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-d81a06f6316a36d0358e177b4525872c566c371b96d4f732d2b36b812510a7b3 | A fair 6-sided die is rolled once. If I roll $n$, then I win $6-n$ dollars. What is the expected value of my win, in dollars? | [
"2.50"
] | [
"A fair 6-sided die is rolled once. If I roll $n$, then I win $6-n$ dollars. What is the expected value of my win, in dollars?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8b96148879bfbe19cd6f02332e16c1a30862364bb1f89554aaca6f52ddef3239 | A fair 6-sided die is rolled repeatedly, forming a sequence of values under the following rules: If any even value is rolled, add it to the current sequence. If a 3 or 5 is rolled, discard the entire sequence and don't add the 3 or 5 to the start of the new sequence. If a 1 is rolled, add the 1 to the current sequence ... | [
"\\(\\frac{4}{3}\\)"
] | [
"A fair 6-sided die is rolled repeatedly, forming a sequence of values under the following rules: If any even value is rolled, add it to the current sequence. If a 3 or 5 is rolled, discard the entire sequence and don't add the 3 or 5 to the start of the new sequence. If a 1 is rolled, add the 1 to the current sequ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b2510cae6355ba8410c83d124545e4e004881c3f473957b87529348a0205f80c | A fair 6-sided die is rolled six times. If the expected value of the smallest number rolled is \( x \), determine the value of \( 1000x \). | [
"1439.707"
] | [
"A fair 6-sided die is rolled six times. If the expected value of the smallest number rolled is \\( x \\), determine the value of \\( 1000x \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 8,
"tir": 6
} | 26 | 0 | 0 |
omr-ws-2497e2c468c40b2dbd42373f459349080856b60849fa3bb59200b9f04360fbce | A fair 6-sided die is rolled. If I roll $n$, then I win $n^2$ dollars. What is the expected value of my win? Express your answer as a dollar value rounded to the nearest cent. | [
"\\$15.17"
] | [
"A fair 6-sided die is rolled. If I roll $n$, then I win $n^2$ dollars. What is the expected value of my win? Express your answer as a dollar value rounded to the nearest cent."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-97b9b88da98a4832cb8153250bae4a553e5a39300a76aa84c0aa4d15ceebe8b3 | A fair 6-sided die is rolled. If the roll is even, then you win that amount of dollars (so that, for example, if you roll 4, then you win $\$4$). If the roll is odd, you win nothing. What is the expected value of your winnings? Express your answer as a dollar value. | [
"\\$2"
] | [
"A fair 6-sided die is rolled. If the roll is even, then you win that amount of dollars (so that, for example, if you roll 4, then you win $\\$4$). If the roll is odd, you win nothing. What is the expected value of your winnings? Express your answer as a dollar value."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c37eecbe33d264f9eafb096bc838225b20d744a8656f7f2f703b69123c244b79 | A fair 6-sided die is rolled. What is the probability that the number rolled is a divisor of 6? | [
"\\dfrac23"
] | [
"A fair 6-sided die is rolled. What is the probability that the number rolled is a divisor of 6?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-374bfd26d104aaf00602de796989260b4c43f6795dc186cdde72148bc4c37f5d | A fair 6-sided die is thrown 3 times. If the sum of the numbers obtained is less than or equal to 10 and no repetition, then we get 2 marks. Otherwise, we get 0 marks. Find the expected value of the marks. | [
"\\(\\frac{5}{9}\\)"
] | [
"A fair 6-sided die is thrown 3 times. If the sum of the numbers obtained is less than or equal to 10 and no repetition, then we get 2 marks. Otherwise, we get 0 marks. Find the expected value of the marks."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-3951366ec881c9bbad1289c12efce0da680a16523cbf9d51bef28e830bec33e2 | A fair 6-sided die is tossed 8 times. The sequence of results forms an 8-digit number. What is the probability that the number formed is a multiple of 8? | [
"$\\frac{1}{8}$"
] | [
"A fair 6-sided die is tossed 8 times. The sequence of results forms an 8-digit number. What is the probability that the number formed is a multiple of 8?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 32
} | 47 | 0 | 0 |
omr-ws-04b6d246c860f583831d554c57bca9967b08a7d86504e4cdc748cc88f63c4137 | A fair coin is flipped $7$ times. What is the probability that at least $5$ consecutive flips come up heads? | [
"\\frac{1}{16}"
] | [
"A fair coin is flipped $7$ times. What is the probability that at least $5$ consecutive flips come up heads?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f9fd8f270a37606351880428d5bec6c81b84a672d0036bbf3850028db5775320 | A fair coin is flipped $n$ times in a row. Let $p_i$ denote the number of ways in which $i$ of the $n$ coin flips land on heads. Find the number of composite values of $n < 1000$ such that the greatest common factor of all $p_i$ for $0 < i < n$ is greater than $1$. | [
"25"
] | [
"A fair coin is flipped $n$ times in a row. Let $p_i$ denote the number of ways in which $i$ of the $n$ coin flips land on heads. Find the number of composite values of $n < 1000$ such that the greatest common factor of all $p_i$ for $0 < i < n$ is greater than $1$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-aa84d6b216664bacc1597d6aba80f4aed28be83e3cc024e2d7ff8c35f33f0313 | A fair coin is flipped $n$ times. What is the expected length of the longest streak of heads that is achieved? | [
"\\( 2 - \\frac{n+2}{2^n} \\)"
] | [
"A fair coin is flipped $n$ times. What is the expected length of the longest streak of heads that is achieved?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e7408167803038422e0f4b6bb6e82b036e302736c44e5320ddf7c97438a51589 | A fair coin is flipped 10 times. Let \( \frac{i}{j} \), in lowest terms, be the probability that heads never occur on consecutive tosses. Find \( i+j \). | [
"73"
] | [
"A fair coin is flipped 10 times. Let \\( \\frac{i}{j} \\), in lowest terms, be the probability that heads never occur on consecutive tosses. Find \\( i+j \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8a12a20338f0e8ff310e7b0327bdee73f8e2b3f07ae1fa5b457f92b6b24b6a51 | A fair coin is flipped 3 times, and the result is written on a slip of paper and thrown into a hat. (a) Find the probability that all 3 tosses were heads, given that at least two of them were heads. (b) Two of the slips of paper are drawn at random and both say heads. Given this, find the probability that all 3 tosses ... | [
"\\dfrac{1}{2}"
] | [
"A fair coin is flipped 3 times, and the result is written on a slip of paper and thrown into a hat.\n(a) Find the probability that all 3 tosses were heads, given that at least two of them were heads.\n(b) Two of the slips of paper are drawn at random and both say heads. Given this, find the probability that all 3 ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8855330e99b9867a581077ee8a45bf53bb46451d7d4b3864750f61b9ad4e230e | A fair coin is flipped 3 times, and the result is written on a slip of paper and thrown into a hat. (a) Find the probability that all 3 tosses were heads, given that at least two of them were heads. (b) Two of the slips of paper are drawn at random and both say heads. Given this, find the probability that all three tos... | [
