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-1efeb3915d8e52d34edf939a8291aaadf0657b816fee7f9e5374d6a04a1a301f | Using the letters $A$, $G$, $M$, $O$, $S$, and $U$, we can form six-letter "words". If these "words" are arranged in alphabetical order, then the "word" $AMOGUS$ occupies which position? | [
"32"
] | [
"Using the letters $A$, $G$, $M$, $O$, $S$, and $U$, we can form six-letter \"words\". If these \"words\" are arranged in alphabetical order, then the \"word\" $AMOGUS$ occupies which position?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-3bd05a4e34ba76a5bb7277ec69e42461fc73430bbcdd69d971fc6be1c2a9a0d7 | Using the letters $A$, $M$, $O$, $S$, and $U$, we can form 120 five-letter "words". If these "words" are arranged in alphabetical order, then what position does the "word" $USAMO$ occupy? | [
"115"
] | [
"Using the letters $A$, $M$, $O$, $S$, and $U$, we can form 120 five-letter \"words\". If these \"words\" are arranged in alphabetical order, then what position does the \"word\" $USAMO$ occupy?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0d350660700a5e2b38b51a2ffc4d65eed0e637d8244c2aaf9361e15c0c1239e2 | Using the letters $A$, $M$, $O$, $S$, and $U$, we can form five-letter "words". If these "words" are arranged in alphabetical order, what position does the "word" $USAMO$ occupy? | [
"115"
] | [
"Using the letters $A$, $M$, $O$, $S$, and $U$, we can form five-letter \"words\". If these \"words\" are arranged in alphabetical order, what position does the \"word\" $USAMO$ occupy?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-e3cc91acda3f9c20c1900fed8da040b45153c0f2496e510d82e5583fe9ccc6c4 | Using the letters in the word ‘IMPORTANCE’ without repetition, find the probability that a seven-letter word consisting of 3 vowels and 4 consonants will start with a P or end in a C but not both. | [
"\\(\\frac{19}{105}\\)"
] | [
"Using the letters in the word ‘IMPORTANCE’ without repetition, find the probability that a seven-letter word consisting of 3 vowels and 4 consonants will start with a P or end in a C but not both."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fe57b265d5c4cd7f2f5497d4e025664454e574e947b9ca894879f86fbd0e6ce0 | Using the letters k, a, l, a, m, a, z, o, o, how many words can be written such that the letter 'o' appears before the letter 'k'? | [
"10080"
] | [
"Using the letters k, a, l, a, m, a, z, o, o, how many words can be written such that the letter 'o' appears before the letter 'k'?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 3
} | 18 | 0 | 0 |
omr-ws-7ec7381c92b9c43220125b6ca55760e94e4d7b09f5e104559acbfd923d3f52a5 | Using the limit definition of the derivative, calculate \( \frac{d}{dx} e^x \) at \( x = 0 \) and express your answer as a limit. Then, use this result to find the value of \( \frac{d}{dx} e^x \) at \( x = 1 \). | [
"e"
] | [
"Using the limit definition of the derivative, calculate \\( \\frac{d}{dx} e^x \\) at \\( x = 0 \\) and express your answer as a limit. Then, use this result to find the value of \\( \\frac{d}{dx} e^x \\) at \\( x = 1 \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5955e3cb05434dd271194f6533a32f8fe847dfbc513ebedb3b02b5a19a3fcb3d | Using the limit definition of the derivative, find \( f'(x) \) for \( f(x) = \frac{x+1}{x-1} \). | [
"-\\dfrac{2}{(x - 1)^2}"
] | [
"Using the limit definition of the derivative, find \\( f'(x) \\) for \\( f(x) = \\frac{x+1}{x-1} \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6ce4ad1196db0346a8082438acb2a68ab4f5a805335fe8367f3a03aee43ef454 | Using the logarithmic test, determine for which values of \( p \in \mathbb{R} \) the series \(\sum_{n=2}^{\infty} \frac{1}{(\log n)^p}\) converges. | [
"\\text{The series diverges for all } p \\in \\mathbb{R}"
] | [
"Using the logarithmic test, determine for which values of \\( p \\in \\mathbb{R} \\) the series \\(\\sum_{n=2}^{\\infty} \\frac{1}{(\\log n)^p}\\) converges."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-102ff00f3354b78020cbc1b48e080e85a9ac012ff96bc664ff25f71bc8fdd0f9 | Using the logistic equation \( \frac{dP}{dt} = kP(1 - \frac{P}{K}) \) with the initial population \( P(0) = 400 \), carrying capacity \( K = 8400 \), and the population doubling in the first year, how long will it take for the population to increase to 4200 (half of the carrying capacity)? | [
"4.02"
] | [
"Using the logistic equation \\( \\frac{dP}{dt} = kP(1 - \\frac{P}{K}) \\) with the initial population \\( P(0) = 400 \\), carrying capacity \\( K = 8400 \\), and the population doubling in the first year, how long will it take for the population to increase to 4200 (half of the carrying capacity)?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 7
} | 19 | 0 | 0 |
omr-ws-3380d7da5bbc2dd23c7a7bc326c978eb8fbb94957fcec4381c69e48718c3a075 | Using the maximum principle for holomorphic functions, find the maximum and minimum values of $\operatorname{Im}(f(z))$ where $f(z) = iz^2 + z$ in the set $\overline{B(0,2)}$. | [
"Maximum value is \\(\\frac{33}{8}\\) and minimum value is \\(-6\\)."
