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-88418fbcd3895a3a489afad8f6a0667ff8d983c926b1332bfb71a711ff6e2fe0 | What is the probability that the 5th strike of oil comes on the 8th well drilled, given that each well has a probability of 0.4 of striking oil? | [
"\\(0.4\\binom{7}{4}(0.4)^4(0.6)^3\\) or approximately \\(0.0774\\)."
] | [
"What is the probability that the 5th strike of oil comes on the 8th well drilled, given that each well has a probability of 0.4 of striking oil?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
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"cot",
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omr-ws-9c6578e0d58f18cb4b76a1e3346e612925368a460db6c41c2e604a3aaffc8983 | What is the probability that the Exeter Reds will win the series in the seventh game, given that each team has a 50 percent chance of winning any game they play? | [
"\\(\\frac{5}{32}\\)"
] | [
"What is the probability that the Exeter Reds will win the series in the seventh game, given that each team has a 50 percent chance of winning any game they play?"
] | [
"aops_c6_high_school_olympiads"
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"cot",
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"cot": 17,
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omr-ws-6b7612b734625e1fc132e6b6cfcfcc0c3581b43c230a4ca94ebd889d896a2f84 | What is the probability that the area of $\bigtriangleup PBC$ is greater than half of the area of $\bigtriangleup ABC$ when point $P$ is in the interior of $\bigtriangleup ABC$? | [
"\\(\\frac{1}{4}\\)"
] | [
"What is the probability that the area of $\\bigtriangleup PBC$ is greater than half of the area of $\\bigtriangleup ABC$ when point $P$ is in the interior of $\\bigtriangleup ABC$?"
] | [
"aops_c4_high_school_math"
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"cot",
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] | {
"cot": 16,
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} | 43 | 0 | 0 |
omr-ws-5716f732c84788ce6378e881120288ae2514b8865b827ff2c53192e3a83e71aa | What is the probability that the area of $\bigtriangleup PBC$ is half of the area of $\bigtriangleup ABC$ when point $P$ is in the interior of $\bigtriangleup ABC$? | [
"0"
] | [
"What is the probability that the area of $\\bigtriangleup PBC$ is half of the area of $\\bigtriangleup ABC$ when point $P$ is in the interior of $\\bigtriangleup ABC$?"
] | [
"aops_c4_high_school_math"
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"cot"
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"cot": 16
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omr-ws-49d398fcc935081f0fdd7f200acac994e41f944957be5f460b5b1673eb9b9818 | What is the probability that the area of a triangle formed by three random points on a circle is greater than $\frac{1}{3}$ of the circle's area? | [
"\\dfrac{1}{4}"
] | [
"What is the probability that the area of a triangle formed by three random points on a circle is greater than $\\frac{1}{3}$ of the circle's area?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
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"cot": 15
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omr-ws-99fa73a66ceb6b495a6d1cd3cc53e0b0decb26dae3b924a0c513159b58963263 | What is the probability that the ball lands in the same color of pocket in each of the two turns on a roulette wheel? | [
"\\dfrac{163}{361}"
] | [
"What is the probability that the ball lands in the same color of pocket in each of the two turns on a roulette wheel?"
] | [
"aops_c7_college_math"
] | [
"cot"
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"cot": 14
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omr-ws-fe9f684fae0dc849ecf23e3bf870fc39ab2d145dd3dc228e07d074e91aa39bb0 | What is the probability that the birthdays of 6 different persons will fall in exactly two different months? | [
"\\(\\frac{\\binom{12}{2} \\times (2^6 - 2)}{12^6}\\)"
] | [
"What is the probability that the birthdays of 6 different persons will fall in exactly two different months?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
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"cot": 16,
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} | 35 | 0 | 0 |
omr-ws-8965789a78740c2f1f71fbb641c1aa6ca792f7193c3660ccf4be0bdfd5ab6ff2 | What is the probability that the birthdays of two people in the month of November are exactly 5 days apart? | [
"\\(\\frac{1}{18}\\)"
] | [
"What is the probability that the birthdays of two people in the month of November are exactly 5 days apart?"
] | [
"aops_c4_high_school_math"
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"cot",
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] | {
"cot": 14,
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} | 45 | 0 | 0 |
omr-ws-626a962d775b1bc17b1e8cb29d44da39ff04731f3eeb1c0465b6c1d1bf96eb15 | What is the probability that the cards with numbers 11 and 19 will be adjacent when all prime numbers less than 24 are written on cards, mixed, and placed in a row? | [
"\\(\\frac{2}{9}\\)"
] | [
"What is the probability that the cards with numbers 11 and 19 will be adjacent when all prime numbers less than 24 are written on cards, mixed, and placed in a row?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-380919134188279c08e49c1da82e6990c53249c30e1a6ccf271fb20cc9d146de | What is the probability that the center of a sphere is contained within the tetrahedron formed by 4 randomly chosen points on the sphere? | [
"\\(\\frac{1}{8}\\)"
] | [
"What is the probability that the center of a sphere is contained within the tetrahedron formed by 4 randomly chosen points on the sphere?"
] | [
"aops_c6_high_school_olympiads",
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"cot",
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"cot": 16,
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omr-ws-82f44e90724f133d5e7e1be99e4d5e92776b952bd05ef129851e74b7d65db7be | What is the probability that the center of a sphere lies inside the tetrahedron formed by four randomly chosen points on the sphere's surface, assuming each point is independently chosen according to a uniform distribution? | [
"\\(\\frac{1}{8}\\)"
] | [
"What is the probability that the center of a sphere lies inside the tetrahedron formed by four randomly chosen points on the sphere's surface, assuming each point is independently chosen according to a uniform distribution?"
