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-1ce75a5704568221df4c6fea867fc9100a2db8ac93354859bcc5d618c4aee5da | What is the smallest positive integer that ends in 9 and is divisible by 7? | [
"49"
] | [
"What is the smallest positive integer that ends in 9 and is divisible by 7?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ced4b60a6a778e3fe831b09261e35c15c1740e9ac3d056824255fd20cac28751 | What is the smallest positive integer that has $24$ positive divisors? | [
"360"
] | [
"What is the smallest positive integer that has $24$ positive divisors?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 6
} | 21 | 0 | 0 |
omr-ws-27f87e68803cbe8565d75dcb66ba6a2c20c3515154d4d5be2aaa57bf34baa37a | What is the smallest positive integer that has at least $25$ different positive divisors? | [
"\\( 2^{24} \\)"
] | [
"What is the smallest positive integer that has at least $25$ different positive divisors?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0816ade4f1f9b06a6d94ecb002f27d674e065087bcd200913153edfdf759cd1a | What is the smallest positive integer that has exactly 18 positive integer factors and leaves a remainder of 9 when divided by 31? | [
"288"
] | [
"What is the smallest positive integer that has exactly 18 positive integer factors and leaves a remainder of 9 when divided by 31?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-3af190f51c5826aa10224ac4225e919b8db238be96ddea12d6394d7317288abc | What is the smallest positive integer that has exactly eight distinct positive factors? | [
"24"
] | [
"What is the smallest positive integer that has exactly eight distinct positive factors?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ef7fc662df970d4b4f335642cf448f490bccd0939b2ea8125253e71dd52bafda | What is the smallest positive integer that is a multiple of both 30 and 40 but not a multiple of 16? | [
"120"
] | [
"What is the smallest positive integer that is a multiple of both 30 and 40 but not a multiple of 16?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3e8acf33382690de8cbb2f29e23c1de7a2097883e1a05c253964f6f83cbc4d87 | What is the smallest positive integer that is an upper bound of the zeros of the polynomial \( P(x) = 3x^3 + 2x^2 + 4x - 6 \)? | [
"1"
] | [
"What is the smallest positive integer that is an upper bound of the zeros of the polynomial \\( P(x) = 3x^3 + 2x^2 + 4x - 6 \\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-3cea259311eb16c7228ec10b9415270fc0e6b653685c28f6c399cfc5de707f19 | What is the smallest positive integer that is both a multiple of $7$ and a multiple of $4$? | [
"28"
] | [
"What is the smallest positive integer that is both a multiple of $7$ and a multiple of $4$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-223b0f05ac7adf8f793aa50aee68caf6442480b347efe5f1d5c2597129983c5c | What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 50? | [
"3127"
] | [
"What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 50?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-baff85ed94ca491b09012c80928678e546a7b83a15350a8b138b159d7c39be1e | What is the smallest positive integer that is one-ninth of its reverse? | [
"1089"
] | [
"What is the smallest positive integer that is one-ninth of its reverse?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ee8cbdff343ff7cc2f02594b1588b4efbe95bd29a4df3d284361fabc9fcff870 | What is the smallest positive integer that leaves a remainder of 4 when divided by 5 and a remainder of 6 when divided by 7? | [
"34"
] | [
"What is the smallest positive integer that leaves a remainder of 4 when divided by 5 and a remainder of 6 when divided by 7?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0059827421a57941920f50d9c2f8406e88dbca5f98e851855f25a323d556d792 | What is the smallest positive integer that leaves a remainder of 7 when divided by 11, a remainder of 17 when divided by 21, and a remainder of 26 when divided by 40? | [
"2306"
] | [
"What is the smallest positive integer that leaves a remainder of 7 when divided by 11, a remainder of 17 when divided by 21, and a remainder of 26 when divided by 40?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-52a04a6de4a4e9b51fded7443d4612730ac5e7ae48875b26d7c6d7c3a303fa8a | What is the smallest positive integer that satisfies the congruence $4x \equiv 13 \pmod{27}$? | [
"10"
] | [
"What is the smallest positive integer that satisfies the congruence $4x \\equiv 13 \\pmod{27}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ed6f7e1486f13be966ad3f3ca17232df5bc4a9611053fa9f3872ab086f4cb628 | What is the smallest positive integer value of $n$ so that any $n$-element subset of $\{1,2,3,...,1024\}$ contains two integers such that one is twice the other? | [
"684"
] | [
"What is the smallest positive integer value of $n$ so that any $n$-element subset of $\\{1,2,3,...,1024\\}$ contains two integers such that one is twice the other?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9a3c02f49f34c15af57217e83b45be8256b010da7e8eb3f1d55522dd04bbc067 | What is the smallest positive integer value of $n$ such that $10^n - 1$ is divisible by $81$? | [
"9"
] | [
"What is the smallest positive integer value of $n$ such that $10^n - 1$ is divisible by $81$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-30e5d8439314a2812ffdd3c11f0b4c5d69dd701e3889f0e214d9ed5be32d89cf | What is the smallest positive integer value of \( n \) such that \( 2^n - n^2 \) is a non-zero multiple of 16? | [
"8"
] | [
"What is the smallest positive integer value of \\( n \\) such that \\( 2^n - n^2 \\) is a non-zero multiple of 16?"
