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-372bb8d485efc7af6a8f472441ff286ec3c5aa194e4f2fdcc2d7b811c9c5e0ad | What is the least positive integer that has a remainder of 0 when divided by 2, a remainder of 1 when divided by 3, and a remainder of 2 when divided by 4? | [
"10"
] | [
"What is the least positive integer that has a remainder of 0 when divided by 2, a remainder of 1 when divided by 3, and a remainder of 2 when divided by 4?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-321b49476e670e186bddeef6afae7d9533ebc228ce4a2e9fbbf08548148a2928 | What is the least positive integer that is divisible by three distinct primes? | [
"30"
] | [
"What is the least positive integer that is divisible by three distinct primes?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e16ffb08050f485717fb4f16048ecdd6fe8fff1f6d714ee5ccd298796eb6f5df | What is the least positive integer that satisfies the following conditions? a) When divided by 2, the remainder is 1. b) When divided by 3, the remainder is 2. c) When divided by 4, the remainder is 3. d) When divided by 5, the remainder is 4. | [
"59"
] | [
"What is the least positive integer that satisfies the following conditions?\n\na) When divided by 2, the remainder is 1.\n\nb) When divided by 3, the remainder is 2.\n\nc) When divided by 4, the remainder is 3.\n\nd) When divided by 5, the remainder is 4."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-33edfefd49cbb56c7c5f61b2b6cd21d709a39d00c280f323e0751337f769aae0 | What is the least positive integer value of $N$ for which $\frac{N-12}{5N+23}$ is a non-zero reducible fraction? | [
"95"
] | [
"What is the least positive integer value of $N$ for which $\\frac{N-12}{5N+23}$ is a non-zero reducible fraction?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-1dca2af9de49b9c80e6feeefce446027b6a48bc0029f49b40887ebc8248be490 | What is the least positive integer value of $x$ such that $(2x)^2 + 2\cdot 37\cdot 2x + 37^2$ is a multiple of 47? | [
"5"
] | [
"What is the least positive integer value of $x$ such that $(2x)^2 + 2\\cdot 37\\cdot 2x + 37^2$ is a multiple of 47?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e80fc85583e1b2a186b38ffec45f6b7f828f6f5b2d6fc3f271b5c8bee76fa152 | What is the least positive integer which when divided by 5 gives a remainder of 4, when divided by 6 gives a remainder of 5, when divided by 7 gives a remainder of 6, when divided by 8 gives a remainder of 7, when divided by 9 gives a remainder of 8, and when divided by 10 gives a remainder of 9? | [
"2519"
] | [
"What is the least positive integer which when divided by 5 gives a remainder of 4, when divided by 6 gives a remainder of 5, when divided by 7 gives a remainder of 6, when divided by 8 gives a remainder of 7, when divided by 9 gives a remainder of 8, and when divided by 10 gives a remainder of 9?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-675afa8ca59d85736dd0a6a7118f3e3199ef51e555808faba1841fe822df9486 | What is the least positive integer with exactly $10$ positive factors? | [
"48"
] | [
"What is the least positive integer with exactly $10$ positive factors?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-10436b3e6dae26949eabae1db7cfbcf75dedb473a5f34ed595621bad287e91eb | What is the least positive integer with exactly five distinct positive factors? | [
"16"
] | [
"What is the least positive integer with exactly five distinct positive factors?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-12cda38c2157f40d32b1c39e0d772d91f12f46668f0e4b33509c08f97a3d88b2 | What is the least positive multiple of 21 that is greater than 380? | [
"399"
] | [
"What is the least positive multiple of 21 that is greater than 380?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b7c697fe59b62a0821fc68ce0def1b423cc506487f3719dac101c193f283e1ef | What is the least positive multiple of 25 for which the product of its digits is also a positive multiple of 25? | [
"525"
] | [
"What is the least positive multiple of 25 for which the product of its digits is also a positive multiple of 25?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8bc75f5807129d7228484ae89ad2b12437bf34c82e8faac91096c6ffd93c97ed | What is the least positive number that, when divided by 7, gives a remainder of 4, when divided by 8, gives a remainder of 5, and when divided by 9, gives a remainder of 6? | [
"501"
] | [
"What is the least positive number that, when divided by 7, gives a remainder of 4, when divided by 8, gives a remainder of 5, and when divided by 9, gives a remainder of 6?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 8
} | 40 | 0 | 0 |
omr-ws-bd6bf2f511c48b8cd45458731fb24a641331c44c3ce7732514ab26fbbf056fd2 | What is the least positive three-digit multiple of 7? | [
"105"
] | [
"What is the least positive three-digit multiple of 7?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5f05f80ff830e5b5f7ad0db7f2d491063e4178124771100c16be9a18415eb912 | What is the least positive value of \( a \) such that \[ (1 + \tan(a \cdot \sin^2(\frac{17\pi}{32})))(1 + \tan(a \cdot \cos^2(\frac{17\pi}{32}))) = 2? \] | [
