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-a656c2dfa60f38672c4c5e304ee038ab21d993ea59e36ee83db955823dc793ce | A student observed that during a vacation of $d$ days, it rained 7 times (either in the morning or afternoon), there were 5 clear afternoons, and 6 clear mornings. Additionally, whenever it rained in the afternoon, the morning was clear. Determine the value of $d$. | [
"9"
] | [
"A student observed that during a vacation of $d$ days, it rained 7 times (either in the morning or afternoon), there were 5 clear afternoons, and 6 clear mornings. Additionally, whenever it rained in the afternoon, the morning was clear. Determine the value of $d$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-54c4429e5965c09ccad5b80ce2148b5512c7a53e247b0fb9200be4ac78d48dc6 | A student partitions 2000 consecutive integers into 1000 pairs and substitutes each pair with their difference and sum. Let $S$ be the set of all possible sums of the 2000 resulting integers after one operation. Find the smallest positive integer $k$ such that $S$ contains at least $k$ distinct even numbers. | [
"499501"
] | [
"A student partitions 2000 consecutive integers into 1000 pairs and substitutes each pair with their difference and sum. Let $S$ be the set of all possible sums of the 2000 resulting integers after one operation. Find the smallest positive integer $k$ such that $S$ contains at least $k$ distinct even numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 5
} | 7 | 0 | 0 |
omr-ws-c0ac6c8abf9dda7b6a57e2805f20d6ef3de0fb38325d37a800b9ae4d46630f95 | A student puts $2010$ red balls and $1957$ blue balls into a box. Weiqing draws randomly from the box one ball at a time without replacement. She wins if, at any time, the total number of blue balls drawn is more than the total number of red balls drawn. Assuming Weiqing keeps drawing balls until she either wins or run... | [
"\\dfrac{1957}{2011}"
] | [
"A student puts $2010$ red balls and $1957$ blue balls into a box. Weiqing draws randomly from the box one ball at a time without replacement. She wins if, at any time, the total number of blue balls drawn is more than the total number of red balls drawn. Assuming Weiqing keeps drawing balls until she either wins o... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-1e0d5ed1d6159523d6d95450e6188ae0777356ac6e228f5a7704d16323639909 | A student puts 2010 red balls and 1957 blue balls into a box. Weiqing draws balls one at a time without replacement. She wins if, at any point, the total number of blue balls drawn exceeds the total number of red balls drawn. Compute the probability that Weiqing eventually wins. | [
"\\(\\frac{1957}{2011}\\)"
] | [
"A student puts 2010 red balls and 1957 blue balls into a box. Weiqing draws balls one at a time without replacement. She wins if, at any point, the total number of blue balls drawn exceeds the total number of red balls drawn. Compute the probability that Weiqing eventually wins."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 17
} | 38 | 0 | 0 |
omr-ws-a9217877b9e9dcaaf6d580865cd30cad13bb83ace582a412cb70bb0e616a6b1f | A student read advanced mathematics for 37 days, adhering to the following rules: - He read at least one hour every day. - He read an integer number of hours, but not more than 12, each day. - He read at most 60 hours in total. Let \( S(n) \) be the total number of hours read by the end of day \( n \). Determine the mi... | [
"14"
] | [
"A student read advanced mathematics for 37 days, adhering to the following rules:\n- He read at least one hour every day.\n- He read an integer number of hours, but not more than 12, each day.\n- He read at most 60 hours in total.\n\nLet \\( S(n) \\) be the total number of hours read by the end of day \\( n \\). D... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-2f76202fd30c83b49a8870c8858483bfb0039876604f17d4b87e8d3db03411fc | A student read for his exams for 37 days such that every day he reads at least one hour, every day he reads an integer number of hours, and he must read exactly 60 hours in total. Calculate the minimum number of pairs of successive days where the total number of hours read is exactly 13. | [
"0"
] | [
"A student read for his exams for 37 days such that every day he reads at least one hour, every day he reads an integer number of hours, and he must read exactly 60 hours in total. Calculate the minimum number of pairs of successive days where the total number of hours read is exactly 13."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8f923fa925e91dd68a6e97f54a6293168c7be00d77b03dc8c520d50fac2eec53 | A student rolled a fair die until the sum of her scores was a prime number. She had to roll three times. What is the probability that her last throw was a 6? | [
"0"
] | [
"A student rolled a fair die until the sum of her scores was a prime number. She had to roll three times. What is the probability that her last throw was a 6?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 7
} | 22 | 0 | 0 |
omr-ws-350a113451e421a3aafb788595acb2b81762ef692e37bd678cc5b3f7b3c0710f | A student saves RM50 each month in a bank that pays 0.5% interest per month. How many months does he need to save to afford an encyclopedia set costing RM1200? | [
"23"
] | [
"A student saves RM50 each month in a bank that pays 0.5% interest per month. How many months does he need to save to afford an encyclopedia set costing RM1200?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5734c6f2621e684be62d92ba0558da94116e8ca64acaf38d9286e2817b805a59 | A student scores points by tossing a fair coin, gaining 1 point for each head and 2 points for each tail. Let \( P(n) \) be the probability that the student scores exactly \( n \) points. Find the value of \( P(5) + P(6) \). | [
"\\dfrac{85}{64}"
] | [
"A student scores points by tossing a fair coin, gaining 1 point for each head and 2 points for each tail. Let \\( P(n) \\) be the probability that the student scores exactly \\( n \\) points. Find the value of \\( P(5) + P(6) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-8e6a8911b398e58171bb7392d7357e703cc47757781026d85c1f17bf3dd6ef46 | A student solves 1 problem in the morning and \(\frac{1}{7}\) of the remaining problems in the afternoon each day. The student continues this pattern and solves all the problems in \(n\) days. What are the possible values of \(n\)? | [
"1"
] | [
"A student solves 1 problem in the morning and \\(\\frac{1}{7}\\) of the remaining problems in the afternoon each day. The student continues this pattern and solves all the problems in \\(n\\) days. What are the possible values of \\(n\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-58a07fa95a813536ba872054e26282664dfa4e27c1d79007ce0bedb3ba484212 | A student starts with a quadratic equation $x^2 + ax + b = 0$ where $a$ and $b$ are both non-zero. If this equation has solutions $p$ and $q$ with $p \leq q$, he forms a second quadratic equation $x^2 + px + q = 0$. If this second equation has solutions, he forms a third quadratic equation in the same manner. Let $N$ b... | [
"no_valid_answer_found"
] | [
"A student starts with a quadratic equation $x^2 + ax + b = 0$ where $a$ and $b$ are both non-zero. If this equation has solutions $p$ and $q$ with $p \\leq q$, he forms a second quadratic equation $x^2 + px + q = 0$. If this second equation has solutions, he forms a third quadratic equation in the same manner. Let... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1094bfa415906c93112b0d866de15c08ac10b0ab0d721a32ea277d89fea2bcb8 | A student starts with a quadratic equation \(x^2 + ax + b = 0\) where \(a\) and \(b\) are both non-zero. If this equation has solutions \(p\) and \(q\) with \(p \le q\), he forms a second quadratic equation \(x^2 + px + q = 0\). If this second equation has solutions, he forms a third quadratic equation in the same mann... | [
"3"
] | [
"A student starts with a quadratic equation \\(x^2 + ax + b = 0\\) where \\(a\\) and \\(b\\) are both non-zero. If this equation has solutions \\(p\\) and \\(q\\) with \\(p \\le q\\), he forms a second quadratic equation \\(x^2 + px + q = 0\\). If this second equation has solutions, he forms a third quadratic equat... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-bc491b40ff737c1af3cdab245785d4d8446f9a1065395be1609a5d99c0f28827 | A student starts with a quadratic equation \(x^2 + ax + b = 0\) where \(a\) and \(b\) are both non-zero. If this equation has solutions \(p\) and \(q\) with \(p \leq q\), he forms a second quadratic equation \(x^2 + px + q = 0\). If this second equation has solutions, he forms a third quadratic equation in the same man... | [
"3"
] | [
"A student starts with a quadratic equation \\(x^2 + ax + b = 0\\) where \\(a\\) and \\(b\\) are both non-zero. If this equation has solutions \\(p\\) and \\(q\\) with \\(p \\leq q\\), he forms a second quadratic equation \\(x^2 + px + q = 0\\). If this second equation has solutions, he forms a third quadratic equa... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-d8061907f116a10998e9473276f8e223a3daa9f21de83544a8722c904317a537 | A student starts with the number \( x = 3 \) on the board. In each process, the student deletes \( x \) and replaces it with either \( 2x + 4 \), \( 3x + 8 \), or \( x^2 + 5x \). Is it possible to make the number \( 20^{17} + 2016 \) appear on the board? (Explain your answer.) | [
"No"
] | [
"A student starts with the number \\( x = 3 \\) on the board. In each process, the student deletes \\( x \\) and replaces it with either \\( 2x + 4 \\), \\( 3x + 8 \\), or \\( x^2 + 5x \\). Is it possible to make the number \\( 20^{17} + 2016 \\) appear on the board? (Explain your answer.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 18,
"tir": 26
} | 44 | 0 | 0 |
omr-ws-3fccea2fabe9892fb15b0a81835ae1d1b405699f6892067b308bfd3141b6b31c | A student studies at least 1 hour a day for 35 days, but no more than 60 hours in total. Let $S_n$ represent the cumulative number of hours studied by the end of day $n$. Find the maximum number of distinct values of $k$ for which there exist integers $i$ and $j$ such that $S_j - S_i = 13$ and $1 \leq i < j \leq 35$. | [
"59"
] | [