] | [
"Using the maximum principle for holomorphic functions, find the maximum and minimum values of $\\operatorname{Im}(f(z))$ where $f(z) = iz^2 + z$ in the set $\\overline{B(0,2)}$."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 29
} | 29 | 0 | 0 |
omr-ws-05eda038ae3d4d13f4ea79ec2b2ca40fea866e8ae8a3f599140eeccfc6f3126f | Using the method described above, derive the formula for the terms of the Fibonacci sequence $0, 1, 1, 2, 3, 5, 8, \ldots$. | [
"F_n = \\frac{\\left( \\frac{1 + \\sqrt{5}}{2} \\right)^n - \\left( \\frac{1 - \\sqrt{5}}{2} \\right)^n}{\\sqrt{5}}"
] | [
"Using the method described above, derive the formula for the terms of the Fibonacci sequence $0, 1, 1, 2, 3, 5, 8, \\ldots$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8fdc0fa89178eb87d44788332fb6a84c034a0609d3a0ed6434ad28b55db5f539 | Using the method of Lagrange multipliers, find the minimum value of the function \( f(x) = \sum_{i=1}^{n} a_i g(x_i) \) subject to the constraint \( \sum_{i=1}^{n} a_i x_i = b \), where \( g \) is a convex function, \( a_i > 0 \) for all \( i \), and \( b \) is a constant. Express your answer in terms of \( g \), \( a_... | [
"\\left( \\sum_{i=1}^{n} a_i \\right) g\\left( \\dfrac{b}{\\sum_{i=1}^{n} a_i} \\right)"
] | [
"Using the method of Lagrange multipliers, find the minimum value of the function \\( f(x) = \\sum_{i=1}^{n} a_i g(x_i) \\) subject to the constraint \\( \\sum_{i=1}^{n} a_i x_i = b \\), where \\( g \\) is a convex function, \\( a_i > 0 \\) for all \\( i \\), and \\( b \\) is a constant. Express your answer in term... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-37c4fd33cee4c860a858ff8e7598e2cf777d1cdd8379d21cb152020b85632ef2 | Using the notations \[ B^{1}_{0}(x) = \frac{1}{2} \] \[ B^{1}_{k}(x) = \frac{1}{2} + \sum_{n=1}^{k} \cos(nx) \] \[ B^{p}_{k}(x) = \sum_{v=0}^{k} B^{p-1}_{v}(x), \quad p=2,3,\ldots \text{ and } k \geq 0 \] find the smallest integer \( p \) such that \( B^{p}_{k}(x) \leq \frac{1}{x^{p}} \) for all \( x \in (0, \pi] \) an... | [
"no_valid_answer_found"
] | [
"Using the notations\n\\[ B^{1}_{0}(x) = \\frac{1}{2} \\]\n\\[ B^{1}_{k}(x) = \\frac{1}{2} + \\sum_{n=1}^{k} \\cos(nx) \\]\n\\[ B^{p}_{k}(x) = \\sum_{v=0}^{k} B^{p-1}_{v}(x), \\quad p=2,3,\\ldots \\text{ and } k \\geq 0 \\]\nfind the smallest integer \\( p \\) such that \\( B^{p}_{k}(x) \\leq \\frac{1}{x^{p}} \\) f... | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-67100bd5b29d7de75b31cc6172d5c2198eb0e5b659ef00b9a4fa8edb174768f3 | Using the notion of dimension introduced, where scaling an object by a factor of $n$ yields $n^d$ copies of the original object, determine the dimension of the Sierpinski triangle. Express your answer as a decimal rounded to the nearest tenth. | [
"1.6"
] | [
"Using the notion of dimension introduced, where scaling an object by a factor of $n$ yields $n^d$ copies of the original object, determine the dimension of the Sierpinski triangle. Express your answer as a decimal rounded to the nearest tenth."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-59777eb4a2ca9c53714190590376ecef4aa51c000842417cca62b4bbdae4624e | Using the nth roots of unity, find the value of the product $\prod_{k=1}^{n-1} \cot\left(\frac{k\pi}{2n}\right)$. | [
"1"
] | [
"Using the nth roots of unity, find the value of the product $\\prod_{k=1}^{n-1} \\cot\\left(\\frac{k\\pi}{2n}\\right)$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-41bba38c27aeab626a9e732b1137da382a099a6e7bc61b51b3f6dcf1b7050bcf | Using the number 2 and any combination of operations (+, -, ×, /, ln, log, nth root, etc.), construct an expression that evaluates to 2008. | [
"2222 - 222 + 2 \\times 2 \\times 2"
] | [
"Using the number 2 and any combination of operations (+, -, ×, /, ln, log, nth root, etc.), construct an expression that evaluates to 2008."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-2e52ac987a32ccde00f1d819ea080ffac8f6685e57bffe871f55e83bdd8d51fd | Using the numbers $1, 2, 3, 4$ once each and the operations $+, -, \times, \div$ no more than once each, what are the minimum and maximum of the absolute value of the resulting quantity? | [
"Minimum = 0, Maximum = 20"
] | [
"Using the numbers $1, 2, 3, 4$ once each and the operations $+, -, \\times, \\div$ no more than once each, what are the minimum and maximum of the absolute value of the resulting quantity?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 6
} | 6 | 0 | 0 |
omr-ws-c13a26b2ee8e40f1f83d3be2624bec8bcabc22f97705bf679cc04c3e0dbc5d33 | Using the numbers $1, 2, 3, 4, 5,$ and $6$ and the operations addition, subtraction, multiplication, division, and exponentiation, create the number $1234321$. | [
"\\(\\left(\\frac{6^5 + 1}{3 + 4}\\right)^2\\)"
] | [
"Using the numbers $1, 2, 3, 4, 5,$ and $6$ and the operations addition, subtraction, multiplication, division, and exponentiation, create the number $1234321$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 1
} | 7 | 0 | 0 |
omr-ws-bd83a2ca73875359f018ad423f0b1d901a7c66b05c9a3187d3c3569eed5c5324 | Using the numbers $6, 6, 6, 6, 8, 8$ and the operations addition, subtraction, multiplication, and division, how can you make the number $163$? | [
"\\(\\frac{6 \\times 6 \\times 6 \\times 6 + 8}{8}\\)"
] | [
"Using the numbers $6, 6, 6, 6, 8, 8$ and the operations addition, subtraction, multiplication, and division, how can you make the number $163$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 5,
"tir": 3
} | 16 | 0 | 0 |
omr-ws-72872f3a6f279a314d731c2b4884a20627e8127eace03e92ee5256abf30e7337 | Using the numbers 1, 1, 4, and 5 along with the arithmetic operations $+, -, \cdot, :$ and brackets, construct an expression that evaluates to 5. | [
"\\((1 - 1) \\times 4 + 5\\)"
] | [
"Using the numbers 1, 1, 4, and 5 along with the arithmetic operations $+, -, \\cdot, :$ and brackets, construct an expression that evaluates to 5."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 3,
"tir": 20
} | 34 | 0 | 0 |
omr-ws-29fba43c7108f2ddb6ba55390f5740ab6553d627d5b49ca35bbeae626285d4a6 | Using the numbers 1, 5, 5, and 7 exactly once each, create the integers from 1 to 100 using the four basic arithmetic operations, decimal points, exponents, and the square root sign. Solve for the following "Dirty Dozen" numbers without using factorial notation or the repeating-bar symbol: 46, 66, 67, 78, 83, 87, 89, 9... | [
"7 \\times (5 + 5) - \\sqrt{5 + 5 - 1}"
] | [
"Using the numbers 1, 5, 5, and 7 exactly once each, create the integers from 1 to 100 using the four basic arithmetic operations, decimal points, exponents, and the square root sign. Solve for the following \"Dirty Dozen\" numbers without using factorial notation or the repeating-bar symbol: 46, 66, 67, 78, 83, 87... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-40ebf2ca128483047e25da3094f6e2fa15a39936361e5b722a062d2ea4654e44 | Using the numbers 1, 5, 5, and 7 once and only once, create the integer 67 by using the four basic arithmetic operations and (if needed) decimal points, exponents, and the square root sign. | [