] | [
"aops_c7_college_math"
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"cot",
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"cot": 16,
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omr-ws-e418781e9da118e9de2d755bbe1ebd7a789f6884ac6f5fb5858e950d9f8f2317 | What is the probability that the center of a sphere lies inside the tetrahedron formed by four randomly chosen points on the sphere's surface? | [
"\\(\\frac{1}{8}\\)"
] | [
"What is the probability that the center of a sphere lies inside the tetrahedron formed by four randomly chosen points on the sphere's surface?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 22
} | 37 | 0 | 0 |
omr-ws-9ea77667424d2aa3a37fd6c89be71d64efdff99010083da62a29600be5c7461e | What is the probability that the cockroach will be at vertex $A$ after exactly $2023$ seconds? | [
"0"
] | [
"What is the probability that the cockroach will be at vertex $A$ after exactly $2023$ seconds?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
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omr-ws-5d434a85ef422637ce5dbf2b5310a2bc46ae2511893e8846c61bc0eb9f87c9f3 | What is the probability that the cockroach will be at vertex $A$ after exactly $2024$ seconds? | [
"\\(\\frac{1}{4}(1 + 3^{-1011})\\)"
] | [
"What is the probability that the cockroach will be at vertex $A$ after exactly $2024$ seconds?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
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"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-023685b42d53f11843a69cc4e3ab7138916a797f4d730c0e28f88f4173b92545 | What is the probability that the coin is facing the same way three times in a row (three heads in a row or three tails in a row) exactly once in 21 tosses? | [
"\\dfrac{59155}{1048576}"
] | [
"What is the probability that the coin is facing the same way three times in a row (three heads in a row or three tails in a row) exactly once in 21 tosses?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-51ec050cc5a6463c155e68c6d2a40447a2c6a374b1f6080c1a03d09a3cf39cce | What is the probability that the combined total number of bleats and neighs in one minute is exactly $10$, given that the sheep bleats $4$ times with probability $\frac{2}{3}$ and $2$ times with probability $\frac{1}{3}$, and the horse neighs $6$ times with probability $\frac{4}{5}$ and $3$ times with probability $\fra... | [
"\\(\\frac{8}{15}\\)"
] | [
"What is the probability that the combined total number of bleats and neighs in one minute is exactly $10$, given that the sheep bleats $4$ times with probability $\\frac{2}{3}$ and $2$ times with probability $\\frac{1}{3}$, and the horse neighs $6$ times with probability $\\frac{4}{5}$ and $3$ times with probabili... | [
"aops_c4_high_school_math"
] | [
"cot",
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] | {
"cot": 16,
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} | 40 | 0 | 0 |
omr-ws-62680c9cbc3ffddad7d22cb8f0e1430a8501e376e60a194e26fdda075285ca2d | What is the probability that the cross section of a cube, formed by a random plane, is a quadrilateral? | [
"\\(\\frac{2}{3}\\)"
] | [
"What is the probability that the cross section of a cube, formed by a random plane, is a quadrilateral?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
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"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-362d68b625ce47bd686034f0e7122c85f2b58b1802091f9072234e62b7ec0558 | What is the probability that the cross section of a cube, formed by a random plane, is a triangle? | [
"\\(\\frac{1}{3}\\)"
] | [
"What is the probability that the cross section of a cube, formed by a random plane, is a triangle?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-8d6ef55c0e1593f7754d637f9386150fc85258e802bb1a4b53a15194a78030bb | What is the probability that the dealer will draw a blackjack in a game of Blackjack using a standard 52-card deck? A blackjack is defined as drawing an ace and any card with a point value of 10 (i.e., a 10, jack, queen, or king). | [
"\\(\\frac{6}{221}\\)"
] | [
"What is the probability that the dealer will draw a blackjack in a game of Blackjack using a standard 52-card deck? A blackjack is defined as drawing an ace and any card with a point value of 10 (i.e., a 10, jack, queen, or king)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bc66cdf180ad18e3a987552dac66c52d585775942e6281e03c1e66e32f5a608b | What is the probability that the determinant of a random \( n \times n \) integer matrix is even? | [
"0.71121"
] | [
"What is the probability that the determinant of a random \\( n \\times n \\) integer matrix is even?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fd0d9fc3642860032bd8af22a7d1a053a3a07eb7d96b0fea85f150d44a350ffd | What is the probability that the difference between the greatest and the least of three randomly chosen numbers between 0 and 1 is $\frac{1}{3}$? | [
"0"
] | [
"What is the probability that the difference between the greatest and the least of three randomly chosen numbers between 0 and 1 is $\\frac{1}{3}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-a8f62ab6279d1f64130039c0b4dd2d462404ffdd583a7ccca0a7cf16996803f7 | What is the probability that the difference between the largest and smallest of three numbers chosen between 0 and 1 is less than 1/2? | [
"\\(\\frac{1}{2}\\)"
] | [
"What is the probability that the difference between the largest and smallest of three numbers chosen between 0 and 1 is less than 1/2?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
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} | 46 | 0 | 0 |
omr-ws-33f8bd43f44899c628b356577b533389482388e8ddc5f847e0e40ab8f44ec1d6 | What is the probability that the difference between two randomly selected numbered balls (without replacement) from a jar of 50 balls (numbered 1 to 50) is 10 or less? | [
"\\(\\frac{89}{245}\\)"
] | [