] | [
"aops_c5_contests_amp_programs",
"c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-72a5b0f9f2d4abae830cff6234a572b5e5fcb9503815eb58005f5f50b6a147d3 | What is the smallest positive integer whose square ends in $444$? | [
"38"
] | [
"What is the smallest positive integer whose square ends in $444$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 3
} | 20 | 0 | 0 |
omr-ws-89b7918f8e774afb1cb6372539f6936ba6a7a53bbc6e7956f5c7084e53c53742 | What is the smallest positive integer with 2010 distinct positive factors? | [
"\\(2^{66} \\cdot 3^4 \\cdot 5^2 \\cdot 7\\)"
] | [
"What is the smallest positive integer with 2010 distinct positive factors?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 26,
"tir": 3
} | 29 | 0 | 0 |
omr-ws-2f5127ea8bd6f2e53dc7f07f60bf1edb7ddc69533bb301fb9d2a8fec45873bd8 | What is the smallest positive integer with 6 positive odd integer divisors and 12 positive even integer divisors? | [
"180"
] | [
"What is the smallest positive integer with 6 positive odd integer divisors and 12 positive even integer divisors?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-5b71c420c7d5fa3c3d54d987cb576822c11c6619cf1ba462ed539e26a5b92b65 | What is the smallest positive integer with exactly $768$ divisors? Your answer may be written in its prime factorization. | [
"\\( 2^7 \\times 3^7 \\times 5^3 \\times 7^2 \\)"
] | [
"What is the smallest positive integer with exactly $768$ divisors? Your answer may be written in its prime factorization."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b6c776e1c0586aaa6defd34c759a573b5e68d1d40184ee5e839b483b9c0ea2b8 | What is the smallest positive integer with exactly 10 positive integer divisors? | [
"48"
] | [
"What is the smallest positive integer with exactly 10 positive integer divisors?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e65f4c9c128c6c820dfad291c521058b1422d336511a59b141cdec2d7b1fe22b | What is the smallest positive integer with exactly 14 positive divisors? | [
"192"
] | [
"What is the smallest positive integer with exactly 14 positive divisors?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-88de582b76e47d5238daff54bbc7e32e00e17edc7f9cbd0c515f8b4d77203676 | What is the smallest positive integer with six positive odd integer divisors and twelve positive even integer divisors? | [
"180"
] | [
"What is the smallest positive integer with six positive odd integer divisors and twelve positive even integer divisors?",
"What is the smallest positive integer with six positive odd integer divisors and twelve positive even integer divisors?\n"
] | [
"MATH_training_set",
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 1
} | 48 | 0 | 0 |
omr-ws-cb8a554a4a5fd38c039d388fe8a018462bfd0b5cf2327d1a095c273ed4709071 | What is the smallest positive multiple of $23$ that is $4$ more than a multiple of $89$? | [
"805"
] | [
"What is the smallest positive multiple of $23$ that is $4$ more than a multiple of $89$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d365938efd1dfbf1bb6d1dd88c5fa516467f60d6eb60375312363ed57a705f09 | What is the smallest positive multiple of $32$? | [
"32"
] | [
"What is the smallest positive multiple of $32$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-22d9b0e4e9c0c6981b27e0de379fd341b8c4632eeabd1be9e2c92cd5e04adef7 | What is the smallest positive multiple of 24 that can be written using digits 4 and 5 only? | [
"5544"
] | [
"What is the smallest positive multiple of 24 that can be written using digits 4 and 5 only?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-5cfe14592eab569c5b6ce19c13ad47e81d7255948f5e3c0a637eb2960e1a7f59 | What is the smallest positive multiple of 99 such that the sum of its digits is odd? | [
"10989"
] | [
"What is the smallest positive multiple of 99 such that the sum of its digits is odd?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-046468e4aed784f734ca61d9568ef940d6c13eb25ac5e2189551e8e8486bd9e5 | What is the smallest positive number $y$ for which $(\sqrt{2})(\sqrt{4})y$ and $(\sqrt{3})(\sqrt{6})y$ both have positive integral values? | [
"\\( \\frac{1}{\\sqrt{2}} \\)"
] | [
"What is the smallest positive number $y$ for which $(\\sqrt{2})(\\sqrt{4})y$ and $(\\sqrt{3})(\\sqrt{6})y$ both have positive integral values?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-c035c3b57f9e7757fad24316cf656f946c999fc9d3b5b4f31f3efce5b67c8ec8 | What is the smallest positive number \( n \) such that \( x^5 \equiv 1 \pmod{361179n} \) has five integer solutions for \( 0 \le x < 361179n \)? Note: \( 361179 = 3^4 \cdot 7^3 \cdot 13 \). | [
"11"
] | [
"What is the smallest positive number \\( n \\) such that \\( x^5 \\equiv 1 \\pmod{361179n} \\) has five integer solutions for \\( 0 \\le x < 361179n \\)? Note: \\( 361179 = 3^4 \\cdot 7^3 \\cdot 13 \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3c1e984f67403d4b7e101d9319b674619150b913259acec4e48cea22eb9686b7 | What is the smallest positive number that is either one larger than a perfect cube and one less than a perfect square, or vice versa? | [