"\\(\\frac{\\pi}{4}\\)"
] | [
"What is the least positive value of \\( a \\) such that\n\\[\n(1 + \\tan(a \\cdot \\sin^2(\\frac{17\\pi}{32})))(1 + \\tan(a \\cdot \\cos^2(\\frac{17\\pi}{32}))) = 2?\n\\]"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 13
} | 28 | 0 | 0 |
omr-ws-98a3c184367e09371d364e8cbb7f287a13acb560eecde03f0a980d79592e5065 | What is the least positive whole number divisible by four different prime numbers? | [
"210"
] | [
"What is the least positive whole number divisible by four different prime numbers?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3eab2ea11b6d77ded0e8a6de61c4924cb53892c2de370dec52bd0fc65adb6365 | What is the least possible area of a triangle with altitudes satisfying $h_a \ge 3$, $h_b \ge 4$, and $h_c \ge 5$? | [
"10"
] | [
"What is the least possible area of a triangle with altitudes satisfying $h_a \\ge 3$, $h_b \\ge 4$, and $h_c \\ge 5$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-399d1a17f40078a346bd980e4c77a3e0e65ded978b89a696111fb79b44672d86 | What is the least possible integer value of \( a \) such that \( 10^{2n+1} + a \cdot 7^{2n+1} \) is divisible by 51? | [
"35"
] | [
"What is the least possible integer value of \\( a \\) such that \\( 10^{2n+1} + a \\cdot 7^{2n+1} \\) is divisible by 51?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 3
} | 5 | 0 | 0 |
omr-ws-84f9a994e6cebf7e078f325c9c9d9bb9ec263baf24c1fc625f498c4e04f2e0f5 | What is the least possible measure of the greatest angle of a triangle? | [
"\\(\\frac{\\pi}{3}\\)"
] | [
"What is the least possible measure of the greatest angle of a triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-f508522479f7b1de2aa5cd6b049f7626699afacaf97b1b269db458c12a1fc28f | What is the least possible number of cells that can be marked on an $n \times n$ board such that for each $m > \frac{n}{2}$, both diagonals of any $m \times m$ sub-board contain a marked cell? | [
"\\( n \\)"
] | [
"What is the least possible number of cells that can be marked on an $n \\times n$ board such that for each $m > \\frac{n}{2}$, both diagonals of any $m \\times m$ sub-board contain a marked cell?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-220119ce1fd6ec139601ba81305d2ead1bf5dcaf62e163f11ce41641b202235d | What is the least possible number of cells that can be marked on an $n \times n$ board such that for each $m > \frac{n}{2}$, both diagonals of any $m \times m$ sub-board contain a marked cell? Provide an example. | [
"\\( 2n - 1 \\)"
] | [
"What is the least possible number of cells that can be marked on an $n \\times n$ board such that for each $m > \\frac{n}{2}$, both diagonals of any $m \\times m$ sub-board contain a marked cell? Provide an example."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6fe0315bad21367a2aba0a47d77d3885adddf331aa1210ac77de44ab3d490aad | What is the least possible number of control weights, each weighing a non-integer number of grams, such that any integer weight from 1 g to 40 g can be balanced by some combination of these weights placed on one pan of a balance, with the measured weight on the other pan? | [
"7"
] | [
"What is the least possible number of control weights, each weighing a non-integer number of grams, such that any integer weight from 1 g to 40 g can be balanced by some combination of these weights placed on one pan of a balance, with the measured weight on the other pan?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-d22788db5706fc38363416bd0f9c2045198ba6e7733ddaa28e652c601c702bcd | What is the least possible number of positive integers such that the sum of their squares equals 1995? | [
"3"
] | [
"What is the least possible number of positive integers such that the sum of their squares equals 1995?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 62,
"tir": 33
} | 95 | 0 | 0 |
omr-ws-52af440168e629cb1643aa21aa435db95fb9d2d17e548afedf2e4e11e6867230 | What is the least possible number of unit squares $(1 \times 1)$ that must be drawn to form a $25 \times 25$ square divided into $625$ unit squares? | [
"360"
] | [
"What is the least possible number of unit squares $(1 \\times 1)$ that must be drawn to form a $25 \\times 25$ square divided into $625$ unit squares?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-40a925bca4918fffde00d406a3587e948f526db2523e0271ecca5de4812b1ed9 | What is the least possible positive integer value of $b$ such that there exists a non-negative integer $a$ for which the set $$\{P(a+1),P(a+2),\ldots,P(a+b)\}$$ is fragrant, where $P(n)=n^2+n+1$ and a set of positive integers is called fragrant if it contains at least two elements and each of its elements has a prime f... | [
"6"
] | [
"What is the least possible positive integer value of $b$ such that there exists a non-negative integer $a$ for which the set $$\\{P(a+1),P(a+2),\\ldots,P(a+b)\\}$$ is fragrant, where $P(n)=n^2+n+1$ and a set of positive integers is called fragrant if it contains at least two elements and each of its elements has a... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 1
