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-698872eb02abc9c8f27b4adc539b71a620ab22cfeaf713689d13a77a709e3362 | What is the largest prime factor of $15! - 13!$? | [
"19"
] | [
"What is the largest prime factor of $15! - 13!$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-3e8a75aed299263adfa782bd312b9349b0f2bc9815b7bb507a9acacc014161e5 | What is the largest prime factor of $3328$? | [
"13"
] | [
"What is the largest prime factor of $3328$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-29088472d0ebd71b5fc1e2bfe93c380366492b77ca26d31d0725d7b4f4b21181 | What is the largest prime factor of $33^4 + 32^4 - 1$? | [
"23"
] | [
"What is the largest prime factor of $33^4 + 32^4 - 1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-6f94fb7335b237de65ef48eeacc541f69c3bcecd68ac0d2eb53d934b6260754b | What is the largest prime factor of $5! + 6!$? | [
"7"
] | [
"What is the largest prime factor of $5! + 6!$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fa23d83f94064b487c26e4e845d5f7ae7924937a2fa3c5c12c75adcdd7ae69ce | What is the largest prime factor of $729(64) + 27(16)$? | [
"109"
] | [
"What is the largest prime factor of $729(64) + 27(16)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-d2a1e1a7e4712f26af2df9844f7e070f08b1560278a0b35068bd01fb97eabd9b | What is the largest prime factor of 2323? | [
"101"
] | [
"What is the largest prime factor of 2323?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c0617136c49e55ddf514e36511bfbc09fa6063128b779f227b208c9f55dbd907 | What is the largest prime factor of 8091? | [
"31"
] | [
"What is the largest prime factor of 8091?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-9bda70846f6db993479e6e060fcf3e96fef3d6dd1d6e8aafd3c0e3fe2510d8aa | What is the largest prime factor of 95!? | [
"89"
] | [
"What is the largest prime factor of 95!?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-37b01e1f694b0f54538b386dc7d8908c53c6d10e0293838d25ea8b63bb627df2 | What is the largest prime factor of 999? | [
"37"
] | [
"What is the largest prime factor of 999?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9ec23c13bc4bfd081e831c87a6fdc704191e53b2cb91c45735901bb0e5d4dce4 | What is the largest prime factor of \(1^3 + 2^3 + 3^3 + \ldots + 2004^3\)? | [
"401"
] | [
"What is the largest prime factor of \\(1^3 + 2^3 + 3^3 + \\ldots + 2004^3\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 22
} | 45 | 0 | 0 |
omr-ws-bb3a4615608b350617186faa53d02113381a8f9c0a59e65baa3033af5e83484f | What is the largest prime number that divides \(3^{22} + 3\)? | [
"2269"
] | [
"What is the largest prime number that divides \\(3^{22} + 3\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-0b6811630b5688fa28064246fd97d4e0855d0cf5b501e7fc33a2302255b9bce5 | What is the largest prime number that divides the difference between a five-digit integer and its reverse, given that the difference is not zero? | [
"929"
] | [
"What is the largest prime number that divides the difference between a five-digit integer and its reverse, given that the difference is not zero?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 3,
"tir": 2
} | 10 | 0 | 0 |
omr-ws-e0574aeec12359920c04de1cb707f013a7580ed695a5d30f270bcbb2db404908 | What is the largest prime number which divides \(\frac{2 + 2 \cdot 2! + 3 \cdot 3! + 4 \cdot 4! + \cdots + 2023 \cdot 2023!}{2022!}\)? | [
"23"
] | [
"What is the largest prime number which divides \\(\\frac{2 + 2 \\cdot 2! + 3 \\cdot 3! + 4 \\cdot 4! + \\cdots + 2023 \\cdot 2023!}{2022!}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-d10e691f8ef2e8a1e1c95be08b5f04e776e6829f7320b0c5eedd477ce63e6c1c | What is the largest prime that divides both \(2x + 3y\) and \(9x + 5y\)? For which values of \(x\) and \(y\) does this occur? | [
"17"
] | [
"What is the largest prime that divides both \\(2x + 3y\\) and \\(9x + 5y\\)? For which values of \\(x\\) and \\(y\\) does this occur?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-770bdc804b9b0c755f11f429606ef72552d999f3a9035fda952fa649ea20a307 | What is the largest product of a 3-digit and 4-digit number, if they are formed using each of the digits 1-7 exactly once? | [
"4835013"
] | [
"What is the largest product of a 3-digit and 4-digit number, if they are formed using each of the digits 1-7 exactly once?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c4d0f9743589b2d14aaf4c5523be0f41151f502793a8f5fbdaaa1260a29e941d | What is the largest radius \( r \) for which a ball can touch the lowest point of the bottom in a round goblet whose section is the graph of the function \( y = x^4 \)? (In mathematical terms: what is the maximal radius of the disc lying in the domain \( y \geq x^4 \) and containing the origin?) | [