"26"
] | [
"What is the smallest positive number that is either one larger than a perfect cube and one less than a perfect square, or vice versa?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-f9e8463f2534c79908459d16e886be50df342add0e2f4defea5b1523c66aa09c | What is the smallest positive number that is prime and $10$ less than a perfect square? | [
"71"
] | [
"What is the smallest positive number that is prime and $10$ less than a perfect square?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c32bf604ec971c93b65b5052c9eadb92707bda2c1e7c61c9b4295f0c462b01c0 | What is the smallest positive number which is equal to the cube of one positive integer and also equal to the fourth power of a different positive integer? | [
"4096"
] | [
"What is the smallest positive number which is equal to the cube of one positive integer and also equal to the fourth power of a different positive integer?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-36e73da0597e591ec8b2c6b8ff0479b870d27a95e2b5b11a2bc36e0ab74b4e47 | What is the smallest positive odd integer that has the same number of positive divisors as $360$? | [
"3465"
] | [
"What is the smallest positive odd integer that has the same number of positive divisors as $360$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-b8c9afe601832d43e8156f0880318a5350e9d13e7025d22d5cb6412cdd72d053 | What is the smallest positive perfect cube that has $pq^{2}r^{4}$ as a factor, where $p$, $q$, and $r$ are three distinct primes? | [
"\\( 60^3 \\)"
] | [
"What is the smallest positive perfect cube that has $pq^{2}r^{4}$ as a factor, where $p$, $q$, and $r$ are three distinct primes?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 30
} | 50 | 0 | 0 |
omr-ws-0ad1f0082706a535a6f0b27aa2642b2211e7ee181570f00b00bc82c122c4b260 | What is the smallest positive perfect square that is divisible by both 2 and 3? | [
"36"
] | [
"What is the smallest positive perfect square that is divisible by both 2 and 3?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-845daad94aab2fcbd11a51b353f10eb0b1236d0da8183914b09c99e8e69c9922 | What is the smallest positive real number \( r \) such that a square of side length 1 can be covered by 3 disks of radius \( r \)? | [
"\\( \\frac{\\sqrt{65}}{16} \\)"
] | [
"What is the smallest positive real number \\( r \\) such that a square of side length 1 can be covered by 3 disks of radius \\( r \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-4de29793b60174f126fbffdeee0acdca82b4fcc4ecbb56f6916ba3726d898355 | What is the smallest positive root of the equation \(\sqrt{\sin(1-x)} = \sqrt{\cos(x)}\)? | [
"\\(\\frac{2 + 7\\pi}{4}\\)"
] | [
"What is the smallest positive root of the equation \\(\\sqrt{\\sin(1-x)} = \\sqrt{\\cos(x)}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 35,
"genselect": 8,
"tir": 8
} | 51 | 0 | 0 |
omr-ws-6e5990d66dcc1d6b4212be18c43b953b3732fee950924cd3f76ea57e7a6ff000 | What is the smallest positive value of $\theta$ such that $\sin \theta$ is the same whether $\theta$ is measured in degrees or radians? | [
"\\(\\frac{180 \\pi}{180 + \\pi}\\)"
] | [
"What is the smallest positive value of $\\theta$ such that $\\sin \\theta$ is the same whether $\\theta$ is measured in degrees or radians?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 21
} | 43 | 0 | 0 |
omr-ws-f6db4a7fecb1224d23adaf33cd10532e2eac693b0b228211bc5dcad1e880b89f | What is the smallest positive value of $k_{10}$ such that $9_{10}$ evenly divides $(12345678910...2016_{12345...k} - 2016201520142013...321_{12345...k})_{10}$ where $a_n$ signifies the value of $a$'s expansion in base $n$ number system? | [
"7"
] | [
"What is the smallest positive value of $k_{10}$ such that $9_{10}$ evenly divides $(12345678910...2016_{12345...k} - 2016201520142013...321_{12345...k})_{10}$ where $a_n$ signifies the value of $a$'s expansion in base $n$ number system?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 3
} | 5 | 0 | 0 |
omr-ws-eef71f9820d3895d0d93994373f1aa04e20483349f95edce8e654bcba73335de | What is the smallest positive value of $m$ so that the equation $10x^2 - mx + 420 = 0$ has integral solutions? | [
"130"
] | [
"What is the smallest positive value of $m$ so that the equation $10x^2 - mx + 420 = 0$ has integral solutions?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c23df35bab4e4ccfbcbbaa1d583ad60712364dd8f7a09905d825f8597ef610d0 | What is the smallest positive value of $x$ such that $x + 5678$ results in a palindrome? | [
"97"
] | [
"What is the smallest positive value of $x$ such that $x + 5678$ results in a palindrome?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-83ba5722bd7dca388f144d52c15072fc96bd5de2605d4971b2955eeb8df544f6 | What is the smallest positive value of \( 33^a - 7^b \) where \( a \) and \( b \) are non-negative integers? | [
"26"
] | [