} | 3 | 0 | 0 |
omr-ws-f8b30fdb208688df44e131616b2fa9053ce47b214514edf072e622a79763f193 | What is the least possible sum of two positive integers $a$ and $b$ where $ab = 10!$? | [
"3810"
] | [
"What is the least possible sum of two positive integers $a$ and $b$ where $ab = 10!$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-141c3a129c8b002b71278c571278f9a904e24d278647fab3cd49140682ccc1e0 | What is the least possible surface area of a rectangular prism with a volume of 52? | [
"\\( 12\\sqrt[3]{2 \\cdot 13^2} \\)"
] | [
"What is the least possible surface area of a rectangular prism with a volume of 52?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 11
} | 11 | 0 | 0 |
omr-ws-19e22b2ece40641ab4ba1f64c439b252a44e513fd10b038f931a4a52e2783207 | What is the least possible surface area when four cubes with volumes of $1cm^3$, $8cm^3$, $27cm^3$, and $125cm^3$ are combined into one solid? | [
"195 cm²"
] | [
"What is the least possible surface area when four cubes with volumes of $1cm^3$, $8cm^3$, $27cm^3$, and $125cm^3$ are combined into one solid?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-e60905f356ee55ad9a7fa65d441cc907e4e0d1528e368a8bc123aab37f8f687f | What is the least possible value for \( x^2 + 3x + 2 \) if \( x^2 - 3x - 2 \leq 0 \)? | [
"13 - 3\\sqrt{17}"
] | [
"What is the least possible value for \\( x^2 + 3x + 2 \\) if \\( x^2 - 3x - 2 \\leq 0 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d297b433e2754e3831c8194471a813a20ef761a19e12952d2ccc11d7ebe47022 | What is the least possible value of $(x+1)(x+2)(x+3)(x+4)+2019$ where $x$ is a real number? | [
"2018"
] | [
"What is the least possible value of $(x+1)(x+2)(x+3)(x+4)+2019$ where $x$ is a real number?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-db8b12d63c72b507454cc23f7876406399e9881b83b131f0e24379640a788afb | What is the least possible value of $(xy-1)^2+(x+y)^2$ for real numbers $x$ and $y$? | [
"1"
] | [
"What is the least possible value of $(xy-1)^2+(x+y)^2$ for real numbers $x$ and $y$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 96
} | 96 | 0 | 0 |
omr-ws-fca45b4b60fb1b6a67d6654e4b3fce81a088ac4b2b653617786afeb213d8fff9 | What is the least possible value of $ab + bc + ac$ such that $a^2 + b^2 + c^2 = 1$ where $a, b, c$ are real numbers? | [
"\\(-\\frac{1}{2}\\)"
] | [
"What is the least possible value of $ab + bc + ac$ such that $a^2 + b^2 + c^2 = 1$ where $a, b, c$ are real numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-e579fc0df3b56b2da7498d81f821ace7d9da4ece2bb30de3c506538f5c9bd78c | What is the least possible value of $n$ so that no matter how $n$ real numbers between 2.048 and 2048 are selected, there will always be at least one triple of them that could form a triangle with positive area? | [
"17"
] | [
"What is the least possible value of $n$ so that no matter how $n$ real numbers between 2.048 and 2048 are selected, there will always be at least one triple of them that could form a triangle with positive area?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 7,
"tir": 8
} | 28 | 0 | 0 |
omr-ws-b10afc7c93ce55868e4b0f86956725d90af7eae3901c09ff71b2c5f40ecfaa90 | What is the least possible value of $n$ such that among $n$ randomly spaced points in a 12x12 square, there exists a set of 7 points that can be contained within a 1x1 square? | [
"865"
] | [
"What is the least possible value of $n$ such that among $n$ randomly spaced points in a 12x12 square, there exists a set of 7 points that can be contained within a 1x1 square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-6d4ac6b2a9d20503b8c026c6af82ec0f39aa63e41a7442c1a5c86296806fb076 | What is the least possible value of $x^2 + y^2 - x - y - xy$ where $x, y$ are real numbers? | [
"-1"
] | [
"What is the least possible value of $x^2 + y^2 - x - y - xy$ where $x, y$ are real numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-9c2ed624059524ed2af9eed87876e5db18ff0836657535bc4a37376587bb01c5 | What is the least possible value of \( m \) such that a hypercube \( Q(n,k) = \{1,2,...,k\}^n \) can be covered by the surfaces of \( m \) \( n \)-dimensional hyperspheres? Given bounds are: \[ 1 + n\left\lfloor{\frac{k - 1}{2}}\right\rfloor \leq m(n,k) \leq 1 + n\left\lfloor{\left(\frac{k - 1}{2}\right)^2}\right\rfloo... | [
"1 + n\\left\\lfloor \\frac{k - 1}{2} \\right\\rfloor"
] | [
"What is the least possible value of \\( m \\) such that a hypercube \\( Q(n,k) = \\{1,2,...,k\\}^n \\) can be covered by the surfaces of \\( m \\) \\( n \\)-dimensional hyperspheres?\n\nGiven bounds are:\n\\[ 1 + n\\left\\lfloor{\\frac{k - 1}{2}}\\right\\rfloor \\leq m(n,k) \\leq 1 + n\\left\\lfloor{\\left(\\frac{... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-040dd55a59aae74c837b8fbdb6313fc8f6883f610e0cd5b32ae53737127df618 | What is the least possible value of \( n \) for a convex non-equilateral \( n \)-gon such that segments connecting an inner point to its vertices divide the \( n \)-gon into \( n \) equal triangles? | [
