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-bedf6b9be47f7ab479a3dc9ff236586b9278e473f54aeb9b8eae53f6716d8da5 | What is the sum of all three-digit numbers with different digits? | [
"355680"
] | [
"What is the sum of all three-digit numbers with different digits?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
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} | 44 | 0 | 0 |
omr-ws-40a96f6c64559b96e5edd72f3bae3b10fbd773ebc2a9dcbdb7955e6609816713 | What is the sum of all two-digit odd numbers whose digits are all greater than $6$? | [
"528"
] | [
"What is the sum of all two-digit odd numbers whose digits are all greater than $6$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 18
} | 33 | 0 | 0 |
omr-ws-629b0f6476cc4056080c4b5bc5f50fa4ad700371c020d54e0943c581171d2d63 | What is the sum of all two-digit odd numbers whose digits are all greater than 6? | [
"528"
] | [
"What is the sum of all two-digit odd numbers whose digits are all greater than 6?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-2c7d7b3936ac6b62baaffcd533791c1d6cd97b80d99eae3614b3f86547795bdd | What is the sum of all two-digit positive integers \( n < 50 \) for which the sum of the squares of the first \( n \) positive integers is not a divisor of \( (2n)! \)? | [
"498"
] | [
"What is the sum of all two-digit positive integers \\( n < 50 \\) for which the sum of the squares of the first \\( n \\) positive integers is not a divisor of \\( (2n)! \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
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"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3aca6bb93633719d53351fce79793170e7fa9e606ee7c6a1065b1a50fc0b2fbc | What is the sum of all two-digit positive integers whose squares end with the digits 01? | [
"199"
] | [
"What is the sum of all two-digit positive integers whose squares end with the digits 01?"
] | [
"MATH_training_set"
] | [
"cot"
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"cot": 16
} | 16 | 0 | 0 |
omr-ws-93d0b91adab9eada2cb4ef682133450c44ac4cfb6ccd7b294a3abfe48ca33e1d | What is the sum of all values of $k$ such that the equation $2x^2-kx+8=0$ has two distinct integer solutions? | [
"0"
] | [
"What is the sum of all values of $k$ such that the equation $2x^2-kx+8=0$ has two distinct integer solutions?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f230ce29502af26a43c5b667589eda53fe3099fbfbb74d948997b27e029c5244 | What is the sum of all values of $x$ for which $\sqrt{(x+3)^{2}} = 7$? | [
"-6"
] | [
"What is the sum of all values of $x$ for which $\\sqrt{(x+3)^{2}} = 7$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-819ab696841d31bd72fb2719bfb23c7bd554991527fd678ed1622ab7b003e841 | What is the sum of all values of $x$ that are solutions to the equation $x^2 = 7x - 12$? | [
"7"
] | [
"What is the sum of all values of $x$ that are solutions to the equation $x^2 = 7x - 12$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-65fc48c7b89c3acd2903d790056b66876d44ae40e0638fdc099ccaa91d20a156 | What is the sum of all values of $y$ for which the expression $\frac{y+6}{y^2-5y+4}$ is undefined? | [
"5"
] | [
"What is the sum of all values of $y$ for which the expression $\\frac{y+6}{y^2-5y+4}$ is undefined?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ee6c6b68fd23bee0ffb89326a9bc5435fd5cc0aeb5f72528d8228c8651bab186 | What is the sum of all values of $z$ for which $f(5z) = 21$ and $f$ is a function defined by $f\left(\frac{x}{5}\right) = x^2 + x + 1$? | [
"-\\dfrac{1}{25}"
] | [