"What is the smallest positive value of \\( 33^a - 7^b \\) where \\( a \\) and \\( b \\) are non-negative integers?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2bd3f50aa274fd2cc41016ed22864c72ae715d7a1790a4d155ef4118eed70a58 | What is the smallest positive value of \( x \) (in radians) such that: \[ \frac{\sin x - \sin 7x}{\cos 7x - \cos x} = \tan 6x \] | [
"\\(\\pi/20\\)"
] | [
"What is the smallest positive value of \\( x \\) (in radians) such that:\n\\[\n\\frac{\\sin x - \\sin 7x}{\\cos 7x - \\cos x} = \\tan 6x\n\\]"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 31
} | 46 | 0 | 0 |
omr-ws-41693e51a9f8d0aa7daf5e46aebbcb2f1bfab03bac5dd93bc9629f76c59b0346 | What is the smallest positive value of \( x \) that satisfies the following system of congruences: \[ x \equiv 8 \pmod{11} \] \[ x \equiv 4 \pmod{15} \] \[ 2x \equiv 7 \pmod{13} \] | [
"1999"
] | [
"What is the smallest positive value of \\( x \\) that satisfies the following system of congruences:\n\\[ x \\equiv 8 \\pmod{11} \\]\n\\[ x \\equiv 4 \\pmod{15} \\]\n\\[ 2x \\equiv 7 \\pmod{13} \\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a2f2d8a668106828691532067d4b5d4b0c9c2b53e75ceba4e10b78896a79038f | What is the smallest possible area of a pentagon with integral coordinates? | [
"\\(\\frac{3}{2}\\)"
] | [
"What is the smallest possible area of a pentagon with integral coordinates?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 24,
"genselect": 7,
"tir": 10
} | 41 | 0 | 0 |
omr-ws-7f69fa802d1dec02041ab49198473bdabc5ccbfe277b71f8b5fcad6857b6146e | What is the smallest possible area of a square that can contain both a 2×3 rectangle and a 3×4 rectangle without overlapping, given that the sides of the square are parallel to the sides of the rectangles? | [
"25"
] | [
"What is the smallest possible area of a square that can contain both a 2×3 rectangle and a 3×4 rectangle without overlapping, given that the sides of the square are parallel to the sides of the rectangles?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-d9257f92d6a24ee35d0c811d00aa7c089517973425d7c3ca37ada65585e803e2 | What is the smallest possible area of a triangle where all three sides and the area are integers? | [
"6"
] | [
"What is the smallest possible area of a triangle where all three sides and the area are integers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-83d3b6ff0dd194a30aa02a55532357195c36b3e8aac8279109bc5794a11aa677 | What is the smallest possible area of a triangle with integer side lengths? | [
"6"
] | [
"What is the smallest possible area of a triangle with integer side lengths?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-65fedc1c920fd94c42997ee2ac71bd843cdd7f40cd10918db42548b01967d43b | What is the smallest possible area of an ellipse containing the points $(4,1)$, $(4,-1)$, $(-4,1)$, and $(-4,-1)$? | [
"\\(8\\pi\\)"
] | [
"What is the smallest possible area of an ellipse containing the points $(4,1)$, $(4,-1)$, $(-4,1)$, and $(-4,-1)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-b13293a2abf763ea837c33d653b9e1b88b223818a64add996680940651198950 | What is the smallest possible area, in square units, of a right triangle with two sides measuring $4$ units and $5$ units? | [
"6"
] | [
"What is the smallest possible area, in square units, of a right triangle with two sides measuring $4$ units and $5$ units?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-05399d7a52bb10beccd41f46cd5508a0555e7c430ef9662fe5bb1a253b26133b | What is the smallest possible cardinality of the set of limit points of an uncountable subset of $\mathbb{R}$? | [
"$\\mathfrak{c}$"
] | [
"What is the smallest possible cardinality of the set of limit points of an uncountable subset of $\\mathbb{R}$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 4,
"tir": 9
} | 28 | 0 | 0 |
omr-ws-683bd60850b99edc7e2a8d2a58e4faa796bc069e274841944075d23a55d714a4 | What is the smallest possible digit sum of a positive integer $A$ that is divisible by $2024$? | [
"4"
] | [
"What is the smallest possible digit sum of a positive integer $A$ that is divisible by $2024$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 5
} | 19 | 0 | 0 |
omr-ws-f65dcd3c22a340176b62f4a92ffb8f3e7580ab5b0a5eaf99c213bdb0d3cf3d46 | What is the smallest possible integer $n > 1$ such that $3^n$ ends with 003? | [
"101"
] | [
"What is the smallest possible integer $n > 1$ such that $3^n$ ends with 003?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-d36f90e529086c697556937036f82bfcab8afca8799006db954ea90f53a67e21 | What is the smallest possible integer value of $n$ such that in any group of $2n-10$ persons, there are always at least $10$ persons who have the same birthday? Assume there are exactly $365$ different possible birthdays. | [
"1648"
] | [
"What is the smallest possible integer value of $n$ such that in any group of $2n-10$ persons, there are always at least $10$ persons who have the same birthday? Assume there are exactly $365$ different possible birthdays."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-011a0c706c23fe2b432db9b57675440e9540c05295909814fed1d3730d3265d3 | What is the smallest possible length of the diagonal of a rectangle inscribed in a triangle with known side lengths, where the rectangle has two neighboring vertices on one side of the triangle and the other two vertices on the remaining two sides? | [