"4"
] | [
"What is the least possible value of \\( n \\) for a convex non-equilateral \\( n \\)-gon such that segments connecting an inner point to its vertices divide the \\( n \\)-gon into \\( n \\) equal triangles?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 18,
"genselect": 4
} | 22 | 0 | 0 |
omr-ws-435c3060d73ebd132628aad0e27556effbfdf3fc74b3b9efe8776e6a6a2927d9 | What is the least possible value of \( n \) such that no matter how \( n \) real numbers on the interval \([1, 2013]\) are selected, it is always possible to find a scalene polygon whose sides are equal to some of the selected numbers? | [
"13"
] | [
"What is the least possible value of \\( n \\) such that no matter how \\( n \\) real numbers on the interval \\([1, 2013]\\) are selected, it is always possible to find a scalene polygon whose sides are equal to some of the selected numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 23,
"genselect": 3
} | 26 | 0 | 0 |
omr-ws-0228a442aeeafc14b9e746ccc6f99114f14bc16a82156b543385ba569f03401c | What is the least possible value of \((xy - 1)^2 + (x + y)^2\) for all real numbers \(x\) and \(y\)? | [
"1"
] | [
"What is the least possible value of \\((xy - 1)^2 + (x + y)^2\\) for all real numbers \\(x\\) and \\(y\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1d850d8368ce774be6ee622a454795bb7399e1a3c703fc20ccff918a975a3a20 | What is the least possible value of \((xy-1)^2+(x+y)^2\) for real numbers \(x\) and \(y\)? | [
"1"
] | [
"What is the least possible value of \\((xy-1)^2+(x+y)^2\\) for real numbers \\(x\\) and \\(y\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ab9b2c25a098842f61b45ad74f777faa5383fff45ab9f7023e06ff4e3add21a4 | What is the least possible value of integer \( a \) such that \( 10^{2n+1} + a \cdot 7^{2n+1} \) is divisible by 51 for any natural number \( n \)? | [
"35"
] | [
"What is the least possible value of integer \\( a \\) such that \\( 10^{2n+1} + a \\cdot 7^{2n+1} \\) is divisible by 51 for any natural number \\( n \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 1,
"tir": 6
} | 20 | 0 | 0 |
omr-ws-11d0877080e4c66490972be548bb91ef3e4dab259a65b160096d27207aa8a6f8 | What is the least possible value of the greatest term in an arithmetic sequence of 7 prime numbers? | [
"907"
] | [
"What is the least possible value of the greatest term in an arithmetic sequence of 7 prime numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-d7e8287c705cb2578c0927661628fc27acb15150ea36ea9f0db5b76055853310 | What is the least prime factor of $7^4 - 7^3$? | [
"2"
] | [
"What is the least prime factor of $7^4 - 7^3$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-facf41bef94ae3dc522e3e47814d1588ec52faea36b9f6d11c2214d32e8ebf12 | What is the least real number $C$ that satisfies $\sin x \cos x \leq C(\sin^6 x + \cos^6 x)$ for every real number $x$? | [
"2"
] | [
"What is the least real number $C$ that satisfies $\\sin x \\cos x \\leq C(\\sin^6 x + \\cos^6 x)$ for every real number $x$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 18
} | 18 | 0 | 0 |
omr-ws-d0a04a924d295b4b46f5ad4e39c8bfd3736e2e9e6c2552f59b69a88422a82a88 | What is the least real number $C$ that satisfies $\sin x \cos x \leq C(\sin^6x+\cos^6x)$ for every real number $x$? | [
"2"
] | [
"What is the least real number $C$ that satisfies $\\sin x \\cos x \\leq C(\\sin^6x+\\cos^6x)$ for every real number $x$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-782af1208e70fd9a10ca887fe2e137f6745bc3b994fd6073777cb82a80c62831 | What is the least real value of the expression $\sqrt{x^2-6x+13} + \sqrt{x^2-14x+58}$ where $x$ is a real number? | [
"\\(\\sqrt{41}\\)"
] | [
"What is the least real value of the expression $\\sqrt{x^2-6x+13} + \\sqrt{x^2-14x+58}$ where $x$ is a real number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-bbf626398b8d0ce9a6655b3b7e6065e86e4d7f93de72621fec9abef87ea2c549 | What is the least surface area of a box that can contain 24 cylindrical candies, each with a diameter of 1 and height of 1? The box should be constructible out of paper. | [
"52"
] | [
"What is the least surface area of a box that can contain 24 cylindrical candies, each with a diameter of 1 and height of 1? The box should be constructible out of paper."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 2
} | 16 | 0 | 0 |
omr-ws-058af31a19e26f4fe2bb610fea45fed036f569481669c708542fd7cf5aa67bb0 | What is the least three-digit positive integer that has 2, 5 and 7 as factors? | [
"140"
] | [
"What is the least three-digit positive integer that has 2, 5 and 7 as factors?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-00762e640f68cc7ce20a5623e246a04db64d0a01c54be6e5722eb0377424b1a1 | What is the least three-digit whole number, the product of whose digits is 6? | [
"116"
] | [
"What is the least three-digit whole number, the product of whose digits is 6?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8dc9a35c205ccaa558114d19097cd40e58609e65bc6603ddb41f7d6db6979cff | What is the least total number of pens that can be distributed into 49 packages such that each package contains a different number of pens, and no package is left empty? | [