"What is the sum of all values of $z$ for which $f(5z) = 21$ and $f$ is a function defined by $f\\left(\\frac{x}{5}\\right) = x^2 + x + 1$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f187803ba740576ab2b4da6d5d12a3ef4152cbf20896fd72a3a1c2496e969c0c | What is the sum of all values of $z$ such that $z^2=12z-7$? | [
"12"
] | [
"What is the sum of all values of $z$ such that $z^2=12z-7$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-985706fe95bc3f36fd6bde03f543da6a6401bce5473f27d770d549eb52433a50 | What is the sum of cubes of real roots of the equation $x^3 - 2x^2 - x + 1 = 0$? | [
"11"
] | [
"What is the sum of cubes of real roots of the equation $x^3 - 2x^2 - x + 1 = 0$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8348b053bbde55aab6c9b770103909afef1bd270ff395afe692c27669504bb1f | What is the sum of distinct real roots of \( x^5 - 2x^2 - 9x - 6 \)? | [
"1"
] | [
"What is the sum of distinct real roots of \\( x^5 - 2x^2 - 9x - 6 \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b5261045e43a83406c0c63e4d8ac899bc52c6c1ff787c5b4545c7b9bdf227dd1 | What is the sum of distinct remainders when $(2n-1)^{502} + (2n+1)^{502} + (2n+3)^{502}$ is divided by $2012$ where $n$ is a positive integer? | [
"1514"
] | [
"What is the sum of distinct remainders when $(2n-1)^{502} + (2n+1)^{502} + (2n+3)^{502}$ is divided by $2012$ where $n$ is a positive integer?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 10
} | 57 | 0 | 0 |
omr-ws-f494778603b926a51e2846335c3c224d8a7b2344a76b192051e47893f7fe31c8 | What is the sum of distinct remainders when $(2n-1)^{502} + (2n+1)^{502} + (2n+3)^{502}$ is divided by $2012$, where $n$ is a positive integer? | [
"1514"
] | [
"What is the sum of distinct remainders when $(2n-1)^{502} + (2n+1)^{502} + (2n+3)^{502}$ is divided by $2012$, where $n$ is a positive integer?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 11
} | 59 | 0 | 0 |
omr-ws-50ff334c9a61a461010d74651a88beebc7bbaa6ca2288afd95650860ba5603c1 | What is the sum of every other odd number starting from 1 and included in the first one million counting numbers? That is, find the sum \(1 + 5 + 9 + \ldots\) up to the largest term less than or equal to 1,000,000. | [
"124999750000"
] | [
"What is the sum of every other odd number starting from 1 and included in the first one million counting numbers? That is, find the sum \\(1 + 5 + 9 + \\ldots\\) up to the largest term less than or equal to 1,000,000."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-c8f19115a254fe320d7a29beb4fde3963a53d45fdbe010c3a652b711d823c2ab | What is the sum of powers of all triples $(x,y,z)$ where $x,y,z \leq 9$, where the power of the triple is defined as the sum of the largest number and the smallest number of the triple? | [
"7290"
] | [
"What is the sum of powers of all triples $(x,y,z)$ where $x,y,z \\leq 9$, where the power of the triple is defined as the sum of the largest number and the smallest number of the triple?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 1,
"tir": 23
} | 33 | 0 | 0 |
omr-ws-821bac53d378773968cab9f61d69c18771aa1b77f1615d1a2e57cd8581f2a70f | What is the sum of real numbers satisfying the equation $\left \lfloor \frac{6x+5}{8} \right \rfloor = \frac{15x-7}{5}$? | [
"\\(\\frac{19}{15}\\)"
] | [
"What is the sum of real numbers satisfying the equation $\\left \\lfloor \\frac{6x+5}{8} \\right \\rfloor = \\frac{15x-7}{5}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-7691f9803681d04e4960422228318be9693bf234350c13f85f1efa51b64d1092 | What is the sum of real numbers satisfying the equation \(\left \lfloor \frac{6x+5}{8} \right \rfloor = \frac{15x-7}{5}\)? | [