"\\( 3\\cdot2^{-\\frac{5}{3}} \\)"
] | [
"What is the largest radius \\( r \\) for which a ball can touch the lowest point of the bottom in a round goblet whose section is the graph of the function \\( y = x^4 \\)? (In mathematical terms: what is the maximal radius of the disc lying in the domain \\( y \\geq x^4 \\) and containing the origin?)"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 25
} | 47 | 0 | 0 |
omr-ws-3080c0c2602ec3198471696a2536ee76f9ed89f360b6d75e260575750c65204a | What is the largest real number $C$ that satisfies the inequality $x^2 \geq C \lfloor x \rfloor (x-\lfloor x \rfloor)$ for every real $x$? | [
"4"
] | [
"What is the largest real number $C$ that satisfies the inequality $x^2 \\geq C \\lfloor x \\rfloor (x-\\lfloor x \\rfloor)$ for every real $x$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-42b4b768b33aa3daa4ecd20b3eb624212173901ede20ebed57e6bd13e6a2270e | What is the largest real number $\alpha$ (if it exists) such that for any real function $u$ with $u(0)=0$, $u(1)=1$, and $u'$ existing, the inequality $\int_0^1 |u' - u| \, dx \ge \alpha$ holds? | [
"\\(\\frac{1}{e}\\)"
] | [
"What is the largest real number $\\alpha$ (if it exists) such that for any real function $u$ with $u(0)=0$, $u(1)=1$, and $u'$ existing, the inequality $\\int_0^1 |u' - u| \\, dx \\ge \\alpha$ holds?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 6,
"tir": 6
} | 20 | 0 | 0 |
omr-ws-3de43b4f05ea81912baea50c5adac8ea59bed45b312caaf1cddd83d8f05bc3bc | What is the largest real number $\theta$ less than $\pi$ such that $$\prod_{k=0}^{10}\cos\left(2^k \theta\right) \neq 0$$ and $$\prod_{k=0}^{10}\left(1 + \frac{1}{\cos\left(2^k\theta\right)}\right)=1?$$ | [
"\\dfrac{2046}{2047}\\pi"
] | [
"What is the largest real number $\\theta$ less than $\\pi$ such that \n$$\\prod_{k=0}^{10}\\cos\\left(2^k \\theta\\right) \\neq 0$$ \nand \n$$\\prod_{k=0}^{10}\\left(1 + \\frac{1}{\\cos\\left(2^k\\theta\\right)}\\right)=1?$$"
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 7
} | 7 | 0 | 0 |
omr-ws-8bd8ea82f93731db456047492bf1aa0432f2a8c96aa6cd2a4ffedc4a4c51dcc4 | What is the largest real 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 real 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": 2,
"genselect": 4
} | 6 | 0 | 0 |
omr-ws-c9be5178ee895519c6eb426f1e8974b263c938fea5caab3c25fc37f2102ab6a3 | What is the largest real value of $r$ for which the ranges of the functions $f(x) = 2x^2 - 3x + r$ and $f(f(x))$ are the same set? | [
"\\(\\frac{15}{8}\\)"
] | [
"What is the largest real value of $r$ for which the ranges of the functions $f(x) = 2x^2 - 3x + r$ and $f(f(x))$ are the same set?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-0fdaeaebedb4f87e4489cf8599a0e54bfb10e699f0772a0f667114961fb7f7e2 | What is the largest remainder obtained by dividing 2015 by each of the numbers from 1 to 1000? | [
"671"
] | [
"What is the largest remainder obtained by dividing 2015 by each of the numbers from 1 to 1000?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 5
} | 20 | 0 | 0 |
omr-ws-416870dac9677a15fe3336a4b44b7d4dcdf46811d3b74cca0188ffb812e9d67f | What is the largest remainder obtained when 2015 is divided by each of the numbers from 1 to 1000? | [
"671"
] | [
"What is the largest remainder obtained when 2015 is divided by each of the numbers from 1 to 1000?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-a619e74bc02ef94b39d39bb3c70870d5e11f01513500d46cbf46fa449aee33cc | What is the largest repunit that is less than 2009? | [
"2008"
] | [
"What is the largest repunit that is less than 2009?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-e7dbd1e7f269410af8bdca5c165eca73e3f93e0326a5fb2e68f4707161cf7d4a | What is the largest score that is not attainable in an American football game, given the possible scores are 2, 3, 6, 7, and 8 points? | [
"1"
] | [
"What is the largest score that is not attainable in an American football game, given the possible scores are 2, 3, 6, 7, and 8 points?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-416ca6d1f57dae6f6cd91d2321f20b5d60c8bd78a693ae311ad4dce8f82f08b5 | What is the largest six-digit number whose digits are all different such that the sum of the squares of the digits in the even-numbered positions equals the sum of the squares of the digits in the odd-numbered positions? | [
"986702"
] | [
"What is the largest six-digit number whose digits are all different such that the sum of the squares of the digits in the even-numbered positions equals the sum of the squares of the digits in the odd-numbered positions?"