"285"
] | [
"What is the least total number of pens that can be distributed into 49 packages such that each package contains a different number of pens, and no package is left empty?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c3f45ad68dd6bacd1db44c66c18904bce066999bbcd0a2c837f0b357ed1f80c8 | What is the least total surface area of a disphenoid whose faces are scalene triangles with integer side lengths? | [
"\\( 15\\sqrt{7} \\)"
] | [
"What is the least total surface area of a disphenoid whose faces are scalene triangles with integer side lengths?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 6,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-8eed8ccd8b6a7f3fb238fb22257fd8c2fd6ce6f7b88199bbbd7a9982e9209f93 | What is the least upper bound of the real numbers $b$ such that there exists a continuous real-valued function $f(x)$ for which the expression \[ xf(x) - \int_1^x (f(t) + t(f(t))^2) \, dt - b(\ln x) \] tends to infinity as $x \to \infty$? | [
"\\dfrac{1}{4}"
] | [
"What is the least upper bound of the real numbers $b$ such that there exists a continuous real-valued function $f(x)$ for which the expression\n\\[ xf(x) - \\int_1^x (f(t) + t(f(t))^2) \\, dt - b(\\ln x) \\]\ntends to infinity as $x \\to \\infty$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-c0ffd8fa37b4d5ea9ab563fda6bea7230bd70dfe39a66c6dc738ac05ffbe4ba0 | What is the least value of $K > 0$ such that $|\sin^2 x - \sin^2 y| \le K |x - y|$ for all real numbers $x$ and $y$? | [
"1"
] | [
"What is the least value of $K > 0$ such that $|\\sin^2 x - \\sin^2 y| \\le K |x - y|$ for all real numbers $x$ and $y$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-5b600c9f595425b4a72d76ccf42cb51d458425d444e17a9406d7f8ff33675c2b | What is the least value of $N$ such that in a $3 \times 7 \times N$ set of lattice points, where each lattice point is colored either black or white, there exists a rectangular prism with all vertices colored the same color? | [
"127"
] | [
"What is the least value of $N$ such that in a $3 \\times 7 \\times N$ set of lattice points, where each lattice point is colored either black or white, there exists a rectangular prism with all vertices colored the same color?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 16,
"genselect": 6
} | 22 | 0 | 0 |
omr-ws-bdac9308ef352b9653c0cd39fb088574da7f3b559080af29dfc6eb31c31ad390 | What is the least value of $\sin^{14}\theta + \cos^{20}\theta$? | [
"0.0062"
] | [
"What is the least value of $\\sin^{14}\\theta + \\cos^{20}\\theta$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 8,
"tir": 3
} | 15 | 0 | 0 |
omr-ws-c3680788acc6d2b48b23819de0f1689311313649e1ef5e9a7ae9b218b1427025 | What is the least value of $\sqrt{x^2 + (1-x)^2} + \sqrt{(1-x)^2 + (1+x)^2}$? | [
"\\(\\sqrt{5}\\)"
] | [
"What is the least value of $\\sqrt{x^2 + (1-x)^2} + \\sqrt{(1-x)^2 + (1+x)^2}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-cc20c056b303e80ddfd5889ca1de97b3f027517a0f9352365a18aebc8b2f40b4 | What is the least value of $k$ such that when Anna, John, and Paul each take $k$ blocks from a set of $k$ red, $k$ blue, and $k$ green blocks, it is certain that one player has at least $10$ red blocks, another has at least $10$ blue blocks, and the third has at least $10$ green blocks? | [
"27"
] | [
"What is the least value of $k$ such that when Anna, John, and Paul each take $k$ blocks from a set of $k$ red, $k$ blue, and $k$ green blocks, it is certain that one player has at least $10$ red blocks, another has at least $10$ blue blocks, and the third has at least $10$ green blocks?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-77d26771a9be3276f56f757111778a734de376d76d5b321c608ef864a0c61650 | What is the least value of $n$ such that the last three non-zero digits of ${n!}^{9}$ are $888$? | [
"30"
] | [
"What is the least value of $n$ such that the last three non-zero digits of ${n!}^{9}$ are $888$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-864ad1cbf6d0412de591fdd69dabf6214fec068eb8ad344e5e8f85dfeeecae95 | What is the least value of $x$ so that $31^n + x \cdot 96^n$ is a multiple of 2015 for every positive integer $n$? | [
"1364"
] | [
"What is the least value of $x$ so that $31^n + x \\cdot 96^n$ is a multiple of 2015 for every positive integer $n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-8283351a04c6deef332f668f8307eebdd5a88ada9ea0ef6a4b1e5a939bb85482 | What is the least value of $y$ such that $3y^2 + 5y + 2 = 4$? | [
"-2"
] | [
"What is the least value of $y$ such that $3y^2 + 5y + 2 = 4$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3aa7f677931310610a81386dcaf3e1acb599980c348120f31b6fb91122611af7 | What is the least value of $|ax - by|$, where $a$ and $b$ are positive constant real numbers and $x, y$ are positive integers? | [
"\\( 0 \\) if \\( \\frac{a}{b} \\) is rational, otherwise no positive minimum value."