"\\(\\frac{19}{15}\\)"
] | [
"What is the sum of real numbers satisfying the equation \\(\\left \\lfloor \\frac{6x+5}{8} \\right \\rfloor = \\frac{15x-7}{5}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-d7c58146d83a138d4464a34d9003eb29552d01b49a593d13b0575afc764b307d | What is the sum of real roots of the equation $(x\sqrt{x})^x = x^{x\sqrt{x}}$? | [
"\\(\\frac{13}{4}\\)"
] | [
"What is the sum of real roots of the equation $(x\\sqrt{x})^x = x^{x\\sqrt{x}}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-6b652f68e938d4f20dc6ee04c5db77744435fab90b9c6273a52f2c23b3e5ea75 | What is the sum of real roots of the equation $\left(x + 1\right)\left(x + \frac{1}{4}\right)\left(x + \frac{1}{2}\right)\left(x + \frac{3}{4}\right) = \frac{45}{32}$? | [
"-\\dfrac{5}{4}"
] | [
"What is the sum of real roots of the equation $\\left(x + 1\\right)\\left(x + \\frac{1}{4}\\right)\\left(x + \\frac{1}{2}\\right)\\left(x + \\frac{3}{4}\\right) = \\frac{45}{32}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3f497bf7cf6899c609738c61a062ffc9b88c159dd44b92b6611cf736ea6bc70f | What is the sum of real roots of the equation $x^4 - 4x^3 + 5x^2 - 4x + 1 = 0$? | [
"3"
] | [
"What is the sum of real roots of the equation $x^4 - 4x^3 + 5x^2 - 4x + 1 = 0$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-695b30425d25043177874e16e8943280d1ae16090b6ab712a7379502a43ab471 | What is the sum of real roots of the equation \((x + 1)(x + \frac{1}{4})(x + \frac{1}{2})(x + \frac{3}{4}) = \frac{45}{32}\)? | [
"\\(-\\frac{5}{4}\\)"
] | [
"What is the sum of real roots of the equation \\((x + 1)(x + \\frac{1}{4})(x + \\frac{1}{2})(x + \\frac{3}{4}) = \\frac{45}{32}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-9ff2864d3b708bd6c0f18a7f35e2822ccd590f5334ba35029e37b589df15d3b0 | What is the sum of real roots of the equation \(x^4 - 8x^3 + 13x^2 - 24x + 9 = 0\)? | [
"7"
] | [
"What is the sum of real roots of the equation \\(x^4 - 8x^3 + 13x^2 - 24x + 9 = 0\\)?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-5420e7ff18165a650fdf48800f09d1609faa227a9138dfebc78f834f7a83bc3d | What is the sum of the $15$th term and the $19$th term in the sequence $4, 16, 14, 20, \ldots$? | [
"624"
] | [
"What is the sum of the $15$th term and the $19$th term in the sequence $4, 16, 14, 20, \\ldots$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-86ffbee120d2ec0ae2fe3ff29b1ea0a99943bf96cdb5c733dd3283db64d8f352 | What is the sum of the $x$-values that satisfy the equation $5=\frac{x^3-2x^2-8x}{x+2}$? | [
"4"
] | [
"What is the sum of the $x$-values that satisfy the equation $5=\\frac{x^3-2x^2-8x}{x+2}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-53279182a971456260044e4d445529afa965fd81a50105331c5082dfba2229f5 | What is the sum of the 2009 fractions of the form $\frac{2}{n(n+2)}$ if the values of $n$ are the positive integers from 1 through 2009? Express your answer as a decimal to the nearest thousandth. | [
"1.499"
] | [
"What is the sum of the 2009 fractions of the form $\\frac{2}{n(n+2)}$ if the values of $n$ are the positive integers from 1 through 2009? Express your answer as a decimal to the nearest thousandth."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1b61f6e95958398bf8008c171eeb93fb34c22e0a6ebdfa97a63b7b468eeece13 | What is the sum of the 25 smallest perfect squares which are not perfect cubes? | [
"6920"
] | [
"What is the sum of the 25 smallest perfect squares which are not perfect cubes?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 25
} | 40 | 0 | 0 |