"\\dfrac{abc}{\\sqrt{a^2b^2 + b^2c^2 + c^2a^2}}"
] | [
"What is the smallest possible length of the diagonal of a rectangle inscribed in a triangle with known side lengths, where the rectangle has two neighboring vertices on one side of the triangle and the other two vertices on the remaining two sides?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-3581205e45bfa1bbc549b969f0523f5daa7182ebdea707b32018a2710e9f8303 | What is the smallest possible natural number $n$ for which the quadratic equation $x^2 - nx + 2014 = 0$ has integer roots? | [
"91"
] | [
"What is the smallest possible natural number $n$ for which the quadratic equation $x^2 - nx + 2014 = 0$ has integer roots?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 23
} | 55 | 0 | 0 |
omr-ws-8410833068d947bc217741841c3c4313bac12bf4536c1add93e81ceba6cc3703 | What is the smallest possible number $p$ such that from any set of $p$ integers, we can always choose $n+2$ differences, all of which are divisible by $n$? | [
"2n + 1"
] | [
"What is the smallest possible number $p$ such that from any set of $p$ integers, we can always choose $n+2$ differences, all of which are divisible by $n$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-9982e6380e1d814d748b7ee0148cfe13fd04842b95e5ef9bf5eab8c67f72153d | What is the smallest possible number of whole 2-by-3 non-overlapping rectangles needed to cover a square region exactly, without extra over-hangs and without gaps? | [
"6"
] | [
"What is the smallest possible number of whole 2-by-3 non-overlapping rectangles needed to cover a square region exactly, without extra over-hangs and without gaps?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4af391296ee4872499600babdd9a669fc585c341f9e6bb701f35c19e8da6077b | What is the smallest possible perimeter of a triangle with integer coordinate vertices, area $\frac{1}{2}$, and no side parallel to an axis? | [
"\\(\\sqrt{2} + \\sqrt{5} + \\sqrt{13}\\)"
] | [
"What is the smallest possible perimeter of a triangle with integer coordinate vertices, area $\\frac{1}{2}$, and no side parallel to an axis?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-ebe3340303e0ae0ae7d5954cc784c61cfcc4a8871c830e6e90f72cc1566fb437 | What is the smallest possible perimeter of an isosceles triangle with integer side lengths if one side has a length of 4? | [
"9"
] | [
"What is the smallest possible perimeter of an isosceles triangle with integer side lengths if one side has a length of 4?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8b6c680afe60f40576641b5547a86d2c0e126b7df6d6e4e4677ae9ef89cf367d | What is the smallest possible perimeter, in units, of a triangle whose side-length measures are consecutive integer values? | [
"9"
] | [
"What is the smallest possible perimeter, in units, of a triangle whose side-length measures are consecutive integer values?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-181baf9d834ee9acd95dfc3eb33588d9a1d1ad32badfee53f2e560497b9a84c2 | What is the smallest possible product for a 3 by 3 magic square where the product of the numbers in each horizontal and vertical row, and each diagonal are the same, and each number is a distinct integer? | [
"216"
] | [
"What is the smallest possible product for a 3 by 3 magic square where the product of the numbers in each horizontal and vertical row, and each diagonal are the same, and each number is a distinct integer?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 3
} | 16 | 0 | 0 |
omr-ws-c2beccb4c071467ab1fb864caa8b82176c3d1080d58c0a4b393045b808c3c005 | What is the smallest possible radius of a circle circumscribed around a triangle when two sides are fixed and the third side can vary? | [
"\\dfrac{\\max(a, b)}{2}"
] | [
"What is the smallest possible radius of a circle circumscribed around a triangle when two sides are fixed and the third side can vary?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a107992acbf12c8678386a1f163a560fc61b1d09fae1e15d6b38a031a7c57643 | What is the smallest possible radius of a circle that can enclose a symmetrical figure composed of 3 squares, each with a side length of 1? | [
"\\(\\frac{5\\sqrt{17}}{16}\\)"
] | [
"What is the smallest possible radius of a circle that can enclose a symmetrical figure composed of 3 squares, each with a side length of 1?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 5,
"tir": 1
} | 8 | 0 | 0 |
omr-ws-93e835a996283c4b09f870d8e646f34fc1a5a86eb07ccf8aba577c6d9b1a0535 | What is the smallest possible radius of a circle that passes through at least 16 lattice points on the coordinate grid? (A lattice point is defined as one whose coordinates are both integers.) | [
"\\(5\\sqrt{2}\\)"
] | [
