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-da0fafcf87aad1a6d6c9ba02026d3948851c8d8194f3e5f5c889519ffbbe814b | What is the minimum width of an infinite strip from which any triangle of area $S$ can be cut? | [
"\\(\\sqrt{\\frac{S}{8}}\\)"
] | [
"What is the minimum width of an infinite strip from which any triangle of area $S$ can be cut?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-189fd9c84220eaa388bc05af1274cca6d222a2a235df42472e7edeb6a473c48b | What is the missing number in the sequence $20, 22, 24, 26, 30, 33, 40, 44, 120, \ldots, 11000$? | [
"220"
] | [
"What is the missing number in the sequence $20, 22, 24, 26, 30, 33, 40, 44, 120, \\ldots, 11000$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e31fc50332209365af929bc2b651e59ab5899676f75367a1788e095638ad8092 | What is the missing number in the sequence 7, 84, 951, ?, 3? | [
"62"
] | [
"What is the missing number in the sequence 7, 84, 951, ?, 3?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5a5bc3db3d6c21cd13b036ca24ee596d6079df6f8ba5e66668c139da7b1a7fcb | What is the missing term of the sequence $1, 4, 15, 40, \_\_\_, 156, 259$? | [
"85"
] | [
"What is the missing term of the sequence $1, 4, 15, 40, \\_\\_\\_, 156, 259$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-50ed6324807298a72a750ed4c76e3d1a1f9ead14a1bc6231bbabe42a6227c665 | What is the modular inverse of $11$, modulo $1000$? Express your answer as an integer from $0$ to $999$, inclusive. | [
"91"
] | [
"What is the modular inverse of $11$, modulo $1000$?\n\nExpress your answer as an integer from $0$ to $999$, inclusive."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-acab646bb76f813959f286c31ea6ad31f55b0b8a99c615f3dacec36092138256 | What is the modulo $13$ residue of $247+5 \cdot 39 + 7 \cdot 143 +4 \cdot 15?$ | [
"8"
] | [
"What is the modulo $13$ residue of $247+5 \\cdot 39 + 7 \\cdot 143 +4 \\cdot 15?$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5ace96ecf7b04d140a68039fe7b0f70d6939177536f1109db8592fda716b6608 | What is the modulo $5$ remainder of the sum $1+2+3+4+5+ \ldots + 120+121+122+123?$ | [
"1"
] | [
"What is the modulo $5$ remainder of the sum $1+2+3+4+5+ \\ldots + 120+121+122+123?$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-62f3997746ddb42eaf819c41052555cc79dfb42fdb0eaa8394d657071dd016ee | What is the modulus of $\frac{1+\sqrt{3}i}{\sqrt{3}-i}$? | [
"\\(\\frac{1}{2}\\)"
] | [
"What is the modulus of $\\frac{1+\\sqrt{3}i}{\\sqrt{3}-i}$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9ef646a377878206803a1e5268cde709cae3757a353d2f1400242b4727d684c0 | What is the most exact approximate formula for the perimeter of the ellipse given by $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$? | [
"\\(\\pi (a+b)\\)"
] | [
"What is the most exact approximate formula for the perimeter of the ellipse given by $\\frac{x^{2}}{a^{2}}+\\frac{y^{2}}{b^{2}}=1$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6e23d8cc26157fe1a59f1b133d31160d4dfb3070229db3c8bb78ae7f482b402f | What is the most general form into which we can factor a polynomial of 2 variables, \( p(x, y) = \sum_{j, k} c_{jk} x^j y^k \)? | [
"p(x, y) = c \\prod_{i=1}^n f_i(x, y)^{k_i}"
] | [
"What is the most general form into which we can factor a polynomial of 2 variables, \\( p(x, y) = \\sum_{j, k} c_{jk} x^j y^k \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-f30ebaae1e5637b79214afc258e9601bcb44ce92ac6448de0964c0b32d0035b7 | What is the most general form of the rational function which has modulus one on the unit circle? | [
"\\( f(z)g(1/z) \\), where \\( f \\) and \\( g \\) are finite Blaschke products."
] | [
"What is the most general form of the rational function which has modulus one on the unit circle?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-440977860b7bbb83840b3de3880849e94e3739e37fdf2bd2271493cde5af746b | What is the most general value of \( x \) satisfying the equation \( \sin^2 x = \frac{1}{4} \)? Explanation: The forum post provides a solution to the equation but asks for clarification on the final step. The problem itself is to find the general solution for \( x \) given \( \sin^2 x = \frac{1}{4} \). | [
"\\( x = n\\pi \\pm \\frac{\\pi}{6} \\)"
] | [
"What is the most general value of \\( x \\) satisfying the equation \\( \\sin^2 x = \\frac{1}{4} \\)? \n\nExplanation: The forum post provides a solution to the equation but asks for clarification on the final step. The problem itself is to find the general solution for \\( x \\) given \\( \\sin^2 x = \\frac{1}{4}... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-b6bf306b6b2d1b78f2bbecebf2001b2750cea45548fca79a7a009ce1d083c70b | What is the most likely number of individuals who will be audited by the IRS in a group of 15 individuals, each with an annual income exceeding $200,000$ dollars, given that the IRS audits 5% of such individuals? | [
"0"
] | [
"What is the most likely number of individuals who will be audited by the IRS in a group of 15 individuals, each with an annual income exceeding $200,000$ dollars, given that the IRS audits 5% of such individuals?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-d7de3cc254cea28993877bb195806d916d0200e7be1b94cd66715cc7f93c0b51 | What is the name of the limit \( S = \sum_{n=1}^{\infty} \frac{1}{n^n} \)? | [
"Sophomore's Dream."