"(a) \\(\\frac{1}{2}\\), (b) \\(\\frac{2}{5}\\)"
] | [
"A fair coin is flipped 3 times, and the result is written on a slip of paper and thrown into a hat.\n(a) Find the probability that all 3 tosses were heads, given that at least two of them were heads.\n(b) Two of the slips of paper are drawn at random and both say heads. Given this, find the probability that all th... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5c4e46840cf8b4b6edc5f06203a9075f80dadb7318a5730a927f8d93021bb550 | A fair coin is flipped 7 times. What is the probability that at least 5 of the flips come up heads? | [
"\\frac{29}{128}"
] | [
"A fair coin is flipped 7 times. What is the probability that at least 5 of the flips come up heads?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f4c70e18d3cf6b1046b4002eab6a6379b7a33e0816118c3600675dee14bd59f4 | A fair coin is flipped \( n \) times where \( n \geq 2 \). Let \( X \) be the number of isolated heads and \( Y \) be the number of runs. Calculate the value of \( \text{Cov}(X, Y) \). | [
"\\dfrac{n}{8}"
] | [
"A fair coin is flipped \\( n \\) times where \\( n \\geq 2 \\). Let \\( X \\) be the number of isolated heads and \\( Y \\) be the number of runs. Calculate the value of \\( \\text{Cov}(X, Y) \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-3d7c546aeef2287d74d24053872b41cdcd7f16347462034a68c48ac4a43cacfe | A fair coin is flipped repeatedly until it turns up tails five times. What is the expected number of heads before that happens? | [
"5"
] | [
"A fair coin is flipped repeatedly until it turns up tails five times. What is the expected number of heads before that happens?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-419a4c36f48ea60dcae3c744ccb817643b6f0ab660a5b7a1cf33ff3e134d929b | A fair coin is flipped repeatedly. Compute the probability that two tails are obtained before three heads are obtained. | [
"\\(\\frac{11}{16}\\)"
] | [
"A fair coin is flipped repeatedly. Compute the probability that two tails are obtained before three heads are obtained."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-aff12cfd407f026d8d5695493c7585a0424999cae1a6bac542ccd7366a5286ba | A fair coin is repeatedly flipped until 2019 consecutive coin flips are the same. Compute the probability that the first and last flips of the coin come up differently. | [
"\\(\\frac{2^{2018}-1}{2^{2019}-1}\\)",
"\\dfrac{2^{2018} - 1}{2^{2019} - 1}"
] | [
"A fair coin is repeatedly flipped until 2019 consecutive coin flips are the same. Compute the probability that the first and last flips of the coin come up differently."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 28,
"tir": 14
} | 42 | 0 | 0 |
omr-ws-01402b4c13b7d3ff4985d6a0e4289a185bc3806a9a90fa63fc8faf41024b38b4 | A fair coin is repeatedly tossed. We receive one marker for every "head" and two markers for every "tail". We win the game if, at some moment, we possess exactly 100 markers. Is the probability of winning the game greater than, equal to, or less than $2/3$? | [
"\\text{Greater than } \\dfrac{2}{3}"
] | [
"A fair coin is repeatedly tossed. We receive one marker for every \"head\" and two markers for every \"tail\". We win the game if, at some moment, we possess exactly 100 markers. Is the probability of winning the game greater than, equal to, or less than $2/3$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 5
} | 19 | 0 | 0 |
omr-ws-99c651a7a7199d06f8c0f7e55c14769db4a9d3ee88058428c6553843c530e846 | A fair coin is to be tossed 10 times. Let \( \frac{i}{j} \) in lowest terms be the probability that heads never occur on consecutive tosses. Find \( i + j \). | [
"73"
] | [
"A fair coin is to be tossed 10 times. Let \\( \\frac{i}{j} \\) in lowest terms be the probability that heads never occur on consecutive tosses. Find \\( i + j \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-ae16c7b4b940e7d6caa20f565d6472fc52acb33d17c91e39e7fea5b74bf5b394 | A fair coin is to be tossed 10 times. Let \( \frac{i}{j} \), in lowest terms, be the probability that heads never occur on consecutive tosses. Find \( i+j \). | [
"73"
] | [
"A fair coin is to be tossed 10 times. Let \\( \\frac{i}{j} \\), in lowest terms, be the probability that heads never occur on consecutive tosses. Find \\( i+j \\)."
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 9
} | 40 | 0 | 0 |
omr-ws-04db63b94d4a99108412676188c5d6a3c07c2e5e1de796ff8674f1a8d264c3cd | A fair coin is to be tossed ten times. Let \( \frac{i}{j} \), in lowest terms, be the probability that heads never occur on consecutive tosses. Find \( i+j \). | [
"793"
] | [
"A fair coin is to be tossed ten times. Let \\( \\frac{i}{j} \\), in lowest terms, be the probability that heads never occur on consecutive tosses. Find \\( i+j \\)."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d8a7cdaedeb245d8601170172beb8ecb40cf2118a789f3a9401efaec79f6e5cc | A fair coin is tossed $10$ times. Find the probability that two tails do not appear in succession. | [
"\\(\\frac{9}{128}\\)"
] | [
"A fair coin is tossed $10$ times. Find the probability that two tails do not appear in succession."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-170d9c713c810beda3ad869d38d643491fa6f692f045cd6943ea87698100eefe | A fair coin is tossed $9$ times. What is the probability that no two consecutive heads appear? | [
"$\\frac{89}{512}$"
] | [
"A fair coin is tossed $9$ times. What is the probability that no two consecutive heads appear?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-2fa37e22558417ef94da5123dd9b63f009ee3ab7455c3113ceade6d106ccf118 | A fair coin is tossed 10 times. Compute the probability that heads does not occur on consecutive tosses. | [
"\\dfrac{9}{64}"
] | [
"A fair coin is tossed 10 times. Compute the probability that heads does not occur on consecutive tosses."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ba4d07a60871c6eee6eb30a3172fafee6454e5804772e396d448591510ff8567 | A fair coin is tossed 10 times. What is the probability that it comes up heads 5 times in a row, but not 6 times in a row? | [
"\\dfrac{1}{16}"
] | [
"A fair coin is tossed 10 times. What is the probability that it comes up heads 5 times in a row, but not 6 times in a row?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4f8c2ae7e752ddb86cec9baf4ca4ebd0edc977b0945288a981461ad271d2ab7d | A fair coin is tossed 1997 times. Find the probability that at no point during the tossing are two heads flipped consecutively. | [
"\\(\\frac{F_{1997}}{2^{1997}}\\)"
] | [
