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-a66cff87f543efc954e897128fa830460bc85698c1237e54dbad63396904483c | What is the largest integer \(n\) that satisfies \((100^2 - 99^2)(99^2 - 98^2) \cdots (3^2 - 2^2)(2^2 - 1^2)\) is divisible by \(3^n\)? | [
"49"
] | [
"What is the largest integer \\(n\\) that satisfies \\((100^2 - 99^2)(99^2 - 98^2) \\cdots (3^2 - 2^2)(2^2 - 1^2)\\) is divisible by \\(3^n\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-f6817fff23a21b7efaa0847c27e8b0e35d86465d1c90da2461b86717152cd18d | What is the largest integer divisible by all positive integers less than its cube root? | [
"420"
] | [
"What is the largest integer divisible by all positive integers less than its cube root?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-acc5878aa1214a28f78c7d814bbf7a3ea77fee310c6af95092178449ab17d9cd | What is the largest integer less than $\log_2 \frac{2}{1} + \log_2 \frac{3}{2} + \cdots + \log_2 \frac{2009}{2008} + \log_2 \frac{2010}{2009}$? | [
"10"
] | [
"What is the largest integer less than $\\log_2 \\frac{2}{1} + \\log_2 \\frac{3}{2} + \\cdots + \\log_2 \\frac{2009}{2008} + \\log_2 \\frac{2010}{2009}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-55060464504f7da5cca4e0b97c878d4f548b5d8b945a1168228091efcdf5b60b | What is the largest integer less than $\sqrt[3]{(2011)^3 + 3 \times (2011)^2 + 4 \times 2011 + 5}$? | [
"2012"
] | [
"What is the largest integer less than $\\sqrt[3]{(2011)^3 + 3 \\times (2011)^2 + 4 \\times 2011 + 5}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-06a2b0e682f35fb33333ce179adbc4e32610166dc05b1cde5d77a77ecf07f7ad | What is the largest integer less than or equal to $4x^3 - 3x$, where $x=\frac{\sqrt[3]{2+\sqrt3}+\sqrt[3]{2-\sqrt3}}{2}$? | [
"2"
] | [
"What is the largest integer less than or equal to $4x^3 - 3x$, where $x=\\frac{\\sqrt[3]{2+\\sqrt3}+\\sqrt[3]{2-\\sqrt3}}{2}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8c787e0bd494baabc9e0ab7d816f3d984c4a48a9191c9042629101a4e68105f2 | What is the largest integer less than or equal to \[ \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \frac{1}{\sqrt{4}} + \cdots + \frac{1}{\sqrt{10000}}? \] | [
"197"
] | [
"What is the largest integer less than or equal to \n\\[\n\\frac{1}{\\sqrt{2}} + \\frac{1}{\\sqrt{3}} + \\frac{1}{\\sqrt{4}} + \\cdots + \\frac{1}{\\sqrt{10000}}?\n\\]"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 8
} | 23 | 0 | 0 |
omr-ws-6a4d6c493624634020013fb43d995600ae4030b4f642201f0281e1f165e5b711 | What is the largest integer not exceeding $8x^3 + 6x - 1$, where $x = \frac{1}{2} \left(\sqrt[3]{2+\sqrt{5}} + \sqrt[3]{2-\sqrt{5}}\right)$? | [
"3"
] | [
"What is the largest integer not exceeding $8x^3 + 6x - 1$, where $x = \\frac{1}{2} \\left(\\sqrt[3]{2+\\sqrt{5}} + \\sqrt[3]{2-\\sqrt{5}}\\right)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-4d62b00a3faf7671d199fd656bf68564411536427d6d98c403f41e34812e82c2 | What is the largest integer not exceeding $[(n+1)a] - [na]$ where $n$ is a natural number and $a = \frac{\sqrt{2013}}{\sqrt{2014}}$? | [
"1"
] | [
"What is the largest integer not exceeding $[(n+1)a] - [na]$ where $n$ is a natural number and $a = \\frac{\\sqrt{2013}}{\\sqrt{2014}}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3d591f7b3246607ffc74a491f7e06d55f1aa5ffdbb3c4bdf788e0ea4993bd41c | What is the largest integer of the form $3a + 5b + 7c$, where $a, b, c \in \mathbb{Z}^+$, that cannot be obtained? | [
"19"
] | [
"What is the largest integer of the form $3a + 5b + 7c$, where $a, b, c \\in \\mathbb{Z}^+$, that cannot be obtained?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 21
} | 43 | 0 | 0 |
omr-ws-e0bd6741c8143bc6c0ed8262121e98faa018fef219e17377a70904693fb0beac | What is the largest integer such that every pair of neighbouring digits, taken in order, forms a different prime number? | [
"83797173119"
] | [
"What is the largest integer such that every pair of neighbouring digits, taken in order, forms a different prime number?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-65a944e73764666321a7dbc33f0846a025b724a37556bd36f5da8566ff8021d6 | What is the largest integer such that you can determine the chosen number with twenty "yes" or "no" questions? | [
"1,048,576\n\nExplanation:\nThe forum discussion provides a clear explanation that with twenty \"yes\" or \"no\" questions, the largest integer that can be determined is 1,048,576. This is based on the binary representation of numbers and the divide-and-conquer strategy, where each question ideally halves the remai... | [
"What is the largest integer such that you can determine the chosen number with twenty \"yes\" or \"no\" questions?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 34,
"genselect": 3,
"tir": 26
} | 63 | 0 | 0 |
omr-ws-a50751fa54b54a25910a2308fa20e29df0aba82a234d706a40938e8b177ee6da | What is the largest integer that cannot be expressed as $5a + 7b$ for positive integers $a$ and $b$? | [
"17"
] | [
"What is the largest integer that cannot be expressed as $5a + 7b$ for positive integers $a$ and $b$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4354b20c5357e6299a91567d50be7bbe091b6cd3a90e37ca7a51321d416381f2 | What is the largest integer that cannot be obtained by adding as many 7's and 9's as you like? What if you are allowed to add as many 6's and 9's as you like? | [
"47 (for 7's and 9's), and no largest integer (for 6's and 9's)"
] | [
