balls are drawn with replacement. What is the probability that the first person to draw a ball gets a white ball?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 2
} | 9 | 0 | 0 |
omr-ws-827996656f16b939b2c6a982ca6bd3592c536082681ad9f381d898e9a33a6597 | A box contains $n$ balls: $a$ are white and $n - a$ are red. We draw the balls from the box in some order without replacement. Before each draw, we guess the color of the ball being drawn, always guessing the color more likely to occur (if one is more likely than the other). What is the expected number $E[N(n,a)]$ of c... | [
"\\( \\max(a, n-a) \\)"
] | [
"A box contains $n$ balls: $a$ are white and $n - a$ are red. We draw the balls from the box in some order without replacement. Before each draw, we guess the color of the ball being drawn, always guessing the color more likely to occur (if one is more likely than the other). What is the expected number $E[N(n,a)]$... | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-f3a6db7d13030e825af6372e17bcdc3c661aa97ed0f6ad9d8e01fc5ca8af7473 | A box contains $p$ white balls and $q$ black balls. Beside the box, there is a pile of black balls. Two balls are taken out of the box. If they have the same color, a black ball from the pile is put into the box. If they have different colors, the white ball is put back into the box. This procedure is repeated until th... | [
"\\( \\begin{cases} \n1 & \\text{if } p \\text{ is odd} \\\\\n0 & \\text{if } p \\text{ is even} \n\\end{cases} \\)"
] | [
"A box contains $p$ white balls and $q$ black balls. Beside the box, there is a pile of black balls. Two balls are taken out of the box. If they have the same color, a black ball from the pile is put into the box. If they have different colors, the white ball is put back into the box. This procedure is repeated unt... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 18,
"tir": 19
} | 37 | 0 | 0 |
omr-ws-06ad69ad9ed803405e98ca279339c7059793eb8b25ba1b053825e9b9f4c83fbd | A box contains 'p' white balls and 'q' black balls. Two balls are taken out at random from the box. If the balls are different, then the white ball is replaced and the black ball is removed from the box. If the balls are similar, then both balls are removed and replaced with a black ball from a pile. The process is con... | [
"\\( \\begin{cases} \n0 & \\text{if } p \\text{ is even} \\\\\n1 & \\text{if } p \\text{ is odd} \n\\end{cases} \\)"
] | [
"A box contains 'p' white balls and 'q' black balls. Two balls are taken out at random from the box. If the balls are different, then the white ball is replaced and the black ball is removed from the box. If the balls are similar, then both balls are removed and replaced with a black ball from a pile. The process i... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-0cb47ab9e215e9c2f098c0bd049a888b70a6b0e06383d8b2e7b214a4e2254271 | A box contains 1,000,000 marbles: 940,000 marbles have the number 5, 50,000 marbles have the number 6, 5,000 marbles have the number 7, and 1,000 marbles have the number 8. Marbles are extracted two at a time. What is the probability of extracting two marbles with the number 8? | [
"\\dfrac{1}{1001000}"
] | [
"A box contains 1,000,000 marbles: 940,000 marbles have the number 5, 50,000 marbles have the number 6, 5,000 marbles have the number 7, and 1,000 marbles have the number 8. Marbles are extracted two at a time. What is the probability of extracting two marbles with the number 8?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3fd39612e95d946b2ce52270fde374d387b4ae800bdde5f02479f8860de62b9c | A box contains 10 dice, 5 of which are crooked and produce a '6' with probability 0.4, while the other 5 are fair. A die is selected at random from the box and thrown twice, producing a '6' each time. Find the probability that the die is crooked. | [
"\\(\\frac{144}{169}\\)"
] | [