"What is the probability that the difference between two randomly selected numbered balls (without replacement) from a jar of 50 balls (numbered 1 to 50) is 10 or less?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-cd58c622320f705343b12d6fdeed581e0ca0c5bda2bb7e34fd00bdedb198d817 | What is the probability that the difference between two randomly selected numbers between $0$ and $1$ is less than $\frac{1}{3}$? | [
"\\(\\frac{5}{9}\\)"
] | [
"What is the probability that the difference between two randomly selected numbers between $0$ and $1$ is less than $\\frac{1}{3}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-f4771dece83ce82c2b72515497444b9925caacd4243de148f5c7dd3df6e4e9e6 | What is the probability that the distance between two randomly selected points in a unit square is less than $1$? | [
"\\(\\pi - \\frac{13}{6}\\)"
] | [
"What is the probability that the distance between two randomly selected points in a unit square is less than $1$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 6,
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} | 37 | 0 | 0 |
omr-ws-f1647a19bf1cff24740b26bbdc902bc35165dba071dd09bf776aea19d3242e2b | What is the probability that the distance between two randomly selected points inside a given circle is greater than the radius of the circle? | [
"\\(\\frac{3\\sqrt{3}}{4\\pi}\\)"
] | [
"What is the probability that the distance between two randomly selected points inside a given circle is greater than the radius of the circle?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 14
} | 28 | 0 | 0 |
omr-ws-0a00157c25135f7efe38770f8eefed658e782e0ea9ef9aba9cb9f66543ff95ea | What is the probability that the fifth roll of a standard 6-sided die will match one or more of the previous four rolls? | [
"\\(\\frac{671}{1296}\\)"
] | [
"What is the probability that the fifth roll of a standard 6-sided die will match one or more of the previous four rolls?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-70f4b52134658e195b3c6f2eafa22e801bd2faf90be18f36616268183bf1c99f | What is the probability that the first card drawn from a standard deck of 52 cards is a king and the second card is a heart? | [
"\\(\\frac{1}{52}\\)"
] | [
"What is the probability that the first card drawn from a standard deck of 52 cards is a king and the second card is a heart?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-1fc531c926e7207c6910a3b4a5418431d51cb2c6a40b4271342a127cdbfc607f | What is the probability that the first card drawn is a King and the second card drawn is a heart? | [
"\\(\\frac{1}{52}\\)"
] | [
"What is the probability that the first card drawn is a King and the second card drawn is a heart?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-a3ce3cfb8e8ef2ba450b5f0259d8d30c47ef5a6314bdeccb417af9e675d455e3 | What is the probability that the first die (with 3 dots on 3 sides and 4 dots on 3 sides) will roll a higher number than a normal die? | [
"\\dfrac{5}{12}"
] | [
"What is the probability that the first die (with 3 dots on 3 sides and 4 dots on 3 sides) will roll a higher number than a normal die?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-53fcc2eca9ac4b0ebe353189f8caa2fe51565f124016182aa160cae1ba0ede7c | What is the probability that the first digit of \(2^n\) is a 1? | [
"\\(\\log{2}\\)"
] | [
"What is the probability that the first digit of \\(2^n\\) is a 1?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 12
} | 12 | 0 | 0 |
omr-ws-c98674a42b52967729e5cc39d5f77c03b2d579fc85499aa0f208f1ded85046bf | What is the probability that the first player wins at tic-tac-toe if all moves are random? | [
"0.381976"
] | [
"What is the probability that the first player wins at tic-tac-toe if all moves are random?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 5
} | 6 | 0 | 0 |
omr-ws-0fd05c201d5ef3befc581d8e8d3973a7a4f5725a7b49bc8ff78ab52f9245eb1c | What is the probability that the first student to sit down ends up being the only person at their table, given that each of the $n$ students independently decides to sit at table $X$ with probability $p$ or table $Y$ with probability $1 - p$? | [
"\\( p \\cdot (1-p)^{n-1} + (1-p) \\cdot p^{n-1} \\)"
] | [
"What is the probability that the first student to sit down ends up being the only person at their table, given that each of the $n$ students independently decides to sit at table $X$ with probability $p$ or table $Y$ with probability $1 - p$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
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} | 36 | 0 | 0 |
omr-ws-a135de0d5d28e6ccb653f8c2aa1bedaae827ed99df9dc2f0f9de5e3a2b1c1a70 | What is the probability that the first three boxes in which a man doodles on a tic-tac-toe board correspond to a winning path? | [
"\\(\\frac{2}{21}\\)"
] | [
"What is the probability that the first three boxes in which a man doodles on a tic-tac-toe board correspond to a winning path?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-50a72d843f00edb64962a144427054da993593743b0fdf8dd4858445b96715bf | What is the probability that the first time heads is flipped twice in a row is on the seventh and eighth flips of a fair coin? | [
"\\(\\frac{17}{256}\\)"
] | [
"What is the probability that the first time heads is flipped twice in a row is on the seventh and eighth flips of a fair coin?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
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omr-ws-c2c81386a7f17692785a5704c7b3c26182dc896e24008ab1a1b4b0e9e261c9ae | What is the probability that the fourth coin pulled out is a nickel and the seventh coin pulled out is a quarter, given that there are 4 quarters, 7 nickels, and 2 pennies in a wallet and the coins are taken out one by one without replacement? | [
"\\(\\frac{7}{39}\\)"
] | [
"What is the probability that the fourth coin pulled out is a nickel and the seventh coin pulled out is a quarter, given that there are 4 quarters, 7 nickels, and 2 pennies in a wallet and the coins are taken out one by one without replacement?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-059502faa6698a470b0041a2399542cff354a06d5aca03209a2eea0698df03fc | What is the probability that the fourth head will appear on the sixth toss of a fair coin? | [
"\\dfrac{5}{32}"
] | [