"What is the largest integer that cannot be obtained by adding as many 7's and 9's as you like? What if you are allowed to add as many 6's and 9's as you like?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 10
} | 10 | 0 | 0 |
omr-ws-2c2e1a03a10170d669c17cf39ed35dcf54e3cfef21b3b71a9d5cb73cc0f05087 | What is the largest integer that cannot be obtained from adding as many 11's, 12's, 13's, and 14's as you like? What is the largest integer that cannot be obtained from adding as many x's, x+1's, x+2's, and x+3's as you like? | [
"4x - 1"
] | [
"What is the largest integer that cannot be obtained from adding as many 11's, 12's, 13's, and 14's as you like? What is the largest integer that cannot be obtained from adding as many x's, x+1's, x+2's, and x+3's as you like?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 11
} | 11 | 0 | 0 |
omr-ws-b51e7e941c7a2ac8be762b0586ac9eb16edfc9d7de4ebc71a195c938dbe9e943 | What is the largest integer that cannot be obtained from adding as many 11's, 17's, and 34's as you like? What is the largest integer that cannot be obtained from adding as many 11's, 17's, and 37's as you like? | [
"97"
] | [
"What is the largest integer that cannot be obtained from adding as many 11's, 17's, and 34's as you like? What is the largest integer that cannot be obtained from adding as many 11's, 17's, and 37's as you like?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-69046143941157ba359f207fe2c27dd0e604840adaf0bea442e254551c82300f | What is the largest integer that cannot be obtained from adding as many 9's, 10's, and 11's as you like? What is the largest integer that cannot be obtained from adding as many x's, x+1's, and x+2's as you like? | [
"\\left\\lfloor \\dfrac{x}{2} \\right\\rfloor \\cdot x - 1"
] | [
"What is the largest integer that cannot be obtained from adding as many 9's, 10's, and 11's as you like? What is the largest integer that cannot be obtained from adding as many x's, x+1's, and x+2's as you like?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-2dce90cef6d90feee79e3e88c262647028c6417cd3e5b733e4dd2167bc908f45 | What is the largest integer that cannot be obtained from adding as many as you like of n consecutive integers starting with x? | [
"\\frac{n(2x + n - 1)}{2} - 1"
] | [
"What is the largest integer that cannot be obtained from adding as many as you like of n consecutive integers starting with x?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 2
} | 9 | 0 | 0 |
omr-ws-9b492907de5bc74c9f3e73703ad36821854167440e84bd1f121129953fa4a36f | What is the largest integer that cannot be written in the form $6a + 9b + 20c$? | [
"37"
] | [
"What is the largest integer that cannot be written in the form $6a + 9b + 20c$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4a9c79729711094e8e44ba5dc69cb0c2d11fa030d77366a002fa5f772b55f00d | What is the largest integer that divides $(n+1)(n+3)(n+7)(n+9)$ for all positive even integers $n$? | [
"15"
] | [
"What is the largest integer that divides $(n+1)(n+3)(n+7)(n+9)$ for all positive even integers $n$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-112f00269bf0252ca36d0371629e20797fbc95a0c27d989c4ac95449baec8007 | What is the largest integer that divides \( n^2 \cdot (n^2 - 1) \cdot (n^2 - 4) \) for all integers \( n \)? | [
"360"
] | [
"What is the largest integer that divides \\( n^2 \\cdot (n^2 - 1) \\cdot (n^2 - 4) \\) for all integers \\( n \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 13
} | 28 | 0 | 0 |
omr-ws-b9eed9d895eb86edcfcbbd51dafb84d2b1a6440141b755f809f3d599020994fd | What is the largest integer that divides \( n^4 - n^2 \) for all integers \( n \)? | [
"12"
] | [
"What is the largest integer that divides \\( n^4 - n^2 \\) for all integers \\( n \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 17
} | 32 | 0 | 0 |
omr-ws-e45213afc671378c117db210b73f601e609b4fba45c61e021088a7387ec882c8 | What is the largest integer that divides \((n+1)(n+3)(n+5)(n+7)(n+9)\) for all even integers \(n\)? | [
"15"
] | [
"What is the largest integer that divides \\((n+1)(n+3)(n+5)(n+7)(n+9)\\) for all even integers \\(n\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-a5012d8b9b44787f5cf81b91e01ea5b14fd8d5bc3323c4de9f27c4d04d27cf00 | What is the largest integer that is a divisor of \((n+1)(n+3)(n+5)(n+7)(n+9)\) for all positive even integers \(n\)? | [
"5"
] | [
"What is the largest integer that is a divisor of \\((n+1)(n+3)(n+5)(n+7)(n+9)\\) for all positive even integers \\(n\\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fb4420d0ccbaba8d4204126dc33c379e68d7b150e747d6dda8ff31db691ae154 | What is the largest integer that is a divisor of \((n+1)(n+3)(n+5)(n+7)(n+9)\)? | [
"15"
] | [
"What is the largest integer that is a divisor of \\((n+1)(n+3)(n+5)(n+7)(n+9)\\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-b5479cd51ca7b5cca99a04bc467a87af02352f3e18e02832e35ba04239eabe70 | What is the largest integer that is a divisor of \[ (n+1)(n+3)(n+5)(n+7)(n+9) \]for all positive even integers $n$? | [
"15"
] | [
"What is the largest integer that is a divisor of \\[\n(n+1)(n+3)(n+5)(n+7)(n+9)\n\\]for all positive even integers $n$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f81c5db3fff04eba6611b339bb676543899a9c8dafa9c162b82768490fa987e7 | What is the largest integer that is equal to the product of its digits? | [
"9"
] | [