] | [
"What is the least value of $|ax - by|$, where $a$ and $b$ are positive constant real numbers and $x, y$ are positive integers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 7
} | 7 | 0 | 0 |
omr-ws-70dab588c5c3a38c2ee461b5e2777006f9092483d32083b18947650604e45868 | What is the least value of \( N \in \mathbb{N} \) such that \( 999 \times N = 11111\ldots1111 \) (where all digits are 1)? | [
"111222333444555666777889"
] | [
"What is the least value of \\( N \\in \\mathbb{N} \\) such that \\( 999 \\times N = 11111\\ldots1111 \\) (where all digits are 1)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 7,
"tir": 19
} | 38 | 0 | 0 |
omr-ws-48dc5d84f4649d99978c8b71ea7dd83b31b3f5e10f3d5da40f2d8073e317f834 | What is the least value of \( n \) such that every subset of \(\{1, 2, \ldots, 100\}\) with \( n \) elements has the property that we can choose 4 distinct elements \( a, b, c, d \) so that \( a + b = c + d \)? | [
"16"
] | [
"What is the least value of \\( n \\) such that every subset of \\(\\{1, 2, \\ldots, 100\\}\\) with \\( n \\) elements has the property that we can choose 4 distinct elements \\( a, b, c, d \\) so that \\( a + b = c + d \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-083fbe2b4db45ed10d87fe59da6833f6c8411145b85312a904253b4a43e4d6f4 | What is the least value of \( n \) such that from every set of \( n \) elements, it is possible to find six elements whose sum is divisible by 6? | [
"11"
] | [
"What is the least value of \\( n \\) such that from every set of \\( n \\) elements, it is possible to find six elements whose sum is divisible by 6?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9a8cf25bef61f7495296c9f13260d6e46b3702168eaf117c7b046edd2c5979ab | What is the least value of \( n \) such that it is always possible to contain 2 points in a 1x1 square when \( n \) points are placed on a 12x12 square? | [
"145"
] | [
"What is the least value of \\( n \\) such that it is always possible to contain 2 points in a 1x1 square when \\( n \\) points are placed on a 12x12 square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 4
} | 15 | 0 | 0 |
omr-ws-6abe5b9165f2042643543160b6915b16a901140a6800562d6ad48e5d9f45ad35 | What is the least value of \( n \) such that the sum \(\sum_{k=0}^{n} k^3 \binom{n}{k}\) is divisible by 2021? | [
"470"
] | [
"What is the least value of \\( n \\) such that the sum \\(\\sum_{k=0}^{n} k^3 \\binom{n}{k}\\) is divisible by 2021?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 5
} | 20 | 0 | 0 |
omr-ws-6fcabfe976d538c8778c7afac73fc8c67d7f82a25118527f779d37acd7a3e10a | What is the least value of \( y \) that satisfies the inequality \( |4 + x| + |5 + y| \leq 100 \)? | [
"-105"
] | [
"What is the least value of \\( y \\) that satisfies the inequality \\( |4 + x| + |5 + y| \\leq 100 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-c8832b6e6cdb97d279818c66e9326e0de443abec341b134af4b052069530098a | What is the least value of \(\frac{(x^3 + 3)(y^3 + 6)(z^3 + 12)}{(x + y + z)^3}\) where \(x, y,\) and \(z\) are positive real numbers? | [
"8"
] | [
"What is the least value of \\(\\frac{(x^3 + 3)(y^3 + 6)(z^3 + 12)}{(x + y + z)^3}\\) where \\(x, y,\\) and \\(z\\) are positive real numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1e326eada9915569586dc3f385aaa653cc027cd3a2197f1f204d1c481e402df0 | What is the least whole number that is divisible by 7, but leaves a remainder of 1 when divided by any integer 2 through 6? | [
"301"
] | [
"What is the least whole number that is divisible by 7, but leaves a remainder of 1 when divided by any integer 2 through 6?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0b410fec32ee247b7f93ac0590b08ba742927fcabc4af7c30ba0f80038397968 | What is the leftmost (most significant) digit of $2^{400}$? | [
"2"
] | [
"What is the leftmost (most significant) digit of $2^{400}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-36a8ac87d0180c6bac36bbc861add9b6c430a4878d00955f4cce256a71bf43a1 | What is the leftmost digit of $2006^{2006}$? | [
"2"
] | [
"What is the leftmost digit of $2006^{2006}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 4
} | 19 | 0 | 0 |
omr-ws-32f99b3c8a7c2943df08f2d490cf8d135b23d0e0f4922982856c1d9c86b95861 | What is the leftmost digit of \(12^{60}\)? You may use the values of \(\log_{10} 2 = 0.3010\) and \(\log_{10} 3 = 0.4771\). | [
"5"
] | [
"What is the leftmost digit of \\(12^{60}\\)?\n\nYou may use the values of \\(\\log_{10} 2 = 0.3010\\) and \\(\\log_{10} 3 = 0.4771\\)."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 14
} | 29 | 0 | 0 |
omr-ws-60bbb392776c887cd1a365a53d790ae87bd6a586122a5bf8bc481b222d8e5689 | What is the length and location of the shortest straight line cut which divides an isosceles right-angled triangle with base length 1 into two pieces of equal area? | [
"\\(\\sqrt{\\sqrt{2}-1}\\)"
] | [
"What is the length and location of the shortest straight line cut which divides an isosceles right-angled triangle with base length 1 into two pieces of equal area?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 5,
"tir": 3
} | 12 | 0 | 0 |
omr-ws-5b1b171a04f98b535fc9ba661da5fbfa15267e2817467caf8dd2dddfeb932793 | What is the length of a median to side \( c \) of a triangle with sides 5, 12, and \( c = 8 \)? | [
"\\dfrac{\\sqrt{274}}{2}"
] | [
"What is the length of a median to side \\( c \\) of a triangle with sides 5, 12, and \\( c = 8 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-cb0bcaa4fe78ba2e4db2694ad8f2767626cfb3bc70efd4695f6bf6e5f1f1aa09 | What is the length of each side of a regular hexagon if opposite sides are 18 inches apart? | [
"\\( 6\\sqrt{3} \\)"
] | [
"What is the length of each side of a regular hexagon if opposite sides are 18 inches apart?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-f86c8693d84f81a421711af4daec7356a4796a43969547d279493dd1b60a30bd | What is the length of each side of a regular hexagon if the distance between opposite sides is 12 inches? | [