] | [
"aops_c4_high_school_math",
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 26,
"genselect": 23,
"tir": 50
} | 99 | 0 | 0 |
omr-ws-afa5333190cccd88396722e981726c6281a41c6a06be352b4a74ead2a029e522 | What is the largest square that can fit inside a triangle with sides of lengths 2, 3, and 4? | [
"\\(\\frac{270 - 48\\sqrt{15}}{71}\\)"
] | [
"What is the largest square that can fit inside a triangle with sides of lengths 2, 3, and 4?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 4,
"tir": 6
} | 18 | 0 | 0 |
omr-ws-5f8e47b18451de7de9cd3a7228a72c95ee4f915226d766498440a5620ffee345 | What is the largest square you can paint onto a chessboard such that you can still place five queens on unpainted squares, where all 64 squares are attacked? | [
"6"
] | [
"What is the largest square you can paint onto a chessboard such that you can still place five queens on unpainted squares, where all 64 squares are attacked?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 2
} | 8 | 0 | 0 |
omr-ws-5005011e12480b00830e3b3b9c8342b0761cd9a9aab6a18690e4432946c8525b | What is the largest subset \( A \) of \(\{1, 2, \ldots, 2002\}\) such that no element of \( A \) is a factor of any other element of \( A \)? | [
"1001"
] | [
"What is the largest subset \\( A \\) of \\(\\{1, 2, \\ldots, 2002\\}\\) such that no element of \\( A \\) is a factor of any other element of \\( A \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-456a5d7ba6a3b21fdfd9fd2359bffcf363684a67e30faf462969c8871eec8883 | What is the largest subset of the integers \(1, 2, \ldots, n\) such that no two elements sum to a third element? | [
"\\( \\left\\lceil \\frac{n}{2} \\right\\rceil \\)"
] | [
"What is the largest subset of the integers \\(1, 2, \\ldots, n\\) such that no two elements sum to a third element?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b70afac77f1be7ff40af6844c6e5a274cbbf6e533b3190233100598f3ee73e42 | What is the largest sum of the digits that can be obtained from a date written in the form day/month (e.g., 01/09)? | [
"20"
] | [
"What is the largest sum of the digits that can be obtained from a date written in the form day/month (e.g., 01/09)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3e0b4668c318c5ed2a5a12bbc9f8ec04dd4c93d14fda298b59c8077cf096708b | What is the largest sum that cannot be obtained using the numbers 2, 3, 5, and 7? | [
"18"
] | [
"What is the largest sum that cannot be obtained using the numbers 2, 3, 5, and 7?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-22a4a8a536ee01adafdcc806e27963a3159978acb6e07802022ada34426c41b8 | What is the largest sum that results when one of the arrangements of the digits of 2017 is added to one of the arrangements of the digits of 2016, if none of the digits 0, 1, or 2 can occupy the same position in both numbers? | [
"13321"
] | [
"What is the largest sum that results when one of the arrangements of the digits of 2017 is added to one of the arrangements of the digits of 2016, if none of the digits 0, 1, or 2 can occupy the same position in both numbers?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-2f147116a638834512947375fc34c71240c3efb45bf5281b9e63c0f7cc6332c5 | What is the largest three-digit integer $n$ such that the hundreds digit of $n^{302}$ is $9$? | [
"978"
] | [
"What is the largest three-digit integer $n$ such that the hundreds digit of $n^{302}$ is $9$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 7
} | 13 | 0 | 0 |
omr-ws-8a672ee05fecc35dcc4c14179039858f2bc8e4c6bd26f47db17c2cf98ce775a5 | What is the largest three-digit integer $n$ that satisfies $$55n\equiv 165\pmod{260}~?$$ | [
"991"
] | [
"What is the largest three-digit integer $n$ that satisfies $$55n\\equiv 165\\pmod{260}~?$$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-67451cea213ea87197bc01c8c903d82e47386a3f59359f45f5f55f6ede68dba5 | What is the largest three-digit integer whose digits are distinct and form a geometric sequence? | [
"964"
] | [
"What is the largest three-digit integer whose digits are distinct and form a geometric sequence?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d80db8b73774a45cfdf39805b8817bc1cee6ef4210d67ed06019e588abb71d4e | What is the largest three-digit multiple of 9 whose digits' sum is 18? | [
"990"
] | [
"What is the largest three-digit multiple of 9 whose digits' sum is 18?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0213517b9f83ebfef84767f121e726ab741c7922821ce4e7d881d24d576960e6 | What is the largest three-digit number such that the number is equal to the sum of its hundreds digit, the square of its tens digit, and the cube of its units digit? | [
"598"
] | [