"What is the largest integer that is equal to the product of its digits?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-87c1b6739495f1177bc376fec251514cf77547dbbc661c43bc0309a9a302ea01 | What is the largest integer that must divide $n^5 - 5n^3 + 4n$ for all integers $n$? | [
"120"
] | [
"What is the largest integer that must divide $n^5 - 5n^3 + 4n$ for all integers $n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-ca6809efe8de45861196d70ccee293d098287426cb8f49d1a46df277c0bc7756 | What is the largest integer that must divide all integers of the form $n^5 - n$? | [
"30"
] | [
"What is the largest integer that must divide all integers of the form $n^5 - n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-ccb254a002afe90fa3a1d0b56595a2bc9acfd4458856e8b222d5555a0d810d05 | What is the largest integer that must divide the product of any $4$ consecutive integers? | [
"24"
] | [
"What is the largest integer that must divide the product of any $4$ consecutive integers?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-59868b467804b734c6f25bb7db59195f7f7717f17b39244154985d28fe7a42b8 | What is the largest integer value of $x$ for which $5-4x>17$? | [
"-4"
] | [
"What is the largest integer value of $x$ for which $5-4x>17$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-562a30364c106ca4a2bf0a3d18a1dc41fdaca1c7937d73a60ae8d5751ca8a74a | What is the largest integer which cannot be expressed as $2008x + 2009y + 2010z$ for some positive integers $x, y,$ and $z$? | [
"6031"
] | [
"What is the largest integer which cannot be expressed as $2008x + 2009y + 2010z$ for some positive integers $x, y,$ and $z$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bfee99e4fd26f0899ef439f797efcfef86f67fbf7f147d90d0c1a43871c4c2b7 | What is the largest integer which cannot be expressed as \(2008x + 2009y + 2010z\) for some positive integers \(x\), \(y\), and \(z\)? | [
"2022058"
] | [
"What is the largest integer which cannot be expressed as \\(2008x + 2009y + 2010z\\) for some positive integers \\(x\\), \\(y\\), and \\(z\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 6
} | 11 | 0 | 0 |
omr-ws-749a973e04985b272cb6fab3043c0fb339474ddcec0ef11f850d95b1448fd53e | What is the largest integer which cannot be written as a positive multiple of $5$ plus a positive multiple of $9$? | [
"45"
] | [
"What is the largest integer which cannot be written as a positive multiple of $5$ plus a positive multiple of $9$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 7
} | 55 | 0 | 0 |
omr-ws-8e991d64b2bb4fb52f450bd066e29e3dae50b7e00f6374f6e1da18161426b086 | What is the largest integer which must evenly divide all integers of the form \( n^5 - n \)? | [
"30"
] | [
"What is the largest integer which must evenly divide all integers of the form \\( n^5 - n \\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 45
} | 76 | 0 | 0 |
omr-ws-3d65edcfd6270e22b24f28dd49fb72c57cbf14a756236702a1eba53db58e9669 | What is the largest integer whose prime factors add up to 14? | [
"210"
] | [
"What is the largest integer whose prime factors add up to 14?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4be78e740e790b780c5d2d433054c52df97d9011c9ce6ac37b4fd38dc4524ad7 | What is the largest interior angle sum possible for a triangle on a sphere that has 2 right angles? | [
"\\(360^\\circ\\)"
] | [
"What is the largest interior angle sum possible for a triangle on a sphere that has 2 right angles?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 9
} | 32 | 0 | 0 |
omr-ws-538d34428c14d96532f975b6d4a70d79afedfd0ce328684684bb4d014393ba83 | What is the largest interval on which the floor function has an antiderivative? | [
"\\((n, n+1)\\) for any integer \\(n\\)"
] | [
"What is the largest interval on which the floor function has an antiderivative?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 7,
"tir": 9
} | 16 | 0 | 0 |
omr-ws-eab6bf1b7a15eff2da18b01a089065f5c979f3a3feaadc55c005104690ce8b20 | What is the largest length of a sequence of consecutive natural numbers where the difference between the largest and smallest digits of each number is 5? | [
"12"
] | [
"What is the largest length of a sequence of consecutive natural numbers where the difference between the largest and smallest digits of each number is 5?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 8,
"tir": 3
} | 23 | 0 | 0 |
omr-ws-19df0331a7f32a0ad38dde7fea0294fcc39cf8a6858212316eba3408d325c6fd | What is the largest multiple of $9$ which is smaller than $-70$? | [
"-72"
] | [
"What is the largest multiple of $9$ which is smaller than $-70$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-36a0acfd9720f1277589dc3a613f3e2e517eb1f724efa1babf18a695554c4ec3 | What is the largest multiple of $9$ whose negation is greater than $-100$? | [
"99"
] | [
"What is the largest multiple of $9$ whose negation is greater than $-100$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a424c2b1b781850e8679897963bc343824502cc244905e641dc5fb6d6978894d | What is the largest multiple of 7 less than 10,000 which can be expressed as the sum of squares of three consecutive numbers? | [
"8750"
] | [
"What is the largest multiple of 7 less than 10,000 which can be expressed as the sum of squares of three consecutive numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0089352762168dbb47c47a715cc3a53f0ab839179f972b67d951ab765f84529b | What is the largest multiple of 7 less than 10000 which can be expressed as the sum of squares of three consecutive numbers? | [
"8750"
] | [