] | [
"What is the name of the limit \\( S = \\sum_{n=1}^{\\infty} \\frac{1}{n^n} \\)?"
] | [
"c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-72d25df14bfb172fe6909aa69c1fd3b1bbbeec1fe33723c6fd83d77132800705 | What is the natural number whose thirteenth power is 21982145917308330487013369? | [
"89"
] | [
"What is the natural number whose thirteenth power is 21982145917308330487013369?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-9c06ebc35a843329f570f8cb95462b10c8374b9abc347ddca151cb908b7d3bd9 | What is the nature of a triangle if \( \frac{AB}{AC} = \frac{1}{2} \) and \( \angle BAC = 60^\circ \)? | [
"Right-angled triangle"
] | [
"What is the nature of a triangle if \\( \\frac{AB}{AC} = \\frac{1}{2} \\) and \\( \\angle BAC = 60^\\circ \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-ac0a79fbd532f42aa906078f1cce41a97b31ddb89dbae8e2d187ae29da1ba156 | What is the nature of the series $\sum_{n\geq 1} (\sin(n))^{n}$? | [
"The series diverges."
] | [
"What is the nature of the series $\\sum_{n\\geq 1} (\\sin(n))^{n}$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-aea7af6f31f4f294a9b844ebc1fec9c42e43c1a95f8a54b448a2003e4eefe11f | What is the nearest integer to $(2+\sqrt3)^4$? | [
"194"
] | [
"What is the nearest integer to $(2+\\sqrt3)^4$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-bc9d96c1168a1c057fbfb6b4b4fcc448ce8b1d2e5336a0da7bd04c66e31cbca3 | What is the necessary and sufficient condition for two vectors in $Z^n$ to exist an integer matrix in $GL_n(Z)$ having the first two rows these vectors? | [
"The two vectors must be linearly independent modulo \\( p \\) for every prime \\( p \\)."
] | [
"What is the necessary and sufficient condition for two vectors in $Z^n$ to exist an integer matrix in $GL_n(Z)$ having the first two rows these vectors?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 5,
"tir": 7
} | 14 | 0 | 0 |
omr-ws-62634912dd68d932005b2ef00516ea987d7cb09ffddc88ee99bbba8391b2c3b4 | What is the necessary and sufficient condition on the complex numbers $a_1, \ldots, a_k, b_1, \ldots, b_k$ such that the function $f(x) = \sum_{i=1}^{k} a_i e^{b_i x}$ has a limit as $x$ approaches infinity? | [
"\\text{All } b_i \\text{ have } \\operatorname{Re}(b_i) \\leq 0 \\text{ and for each non-zero purely imaginary } \\lambda, \\sum_{j: b_j = \\lambda} a_j = 0"
] | [
"What is the necessary and sufficient condition on the complex numbers $a_1, \\ldots, a_k, b_1, \\ldots, b_k$ such that the function $f(x) = \\sum_{i=1}^{k} a_i e^{b_i x}$ has a limit as $x$ approaches infinity?"
] | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 3
} | 4 | 0 | 0 |
omr-ws-42c0e1b7fe155663c22098d0626886fd8995fd51292f2bd82b50c055227d475a | What is the necessary condition for a cubic equation $f(x) = x^3 + ax^2 + bx + c = 0$ to have three positive integer roots? | [
"\\( a = -(m + n + p) \\), \\( b = mn + mp + np \\), \\( c = -mnp \\)"
] | [
"What is the necessary condition for a cubic equation $f(x) = x^3 + ax^2 + bx + c = 0$ to have three positive integer roots?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 7,
"tir": 7
} | 28 | 0 | 0 |
omr-ws-ef0b11235dfe08cd39840665f97e8f2c10c6bdeba39fea4aed4e1ed487046c1b | What is the necessary condition for the equality \[ \limsup_{n \rightarrow \infty } (a_n + b_n) = \limsup_{n \rightarrow \infty }a_n + \limsup_{n \rightarrow \infty }b_n \] to hold for two real sequences \(\{a_n\}\) and \(\{b_n\}\)? | [
"Every increasing sequence of indices \\( n_j \\) for which \\( b_{n_j} \\to \\limsup_{n \\to \\infty} b_n \\) contains an increasing subsequence of indices \\( n_{j_k} \\) such that \\( a_{n_{j_k}} \\to \\limsup_{n \\to \\infty} a_n \\)."