"\\( 4\\sqrt{3} \\)"
] | [
"What is the length of each side of a regular hexagon if the distance between opposite sides is 12 inches?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-9226c729431118109c929e73e03ce22ed0ade3b723201d5572bb35676bca4a0f | What is the length of the angle bisector to side \( C \) in a triangle where \( A = 5 \), \( B = 14 \), and \( C = 17 \)? | [
"3.8\\overline{63}"
] | [
"What is the length of the angle bisector to side \\( C \\) in a triangle where \\( A = 5 \\), \\( B = 14 \\), and \\( C = 17 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 13
} | 28 | 0 | 0 |
omr-ws-1d0ac53be910f0866468d079efeafa4ce47d8dd314d179e9df2614ccff19014f | What is the length of the angle bisector to the side with length 16 in a triangle with sides 15, 16, and 17? | [
"\\(\\frac{3\\sqrt{85}}{2}\\)"
] | [
"What is the length of the angle bisector to the side with length 16 in a triangle with sides 15, 16, and 17?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-59b7b26af0b5d8184b22f87cabd87231ad62af2c5c5fdae4a02031e899fa51ff | What is the length of the boundary of the region in the $xy$ plane consisting of points of the form $(2u – 3w, v+4w)$ where $0 \leq u \leq 1$, $0 \leq v \leq 1$, and $0 \leq w \leq 1$? | [
"16"
] | [
"What is the length of the boundary of the region in the $xy$ plane consisting of points of the form $(2u – 3w, v+4w)$ where $0 \\leq u \\leq 1$, $0 \\leq v \\leq 1$, and $0 \\leq w \\leq 1$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 4
} | 19 | 0 | 0 |
omr-ws-3592f306ed4dff139d4efae2063a993112415a9f23bb95e861a57ece03626553 | What is the length of the common tangent between two non-intersecting circles with radii $r$ and $3r$, where the shortest distance between their centers is equal to the radius of the first circle? | [
"\\( r\\sqrt{21} \\) or \\( 3r \\)"
] | [
"What is the length of the common tangent between two non-intersecting circles with radii $r$ and $3r$, where the shortest distance between their centers is equal to the radius of the first circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 1
} | 1 | 0 | 0 |
omr-ws-b19ec0750bcd3f258411ecf7b5a147ad4f9d6f755bbd2b158412bdefc34acc06 | What is the length of the curved portion of the parabola $y=x^2$ that lies between the points $(0,0)$ and $(1,1)$? | [
"\\(\\frac{\\ln(\\sqrt{5}+2)+2\\sqrt{5}}{4}\\)"
] | [
"What is the length of the curved portion of the parabola $y=x^2$ that lies between the points $(0,0)$ and $(1,1)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-e9c330f38801f8b36948733ebf17df6d04486bf74aac1710ab888fe1cb8ce602 | What is the length of the diagonal of a square with side length $50\sqrt{2}$ cm? Express your answer in simplest form. | [
"100"
] | [
"What is the length of the diagonal of a square with side length $50\\sqrt{2}$ cm? Express your answer in simplest form."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6ce877822b1e42139eac442a78b19485d563283b240c31cd1174b95ac956a781 | What is the length of the largest altitude of a triangle with side lengths $10$, $12$, and $18$? | [
"\\(8\\sqrt{12}\\)"
] | [
"What is the length of the largest altitude of a triangle with side lengths $10$, $12$, and $18$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-648c967cae85e3e8f7f9ff25808a4656bc4e128d3eca4f3ac641c1c3df95f3f4 | What is the length of the largest chord of the outer circle that lies outside the smaller circle, given that two circles touch each other internally with radii of 4 cm and 6 cm? | [
"\\(4\\sqrt{5}\\)"
] | [
"What is the length of the largest chord of the outer circle that lies outside the smaller circle, given that two circles touch each other internally with radii of 4 cm and 6 cm?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-610af345a2441485b5711351d9024e7536974e8e896efb85ead50f1c09894953 | What is the length of the line segment that connects the midpoints of the two diagonals in a cyclic quadrilateral? | [
"\\dfrac{|AB - CD|}{2}"
] | [
"What is the length of the line segment that connects the midpoints of the two diagonals in a cyclic quadrilateral?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
omr-ws-fe0bd9f0c13c6633931572478db633c1b966c57b3cf22bf252bbfb0ab771defc | What is the length of the longer side of a parallelogram with integer side lengths and diagonals of lengths $123$ and $157$? | [
"140"
] | [
"What is the length of the longer side of a parallelogram with integer side lengths and diagonals of lengths $123$ and $157$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 31
} | 62 | 0 | 0 |
omr-ws-ee943bc79fd5b90cd3876cd2ce02560ce4f00a6903559cebc0d2fbd6ccdc3caa | What is the length of the longest "articulated snake" that can fit in a circular cage with a radius of 3, where the snake is composed of length 1 segments that can only be connected at 90, 180, or 270 degrees, and the head of the snake must be at the center of the cage? | [
"20"
] | [
"What is the length of the longest \"articulated snake\" that can fit in a circular cage with a radius of 3, where the snake is composed of length 1 segments that can only be connected at 90, 180, or 270 degrees, and the head of the snake must be at the center of the cage?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 2
} | 7 | 0 | 0 |
omr-ws-10901dfb6e0016e41c9d149af281a377653b3d4b55ac5404bec70d94cbdd0901 | What is the length of the longest "articulated snake" that can fit in a circular cage with a radius of 4, where the snake is composed of length 1 segments that can only be connected at 90, 180, or 270 degrees, and the head of the snake must be at the center of the cage? | [
"32"
] | [