"What is the largest multiple of 7 less than 10000 which can be expressed as the sum of squares of three consecutive numbers?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 44
} | 75 | 0 | 0 |
omr-ws-6325cec00194469f2d10d3108e6b1579d01ee2c4428b983a7f829603e3c73df4 | What is the largest multiple of 7 less than 50? | [
"49"
] | [
"What is the largest multiple of 7 less than 50?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c5f686f783ab458fec1754b64ab2ab3ed102589827c9670283266c2579d094a1 | What is the largest n-cube that can fit inside an n-sphere of radius r? | [
"\\dfrac{2r}{\\sqrt{n}}"
] | [
"What is the largest n-cube that can fit inside an n-sphere of radius r?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f20a31150d58cea74e87a57608525eb25bd623d414d4cac9b30f3f7e750e6801 | What is the largest natural number \( k \) satisfying \[ \frac{1}{2} \cdot \frac{3}{4} \cdot \frac{5}{6} \cdots \frac{2003}{2004} < \frac{1}{k} \]? | [
"56"
] | [
"What is the largest natural number \\( k \\) satisfying\n\\[ \\frac{1}{2} \\cdot \\frac{3}{4} \\cdot \\frac{5}{6} \\cdots \\frac{2003}{2004} < \\frac{1}{k} \\]?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-ffd590fda332ddca72f4d30404969c0fba0d2ddb80f3f8a596c3806f61f21cc3 | What is the largest natural number \( n \) such that \( n \) points can be chosen from the perimeter or the interior of a regular hexagon with sides of unit length, so that the distance between any two of them is not less than \( \sqrt{2} \)? | [
"7"
] | [
"What is the largest natural number \\( n \\) such that \\( n \\) points can be chosen from the perimeter or the interior of a regular hexagon with sides of unit length, so that the distance between any two of them is not less than \\( \\sqrt{2} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-0f78bea2388eb46a2c9de33af16d9fa77a4073f109090f9a234380becc539a7c | What is the largest natural number \( n \) such that the inequality \(\frac{(a+b)^{3n}}{2^{3n-1}} \ge a^nb^n(a^n + b^n)\) holds for all \( a, b \ge 0 \) with \( a + b \le 3 \)? | [
"3"
] | [
"What is the largest natural number \\( n \\) such that the inequality \\(\\frac{(a+b)^{3n}}{2^{3n-1}} \\ge a^nb^n(a^n + b^n)\\) holds for all \\( a, b \\ge 0 \\) with \\( a + b \\le 3 \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-bedcee5bbfb0e7acb2eb746535544cc41024970d817d8aa559bda08515d4df20 | What is the largest number \( M \) such that there exists a number \( n(M) \) such that every binary word of length \( n(M) \) contains an \( M \)-periodic word? | [
"2"
] | [
"What is the largest number \\( M \\) such that there exists a number \\( n(M) \\) such that every binary word of length \\( n(M) \\) contains an \\( M \\)-periodic word?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-04b937110ad10c6488f705b8c4622e219a2cc51f881df01a5652e106f1366e96 | What is the largest number \( N \) for which there exist \( N \) consecutive positive integers such that the sum of the digits in the \( k \)-th integer is divisible by \( k \) for \( 1 \leq k \leq N \)? | [
"21"
] | [
"What is the largest number \\( N \\) for which there exist \\( N \\) consecutive positive integers such that the sum of the digits in the \\( k \\)-th integer is divisible by \\( k \\) for \\( 1 \\leq k \\leq N \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 23,
"genselect": 1,
"tir": 5
} | 29 | 0 | 0 |
omr-ws-4602d38c2d7111b11afb2c232d1a268e366dc5ab863e54f4aa20a553586b5fad | What is the largest number by which the expression $n^3 - n$ is divisible for all possible integral values of $n$? | [
"6"
] | [
"What is the largest number by which the expression $n^3 - n$ is divisible for all possible integral values of $n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-fc96c417c3a7579636d6e4b1a5e778e68c24ac16307a7149bf9ba9518bac58d4 | What is the largest number by which the expression \( n^3 - n \) is divisible for all possible integral values of \( n \)? | [
"6"
] | [
"What is the largest number by which the expression \\( n^3 - n \\) is divisible for all possible integral values of \\( n \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-ec4e9964aecd0214f0a0c9384607bc00c63fadd32f992861fc3fccb7c2eec0d6 | What is the largest number of 0s that can occur at the end of \(1^n + 2^n + 3^n + 4^n\) for any positive integer \(n\)? | [
"2"
] | [
"What is the largest number of 0s that can occur at the end of \\(1^n + 2^n + 3^n + 4^n\\) for any positive integer \\(n\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3b0ff446e90056721fbfdbe3053d9bb7c0ab4a7c12a494739bc688cc24ff690e | What is the largest number of 2x11 rectangles that can be cut from a 140x70 checkered rectangular table? (The cuts should go along the cell lines.) | [
"420"
] | [
"What is the largest number of 2x11 rectangles that can be cut from a 140x70 checkered rectangular table? (The cuts should go along the cell lines.)"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f825a8e054d1e4a2807ad7c18e994fab3879771d4d86d71159fc3e2da05504cf | What is the largest number of 7 m by 9 m by 11 m boxes that can fit inside a box of size 17 m by 37 m by 27 m? | [
"18"
] | [
"What is the largest number of 7 m by 9 m by 11 m boxes that can fit inside a box of size 17 m by 37 m by 27 m?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-574f57f4f2089c93653d7c53a9366c0f706e85cb8eb786b286c5988c1f13bdc5 | What is the largest number of Chicken McNuggets that cannot be bought if they are sold in boxes of 8, 12, or 23? | [