] | [
"What is the necessary condition for the equality\n\\[ \\limsup_{n \\rightarrow \\infty } (a_n + b_n) = \\limsup_{n \\rightarrow \\infty }a_n + \\limsup_{n \\rightarrow \\infty }b_n \\]\nto hold for two real sequences \\(\\{a_n\\}\\) and \\(\\{b_n\\}\\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 1,
"tir": 1
} | 7 | 0 | 0 |
omr-ws-a669fe17245b8ecbb46233463416e0e6cf74fa93b36074b6807bbe974a137cfe | What is the next "funny" date after June 19, 1999, where a date is considered "funny" if the number formed by concatenating the day, month, and year is a perfect square? | [
"19079424"
] | [
"What is the next \"funny\" date after June 19, 1999, where a date is considered \"funny\" if the number formed by concatenating the day, month, and year is a perfect square?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-3c18dc8339d1e1063ce5037715d945b251b893835f999b3886290dba2fb67621 | What is the next exact time after 2:00 that a clock has its hands at an angle of 70 degrees? | [
"\\( 2:23 \\frac{7}{11} \\)"
] | [
"What is the next exact time after 2:00 that a clock has its hands at an angle of 70 degrees?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-29e19f05614f004d5b0d56c93db935dff1ce9dd08f3553b7f5acdbaeed1f35ea | What is the next higher number of equiangular triangles that can be used to form an equiangular hexagon? | [
"24"
] | [
"What is the next higher number of equiangular triangles that can be used to form an equiangular hexagon?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-c2ea3d503b8fc13a7b6594844dc09e7607c8a129cfbdce9ec46bfba06bf797db | What is the next letter/number combination in the sequence S6, A10, R17, E21, H26, R30? | [
"G33"
] | [
"What is the next letter/number combination in the sequence S6, A10, R17, E21, H26, R30?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 3
} | 7 | 0 | 0 |
omr-ws-137fc1dbbc16fc82c727ad005521fb4b00fdda05540137df8b26fbb79e72ed30 | What is the next number in the Heegner number sequence: 1, 2, 3, 7, 11, 19, 43, 67, 163, ??? | [
"163"
] | [
"What is the next number in the Heegner number sequence: 1, 2, 3, 7, 11, 19, 43, 67, 163, ???"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-fb76790a1f3120c2c6b0bf0b0022f2b33786ea3db17f105ddfad33f52ce6a99d | What is the next number in the sequence $1, 2, 4, 7, 8, 11, 14, 16, 17, 19, 22, 26, 28, 29, 41, 44$? | [
"46"
] | [
"What is the next number in the sequence $1, 2, 4, 7, 8, 11, 14, 16, 17, 19, 22, 26, 28, 29, 41, 44$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 21,
"genselect": 6,
"tir": 3
} | 30 | 0 | 0 |
omr-ws-917a77b90edc7e373d02d95ef4325038c71d72daeab977263b3aeb73e15752ce | What is the next number in the sequence $111, 21, 13, 12, 11, 10, 7, 7, ...$? Hint: number bases. | [
"7"
] | [
"What is the next number in the sequence $111, 21, 13, 12, 11, 10, 7, 7, ...$? Hint: number bases."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-e20a56042141ab7bd9c18cdfb6893de0356a6be3c0f0614ba1c266736159f315 | What is the next number in the sequence $7, 33, 29, 59, 51, 86, 73, \_\_$? | [
"114"
] | [
"What is the next number in the sequence $7, 33, 29, 59, 51, 86, 73, \\_\\_$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-9fd5feda6e49d085270fbfdef5e1df560463a1c38a6adcfce8de1c6accbe7950 | What is the next number in the sequence 12, 9, 13, 9, 20 that makes the most sense? | [
"9"
] | [
"What is the next number in the sequence 12, 9, 13, 9, 20 that makes the most sense?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 5
} | 17 | 0 | 0 |
omr-ws-d5963ab42bcfa711cfad7687681f04b956387839917694f10c19b9e0ada4c272 | What is the next number in the sequence: \(1, \frac{4}{3}, 2, \frac{16}{5}, \frac{16}{3}, \frac{64}{7}, \ldots\)? | [
"16"
] | [
"What is the next number in the sequence: \\(1, \\frac{4}{3}, 2, \\frac{16}{5}, \\frac{16}{3}, \\frac{64}{7}, \\ldots\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-42da830d2d1de9e230337b1178d87509a5d37a68364bc387d77e4e46252ee1c3 | What is the next number in the series 1, 3, 3, 11, 19, _? The numbers depend on the preceding two numbers. | [
"59"
] | [