"A student studies at least 1 hour a day for 35 days, but no more than 60 hours in total. Let $S_n$ represent the cumulative number of hours studied by the end of day $n$. Find the maximum number of distinct values of $k$ for which there exist integers $i$ and $j$ such that $S_j - S_i = 13$ and $1 \\leq i < j \\leq... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 7
} | 9 | 0 | 0 |
omr-ws-b5261e669f6f8693ca295478ea80445ee1cafe38e34fb674e71f8fde5c3c4097 | A student takes a test consisting of 30 questions, each with an answer of Yes or No. After each test, the student is told how many questions were answered correctly. How many times must the student take the test to guarantee getting all 30 answers correct? | [
"31"
] | [
"A student takes a test consisting of 30 questions, each with an answer of Yes or No. After each test, the student is told how many questions were answered correctly. How many times must the student take the test to guarantee getting all 30 answers correct?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-7e78a1ba964d68a52f866d4913ccb8ecf98a51774affcf23892ee955451a9412 | A student tosses a fair coin and scores one point for each head that turns up, and two points for each tail. Let $P(n)$ be the probability that the student scores exactly $n$ points at some time in a sequence of $n$ tosses. Calculate $P(10)$. | [
"\\dfrac{683}{1024}"
] | [
"A student tosses a fair coin and scores one point for each head that turns up, and two points for each tail. Let $P(n)$ be the probability that the student scores exactly $n$ points at some time in a sequence of $n$ tosses. Calculate $P(10)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-e3c429a6b43d6fdb8dfc0ea32820c6902e9d4b2520463a90ffaf008d7d7832d4 | A student tosses a fair coin. She gains 1 point for each head and 2 points for each tail. Let \(P(n)\) be the probability of the student scoring exactly \(n\) points. Calculate \(P(10)\). | [
"\\dfrac{683}{1024}"
] | [
"A student tosses a fair coin. She gains 1 point for each head and 2 points for each tail. Let \\(P(n)\\) be the probability of the student scoring exactly \\(n\\) points. Calculate \\(P(10)\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-14fc5e395df8ef65b6e1a8a3d5e9aa466ab5602eb080c9b13ddc0432e56190c0 | A student walks down a hall containing 1024 closed lockers numbered 1 to 1024. He opens locker 1 and then alternates between skipping and opening each closed locker thereafter. When he reaches the end, he turns around and repeats the process until every locker is open. What is the number of the last locker he opens? | [
"342"
] | [
"A student walks down a hall containing 1024 closed lockers numbered 1 to 1024. He opens locker 1 and then alternates between skipping and opening each closed locker thereafter. When he reaches the end, he turns around and repeats the process until every locker is open. What is the number of the last locker he open... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 2,
"tir": 1
} | 5 | 0 | 0 |
omr-ws-ed58618e1805c68b245c55b83292b30d48de794225c46298ba906c8fac97b5ef | A student walks down a hall containing 1024 closed lockers numbered 1 to 1024. He opens the locker numbered 1, and then alternates between skipping and opening each closed locker thereafter. When he reaches the end of the hall, he turns around and starts back, opening the first closed locker he encounters and then alte... | [
"342"
] | [
"A student walks down a hall containing 1024 closed lockers numbered 1 to 1024. He opens the locker numbered 1, and then alternates between skipping and opening each closed locker thereafter. When he reaches the end of the hall, he turns around and starts back, opening the first closed locker he encounters and then... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 6
} | 8 | 0 | 0 |
omr-ws-7ea85917f7689bfd3c614201fc3442133fd5613f93a3e6338f73880254f559f7 | A student walks from her home to school, which is located 10 blocks east and 14 blocks north from home. She meets her best friend at a location 4 blocks east and 5 blocks north of her home and avoids passing the intersection where the park is located 3 blocks east and 6 blocks north of her friend's house. How many diff... | [
"\\(\\binom{9}{4}\\binom{15}{6} - \\binom{9}{4}\\binom{9}{3}\\binom{6}{3}\\)"
] | [
"A student walks from her home to school, which is located 10 blocks east and 14 blocks north from home. She meets her best friend at a location 4 blocks east and 5 blocks north of her home and avoids passing the intersection where the park is located 3 blocks east and 6 blocks north of her friend's house. How many... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-67dfd6e4ad0f38b326f0312e4d8441e4e7213f9d0d991db0e24b7fd6f5536b2f | A student walks from her home to school, which is located 10 blocks east and 14 blocks north from home. She meets her best friend at a location 4 blocks east and 5 blocks north of her home and stops at a park 3 blocks east and 6 blocks north of her friend's house. How many different shortest walks of 24 blocks are poss... | [
"\\(\\binom{9}{4}\\binom{9}{3}\\binom{6}{3}\\)"
] | [
"A student walks from her home to school, which is located 10 blocks east and 14 blocks north from home. She meets her best friend at a location 4 blocks east and 5 blocks north of her home and stops at a park 3 blocks east and 6 blocks north of her friend's house. How many different shortest walks of 24 blocks are... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2e4b7357a7ba74917b760f8a802f548ff34fd50352024b0bf3b4432b108a0e6d | A student walks from her home to school, which is located 10 blocks east and 14 blocks north from home. She meets her best friend at a location 4 blocks east and 5 blocks north of her home. How many different shortest walks of 24 blocks are possible? | [
"\\binom{9}{4}\\binom{15}{6}"
] | [
"A student walks from her home to school, which is located 10 blocks east and 14 blocks north from home. She meets her best friend at a location 4 blocks east and 5 blocks north of her home. How many different shortest walks of 24 blocks are possible?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 3,
"tir": 22
} | 32 | 0 | 0 |
omr-ws-a63a06ffe7328c9bfe87f5eacc41a22f148fad5994c5fef4322b3a0bcba3f10d | A student walks from intersection A to intersection B, always walking along streets shown, and always going east or south. At each intersection where he has a choice, he chooses with probability $\frac{1}{2}$ whether to go east or south. Find the probability that through any given morning, he goes through intersection ... | [
"\\(\\frac{4}{7}\\)"
] | [
"A student walks from intersection A to intersection B, always walking along streets shown, and always going east or south. At each intersection where he has a choice, he chooses with probability $\\frac{1}{2}$ whether to go east or south. Find the probability that through any given morning, he goes through interse... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f36c938fb0ffdd90c8cc345f932d6334e7e82964b95f7eb956452c030373b4fc | A student wants to have a probability of 0.75 of being able to answer all 5 questions on an exam where she must answer 5 out of 10 questions. How many questions out of the 10 must she prepare for? | [
"10"
] | [
"A student wants to have a probability of 0.75 of being able to answer all 5 questions on an exam where she must answer 5 out of 10 questions. How many questions out of the 10 must she prepare for?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-04ceb51743c889304dcae341f613b84958b1b2a03074908ed5be0229a1255e4f | A student was to multiply 72 by a two-place number in which there are thrice as many tens as ones; by mistake he reversed the digits in the second factor and obtained a product which is smaller than the actual one by 2592. What is the actual product equal to? | [
"6464"
] | [
"A student was to multiply 72 by a two-place number in which there are thrice as many tens as ones; by mistake he reversed the digits in the second factor and obtained a product which is smaller than the actual one by 2592. What is the actual product equal to?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7c8b673408d1bfed70189d8e848c11e55398d91d1b4eef275ad0a1de88cc47cb | A student writes $n$ numbers, all equal to $1$, on the board. At each step, he erases two of the numbers and replaces them with half of their arithmetic mean. After repeating this process until only one number remains, calculate the minimum possible value of this final number for $n = 16$. | [
"\\dfrac{1}{16}"
] | [
"A student writes $n$ numbers, all equal to $1$, on the board. At each step, he erases two of the numbers and replaces them with half of their arithmetic mean. After repeating this process until only one number remains, calculate the minimum possible value of this final number for $n = 16$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-9e5c035d924843e8f05ca8c0cd41d53e02596f7814dfcdb7da62be4ea093f5b2 | A student writes down a three-digit room number and repeats it to form a six-digit number (e.g., 123 becomes 123123). The number is then divided successively by three numbers \(a\), \(b\), and \(c\) with no remainder each time. The final quotient is given to the professor, who can correctly predict the original three-d... | [
"\\(a = 7\\), \\(b = 11\\), \\(c = 13\\)"
] | [
"A student writes down a three-digit room number and repeats it to form a six-digit number (e.g., 123 becomes 123123). The number is then divided successively by three numbers \\(a\\), \\(b\\), and \\(c\\) with no remainder each time. The final quotient is given to the professor, who can correctly predict the origi... | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 7
} | 7 | 0 | 0 |
omr-ws-f8940be29aa46e0ba1181e01e3bbdb304975ea21cb7489d0343c7e730c43354f | A student writes the numbers $1000, 1001, 1002, \ldots, 2999$ on the board. He then deletes any two numbers $a$ and $b$ and writes in their place the number $\frac{1}{2} \min(a, b)$. After performing this action 1999 times, a single number $c$ remains on the board. Calculate the maximum possible value of $c$. | [
""
] | [
"A student writes the numbers $1000, 1001, 1002, \\ldots, 2999$ on the board. He then deletes any two numbers $a$ and $b$ and writes in their place the number $\\frac{1}{2} \\min(a, b)$. After performing this action 1999 times, a single number $c$ remains on the board. Calculate the maximum possible value of $c$."
... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 1 |
omr-ws-2caaeaac3dcd5656756e9c9c36fcb3c902678d054ebf9baef073000b020e7eb0 | A student writes the numbers 1 through n in a circle and starts erasing numbers in a pattern: erase 1, leave 2, erase 3, leave 4, and so on. This process continues in subsequent rounds, always starting from the first remaining number. If n = 2016, what is the last remaining number in the circle? | [
"64"
] | [
"A student writes the numbers 1 through n in a circle and starts erasing numbers in a pattern: erase 1, leave 2, erase 3, leave 4, and so on. This process continues in subsequent rounds, always starting from the first remaining number. If n = 2016, what is the last remaining number in the circle?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5236731da31b57a554365d57c7f789b35f6660f393e6010eca872e3903ea75c1 | A student wrote a six-digit number by concatenating two three-digit numbers without a multiplication sign. This six-digit number is 7 times greater than the product of the two three-digit numbers. Find the two three-digit numbers. | [
"143 and 143"
] | [
"A student wrote a six-digit number by concatenating two three-digit numbers without a multiplication sign. This six-digit number is 7 times greater than the product of the two three-digit numbers. Find the two three-digit numbers."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 4,
"tir": 32
} | 43 | 0 | 0 |
omr-ws-344ffb017365783fa7f769af91407ce19149f83559f33e52fe989606659ac4ab | A student wrote down five numbers from a sequence of six positive integers in arithmetic progression: \(113, 137, 149, 155, 173\). One number was omitted, and one was miscopied. Recover the original sequence of six positive integers in arithmetic progression. | [
"\\(113, 125, 137, 149, 161, 173\\)"
] | [
"A student wrote down five numbers from a sequence of six positive integers in arithmetic progression: \\(113, 137, 149, 155, 173\\). One number was omitted, and one was miscopied. Recover the original sequence of six positive integers in arithmetic progression."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 8
} | 17 | 0 | 0 |
omr-ws-625061a873b59e939ee7233755e17deb4cd506d3d0369a43943e750d28e3680c | A student's math exam grades are 90, 96, 82, and 88. If the student receives a passing grade (70 or better) on each of the next two exams, what is the probability that the student will have an average of at least 90? Assume that any score is equally likely to occur and that all exams have 100 possible points. | [
"\\dfrac{153}{961}"
] | [
"A student's math exam grades are 90, 96, 82, and 88. If the student receives a passing grade (70 or better) on each of the next two exams, what is the probability that the student will have an average of at least 90? Assume that any score is equally likely to occur and that all exams have 100 possible points."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-8128385a3e7f428b06feb6296f6b1751c0670a9150919d62feb1161ea981cd76 | A stuntman jumps out of a plane and parachutes to safety within a 100 foot by 100 foot square field, which is entirely surrounded by a wooden fence. There is a flag pole in the middle of the square field. Assuming the stuntman is equally likely to land on any point in the field, the probability that he lands closer to ... | [
"17"
] | [
"A stuntman jumps out of a plane and parachutes to safety within a 100 foot by 100 foot square field, which is entirely surrounded by a wooden fence. There is a flag pole in the middle of the square field. Assuming the stuntman is equally likely to land on any point in the field, the probability that he lands close... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 7,
"tir": 1
} | 20 | 0 | 0 |
omr-ws-9f40b3dff4204583ad5cff1abe223adf067b08bd081f83e5b19dd9dde1c5fca7 | A submarine is lost in one of two regions, $A$ or $B$. The probability of the submarine being in region $A$ is $70\%$, and in region $B$ is $30\%$. The probability of finding the submarine in region $A$ on any given search is $40\%$, and in region $B$ is $80\%$. After searching region $A$ four times and region $B$ once... | [
"B"
] | [
"A submarine is lost in one of two regions, $A$ or $B$. The probability of the submarine being in region $A$ is $70\\%$, and in region $B$ is $30\\%$. The probability of finding the submarine in region $A$ on any given search is $40\\%$, and in region $B$ is $80\\%$. After searching region $A$ four times and region... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8a8d88e6fea57df6e0370729d36e5e303868b841b14c6c490058e2952a6234d9 | A submarine starts at position \( n \) on a real number line at time \( t = 0 \) and moves with a constant velocity \( s \), where \( s \) and \( n \) are integers. You can guess the submarine's position at every integer time \( t \geq 0 \). Let \( g(t) \) be the position you guess at time \( t \). Find the smallest po... | [
"2"
] | [
"A submarine starts at position \\( n \\) on a real number line at time \\( t = 0 \\) and moves with a constant velocity \\( s \\), where \\( s \\) and \\( n \\) are integers. You can guess the submarine's position at every integer time \\( t \\geq 0 \\). Let \\( g(t) \\) be the position you guess at time \\( t \\)... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-da591371a75d1ddeafae553a95749c6bf0fa35c5d23a4cbaf8328bbcb0efb1db | A submarine travels in a straight line beneath the surface of an infinite sea at a constant speed of N kilometers per hour, where N is a positive integer. At the end of every hour, the submarine is directly below an integer on an infinitely long number line laid along the surface. You can drop a depth charge at any int... | [
"\\text{Drop a depth charge at position } y_t = d + k \\cdot t \\text{ at time } t, \\text{ where } (k, d) \\text{ enumerates all pairs of positive integers } k \\text{ and integers } d."