"A fair coin is tossed 1997 times. Find the probability that at no point during the tossing are two heads flipped consecutively."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-181ee2ba6d2606653a4fa90913da6b7a04c6895f47d75ab321c1d8173bc51ce3 | A fair coin is tossed 99 times. If \( x \) is the number of times heads occur, for what value of \( r \) is \( P(x=r) \) maximized? | [
"\\( r = 49 \\) or \\( r = 50 \\)"
] | [
"A fair coin is tossed 99 times. If \\( x \\) is the number of times heads occur, for what value of \\( r \\) is \\( P(x=r) \\) maximized?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 6,
"tir": 6
} | 12 | 0 | 0 |
omr-ws-a455d8ba533480b8dfcbd90c7bd319c67f4a5bf2cc91ae9027f5111fb0933320 | A fair coin is tossed \( n \) times. What is the expected length of the longest chain of flipping the same side face up? | [
"\\(\\lfloor0.5+\\log_{2}{2n}\\rfloor\\)"
] | [
"A fair coin is tossed \\( n \\) times. What is the expected length of the longest chain of flipping the same side face up?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-304249bd9ba8910f46a79337ac19fc51a6885f1f371dfdff1235009967e8809c | A fair coin is tossed \( n \) times. What is the expected length of the longest chain of heads? | [
"\\(0.5 + \\log_{2}{n}\\)"
] | [
"A fair coin is tossed \\( n \\) times. What is the expected length of the longest chain of heads?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-41c07a69dc5a98cc317a54382c5c217ab32d90c32cf35e95016692790d626efb | A fair coin is tossed \( n+1 \) times. For \( 1 \leq i \leq n \), let \( A(i) \) be 1 if the \( i \)-th and the \( (i+1) \)-st outcomes are both heads, and 0 otherwise. Find the mean and variance of \( A(i) \). | [
"Mean = \\(\\frac{1}{4}\\), Variance = \\(\\frac{3}{16}\\)"
] | [
"A fair coin is tossed \\( n+1 \\) times. For \\( 1 \\leq i \\leq n \\), let \\( A(i) \\) be 1 if the \\( i \\)-th and the \\( (i+1) \\)-st outcomes are both heads, and 0 otherwise. Find the mean and variance of \\( A(i) \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 28
} | 32 | 0 | 0 |
omr-ws-60d36840be825a12545b13e8425c9a0df9114f432d49a18c4a12b3ef85e7d434 | A fair coin is tossed eight times. Find the probability that heads comes up more often than tails. | [
"93/256"
] | [
"A fair coin is tossed eight times. Find the probability that heads comes up more often than tails."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-251a18dfb5ecde194a4562d54af296428b72081fd6980feeb31be12a2b732ea5 | A fair coin is tossed once. If it turns up heads, a balanced die is rolled twice, and the sum of the number of dots on the faces that show up is recorded. If the coin turns up tails, the same die is rolled once, and the number of dots on the face that shows up is recorded. What is the probability that the recorded numb... | [
"\\(\\frac{4}{9}\\)"
] | [
"A fair coin is tossed once. If it turns up heads, a balanced die is rolled twice, and the sum of the number of dots on the faces that show up is recorded. If the coin turns up tails, the same die is rolled once, and the number of dots on the face that shows up is recorded. What is the probability that the recorded... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-aa7422bf2cd7749ec935da24d00ff6dc4f2731406da40a5ad65ed3602591e38f | A fair coin is tossed six times and the sequence of heads and tails is recorded. What is the probability that the sequence contains exactly two heads? Express your answer as a common fraction. | [
"\\frac{15}{64}"
] | [
"A fair coin is tossed six times and the sequence of heads and tails is recorded. What is the probability that the sequence contains exactly two heads? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-de28a5106ef41376ede0782f3ee720d8d373365dd0ef12fe19a7c45cadec3ada | A fair coin is tossed until either two consecutive heads or two consecutive tails are obtained. What is the expected number of coin tosses required? | [
"3"
] | [
"A fair coin is tossed until either two consecutive heads or two consecutive tails are obtained. What is the expected number of coin tosses required?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e678903c67bb0985082029c9f42c6ef8cda1c5bac1c4c99e6d1110252542296a | A fair coin is tossed until the fourth tail appears. What is the probability that the fourth tail is preceded by at least 3 heads? | [
"\\(\\frac{21}{32}\\)"
] | [
"A fair coin is tossed until the fourth tail appears. What is the probability that the fourth tail is preceded by at least 3 heads?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-11927e12d3b98800527a9dc8b559149e5d2c6d3727a442508723382dd901d9e1 | A fair coin is tossed until the fourth tail appears. What is the probability that the fourth tail is preceded by exactly 3 heads? | [
"\\(\\frac{5}{16}\\)"
] | [
"A fair coin is tossed until the fourth tail appears. What is the probability that the fourth tail is preceded by exactly 3 heads?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1e0d9d1e5482ee8f3975a2af6f21e890a30e29fd7b39f10cd9223d158179cdf7 | A fair dice is repeatedly rolled until the total exceeds 765. Find the probability that this will require more than 225 rolls. | [
"0.3050"
] | [
"A fair dice is repeatedly rolled until the total exceeds 765. Find the probability that this will require more than 225 rolls."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c8fc320d59f7e2bfa3c732d493110b3e959be4ecf161a5a29c0db7b9cb8a5bf5 | A fair dice is rolled 4 times. Compute the probability that each of the final 3 rolls is at least as large as the roll before it. | [
"\\(\\frac{126}{6^4}\\)"
] | [
"A fair dice is rolled 4 times. Compute the probability that each of the final 3 rolls is at least as large as the roll before it."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-5db4b1133ace50be155cbe594d6e345fced6a4a82edff1b13a12a7df340c417b | A fair dice is thrown 4 times. Let $p$ and $q$ be relatively prime positive integers such that $\frac{p}{q}$ is the probability that no number appears more than twice. Find the value of $p+q$. | [
"71"
] | [
"A fair dice is thrown 4 times. Let $p$ and $q$ be relatively prime positive integers such that $\\frac{p}{q}$ is the probability that no number appears more than twice. Find the value of $p+q$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-63d7e0d8ee19987302123d4f9f23a1a48c2b15256251151a4d904b676a0b0e27 | A fair dice numbered 1-6 is rolled 4 times. Find the probability that two of the rolls are prime numbers, one is a composite number, and another is 1. The probability can be written as $\frac{P}{Q}$ with gcd$(P,Q)=1$. What is $Q-P$? | [
"5"
] | [
"A fair dice numbered 1-6 is rolled 4 times. Find the probability that two of the rolls are prime numbers, one is a composite number, and another is 1. The probability can be written as $\\frac{P}{Q}$ with gcd$(P,Q)=1$. What is $Q-P$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f74b842b28584078d7aee96a3d350d9488451b0d824410e063f3609f5e7d67b8 | A fair die is rolled repeatedly. Let $X$ be the number of rolls needed to get a 5 and $Y$ be the number of rolls needed for a 6. Find $E(X|Y=2)$. | [