"What is the smallest possible radius of a circle that passes through at least 16 lattice points on the coordinate grid? (A lattice point is defined as one whose coordinates are both integers.)"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-21097e437f00c398ff82be8704ca999ded3fbddbe03070876a539a1de80757ae | What is the smallest possible ratio of sides of a harmonic triangle, where a harmonic triangle is defined as a triangle in which the length of one side is the harmonic mean of the lengths of the other two sides? State your answer by giving the largest number smaller than every possible ratio. | [
"\\(\\sqrt{2} - 1\\)"
] | [
"What is the smallest possible ratio of sides of a harmonic triangle, where a harmonic triangle is defined as a triangle in which the length of one side is the harmonic mean of the lengths of the other two sides? State your answer by giving the largest number smaller than every possible ratio."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 30
} | 52 | 0 | 0 |
omr-ws-73d3923c2eacc3ba9248cc4510ca8272953c6460ba5261d2f0aacdb81a575669 | What is the smallest possible real value of $x^2 + 8x$? | [
"-16"
] | [
"What is the smallest possible real value of $x^2 + 8x$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-95baa95a876e4c8878257bfe37fde32e4d5acb451284eccfb127abfb4254fb02 | What is the smallest possible sum of \( n \) and \( k \) if \( \binom{12}{2} + \binom{12}{3} = \binom{n}{k} \)? | [
"16"
] | [
"What is the smallest possible sum of \\( n \\) and \\( k \\) if \\( \\binom{12}{2} + \\binom{12}{3} = \\binom{n}{k} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-d0cc10466c6b30f1ad2636c45cddfce642872705ca916e80d0b0c309356f2c1e | What is the smallest possible sum of all the numbers on the surface of a 3x3x3 cube formed by gluing together 27 standard cubical dice, where opposite sides of each die add to 7? | [
"90"
] | [
"What is the smallest possible sum of all the numbers on the surface of a 3x3x3 cube formed by gluing together 27 standard cubical dice, where opposite sides of each die add to 7?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-f1d6960e2f6aa73a897eb1dce1dd3efde2bd937b8e630bc0a9a5dc68a28d6762 | What is the smallest possible sum of all the numbers showing on the surface of a $3 \times 3 \times 3$ cube formed by gluing together 27 standard cubical dice, where the sum of the numbers on any pair of opposite faces of a die is 7? | [
"90"
] | [
"What is the smallest possible sum of all the numbers showing on the surface of a $3 \\times 3 \\times 3$ cube formed by gluing together 27 standard cubical dice, where the sum of the numbers on any pair of opposite faces of a die is 7?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-4834883904cd3fb971d3cdb454512bc41f93fb302e5f9d83e52c28fb73f85aad | What is the smallest possible sum of all the numbers showing on the surface of a $3 \times 3 \times 3$ cube formed by gluing together 27 standard cubical dice? (On a standard die, the sum of the numbers on any pair of opposite faces is 7.) | [
"90"
] | [
"What is the smallest possible sum of all the numbers showing on the surface of a $3 \\times 3 \\times 3$ cube formed by gluing together 27 standard cubical dice? (On a standard die, the sum of the numbers on any pair of opposite faces is 7.)"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 1
} | 18 | 0 | 0 |
omr-ws-134af1838314aca4b63634c56ad235bde162bb971cc821ca48a2ba740b45307e | What is the smallest possible sum of all the numbers showing on the surface of a 3 by 3 by 3 cube formed by gluing together 27 standard cubical dice? | [
"90"
] | [
"What is the smallest possible sum of all the numbers showing on the surface of a 3 by 3 by 3 cube formed by gluing together 27 standard cubical dice?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-ae41cdd57e1cce4befbb552572c3da995c0ff063105e7546d9e7a33b1d5756e4 | What is the smallest possible sum of six distinct positive integers for which the sum of any five of them is prime? | [
"108"
] | [
"What is the smallest possible sum of six distinct positive integers for which the sum of any five of them is prime?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-1dc49298a1beea16ce8abbbfae1ad6c5afacb6069ec83fd93a63d81c81401bb2 | What is the smallest possible surface area for a rectangular box with sides of integer length and volume $2013$ cubic units? | [
"1774"
] | [
"What is the smallest possible surface area for a rectangular box with sides of integer length and volume $2013$ cubic units?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-5fbb6a9e6afec6efbc152e0dbb0b722a2b74cd6923a8a3d8126b7b5951787df5 | What is the smallest possible value for the area of a triangle where the sides and the area are all integer numbers? | [
"6"
] | [
"What is the smallest possible value for the area of a triangle where the sides and the area are all integer numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-1f2e039a4167a323c05f406b025ba25601285bcade3df5f276192a7a8da9a0ed | What is the smallest possible value for the area of an isosceles trapezoid ABCD, where the length of each side is a positive integer? | [