"A box contains 10 dice, 5 of which are crooked and produce a '6' with probability 0.4, while the other 5 are fair. A die is selected at random from the box and thrown twice, producing a '6' each time. Find the probability that the die is crooked."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-bbd3bbb4e56fd9fccc6a73fea22b96e6ebf02998874bde237f4e4242f3010c5c | A box contains 10 tokens: five with the number 1, three with the number 3, and two with the number 9. If 100 tokens are drawn randomly with replacement, determine the approximate probability that the product of the numbers on the drawn tokens will be between \(3^{60}\) and \(3^{100}\). | [
"0.8997"
] | [
"A box contains 10 tokens: five with the number 1, three with the number 3, and two with the number 9. If 100 tokens are drawn randomly with replacement, determine the approximate probability that the product of the numbers on the drawn tokens will be between \\(3^{60}\\) and \\(3^{100}\\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 1,
"tir": 11
} | 26 | 0 | 0 |
omr-ws-6268c9a3c3b0b8a92bceaf232eb8f4a9ace064d9a72636e887a195121ee170c9 | A box contains 100 Minion toys, of which 20 are Evil Minions and 80 are Regular Minions. If a worker randomly pulls 5 toys out of the box, what is the probability that four or more of the toys are Regular Minions? | [
"0.74034"
] | [
"A box contains 100 Minion toys, of which 20 are Evil Minions and 80 are Regular Minions. If a worker randomly pulls 5 toys out of the box, what is the probability that four or more of the toys are Regular Minions?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 8,
"tir": 25
} | 44 | 0 | 0 |
omr-ws-8858940d9fcca8f1186b60e4a70536ebcfda624a8487fabcde4a0e7a99c77312 | A box contains 100 tickets numbered 1 to 100. A person picks out 3 tickets such that the product of the numbers on two of the tickets yields the number on the third ticket. How many tickets can never be picked up? | [
"11"
] | [
"A box contains 100 tickets numbered 1 to 100. A person picks out 3 tickets such that the product of the numbers on two of the tickets yields the number on the third ticket. How many tickets can never be picked up?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 7
} | 30 | 0 | 0 |
omr-ws-57277e11a61f555d9813b09b4d0176a1642cdcfc338d66f823037af33472e899 | A box contains 11 balls, numbered 1 through 11. If 6 balls are drawn simultaneously at random, what is the probability that the sum of the numbers on the balls drawn is odd? | [
"$\\frac{118}{231}$"
] | [
"A box contains 11 balls, numbered 1 through 11. If 6 balls are drawn simultaneously at random, what is the probability that the sum of the numbers on the balls drawn is odd?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 32
} | 47 | 0 | 0 |
omr-ws-7c984e663b8d47698fb5d710e62f64bd4996a02fcf5bc797535eecf956b3bae7 | A box contains 11 marbles, numbered 1, 2, 3, ..., 11. If 6 marbles are drawn simultaneously at random, and the probability that the sum of the numbers drawn is odd is \(\frac{D}{Q}\), with \(D\) and \(Q\) relatively prime positive integers, find \(D + Q\). | [
"349"
] | [
"A box contains 11 marbles, numbered 1, 2, 3, ..., 11. If 6 marbles are drawn simultaneously at random, and the probability that the sum of the numbers drawn is odd is \\(\\frac{D}{Q}\\), with \\(D\\) and \\(Q\\) relatively prime positive integers, find \\(D + Q\\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-59f9270a0bb136e2a9783b5789c4fc864f2817a8841064f21bff4fe060fe9998 | A box contains 12 batteries, 5 of which are dead. Find the probability that the first four batteries chosen all work. | [
"\\(\\frac{7}{99}\\)"
] | [
"A box contains 12 batteries, 5 of which are dead. Find the probability that the first four batteries chosen all work."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-6a1a54b70d81e0ac7214ae4b9604f2761b7da332db7e5d815888918b07606669 | A box contains 12 batteries, 5 of which are dead. Find the probability that you have to pick 5 batteries in order to find one that works. | [
"\\(\\frac{7}{792}\\)"
] | [