"6.6"
] | [
"A fair die is rolled repeatedly. Let $X$ be the number of rolls needed to get a 5 and $Y$ be the number of rolls needed for a 6. Find $E(X|Y=2)$."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-11fa1e9ad43028805a5120301418accb37be845577e9fb01829a46c797c2467a | A fair die is thrown three times, resulting in outcomes \( x \), \( y \), and \( z \). What is the probability that \( w^x + w^y + w^z = 0 \), where \( w \) is a cube root of unity and \( w \neq 1 \)? | [
"\\(\\frac{2}{9}\\)"
] | [
"A fair die is thrown three times, resulting in outcomes \\( x \\), \\( y \\), and \\( z \\). What is the probability that \\( w^x + w^y + w^z = 0 \\), where \\( w \\) is a cube root of unity and \\( w \\neq 1 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-a040c3fbf38252f9f3496971812acf0ae26dd1208d3383f51fe79d772c4d1dd1 | A fair die is thrown three times, with the results recorded as $x$, $y$, and $z$ respectively. Given that $x + y = z$, what is the probability that at least one of $x$, $y$, and $z$ is 2? | [
"\\(\\frac{8}{15}\\)"
] | [
"A fair die is thrown three times, with the results recorded as $x$, $y$, and $z$ respectively. Given that $x + y = z$, what is the probability that at least one of $x$, $y$, and $z$ is 2?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 32
} | 47 | 0 | 0 |
omr-ws-fc7326884f4bd7b40855a80a170a69fcfdf96a09d535581b721c0dc0bbf606c6 | A fair die is tossed 100 times. What is the most likely number of 6's we would obtain? | [
"17"
] | [
"A fair die is tossed 100 times. What is the most likely number of 6's we would obtain?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 1
} | 2 | 0 | 0 |
omr-ws-d711b5cff6cdd63abbef815cf43fb5912e4811f03711f0241876e2a20896c088 | A fair die is tossed 7 times. Suppose that on tosses 2 through 7, the value seen is different from the value seen on the preceding toss. Let \( F \) denote the sum of the 7 values seen. What is the probability that \( F \) is even? | [
"\\(\\frac{1}{2}\\)"
] | [
"A fair die is tossed 7 times. Suppose that on tosses 2 through 7, the value seen is different from the value seen on the preceding toss. Let \\( F \\) denote the sum of the 7 values seen. What is the probability that \\( F \\) is even?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-f0f4b8fda58f4540a1fa22650f459d769f728d6995e8b85b9dde4605e114c65a | A fair die is tossed. What are the outcomes and the probability that a number less than or equal to 3 appears? | [
"1, 2, 3, \\(\\frac{1}{2}\\)"
] | [
"A fair die is tossed. What are the outcomes and the probability that a number less than or equal to 3 appears?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 27
} | 27 | 0 | 0 |
omr-ws-d105d06d4d86e2a693012d9956dd473c009dbf0ad2208f6423e25389b71b21a6 | A fair die is tossed. What are the outcomes and the probability that an odd number is tossed? | [
"1, 3, 5, \\(\\frac{1}{2}\\)"
] | [
"A fair die is tossed. What are the outcomes and the probability that an odd number is tossed?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 8,
"tir": 21
} | 30 | 0 | 0 |
omr-ws-fb749a404d75eb1d54637ea660e3be9cc05b6703950781d72a9197c8312b7b92 | A fair die with 12 sides numbered 1 through 12 inclusive is rolled \( n \) times. The probability that the sum of the rolls is 2012 is nonzero and is equivalent to the probability that a sum of \( k \) is rolled. Find the minimum value of \( k \). | [
"172"
] | [
"A fair die with 12 sides numbered 1 through 12 inclusive is rolled \\( n \\) times. The probability that the sum of the rolls is 2012 is nonzero and is equivalent to the probability that a sum of \\( k \\) is rolled. Find the minimum value of \\( k \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-f5db56cb9d9fa5e7d0317a6a13c5a855763d3215ab620e57b5a443f61c7ebe93 | A fair six-sided die has four faces labelled $A$ and two faces labelled $B$. The die is rolled repeatedly. What is the probability that the sequence $ABA$ is rolled before the sequence $BBA$? | [
"\\(\\frac{16}{27}\\)"
] | [
"A fair six-sided die has four faces labelled $A$ and two faces labelled $B$. The die is rolled repeatedly. What is the probability that the sequence $ABA$ is rolled before the sequence $BBA$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 3
} | 17 | 0 | 0 |
omr-ws-14de7fa30bbd96a25f12cab22a1de37e52aeeb1e83a574061e98986a1c54145c | A fair six-sided die is labeled 1-6 such that opposite faces sum to 7. The die is rolled, but before you can look at the outcome, the die gets tipped over to an adjacent face. If the new face shows a 4, what is the probability the original roll was a 1? | [
"\\(\\frac{1}{4}\\)"
] | [
"A fair six-sided die is labeled 1-6 such that opposite faces sum to 7. The die is rolled, but before you can look at the outcome, the die gets tipped over to an adjacent face. If the new face shows a 4, what is the probability the original roll was a 1?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-946ff13a5f6673f82f73e42b7904df1140b5ae62410f414f07680cb405d8361f | A fair six-sided die is repeatedly tossed until an odd number appears. What is the probability that each even number (2, 4, 6) appears at least once before the first appearance of an odd number? | [
"\\(\\frac{1}{20}\\)"
] | [
"A fair six-sided die is repeatedly tossed until an odd number appears. What is the probability that each even number (2, 4, 6) appears at least once before the first appearance of an odd number?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 25
} | 72 | 0 | 0 |
omr-ws-05568ff2d48883f0a820f2b5e79e518eca3d768bd7614630f16dfaa97bdaad6f | A fair six-sided die is rolled repeatedly until a 1 appears for the first time. Find the probability that the number of times the die is rolled is an even number. | [
"\\(\\frac{5}{11}\\)"
] | [
"A fair six-sided die is rolled repeatedly until a 1 appears for the first time. Find the probability that the number of times the die is rolled is an even number."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-33697169f5998b93a10b8caec6190187b7e32e8fc648ee7c31cd5305218bda9a | A fair six-sided die is rolled until a sum greater than or equal to 700 is achieved. What is the expected number of rolls needed? | [
"\\dfrac{2805}{14}"
] | [
"A fair six-sided die is rolled until a sum greater than or equal to 700 is achieved. What is the expected number of rolls needed?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-b5034427eba5ffefcaaeca75fd38dbd219e6833889462377a91ddec0cba61d83 | A fair six-sided die is rolled until the sum of the rolls is greater than or equal to 700. What is the expected number of rolls needed? | [
"\\dfrac{4210}{21}"
] | [