"\\(\\frac{3\\sqrt{3}}{2}\\)"
] | [
"What is the smallest possible value for the area of an isosceles trapezoid ABCD, where the length of each side is a positive integer?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-38265fbf98f859629622c83db65de08e6d3d8afe83df3a22e61a743210f0691c | What is the smallest possible value of $\left|12^{m}-5^{n}\right|$, where $m$ and $n$ are positive integers? | [
"7"
] | [
"What is the smallest possible value of $\\left|12^{m}-5^{n}\\right|$, where $m$ and $n$ are positive integers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-c969c62571f4c7dda165fe2118cc51144ffff7091e861745fb5be74d92b0c353 | What is the smallest possible value of $a+b$, if $a$ and $b$ are positive integers such that $\frac{1}{2a} + \frac{1}{3a} + \frac{1}{3a} = \frac{1}{b^2 - 2b}$? | [
"34"
] | [
"What is the smallest possible value of $a+b$, if $a$ and $b$ are positive integers such that $\\frac{1}{2a} + \\frac{1}{3a} + \\frac{1}{3a} = \\frac{1}{b^2 - 2b}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-ef67b8b5ac17bc61274a4ed3d26abab9042fb8a879b7430d36fc23a1355265a0 | What is the smallest possible value of $n$ such that $\displaystyle\prod_{k=1}^n \frac{n}{10} > 1$? | [
"25"
] | [
"What is the smallest possible value of $n$ such that $\\displaystyle\\prod_{k=1}^n \\frac{n}{10} > 1$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-767a90f439977397e75df2ee5dd3df73f2ceab075624694049bea6592b4e8902 | What is the smallest possible value of $t\left(P_{1} \cdot P_{2} \cdot \ldots \cdot P_{2021}\right)$, if $P_{1}, P_{2}, \ldots, P_{2021} \in \mathbb{R}[x]$ are non-constant, monic, pairwise distinct polynomials, none of which is symmetric, and the product of any two (distinct) polynomials is symmetric? | [
"4042"
] | [
"What is the smallest possible value of $t\\left(P_{1} \\cdot P_{2} \\cdot \\ldots \\cdot P_{2021}\\right)$, if $P_{1}, P_{2}, \\ldots, P_{2021} \\in \\mathbb{R}[x]$ are non-constant, monic, pairwise distinct polynomials, none of which is symmetric, and the product of any two (distinct) polynomials is symmetric?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-da73066678efc70610cb660bf3d97b072e5bb6af430f826d3ab221bed76e8385 | What is the smallest possible value of $x$ such that $2x^2+24x-60=x(x+13)$? | [
"-15"
] | [
"What is the smallest possible value of $x$ such that $2x^2+24x-60=x(x+13)$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2cb4675f1dcb04db78a368d75f4b7ca34055a2f0c764d91d06bd036f42d6b2ee | What is the smallest possible value of $x^2 + 2y^2 - x - 2y - xy$? | [
"-\\dfrac{8}{7}"
] | [
"What is the smallest possible value of $x^2 + 2y^2 - x - 2y - xy$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c222a1f9c4520dbab365e240bd0818184a1c939d4f74e6c72f30a6b9b77c4b0a | What is the smallest possible value of $x^2 + y^2 - x - y - xy$? | [
"\\(-1\\)"
] | [
"What is the smallest possible value of $x^2 + y^2 - x - y - xy$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-bc2ea787e47fc6fcb79b04555c55290140a829fb6894e54c0839b4deee37b0d8 | What is the smallest possible value of \( k + m + n \) if \( k \), \( m \), and \( n \) are positive integers such that \( \frac{1}{k} + \frac{1}{m} + \frac{1}{n} < 1 \)? | [
"10"
] | [
"What is the smallest possible value of \\( k + m + n \\) if \\( k \\), \\( m \\), and \\( n \\) are positive integers such that \\( \\frac{1}{k} + \\frac{1}{m} + \\frac{1}{n} < 1 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d2ffa98cc4456a2f9e89d437972f8667d6f98c4ba1afbc2215904d41bb3754fd | What is the smallest possible value of \( n \) for which the sum of the interior angles of a regular polygon with \( n \) sides is less than \( n^2 \) degrees? | [
"178"
] | [
"What is the smallest possible value of \\( n \\) for which the sum of the interior angles of a regular polygon with \\( n \\) sides is less than \\( n^2 \\) degrees?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-fc71301cea8db36fa4fada246bd0072e1dc7b00892380fca8a9200ea937581ca | What is the smallest possible value of \( x^2 + 2y^2 + 2xy + 6x + 2y \)? | [
"-13"
] | [
"What is the smallest possible value of \\( x^2 + 2y^2 + 2xy + 6x + 2y \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-dde17a192cccd40dcc64ffb5765157c4c709b48a35a41cd4778b4ab5aa1216ab | What is the smallest possible value of \( |S| \), the number of elements in a set \( S \) of positive integers, such that the positive differences between all pairs of integers in \( S \) include every integer from 1 to 10? | [
"6"
] | [
"What is the smallest possible value of \\( |S| \\), the number of elements in a set \\( S \\) of positive integers, such that the positive differences between all pairs of integers in \\( S \\) include every integer from 1 to 10?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bbfc92e2a5167aa7bcab6d2eb1c2a1a8ddddb847cef7a9395f8175294bbc166c | What is the smallest possible value of \(x + y + z\) if \(\frac{1}{x} + \frac{1}{y} + \frac{1}{z} \leq 1\)? | [