"What is the next number in the series 1, 3, 3, 11, 19, _? The numbers depend on the preceding two numbers."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 7,
"tir": 2
} | 14 | 0 | 0 |
omr-ws-56651ff76136e49aa613ca3c0230b7470a048acbcb64bd11c59aa02b34b19556 | What is the next smallest positive integer after $1995$ whose prime factorization uses each odd digit exactly once? | [
"2685"
] | [
"What is the next smallest positive integer after $1995$ whose prime factorization uses each odd digit exactly once?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 14
} | 37 | 0 | 0 |
omr-ws-c1653f2016d117794a9750ee43159785f8c7d2f762591f9a9d8bf53dd82e5fcb | What is the next term in the geometric sequence $$2, 6x, 18x^2, 54x^3, \ldots ?$$ Express your answer in terms of $x$. | [
"162x^4"
] | [
"What is the next term in the geometric sequence $$2, 6x, 18x^2, 54x^3, \\ldots ?$$ Express your answer in terms of $x$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6c836a9149d036fc9ab00605393ac9bfd6a6b60f7423ee09efb57411c59bc5cc | What is the next term in the sequence $5, -4, 13, 41, 79, 130, \ldots$? | [
"191"
] | [
"What is the next term in the sequence $5, -4, 13, 41, 79, 130, \\ldots$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 5
} | 11 | 0 | 0 |
omr-ws-155252eee8a541ff99932556a855c348e5b929ec72ba14c77d702cd530e29e82 | What is the next term in the sequence 2, 12, 1112, 3112, 132112, ...? | [
"1113122112"
] | [
"What is the next term in the sequence 2, 12, 1112, 3112, 132112, ...?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-4c05badd911a96cb7baf9cafdd607646b8957819aad1858277b42ba68dde8026 | What is the next term in the sequence: 2, 6, 18, 144, 2736? | [
"87552"
] | [
"What is the next term in the sequence: 2, 6, 18, 144, 2736?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-f16590d4c9cadaa3eb7f75ad16574dfbe502c70891bd20d2602f8ce4511d9f8e | What is the next term of the sequence $-3, -1, -11, 39, 89, 167$? | [
"279"
] | [
"What is the next term of the sequence $-3, -1, -11, 39, 89, 167$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 1
} | 3 | 0 | 0 |
omr-ws-7d1951f510fa359ddd94e232e13f559b3107ddd4c799e220250ea495977ea780 | What is the next term of the sequence defined by $a_n = 3, 2, 2, 4, 12, 40, 136$? | [
"464"
] | [
"What is the next term of the sequence defined by $a_n = 3, 2, 2, 4, 12, 40, 136$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-13b7ab0c91e454d149e3afb16c5a3936b8ce25a0c0862aabca46ea98bb6f3341 | What is the ninth digit from the right in \(101^{20}\)? | [
"6"
] | [
"What is the ninth digit from the right in \\(101^{20}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 6,
"tir": 17
} | 37 | 0 | 0 |
omr-ws-824e7de80432122e03a97b0017ce0743f125d1294b3d4a26cce5257acc61359c | What is the ninth term in the expansion of $(x+1)^{10}$? | [
"\\(45x^2\\)"
] | [
"What is the ninth term in the expansion of $(x+1)^{10}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-875bc26ad891ed44c3eb36c1106684bdec60aa8a05fc21038fc3709b32c1587b | What is the nonnegative difference between the roots for the equation $x^2+30x+180=-36$? | [
"6"
] | [
"What is the nonnegative difference between the roots for the equation $x^2+30x+180=-36$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3c579a806e740b4a96aa80e414b83cc3c224ffbf9a545e4a0381e7c4f5500574 | What is the norm of the linear functional $f$ defined on $C[-1,1]$ by $f(x) = \int_{-1}^0 x(t) \, dt - \int_0^1 x(t) \, dt$, where $C[-1,1]$ denotes the Banach space of all real-valued functions $x(t)$ on $[-1,1]$ with the norm $\|x\| = \max_{t \in [-1,1]} |x(t)|$? | [
"2"
] | [
"What is the norm of the linear functional $f$ defined on $C[-1,1]$ by $f(x) = \\int_{-1}^0 x(t) \\, dt - \\int_0^1 x(t) \\, dt$, where $C[-1,1]$ denotes the Banach space of all real-valued functions $x(t)$ on $[-1,1]$ with the norm $\\|x\\| = \\max_{t \\in [-1,1]} |x(t)|$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-153ece11c5209403691bf69de0ed9a5b5fbd56ea0ad66d82067f72f2c9eb7dc9 | What is the nth derivative of \( x^x \) at \( x = 1 \)? | [
"n \\cdot (n - 2)!"
] | [
"What is the nth derivative of \\( x^x \\) at \\( x = 1 \\)?"