"A fair six-sided die is rolled until the sum of the rolls is greater than or equal to 700. What is the expected number of rolls needed?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-6154003cb23d7ebff5af9fe512020ae2152ace992a8b91aa2a8f6a81685a2a95 | A fair six-sided die is tossed three times. Given that the sum of the first two tosses equals the third, what is the probability that at least one "2" is tossed? | [
"\\(\\frac{8}{15}\\)"
] | [
"A fair six-sided die is tossed three times. Given that the sum of the first two tosses equals the third, what is the probability that at least one \"2\" is tossed?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-61e1176e9d899be8d8cad967151a3eebd33ca020a535f315402179cedaa31e36 | A fair six-sided number cube with the digits 1-6 on its faces is rolled three times. The positive difference between the values on the first two rolls is equal to the value of the third roll. What is the probability that at least one 3 was rolled? Express your answer as a common fraction. | [
"7/15"
] | [
"A fair six-sided number cube with the digits 1-6 on its faces is rolled three times. The positive difference between the values on the first two rolls is equal to the value of the third roll. What is the probability that at least one 3 was rolled? Express your answer as a common fraction."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 33
} | 48 | 0 | 0 |
omr-ws-3d5c09a4ca5e40a2d50753b67aeb107cde2cc25662398395849f394006361987 | A fair standard die is rolled 6 times. For each roll \( r_i \) with \( 1 \le i \le 6 \), a complex number of magnitude \( r_i \) is randomly chosen as a point on the complex plane. What is the probability that the hexagon formed by connecting the six, not necessarily distinct or non-collinear, points in order will be c... | [
"\\dfrac{1}{32}"
] | [
"A fair standard die is rolled 6 times. For each roll \\( r_i \\) with \\( 1 \\le i \\le 6 \\), a complex number of magnitude \\( r_i \\) is randomly chosen as a point on the complex plane. What is the probability that the hexagon formed by connecting the six, not necessarily distinct or non-collinear, points in or... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 6
} | 15 | 0 | 0 |
omr-ws-a99067386e557493804d7a9862545a82b37dc33261bbb120c8f89cdbfe7f9e4a | A fair standard six-sided die is tossed three times. Given that the sum of the first two tosses equals the third, what is the probability that at least one 2 is tossed? | [
"$\\frac{8}{15}$",
"\\(\\frac{8}{15}\\)"
] | [
"A fair standard six-sided die is tossed three times. Given that the sum of the first two tosses equals the third, what is the probability that at least one 2 is tossed?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 62
} | 93 | 0 | 0 |
omr-ws-dce6a7ead38c06ba38adc92a57af7903fb58e0e56f4f4d2d7794b1a08341eae6 | A fair, standard six-sided die is rolled until a 6 is rolled for the first time. Let A be the expected value of the number of distinct digits that are rolled (including the 6). Let B be the expected value of the sum of all rolled numbers. Find \(60 \times A \times B\). | [
"4410"
] | [
"A fair, standard six-sided die is rolled until a 6 is rolled for the first time. Let A be the expected value of the number of distinct digits that are rolled (including the 6). Let B be the expected value of the sum of all rolled numbers. Find \\(60 \\times A \\times B\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-4046422ceeb0663f0810daa07039eae33f17f19d900a90a088d5159ec74b1df9 | A fair, standard six-sided die is rolled until a 6 is rolled for the first time. What is the expected value of the sum of all rolled numbers? | [
"21"
] | [
"A fair, standard six-sided die is rolled until a 6 is rolled for the first time. What is the expected value of the sum of all rolled numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-39640f0a11c05bad8d144a4f1dcaa98839b5bed0159cf4b8177471fcfa904533 | A fair, twenty-faced die has $19$ of its faces numbered from $1$ through $19$ and has one blank face. Another fair, twenty-faced die has $19$ of its faces numbered from $1$ through $8$ and $10$ through $20$ and has one blank face. When the two dice are rolled, what is the probability that the sum of the two numbers fac... | [
"\\frac{3}{80}"
] | [
"A fair, twenty-faced die has $19$ of its faces numbered from $1$ through $19$ and has one blank face. Another fair, twenty-faced die has $19$ of its faces numbered from $1$ through $8$ and $10$ through $20$ and has one blank face. When the two dice are rolled, what is the probability that the sum of the two number... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6e2daded3490d57da6aeba3e7930916060955d8508766aac4d90f7cc8d1a17f1 | A fair, unbiased coin is tossed three times on Friday, four times on Saturday, and five times on Sunday. What is the probability that exactly one toss shows "tails" on at least one of the three days? | [
"\\(\\frac{619}{1024}\\)"
] | [
"A fair, unbiased coin is tossed three times on Friday, four times on Saturday, and five times on Sunday. What is the probability that exactly one toss shows \"tails\" on at least one of the three days?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 32
} | 52 | 0 | 0 |
omr-ws-f9e95181c916a6aa08755ea03a813db7edbd525a085567b329562941e52ef766 | A family $\{ A_1, A_2, \ldots, A_k \}$ of subsets of $\{ 1, 2, \ldots, n \}$ is called good if for any $1 \le i < j \le k$, if $A_i \subseteq A_j$, then $|A_j \setminus A_i| \ge 3$. Find the maximum value of $k$. | [
"\\left\\lfloor \\dfrac{2^n + 2}{3} \\right\\rfloor"
] | [
"A family $\\{ A_1, A_2, \\ldots, A_k \\}$ of subsets of $\\{ 1, 2, \\ldots, n \\}$ is called good if for any $1 \\le i < j \\le k$, if $A_i \\subseteq A_j$, then $|A_j \\setminus A_i| \\ge 3$. Find the maximum value of $k$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-6c1098f53c4623c5f91a1791103de850682b6e02f04ebc28e233e7726b59c1c9 | A family of curves satisfies the differential equation $\frac{9-y^2}{9-x^2} = \left(\frac{dy}{dx}\right)^2$. Determine the number of ordered pairs $(a, b)$ such that infinitely many members of the family touch the line $ax + by = 1$. | [
"4"
] | [
"A family of curves satisfies the differential equation $\\frac{9-y^2}{9-x^2} = \\left(\\frac{dy}{dx}\\right)^2$. Determine the number of ordered pairs $(a, b)$ such that infinitely many members of the family touch the line $ax + by = 1$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-0c5249b6e8ae4b21eb5ae38611c700fb7e7b82af7b5c8108b13bc3ccd959759d | A family of distinct lines divides the plane into several regions so that the number of bounded regions exceeds the number of unbounded regions by 2022. What is the minimum number of lines? | [
"68"
] | [
"A family of distinct lines divides the plane into several regions so that the number of bounded regions exceeds the number of unbounded regions by 2022. What is the minimum number of lines?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-d756ae7d5a4db392fbfcecb90feb7300a07f9244964b5d09fd36086b133215de | A family of lines is given by $(1 + 2 \lambda)x + (1 - \lambda)y + \lambda = 0$, where $\lambda$ is a parameter. Find the line from this family that is at the maximum distance from the point (1,4). Express this line in the form $ax + by + c = 0$ and determine the value of $\frac{a+b+c}{19}$. | [