"5^2 + 7 \\times (5 + 1)"
] | [
"Using the numbers 1, 5, 5, and 7 once and only once, create the integer 67 by using the four basic arithmetic operations and (if needed) decimal points, exponents, and the square root sign."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-74aaf4dee7bf70b6d39e2d0df200cd592cc1afc01d698c399d5a9f29816aa118 | Using the numbers 1, 5, 5, and 7 once and only once, create the integer 78 by using the four basic arithmetic operations and (if needed) decimal points, exponents, and the square root sign. | [
"78"
] | [
"Using the numbers 1, 5, 5, and 7 once and only once, create the integer 78 by using the four basic arithmetic operations and (if needed) decimal points, exponents, and the square root sign."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-407019eed898769b96b0e8c5d9a915acf012dc505437f98be8ff0c257b089d83 | Using the numbers 1, 5, 5, and 7 once and only once, create the integer 83 by using the four basic arithmetic operations and (if needed) decimal points, exponents, and the square root sign. | [
"15 \\times 5 + 7 + 1"
] | [
"Using the numbers 1, 5, 5, and 7 once and only once, create the integer 83 by using the four basic arithmetic operations and (if needed) decimal points, exponents, and the square root sign."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 8
} | 16 | 0 | 0 |
omr-ws-e5db35b7ca5c9c7a48b21ddc99e77b5fd865885827ceb6cbec8b31c62f10c0f7 | Using the numbers 1, 5, 5, and 7 once and only once, create the integer 87 by using the four basic arithmetic operations and (if needed) decimal points, exponents, and the square root sign. | [
"87"
] | [
"Using the numbers 1, 5, 5, and 7 once and only once, create the integer 87 by using the four basic arithmetic operations and (if needed) decimal points, exponents, and the square root sign."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-f709d65cf7602a7a51a574328fe3cf44403ddbda9a217079e4bb465a4a60327a | Using the numbers 1, 5, 5, and 7 once and only once, create the integer 89 by using the four basic arithmetic operations and (if needed) decimal points, exponents, and the square root sign. | [
"89"
] | [
"Using the numbers 1, 5, 5, and 7 once and only once, create the integer 89 by using the four basic arithmetic operations and (if needed) decimal points, exponents, and the square root sign."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-fc95f2a6c8647292f8a76ee28114919ba73461d1860eea6550d5fe619b735c15 | Using the numbers 1, 5, 5, and 7 once and only once, create the integer 91 by using the four basic arithmetic operations and (if needed) decimal points, exponents, and the square root sign. | [
"91"
] | [
"Using the numbers 1, 5, 5, and 7 once and only once, create the integer 91 by using the four basic arithmetic operations and (if needed) decimal points, exponents, and the square root sign."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 3
} | 9 | 0 | 0 |
omr-ws-8b1753aa955b5f4e351529bee3f8057d2f535e525f6e77244300056806beb253 | Using the numbers 1, 5, 5, and 7 once and only once, create the integer 92 by using the four basic arithmetic operations and (if needed) decimal points, exponents, and the square root sign. | [
"92"
] | [
"Using the numbers 1, 5, 5, and 7 once and only once, create the integer 92 by using the four basic arithmetic operations and (if needed) decimal points, exponents, and the square root sign."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 8
} | 21 | 0 | 0 |
omr-ws-bdb6c223ed5200aa294c8b4936e0869039c3ce132a7308180b618d60f9c43353 | Using the numbers 1, 5, 5, and 7 once and only once, create the integer 94 by using the four basic arithmetic operations and (if needed) decimal points, exponents, and the square root sign. | [
"94"
] | [
"Using the numbers 1, 5, 5, and 7 once and only once, create the integer 94 by using the four basic arithmetic operations and (if needed) decimal points, exponents, and the square root sign."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 5
} | 17 | 0 | 0 |
omr-ws-a0a6b96b59421b760874ba62fc6d8140e2cf5d62d8656edc73be009091ec3d83 | Using the numbers 1, 5, 5, and 7 once and only once, create the integer 97 by using the four basic arithmetic operations and (if needed) decimal points, exponents, and the square root sign. | [
"97"
] | [
"Using the numbers 1, 5, 5, and 7 once and only once, create the integer 97 by using the four basic arithmetic operations and (if needed) decimal points, exponents, and the square root sign."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 6
} | 6 | 0 | 0 |
omr-ws-fef7bc894ba90baff8a5076c6856939f19623f18677922b87f577496f6869e17 | Using the numbers 1, 5, 5, and 7 once and only once, create the integer 99 by using the four basic arithmetic operations and (if needed) decimal points, exponents, and the square root sign. | [
"99"
] | [
"Using the numbers 1, 5, 5, and 7 once and only once, create the integer 99 by using the four basic arithmetic operations and (if needed) decimal points, exponents, and the square root sign."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-67545862b5f707ed69641958eb6bc538cef256de68f931d9947eea5ffb9ffdf4 | Using the numbers 1-6, how many ways are there to label the six sides of a fair die? | [
"30"
] | [
"Using the numbers 1-6, how many ways are there to label the six sides of a fair die?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-8af3252ceb58dcf37cc2520017b7812edf3425e6b7155df1ad2743dc1198579d | Using the numbers 10, 4, 2, and 1 exactly once, and the operations of addition, subtraction, multiplication, division, exponentiation, and factorial, how can you obtain the number 93? | [
"10^2 - 4 - 2 - 1"
] | [
"Using the numbers 10, 4, 2, and 1 exactly once, and the operations of addition, subtraction, multiplication, division, exponentiation, and factorial, how can you obtain the number 93?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 8
} | 17 | 0 | 0 |
omr-ws-aff3fa29aba42c9f5cb781f98fdcf5e40ae9ccdc08f003c2358902a0331db161 | Using the numbers 10, 9, 1, and 3 exactly once, form two fractions such that their sum, difference, product, or quotient equals \( \frac{23}{20} \). | [
"\\frac{9}{10} + \\frac{1}{3} = \\frac{37}{30}"
] | [
"Using the numbers 10, 9, 1, and 3 exactly once, form two fractions such that their sum, difference, product, or quotient equals \\( \\frac{23}{20} \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-b983502a2834956f04e0ce304aa7615db241d90042235d0db65cda0c62f7fc55 | Using the numbers 12, 12, 12 and any known operations, achieve the number 6. (Note: The operations used must involve only the given numbers, and no external numbers or operations not derived from the given numbers should be used.) | [
"6"
] | [
"Using the numbers 12, 12, 12 and any known operations, achieve the number 6.\n\n(Note: The operations used must involve only the given numbers, and no external numbers or operations not derived from the given numbers should be used.)"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 1,