"What is the length of the longest \"articulated snake\" that can fit in a circular cage with a radius of 4, where the snake is composed of length 1 segments that can only be connected at 90, 180, or 270 degrees, and the head of the snake must be at the center of the cage?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-8caa6df2d5ecb0860ef92c3c1cec059e0b4bc21269a18fb93370fb586dcbae1d | What is the length of the longest altitude in triangle ABC if the side lengths are $a=7$, $b=8$, and $c=9$? | [
"\\(\\frac{24\\sqrt{5}}{7}\\)"
] | [
"What is the length of the longest altitude in triangle ABC if the side lengths are $a=7$, $b=8$, and $c=9$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-47f86ac8c75c47b963fd1ace94a2e9b32bc8fd65885136d8c9a037c2585db043 | What is the length of the longest altitude of a triangle with side lengths 9, 10, and 11? | [
"\\dfrac{20\\sqrt{2}}{3}"
] | [
"What is the length of the longest altitude of a triangle with side lengths 9, 10, and 11?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-917e56de2af383ec4b13d31d1cd806f382d3b498dc992f957b207d8257659155 | What is the length of the longest arithmetic progression of positive integers whose sum is $2102024$? | [
"16"
] | [
"What is the length of the longest arithmetic progression of positive integers whose sum is $2102024$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-361990912618b3885428954c528ac120b508c70549f0c162936a710e9d81b8c2 | What is the length of the longest line that could be inscribed in a 30-dimensional figure with dimensions 1x1x1x...x1 (30 times)? | [
"$\\sqrt{30}$"
] | [
"What is the length of the longest line that could be inscribed in a 30-dimensional figure with dimensions 1x1x1x...x1 (30 times)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 22
} | 37 | 0 | 0 |
omr-ws-3e9febc3f122e5b6951db7b85ffc2b6ec51b98dd90a963e5a7ee06d5284c39cb | What is the length of the longest pipe that can be carried level around a right-angled corner at the intersection of two corridors with widths of 4 feet and 6 feet? | [
"(4^{2/3} + 6^{2/3})^{3/2}"
] | [
"What is the length of the longest pipe that can be carried level around a right-angled corner at the intersection of two corridors with widths of 4 feet and 6 feet?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-41012491021f9f664620d77d5e63a19a51aba7952a2e739a2d4259f51eab150b | What is the length of the longest sequence Alex can write if any 20 consecutive letters must have an equal number of $A$'s and $B$'s, but any 22 consecutive letters must have a different number of $A$'s and $B$'s? | [
"40"
] | [
"What is the length of the longest sequence Alex can write if any 20 consecutive letters must have an equal number of $A$'s and $B$'s, but any 22 consecutive letters must have a different number of $A$'s and $B$'s?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 2
} | 6 | 0 | 0 |
omr-ws-f2a7e14aed2c60c2b1568244d37965f109136be7c4a8b97a236dcd06e730facd | What is the length of the longest string of consecutive prime numbers that divide $224444220$? | [
"6"
] | [
"What is the length of the longest string of consecutive prime numbers that divide $224444220$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-5bcba3ca0b10e922036401b6441ed94a77708c1b8d6427930b6331a9455cd384 | What is the length of the major axis of the ellipse given by the equation 100x² + 25y² = 175? | [
"2\\sqrt{7}"
] | [
"What is the length of the major axis of the ellipse given by the equation 100x² + 25y² = 175?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fcf7deaec0bc8070124f3a7add3d6b06e1de96edf0058812a690f6b616b587e6 | What is the length of the maximal chain in the power set $ P(\mathbb{R})$ (with respect to the inclusion relation)? | [
"2^{\\mathfrak{c}}"
] | [
"What is the length of the maximal chain in the power set $ P(\\mathbb{R})$ (with respect to the inclusion relation)?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fe6b00edfddfa13dcba5b1d61674fafd3853e47ec737c85672747944e3ffbf11 | What is the length of the minimal special sequence in terms of \( n \) in general, where a special sequence is defined as a sequence of numbers from the set \(\{1, 2, \ldots, n\}\) such that all permutations of the set appear in the sequence? | [
"\\sum_{k=1}^n k!"
] | [
"What is the length of the minimal special sequence in terms of \\( n \\) in general, where a special sequence is defined as a sequence of numbers from the set \\(\\{1, 2, \\ldots, n\\}\\) such that all permutations of the set appear in the sequence?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 4
} | 14 | 0 | 0 |
omr-ws-d59adefa935398078517c93bd7801ef5b7ce54e4b8a1087a002026b6d39d0c66 | What is the length of the outer handrail of a spiral staircase with a radius of 3 feet that turns 270° while rising 10 feet? | [
"\\(17.32\\)"
] | [
"What is the length of the outer handrail of a spiral staircase with a radius of 3 feet that turns 270° while rising 10 feet?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 26
} | 30 | 0 | 0 |
omr-ws-b9e50ec908f251b9a304a5d95234e2112938eec8432749d2e9c61781015e84cf | What is the length of the portion of the parabola defined by $f(x) = x^2$ between $x=0$ and $x=2$ when it is straightened out? | [
"\\(\\sqrt{17}+\\frac{1}{4}\\ln(4+\\sqrt{17})\\)"
] | [
"What is the length of the portion of the parabola defined by $f(x) = x^2$ between $x=0$ and $x=2$ when it is straightened out?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 31
} | 47 | 0 | 0 |
omr-ws-9c4e967ae2064a7bd23ef064799dc31f450dc04017b0e98fbd92c3bc70cd246e | What is the length of the segment of the number line whose endpoints satisfy $|x-\sqrt[5]{16}|=3$? | [
"6"
] | [
"What is the length of the segment of the number line whose endpoints satisfy $|x-\\sqrt[5]{16}|=3$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-701b2d78b92440e16e523fe8a67c2784f9392d236320a452ed934994cdbc45ca | What is the length of the set \(\{a\}\), where \(a\) is any real number? | [
"0"
] | [
"What is the length of the set \\(\\{a\\}\\), where \\(a\\) is any real number?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
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