"73"
] | [
"What is the largest number of Chicken McNuggets that cannot be bought if they are sold in boxes of 8, 12, or 23?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 29
} | 52 | 0 | 0 |
omr-ws-6abb7a92d238048358998df7cfade28331431ea47a67302bc5a72beaa3dbd9f0 | What is the largest number of Dalmatians that can chase 2012 squirrels in a way that each Dalmatian chases at least one squirrel and no two Dalmatians chase the same number of squirrels? | [
"62"
] | [
"What is the largest number of Dalmatians that can chase 2012 squirrels in a way that each Dalmatian chases at least one squirrel and no two Dalmatians chase the same number of squirrels?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-598f1a90d0c5ea86d026cfb59cb3d25ee2076d29d4399864b568c19517fc20ec | What is the largest number of acute angles that a convex hexagon can have? | [
"3"
] | [
"What is the largest number of acute angles that a convex hexagon can have?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-eca5ca48c070fefdb49b7dffcbcde46fbbc0d960d3140d0c43951db348111f0b | What is the largest number of acute angles that can go in a row in a hexagon? | [
"4",
"5"
] | [
"What is the largest number of acute angles that can go in a row in a hexagon?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 16,
"genselect": 15,
"tir": 2
} | 33 | 0 | 0 |
omr-ws-c55b49bfac1fdfc50ceea85b274a0a1687dfa8919f3364870a4e88997cd82338 | What is the largest number of acute triangles of positive area that can be formed using three of the 20 points as vertices, where 10 points are drawn on each of two parallel lines? | [
"0"
] | [
"What is the largest number of acute triangles of positive area that can be formed using three of the 20 points as vertices, where 10 points are drawn on each of two parallel lines?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-536d8c2c771cd8a5fbe59bbdf1fd8d07e4f71adee86b56549f1ed88a270cc7c9 | What is the largest number of binary sequences of length 10 that can be sent to a receiver without any risk of two signals getting mixed up, given that the receiver confuses two signals if they resemble each other at five or more places? | [
"18"
] | [
"What is the largest number of binary sequences of length 10 that can be sent to a receiver without any risk of two signals getting mixed up, given that the receiver confuses two signals if they resemble each other at five or more places?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-44a036143f81f9671ace19760610a2ba83dd30537019cfc4fb64d1448c1df0fe | What is the largest number of books at $3.50 each that can be bought with $30? How much will be left over? | [
"8 books, $1.07 left over"
] | [
"What is the largest number of books at $3.50 each that can be bought with $30? How much will be left over?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-42c7fb1a32563262ffd71757f96df0246abd9aa3d32f6f0149393b5e1392b1fc | What is the largest number of cans that Amos cannot pack using boxes that hold 5, 7, or 11 cans each, if Amos must use at least one of each box size? | [
"36"
] | [
"What is the largest number of cans that Amos cannot pack using boxes that hold 5, 7, or 11 cans each, if Amos must use at least one of each box size?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 20
} | 35 | 0 | 0 |
omr-ws-4bece6735a22e3f1cdafef2475019a3d39f17add7e23272a519abedc9a681069 | What is the largest number of cards from the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$ that can be arranged in a row such that for any two adjacent cards, one number is divisible by the other? | [
"8"
] | [
"What is the largest number of cards from the set $\\{1, 2, 3, 4, 5, 6, 7, 8, 9\\}$ that can be arranged in a row such that for any two adjacent cards, one number is divisible by the other?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-ba1572d80ec0a85d72e619a8d2883142005086604d71180807400fb0c6af9c48 | What is the largest number of cards that can be selected from a set of 2016 cards numbered $1, 2, 3, \ldots, 2016$ such that no two selected cards have a sum divisible by 8? | [
"758"
] | [
"What is the largest number of cards that can be selected from a set of 2016 cards numbered $1, 2, 3, \\ldots, 2016$ such that no two selected cards have a sum divisible by 8?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-7f11419f30cc8934a7d50226aec7bf9a9af4f4293410840d6a050bef4bc67100 | What is the largest number of cells that can be colored in a $7 \times 7$ table such that any $2 \times 2$ subtable has at most 2 colored cells? | [
"28"
] | [
"What is the largest number of cells that can be colored in a $7 \\times 7$ table such that any $2 \\times 2$ subtable has at most 2 colored cells?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-5bf226cfecea4dd459e55eaf783b01308f59d710f9724abb9459af088f8f49ae | What is the largest number of cells that can be colored in a $7\times7$ table such that any $2\times2$ subtable has at most $2$ colored cells? | [
"28"
] | [
"What is the largest number of cells that can be colored in a $7\\times7$ table such that any $2\\times2$ subtable has at most $2$ colored cells?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 25,
"genselect": 8,
"tir": 3
} | 36 | 0 | 0 |