"tir": 23
} | 36 | 0 | 0 |
omr-ws-468969e03724398d2b13fe3cb9254cfca717c43d3f0da396a40c693ef0fa6fc7 | Using the numbers 16, 17, 23, 24, 39, and 40 (with repetition allowed), find four numbers that add up to 100. | [
"16, 17, 23, 44"
] | [
"Using the numbers 16, 17, 23, 24, 39, and 40 (with repetition allowed), find four numbers that add up to 100."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 3
} | 12 | 0 | 0 |
omr-ws-6ce44adb8e24fd89abf6822f3198cc404bcc5dbccecc59669e410907c3a8fd27 | Using the numbers 2, 2, 2 and any known operations, achieve the number 6. | [
"2 + 2 + 2"
] | [
"Using the numbers 2, 2, 2 and any known operations, achieve the number 6."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-897f923468e5aab63d2b31b5fa57bcf0a2b5ccce107a24c2fe1b28e35eadde03 | Using the numbers 2, 3, 4, and 6 each exactly once, and using only the four basic operations and the following standard calculator operations $(\sqrt{x}$, $log$, $ln$, $sin$, $cos$, $tan$, $e^x$, $!$ and $x^y)$, obtain every integer. | [
"24"
] | [
"Using the numbers 2, 3, 4, and 6 each exactly once, and using only the four basic operations and the following standard calculator operations $(\\sqrt{x}$, $log$, $ln$, $sin$, $cos$, $tan$, $e^x$, $!$ and $x^y)$, obtain every integer."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-7ec37847f8ed3098d1d67fdad528673a4e6ba5dd943717ba5290c1588f4569cd | Using the numbers 2, 3, 4, and 6 each exactly once, and using only the four basic operations and the following standard calculator operations $(\sqrt{x}$, $log$, $ln$, $sin$, $cos$, $tan$, $e^x$, $!$ and $x^y)$, obtain the integer 1337. | [
"1337"
] | [
"Using the numbers 2, 3, 4, and 6 each exactly once, and using only the four basic operations and the following standard calculator operations $(\\sqrt{x}$, $log$, $ln$, $sin$, $cos$, $tan$, $e^x$, $!$ and $x^y)$, obtain the integer 1337."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-03e5da9aa3d45c776aa687e161dce9fa3b16fa38ab477f0828ab9b839c8b0ec3 | Using the numbers 2, 3, 4, and 6 each exactly once, and using only the four basic operations and the following standard calculator operations $(\sqrt{x}$, $log$, $ln$, $sin$, $cos$, $tan$, $e^x$, $!$ and $x^y)$, obtain the integer 53. | [
"53"
] | [
"Using the numbers 2, 3, 4, and 6 each exactly once, and using only the four basic operations and the following standard calculator operations $(\\sqrt{x}$, $log$, $ln$, $sin$, $cos$, $tan$, $e^x$, $!$ and $x^y)$, obtain the integer 53."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 3
} | 16 | 0 | 0 |
omr-ws-4c246a89aab024bf53fc3137258d806ee4da05d73a173890c72a176926798404 | Using the numbers 2010, 73, 4, 10, and 1 exactly once, and the operations +, /, *, and -(minus), without rearranging the numbers, form an expression that equals 0. | [
"2010 - 73 \\times 4 \\times 10 / 1"
] | [
"Using the numbers 2010, 73, 4, 10, and 1 exactly once, and the operations +, /, *, and -(minus), without rearranging the numbers, form an expression that equals 0."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 12
} | 12 | 0 | 0 |
omr-ws-eb55a03782fa74b236604667baba7df5f3bb71b6d332a002e838314afc65d60e | Using the numbers 3, 3, 3 and any known operations, achieve the number 6. | [
"3 \\times 3 - 3"
] | [
"Using the numbers 3, 3, 3 and any known operations, achieve the number 6."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 6
} | 21 | 0 | 0 |
omr-ws-b3916192772512efe4ab005cc2f155fda7e08dbbae07a1dbe36737407d91aec6 | Using the numbers 3, 3, 5, and 9 exactly once, along with any operations, how can you get the number 7? | [
"\\(5 + \\frac{9 - 3}{3}\\)\n\nThis expression simplifies to:\n\\[ 5 + \\frac{6}{3} = 5 + 2 = 7 \\]"
] | [
"Using the numbers 3, 3, 5, and 9 exactly once, along with any operations, how can you get the number 7?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 6,
"tir": 26
} | 47 | 0 | 0 |
omr-ws-0be81968fd6acade7fb35bb3a1beacf15efa8d0cc2735a7511f40b119c7edf0d | Using the numbers 429988021, 6325, 983, and 4294968279, make the number 24 through arithmetic operations. | [
"24"
] | [
"Using the numbers 429988021, 6325, 983, and 4294968279, make the number 24 through arithmetic operations."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 1,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-c076ef3f889669f25cbf28c844d6cea4f5ce7773289a19c179b7aa8e5efcf2d4 | Using the numbers in the set $\{1, 2, 3, 5, 9, 26, 86, 105, 186, 300, 500, 1000\}$ exactly once and only addition, can you obtain a total of $1060$? | [
"\\text{No}"
] | [
"Using the numbers in the set $\\{1, 2, 3, 5, 9, 26, 86, 105, 186, 300, 500, 1000\\}$ exactly once and only addition, can you obtain a total of $1060$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-67ff00bebb7516066f445b1bcfd0584d8763af00d0de4d15bf10c3b5e6668f82 | Using the numbers in the set \( S = \{1, 2, 3, 4\} \), form all possible four-digit numbers that are divisible by 11. Each number from \( S \) can be repeated more than once. | [
"44"
] | [
"Using the numbers in the set \\( S = \\{1, 2, 3, 4\\} \\), form all possible four-digit numbers that are divisible by 11. Each number from \\( S \\) can be repeated more than once."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-ac8bb34d33123da9fb774c6b3c8bda7f48fdea58953f6f5983db51deb5f8fca5 | Using the operations $+, -, \times, \div$ and the order of operations, fill in the squares to make the equation true: \[ 9 \square 9 \square 9 \square 9 = 100 \] | [
"\\(99 + \\frac{9}{9}\\)"
] | [
"Using the operations $+, -, \\times, \\div$ and the order of operations, fill in the squares to make the equation true:\n\n\\[\n9 \\square 9 \\square 9 \\square 9 = 100\n\\]"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 7,
"tir": 7
} | 22 | 0 | 0 |
omr-ws-b40510d0966e1afd248a6d5c21912d2104705a1b02e60a2df3dfaacf567ace6d | Using the orthonormal basis obtained from the Gram-Schmidt procedure, how would you find the adjoint of a given linear map on $P_4([0, 1])$? | [
"\\text{Transpose the matrix representation of the linear map with respect to the orthonormal basis.}"
] | [
"Using the orthonormal basis obtained from the Gram-Schmidt procedure, how would you find the adjoint of a given linear map on $P_4([0, 1])$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c1ac9068209a81425c16ed29ab11ddd67a8539dc5404d904258d03d7226aa06c | Using the population function \( P(t) = 30 - \frac{4}{t} \) (where \( t \) is the number of years after January 1, 1997, and \( P(t) \) is the population in thousands), calculate the increase in population during the year 2002. | [
"133 people"
] | [
"Using the population function \\( P(t) = 30 - \\frac{4}{t} \\) (where \\( t \\) is the number of years after January 1, 1997, and \\( P(t) \\) is the population in thousands), calculate the increase in population during the year 2002."