"What is the largest three-digit number such that the number is equal to the sum of its hundreds digit, the square of its tens digit, and the cube of its units digit?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 32
} | 54 | 0 | 0 |
omr-ws-1eec74674d504cf814508a7fdfcc7b6ef71de5682e2ad5c6726edff36852101f | What is the largest two-digit integral value of \( x \lfloor x \rfloor \), where \( \lfloor x \rfloor \) denotes the largest integer less than or equal to the real number \( x \)? | [
"89"
] | [
"What is the largest two-digit integral value of \\( x \\lfloor x \\rfloor \\), where \\( \\lfloor x \\rfloor \\) denotes the largest integer less than or equal to the real number \\( x \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-87c945587ea0303ff6f1fc1fafd9c1e7bb6b73988b64c1f45cfb614dcb48d8a9 | What is the largest two-digit number that can serve as the hypotenuse of a right triangle with integer side lengths? | [
"97"
] | [
"What is the largest two-digit number that can serve as the hypotenuse of a right triangle with integer side lengths?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-5171528fb30cb069685b7349055fddf50acf99e2857d0c095d99ccf73bdd8ff5 | What is the largest two-digit prime factor of $\binom{200}{100}$? | [
"61"
] | [
"What is the largest two-digit prime factor of $\\binom{200}{100}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-a871bf7dc09624d0af67280562db83bc53d9a0347fb2ed9d17c3b237f1ba017a | What is the largest two-digit prime factor of the binomial coefficient $C_{200}^{100}$? | [
"61"
] | [
"What is the largest two-digit prime factor of the binomial coefficient $C_{200}^{100}$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-92fceeb50a91f2c0560b2fb4f9670477107229fb0e10c10618d05296b0eb181c | What is the largest two-digit prime number that divides $\frac{198!}{99! \times 99!}$? | [
"61"
] | [
"What is the largest two-digit prime number that divides $\\frac{198!}{99! \\times 99!}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 23
} | 38 | 0 | 0 |
omr-ws-2574f4a89762332302666c8d2c07de64db09012dbe518fd5a2ace76ecace6294 | What is the largest value among $\operatorname{lcm}[12,2],$ $\operatorname{lcm}[12,4],$ $\operatorname{lcm}[12,6],$ $\operatorname{lcm}[12,8],$ $\operatorname{lcm}[12,10],$ and $\operatorname{lcm}[12,12]?$ Express your answer as an integer. | [
"60"
] | [
"What is the largest value among $\\operatorname{lcm}[12,2],$ $\\operatorname{lcm}[12,4],$ $\\operatorname{lcm}[12,6],$ $\\operatorname{lcm}[12,8],$ $\\operatorname{lcm}[12,10],$ and $\\operatorname{lcm}[12,12]?$ Express your answer as an integer."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a9e426217161f36609404a27326d60f9322dbce3a606b7e320073c360a5af8ad | What is the largest value of $6\cos^2(x) - 24\sin(x)\cos(x) - 4\sin^2(x)$ where $0^\circ \leq x \leq 360^\circ$? | [
"14"
] | [
"What is the largest value of $6\\cos^2(x) - 24\\sin(x)\\cos(x) - 4\\sin^2(x)$ where $0^\\circ \\leq x \\leq 360^\\circ$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-27dd1b10840b51a900644726f931581d7e1750fd61401170ce925ca6ae1b194c | What is the largest value of $\prod_{i=1}^{n} \cos t_{i}$ given that $\sum_{i=1}^{n} t_{i}^2 = 1$? | [
"$\\cos^n (1/\\sqrt{n})$"
] | [
"What is the largest value of $\\prod_{i=1}^{n} \\cos t_{i}$ given that $\\sum_{i=1}^{n} t_{i}^2 = 1$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-8c3e36b11d6f392b0cceb474dd7251b1bc9356c26c6044cf2795acc3f33da17f | What is the largest value of $k$ for which the polynomials $Q_1(x) = x^2 + (k-29)x - k$ and $Q_2(x) = 2x^2 + (2k-43)x + k$ are both factors of a cubic polynomial $P(x)$? | [
"30"
] | [
"What is the largest value of $k$ for which the polynomials $Q_1(x) = x^2 + (k-29)x - k$ and $Q_2(x) = 2x^2 + (2k-43)x + k$ are both factors of a cubic polynomial $P(x)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-eecb5cf04932c73c6ccdc17e91ebb6ad94f9c9d10689289f144f820a5e4a97b2 | What is the largest value of $m$ such that the number $2013$ can be written as a sum of $m$ composite numbers? | [
"502"
] | [
"What is the largest value of $m$ such that the number $2013$ can be written as a sum of $m$ composite numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-27b177b8755b9124679f404371f101743fbb79ee23d4d7ef04d7acf8cced98bf | What is the largest value of $n$ less than 100,000 for which the expression $8(n-2)^5-n^2+14n-24$ is a multiple of 5? | [
"99997"
] | [
"What is the largest value of $n$ less than 100,000 for which the expression $8(n-2)^5-n^2+14n-24$ is a multiple of 5?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-05015901b5e1a69492069619616695ccf64855a8720208f47a0af8e3f93f2619 | What is the largest value of $x$ if \[ \left| \begin{array}{ccc} x & 0 & y \\ 3 & 1 & 2 \\ -y & 2 & x \end{array} \right| = 156?\] | [