"A box contains 12 batteries, 5 of which are dead. Find the probability that you have to pick 5 batteries in order to find one that works."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-26c60e02bbd4de00fd9a468a512fb2d8d706fa8da2b882669ebb65afc90d3985 | A box contains 12 red and 6 white balls. Balls are drawn from the box one at a time without replacement. If in 6 draws, there are at least 4 white balls, find the probability that exactly one white ball is drawn in the next two draws. | [
"\\dfrac{312}{1063}"
] | [
"A box contains 12 red and 6 white balls. Balls are drawn from the box one at a time without replacement. If in 6 draws, there are at least 4 white balls, find the probability that exactly one white ball is drawn in the next two draws."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3e881aa16076dce886779cf89f10a4a80cac68c63367eec280948ac755b27c41 | A box contains 12 red and 6 white balls. Balls are drawn one at a time without replacement. If in the first 6 draws there are at least 4 white balls, find the probability that exactly one white ball is drawn in the next two draws. | [
"\\(\\frac{17}{57}\\)"
] | [
"A box contains 12 red and 6 white balls. Balls are drawn one at a time without replacement. If in the first 6 draws there are at least 4 white balls, find the probability that exactly one white ball is drawn in the next two draws."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-744043fbc50d1d093dae45fd41c6d4784a3017591d797d0e83ac4ed492b311ae | A box contains 14 balls: 7 black balls numbered from 1 to 7 and 7 white balls numbered from 1 to 7. A person picks 2 balls at a time and repeats the process until the box is empty. Let \( n \) be the total number of ways to draw the balls such that the difference in the numbers on the balls picked each time is at most ... | [
"595"
] | [
"A box contains 14 balls: 7 black balls numbered from 1 to 7 and 7 white balls numbered from 1 to 7. A person picks 2 balls at a time and repeats the process until the box is empty. Let \\( n \\) be the total number of ways to draw the balls such that the difference in the numbers on the balls picked each time is a... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-eba0a529ded3c259122c382a02dcd9aa9fb419d726f04c3c2daf9a8f1b050345 | A box contains 15 red pencils, 13 blue pencils, and 8 green pencils. What is the minimum number of pencils Constanza should take out blindfolded to ensure she gets at least 1 red, 2 blue, and 3 green pencils? | [
"22"
] | [
"A box contains 15 red pencils, 13 blue pencils, and 8 green pencils. What is the minimum number of pencils Constanza should take out blindfolded to ensure she gets at least 1 red, 2 blue, and 3 green pencils?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-c08f2bd2cbeb2ad6cfc33e9f504131cda4d7786b271d443776a3b0e89f0e790b | A box contains 18 green and 12 blue balls. Three balls are drawn without replacement, and then another three balls are drawn without replacement. What is the probability that at least 2 of the first 3 balls drawn are green and at least 2 of the second 3 balls drawn are blue? | [
"\\dfrac{6919}{28275}"
] | [
"A box contains 18 green and 12 blue balls. Three balls are drawn without replacement, and then another three balls are drawn without replacement. What is the probability that at least 2 of the first 3 balls drawn are green and at least 2 of the second 3 balls drawn are blue?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-34497414e2984dff667576d85a8a432686bc723a8f76041929746c352ba4ca41 | A box contains 2 coins with heads on both sides, 1 standard coin, and 1 coin with tails on both sides. If one coin is picked from the box and flipped twice, what is the probability that both flips will be heads? | [
"\\(\\frac{9}{16}\\)"
] | [
"A box contains 2 coins with heads on both sides, 1 standard coin, and 1 coin with tails on both sides. If one coin is picked from the box and flipped twice, what is the probability that both flips will be heads?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-51052232d95753d80580f35b7d443ec80b97d91007b130be662889c292d6b3a5 | A box contains 2 pennies, 4 nickels, and 6 dimes. Six coins are drawn randomly and without replacement. Given that the probability that the value of the coins drawn is at least 50 cents is \(\frac{m}{n}\), with \(m\) and \(n\) relatively prime positive integers, find \(m + n\). | [