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 4
} | 4 | 0 | 0 |
omr-ws-a3e35c0681120032ce76bc52130ebb2470a0ce03bb5302ffd9f90d466efb8833 | Using the positive integers from $1$ to $7$, create the integer $8$ using any mathematical operations. | [
"2^{3} \\times (7 - 6) \\times (5 - 4) \\times 1"
] | [
"Using the positive integers from $1$ to $7$, create the integer $8$ using any mathematical operations."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 6
} | 19 | 0 | 0 |
omr-ws-63a700201cfb7f7254e677d37e84ff05a078aba47077c6b9ac60d6cbb117f7c2 | Using the previous exercise, compute \( L(1, \chi) \) where \( \chi \) is the nontrivial quadratic character modulo \( 5 \). | [
"\\(\\frac{1}{\\sqrt{5}} \\ln\\left(\\frac{3 + \\sqrt{5}}{2}\\right)\\)"
] | [
"Using the previous exercise, compute \\( L(1, \\chi) \\) where \\( \\chi \\) is the nontrivial quadratic character modulo \\( 5 \\)."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-9063842d1f291f8940d3b4f82ca33302c739f60de820640264fe5f19fe71eeb3 | Using the principle of contraction mappings, find a sufficient condition for the parameter $\lambda$ such that the equation $\varphi(x) = \lambda \int\limits_{a}^{b} K(x,y) \varphi(y) \, dy + f(x)$ has a unique solution $\varphi \in C[a,b]$, where $K \in C[a,b]^2$ and $f \in C[a,b]$. | [
"\\(\\lambda = \\frac{\\alpha}{(b-a)\\max K}\\)"
] | [
"Using the principle of contraction mappings, find a sufficient condition for the parameter $\\lambda$ such that the equation $\\varphi(x) = \\lambda \\int\\limits_{a}^{b} K(x,y) \\varphi(y) \\, dy + f(x)$ has a unique solution $\\varphi \\in C[a,b]$, where $K \\in C[a,b]^2$ and $f \\in C[a,b]$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6da2086b15e71adf218154c80eac19d15d58052cdfa8172709a4b3e8a030d7b4 | Using the principle of inclusion and exclusion, find the number of ways to arrange the digits 0, 0, 0, 1, 1, 1, 2, 2, 2 so that no three consecutive digits are the same. | [
"1314"
] | [
"Using the principle of inclusion and exclusion, find the number of ways to arrange the digits 0, 0, 0, 1, 1, 1, 2, 2, 2 so that no three consecutive digits are the same."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 22
} | 38 | 0 | 0 |
omr-ws-5ceed75bdf74969dace1f75f9eea206e9221d3e6c5fc3cab9e1bf2a41a590543 | Using the property that \((a^{-1})^n \equiv (a^n)^{-1} \pmod{p}\), find the inverse of \(256\) modulo \(47\). | [
"9"
] | [
"Using the property that \\((a^{-1})^n \\equiv (a^n)^{-1} \\pmod{p}\\), find the inverse of \\(256\\) modulo \\(47\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 13
} | 29 | 0 | 0 |
omr-ws-71d24516aa6c1313a2b4cb336e683e5c1de5fbc8f74781214c14ca391ff25a20 | Using the quadratic formula, solve the equation \(Pqx^2 + (q^2 - pr)x - qr = 0\) for \(x\). | [
"\\frac{pr - q^2 \\pm \\sqrt{(q^2 - pr)^2 + 4Pq^2r}}{2Pq}"
] | [
"Using the quadratic formula, solve the equation \\(Pqx^2 + (q^2 - pr)x - qr = 0\\) for \\(x\\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 8
} | 22 | 0 | 0 |
omr-ws-2a5d9522b4d26c364c72c4c7bb69c1085f50388821f12368327c014282b6b57f | Using the quadratic formula, solve the equation: \( a^2b^2x^2 - (4b^4 - 3a^4)x - 12a^2b^2 = 0 \). | [
"\\( x = \\frac{4b^2}{a^2} \\) or \\( x = \\frac{-3a^2}{b^2} \\)"
] | [
"Using the quadratic formula, solve the equation: \\( a^2b^2x^2 - (4b^4 - 3a^4)x - 12a^2b^2 = 0 \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 4,
"tir": 22
} | 26 | 0 | 0 |
omr-ws-49c0d0468ce4487bca8f1feb2e6b661e46ffb9ea2a07ab23f912d9537f62b039 | Using the quadratic reciprocity theorem, determine if the congruence \(x^2 \equiv 37 \pmod{101}\) has a solution. Given that \(x = 21\) is a solution, find the smallest positive integer \(k\) such that \(y^2 \equiv 101 \pmod{37}\) has no solution for \(y \leq k\). | [
"7"
] | [
"Using the quadratic reciprocity theorem, determine if the congruence \\(x^2 \\equiv 37 \\pmod{101}\\) has a solution. Given that \\(x = 21\\) is a solution, find the smallest positive integer \\(k\\) such that \\(y^2 \\equiv 101 \\pmod{37}\\) has no solution for \\(y \\leq k\\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-17c244827e321ee7cec90f475bc63e8b372c0c0af8b897492bbbe28c1584be14 | Using the rate of change of the area with respect to θ found in part c) (which is 50), calculate the approximate change in area if the angle θ is increased by one degree. | [
"\\(\\frac{5\\pi}{18}\\)"
] | [
"Using the rate of change of the area with respect to θ found in part c) (which is 50), calculate the approximate change in area if the angle θ is increased by one degree."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 20
} | 34 | 0 | 0 |
omr-ws-239e41a47502285d7da920d22ff1411d786ddfbaae3a88962cb77bc3f8b18c6f | Using the real numbers \( x_t = t\sqrt{2} - \lfloor t\sqrt{2} \rfloor \), \( t \in \mathbb{N} \). For \( n = 100 \), find the smallest positive integer \( k \) such that there exists an integer \( h \) satisfying \( |k\sqrt{2} - h| < \frac{1}{100} \). | [
"70"
] | [
"Using the real numbers \\( x_t = t\\sqrt{2} - \\lfloor t\\sqrt{2} \\rfloor \\), \\( t \\in \\mathbb{N} \\). For \\( n = 100 \\), find the smallest positive integer \\( k \\) such that there exists an integer \\( h \\) satisfying \\( |k\\sqrt{2} - h| < \\frac{1}{100} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3bf7a446b9473b7cbc898a82d675a9e5b0dcf3aead853e755197536960f7754e | Using the reduction formula \( 2nI_{n+1} = (2n-1)I_{n} \), find a formula for \( I_{n} \) if \( n = m + \frac{1}{2} \) where \( m \) is a positive integer. | [
"\\( \\frac{2^{m} (m-1)!}{(2m-1)!!} \\)"
] | [
"Using the reduction formula \\( 2nI_{n+1} = (2n-1)I_{n} \\), find a formula for \\( I_{n} \\) if \\( n = m + \\frac{1}{2} \\) where \\( m \\) is a positive integer."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e8ee0f7b832811f5357936a51a6f468bd89bf32772c281443516162293360da6 | Using the reduction formula \( 2nI_{n+1} = (2n-1)I_{n} \), find a formula for \( I_{n} \) if \( n \) is a positive integer. | [
"\\( I_{n} = \\frac{(2n-3)!!}{(2n-2)!!} \\cdot \\frac{\\pi}{2} \\)"
] | [
"Using the reduction formula \\( 2nI_{n+1} = (2n-1)I_{n} \\), find a formula for \\( I_{n} \\) if \\( n \\) is a positive integer."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bdc58313bb7465ccd3fa00386cf0c2380fe2ac2f7a75373ea56266202d87fd37 | Using the reduction formula, find \( I_{\frac{5}{2}} \), where \( I_{n} = \int \frac{x^{n}}{\sqrt{1+x}} \, dx \). | [
"\\frac{(8x^2 - 10x + 15)\\sqrt{x(1 + x)}}{24} - \\frac{5}{8}\\ln\\left(\\sqrt{x} + \\sqrt{1 + x}\\right) + C"
] | [
"Using the reduction formula, find \\( I_{\\frac{5}{2}} \\), where \\( I_{n} = \\int \\frac{x^{n}}{\\sqrt{1+x}} \\, dx \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-69e2aa83e1e15634a2952c6daa4921db4fbdf2587666824693ad4510f21544b3 | Using the relation \( t^k \sqrt{1-t^2} = t^k (1-t^2) \frac{1}{\sqrt{1-t^2}} \), find \( a_{2n} \) such that \( \lim_{c \rightarrow +0} (I_{2n+2, c} - a_{2n} I_{2n, c}) = 0 \). | [