] | [
"aops_c7_college_math"
] | [
"additional_problems",
"genselect"
] | {
"additional_problems": 1,
"genselect": 2
} | 3 | 0 | 0 |
omr-ws-bf4a3507f98bf4eea8c5d73a1b437a7fbbe47a726dade15f832615bdcf6e5f8d | What is the nth term in the sequence 12, 18, 20, 28, ...? | [
"63"
] | [
"What is the nth term in the sequence 12, 18, 20, 28, ...?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-115aee5e2561d00845b52c0cd628ba59ab388fbeae569f7416d07a31a13ede36 | What is the number (order) of the middle term in the expansion of $(x+1)^{14}$? | [
"8"
] | [
"What is the number (order) of the middle term in the expansion of $(x+1)^{14}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-65177c7b58ba520ee4720de179d0b077467320c04fb5233c8d7f788747dca989 | What is the number 14987 in base 8? | [
"35213"
] | [
"What is the number 14987 in base 8?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f3a1a47c18a7a777a7c617608481095ca2534a47a7e2be6b79074050840bf1e6 | What is the number \( R(n) \) of rooks required to dominate an \( n \times n \times n \) chessboard? | [
"n"
] | [
"What is the number \\( R(n) \\) of rooks required to dominate an \\( n \\times n \\times n \\) chessboard?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-8641ab802bb88865085a5bd465820cd1ee0b84b16fd6c939abf5eadd4a15a72c | What is the number of (labelled) subgraphs of the (labelled) complete bipartite graph $K_{m,n}$? | [
"\\(2^{m+n + m \\times n}\\)"
] | [
"What is the number of (labelled) subgraphs of the (labelled) complete bipartite graph $K_{m,n}$?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-aef81f47098862201c1df32191e9311efe01915f6d46c710dad1664c79fca315 | What is the number of 0-1 strings of length \( n \) such that there are no two consecutive ones? | [
"\\( F_{n+2} \\)"
] | [
"What is the number of 0-1 strings of length \\( n \\) such that there are no two consecutive ones?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-e24ca57c5a5e3f5a26d52dce4d81dc9900ae370ed7354b5e552a659c7c2f5f19 | What is the number of 0s in the base-48 expression of \(2018!\)? | [
"502"
] | [
"What is the number of 0s in the base-48 expression of \\(2018!\\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 30
} | 45 | 0 | 0 |
omr-ws-33e55b50761e40503b314244af39ff4dec829f3d7414286380020a86fc97688d | What is the number of 5-Sylow subgroups in the symmetric group $S_7$? | [
"126"
] | [
"What is the number of 5-Sylow subgroups in the symmetric group $S_7$?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-732b77188c3b4cbe87548c1382aa1bcea2ca5db0920931ebacbf866063bec579 | What is the number of Boole subalgebras of $ \mathbb{L}_{2}^{n}$? | [
"B_n"
] | [
"What is the number of Boole subalgebras of $ \\mathbb{L}_{2}^{n}$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-2672238fa236c2e38ee514902c9996b762f14fc7ec8642abd5214966e56d764e | What is the number of Sylow 127-subgroups of PSL(7,2)? | [
"184308203520"
] | [
"What is the number of Sylow 127-subgroups of PSL(7,2)?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 4,
"genselect": 3,
"tir": 7
} | 14 | 0 | 0 |
omr-ws-d383853a24ce495e16fcab94cbf72cbfcd65c8a505e47e6b82e79fc24f502c8b | What is the number of Sylow 127-subgroups of PSL(8,2)? | [
"6015819762892800"
] | [
"What is the number of Sylow 127-subgroups of PSL(8,2)?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6ebe7c45ff949e4db14dea871242478061ca65fb0846d8688e90ad7ba5262d68 | What is the number of Sylow 31-subgroups of PSL(6,2)? | [
"130056192"
] | [
"What is the number of Sylow 31-subgroups of PSL(6,2)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 7,
"tir": 2
} | 15 | 0 | 0 |
omr-ws-43ffd68612e944d26130d52b388f74ec107c982c2125fcb36e7cb3ed6fdc33be | What is the number of arrangements of three circles A, B, and C on a plane, where A and B have the same radius and C has a smaller radius, given that A and B have no common point? The arrangements are identified by the number of their intersections. | [
"9"
] | [
"What is the number of arrangements of three circles A, B, and C on a plane, where A and B have the same radius and C has a smaller radius, given that A and B have no common point? The arrangements are identified by the number of their intersections."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 6
} | 14 | 0 | 0 |
omr-ws-412076d7d072443c25d6769cbb0362ab24f14c1becce8fbf6585ebcb54892166 | What is the number of binary strings of length \( n \) (where \( n \geq 4 \)) in which the substring "01" occurs exactly twice? | [
"\\dbinom{n+1}{5}"
] | [
"What is the number of binary strings of length \\( n \\) (where \\( n \\geq 4 \\)) in which the substring \"01\" occurs exactly twice?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 20,
"genselect": 8
} | 28 | 0 | 0 |
omr-ws-81b951d6da9737ef0b392f7acd607230697153a69ca8fbccb9877348d91c9bdf | What is the number of circular permutations of eight identical apples and four identical melons? | [
"43"
] | [
"What is the number of circular permutations of eight identical apples and four identical melons?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-a3f44114c2388d73055de907bdd4f3895d59c1ea658f3f6fb142ca15c14cb30c | What is the number of common terms in the two sequences $\{3, 7, 11, \ldots, 407\}$ and $\{2, 9, 16, \ldots, 709\}$? | [
"14"
] | [
"What is the number of common terms in the two sequences $\\{3, 7, 11, \\ldots, 407\\}$ and $\\{2, 9, 16, \\ldots, 709\\}$?"