] | [
"A submarine travels in a straight line beneath the surface of an infinite sea at a constant speed of N kilometers per hour, where N is a positive integer. At the end of every hour, the submarine is directly below an integer on an infinitely long number line laid along the surface. You can drop a depth charge at an... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 8
} | 9 | 0 | 0 |
omr-ws-26c73768bdab0f0ac93c5da91417981d10108d95567de59c98348fc0ad849800 | A subset $B$ of $\{1, 2, \dots, 2017\}$ is said to have property $T$ if any three elements of $B$ can form the sides of a nondegenerate triangle. Find the maximum number of elements that a set with property $T$ may contain. | [
"1009"
] | [
"A subset $B$ of $\\{1, 2, \\dots, 2017\\}$ is said to have property $T$ if any three elements of $B$ can form the sides of a nondegenerate triangle. Find the maximum number of elements that a set with property $T$ may contain."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a587f40078c9876612a4b5bede8ac7808fbfd0d1427edf445d863aff8d666f05 | A subset $S$ of the first $2014$ positive integers is chosen so that no two different elements in $S$ sum up to a positive power of $2$. What is the maximum possible number of elements $S$ could contain? | [
"1009"
] | [
"A subset $S$ of the first $2014$ positive integers is chosen so that no two different elements in $S$ sum up to a positive power of $2$. What is the maximum possible number of elements $S$ could contain?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 4,
"tir": 6
} | 11 | 0 | 0 |
omr-ws-5a0e71ebc12c1f516675a18d7dd8fabaadd95b8ee8e676c0ed624d3882054833 | A subset B of the set of integers from $1$ to $100$, inclusive, has the property that no two elements of B sum to $125$. What is the maximum possible number of elements in B? | [
"62"
] | [
"A subset B of the set of integers from $1$ to $100$, inclusive, has the property that no two elements of B sum to $125$. What is the maximum possible number of elements in B?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b40b326a9efb016239a9a8d54f24833a7bbd3a4fa47fd60e25e87c5174f2aca9 | A subset X is chosen from the set {1,2,3,…,20} such that the sum of the elements of X is 25. If an element of X is picked at random, what is the expected value of that element? | [
"5"
] | [
"A subset X is chosen from the set {1,2,3,…,20} such that the sum of the elements of X is 25. If an element of X is picked at random, what is the expected value of that element?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-cc3fb773e9dcd82eec197eb9bf63c6700c7e68ac6b86e09e5ee63079ef67fb0a | A subset \( A \) of \( \{1, 2, \ldots, 15\} \) is called "good" if for every \( a \in A \), either \( a-1 \in A \) or \( a+1 \in A \). Find the number of good subsets with five elements of \( \{1, 2, \ldots, 15\} \). | [
"121"
] | [
"A subset \\( A \\) of \\( \\{1, 2, \\ldots, 15\\} \\) is called \"good\" if for every \\( a \\in A \\), either \\( a-1 \\in A \\) or \\( a+1 \\in A \\). Find the number of good subsets with five elements of \\( \\{1, 2, \\ldots, 15\\} \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-a4c26140d74d6d7efd0d90e5124947ee8a0d785367cfef97613fa421f243fab0 | A subset \( B \) of the set of integers from \( 1 \) to \( 100 \), inclusive, has the property that no two elements of \( B \) sum to \( 125 \). What is the maximum possible number of elements in \( B \)? | [
"62"
] | [
"A subset \\( B \\) of the set of integers from \\( 1 \\) to \\( 100 \\), inclusive, has the property that no two elements of \\( B \\) sum to \\( 125 \\). What is the maximum possible number of elements in \\( B \\)?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 63,
"tir": 5
} | 68 | 0 | 0 |
omr-ws-20dd156785d3b5eaf5afc1b98dbe1c28501ca8c58b70be6652693bb0b8c7fe0c | A subset \( B \) of the set of the first 100 positive integers has the property that no two elements of \( B \) sum to 125. What is the maximum possible number of elements in \( B \)? | [
"62"
] | [
"A subset \\( B \\) of the set of the first 100 positive integers has the property that no two elements of \\( B \\) sum to 125. What is the maximum possible number of elements in \\( B \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-848cb8da4ca63d98d9564c4c812e89b41da5485f2c05e9d488b550067d3998dd | A subset \( E \) of the set \( \{1, 2, 3, \ldots, 50\} \) is said to be special if it does not contain any pair of the form \( \{x, 3x\} \). A special set \( E \) is superspecial if it contains as many elements as possible. Determine the number of elements in a superspecial set and the number of such superspecial sets. | [
"38 elements in a superspecial set and 384 superspecial sets."
] | [
"A subset \\( E \\) of the set \\( \\{1, 2, 3, \\ldots, 50\\} \\) is said to be special if it does not contain any pair of the form \\( \\{x, 3x\\} \\). A special set \\( E \\) is superspecial if it contains as many elements as possible. Determine the number of elements in a superspecial set and the number of such ... | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 15
} | 15 | 0 | 0 |
omr-ws-3ab00d172e3f972f4c5453fbbe46c55a98d2aed7056f5c902e25939a28b9249b | A subset \( P \subseteq \mathbb{R}_{>0} \) is defined as nice if \( 1 \in P \) and, for all \( x, y, z \in P \), if there is a triangle with sides \( x, y, z \), then its perimeter is in \( P \). What is the intersection of all nice sets? | [
"\\(\\{2n+1:n \\in \\mathbb{Z}^{\\ge 0}\\}-\\{5,11,13,29\\}\\)"
] | [
"A subset \\( P \\subseteq \\mathbb{R}_{>0} \\) is defined as nice if \\( 1 \\in P \\) and, for all \\( x, y, z \\in P \\), if there is a triangle with sides \\( x, y, z \\), then its perimeter is in \\( P \\). What is the intersection of all nice sets?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1829357de1670d9137853fddad1843cd5871438d9d422e63f63c798ee5defd00 | A subset \( S \) is randomly selected from the set \(\{1, 2, 3, 4, 5\}\). If \( S \) may contain any number of terms, what is the expected value for the smallest term in \( S \)? Do not count the empty set. | [
"\\(\\frac{57}{31}\\)"
] | [
"A subset \\( S \\) is randomly selected from the set \\(\\{1, 2, 3, 4, 5\\}\\). If \\( S \\) may contain any number of terms, what is the expected value for the smallest term in \\( S \\)? Do not count the empty set."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-0026dc1f24c3ce243af62df63236763aa4fed6938d84ac35cdecbc17bdb5360e | A subset \( S \) of \( \{ 1, 2, \ldots, n \} \) is called selfish if \( |S| \in S \). A selfish subset is called minimal if none of its proper subsets are selfish. Find the number of minimal selfish subsets of \( \{ 1, 2, \ldots, n \} \). | [
"\\(F_n\\) (where \\(F_n\\) is the \\(n\\)-th Fibonacci number)"
] | [
"A subset \\( S \\) of \\( \\{ 1, 2, \\ldots, n \\} \\) is called selfish if \\( |S| \\in S \\). A selfish subset is called minimal if none of its proper subsets are selfish. Find the number of minimal selfish subsets of \\( \\{ 1, 2, \\ldots, n \\} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 27,
"genselect": 7,
"tir": 1
} | 35 | 0 | 0 |
omr-ws-9369e5e854b60f3bfdd7c1583e7d50b902fb28c818b63f0215b822d3b42dce6e | A subset \( S \) of \(\{1, 2, \ldots, 49\}\) is called dense if \( |S| > 40 \) and it contains no six consecutive numbers. How many dense sets exist? | [
"5"
] | [
"A subset \\( S \\) of \\(\\{1, 2, \\ldots, 49\\}\\) is called dense if \\( |S| > 40 \\) and it contains no six consecutive numbers. How many dense sets exist?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-174deb5077dcfaf8dc0f638f0e724f3a6b265335ae2ec60964281154edb1ffa4 | A subset \( T \) of \( S = \{1, 2, \ldots, 2006\} \) is called naughty if it does not contain any pair of integers \( (a, b) \) such that \( a \) divides \( b \). Let \( f(S) \) be the maximum number of elements in a naughty subset of \( S \). Find \( f(S) \). | [
"1003"
] | [