"\\( \\frac{2n+1}{2n+2} \\)"
] | [
"Using the relation \\( t^k \\sqrt{1-t^2} = t^k (1-t^2) \\frac{1}{\\sqrt{1-t^2}} \\), find \\( a_{2n} \\) such that \\( \\lim_{c \\rightarrow +0} (I_{2n+2, c} - a_{2n} I_{2n, c}) = 0 \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f83624560ac3d89daf8a29bcc3940386e890fd859d32cb65631beb2a47505287 | Using the residue theorem, compute the integral \(\int_{-\infty}^{+\infty} \frac{1}{x^6 - x} \, dx\). | [
"\\(0\\)"
] | [
"Using the residue theorem, compute the integral \\(\\int_{-\\infty}^{+\\infty} \\frac{1}{x^6 - x} \\, dx\\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-e5b965f8bceaa360fad2da939be547211fbbc9198466af905ccc4ec52377c076 | Using the residue theorem, solve the integral $\int_0^{\infty} \frac{dx}{(x^2+1)^2}$. | [
"\\(\\frac{\\pi}{2}\\)"
] | [
"Using the residue theorem, solve the integral $\\int_0^{\\infty} \\frac{dx}{(x^2+1)^2}$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3da87dd9a1abc78a3635a84f7adb76e45d488063d8d629f49828ebf77b06ec06 | Using the result established in Problem 238(a) derive the general formula for the solutions of the cubic equation $$x^3+px+q=0.$$ | [
"The expression $a^3+b^3+c^3-3abc$ is divisible by $a+b+c$"
] | [
"Using the result established in Problem 238(a) derive the general formula for the solutions of the cubic equation $$x^3+px+q=0.$$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ced58b35fd48b1c12688d2f548ef8512ef5eb517d308a2666d952aa3c5170560 | Using the result from Problem 1 or by any other means, express \( x \) as a rational function of \( x^3 \) and \( x + \frac{1}{x} \). | [
"\\( x = \\frac{x^3 + \\left(x + \\frac{1}{x}\\right)}{\\left(x + \\frac{1}{x}\\right)^2 - 1} \\)"
] | [
"Using the result from Problem 1 or by any other means, express \\( x \\) as a rational function of \\( x^3 \\) and \\( x + \\frac{1}{x} \\)."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 12
} | 25 | 0 | 0 |
omr-ws-e13c545d13a17239665603b8683a83d382efbab64ee1a2debfdf711fd376927e | Using the result from Problem 1, find \( w \) and \( C \) such that \[ \frac{\frac{B_{2n}}{(2n)!} - \frac{C}{w^n}}{\frac{C}{w^n}} \to 0 \text{ as } n \to \infty, \] where \( B_{2n} \) are Bernoulli numbers. | [
"2"
] | [
"Using the result from Problem 1, find \\( w \\) and \\( C \\) such that \n\\[ \\frac{\\frac{B_{2n}}{(2n)!} - \\frac{C}{w^n}}{\\frac{C}{w^n}} \\to 0 \\text{ as } n \\to \\infty, \\]\nwhere \\( B_{2n} \\) are Bernoulli numbers."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-3907e89e0d6b248739705a621ce1535c427df7eb48a60f54898e66275871dc60 | Using the result from Problem 2, compute the integrals: \[ \int_{-\infty}^{\infty} e^{-t x^2} \, dx \quad \text{and} \quad \int_{-\infty}^{\infty} x^2 \, e^{-t x^2} \, dx \quad (t > 0). \] | [
"\\(\\frac{\\sqrt{\\pi}}{\\sqrt{t}}\\) and \\(\\frac{\\sqrt{\\pi}}{2t^{3/2}}\\)"
] | [
"Using the result from Problem 2, compute the integrals:\n\\[\n\\int_{-\\infty}^{\\infty} e^{-t x^2} \\, dx \\quad \\text{and} \\quad \\int_{-\\infty}^{\\infty} x^2 \\, e^{-t x^2} \\, dx \\quad (t > 0).\n\\]"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 29
} | 29 | 0 | 0 |
omr-ws-e1ea68785fa6db1c7c41359df3b9cd2063c2d55cdffc54b08c60867c7ee32e5c | Using the result from part (a), find \( \lim_{n \to \infty} x_n \). | [
"\\(\\sqrt{5}\\)"
] | [
"Using the result from part (a), find \\( \\lim_{n \\to \\infty} x_n \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9ea6bde00ad6a41ca8ffd7195b4b3e6eb1f2ac650765d1dee8c40055be6df55d | Using the result from part (a), find \(\lim_{n \to \infty} x_n\). | [
"\\(\\sqrt{5}\\)"
] | [
"Using the result from part (a), find \\(\\lim_{n \\to \\infty} x_n\\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f814bfa57b28b943ec682c53e7a13bf903be9a87d4b8717e71ae2fd61de6cd4a | Using the result from part (a), find all positive integers \( m \) for which \[ p_{m+1}^2 < p_1 p_2 \cdots p_m. \] | [
"\\( m \\ge 4 \\)"
] | [
"Using the result from part (a), find all positive integers \\( m \\) for which \n\\[ p_{m+1}^2 < p_1 p_2 \\cdots p_m. \\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-9faedbdd723c7c7e5aa37fd761ead0101f54caf38cfe79ba691eac8f18940057 | Using the result from part (a), find the general solution of \[ \sin x - \sin 3x + \sin 5x - \ldots + (-1)^{n+1} \sin (2n-1)x = 0. \] | [
"\\(x = \\frac{k\\pi}{2n}\\) for integer \\(k\\)."
] | [
"Using the result from part (a), find the general solution of\n\\[\n\\sin x - \\sin 3x + \\sin 5x - \\ldots + (-1)^{n+1} \\sin (2n-1)x = 0.\n\\]"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-89639f237952a3ec84c50d88dc8e0a0743edfc872630eb38f97ec308f2945fae | Using the result from part (a), find the three roots of the equation \( x^3 - 6x + 2 = 0 \). Give your answers to 4 decimal places. | [
"\\( 2, 0.7321, -2.7321 \\)"
] | [
"Using the result from part (a), find the three roots of the equation \\( x^3 - 6x + 2 = 0 \\). Give your answers to 4 decimal places."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bbb0e5790ec4da5b34be5ea6f51d928d0497535e28715279c8ab8be0d63f6d35 | Using the result from part a), find \( E(X | X^2) \). | [
"\\(0\\)"
] | [
"Using the result from part a), find \\( E(X | X^2) \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-36853dbd40d287f6551222e15e77f2f910f389a12420b337a8822a98e96c19db | Using the result from part c), find \( E(|X| | X^2) \) and \( E(T(X) | X^2) \) where \( T(X) = 1_{\{X > 0\}} - 1_{\{X < 0\}} \). | [
"\\( E(|X| | X^2) = |X| \\) and \\( E(T(X) | X^2) = 0 \\)"
] | [
"Using the result from part c), find \\( E(|X| | X^2) \\) and \\( E(T(X) | X^2) \\) where \\( T(X) = 1_{\\{X > 0\\}} - 1_{\\{X < 0\\}} \\)."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 21
} | 21 | 0 | 0 |
omr-ws-5e02b8df463d24f95cfb41dbc02f1037ef38c131a24cdcf5b2a3e01ceeda5e4b | Using the result from the previous problem, find $\int{\frac{x^{3}}{x^{2}+2}}\,dx$ without performing any further division. | [
"\\(\\frac{x^2}{2} - \\ln(x^2 + 2) + C\\)"
] | [
"Using the result from the previous problem, find $\\int{\\frac{x^{3}}{x^{2}+2}}\\,dx$ without performing any further division."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-b1f382643ee9826874a622eab3451e9a351448fb0f046605981ef840cdd268ae | Using the result of problem 2, determine the least number of weights required to weigh any integral number of pounds up to 80 pounds if one is allowed to put weights in both pans of a balance. | [
"5"
] | [
"Using the result of problem 2, determine the least number of weights required to weigh any integral number of pounds up to 80 pounds if one is allowed to put weights in both pans of a balance."