"What is the probability that the fourth head will appear on the sixth toss of a fair coin?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0b76fc9711e10e07068169a4a73d478dac8492136264dcdfb4d8f57a396285bb | What is the probability that the fourth six appears on the tenth throw when a fair die is thrown 20 times? | [
"\\(\\frac{109375}{5038848}\\)"
] | [
"What is the probability that the fourth six appears on the tenth throw when a fair die is thrown 20 times?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
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] | {
"cot": 15,
"genselect": 7,
"tir": 28
} | 50 | 0 | 0 |
omr-ws-1588e22faca5934aea34b0b747d7500a5c569d80f6439b0a468557f9371357ed | What is the probability that the fourth six appears on the twentieth roll when a single fair die is rolled repeatedly? | [
"\\(\\binom{19}{3} \\left(\\frac{1}{6}\\right)^4 \\left(\\frac{5}{6}\\right)^{19}\\)"
] | [
"What is the probability that the fourth six appears on the twentieth roll when a single fair die is rolled repeatedly?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
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"cot": 8,
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} | 10 | 0 | 0 |
omr-ws-2c815729bee7123a5f681a67e8491787d9b7c8aad7a861271b3a24c91afef28f | What is the probability that the game goes into overtime given that Kobe has an 82% chance of making each free throw and a 50% chance of missing the second shot if he misses the first? | [
"0.2376"
] | [
"What is the probability that the game goes into overtime given that Kobe has an 82% chance of making each free throw and a 50% chance of missing the second shot if he misses the first?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
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] | {
"cot": 13,
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} | 24 | 0 | 0 |
omr-ws-564bb53f8168af6d2428bf0cb05ebeaa9a9fd21446f9191681043dd471b0d3a0 | What is the probability that the gestapo will find the Franks by correctly choosing through a series of doors? The sequence involves choosing the correct door out of 2, then out of 17, and finally out of 69. Express the probability as a common fraction. | [
"\\(\\frac{1}{2318}\\)"
] | [
"What is the probability that the gestapo will find the Franks by correctly choosing through a series of doors? The sequence involves choosing the correct door out of 2, then out of 17, and finally out of 69. Express the probability as a common fraction."
] | [
"aops_c4_high_school_math"
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"additional_problems": 1
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omr-ws-a8a6bd10d980551761a0e2d7bbada6e6c8f07f16488dcc2751a2544c45836853 | What is the probability that the greatest common divisor (G.C.D.) of two 1000-digit numbers is greater than a 700-digit number? | [
"\\dfrac{6}{\\pi^2 \\cdot 10^{699}}"
] | [
"What is the probability that the greatest common divisor (G.C.D.) of two 1000-digit numbers is greater than a 700-digit number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
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"cot": 5,
"genselect": 3
} | 8 | 0 | 0 |
omr-ws-9ec1a4c55d9bcbc9dba7f49e4c5a99be95fd16b5cabee2625936967d1bd35fe6 | What is the probability that the greatest common factor (GCF) of \( n \) and 30 is greater than 1, if \( n \) is a randomly picked positive integer? | [
"\\(\\frac{23}{30}\\)"
] | [
"What is the probability that the greatest common factor (GCF) of \\( n \\) and 30 is greater than 1, if \\( n \\) is a randomly picked positive integer?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
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"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6521b498dae044a5d801b31230bcf8969c4104d37342f0242988030c5b18d04e | What is the probability that the integer $3^a + 7^b$ has a units digit of 8, where $a$ and $b$ are chosen at random from the set $\{1, 2, 3, \ldots, 99, 100\}$? | [
"\\(\\frac{3}{16}\\)"
] | [
"What is the probability that the integer $3^a + 7^b$ has a units digit of 8, where $a$ and $b$ are chosen at random from the set $\\{1, 2, 3, \\ldots, 99, 100\\}$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
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"cot": 15,
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omr-ws-581e8d4fd319c2c8eca3c93b2651eb485619bdc2e828dea794c531812b04fb6c | What is the probability that the integer \(3^a + 7^b\) has a units digit of 8, where \(a\) and \(b\) are chosen at random from the set \(\{1, 2, 3, \ldots, 99, 100\}\)? | [
"\\(\\frac{3}{16}\\)"
] | [
"What is the probability that the integer \\(3^a + 7^b\\) has a units digit of 8, where \\(a\\) and \\(b\\) are chosen at random from the set \\(\\{1, 2, 3, \\ldots, 99, 100\\}\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
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"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-fbaf2edb88818505473d5e59d1e3144f88a31f4686f622fc3fb328106296254d | What is the probability that the island is inhabited by time 3 months, given that creatures arrive as a Poisson process with rate \( a \) per month and each creature produces offspring as a Poisson process with rate \( b \) per month? Assume there are no deaths and no creatures leave the island. | [
"1 - e^{-3a}"
] | [
"What is the probability that the island is inhabited by time 3 months, given that creatures arrive as a Poisson process with rate \\( a \\) per month and each creature produces offspring as a Poisson process with rate \\( b \\) per month? Assume there are no deaths and no creatures leave the island."
] | [
"aops_c7_college_math"
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"cot"
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"cot": 15
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omr-ws-17a30a6f389fabbd6a8e90911e5acaf891ce9fba318efc9d898f72b560d031e6 | What is the probability that the largest number rolled is $5$ when a fair die is rolled $5$ times? | [
"\\dfrac{2101}{7776}"
] | [
"What is the probability that the largest number rolled is $5$ when a fair die is rolled $5$ times?"