"A subset \\( T \\) of \\( S = \\{1, 2, \\ldots, 2006\\} \\) is called naughty if it does not contain any pair of integers \\( (a, b) \\) such that \\( a \\) divides \\( b \\). Let \\( f(S) \\) be the maximum number of elements in a naughty subset of \\( S \\). Find \\( f(S) \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c8a5d0f9c0cee74d786a893b73d6b959959b478f0c1334a63d15647dd47d6510 | A subset \( T \) of \(\{1, 2, \dots, 2024\}\) is called [b]kawaii[/b] if it satisfies the following properties: 1. \( T \) has at least two distinct elements. 2. For any two distinct elements \( x \) and \( y \) of \( T \), \( x - y \) does not divide \( x + y \). What is the largest possible number of elements that a ... | [
"675"
] | [
"A subset \\( T \\) of \\(\\{1, 2, \\dots, 2024\\}\\) is called [b]kawaii[/b] if it satisfies the following properties:\n1. \\( T \\) has at least two distinct elements.\n2. For any two distinct elements \\( x \\) and \\( y \\) of \\( T \\), \\( x - y \\) does not divide \\( x + y \\).\n\nWhat is the largest possib... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 3
} | 18 | 0 | 0 |
omr-ws-7d77df4a9e91984eb70b6dfbeee47fba7b4ed3793d772af653c51ac937038087 | A subset \( T \) of \(\{1, 2, \dots, 2024\}\) is called [i]kawaii[/i] if it satisfies the following conditions: 1. \( T \) has at least two elements. 2. For any two elements \( x \) and \( y \) in \( T \), \( x - y \) does not divide \( x + y \). For example, \( T = \{31, 71, 2024\} \) is [i]kawaii[/i], but \( T = \{5,... | [
"675"
] | [
"A subset \\( T \\) of \\(\\{1, 2, \\dots, 2024\\}\\) is called [i]kawaii[/i] if it satisfies the following conditions:\n1. \\( T \\) has at least two elements.\n2. For any two elements \\( x \\) and \\( y \\) in \\( T \\), \\( x - y \\) does not divide \\( x + y \\).\n\nFor example, \\( T = \\{31, 71, 2024\\} \\) ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-392f097a0a2407e81382a0d50bd456e1f84545daa043a5f92e63e5a41c62638b | A subset \( X \) of \( \{1, 2, 3, \ldots, 10000\} \) has the property that if \( a \) and \( b \) are distinct elements of \( X \), then \( ab \not\in X \). What is the maximal number of elements in \( X \)? | [
"9901"
] | [
"A subset \\( X \\) of \\( \\{1, 2, 3, \\ldots, 10000\\} \\) has the property that if \\( a \\) and \\( b \\) are distinct elements of \\( X \\), then \\( ab \\not\\in X \\). What is the maximal number of elements in \\( X \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 5
} | 27 | 0 | 0 |
omr-ws-59436e68c9f00353341566f8c351d70925d416210790bc8fd3d98f73bb03cefd | A subset \( X \) of \(\{1, 2, 3, \ldots, 10000\}\) has the property that if \( a \) and \( b \) are distinct elements of \( X \), then \( ab \notin X \). What is the maximal number of elements in \( X \)? | [
"9901"
] | [
"A subset \\( X \\) of \\(\\{1, 2, 3, \\ldots, 10000\\}\\) has the property that if \\( a \\) and \\( b \\) are distinct elements of \\( X \\), then \\( ab \\notin X \\). What is the maximal number of elements in \\( X \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 3
} | 24 | 0 | 0 |
omr-ws-0976e722bd0fe2ff5bc4dea89e0f7fedb59b647892f93775e0e75363c0faad4e | A subset \(B\) of the set of integers from 1 to 2008 has the property that no two elements of \(B\) sum to 3000. What is the maximum possible number of elements in \(B\)? | [
"1500"
] | [
"A subset \\(B\\) of the set of integers from 1 to 2008 has the property that no two elements of \\(B\\) sum to 3000. What is the maximum possible number of elements in \\(B\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-ee30112c045a35f850b0ae604e3e207b944200b898edf090ac4e945fb5260c6a | A subset \(E\) of \(\{1, 2, 3, \ldots, 50\}\) is called special if it does not contain any pair of the form \(\{x, 3x\}\). A special set \(E\) is superspecial if it contains as many elements as possible. Determine the number of elements in a superspecial set and the number of such superspecial sets. | [
"38 elements, 384 superspecial sets."
] | [
"A subset \\(E\\) of \\(\\{1, 2, 3, \\ldots, 50\\}\\) is called special if it does not contain any pair of the form \\(\\{x, 3x\\}\\). A special set \\(E\\) is superspecial if it contains as many elements as possible. Determine the number of elements in a superspecial set and the number of such superspecial sets."
... | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 12
} | 12 | 0 | 0 |
omr-ws-fd27ba01fcfe147cdfb18ffdf2c4652533e5784417fa0b90b3de83595b1a7652 | A subset of $\{1, 2, 3, 4, 5, 6, 7, 8\}$ of size $3$ is called special if whenever $a$ and $b$ are in the set, the remainder when $a + b$ is divided by $8$ is not in the set. ($a$ and $b$ can be the same.) How many special subsets exist? | [
"6"
] | [
"A subset of $\\{1, 2, 3, 4, 5, 6, 7, 8\\}$ of size $3$ is called special if whenever $a$ and $b$ are in the set, the remainder when $a + b$ is divided by $8$ is not in the set. ($a$ and $b$ can be the same.) How many special subsets exist?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 8
} | 16 | 0 | 0 |
omr-ws-82e59da20aa8923aea8810bfa6d57ab20bd0e896c8ec99267dddbdd30a6f0394 | A subset of $\{1, 2, 3, \ldots, 30\}$ is called delicious if it does not contain an element that is $3$ times another element. A subset is called super delicious if it is delicious and no delicious set has more elements than it has. Determine the number of super delicious subsets. | [
"96"
] | [
"A subset of $\\{1, 2, 3, \\ldots, 30\\}$ is called delicious if it does not contain an element that is $3$ times another element. A subset is called super delicious if it is delicious and no delicious set has more elements than it has. Determine the number of super delicious subsets."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-5d83bd9a91207a85f93b23c2e03593cc6bb297106d353ff437d8af6b128b63c7 | A subset of $\{1, 2, \ldots, 12\}$ is called trivoidant if no two elements of the subset differ by $3$. Determine the number of trivoidant subsets of $\{1, 2, \ldots, 12\}$. | [
"$8^3$ = 512"
] | [
"A subset of $\\{1, 2, \\ldots, 12\\}$ is called trivoidant if no two elements of the subset differ by $3$. Determine the number of trivoidant subsets of $\\{1, 2, \\ldots, 12\\}$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9c2cf41e6e54aa9b9402ddaa12174a01a22e54be146bb331d89ab5c3c6cdba40 | A subset of $\{1,2,\cdots,n\}$ is called "good" if all the elements in that subset are greater than or equal to the cardinality of the subset. Let $T_n$ be the number of "good" subsets of $\{1,2,\cdots,n\}$. Find a recursion for $T_n$. | [
"\\( T_n = T_{n-1} + T_{n-2} \\)"
] | [
"A subset of $\\{1,2,\\cdots,n\\}$ is called \"good\" if all the elements in that subset are greater than or equal to the cardinality of the subset. Let $T_n$ be the number of \"good\" subsets of $\\{1,2,\\cdots,n\\}$. Find a recursion for $T_n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 15
} | 63 | 0 | 0 |
omr-ws-1c8e222599773fc53b2a881aa4d5428cb4efa2932741dc17390d72ec55524e76 | A subset of five distinct positive integers is chosen uniformly at random from the set $\{1, 2, \ldots, 11\}$. The probability that the subset does not contain three consecutive integers can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | [
"52"
] | [
"A subset of five distinct positive integers is chosen uniformly at random from the set $\\{1, 2, \\ldots, 11\\}$. The probability that the subset does not contain three consecutive integers can be written as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 6,
"tir": 17
} | 36 | 0 | 0 |
omr-ws-bf7e367c2a0b31e495ed22523e0c251792a884fdaf855b6af9eb62c281e9e4ac | A subset of the first $4044$ integers $S = \{1, 2, \cdots, 4044\}$ is colored with black and white, such that $2022$ of them are colored black and $2022$ are colored white. For an integer $1 \leq i \leq 4044$, denote the weight of that integer to be the number of all white numbers smaller than $i$ and all black numbers... | [
"2044242"
] | [
"A subset of the first $4044$ integers $S = \\{1, 2, \\cdots, 4044\\}$ is colored with black and white, such that $2022$ of them are colored black and $2022$ are colored white. For an integer $1 \\leq i \\leq 4044$, denote the weight of that integer to be the number of all white numbers smaller than $i$ and all bla... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0bfcba15fa9ce28797d276017cee1c1c4c5c4633c847496f7949133e8154a272 | A subset of the positive integers \( S \) is said to be a configuration if \( 200 \notin S \) and for all nonnegative integers \( x \), \( x \in S \) if and only if both \( 2x \in S \) and \( \left \lfloor \frac{x}{2} \right \rfloor \in S \). Let \( k \) be the number of subsets of \( \{1, 2, 3, \dots, 130\} \) that ar... | [