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-52f3aa7f9f7e7960bdb8cd94174072248480f7cfe59ee9055b16a4aab989b5ea | Using the results from Problem 1, find the generating function for the sequence $u_{n} = n^{2}$. Simplify your answer. | [
"\\(\\frac{x(1+x)}{(1-x)^3}\\)"
] | [
"Using the results from Problem 1, find the generating function for the sequence $u_{n} = n^{2}$. Simplify your answer."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-2b0f150ca6c9123622ef821be2bce510b69d545b1c34ae5359af7d366cb72019 | Using the results from Problems 1 and 2, let \( R \) be any ring. Suppose that \( f(X) \in R[X] \) has a repeated root, in the sense that \( f(X) = g(X)(X-r)^2 \) for some \( r \in R \) and \( g(X) \in R[X] \). Calculate the value of \( D[f(X)] \) at \( r \). | [
"0"
] | [
"Using the results from Problems 1 and 2, let \\( R \\) be any ring. Suppose that \\( f(X) \\in R[X] \\) has a repeated root, in the sense that \\( f(X) = g(X)(X-r)^2 \\) for some \\( r \\in R \\) and \\( g(X) \\in R[X] \\). Calculate the value of \\( D[f(X)] \\) at \\( r \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d893471ae876cc8ae3778e4c8d4cbd738c8a6d3f150f55b597408846175d43f2 | Using the results from part (i): (a) Express $\frac{300^{3} + 1}{301}$ as the product of two factors (neither of which is $1$). (b) Express $\frac{7^{49} + 1}{7^{7} + 1}$ as the product of two factors (neither of which is $1$), each written in terms of various powers of $7$ which you should not attempt to calculate exp... | [
"(a) $271 \\times 331$; (b) $(7^{21} + 7^{18} + 3 \\cdot 7^{14} + 7^{11} + 3 \\cdot 7^7 + 7^4 + 1) \\times (7^{21} - 7^{18} + 3 \\cdot 7^{14} - 7^{11} + 3 \\cdot 7^7 - 7^4 + 1)$"
] | [
"Using the results from part (i):\n\n(a) Express $\\frac{300^{3} + 1}{301}$ as the product of two factors (neither of which is $1$).\n\n(b) Express $\\frac{7^{49} + 1}{7^{7} + 1}$ as the product of two factors (neither of which is $1$), each written in terms of various powers of $7$ which you should not attempt to ... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c856f2e9c5d9f98e1403c81a39b65e8d6c7ccef83fcff287770f55c4d9e759b0 | Using the results from the previous problems and the values of \(p\) and \(r\) from the quartic equation \(x^{4} + 3x^{2} - 6x + 10 = 0\), solve the quartic equation. | [
"\\(x = -1+2i, -1-2i, 1+i, 1-i\\)"
] | [
"Using the results from the previous problems and the values of \\(p\\) and \\(r\\) from the quartic equation \\(x^{4} + 3x^{2} - 6x + 10 = 0\\), solve the quartic equation."
] | [
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-f2fb6db61d20b4a96e1356415ef71e359023a4740e22635ba8aed1808242bba9 | Using the roots of the polynomial \(x^4 + x^3 + x^2 + x + 1\), find \(\sin 18^\circ\) and \(\cos 18^\circ\). No other problems are identified in the post. | [
"\\(\\sin 18^\\circ = \\frac{\\sqrt{5} - 1}{4}\\), \\(\\cos 18^\\circ = \\frac{\\sqrt{10 + 2\\sqrt{5}}}{4}\\)"
] | [
"Using the roots of the polynomial \\(x^4 + x^3 + x^2 + x + 1\\), find \\(\\sin 18^\\circ\\) and \\(\\cos 18^\\circ\\).\n\nNo other problems are identified in the post."
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 22
} | 22 | 0 | 0 |
omr-ws-b86c127837f903162d76241677f792215e3f866f6c0357825d6bcacbda72df75 | Using the rule \( y = 96.64 \log x - 460.34 \), calculate how much time it would take to reach $65,000 and $85,000. | [
"4.78 and 16.04"
] | [
"Using the rule \\( y = 96.64 \\log x - 460.34 \\), calculate how much time it would take to reach $65,000 and $85,000."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 22
} | 23 | 0 | 0 |
omr-ws-8a1177057592acaf4eb6d4dc9e1a286b6cc4ae1817d92fdf4357fdc9549bb08c | Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\int -4\sin \frac{x}{2}\sin x\sin \frac{11}{2}xdx$. | [
"$-\\frac{1}{4}\\cos 4x+\\frac{1}{5}\\cos 5x+\\frac{1}{6}\\cos 6x -\\frac{1}{7}\\cos 7x+c$"
] | [
"Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\\int -4\\sin \\frac{x}{2}\\sin x\\sin \\frac{11}{2}xdx$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b5b087dc66d59bfe25bc796f5a385f8938acca746bb5cbd9d97b548f35ec05e7 | Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\int 2\sqrt 2\cos \alpha \sin \left(\frac{\pi }{4}+2\alpha \right)d\alpha $. | [
"$\\sin \\alpha -\\cos \\alpha +\\frac{1}{3}\\sin 3\\alpha -\\frac{1}{3}\\cos 3\\alpha +c$"
] | [
"Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\\int 2\\sqrt 2\\cos \\alpha \\sin \\left(\\frac{\\pi }{4}+2\\alpha \\right)d\\alpha $."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4edcf22daf891372916d7093103cb9bc63f5c523795f5e882663dd6a50746faf | Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\int 4\cos \frac{x}{2}\cos x\sin \frac{21}{2}xdx$. | [
"$-\\frac{1}{12}\\cos 12x-\\frac{1}{10}\\cos 10x-\\frac{1}{9}\\cos 9x -\\frac{1}{11}\\cos 11x+c$"
] | [
"Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\\int 4\\cos \\frac{x}{2}\\cos x\\sin \\frac{21}{2}xdx$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e1cf568d4841fb71852b3b4f88fe473b163bfd1d8c046c2a3e07f0bd39c14704 | Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\int [\cos ^2 (45^\circ-x)\cos ^2 (60^\circ+x)-\cos 75^\circ \sin (75^\circ-2x)]dx$. | [
"$-\\frac{1}{2}\\cos 2x+c$"
] | [
"Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\\int [\\cos ^2 (45^\\circ-x)\\cos ^2 (60^\\circ+x)-\\cos 75^\\circ \\sin (75^\\circ-2x)]dx$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ee7e65680e045de78381b32628dbabe939ea7c4fcfb668eec80f253a3bd43540 | Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\int \cot\left(\frac{3}{4}\pi -2x\right)\cos 4xdx$. | [