"1051"
] | [
"A box contains 2 pennies, 4 nickels, and 6 dimes. Six coins are drawn randomly and without replacement. Given that the probability that the value of the coins drawn is at least 50 cents is \\(\\frac{m}{n}\\), with \\(m\\) and \\(n\\) relatively prime positive integers, find \\(m + n\\)."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-88a3460858cfdb963008239a28ced7cd5781fc41fd648cfb4c76cfbf1727c19f | A box contains 2 pennies, 4 nickels, and 6 dimes. Six coins are drawn without replacement, with each coin having an equal probability of being chosen. What is the probability that the value of the coins drawn is at least 50 cents? | [
"$\\frac{127}{924}$"
] | [
"A box contains 2 pennies, 4 nickels, and 6 dimes. Six coins are drawn without replacement, with each coin having an equal probability of being chosen. What is the probability that the value of the coins drawn is at least 50 cents?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 16,
"genselect": 6,
"tir": 34
} | 56 | 0 | 0 |
omr-ws-d77b236e13565e108aee6b9af283566fe01b1e3738493af7075c01083d206c03 | A box contains 2 red balls and 3 blue balls. Two balls are randomly taken out without replacement. (a) What is the probability that the second ball is red? (b) What is the probability that both balls are red? (c) What is the probability that the first ball is red and the second ball is blue? | [
"(a) \\( \\frac{2}{5} \\), (b) \\( \\frac{1}{10} \\), (c) \\( \\frac{3}{10} \\)"
] | [
"A box contains 2 red balls and 3 blue balls. Two balls are randomly taken out without replacement.\n\n (a) What is the probability that the second ball is red?\n\n (b) What is the probability that both balls are red?\n\n (c) What is the probability that the first ball is red and the second ball is blue?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 17
} | 17 | 0 | 0 |
omr-ws-3d809a7b88fd64ad61dbb87514864ae9f4b7967ffcd02a5e5b0d49fb7d61db06 | A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last two marbles. What is the probability that Cheryl gets 2 marbles of the same color? | [
"\\(\\frac{1}{5}\\)"
] | [
"A box contains 2 red marbles, 2 green marbles, and 2 yellow marbles. Carol takes 2 marbles from the box at random; then Claudia takes 2 of the remaining marbles at random; and then Cheryl takes the last two marbles. What is the probability that Cheryl gets 2 marbles of the same color?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-0d8fe3b083f7aafa8df48dcf1fc3af830023ac5adaabc6b8561f3c2244d4d795 | A box contains 20 tubes, of which 16 are good and 4 are defective. What is the probability that the third tube is defective, given that one of the other two tubes is good and the other is defective? | [
"\\(\\frac{1}{6}\\)"
] | [
"A box contains 20 tubes, of which 16 are good and 4 are defective. What is the probability that the third tube is defective, given that one of the other two tubes is good and the other is defective?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-2e922a38462a1f34917b5018e92e864e91ec27b265117aa577749ac3cf1e9005 | A box contains 3 black balls, 4 red balls, and 7 green balls. Two balls are drawn at random. What is the probability of drawing at least one red ball? | [
"\\(\\frac{46}{91}\\)"
] | [
"A box contains 3 black balls, 4 red balls, and 7 green balls. Two balls are drawn at random. What is the probability of drawing at least one red ball?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-32d35dd6e43c3441ae9f287be7f02e968af18321ca353f9a2199107d20227595 | A box contains 3 black balls, 4 red balls, and 7 green balls. Two balls are drawn at random. What is the probability of drawing balls of different colors? | [
"\\(\\frac{61}{91}\\)"
] | [
"A box contains 3 black balls, 4 red balls, and 7 green balls. Two balls are drawn at random. What is the probability of drawing balls of different colors?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
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