"9"
] | [
"What is the smallest possible value of \\(x + y + z\\) if \\(\\frac{1}{x} + \\frac{1}{y} + \\frac{1}{z} \\leq 1\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-06ca39cd2b5ab511e575ad4607a76864365df7521bddb14775b6e4d5cb70a18e | What is the smallest possible value of the expression $$\sqrt{a^{2} + 9} + \sqrt{(b-a)^{2}+4} + \sqrt{(8-b)^{2} + 16}$$ for real numbers $a$ and $b$? | [
"\\(\\sqrt{145}\\)"
] | [
"What is the smallest possible value of the expression $$\\sqrt{a^{2} + 9} + \\sqrt{(b-a)^{2}+4} + \\sqrt{(8-b)^{2} + 16}$$ for real numbers $a$ and $b$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 30
} | 53 | 0 | 0 |
omr-ws-2c99e590a62d25918ae4ae3a37db65ce34b70c9a9150652b453fee90a0b94d49 | What is the smallest possible value of the expression $\sqrt{a^2+8}+\sqrt{(b-a)^2+4}+\sqrt{(8-b)^2+16}$? | [
"\\sqrt{108 + 24\\sqrt{2}}"
] | [
"What is the smallest possible value of the expression $\\sqrt{a^2+8}+\\sqrt{(b-a)^2+4}+\\sqrt{(8-b)^2+16}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-2e1aae4f147f6f19e2e9a4f24513da4ab85de16bd69791ddf339a3a2435a852c | What is the smallest possible value of the expression \(\sqrt{x^2+y^2}+\sqrt{(x-1)^2+y^2}+\sqrt{x^2+(y-1)^2}+\sqrt{(x-3)^2+(y-4)^2}\)? | [
"\\(5 + \\sqrt{2}\\)"
] | [
"What is the smallest possible value of the expression \\(\\sqrt{x^2+y^2}+\\sqrt{(x-1)^2+y^2}+\\sqrt{x^2+(y-1)^2}+\\sqrt{(x-3)^2+(y-4)^2}\\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 27
} | 41 | 0 | 0 |
omr-ws-2520105b0621c331659e0cab177693c02e59c85fce8772cfa27f2b5579aa8d0e | What is the smallest possible value of the expression \[ \sqrt {a^2 + 9} + \sqrt {(b - a)^2 + 4} + \sqrt {(8 - b)^2 + 16} \] for real numbers \( a \) and \( b \)? | [
"\\( \\sqrt{145} \\)"
] | [
"What is the smallest possible value of the expression\n\\[ \\sqrt {a^2 + 9} + \\sqrt {(b - a)^2 + 4} + \\sqrt {(8 - b)^2 + 16} \\]\nfor real numbers \\( a \\) and \\( b \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-24ae5ee7b42282eead1566b96f4170dc801f3ff32e4cffd215c04e9aab0687b5 | What is the smallest possible value of the expression \[ \sqrt{a^2+9} + \sqrt{(b-a)^2 +4} + \sqrt{(8-b)^2+16} \] for real numbers \(a\) and \(b\)? | [
"\\(\\sqrt{65}\\)"
] | [
"What is the smallest possible value of the expression\n\n\\[\n\\sqrt{a^2+9} + \\sqrt{(b-a)^2 +4} + \\sqrt{(8-b)^2+16}\n\\]\n\nfor real numbers \\(a\\) and \\(b\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-1fe7c912ed8a0668bf810b55f763ec496ddcb56bff576ff2cc6783a1546f936f | What is the smallest possible value of the first term in a sequence of five positive integers that are increasing by a constant difference, given that their sum is 540? | [
"2"
] | [
"What is the smallest possible value of the first term in a sequence of five positive integers that are increasing by a constant difference, given that their sum is 540?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5e0d3da72c0993ecc2a71ff202d383f65f64dfe1d288bdcd0ef4cd04ef3dd51e | What is the smallest possible value of the length of the midline of a trapezoid whose diagonals are perpendicular and whose height is $h$? | [
"\\( h \\)"
] | [
"What is the smallest possible value of the length of the midline of a trapezoid whose diagonals are perpendicular and whose height is $h$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-598c9e720509043eb38998b53eba3a9e91da48dc270d6cb1a7515cf5295aed72 | What is the smallest possible value of the sum $\lvert x + 2\rvert + \lvert x + 4\rvert + \lvert x + 5\rvert$? | [
"3"
] | [
"What is the smallest possible value of the sum $\\lvert x + 2\\rvert + \\lvert x + 4\\rvert + \\lvert x + 5\\rvert$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9aba95503560286658defadfd432371a64575b36669adc956414da9e0ba71939 | What is the smallest prime divisor of $5^{23} + 7^{17}$? | [
"2"
] | [
"What is the smallest prime divisor of $5^{23} + 7^{17}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-53b983a6b5d3dc8586d94f9472baea93766305dfee69870a96c920a3be5fee39 | What is the smallest prime factor of 1821? | [
"3"
] | [
"What is the smallest prime factor of 1821?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5ba4c91ac5d9789492dcaa1d2182342488fbec8293aa71f7ccb788ead9afb959 | What is the smallest prime number $p$ such that $p^3 + 4p^2 + 4p$ has exactly 30 positive divisors? | [
"43"
] | [
"What is the smallest prime number $p$ such that $p^3 + 4p^2 + 4p$ has exactly 30 positive divisors?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-5c97c7e9ac5e671d43497013f6e0002944f2aeada00721980a01fd4c6f86b632 | What is the smallest prime number greater than 9000? Find two different methods to determine this prime number. | [
"9001"
] | [
"What is the smallest prime number greater than 9000? Find two different methods to determine this prime number."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.