"$-\\left(x+\\frac{1}{\\pi }\\cos 4x\\right)+c$"
] | [
"Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\\int \\cot\\left(\\frac{3}{4}\\pi -2x\\right)\\cos 4xdx$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c496b339cd967ae1670377db0986dec03eb0abc1a038fba455f91258e9e70fc4 | Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\int \frac{(x^{2/m}-9x^{2/n})(x^{(1-m)/m}-3x^{(1-n)/n})} {(x^{1/m}+3x^{1/m})^2-12x^{(m+n)/(mn)}}dx$. | [
"$mx^{1/m}+3nx^{1/n}+c$"
] | [
"Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\\int \\frac{(x^{2/m}-9x^{2/n})(x^{(1-m)/m}-3x^{(1-n)/n})} {(x^{1/m}+3x^{1/m})^2-12x^{(m+n)/(mn)}}dx$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-01f3c677aa34ddca920fd4b31f3bed443cf76a84e09eb642721ba8f5d2c87a50 | Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\int \frac{2\sqrt {1+\frac{1}{4}\left(\sqrt {\frac{1}{t}}-\sqrt t\right)^2}} {\sqrt {1+\frac{1}{4}\left(\sqrt {\frac{1}{t}}-\sqrt t\right)} -\frac{1}{2}\left(\sqrt {\frac{1}{t}}-\sqrt t\right)}dt$... | [
"$t+\\ln|t|+c$"
] | [
"Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\\int \\frac{2\\sqrt {1+\\frac{1}{4}\\left(\\sqrt {\\frac{1}{t}}-\\sqrt t\\right)^2}} {\\sqrt {1+\\frac{1}{4}\\left(\\sqrt {\\frac{1}{t}}-\\sqrt t\\right)} -\\frac{1}{2}\\left(\\sqrt {\\frac{... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-413ca727543f132f0534b6cdfc8d1f6f071d423274d561b5007e4e9db3e2417b | Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\int \frac{\cos 4x+1}{\cot x-\tan x}dx$. | [
"$-\\frac{1}{8}\\cos 4x+c$"
] | [
"Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\\int \\frac{\\cos 4x+1}{\\cot x-\\tan x}dx$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9f119f102785e315682697e8205e048f86d142cbf6a43318b29b90eadc99971a | Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\int \frac{\frac{x^4+5x^3+15x-9}{x^6+3x^4}+\frac{9}{x^4}} {(x^3-4x+3x^2-12)/x^5}dx$. | [
"$x+2\\ln |x-2|+c$"
] | [
"Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\\int \\frac{\\frac{x^4+5x^3+15x-9}{x^6+3x^4}+\\frac{9}{x^4}} {(x^3-4x+3x^2-12)/x^5}dx$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d0b77216f2efecd7056eb0c3a31ba0db6759f2da032039c8cac0ed4415a1084f | Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\int \frac{\left((1-x^2)^{-1/2}+1+\frac{1}{(1-x^2)^{-1/2}-1}\right)^{-2}} {\left(2-x^2-2\sqrt {1-x^2}\right)}dx$. | [
"$x-\\frac{x^3}{3}+c$"
] | [
"Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\\int \\frac{\\left((1-x^2)^{-1/2}+1+\\frac{1}{(1-x^2)^{-1/2}-1}\\right)^{-2}} {\\left(2-x^2-2\\sqrt {1-x^2}\\right)}dx$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bde7d03643cbb3fec41e1d44d146514b8f6ec0ec94fcfc922db324bca5b3f6e1 | Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\int \frac{\left(2-x+4x^2+\frac{5x^2-6x+3}{x-1}\right)} {2x+1+2x/(x-1)}dx$. | [
"$x^2-x+c$"
] | [
"Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\\int \\frac{\\left(2-x+4x^2+\\frac{5x^2-6x+3}{x-1}\\right)} {2x+1+2x/(x-1)}dx$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-05ad55ab295642c240b006ac96ee24a6ad23eff542f30795bfff1f249036626b | Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\int \frac{\left(\sqrt x+1\right)\left(x^2-\sqrt x\right)}{x\sqrt x+x+\sqrt x}dx$. | [
"$\\frac{x^2}{2}-x+c$"
] | [
"Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\\int \\frac{\\left(\\sqrt x+1\\right)\\left(x^2-\\sqrt x\\right)}{x\\sqrt x+x+\\sqrt x}dx$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3298abcb650a04f3d2bc0ccfad955b87e1159b5e9efa92c8e4bc9ed7bb5d4b86 | Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\int \frac{\left(\sqrt x+2\right)\left(\frac{2}{\sqrt x}-1\right) -\left(\sqrt x-2\right)\left(\frac{2}{\sqrt x}+1\right)} {\left(2-\sqrt {x-2}\right):\left(\sqrt {\frac{2}{x}+1} -\frac{1}{\sqrt x... | [
"$2x-8\\ln |x|+c$"
] | [
"Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\\int \\frac{\\left(\\sqrt x+2\\right)\\left(\\frac{2}{\\sqrt x}-1\\right) -\\left(\\sqrt x-2\\right)\\left(\\frac{2}{\\sqrt x}+1\\right)} {\\left(2-\\sqrt {x-2}\\right):\\left(\\sqrt {\\frac... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0ca756a85d3a09d5f1d0e2f8ad82db978bd48f265d8baa59a326cee70d588c2f | Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\int \frac{\sin 2x+\sin 5x-\sin 3x}{\cos x+1-2\sin ^2 2x}dx$. | [
"$-2\\cos x+c$"
] | [
"Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\\int \\frac{\\sin 2x+\\sin 5x-\\sin 3x}{\\cos x+1-2\\sin ^2 2x}dx$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3bb5112c9820ad2dc708cc926756d8ef8daa83b54f2de0d3a89e204752d5d3b3 | Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\int \frac{\sqrt {1-x^2}+1}{\sqrt {1-x}+1/\sqrt {1+x}}dx$. | [
"$\\frac{2(1+x)^{3/2}}{3}+c$"
] | [
"Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\\int \\frac{\\sqrt {1-x^2}+1}{\\sqrt {1-x}+1/\\sqrt {1+x}}dx$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7a92e0ce03ada04a9141739f5f96d8bffb33e062471eca1de88b35e3d854d11b | Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\int \frac{\sqrt {1-x^2}-1}{x} \left(\frac{1-x}{\sqrt {1-x^2}+x-1}+\frac{\sqrt {1+x}}{\sqrt {1+x}-\sqrt {1-x}} \right)dx$. | [
"$-x+c$"
] | [
"Using the rules of integration and the table of indefinite integrals, find the antiderivatives of the following functions. $\\int \\frac{\\sqrt {1-x^2}-1}{x} \\left(\\frac{1-x}{\\sqrt {1-x^2}+x-1}+\\frac{\\sqrt {1+x}}{\\sqrt {1+x}-\\sqrt {1-x}} \\right)dx$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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