] | [
"aops_c4_high_school_math",
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 26
} | 57 | 0 | 0 |
omr-ws-6938c0fbf25cf346d03c8f8656ac65c34d9bbfa46184d882669c0c06f67b73d2 | What is the number of cycles in a complete graph with $n$ vertices? | [
"\\(\\sum_{k=3}^{n} \\frac{(n-1)!}{(n-k)! \\cdot k}\\)"
] | [
"What is the number of cycles in a complete graph with $n$ vertices?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 6,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-6ab80fdd1181267ef8c0185ed05a6483142cca30a42b4d3ad7edb8143bda4ad7 | What is the number of degrees in the acute angle \( \theta \) satisfying \( \cos(\theta) = \frac{1}{2}\sqrt{2 + \sqrt{2}} \)? | [
"22.5 or 67.5"
] | [
"What is the number of degrees in the acute angle \\( \\theta \\) satisfying \\( \\cos(\\theta) = \\frac{1}{2}\\sqrt{2 + \\sqrt{2}} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 1
} | 1 | 0 | 0 |
omr-ws-3c3482f2453da105906cb9700544e30b1d06db267ac669dd23a09062ae11eead | What is the number of degrees in the acute angle formed by the hands of a clock at 6:44? | [
"62^\\circ"
] | [
"What is the number of degrees in the acute angle formed by the hands of a clock at 6:44?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-727c274cb24fbd636b5b05ceffb63fabb05ed4fa8363a2a008120b69b56ceead | What is the number of degrees in the angle at each point of the star formed by extending the sides of a regular polygon with $n$ sides $(n>4)$? | [
"\\( 180 - \\frac{720}{n} \\)"
] | [
"What is the number of degrees in the angle at each point of the star formed by extending the sides of a regular polygon with $n$ sides $(n>4)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-7515ef70cd58ec96bac0baf444ed8cb725eea4c07156bce8794029611174932a | What is the number of different colorings of the K2,3 graph using two colors? Use Burnside's lemma. | [
"12"
] | [
"What is the number of different colorings of the K2,3 graph using two colors? Use Burnside's lemma."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 5
} | 18 | 0 | 0 |
omr-ws-bee793aacf726575c5a19d1fc5d57ac1ba1b0c4bb7a712a727daa0f2108bd34d | What is the number of digits in the decimal expansion of $16^5 \cdot 5^{16}$? | [
"18"
] | [
"What is the number of digits in the decimal expansion of $16^5 \\cdot 5^{16}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-b5e68a8c45ab1cfb903ac3cf3f285d00c77299f3527981176a09d51c8ade6248 | What is the number of digits in the product of \(3659893456789325678 \cdot 342973489379256\)? | [
"35"
] | [
"What is the number of digits in the product of \\(3659893456789325678 \\cdot 342973489379256\\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4c12568aa99dfdc9670def70fcd737ead7bbaa215ccfc7b3e2eef69ec74835dc | What is the number of digits in the smallest string that contains all of the two-digit numbers from $10$ to $99$ inclusive? | [
"99"
] | [
"What is the number of digits in the smallest string that contains all of the two-digit numbers from $10$ to $99$ inclusive?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 7
} | 15 | 0 | 0 |
omr-ws-4e050c4f674a18caf12a155b2be2cf915cb9d343c45b0f301071b7378861ef3a | What is the number of digits of \( N \), where \( N = (1^5 + 2^5 + 3^5 + \ldots + 2012^5) + (1^7 + 2^7 + 3^7 + \ldots + 2012^7) \)? | [
"26"
] | [
"What is the number of digits of \\( N \\), where \\( N = (1^5 + 2^5 + 3^5 + \\ldots + 2012^5) + (1^7 + 2^7 + 3^7 + \\ldots + 2012^7) \\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-2e6cbf5410b557ec10ab50ab571a591d79578c93d6bac17fb9b16218426a0911 | What is the number of digits of the number \(2005^{2006} + 2^{2004}\)? | [
"6625"
] | [
"What is the number of digits of the number \\(2005^{2006} + 2^{2004}\\)?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 4,
"tir": 32
} | 41 | 0 | 0 |
omr-ws-19656c94e4e0150a1ef671f0762c7b759205b5696947fbc97eb186f8898130b2 | What is the number of distinct lines in the plane which contain the origin and at least one point of integer coordinates (x,y) with both x and y in {1,2,3,4,5,6}? | [
"23"
] | [
"What is the number of distinct lines in the plane which contain the origin and at least one point of integer coordinates (x,y) with both x and y in {1,2,3,4,5,6}?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-0a984777840354462f40cdce973964293c636994562c93c1711b7770294160d2 | What is the number of distinct lines representing the altitudes, medians, and interior angle bisectors of an isosceles triangle that is not equilateral? | [
"7"
] | [
"What is the number of distinct lines representing the altitudes, medians, and interior angle bisectors of an isosceles triangle that is not equilateral?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c359dff0ef207b626ab6c6e6c00ef672711a2428dcdb6b6438197eb0b38ab5bb | What is the number of distinct prime factors of the smallest positive integer that has exactly 2000 proper factors (excluding 1 and itself)? | [
"4"
] | [