] | [
"aops_c7_college_math"
] | [
"cot"
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"cot": 16
} | 16 | 0 | 0 |
omr-ws-78a8fc09281773047f99518e88dce43d5b65b8e1e46a2401761b3c4a00bf2df6 | What is the probability that the largest outcome of three throws of a fair die is 3? | [
"\\(\\frac{19}{216}\\)"
] | [
"What is the probability that the largest outcome of three throws of a fair die is 3?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-96e429999b094bf93c00f09d17a1083b585ff05a33b8c491f229cefa08a1f6e2 | What is the probability that the largest outcome of three throws of a fair die is a 3? | [
"\\(\\frac{19}{216}\\)"
] | [
"What is the probability that the largest outcome of three throws of a fair die is a 3?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-657c5a24b3dcc4a053c429a13626a98011b0164b46438af34fa9e020852f359f | What is the probability that the last defective radio is found on the fourth test? | [
"\\(\\frac{1}{7}\\)"
] | [
"What is the probability that the last defective radio is found on the fourth test?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b988880012cab95b6b007adf82311f6bad944d72068ff0081aa63f17fae39afb | What is the probability that the last digit of the product of four randomly chosen positive integers is 1, 3, 7, or 9? | [
"\\(\\left(\\frac{2}{5}\\right)^4\\)"
] | [
"What is the probability that the last digit of the product of four randomly chosen positive integers is 1, 3, 7, or 9?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-36aea4f0c27ae28bcbad00e02f8b7af2bc34ce554c64e1e59bf338177c833595 | What is the probability that the last digit of the product of four randomly chosen positive integers is only 1, 3, 5, or 9? | [
"\\(\\frac{561}{10000}\\)"
] | [
"What is the probability that the last digit of the product of four randomly chosen positive integers is only 1, 3, 5, or 9?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-6423ba4bb8417a1715753901e1d42af314cee5f3aa9cd5fcf392b8803c3b61f1 | What is the probability that the leftmost part is the smallest when a line segment is divided into three parts by two randomly chosen points? | [
"\\(\\frac{1}{3}\\)"
] | [
"What is the probability that the leftmost part is the smallest when a line segment is divided into three parts by two randomly chosen points?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-e80a7ebedc7190098c438ff5b39c310fdbadb6f6f651015944789bfeec81ac43 | What is the probability that the length of a chord randomly drawn in a circle will be smaller than the circle's radius? | [
"\\dfrac{1}{3}"
] | [
"What is the probability that the length of a chord randomly drawn in a circle will be smaller than the circle's radius?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-f3c449cecd64b26662bfce9db744f46fbb04a6dfacfc4681a5149a12667902ce | What is the probability that the length of a randomly chosen chord of a circle lies between $\frac{2}{3}$ and $\frac{5}{6}$ of its diameter? | [
"\\dfrac{1}{4}"
] | [
"What is the probability that the length of a randomly chosen chord of a circle lies between $\\frac{2}{3}$ and $\\frac{5}{6}$ of its diameter?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 1
} | 16 | 0 | 0 |
omr-ws-4e379e24a099e614d1d597660ba8dee881984aaa661ce16be6cdbb4a6ca70624 | What is the probability that the length of a randomly chosen chord of a circle lies between \(\frac{2}{3}\) and \(\frac{5}{6}\) of its diameter? | [
"\\dfrac{1}{4}"
] | [
"What is the probability that the length of a randomly chosen chord of a circle lies between \\(\\frac{2}{3}\\) and \\(\\frac{5}{6}\\) of its diameter?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-a471115f38f9b1637b5043fd1f2e9034d19fc7b96040b3d9fc5900a7c43dbd75 | What is the probability that the letters USAMO appear in that exact order in a randomly chosen permutation of the letters USAMO2009? | [
"\\(\\frac{1}{120}\\)"
] | [
"What is the probability that the letters USAMO appear in that exact order in a randomly chosen permutation of the letters USAMO2009?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 29
} | 77 | 0 | 0 |
omr-ws-0aa609f967b7a5e8892a0247823e80650f6853ae8e0b51249509e97afbb6804d | What is the probability that the longer piece of a randomly cut string is at least x times as large as the shorter piece? | [
"\\(\\frac{2}{x+1}\\)"
] | [
"What is the probability that the longer piece of a randomly cut string is at least x times as large as the shorter piece?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 8
} | 10 | 0 | 0 |
omr-ws-7017dcfc59418372a3c329bedb5be10a898cc04bb082ca0bec535510f6b53538 | What is the probability that the minimum value of three red six-sided dice is greater than the maximum value of three yellow six-sided dice? | [
"\\(\\frac{2003}{46656}\\)"
] | [
"What is the probability that the minimum value of three red six-sided dice is greater than the maximum value of three yellow six-sided dice?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-2f3892aaa4ddb5e45252cbf5f8a546093cbe2a51fa2015e36cbf2aa13d913477 | What is the probability that the next eight flights of Pythag-Air-US airlines will be on time, given that the odds in favor of a flight being on time are 3:1? | [
"\\(\\left(\\frac{3}{4}\\right)^8\\)"
] | [
"What is the probability that the next eight flights of Pythag-Air-US airlines will be on time, given that the odds in favor of a flight being on time are 3:1?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-940df7ab3248ada5aebc5a61b57e5d2fa7ecf50402b10fcb9d9c45032f49055f | What is the probability that the notebook received by Alice has all its page numbers written in the correct order (i.e., $1,2,.....,N$)? | [
"\\dfrac{1}{N!}"
] | [