"15"
] | [
"What is the largest value of $x$ if\n\\[ \\left| \\begin{array}{ccc}\n x & 0 & y \\\\\n 3 & 1 & 2 \\\\\n-y & 2 & x \\end{array} \\right| = 156?\\]"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e9d4aeb7693e8f4ba11e2bdc5108d8fd14b3a0c90f4c9a4d152cdabe8b036b8f | What is the largest value of $x$ such that the expression \[\dfrac{x+1}{8x^2-65x+8}\] is not defined? | [
"8"
] | [
"What is the largest value of $x$ such that the expression \\[\\dfrac{x+1}{8x^2-65x+8}\\] is not defined?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b058a4715c05f7e7eb5eabf48b7c2fc8bef77ff6cbdeb307c1f0e3b2e723235c | What is the largest value of $x$ that satisfies the equation $\sqrt{2x}=4x$? Express your answer in simplest fractional form. | [
"\\frac18"
] | [
"What is the largest value of $x$ that satisfies the equation $\\sqrt{2x}=4x$? Express your answer in simplest fractional form."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-91daf1cb5b45fef26487627bd6ac5c162a973a8f54f05a9564dc0a677f61fb72 | What is the largest value of $x$, if $\frac{x}{5} + \frac{1}{5x} = \frac{1}{2}$? | [
"2"
] | [
"What is the largest value of $x$, if $\\frac{x}{5} + \\frac{1}{5x} = \\frac{1}{2}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ce0ed7107d1d430c0b4e814151af213a9c86be74565e66a4f980c3ac516f8877 | What is the largest value of $|x|$ that makes $\dfrac{x^2 + 2020}{x+20}$ an integer? | [
"2440"
] | [
"What is the largest value of $|x|$ that makes $\\dfrac{x^2 + 2020}{x+20}$ an integer?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 26
} | 73 | 0 | 0 |
omr-ws-9f1bf1072449afe703f7b71aeb8ef310f6269b928c3d21bcc23e656f8084183b | What is the largest value of \( \frac{mn-(m+n)}{mn(mn+1)} \) for relatively prime \( m, n \in \mathbb{N} \) with \( m, n \ge 2 \)? | [
"\\(\\frac{7}{240}\\)"
] | [
"What is the largest value of \\( \\frac{mn-(m+n)}{mn(mn+1)} \\) for relatively prime \\( m, n \\in \\mathbb{N} \\) with \\( m, n \\ge 2 \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fc31911e0a6ab6317a408ba39998c3a32f580caa60418567aa7a524b5d787768 | What is the largest value of \( k \) for which one root of the equation \( x^2 - 2x - kx + 2k - 12 = 0 \) will be twice the other? | [
"10"
] | [
"What is the largest value of \\( k \\) for which one root of the equation \\( x^2 - 2x - kx + 2k - 12 = 0 \\) will be twice the other?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 5
} | 14 | 0 | 0 |
omr-ws-b975a55b0f4884c47e81fec36b5f0acf8b76ce3186121cfc5ca374b7e08bc0a3 | What is the largest value of \( k \) such that the inequality \( x^2 + y^2 + z^2 + k (xyz)^{\frac{2}{3}} \geq \frac{k + 3}{3} (xy + yz + zx) \) holds for all non-negative real numbers \( x, y, \) and \( z \)? | [
"3"
] | [
"What is the largest value of \\( k \\) such that the inequality \\( x^2 + y^2 + z^2 + k (xyz)^{\\frac{2}{3}} \\geq \\frac{k + 3}{3} (xy + yz + zx) \\) holds for all non-negative real numbers \\( x, y, \\) and \\( z \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-2a5477106418165a31adb38e7164d93949590122bc3068fa6fa5baf6680fa633 | What is the largest value of \( n \) such that \( n! + 1 \) is a perfect square? | [
"7"
] | [
"What is the largest value of \\( n \\) such that \\( n! + 1 \\) is a perfect square?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ca9f195d9ed4be9bca43876112bd5fadf3516db147a4b256c805cb2e546cf992 | What is the largest value of \( n \) such that \( n! + 1 \) is a perfect square? Is this an open question? | [
"7"
] | [
"What is the largest value of \\( n \\) such that \\( n! + 1 \\) is a perfect square? Is this an open question?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-199a14836c171c76a87b8ed7839a5e26ecc7f7700e44bf4a31d53f321458a7c2 | What is the largest value of \( n \in \mathbb{N} \) such that \(\sum_{i=1}^{n} \frac{1}{a_i} = 120\) and \(\sum_{i=1}^{n} \sqrt{a_i} = 15\), where \(a_1, a_2, \ldots, a_n \in \mathbb{R}_+\)? | [
"30"
] | [
"What is the largest value of \\( n \\in \\mathbb{N} \\) such that \\(\\sum_{i=1}^{n} \\frac{1}{a_i} = 120\\) and \\(\\sum_{i=1}^{n} \\sqrt{a_i} = 15\\), where \\(a_1, a_2, \\ldots, a_n \\in \\mathbb{R}_+\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-9f4dbb68e5e191005f353d0115a728fde353546c88e7f2156825b923ca12ef8b | What is the largest value of \( x \) for which \( 4^{27} + 4^{1000} + 4^x \) equals the square of a whole number? | [
"1972"
] | [