"1359"
] | [
"A subset of the positive integers \\( S \\) is said to be a configuration if \\( 200 \\notin S \\) and for all nonnegative integers \\( x \\), \\( x \\in S \\) if and only if both \\( 2x \\in S \\) and \\( \\left \\lfloor \\frac{x}{2} \\right \\rfloor \\in S \\). Let \\( k \\) be the number of subsets of \\( \\{1,... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b0c49a208f9ee5e0a9c2283d5dd2e018ad583badc6ec1ffe4e7a3998c9587c5f | A subset of the real numbers has the property that for any two distinct elements \( x \) and \( y \), we have \((x + y - 1)^2 = xy + 1\). What is the maximum number of elements in this set? | [
"3"
] | [
"A subset of the real numbers has the property that for any two distinct elements \\( x \\) and \\( y \\), we have \\((x + y - 1)^2 = xy + 1\\). What is the maximum number of elements in this set?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-d2e6bf753d651d29249512f47b68f8cfb969145cd2b09e7271b3aac2af497891 | A subset of the set $\{1, 2, 3, \ldots, 2020\}$ is chosen randomly, and the product $P$ of the numbers in that subset is taken (if the set is empty, $P=1$). The value of the expected value of $P$ can be expressed as $\frac{m}{2^n}$, where $m$ and $n$ are positive integers and $m$ is odd. Find the value of $n$. | [
"7"
] | [
"A subset of the set $\\{1, 2, 3, \\ldots, 2020\\}$ is chosen randomly, and the product $P$ of the numbers in that subset is taken (if the set is empty, $P=1$). The value of the expected value of $P$ can be expressed as $\\frac{m}{2^n}$, where $m$ and $n$ are positive integers and $m$ is odd. Find the value of $n$.... | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-66a85aa099022cec51ac5b5408c1203e657e39c78de8fe3dabff924105ed6e6f | A subset of the set $\{1, 2, 3, \ldots, 2020\}$ is chosen randomly, and the product $P$ of the numbers in that subset is taken (if the subset is empty, $P$ is defined as $1$). The value of the expected value of $P$ can be expressed as $\frac{m}{2^n}$ where $m$ and $n$ are positive integers and $m$ is odd. Find the valu... | [
"7"
] | [
"A subset of the set $\\{1, 2, 3, \\ldots, 2020\\}$ is chosen randomly, and the product $P$ of the numbers in that subset is taken (if the subset is empty, $P$ is defined as $1$). The value of the expected value of $P$ can be expressed as $\\frac{m}{2^n}$ where $m$ and $n$ are positive integers and $m$ is odd. Find... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a1ac614157bc4071f472f7262aa8581d3cdaf7e527a7ca1349efecacecf11b5b | A subset of the set $\{1, 2, 3, \ldots, 30\}$ is called *delicious* if it does not contain elements $a$ and $b$ such that $a = 3b$. A *delicious* subset is called *super delicious* if no other *delicious* subset has more elements than it. Determine the number of *super delicious* subsets. | [
"32"
] | [
"A subset of the set $\\{1, 2, 3, \\ldots, 30\\}$ is called *delicious* if it does not contain elements $a$ and $b$ such that $a = 3b$. A *delicious* subset is called *super delicious* if no other *delicious* subset has more elements than it. Determine the number of *super delicious* subsets."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-53c196f1fa81d8f3ce6a539157753450a5d33cf97d6d3f2a12f57658554ce640 | A substance contains two Radon isotopes, 210Rn (with a half-life of 2.42 hours) and 211Rn (with a half-life of 15 hours). Initially, 20% of the decays come from 211Rn. How long must one wait until 80% of the decays come from 211Rn? | [
"11.54"
] | [
"A substance contains two Radon isotopes, 210Rn (with a half-life of 2.42 hours) and 211Rn (with a half-life of 15 hours). Initially, 20% of the decays come from 211Rn. How long must one wait until 80% of the decays come from 211Rn?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-17dfd2e1bd1b0fd9d37392c1acef3b60814eb07e47770a0ec798da19c22ec70e | A substance emits particles at an average rate of 200 particles per minute. What is the probability that exactly 200 particles will be emitted in a 1-minute count? | [
"\\(\\frac{200^{200} e^{-200}}{200!}\\)"
] | [
"A substance emits particles at an average rate of 200 particles per minute. What is the probability that exactly 200 particles will be emitted in a 1-minute count?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-f6284c4e12b01115dceaa3387f61f574225823cbfacf988b16492e8396e7a38e | A suitcase lock has 3 dials with the digits $0, 1, 2,..., 9$ on each. How many different settings are possible if all three digits have to be different? | [
"720"
] | [
"A suitcase lock has 3 dials with the digits $0, 1, 2,..., 9$ on each. How many different settings are possible if all three digits have to be different?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-96b11dc01c27e0a1c91d2d3d5fcc6a1a2034c9afce973848ee237ebd828f453d | A sum of $5000 is invested at an interest rate of 9% per year, compounded semi-annually. Find the time required for the money to double. | [
"7.88"
] | [
"A sum of $5000 is invested at an interest rate of 9% per year, compounded semi-annually. Find the time required for the money to double."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 3,
"tir": 11
} | 18 | 0 | 0 |
omr-ws-4b0be57e0f161edf7188eaf9178135ebf9fdaff9061e652cc28232398e5b8a29 | A sum of 10 roubles and 56 kopecks was paid for several textbooks and 56 kopecks for several exercise books. Six more textbooks than exercise hooks were bought. How many textbooks were purchased if the price of one textbook is over a rouble more than that of an exercise book? | [
"8 books"
] | [
"A sum of 10 roubles and 56 kopecks was paid for several textbooks and 56 kopecks for several exercise books. Six more textbooks than exercise hooks were bought. How many textbooks were purchased if the price of one textbook is over a rouble more than that of an exercise book?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cd964f9166976e95e8807c02bf72ae44ceca73ba93a161914036524e331b111f | A sum of a two-digit number and a number represented with the aid of the same digits but written in the reverse order is a perfect square. Find all such numbers. | [
"7744"
] | [
"A sum of a two-digit number and a number represented with the aid of the same digits but written in the reverse order is a perfect square. Find all such numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1403a8d3564ae7a3c81d6933819909fdaff2b4efeea24861461ce28275aa141a | A sum of money is invested in the Mullionaire's Club Bank at 10% compounded continuously. Determine after how many years there will be a 150% increase in the original investment. | [
"\\( 10 \\ln 1.5 \\) years"
] | [
"A sum of money is invested in the Mullionaire's Club Bank at 10% compounded continuously. Determine after how many years there will be a 150% increase in the original investment."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-ac2bc7fc27985f34c5c5de8d55f7e313d0a1a91f1c2b1687323ec8b2bbdf6b06 | A sum of several consecutive positive integers is equal to 1000. Find these integers. | [
"198, 199, 200, 201, 202; 28, 29, ..., 52; 55, 56, ..., 70\n\nHowever, since the format should strictly follow the extraction of final answers as shown in examples, the most appropriate format would be:\n\nm=198, k=4; m=28, k=24; m=55, k=15\n\nBut if we need to strictly adhere to the integers themselves as the fina... | [
"A sum of several consecutive positive integers is equal to 1000. Find these integers."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8d32cc944eec9db9433b544c1c236c9c5b3c26784632f35fa1fe74338be3e2d5 | A sum-day is a date in which the day and the month add up to the year. For example, January 1, 2002 is a sum day because 01/01/02 would be 1+1=2. How many sum-days are there in the 21st century? | [
"365"
] | [