"2"
] | [
"A family of lines is given by $(1 + 2 \\lambda)x + (1 - \\lambda)y + \\lambda = 0$, where $\\lambda$ is a parameter. Find the line from this family that is at the maximum distance from the point (1,4). Express this line in the form $ax + by + c = 0$ and determine the value of $\\frac{a+b+c}{19}$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8cde08a6ac62f9d2cb2b73b7b7b50ff5949f128e677c544bdf55b90d1cb73a10 | A family of sets \( A_1, A_2, \ldots, A_n \) has the following properties: - Each \( A_i \) contains 30 elements. - \( A_i \cap A_j \) contains exactly one element for all \( i, j \) with \( 1 \leq i < j \leq n \). - The intersection of all these sets is empty. Determine the largest possible value of \( n \). | [
"871"
] | [
"A family of sets \\( A_1, A_2, \\ldots, A_n \\) has the following properties:\n- Each \\( A_i \\) contains 30 elements.\n- \\( A_i \\cap A_j \\) contains exactly one element for all \\( i, j \\) with \\( 1 \\leq i < j \\leq n \\).\n- The intersection of all these sets is empty.\n\nDetermine the largest possible va... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-094a965f7b1a08f17b61e18632855bfe970b77f7eeb5e61d9351ada822935f1b | A family of six has six Christmas crackers to pull. Each person will pull two crackers, each with a different person. In how many different ways can this be done? | [
"70"
] | [
"A family of six has six Christmas crackers to pull. Each person will pull two crackers, each with a different person. In how many different ways can this be done?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 7
} | 16 | 0 | 0 |
omr-ws-860454cb7cca8c9b4a184aaaa66af892c372bafd357ef6dfcef187d8f280e0f7 | A family of six washes their ski-wear including their gloves. Each family member grabs any two gloves. In how many ways can they do this so that no one has both of their own gloves? | [
"6840085"
] | [
"A family of six washes their ski-wear including their gloves. Each family member grabs any two gloves. In how many ways can they do this so that no one has both of their own gloves?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 28,
"genselect": 6,
"tir": 11
} | 45 | 0 | 0 |
omr-ws-b29241b6b600a985abb196e04a14a3887eb75434f6f985cdba8a46110af69e88 | A family of six washes their ski-wear including their gloves. Each family member grabs one left-hand and one right-hand glove. In how many ways can they do this so that no one has both of their own gloves? | [
"439975"
] | [
"A family of six washes their ski-wear including their gloves. Each family member grabs one left-hand and one right-hand glove. In how many ways can they do this so that no one has both of their own gloves?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 14
} | 61 | 0 | 0 |
omr-ws-ea318ec616fbf72eee3d7e94363dcdddee14ba186c8e9b4f2897a85eb4001ac1 | A family wears clothes of three colors: red, blue, and green, with a separate, identical laundry bin for each color. Each week, the family generates a total of 10 kg of laundry, with the proportion of each color varying. The laundry is sorted by color and placed in the bins. Next, the heaviest bin (only one of them, if... | [
"25"
] | [
"A family wears clothes of three colors: red, blue, and green, with a separate, identical laundry bin for each color. Each week, the family generates a total of 10 kg of laundry, with the proportion of each color varying. The laundry is sorted by color and placed in the bins. Next, the heaviest bin (only one of the... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c8d007335e87765e3b89c1433a56ee9369b4a6ea26acc31f3b4d614f3a9cddb7 | A family's electricity consumption is distributed as follows: 28% on electric shower, 15% on washing machine, 18% on electric iron, 24% on freezer, 8% on lamps, and 7% on others. The family decides to reduce the use of the electric iron by 40%. If every 1% reduction in total energy consumption compared to the previous ... | [
"36.00"
] | [
"A family's electricity consumption is distributed as follows: 28% on electric shower, 15% on washing machine, 18% on electric iron, 24% on freezer, 8% on lamps, and 7% on others. The family decides to reduce the use of the electric iron by 40%. If every 1% reduction in total energy consumption compared to the prev... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-9222b64e79a39fcd0f77356dd65ceb0ea606f96c96c7eced9ab761737af53cb0 | A famous theorem states that given any five points in the plane, with no three on the same line, there is a unique conic section (ellipse, hyperbola, or parabola) which passes through all five points. The conic section passing through the five points \[(-\tfrac32, 1), \; (0,0), \;(0,2),\; (3,0),\; (3,2).\]is an ellipse... | [
"\\frac{4\\sqrt3}{3}"
] | [
"A famous theorem states that given any five points in the plane, with no three on the same line, there is a unique conic section (ellipse, hyperbola, or parabola) which passes through all five points. The conic section passing through the five points \\[(-\\tfrac32, 1), \\; (0,0), \\;(0,2),\\; (3,0),\\; (3,2).\\]i... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9af7751106b1dc727011c9a5660607144aaa8c54a0740d1611bafc31f4977793 | A fancy bed and breakfast inn has 5 rooms, each with a distinctive color-coded decor. One day 5 friends arrive to spend the night. There are no other guests that night. The friends can room in any combination they wish, but with no more than 2 friends per room. In how many ways can the innkeeper assign the guests to th... | [
"900"
] | [
"A fancy bed and breakfast inn has 5 rooms, each with a distinctive color-coded decor. One day 5 friends arrive to spend the night. There are no other guests that night. The friends can room in any combination they wish, but with no more than 2 friends per room. In how many ways can the innkeeper assign the guests ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-05b572fbcfcb60aaa6f241614f6a561bbf57f7bf083fa7036e11bfba95e0e56e | A fancy triangle is an equilateral triangular array of integers such that the sum of the three numbers in any unit equilateral triangle is a multiple of $3$. Suppose that a fancy triangle has ten rows and that exactly $n$ of the numbers in the triangle are multiples of $3$. Determine all possible values for $n$. | [
"0, 18, 19, 55"
] | [
"A fancy triangle is an equilateral triangular array of integers such that the sum of the three numbers in any unit equilateral triangle is a multiple of $3$. Suppose that a fancy triangle has ten rows and that exactly $n$ of the numbers in the triangle are multiples of $3$. Determine all possible values for $n$."
... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-fbcef663d5a3b172a3e48a738883ada5f0b269d1a0429295075694dbf9bc9450 | A fancy triangle is an equilateral triangular array of integers such that the sum of the three numbers in any unit equilateral triangle is a multiple of 3. Suppose that a fancy triangle has ten rows and that exactly \( n \) of the numbers in the triangle are multiples of 3. Determine all possible values for \( n \). | [
"n \\equiv 0 \\text{ or } 1 \\pmod{3}"
] | [
"A fancy triangle is an equilateral triangular array of integers such that the sum of the three numbers in any unit equilateral triangle is a multiple of 3. Suppose that a fancy triangle has ten rows and that exactly \\( n \\) of the numbers in the triangle are multiples of 3. Determine all possible values for \\( ... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-8ab01bdb1020362ebf843e9dd621fe27ee105a86a40c37928467df889eafb10f | A farm has tractors of four models, $A,B,C,D$. Four tractors (2 of model $B$ and one of models $C$ and $D$) plough a field in two days. Two model $A$ tractors and one model $C$ tractor take three days to do this job, and three tractors of models $A$, $B$ and $C$ take four days. How long will it take to do the job if a ... | [
"12/7 days"
] | [
"A farm has tractors of four models, $A,B,C,D$. Four tractors (2 of model $B$ and one of models $C$ and $D$) plough a field in two days. Two model $A$ tractors and one model $C$ tractor take three days to do this job, and three tractors of models $A$, $B$ and $C$ take four days. How long will it take to do the job ... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-af4574668b4e8133262e95b0b656aaaa94f39fdc093ddad35e20fbc194b85ea1 | A farm is represented as an m x n grid. A lamp placed at any point (i, j) lights up the entire row i and column j. Lamps can only be placed at positions that are not already lit. Determine the number of ways to light up the entire farm using the minimum number of lamps required. | [
"\\(\\frac{n!}{(n-m)!}\\)"
] | [
"A farm is represented as an m x n grid. A lamp placed at any point (i, j) lights up the entire row i and column j. Lamps can only be placed at positions that are not already lit. Determine the number of ways to light up the entire farm using the minimum number of lamps required."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-1a643b41ac9192efe496b52abb1a7c10ba53fcbafed7ceb7487286b59d74f72e | A farmer $F_1$ has a land in the shape of a triangle with vertices at $P(0,0)$, $Q(1,1)$, and $R(2,0)$. A neighboring farmer $F_2$ takes away the region which lies between the side $PQ$ and a curve of the form $y = x^n$ $(n > 1)$. If the area of the region taken away is exactly $30\%$ of the area of $\triangle PQR$, th... | [
"\\( n = 4 \\)"
] | [
"A farmer $F_1$ has a land in the shape of a triangle with vertices at $P(0,0)$, $Q(1,1)$, and $R(2,0)$. A neighboring farmer $F_2$ takes away the region which lies between the side $PQ$ and a curve of the form $y = x^n$ $(n > 1)$. If the area of the region taken away is exactly $30\\%$ of the area of $\\triangle P... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-9029c9111877cb1f032a6ad247c6022b5ecdf0439d9b0d0bbe4a81968bc69a08 | A farmer \(F_{1}\) has a land in the shape of a triangle with vertices at P(0,0), Q(1, 1), and R(2, 0). A neighboring farmer \(F_{2}\) takes away the region between the side PQ and a curve of the form \(y = x^n\) (\(n > 1\)). If the area of the region taken away by farmer \(F_{2}\) is exactly 30% of the area of triangl... | [
"4"
] | [
"A farmer \\(F_{1}\\) has a land in the shape of a triangle with vertices at P(0,0), Q(1, 1), and R(2, 0). A neighboring farmer \\(F_{2}\\) takes away the region between the side PQ and a curve of the form \\(y = x^n\\) (\\(n > 1\\)). If the area of the region taken away by farmer \\(F_{2}\\) is exactly 30% of the ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-de645b90107f6ed10288126226e8c20d9fd1b947820914e25d95b11b97498cad | A farmer and his son are pulling on ropes attached to a boulder. The ropes make a 45-degree angle with each other. The farmer is pulling with a force of 115 Newtons and his son is pulling with a force of 105 Newtons. Their combined force can be expressed to the nearest integer as \(ABC\) Newtons. Find \(A + B + C\). | [
"5"
] | [
"A farmer and his son are pulling on ropes attached to a boulder. The ropes make a 45-degree angle with each other. The farmer is pulling with a force of 115 Newtons and his son is pulling with a force of 105 Newtons. Their combined force can be expressed to the nearest integer as \\(ABC\\) Newtons. Find \\(A + B +... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cdc0f0ec5da9247d687a6022ecd226aeb6961a0f782e6a4159c38508d4012283 | A farmer bought 749 sheeps. He sold 700 of them for the price he paid for the 749 sheeps. The remaining 49 sheeps were sold at the same price per head as the other 700. Based on the cost, what is the percent gain on the entire transaction? | [
"7"
] | [
"A farmer bought 749 sheeps. He sold 700 of them for the price he paid for the 749 sheeps. The remaining 49 sheeps were sold at the same price per head as the other 700. Based on the cost, what is the percent gain on the entire transaction?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-59679614bd757e69008e9ea55c06a0db87a03cd98b9518f6f6765d49d55b0b92 | A farmer builds a rectangular pen with 60 meters of fence. A dog on a 30 meter leash is leashed to one of the outside corners of the pen. What is the maximum area that the dog can access? | [
"\\( 900\\pi \\)"
] | [
"A farmer builds a rectangular pen with 60 meters of fence. A dog on a 30 meter leash is leashed to one of the outside corners of the pen. What is the maximum area that the dog can access?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 6,
"tir": 17
} | 36 | 0 | 0 |
omr-ws-87a2b794cf14ec6a158f26527a1d3ab64dc360e8f4b8603e15173cdd3103bd2a | A farmer goes to market and buys one hundred animals at a total cost of $10,000. Cows cost $500, sheep cost $100, and rabbits cost $5. Assuming the farmer buys at least one of each type of animal, how many cows do they buy? | [
"19"
] | [
"A farmer goes to market and buys one hundred animals at a total cost of $10,000. Cows cost $500, sheep cost $100, and rabbits cost $5. Assuming the farmer buys at least one of each type of animal, how many cows do they buy?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cc200469f026a3cf2346050f4ef6ccae354ebea4cdd85bef510f6adc581a2eff | A farmer had 20 apple trees, with each tree growing an average of 44 apples. The farmer also had yams, with the number of yams being 2/5 the number of apples. How many apples and yams did the farmer have? | [
"880 apples and 352 yams"
] | [
"A farmer had 20 apple trees, with each tree growing an average of 44 apples. The farmer also had yams, with the number of yams being 2/5 the number of apples. How many apples and yams did the farmer have?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-49dbbcd0bd1949379d92098775a14738283631ed26367421968d93dd6dcda54f | A farmer has 100 pigs and 100 chickens distributed into four square yards arranged in a 2x2 grid. The first row must have 120 heads, the second row must have 300 legs, the first column must have 100 heads, and the second column must have 320 legs. How many different ways can the farmer distribute the animals to satisfy... | [
"341"
] | [
"A farmer has 100 pigs and 100 chickens distributed into four square yards arranged in a 2x2 grid. The first row must have 120 heads, the second row must have 300 legs, the first column must have 100 heads, and the second column must have 320 legs. How many different ways can the farmer distribute the animals to sa... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-6247832291221d804cfa67e10b2ea9bbbb133a115ddf43ba1fc9d9b64c3b03a0 | A farmer has 100 pigs and 100 chickens distributed into four square yards forming a 2x2 grid. The first row has 120 heads, the second row has 300 legs, the first column has 100 heads, and the second column has 320 legs. How many different ways are there to distribute the animals to satisfy these conditions? | [