"What is the probability that the notebook received by Alice has all its page numbers written in the correct order (i.e., $1,2,.....,N$)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-0c7ed1d7b1050e1d961c0c30eef5ef4b9dbbdcd184b6402149292aa84b9c06b7 | What is the probability that the number $1,000,000$ occurs in the sequence of partial sums when a fair die is rolled consecutively and the results are added to the previous sum? | [
"\\(\\frac{2}{7}\\)"
] | [
"What is the probability that the number $1,000,000$ occurs in the sequence of partial sums when a fair die is rolled consecutively and the results are added to the previous sum?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 12
} | 27 | 0 | 0 |
omr-ws-10685aa1e50488360f6e16649b5f5fa50aa8ba7b3833280021143f617b2cef11 | What is the probability that the number Bob gets on each roll of five standard 6-sided dice is greater than or equal to the number he got on the previous roll? | [
"\\(\\frac{7}{216}\\)"
] | [
"What is the probability that the number Bob gets on each roll of five standard 6-sided dice is greater than or equal to the number he got on the previous roll?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-a102eafbe02b4cc68c0ee16fb4c742a744442a58caedb3620acb4b5bf6ee7740 | What is the probability that the number formed by randomly placing the digits {1, 3, 5, 7} in the spaces of 2_0_1_0_ is divisible by 7? | [
"1/24"
] | [
"What is the probability that the number formed by randomly placing the digits {1, 3, 5, 7} in the spaces of 2_0_1_0_ is divisible by 7?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-58bb8346e94f13534229f9929eb988ff51ee92b6cbd7376163ac6345ae595e6e | What is the probability that the number of heads from tossing 6 coins matches the number shown on a rolled dice? | [
"\\(\\frac{21}{128}\\)"
] | [
"What is the probability that the number of heads from tossing 6 coins matches the number shown on a rolled dice?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-6cceeb73f7dd40ae4fd5591d24dcd75e0f2f6ce5666607b7e2ef37f7fc9951c3 | What is the probability that the outcome of rolling two fair 6-sided dice can be formed into a two-digit number that is divisible by 7? | [
"\\(\\frac{1}{6}\\)"
] | [
"What is the probability that the outcome of rolling two fair 6-sided dice can be formed into a two-digit number that is divisible by 7?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 25
} | 25 | 0 | 0 |
omr-ws-fda28cdce7d27ae3d3760304da5525988f64d5e2cd01a5ce68ff338fa6b13feb | What is the probability that the outcomes of throwing a fair dice three times can form an isosceles or equilateral triangle? | [
"\\(\\frac{23}{72}\\)"
] | [
"What is the probability that the outcomes of throwing a fair dice three times can form an isosceles or equilateral triangle?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-bb241e3be5da7d891dba8b93b01ca89bdc5f77288bf245b9f2dd8f00f47a3500 | What is the probability that the player will win with a blackjack in a game of Blackjack using a standard 52-card deck? A blackjack is defined as drawing an ace and any card with a point value of 10 (i.e., a 10, jack, queen, or king). | [
"\\dfrac{32}{663}"
] | [
"What is the probability that the player will win with a blackjack in a game of Blackjack using a standard 52-card deck? A blackjack is defined as drawing an ace and any card with a point value of 10 (i.e., a 10, jack, queen, or king)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fc81262a2318454300b342aaf8831f8d0ff02c344f5b9f8106cc4f4c1388dfde | What is the probability that the positive difference between the numbers on a pair of fair six-sided dice is a factor of one of the numbers? | [
"\\(\\frac{4}{9}\\)"
] | [
"What is the probability that the positive difference between the numbers on a pair of fair six-sided dice is a factor of one of the numbers?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-c3c5493cbb691b6d511c689dfcbc8c333f0d6b6a188e158d980eb2675d7e9e28 | What is the probability that the product $a \times b \times c$ is a multiple of 24, where $a$, $b$, and $c$ are integers drawn from the set $S = \{1, 2, 3, \ldots, 2015, 2016\}$? | [
"\\dfrac{19}{54}"
] | [
"What is the probability that the product $a \\times b \\times c$ is a multiple of 24, where $a$, $b$, and $c$ are integers drawn from the set $S = \\{1, 2, 3, \\ldots, 2015, 2016\\}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-2a85f64dcc2cacf5c039c1dc9e44f966f19451ec7282903ff264cec071c92114 | What is the probability that the product \((x-y)(y-z)(z-u)(u-v) = 0\) when a die is thrown 5 times, resulting in the outcomes \(x, y, z, u, v\)? | [
"\\(\\frac{671}{1296}\\)"
] | [
"What is the probability that the product \\((x-y)(y-z)(z-u)(u-v) = 0\\) when a die is thrown 5 times, resulting in the outcomes \\(x, y, z, u, v\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-7c6b1dc866d0176e21b3c13597bbffbe66d82b3e9401afbfcaf2435a91ecf2ea | What is the probability that the product of $n$ randomly generated non-zero digits is divisible by 10? | [
"\\( 1 - \\left(\\frac{8}{9}\\right)^n - \\left(\\frac{5}{9}\\right)^n + \\left(\\frac{4}{9}\\right)^n \\)"
] | [
"What is the probability that the product of $n$ randomly generated non-zero digits is divisible by 10?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-9fc22ed12ca9e1d1aedb1d0afd7155404d41ed69269156298c2468f052ef08b3 | What is the probability that the product of 3 different natural numbers less than 25 is divisible by 4? | [
"\\frac{635}{924}"
] | [
"What is the probability that the product of 3 different natural numbers less than 25 is divisible by 4?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d6fe3a62e79fc2e413514992598f4ca3e77d920a54b368210292df45ffddb19b | What is the probability that the product of \( n \) whole numbers ends with a zero? Provide the general expression and compute the probability when \( n = 4 \). | [
"\\(\\frac{10^n - 8^n - 5^n + 4^n}{10^n}\\) for general \\( n \\), and \\( 0.5535 \\) for \\( n = 4 \\)."