"A sum-day is a date in which the day and the month add up to the year. For example, January 1, 2002 is a sum day because 01/01/02 would be 1+1=2. How many sum-days are there in the 21st century?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 7
} | 16 | 0 | 0 |
omr-ws-7abc1161c1b791f47aaa7adb7a48d6e4b081097cbbc9ccd03521f73c3e596925 | A sun is created by cutting a regular octagon and then cutting eight equilateral triangles from the octagon, each of which has a side that is also a side of the octagon. What is the measure, in degrees, of each of the angles at the tips of the sun's rays? | [
"15 degrees"
] | [
"A sun is created by cutting a regular octagon and then cutting eight equilateral triangles from the octagon, each of which has a side that is also a side of the octagon. What is the measure, in degrees, of each of the angles at the tips of the sun's rays?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6a11144775755dbe4eeff32a50d5c918f321e322fe7d4f83b0b6f97d68cb71de | A super ball dropped from a height of 36 inches rebounds three-fourths of the distance of the previous bounce. Write an expression to represent the distance traveled by the ball in \( n \) bounces. | [
"\\( 36 + 72 \\sum_{i=1}^{n} \\left(\\frac{3}{4}\\right)^i \\)"
] | [
"A super ball dropped from a height of 36 inches rebounds three-fourths of the distance of the previous bounce. Write an expression to represent the distance traveled by the ball in \\( n \\) bounces."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 7,
"genselect": 6,
"tir": 4
} | 17 | 0 | 0 |
omr-ws-2cca21bf344b502e6fe8d4827dc3e19a007d69a63ccb6f9cfb816d51f1f3edfb | A super ball is dropped from 100 feet and rebounds half the distance it falls each time it bounces. How many feet will the ball have traveled when it hits the ground the fourth time? | [
"275"
] | [
"A super ball is dropped from 100 feet and rebounds half the distance it falls each time it bounces. How many feet will the ball have traveled when it hits the ground the fourth time?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-68767b629d623b491ac3e1a2bc7316e386bfefb54f6157d2a3023d2fbf3c9da7 | A superdomino is a domino that can have a positive integer amount of up to 12 dots in each of its two squares. A superdomino is called unique if the number of dots on its top square is relatively prime to the number of dots on its bottom square. What is the sum of the numbers (top and bottom) on all such unique superdo... | [
"882"
] | [
"A superdomino is a domino that can have a positive integer amount of up to 12 dots in each of its two squares. A superdomino is called unique if the number of dots on its top square is relatively prime to the number of dots on its bottom square. What is the sum of the numbers (top and bottom) on all such unique su... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1c8b83ad1f7c7307d4ff0cd91f23c422dabdec9daf46f1500b4ccc0577f62c1f | A supermarket buys 1000 frozen chickens from a supplier, and it is estimated that 30% of frozen chickens contain enough salmonella bacteria to cause illness if improperly cooked. The number of frozen chickens that may be contaminated and are within two standard deviations of the mean is between A and B. Determine the v... | [
"(271, 329)"
] | [
"A supermarket buys 1000 frozen chickens from a supplier, and it is estimated that 30% of frozen chickens contain enough salmonella bacteria to cause illness if improperly cooked. The number of frozen chickens that may be contaminated and are within two standard deviations of the mean is between A and B. Determine ... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 27
} | 28 | 0 | 0 |
omr-ws-661116a211a31be9b0604ca9104d1ed5efcf1177b2b37f3ea00e5161d34531a2 | A surface is given by \( z = f(x, y) \), where \( f(x, y) = xy \). A person stands at the point \( (3, -4) \) and wishes to walk in the direction of the greatest descent. Determine the direction \( \vec{v} \) in the plane in which he will start walking and the slope of the surface in this direction. | [
"\\( \\vec{v} = (4, -3) \\), slope = 5"
] | [
"A surface is given by \\( z = f(x, y) \\), where \\( f(x, y) = xy \\). A person stands at the point \\( (3, -4) \\) and wishes to walk in the direction of the greatest descent. Determine the direction \\( \\vec{v} \\) in the plane in which he will start walking and the slope of the surface in this direction."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b9d127a6110ae6b006c6efbfec184c8187a512297210adbbadec85deffeb3ab9 | A survey carried out over a large number of pop music fans reveals that 95% of them like at least one of Madonna, Sting, and Elvis. A total of 53% like Madonna, 42% like Sting, and 43% like Elvis. If 18% like only Elvis, and 7% like all three, what percentage like Madonna and Sting but not Elvis? | [
"11"
] | [
"A survey carried out over a large number of pop music fans reveals that 95% of them like at least one of Madonna, Sting, and Elvis. A total of 53% like Madonna, 42% like Sting, and 43% like Elvis. If 18% like only Elvis, and 7% like all three, what percentage like Madonna and Sting but not Elvis?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c203c75509948edd423e287c5f8a6eb76f140fba5f78ce2928fdcad933e28129 | A survey of 300 parks showed the following: - 15 had only camping - 20 had only hiking trails - 35 had only picnicking - 185 had camping - 140 had camping and hiking trails - 125 had camping and picnicking - 210 had hiking trails Find the number of parks that: a) had at least one of these features. b) had all three fea... | [
"125"
] | [
"A survey of 300 parks showed the following:\n- 15 had only camping\n- 20 had only hiking trails\n- 35 had only picnicking\n- 185 had camping\n- 140 had camping and hiking trails\n- 125 had camping and picnicking\n- 210 had hiking trails\n\nFind the number of parks that:\na) had at least one of these features.\nb) ... | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ffa51c3ad798f043229c641d0ca22a987e76fe8936f08ce9e9103627aa68e7e1 | A surveyor asks a mother about the ages of her daughters. The mother states that the product of her daughters' ages is 36. The surveyor then learns that the mother has to pick up her daughter Annie from swimming lessons and the twins from daycare. What are the ages of the daughters? | [
"2, 2, 9"
] | [
"A surveyor asks a mother about the ages of her daughters. The mother states that the product of her daughters' ages is 36. The surveyor then learns that the mother has to pick up her daughter Annie from swimming lessons and the twins from daycare. What are the ages of the daughters?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 10,
"tir": 20
} | 30 | 0 | 0 |
omr-ws-91eca63875d97ce7d814ee1698efa350dd861017664487b4b4548d33df77d540 | A surveyor measures the angle DAB to be 35 degrees, walks off a distance of 800 feet to point C, and measures the angle ACB to be 25 degrees. What is the distance from A to B? | [
"390"
] | [
"A surveyor measures the angle DAB to be 35 degrees, walks off a distance of 800 feet to point C, and measures the angle ACB to be 25 degrees. What is the distance from A to B?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f91bb0e9916cbd2002637dea4b6965ae33192ac86bf1c4f9be031262d23c9daa | A surveyor wishes to find the distance across a swamp. The bearing from point A to point B is N 32° W. The surveyor walks 50 meters from A to point C, and at point C, the bearing to B is N 68° W. Find (a) the bearing from A to C and (b) the distance from A to B. | [
"a) 58°, b) 36.325 meters"
] | [
"A surveyor wishes to find the distance across a swamp. The bearing from point A to point B is N 32° W. The surveyor walks 50 meters from A to point C, and at point C, the bearing to B is N 68° W. Find (a) the bearing from A to C and (b) the distance from A to B."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d5946863e6afa6e203801637b68b95bbe290b095c7a3aebeb2237bcf6907325b | A swimmer is at the center of a circular pool. A dog runs along the edge of the pool at a speed 4 times greater than the swimmer's speed in the water. Determine if the swimmer can leave the pool without being caught by the dog. | [
"Yes, the swimmer can leave the pool without being caught by the dog."