"What is the number of distinct prime factors of the smallest positive integer that has exactly 2000 proper factors (excluding 1 and itself)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-0c55121a5c7d513b570e202407ab1863765c43b247b5ce954aad0beebdd70233 | What is the number of distinct solutions $(x,y,z)$ to the equation $x^2+yz=y^2+xz$ if $x, y,$ and $z$ are integers between 1 and 5 inclusive? | [
"33"
] | [
"What is the number of distinct solutions $(x,y,z)$ to the equation $x^2+yz=y^2+xz$ if $x, y,$ and $z$ are integers between 1 and 5 inclusive?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-8202b24a867a37fb909385ddbe27e7890ef4b81a5d079c68f83ce6777514bae5 | What is the number of distinct terms in the expansion of \((x_1 + x_2 + \cdots + x_m)^n\)? | [
"\\dbinom{n + m - 1}{m - 1}"
] | [
"What is the number of distinct terms in the expansion of \\((x_1 + x_2 + \\cdots + x_m)^n\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1bba3198eeeadddaead87117e1d66387194c690155086be3277d7c5e53f3abec | What is the number of distributions of \( n \) objects into different nonempty unordered parts? | [
"q(n)"
] | [
"What is the number of distributions of \\( n \\) objects into different nonempty unordered parts?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 3
} | 12 | 0 | 0 |
omr-ws-a2c4162b78427985b88a184a541f709aff43483e85c8aaef828ba47d3d871910 | What is the number of divisors of $1374 \times 1375 \times 1376$? | [
"448"
] | [
"What is the number of divisors of $1374 \\times 1375 \\times 1376$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-532f3bbfb1391ce621105f5c49906fb91f80460b8ad57bc1beb55b03372e9571 | What is the number of domino tilings of the union of all unit squares that are properly contained in the region $|x| + |y| \le n$? | [
"2^{\\frac{n(n-1)}{2}}"
] | [
"What is the number of domino tilings of the union of all unit squares that are properly contained in the region $|x| + |y| \\le n$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 2
} | 17 | 0 | 0 |
omr-ws-a1e8579984cc3fd1151eeb77b4526c2dd6f90d3c3b623c1afd5d024f68d206b5 | What is the number of elements $r$ in the quotient ring $R=\mathbb{Z}/2020\mathbb{Z}$ such that $r^n=0$ for some integer $n \ge 1$? | [
"2"
] | [
"What is the number of elements $r$ in the quotient ring $R=\\mathbb{Z}/2020\\mathbb{Z}$ such that $r^n=0$ for some integer $n \\ge 1$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7f8b63564fd6eaf6a45c87ffe49f087719684e7655f9640a3c9a963945b0ff02 | What is the number of elements $r$ in the quotient ring $R=\mathbb{Z}/2020\mathbb{Z}$ such that $r^n=1$ for some integer $n \ge 1$? | [
"800"
] | [
"What is the number of elements $r$ in the quotient ring $R=\\mathbb{Z}/2020\\mathbb{Z}$ such that $r^n=1$ for some integer $n \\ge 1$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-f4c057ba8ca3ad2de6be12119b889b7a70b6d7d499de820e66eeedc6cb66e622 | What is the number of elements in the largest subset of \(\{1, 2, 3, \ldots, 2n-1, 2n\}\) such that no number in the subset divides another number? | [
"\\(n\\)"
] | [
"What is the number of elements in the largest subset of \\(\\{1, 2, 3, \\ldots, 2n-1, 2n\\}\\) such that no number in the subset divides another number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-27511fba4070064cff4e80f44495af98adde2e177a3876647f845bfe52260d17 | What is the number of fair coin tosses such that the probability of getting exactly 2 heads is 0.5? Solve for \( n \) in the equation \(\frac{\binom{n}{2}}{2^n} = 0.5\). | [
"\\text{No solution}"
] | [
"What is the number of fair coin tosses such that the probability of getting exactly 2 heads is 0.5? Solve for \\( n \\) in the equation \\(\\frac{\\binom{n}{2}}{2^n} = 0.5\\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7eed6f8c6e36038e91f3a2d3e966f42aba8d110c301f1bce568fc5e0d9fc1afd | What is the number of feet in the longest possible diagonal of a rectangle with a perimeter of 100 feet and integer dimensions? Express your answer as a decimal to the nearest hundredth. | [
"49.01"
] | [
"What is the number of feet in the longest possible diagonal of a rectangle with a perimeter of 100 feet and integer dimensions? Express your answer as a decimal to the nearest hundredth."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-659ecc8e380bab651cf8a6b680e84d00d1c6b2eec1b4be5306a4ffa708f8e492 | What is the number of graphs with the degree sequence $(k,k,k-1,k-1,\ldots,2,2,1,1)$? It is also good to find good bounds for this number. | [
"2^{k-1}"
] | [
"What is the number of graphs with the degree sequence $(k,k,k-1,k-1,\\ldots,2,2,1,1)$? It is also good to find good bounds for this number."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 4
} | 8 | 0 | 0 |
omr-ws-b185fa044713e00fcdc34dc7641ce449bb1b73f7721ac091d4b33d37a5fd477b | What is the number of integers $k$ smaller than $n$ such that $k$ is coprime to both $n - 1$ and $n$? | [
"\\varphi(n) + \\varphi(n-1) - (n - 1)"
] | [
"What is the number of integers $k$ smaller than $n$ such that $k$ is coprime to both $n - 1$ and $n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 10
} | 10 | 0 | 0 |
omr-ws-1c7eddd23f840ce2eae3bcc27ae5e9c9f598a7cec6042ce3ce33a7f8b5bc7d75 | What is the number of integers $n \geq 10$ such that $\binom{n}{10} \cdot \binom{n+1}{10}$ is a perfect square? | [