] | [
"What is the probability that the product of \\( n \\) whole numbers ends with a zero? Provide the general expression and compute the probability when \\( n = 4 \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-7d3b0a5cac00eb524bd8f30122da42104118589a19ec2073f9874d9c6544d5c0 | What is the probability that the product of the numbers obtained from rolling 6 standard 6-sided dice is divisible by 4? | [
"\\(\\frac{59}{64}\\)"
] | [
"What is the probability that the product of the numbers obtained from rolling 6 standard 6-sided dice is divisible by 4?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-c31fdf3c85c59af985e4dbd89e9cfe92846c7bc4aa99f9a724045dc07d18b5a9 | What is the probability that the product of the numbers on four thrown dice equals 36? | [
"\\(\\frac{1}{27}\\)"
] | [
"What is the probability that the product of the numbers on four thrown dice equals 36?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 33
} | 49 | 0 | 0 |
omr-ws-5a53af47e3d134f0eada28054193fb6ef411c18564c7dd9bc186842ee7b42310 | What is the probability that the product of the numbers on the top faces is prime when twelve fair dice are rolled? | [
"\\(\\frac{1}{6^{10}}\\)",
"\\(\\left(\\frac{1}{6}\\right)^{10}\\)"
] | [
"What is the probability that the product of the numbers on the top faces is prime when twelve fair dice are rolled?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 58
} | 90 | 0 | 0 |
omr-ws-f3185ad8d76bfb633c9d20bfae591e0e107e94f9323c771eb843a4a77e06d6d5 | What is the probability that the product of the numbers on two 8-sided dice exceeds 36? | [
"\\(\\frac{5}{32}\\)"
] | [
"What is the probability that the product of the numbers on two 8-sided dice exceeds 36?"
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-10339ea261de45875ebb322ef7381585ec0a8f8b3f0f7e673c4b038dfb286f21 | What is the probability that the product of the numbers on two rolled dice is even, given that both numbers are even? | [
"1"
] | [
"What is the probability that the product of the numbers on two rolled dice is even, given that both numbers are even?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-623908c112a7f5c5629760911a3ac868bc60614f0b3345247c2264e062ba3700 | What is the probability that the product of the numbers on two rolled dice is less than or equal to 12? | [
"\\(\\frac{23}{36}\\)"
] | [
"What is the probability that the product of the numbers on two rolled dice is less than or equal to 12?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-958ec3c65ad27786e1a381fdd023529a71b5fbd45975f3b8ec1b2e74219d73ad | What is the probability that the product of the numbers rolled by Person A (using a fair eight-sided die) and Person B (using a fair six-sided die) is a multiple of 3? | [
"\\(\\frac{1}{2}\\)"
] | [
"What is the probability that the product of the numbers rolled by Person A (using a fair eight-sided die) and Person B (using a fair six-sided die) is a multiple of 3?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-263c8ef41479d1117307a3fb6764a2ccbb8bb6b59b3e16702a9ee4ef1879e4ec | What is the probability that the product of two distinct positive integers chosen from the set $\{1, 2, \ldots, 100\}$ can be expressed as the difference between two perfect squares? | [
"\\(\\frac{74}{99}\\)"
] | [
"What is the probability that the product of two distinct positive integers chosen from the set $\\{1, 2, \\ldots, 100\\}$ can be expressed as the difference between two perfect squares?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-c28c274e39746064e60451c43cb481f17d311eb1ede4571500cd430bddc9ea35 | What is the probability that the product of two independently chosen two-digit positive integers has a units digit of zero? | [
"\\(\\frac{9}{50}\\)"
] | [
"What is the probability that the product of two independently chosen two-digit positive integers has a units digit of zero?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b35872304a0eecac7668c46943060f4bbdc80b203171c3fe73950b581ecb43ef | What is the probability that the product of two randomly chosen integers is a multiple of 8? | [
"\\dfrac{5}{16}"
] | [
"What is the probability that the product of two randomly chosen integers is a multiple of 8?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2a046634699e742544bf829fcc8c030ff7cb97f43f1da47dbb81b0b81a2d7616 | What is the probability that the product of two randomly chosen integers is divisible by 12? | [
"\\(\\frac{5}{18}\\)"
] | [
"What is the probability that the product of two randomly chosen integers is divisible by 12?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 29
} | 50 | 0 | 0 |
omr-ws-1b3e888672e53572acf3481fd2e6b30625c6a721a3b8ae95c75d673b859cc41f | What is the probability that the product of two randomly selected two-digit whole numbers is greater than 100 and less than 200? Express your answer as a common fraction. | [
"\\(\\frac{4}{675}\\)"
] | [
"What is the probability that the product of two randomly selected two-digit whole numbers is greater than 100 and less than 200? Express your answer as a common fraction."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-b36ea5586792306c616cc17d5f211a11e42f184fbd50266a59735cb27f844cd1 | What is the probability that the quadratic equation $x^2 + 2bx + c = 0$ has real roots? | [
"\\dfrac{1}{3}"
] | [
"What is the probability that the quadratic equation $x^2 + 2bx + c = 0$ has real roots?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-096c952f7980f7a043c0085444481cdf356e209a030d74c9d16120032749afd0 | What is the probability that the quadratic equation \( x^{2} + 2bx + c = 0 \) has real roots? | [