] | [
"A swimmer is at the center of a circular pool. A dog runs along the edge of the pool at a speed 4 times greater than the swimmer's speed in the water. Determine if the swimmer can leave the pool without being caught by the dog."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 1
} | 2 | 0 | 0 |
omr-ws-659e3ecc73a305567c0bf1b40f3f52ba676ba91db6d4613294997345c448819d | A swimmer is at the center of a regular hexagon pool and a runner is at the midpoint of one side of the pool. The swimmer moves at a speed not exceeding $1$, while the runner's speed doesn't exceed $v$. Determine the minimum value of $v$ such that the runner can catch the swimmer before the swimmer reaches the edge of ... | [
"2"
] | [
"A swimmer is at the center of a regular hexagon pool and a runner is at the midpoint of one side of the pool. The swimmer moves at a speed not exceeding $1$, while the runner's speed doesn't exceed $v$. Determine the minimum value of $v$ such that the runner can catch the swimmer before the swimmer reaches the edg... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-a88d836c435bf74d53aec07adfbda42bcdd9902c4ce962507666f998bacae3b1 | A swimmer is standing at a corner of a square swimming pool. She swims at a fixed speed and runs at a fixed speed (possibly different). No time is taken entering or leaving the pool. What path should she follow to reach the opposite corner of the pool in the shortest possible time? | [
"Swim along the diagonal if \\(r < \\sqrt{2}s\\), run around the boundary if \\(r > \\sqrt{2}s\\), or either path if \\(r = \\sqrt{2}s\\).",
"\\text{She should follow a path that bends at the pool's edge according to Snell's Law, refracting based on the ratio of her swimming and running speeds.}"
] | [
"A swimmer is standing at a corner of a square swimming pool. She swims at a fixed speed and runs at a fixed speed (possibly different). No time is taken entering or leaving the pool. What path should she follow to reach the opposite corner of the pool in the shortest possible time?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 4,
"tir": 17
} | 26 | 0 | 0 |
omr-ws-ae90675c0fb560697f18b65064772722407af11f9ad78adbb26d4870f97ce03d | A swimmer needs to cross a river of width \( a \) meters. The speed of the river flow is \( w \) m/s, the swimmer's speed in still water is \( v \) m/s (\( v < w \)), and the swimmer's speed on dry land is \( u \) m/s. The swimmer must reach a point directly opposite the starting point on the opposite bank. The swimmer... | [
"\\( a\\frac{\\sqrt{(u+w)^2-v^2}}{uv} \\)"
] | [
"A swimmer needs to cross a river of width \\( a \\) meters. The speed of the river flow is \\( w \\) m/s, the swimmer's speed in still water is \\( v \\) m/s (\\( v < w \\)), and the swimmer's speed on dry land is \\( u \\) m/s. The swimmer must reach a point directly opposite the starting point on the opposite ba... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 46,
"tir": 4
} | 50 | 0 | 0 |
omr-ws-87559dc5fb172673f53f0a871ff926bfd411ee737db39a00cc34da0e6fa08c97 | A swimmer wants to cross a $150$ meter wide river. His swimming speed is $\frac{5}{8}$ meters per second, and the river flows at $\frac{1}{2}$ meters per second. What is the minimum time required for the swimmer to cross the river? (The swimmer can change direction but can only swim at a speed of $\frac{5}{8}$ meters p... | [
"\\(240\\) seconds"
] | [
"A swimmer wants to cross a $150$ meter wide river. His swimming speed is $\\frac{5}{8}$ meters per second, and the river flows at $\\frac{1}{2}$ meters per second. What is the minimum time required for the swimmer to cross the river? (The swimmer can change direction but can only swim at a speed of $\\frac{5}{8}$ ... | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-72d0a1eabc78097b391481a6e5ad952d0ec58af9d7a9bdde0bfd898ae215eaba | A swimming pool can be filled by any of three hoses A, B or C. Hoses A and B together take 4 hours to fill the pool. Hoses A and C together take 5 hours to fill the pool. Hoses B and C together take 6 hours to fill the pool. How many hours does it take hoses A, B and C working together to fill the pool? Express your an... | [
"3.24"
] | [
"A swimming pool can be filled by any of three hoses A, B or C. Hoses A and B together take 4 hours to fill the pool. Hoses A and C together take 5 hours to fill the pool. Hoses B and C together take 6 hours to fill the pool. How many hours does it take hoses A, B and C working together to fill the pool? Express yo... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cfd1d167f080f0b7f4d65283d44287547133f43cd61e5085f816bc63050353f0 | A swimming pool is 12 meters long, 6 meters wide, 1 meter deep at the shallow end, and 3 meters deep at the deep end. Water is being pumped into the pool at 1/4 cubic meter per minute, and there is 1 meter of water at the deep end. At what rate is the water level rising? | [
"\\(\\frac{1}{72} \\frac{\\text{m}}{\\text{min}}\\)"
] | [
"A swimming pool is 12 meters long, 6 meters wide, 1 meter deep at the shallow end, and 3 meters deep at the deep end. Water is being pumped into the pool at 1/4 cubic meter per minute, and there is 1 meter of water at the deep end. At what rate is the water level rising?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4dbd7aac6e8bdfbd3b57889370ad8190dc07acf76aba6da4e61d4503ccda1ca5 | A swimming pool is 12 meters long, 6 meters wide, 1 meter deep at the shallow end, and 3 meters deep at the deep end. Water is being pumped into the pool at 1/4 cubic meter per minute, and there is 1 meter of water at the deep end. What percent of the pool is filled? | [
"12.5%"
] | [
"A swimming pool is 12 meters long, 6 meters wide, 1 meter deep at the shallow end, and 3 meters deep at the deep end. Water is being pumped into the pool at 1/4 cubic meter per minute, and there is 1 meter of water at the deep end. What percent of the pool is filled?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-6507082e764bf032d612a8044444d4cae7ab08e96bc19d6beb412f16e68675e8 | A swimming pool is 15 ft wide, 40 ft long, and has a depth that varies from 3 ft at one end to 10 ft at the other. Water is being added to the pool at a rate of 25 ft³/min. a) How fast is the water level rising when there is 2000 ft³ of water in the pool? b) How fast is the water level rising when there is 3000 ft³ of ... | [
"\\dfrac{1}{24}"
] | [
"A swimming pool is 15 ft wide, 40 ft long, and has a depth that varies from 3 ft at one end to 10 ft at the other. Water is being added to the pool at a rate of 25 ft³/min. \n\na) How fast is the water level rising when there is 2000 ft³ of water in the pool?\n\nb) How fast is the water level rising when there is ... | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-89b87eee2b5f396531ad3ea87c9868ffa81d9d08442bd97a16c576f43a6c0d71 | A swimming pool is 60 ft long and 25 ft wide. Its depth varies uniformly from 3 ft at the shallow end to 15 ft at the deep end. Suppose the pool is being filled with water at the rate of 800 ft³/min. At what rate is the depth of water increasing at the deep end when it is 5 ft deep at that end? | [
"\\dfrac{32}{25}"
] | [
"A swimming pool is 60 ft long and 25 ft wide. Its depth varies uniformly from 3 ft at the shallow end to 15 ft at the deep end. Suppose the pool is being filled with water at the rate of 800 ft³/min. At what rate is the depth of water increasing at the deep end when it is 5 ft deep at that end?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5a2309a14cf134dcf62b57e6d2a99a4d06de9f274d292972d81a97b9937522cf | A swimming pool is filled by two pipes in 6 hours. One pipe alone fills it 5 hours faster than does the other pipe alone. How long will it take for each pipe operating individually to fill the pool? | [
"The first pipe fills the pool in 10 hours, the second, in 15 hours"
] | [
"A swimming pool is filled by two pipes in 6 hours. One pipe alone fills it 5 hours faster than does the other pipe alone. How long will it take for each pipe operating individually to fill the pool?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-81060255d51c8a6acca63259a785197d040373617bd48a3732c2637e8ffbe642 | A swimming pool is to be constructed with a full capacity of 400 m³. The pool is designed to have a semi-circle at one side of a rectangular shape, with the diameter of the circle equal to the length of that side. Find the dimensions of the pool that will have the largest inner area. | [
"\\sqrt[3]{\\dfrac{800}{4 + \\pi}}"
] | [
"A swimming pool is to be constructed with a full capacity of 400 m³. The pool is designed to have a semi-circle at one side of a rectangular shape, with the diameter of the circle equal to the length of that side. Find the dimensions of the pool that will have the largest inner area."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 4
} | 10 | 0 | 0 |
omr-ws-3086a5544f92a51b3fa85822c2d0c6aff16c9e1b795f26a81474686829385da6 | A symmetrical cross with equal arms has an area of 2016 cm^2 and all sides have integer length in centimeters. What is the smallest perimeter the cross can have, in centimeters? | [
"184"
] | [
"A symmetrical cross with equal arms has an area of 2016 cm^2 and all sides have integer length in centimeters. What is the smallest perimeter the cross can have, in centimeters?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 8,
"tir": 23
} | 41 | 0 | 0 |
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