"341"
] | [
"A farmer has 100 pigs and 100 chickens distributed into four square yards forming a 2x2 grid. The first row has 120 heads, the second row has 300 legs, the first column has 100 heads, and the second column has 320 legs. How many different ways are there to distribute the animals to satisfy these conditions?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 7
} | 21 | 0 | 0 |
omr-ws-d8310d1cc92c06bb838f89b0c7ca056fa1d4377c402ccacc94263a8644bcbd38 | A farmer has 100 pigs and 100 chickens. He has four square-shaped yards arranged in a 2×2 grid. The farmer wants to distribute the animals into the yards such that the first row has 120 heads, the second row has 300 legs, the first column has 100 heads, and the second column has 320 legs. How many different ways can th... | [
"341"
] | [
"A farmer has 100 pigs and 100 chickens. He has four square-shaped yards arranged in a 2×2 grid. The farmer wants to distribute the animals into the yards such that the first row has 120 heads, the second row has 300 legs, the first column has 100 heads, and the second column has 320 legs. How many different ways c... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 5
} | 20 | 0 | 0 |
omr-ws-8ceae24f3888b33e89a55aa7f62cf6209c7917faeae23b0122561a755461bbb5 | A farmer has 100 pigs and 100 chickens. He has four square-shaped yards forming a \( 2 \times 2 \) grid. The farmer wants to distribute his animals into the yards such that the first row has 120 heads, the second row has 300 legs, the first column has 100 heads, and the second column has 320 legs. How many different wa... | [
"341"
] | [
"A farmer has 100 pigs and 100 chickens. He has four square-shaped yards forming a \\( 2 \\times 2 \\) grid. The farmer wants to distribute his animals into the yards such that the first row has 120 heads, the second row has 300 legs, the first column has 100 heads, and the second column has 320 legs. How many diff... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-d03bd9ed19994a92e74250558d1a485ac6f15a2cdc0dbfc4471fb404fa8c0239 | A farmer has 1600' of fencing and wants to enclose a maximum area of land next to a river bend with a shoreline radius of 300'. The fence will have one long side and two equal sides at right angles, both ending at the river shore. What is the maximum area of land he can enclose? | [
"320000"
] | [
"A farmer has 1600' of fencing and wants to enclose a maximum area of land next to a river bend with a shoreline radius of 300'. The fence will have one long side and two equal sides at right angles, both ending at the river shore. What is the maximum area of land he can enclose?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 1,
"tir": 2
} | 9 | 0 | 0 |
omr-ws-3428166b810aea764e1a69773aadacb7a028dcb352d60a1bd34107ee1673adf6 | A farmer has 3000 bananas and a camel that eats 1 banana for every 1 km traveled. The market is 1000 km away, and the camel can carry a maximum of 1000 bananas at any time. What is the largest number of bananas the farmer can bring to the market? | [
"500"
] | [
"A farmer has 3000 bananas and a camel that eats 1 banana for every 1 km traveled. The market is 1000 km away, and the camel can carry a maximum of 1000 bananas at any time. What is the largest number of bananas the farmer can bring to the market?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-88c5641b340eb19c6b357f6a2c72460f414dca5d6632f8d99b7906bfbf52904b | A farmer has 32 yards of fencing material and wants to create the largest possible area for a pen, but refuses to build it in the shape of a square. What is the largest possible area he can make, given that each side does not necessarily need to be an integer length? | [
"64"
] | [
"A farmer has 32 yards of fencing material and wants to create the largest possible area for a pen, but refuses to build it in the shape of a square. What is the largest possible area he can make, given that each side does not necessarily need to be an integer length?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-5784b4d9de0285ab231e811f3bb6ba6bd00a4c0b419fc18c90375b7be0dedd7c | A farmer has 4 sons with ages \(X_1, X_2, X_3, X_4\) where \(X_4 \le X_3 \le X_2 \le X_1\), and their ages lie between 2 and 16 inclusive. One year ago, their ages satisfied \((X_1-1)^2 = (X_2-1)^2 + (X_3-1)^2 + (X_4-1)^2\). At present, their ages satisfy \(X_1^2 + X_4^2 = X_2^2 + X_3^2\). Find the possible sets of the... | [
"(16, 12, 11, 3)"
] | [
"A farmer has 4 sons with ages \\(X_1, X_2, X_3, X_4\\) where \\(X_4 \\le X_3 \\le X_2 \\le X_1\\), and their ages lie between 2 and 16 inclusive. One year ago, their ages satisfied \\((X_1-1)^2 = (X_2-1)^2 + (X_3-1)^2 + (X_4-1)^2\\). At present, their ages satisfy \\(X_1^2 + X_4^2 = X_2^2 + X_3^2\\). Find the poss... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-f7f5892721287999202dd775af6abab6664fa17394cf755e5295bdb4233a466e | A farmer has 440 ft of fencing to make three animal pens of equal area next to a barn, so no fence is needed on the barn side. What is the maximum total area of the pens? | [
"12100"
] | [
"A farmer has 440 ft of fencing to make three animal pens of equal area next to a barn, so no fence is needed on the barn side. What is the maximum total area of the pens?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 3,
"tir": 26
} | 44 | 0 | 0 |
omr-ws-1fb5d45508f727a2f81fb28b6eed4c7492064f4e89e97992c09ce8670e0acf84 | A farmer has 480 meters of fencing and wishes to enclose a rectangular plot of land, dividing it into three equal rectangles with two parallel lengths of fence down the middle. What dimensions will maximize the enclosed area? | [
"\\( x = 120 \\) meters, \\( y = 60 \\) meters"
] | [
"A farmer has 480 meters of fencing and wishes to enclose a rectangular plot of land, dividing it into three equal rectangles with two parallel lengths of fence down the middle. What dimensions will maximize the enclosed area?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 28
} | 41 | 0 | 0 |
omr-ws-8135c6766048adf5af396134097e07d4f140c171ab6842aef8a0f39262099969 | A farmer has 76 meters of fencing and wants to build the largest possible pen using an existing wall as one of the sides, without the restriction that the pen must be a quadrilateral. What is the area of the largest possible pen, and what shape does it take? | [
"\\(\\frac{2888 \\pi}{(\\pi + 2)^2}\\)"
] | [
"A farmer has 76 meters of fencing and wants to build the largest possible pen using an existing wall as one of the sides, without the restriction that the pen must be a quadrilateral. What is the area of the largest possible pen, and what shape does it take?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 31,
"genselect": 4,
"tir": 2
} | 37 | 0 | 0 |
omr-ws-a84c4b2a2ad608f05da0feda9d2b300b09c09e47d99f24441d4b89a4e57126a5 | A farmer has 76 meters of fencing and wants to build the largest possible quadrilateral pen using an existing wall as one of the sides. What is the area of the largest possible quadrilateral pen, and what does it look like? | [
"722\n\n(Note: The discussion seems to have drifted towards a hexagon solution for maximizing area without explicitly solving for the quadrilateral case beyond the initial post. Given the problem statement asks specifically for the quadrilateral solution, the boxed answer from the first post is taken as the answer ... | [
"A farmer has 76 meters of fencing and wants to build the largest possible quadrilateral pen using an existing wall as one of the sides. What is the area of the largest possible quadrilateral pen, and what does it look like?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-cf697b3ff12e295a8f4be07e0afc0e50b03e8de90c62c2ce46ef3fc0deedf87d | A farmer has a $20$ft x $20$ft barn. A goat is attached by a $45$ft cable to one of the corners of the barn. Find the area of the region that the goat can graze on. | [
"\\dfrac{3675}{2} \\pi"
] | [
"A farmer has a $20$ft x $20$ft barn. A goat is attached by a $45$ft cable to one of the corners of the barn. Find the area of the region that the goat can graze on."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
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