omr-ws-e4acbd5be0522bf043e2e18a5e5e5ef16a1f539fcc9b15f249e7d325e6f7a401 | What is the largest number of cells that can be marked on a $100 \times 100$ board in such a way that a chess king from any cell attacks no more than two marked ones? (The cell on which a king stands is also considered to be attacked by this king.) | [
"2278"
] | [
"What is the largest number of cells that can be marked on a $100 \\times 100$ board in such a way that a chess king from any cell attacks no more than two marked ones? (The cell on which a king stands is also considered to be attacked by this king.)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-d556242777b2cd99959a10f54f1ea278c423b3bf1e9233c590e05a9e8d47260f | What is the largest number of cells that can be marked on a $3 \times 3 \times 3$ Rubik's cube so that no two marked cells share a common vertex? | [
"8"
] | [
"What is the largest number of cells that can be marked on a $3 \\times 3 \\times 3$ Rubik's cube so that no two marked cells share a common vertex?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-5e6a35ce3be5eb88af2232a5a86b5f6715e3eb0dd9253b2d8ee7ba3cd2e7a342 | What is the largest number of checkerboard cells that can be marked so that there is not a single obtuse triangle with vertices in the centers of the marked cells? | [
"8"
] | [
"What is the largest number of checkerboard cells that can be marked so that there is not a single obtuse triangle with vertices in the centers of the marked cells?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 5
} | 10 | 0 | 0 |
omr-ws-744592cf4b502eb81148db5a1a13d41ece1e07dd5ba416a139c1ce805e9add7f | What is the largest number of chicken nuggets that cannot be bought if chicken nuggets can only be purchased in packs of 6 and 4? | [
"2"
] | [
"What is the largest number of chicken nuggets that cannot be bought if chicken nuggets can only be purchased in packs of 6 and 4?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-4cc9315fb2bf48395a05566f2ccceede4059ede75ae13dd326cf0ae0b443a91c | What is the largest number of chicken nuggets that cannot be obtained using boxes of 9, 13, and 20 nuggets? | [
"50"
] | [
"What is the largest number of chicken nuggets that cannot be obtained using boxes of 9, 13, and 20 nuggets?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-57744c25c89602912ff4497954657ab56d295ae82c889ac9c24af0b1325e358e | What is the largest number of chips that can be placed on an $8 \times 8$ table such that each chip is placed in an empty cell with at least 2 free adjacent cells through sides? | [
"42"
] | [
"What is the largest number of chips that can be placed on an $8 \\times 8$ table such that each chip is placed in an empty cell with at least 2 free adjacent cells through sides?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 3
} | 12 | 0 | 0 |
omr-ws-b447bb0943cf39ed0367ec5a4b216c794ce4d168281a5820013582bb0e8ed15a | What is the largest number of chips that can be placed on an $8 \times 8$ table such that each chip is placed in an empty cell with at least 3 free adjacent cells through sides? | [
"32"
] | [
"What is the largest number of chips that can be placed on an $8 \\times 8$ table such that each chip is placed in an empty cell with at least 3 free adjacent cells through sides?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 1
} | 8 | 0 | 0 |
omr-ws-053e4ef896bf60ff1c704d09c747ebbbc227d445faa2169a0c4ec31c65ad9cd0 | What is the largest number of chips, $n$, that can be placed on a chessboard such that there are exactly 3 chips in any $3 \times 3$ square? | [
"27"
] | [
"What is the largest number of chips, $n$, that can be placed on a chessboard such that there are exactly 3 chips in any $3 \\times 3$ square?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-ac51f5be67c18625f793678e381cdb587ea12efd972e5ea00dea5eefca1099db | What is the largest number of circles with radius $\frac{\sqrt{3}}{4}$ that can be placed without overlapping inside a regular hexagon with side length $1$? (Circles can touch each other and the sides of the hexagon.) | [
"7"
] | [
"What is the largest number of circles with radius $\\frac{\\sqrt{3}}{4}$ that can be placed without overlapping inside a regular hexagon with side length $1$? (Circles can touch each other and the sides of the hexagon.)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 4
} | 7 | 0 | 0 |
omr-ws-1ac102725966353a4b6a02cd8ca7c6821916e7a745826855c9517a2c08291a5b | What is the largest number of circumscribed quadrilaterals that can be formed by splitting a convex hexagon into two quadrilaterals using diagonals connecting opposite vertices? | [
"3"
] | [
"What is the largest number of circumscribed quadrilaterals that can be formed by splitting a convex hexagon into two quadrilaterals using diagonals connecting opposite vertices?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-01f818da0c56d3a6d95e4160dc62bec01eca1a709998fe2d186cbacf05cbabec | What is the largest number of coins from which one can detect a counterfeit in three weighings with a pan balance, if it is known only that the counterfeit coin differs in weight from the others? | [
"27"
] | [
"What is the largest number of coins from which one can detect a counterfeit in three weighings with a pan balance, if it is known only that the counterfeit coin differs in weight from the others?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9159aecc5081c0a7194d4de3aa7f053643fd6a4c17bed7ba7bb83adb96819489 | What is the largest number of coins that a resident of Merryland can have if we know that their total value does not exceed 2021 cents, given that the coins have different integer values starting from 1 cent? | [