"What is the largest value of \\( x \\) for which \\( 4^{27} + 4^{1000} + 4^x \\) equals the square of a whole number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-7534a4061749af8c6c6bcf6e886e7a192ab081362544ee246760027f82a539cf | What is the largest value of \( x \) for which \( 4^{27} + 4^{100} + 4^x \) is a perfect square? | [
"1972"
] | [
"What is the largest value of \\( x \\) for which \\( 4^{27} + 4^{100} + 4^x \\) is a perfect square?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-74f9a426b6dd30097f9594625f0e93e077bb331627fd421767bc6b530ecc5dc2 | What is the largest value of \(\sin x\) which satisfies the equation \(\sin{x} + \sin{2x} + \cos{x} = 0\)? | [
"\\(\\frac{1}{2}\\sqrt{2+\\sqrt{2\\sqrt{5}-2}}\\)"
] | [
"What is the largest value of \\(\\sin x\\) which satisfies the equation \\(\\sin{x} + \\sin{2x} + \\cos{x} = 0\\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e36e47ed6af018f67142014b51e332098859ede9f5cbfbe83c4b61a7cb769708 | What is the largest value that $3x + 4y$ can have if $x^2 + y^2 = 14x + 6y + 6$? | [
"73"
] | [
"What is the largest value that $3x + 4y$ can have if $x^2 + y^2 = 14x + 6y + 6$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c62f532beff73b301b4daf927047b6bbc707314d9ec8adbf0fe3613df2052066 | What is the largest value that the length of the segment cut by the sides of the triangle on the tangent to the inscribed circle parallel to the base can take if the perimeter of the triangle is $2p$? | [
"\\(\\frac{p}{4}\\)"
] | [
"What is the largest value that the length of the segment cut by the sides of the triangle on the tangent to the inscribed circle parallel to the base can take if the perimeter of the triangle is $2p$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-5e0f0ed2cdcd5d62691442dce60f20b24047bad9ebf64e272eb07d21efa1f1cd | What is the largest value that the smallest number can have if twenty-five different positive integers sum up to 2011? | [
"68"
] | [
"What is the largest value that the smallest number can have if twenty-five different positive integers sum up to 2011?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-0602dc19789fd09cc1a8023ebffb51cf2e2118621a89bc48b38f78c37ea5f976 | What is the largest volume of a cylinder that can be made from a rectangular piece of metal with a width of 2 and a height of 10 by cutting two circles for the bases and a rectangle for the sides? (Ignore the thickness of the metal) | [
"\\(\\frac{250\\pi^2}{(\\pi + 2)^3}\\)"
] | [
"What is the largest volume of a cylinder that can be made from a rectangular piece of metal with a width of 2 and a height of 10 by cutting two circles for the bases and a rectangle for the sides? (Ignore the thickness of the metal)"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-90c06239327c817907f28dd218d2583854d9b141416471b3c4023480e5613589 | What is the largest volume of a parallelepiped that can be located inside a tetrahedron with a volume of $1$? | [
"\\(\\frac{4}{9}\\)"
] | [
"What is the largest volume of a parallelepiped that can be located inside a tetrahedron with a volume of $1$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-82db081d08d9d0b6a1097edd642017e1abb77077401ed4b31c05261dee9e9644 | What is the largest volume of a sphere which touches a unit sphere internally and touches externally a regular tetrahedron whose corners are on the unit sphere? | [
"\\(\\frac{1}{3}\\)"
] | [
"What is the largest volume of a sphere which touches a unit sphere internally and touches externally a regular tetrahedron whose corners are on the unit sphere?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-de887f21774adb61f14d29dd6a281eee74fe1ce4f8841908220d3fd3a20f7602 | What is the largest whole number that cannot be expressed as \(6a + 9b + 20c\), where \(a\), \(b\), and \(c\) are non-negative integers? | [
"43"
] | [
"What is the largest whole number that cannot be expressed as \\(6a + 9b + 20c\\), where \\(a\\), \\(b\\), and \\(c\\) are non-negative integers?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e2e6a796f61411ae357e2091870eebd26a62f20b9dfbca013f4ab389d6cd19e5 | What is the largest whole number that is equal to the product of its digits? | [
"9"
] | [
"What is the largest whole number that is equal to the product of its digits?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-582e6408c4d3cae5985f6c6e8a300102bd6f278390d151bb60a4a6f2bde04dd0 | What is the largest whole number value of $n$ that makes the following inequality true? $$\frac13 + \frac{n}7 < 1$$ | [
"4"
] | [
"What is the largest whole number value of $n$ that makes the following inequality true? $$\\frac13 + \\frac{n}7 < 1$$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8a6ecd2b0e15e1b010efcafef11eb440deb182af1dab72165aa709dd9aaef0c0 | What is the last digit in the decimal representation of the number $$7^{\Big(7^{.^{. ^{. ^{\left(7^{(7^7)}\right)}}}}\ldots \Big)}$$ which is written with the aid of 1001 sevens as in Problem (a) but the order of the raising to the power is changed? What are the last two digits of this number? | [