"\\dfrac{2}{3}"
] | [
"What is the probability that the quadratic equation \\( x^{2} + 2bx + c = 0 \\) has real roots?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 3
} | 11 | 0 | 0 |
omr-ws-b703d86648d0eb9e3a217b8089713259ae5a9ab86fc8b1885c079242ad8f125d | What is the probability that the quotient of two randomly chosen integers between 1 and 100 is a terminating decimal? | [
"\\dfrac{437}{2000}"
] | [
"What is the probability that the quotient of two randomly chosen integers between 1 and 100 is a terminating decimal?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-ef1e75348c7b3aa962f45eb2b1f0e03f8cd9466f3f4fd9713862ac7a9e2a45fa | What is the probability that the results of rolling a standard cubical die four times, denoted as \(a, b, c,\) and \(d\) respectively, satisfy \(a \leq b \leq c \leq d\)? | [
"\\dfrac{7}{72}"
] | [
"What is the probability that the results of rolling a standard cubical die four times, denoted as \\(a, b, c,\\) and \\(d\\) respectively, satisfy \\(a \\leq b \\leq c \\leq d\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bd92986a077442a4a09148551498421d9bfc3ff9a22235f454b09331dcf3fd81 | What is the probability that the roots of the quadratic equation $ax^2 + bx + c = 0$ are real, given that $a, b, c$ are any real numbers between $0$ and $1$? | [
"\\(\\frac{5 + 6 \\ln 2}{36}\\)"
] | [
"What is the probability that the roots of the quadratic equation $ax^2 + bx + c = 0$ are real, given that $a, b, c$ are any real numbers between $0$ and $1$?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-6ae37eaffdcb56cffd6d5e17491056b435786fcfc9e52f61bbc7b54993a3b8b7 | What is the probability that the same number will be facing up on each of three standard six-sided dice that are tossed simultaneously? Express your answer as a common fraction. | [
"\\frac{1}{36}"
] | [
"What is the probability that the same number will be facing up on each of three standard six-sided dice that are tossed simultaneously? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-09e236f15ba6032a4f595ed91809502cc817c842edd3e1322d6ac889303389b1 | What is the probability that the same result (heads or tails) shows twice in a row exactly once in ten coin tosses? | [
"\\(\\dfrac{9}{2^9}\\)"
] | [
"What is the probability that the same result (heads or tails) shows twice in a row exactly once in ten coin tosses?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-7b09c8bf94f364b24933372d4f7d220706708a91150325bddd7595574cdc60d5 | What is the probability that the score of a randomly chosen IQ test taker, who has a score that is a normal random variable with mean 100 and standard deviation 15, is between 90 and 110? | [
"0.4972"
] | [
"What is the probability that the score of a randomly chosen IQ test taker, who has a score that is a normal random variable with mean 100 and standard deviation 15, is between 90 and 110?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 1,
"tir": 2
} | 6 | 0 | 0 |
omr-ws-71481e3d67ac127109b60bced8f1af746b5b4115dc4111aac4c746643485af01 | What is the probability that the second die (with 4 dots on 5 sides and 1 dot on 1 side) will roll a higher number than a normal die? | [
"\\dfrac{5}{12}"
] | [
"What is the probability that the second die (with 4 dots on 5 sides and 1 dot on 1 side) will roll a higher number than a normal die?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d82beafd7424a614026e2523f25819ab5a4fcd1dc1acc88095f9f21dff793fa7 | What is the probability that the second hand on a clock is closer to the minute hand than to the hour hand? | [
"\\(\\frac{1}{2}\\)"
] | [
"What is the probability that the second hand on a clock is closer to the minute hand than to the hour hand?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 17
} | 24 | 0 | 0 |
omr-ws-92937d210d031788bb69d0f71b7a2688318382e9315e2cc2b7d8d6b3f9e072ce | What is the probability that the second player wins at tic-tac-toe if all moves are random? | [
"0"
] | [
"What is the probability that the second player wins at tic-tac-toe if all moves are random?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a288dbce646febe3eca7e60ba5ea0b615c4a77dbed67cab75ccaad854fcc817b | What is the probability that the second time heads is flipped is on the eighth flip? | [
"\\(\\frac{7}{256}\\)"
] | [
"What is the probability that the second time heads is flipped is on the eighth flip?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-e0ac38bdbe338dbe234cd1d1c47bab804e9260736d4870c79fc3290052478549 | What is the probability that the sequence THT occurs before HHT when a coin is tossed repeatedly, with heads appearing with probability \( p \) on each toss? | [
"\\( (1 + p)(1 - p)^2 \\)"
] | [
"What is the probability that the sequence THT occurs before HHT when a coin is tossed repeatedly, with heads appearing with probability \\( p \\) on each toss?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-badcaca5d5adf6f982f14754e99347efc304eb756eb56a9d60921faa3c240029 | What is the probability that the series $\sum_{k=1}^{\infty}\frac{i_{k}}{\sqrt{k}}$ converges, given that $i_{k} \in \{-1, 1\}$ and each $i_k$ is chosen randomly for every $k$? | [
"0"
] | [
"What is the probability that the series $\\sum_{k=1}^{\\infty}\\frac{i_{k}}{\\sqrt{k}}$ converges, given that $i_{k} \\in \\{-1, 1\\}$ and each $i_k$ is chosen randomly for every $k$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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