"63"
] | [
"What is the largest number of coins that a resident of Merryland can have if we know that their total value does not exceed 2021 cents, given that the coins have different integer values starting from 1 cent?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2c804fcdb0097eee9c95f39af379bed25b8f27e927bae4a2f058cea6f41bb372 | What is the largest number of coins that can be placed on a $20 \times 13$ board such that no coin has more than two neighbours? Two coins are called neighbours if they are in the same row or column and no other coins lie between them. | [
"260"
] | [
"What is the largest number of coins that can be placed on a $20 \\times 13$ board such that no coin has more than two neighbours? Two coins are called neighbours if they are in the same row or column and no other coins lie between them."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 4
} | 13 | 0 | 0 |
omr-ws-02fbd6ffb769ea72d5cd4ddec752c7b7113debe7cc921ac1882db98417898557 | What is the largest number of coins that can be placed on a 20 x 13 board such that no coin has more than two neighbors, where two coins are neighbors if they are in the same row or column and no other coins lie between them? | [
"260"
] | [
"What is the largest number of coins that can be placed on a 20 x 13 board such that no coin has more than two neighbors, where two coins are neighbors if they are in the same row or column and no other coins lie between them?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-2c9ed3b270b8640031ecc27274dab03b0c4a65ef3e3cbae3f5f44894d036d922 | What is the largest number of colors that can be used to paint an $8 \times 8$ board so that each square is painted with a single color and shares a side with at least two other squares which share its color? | [
"16"
] | [
"What is the largest number of colors that can be used to paint an $8 \\times 8$ board so that each square is painted with a single color and shares a side with at least two other squares which share its color?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-04b02aee48bc2445680ae359d1e1ded3d9a754adb9bf18a47608fb7d2784bd43 | What is the largest number of congruent flat angles that a tetrahedron can have if it is not regular? | [
"8"
] | [
"What is the largest number of congruent flat angles that a tetrahedron can have if it is not regular?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-9b0ae6d7c2732c70c542300a37b949d1979166a4dbb350dcd07c0648b136ba2c | What is the largest number of consecutive values of the expression $(n+1) \cdot 2^n$ that are perfect squares, for $n = 1, 2, 3, \ldots$? | [
"2"
] | [
"What is the largest number of consecutive values of the expression $(n+1) \\cdot 2^n$ that are perfect squares, for $n = 1, 2, 3, \\ldots$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 17
} | 17 | 0 | 0 |
omr-ws-2266973f00fd7206494b0d464fd506af33d8b04b6bef748e8896b10649f6f5ba | What is the largest number of convex 35-gons that can be chosen among the vertices of a given convex 105-gon in such a way that no two 35-gons have a common side? | [
"3"
] | [
"What is the largest number of convex 35-gons that can be chosen among the vertices of a given convex 105-gon in such a way that no two 35-gons have a common side?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-b79a2e3f9d7e7c33f1ccf6b5f493e5dab538c9d534374b01cc7d1d57a2e9b55a | What is the largest number of convex figures that can be glued from two quadrilaterals so that all these figures have a different number of angles? | [
"3"
] | [
"What is the largest number of convex figures that can be glued from two quadrilaterals so that all these figures have a different number of angles?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-3a5bee26a4551828dbca8a1fdec35e3f250d2c3f8375fbd6ea439527c94f4a88 | What is the largest number of diagonals that can be drawn in a convex heptagon so that no three of them form a triangle with vertices at the vertices of the heptagon? | [
"10"
] | [
"What is the largest number of diagonals that can be drawn in a convex heptagon so that no three of them form a triangle with vertices at the vertices of the heptagon?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 8
} | 19 | 0 | 0 |
omr-ws-2fb0526ee6c4d2c95662bb8fd9424cc333e4fa1047a86390e1f6818485d84ecd | What is the largest number of diagonals that can be drawn in a convex nonagon so that no three of them form a triangle with vertices at the vertices of the nonagon? | [
"18"
] | [
"What is the largest number of diagonals that can be drawn in a convex nonagon so that no three of them form a triangle with vertices at the vertices of the nonagon?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-38e6b5604f4a4361a2e029f0a4b4bcc43660e431cfa8649fad3116b30b4d1dfe | What is the largest number of digits that can be erased from the 1000-digit number 20082008...2008 so that the sum of the remaining digits is 2008? | [
"746"
] | [
"What is the largest number of digits that can be erased from the 1000-digit number 20082008...2008 so that the sum of the remaining digits is 2008?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-5efc4781af1c25d05f51e6ddfcc219d89b8f7c22e9efa16c0e9ecfbe81a2a70a | What is the largest number of disjoint diagonals that can be drawn in a convex $n$-gon, where diagonals with a common vertex are allowed? | [