"1"
] | [
"What is the number of integers $n \\geq 10$ such that $\\binom{n}{10} \\cdot \\binom{n+1}{10}$ is a perfect square?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-ef61ad8185cea0647b095331b8d9475f9f27404fb90fa62c648eb02f1bb4b0d0 | What is the number of integers between 1 and 360 which are not divisible by 2, 3, or 5? | [
"96"
] | [
"What is the number of integers between 1 and 360 which are not divisible by 2, 3, or 5?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 14
} | 29 | 0 | 0 |
omr-ws-82b5b259374b8a560b4eb0118feaf6f08d96d48fc4f23d958459eb6c6e715f06 | What is the number of isomorphism classes of groups of order \( n \)? | [
"1"
] | [
"What is the number of isomorphism classes of groups of order \\( n \\)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 2
} | 12 | 0 | 0 |
omr-ws-aff1c6731478add32e30d20c82f404ff8cf5cf23e9279cc7cd4db55c7becb59e | What is the number of linear transformations from a linear space of dimension $m$ into a linear space of dimension $n$? | [
"mn"
] | [
"What is the number of linear transformations from a linear space of dimension $m$ into a linear space of dimension $n$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-29916dfb3550b691a63ee9a86d3928472abd3d41ea9e3cfc960ef531d1c473d7 | What is the number of local maxima of the function $f(x) = x + \sin x$? | [
"0"
] | [
"What is the number of local maxima of the function $f(x) = x + \\sin x$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b91680c6303928de4224e4c34ab2e2e7108e4350b12d22d1d5bc7ac03d79384d | What is the number of mutually non-isomorphic non-ramified coverings of degree 4 of a given elliptic curve? | [
"7"
] | [
"What is the number of mutually non-isomorphic non-ramified coverings of degree 4 of a given elliptic curve?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 3
} | 14 | 0 | 0 |
omr-ws-7caa27f7f2a7c2c952d4e032bb24d2ef189b434fef3456fe9c191cce5d14e552 | What is the number of n-digit numbers which contain the digits 2 and 7 but do not contain the digits 0, 1, 8, or 9? | [
"\\(6^n - 4^n\\)"
] | [
"What is the number of n-digit numbers which contain the digits 2 and 7 but do not contain the digits 0, 1, 8, or 9?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-353c34e1ea3ce1c6f3519a7277136f715e9ce291d78284885cc0a3fcdde7ff87 | What is the number of non-isomorphic abelian groups of order $2^2 \cdot 3^3 \cdot 5^5$ which contain an element of order $2^2 \cdot 3^2 \cdot 5^3$? | [
"8"
] | [
"What is the number of non-isomorphic abelian groups of order $2^2 \\cdot 3^3 \\cdot 5^5$ which contain an element of order $2^2 \\cdot 3^2 \\cdot 5^3$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-5efa5a9e2b8ce8257ec160cfd449af49464d9e798198496c3aff02b267c4bfcb | What is the number of ordered pairs $(A,B)$ where $A$ and $B$ are subsets of $\{1,2,3,4,5\}$ such that neither $A \subseteq B$ nor $B \subseteq A$? | [
"570"
] | [
"What is the number of ordered pairs $(A,B)$ where $A$ and $B$ are subsets of $\\{1,2,3,4,5\\}$ such that neither $A \\subseteq B$ nor $B \\subseteq A$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-db1d5d768830efe5cbfcd0973c01bfee01c7c145e90563afb92a8d8db6a68c41 | What is the number of ordered pairs $(m,n)$ of positive integers such that $3m^2+8mn+3n^2=383$? | [
"2"
] | [
"What is the number of ordered pairs $(m,n)$ of positive integers such that $3m^2+8mn+3n^2=383$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f6022e7c30778d18825de6b03e024230e5c86b633bdf4ccc9e0f4cebf55efcf1 | What is the number of ordered pairs (A, B) where A and B are subsets of {1, 2, ..., 5} such that neither A ⊂ B nor B ⊂ A? | [
"570"
] | [
"What is the number of ordered pairs (A, B) where A and B are subsets of {1, 2, ..., 5} such that neither A ⊂ B nor B ⊂ A?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 9
} | 24 | 0 | 0 |
omr-ws-0327233887cde81431795b943839344c96d815045f44e19cc9b60adcfa10d1e3 | What is the number of ordered pairs (A, B) where A and B are subsets of {1, 2, 3, 4, 5} such that neither $A \subseteq B$ nor $B \subseteq A$? | [
"570"
] | [
"What is the number of ordered pairs (A, B) where A and B are subsets of {1, 2, 3, 4, 5} such that neither $A \\subseteq B$ nor $B \\subseteq A$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-9100d925b46552cfaed7d201b117bc0c91b7281699584a577449a8c4dbb9d684 | What is the number of ordered pairs \((a, b)\) such that \((a + bi)^{2002} = a - bi\)? | [
"2004"
] | [
"What is the number of ordered pairs \\((a, b)\\) such that \\((a + bi)^{2002} = a - bi\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-61a9b61cb0b06ecd9973774c0cff43e46873411278de7eeb7c6d1723852865c7 | What is the number of ordered pairs \((x, y)\), where \(x, y \in \{0, 1, 2, \ldots, 142\}\), such that \(5x^2 + 7y^2 - 1\) is a multiple of 143? | [
"120"
] | [
"What is the number of ordered pairs \\((x, y)\\), where \\(x, y \\in \\{0, 1, 2, \\ldots, 142\\}\\), such that \\(5x^2 + 7y^2 - 1\\) is a multiple of 143?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
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