"7; 07"
] | [
"What is the last digit in the decimal representation of the number $$7^{\\Big(7^{.^{. ^{. ^{\\left(7^{(7^7)}\\right)}}}}\\ldots \\Big)}$$ which is written with the aid of 1001 sevens as in Problem (a) but the order of the raising to the power is changed? What are the last two digits of this number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4ac6e1889402facacf06bfb5371684313a764f3623914416e1394653a04314aa | What is the last digit of $$2^{3^{4^{5^{6^{7^{8^{9^{...^{2013}}}}}}}}} ?$$ | [
"2"
] | [
"What is the last digit of $$2^{3^{4^{5^{6^{7^{8^{9^{...^{2013}}}}}}}}} ?$$"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-42ae546bde7c4d831fbb531834a658f49c18ee3cac1021abdfdaac9517b5dcda | What is the last digit of $1+2+\dots+n$ if the last digit of $1^3+2^3+\dots+n^3$ is $1$? | [
"1"
] | [
"What is the last digit of $1+2+\\dots+n$ if the last digit of $1^3+2^3+\\dots+n^3$ is $1$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a641a38c3b05c2920097e0c6de8c692cc2a6c594ef97a8ac5bbd99bd04f81635 | What is the last digit of $2005^{2005}$? | [
"5"
] | [
"What is the last digit of $2005^{2005}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2bcf6aa19951e3cd14dceb4b67e46b94e615744ad49719dff63dc1936c2c3adb | What is the last digit of $2022 + 2022^{2022} + 2022^{(2022^{2022})}$? | [
"2"
] | [
"What is the last digit of $2022 + 2022^{2022} + 2022^{(2022^{2022})}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d0c77be94ce5d9e7b0f8dbf57acbfb7253a98784a00524c97805ae45fd04a832 | What is the last digit of $\frac{25!}{10^6}$? | [
"4"
] | [
"What is the last digit of $\\frac{25!}{10^6}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 11
} | 32 | 0 | 0 |
omr-ws-e49f97a85885e605858e7595537fa259925023fe999ccf04330b9b3a90d542a0 | What is the last digit of $\left\lfloor{\left(\frac{5+\sqrt{21}}{2}\right)^{2013}\right\rfloor}$? | [
"9"
] | [
"What is the last digit of $\\left\\lfloor{\\left(\\frac{5+\\sqrt{21}}{2}\\right)^{2013}\\right\\rfloor}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-5e3a7b579602473bf0fa1dc473c99ba9f978632787b7a32b285795584899a52b | What is the last digit of $|P(P([2013]))|$, where $[n] = \{1, 2, 3, \ldots, n\}$ and $P(S)$ is the set of non-empty subsets of $S$? | [
"7"
] | [
"What is the last digit of $|P(P([2013]))|$, where $[n] = \\{1, 2, 3, \\ldots, n\\}$ and $P(S)$ is the set of non-empty subsets of $S$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cd01461f4fd8afaf415d4ac815d7f0bdcea8bc5b4d2e3a19ed27c6d9f507c8ab | What is the last digit of \( (...((7)^7)^7)...)^7 \) if there are 1000 7s as exponents and only one 7 in the middle? | [
"3"
] | [
"What is the last digit of \\( (...((7)^7)^7)...)^7 \\) if there are 1000 7s as exponents and only one 7 in the middle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 2
} | 11 | 0 | 0 |
omr-ws-f0bfcc28e4f7a38dbf7af50389918ca6f68ee18cbfa4976189721687e18c038f | What is the last digit of \(17^{19^{23^{29^{\cdots}}}}\), where the exponents are the prime numbers up to 100? | [
"3"
] | [
"What is the last digit of \\(17^{19^{23^{29^{\\cdots}}}}\\), where the exponents are the prime numbers up to 100?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fa4715fe069f4b7d466e5de0b2b393b507b2ba4270d9f7bb1d46a8897c0a8034 | What is the last digit of \(2011!\)? | [
"0"
] | [
"What is the last digit of \\(2011!\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-88660c882b5e7323fb1df3f8000495bfda8be167a26ebb5701cdc00b29dd2593 | What is the last digit of \(7^{6^{5^{4^{3^{2^1}}}}}\)? | [
"1"
] | [
"What is the last digit of \\(7^{6^{5^{4^{3^{2^1}}}}}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-440dd03f6f70e99cde65979cacf99627c9fb3b7f2337d2a96b6bbf7d92110e76 | What is the last digit of \(7^{7^{7^{\cdots^7}}}\) if there are 1000 7s as exponents and only one 7 in the middle? | [
"7"
] | [
"What is the last digit of \\(7^{7^{7^{\\cdots^7}}}\\) if there are 1000 7s as exponents and only one 7 in the middle?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-40a74eaec639304ef89ee326ac0d70148942d67521a339a86a03d260513a256f | What is the last digit of the decimal expansion of $\frac{1}{2^{10}}$? | [
"5"
] | [
"What is the last digit of the decimal expansion of $\\frac{1}{2^{10}}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-02e937e9d2cf098bc1605ce251489cc42da247e0f4d685db152b092cee9850c7 | What is the last digit of the denominator of the product $$\frac12 \cdot \frac24 \cdot \frac38 \cdot \frac{4}{16} \cdot ... \cdot \frac{99}{2^{99}} \cdot \frac{100}{2^{100}}$$ when written in its most simplified form? | [