"\\( n-3 \\)"
] | [
"What is the largest number of disjoint diagonals that can be drawn in a convex $n$-gon, where diagonals with a common vertex are allowed?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-80e365fc7d7431499877d858e27828d49f136523ad439a5b48458db2f472df65 | What is the largest number of distinct elements that can be chosen from the set $\{1, 2, 3, \ldots, 1000\}$ such that no three of them can be the side lengths of a triangle? | [
"15"
] | [
"What is the largest number of distinct elements that can be chosen from the set $\\{1, 2, 3, \\ldots, 1000\\}$ such that no three of them can be the side lengths of a triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-47bbdd3eec8f0324d9c3a6d439a2eda6cafd95c3e83499f38ada04696731bb71 | What is the largest number of distinct positive integers such that no four of them have a sum divisible by 4? | [
"6"
] | [
"What is the largest number of distinct positive integers such that no four of them have a sum divisible by 4?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 62,
"tir": 2
} | 64 | 0 | 0 |
omr-ws-5deddf0294a45b80f85924323aa373fc6f5ce6de9f3359fb069822bf4c3ac3ac | What is the largest number of distinct prime factors under $100$ that the expression $x^4 + x^3 + x^2 + x + 1$ can have if $x$ is a positive integer? | [
"6"
] | [
"What is the largest number of distinct prime factors under $100$ that the expression $x^4 + x^3 + x^2 + x + 1$ can have if $x$ is a positive integer?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 1
} | 23 | 0 | 0 |
omr-ws-1e4dbbb3735df12d810f54ae74b0b4b72eec7d307d678461acfd7ff11d25ab64 | What is the largest number of edges of the pyramid that can be crossed by a plane that does not pass through the vertices, if the base of the pyramid is a 100-gon (not necessarily convex)? | [
"199"
] | [
"What is the largest number of edges of the pyramid that can be crossed by a plane that does not pass through the vertices, if the base of the pyramid is a 100-gon (not necessarily convex)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-888bc3de0bd8956a4530e1e7e2191d6faf05f2dc9721b734cba7458f761d1b0e | What is the largest number of elements that can be contained in a set of vectors such that the set contains the sum of any two different vectors in the set? | [
"3"
] | [
"What is the largest number of elements that can be contained in a set of vectors such that the set contains the sum of any two different vectors in the set?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-d9a62fc576e6881c6965be82e26e156ff0540a1b4431dba4faacd7b925a872cc | What is the largest number of elements that can be selected from the set $\{1, 2, \ldots, 2012, 2013\}$ such that no three distinct elements $a$, $b$, and $c$ exist where $a$ is a divisor or multiple of $b-c$? | [
"672"
] | [
"What is the largest number of elements that can be selected from the set $\\{1, 2, \\ldots, 2012, 2013\\}$ such that no three distinct elements $a$, $b$, and $c$ exist where $a$ is a divisor or multiple of $b-c$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5f08f2873f1b11f0aac2cab0038f09890b3a2d5d9e2cc570dd50c4a2b3b7c4a2 | What is the largest number of empty cells that an incomplete binoku can contain if it can be completed into a binoku in a unique way? A binoku is a $9 \times 9$ grid divided into nine $3 \times 3$ subgrids, where each cell contains either a $0$ or a $1$, each row and column contains at least one $0$ and one $1$, and ea... | [
"18"
] | [
"What is the largest number of empty cells that an incomplete binoku can contain if it can be completed into a binoku in a unique way? A binoku is a $9 \\times 9$ grid divided into nine $3 \\times 3$ subgrids, where each cell contains either a $0$ or a $1$, each row and column contains at least one $0$ and one $1$,... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b6a3ad147713d7883ca17f1ed685bc56601998c780f30685dda80fb74e13d3a6 | What is the largest number of equilateral triangles that can be formed with vertices at the points of intersection of 2018 segments, where each side of the triangles lies on one of these segments? | [
"304368288"
] | [
"What is the largest number of equilateral triangles that can be formed with vertices at the points of intersection of 2018 segments, where each side of the triangles lies on one of these segments?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-bd2a1c29df87860cb24147158bcdd983f3239b451e877055376b52ec6d0f92fb | What is the largest number of faces of a regular dodecahedron that can cross a plane? | [
"10"
] | [
"What is the largest number of faces of a regular dodecahedron that can cross a plane?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-55405b769bdfd7027a5c7a898b9519afaeab2cee475d210c2314bb28e8abc44c | What is the largest number of faces of a regular icosahedron that can cross a plane? | [
"10"
] | [
"What is the largest number of faces of a regular icosahedron that can cross a plane?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-6ff6d5ae77fc6059c21fdd2d1cda102385b1de4c31d93e3c6924e913c0b70f7c | What is the largest number of faces that do not have common edges in pairs that can be chosen from the icosahedron? | [
"8"
] | [
"What is the largest number of faces that do not have common edges in pairs that can be chosen from the icosahedron?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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