"2"
] | [
"What is the last digit of the denominator of the product \n$$\\frac12 \\cdot \\frac24 \\cdot \\frac38 \\cdot \\frac{4}{16} \\cdot ... \\cdot \\frac{99}{2^{99}} \\cdot \\frac{100}{2^{100}}$$\nwhen written in its most simplified form?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-1c09a5505b7dcce5e6a6cb5bd2409a675c4b79f90aaf2ba98006c8e7989241fc | What is the last digit of the expression $2024^{2024} + 2023^{2023} + 2022^{2022} + 2021^{2021}$? | [
"0"
] | [
"What is the last digit of the expression $2024^{2024} + 2023^{2023} + 2022^{2022} + 2021^{2021}$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8c5a5239460b9f0a00ff526b0d094b12d55cda3d412df744b6e407b17a2ac18d | What is the last digit of the floor of \((2 + \sqrt{2})^{2018}\)? | [
"1"
] | [
"What is the last digit of the floor of \\((2 + \\sqrt{2})^{2018}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cb742b1c3a0eca00c5d0259e9ed89d4ff6119a8b5c5b9fadac2a460e668abe36 | What is the last digit of the greatest value of "K" such that \(3^{2011}\) can be written as the sum of "K" consecutive positive integers? | [
"6"
] | [
"What is the last digit of the greatest value of \"K\" such that \\(3^{2011}\\) can be written as the sum of \"K\" consecutive positive integers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-562541d546e3be2c318823d4caebec4c9f7f6f65a9aa5ef08bed804d28423868 | What is the last digit of the number \(2^{2009}\)? | [
"2"
] | [
"What is the last digit of the number \\(2^{2009}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-46a50041507657f6437312a98254fadd81c5e9edb593ef6b1cb6922458597963 | What is the last digit of the product of the numbers from 1 to 2015 after all multiples of 5 are removed? | [
"4"
] | [
"What is the last digit of the product of the numbers from 1 to 2015 after all multiples of 5 are removed?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-3906c2cc97f866118bb9dfb4a5e20d4e06c9e001e8e42c0b09f71c2cc2f9babf | What is the last digit of the sum $1^{2012} + 2^{2012} + 3^{2012} + 4^{2012} + \ldots + 2011^{2012} + 2012^{2012}$? | [
"0"
] | [
"What is the last digit of the sum $1^{2012} + 2^{2012} + 3^{2012} + 4^{2012} + \\ldots + 2011^{2012} + 2012^{2012}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-de0aba76ce61212d8c5f71320581c8ec830a924a824aed0861def0b7295a0352 | What is the last digit of the sum \( S = 22 + 33 + 44 + \ldots + 20192019 \)? | [
"9"
] | [
"What is the last digit of the sum \\( S = 22 + 33 + 44 + \\ldots + 20192019 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1fc5c441c5eb41d7bd91a6390dfa7503f0289f08915bb448befe7eb8ca17c98e | What is the last digit of the sum \(1^{2023} + 2^{2023} + 3^{2023} + \ldots + 2023^{2023}\)? | [
"6"
] | [
"What is the last digit of the sum \\(1^{2023} + 2^{2023} + 3^{2023} + \\ldots + 2023^{2023}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2af2d481a7f3da77bec9de67caef7f931882e7eaf1adf9b7843c41b2357dfa7b | What is the last digit of the sum \(2^0 + 2^1 + 2^2 + \cdots + 2^{2009}\)? | [
"3"
] | [
"What is the last digit of the sum \\(2^0 + 2^1 + 2^2 + \\cdots + 2^{2009}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-fd0409205f5fff595df3199d8e0a0fdc849dfb9bc8b0ef6e9292eb47966b392f | What is the last non-zero digit in the decimal representation of $40!$? | [
"2"
] | [
"What is the last non-zero digit in the decimal representation of $40!$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 1
} | 11 | 0 | 0 |
omr-ws-d3c63d9962005b3d6b0144c8d31ce6f092644fa95dd7f51ca1d1da633f2225cf | What is the last non-zero digit of $(\dots((2018\underbrace{!)!)!\dots)!}_{! \text{ occurs } 1009 \text{ times}}$? | [
"2"
] | [
"What is the last non-zero digit of $(\\dots((2018\\underbrace{!)!)!\\dots)!}_{! \\text{ occurs } 1009 \\text{ times}}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 6
} | 13 | 0 | 0 |
omr-ws-73276eef1777b0b324f6bffe2f4bd57e1a32134b9b652d973f44118a78d9dae5 | What is the last nonzero digit of \(10000!\)? | [
"2"
] | [
"What is the last nonzero digit of \\(10000!\\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1708d213c0d76f306f0c1118a26abe19191f17c787d196aade9cf5f39b4d9d78 | What is the last three digits of the base-4 representation of \(10 \cdot 3^{195} \cdot 49^{49}\)? | [
"232"
] | [
"What is the last three digits of the base-4 representation of \\(10 \\cdot 3^{195} \\cdot 49^{49}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 6
} | 53 | 0 | 0 |
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