omr-ws-97b196c59ce04c1d15e19c830e127779e402a42abc67c16f5f518ec3b944140a | What is the sum of the 3D angles in a tetrahedron? | [
"4\\pi"
] | [
"What is the sum of the 3D angles in a tetrahedron?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3a817e4dcce23bec5bf3ca53c7100d45903d6dcf2605e7b38ad7aecb2e1ca683 | What is the sum of the 4th powers of the lengths of all 21 edges and diagonals of a regular 7-gon inscribed in a unit circle? | [
"147"
] | [
"What is the sum of the 4th powers of the lengths of all 21 edges and diagonals of a regular 7-gon inscribed in a unit circle?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 43,
"genselect": 8,
"tir": 15
} | 66 | 0 | 0 |
omr-ws-7afc15740ab339ee9131e4adfa820b7e6b7d1d6eaec2eb18adfb888e88692bae | What is the sum of the 6 divisors of the integer $2^{48}-1$ that lie between $50$ and $100$? | [
"452"
] | [
"What is the sum of the 6 divisors of the integer $2^{48}-1$ that lie between $50$ and $100$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-2c0185beaa2d986b1dccc564f4e6e01180054d257c59b3a55fc5e2c27888bdf9 | What is the sum of the absolute values of the coefficients of the polynomial \( P(x) = (x-1)\left(x-\frac{1}{2}\right)\left(x-\frac{1}{3}\right) \ldots \left(x-\frac{1}{99}\right) \)? | [
"100"
] | [
"What is the sum of the absolute values of the coefficients of the polynomial \\( P(x) = (x-1)\\left(x-\\frac{1}{2}\\right)\\left(x-\\frac{1}{3}\\right) \\ldots \\left(x-\\frac{1}{99}\\right) \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 11
} | 27 | 0 | 0 |
omr-ws-8888a5859608128f4f8565f77fccbb2939cbf12cb68a3957fa40206ffd2dc98d | What is the sum of the areas of all the circles in a sequence where each circle is tangent to the previous one and has a diameter half that of the previous circle, starting with a circle of radius 1? (Round to the nearest whole number.) | [
"4"
] | [
"What is the sum of the areas of all the circles in a sequence where each circle is tangent to the previous one and has a diameter half that of the previous circle, starting with a circle of radius 1? (Round to the nearest whole number.)"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-2b2e287f7eff1eab18dda07b41e79f9b3f6e0035dc5cc2676258a2a5c0e01c70 | What is the sum of the areas of the three triangles U, G, and K if: - Triangle U has lengths 1 and 4 with an included angle of 90 degrees. - Triangle G has lengths 1 and 2 with an included angle of 90 degrees. - Triangle K has sides \(\sqrt{17}\), \(\sqrt{15}\), and 2. Do this without using a calculator. | [
"4"
] | [
"What is the sum of the areas of the three triangles U, G, and K if:\n- Triangle U has lengths 1 and 4 with an included angle of 90 degrees.\n- Triangle G has lengths 1 and 2 with an included angle of 90 degrees.\n- Triangle K has sides \\(\\sqrt{17}\\), \\(\\sqrt{15}\\), and 2. Do this without using a calculator."... | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-436bf9cad3d66ac68c59c6b155a3ae8aa4836ff5137330c9d2ef56a63850d9ad | What is the sum of the coefficients in the expansion of \((x+y)^k\)? | [
"2^k"
] | [
"What is the sum of the coefficients in the expansion of \\((x+y)^k\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1435f3398e3dcc6435d2f3a63092be3b614a0a07ba74f76b3da86e413487e239 | What is the sum of the coefficients of the polynomial after expanding \((25 - 60x + 22x^2 + 18x^3 - 6x^8)^{2011}\)? | [
"-1"
] | [
"What is the sum of the coefficients of the polynomial after expanding \\((25 - 60x + 22x^2 + 18x^3 - 6x^8)^{2011}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-de2b32f5f85116a5066ebce1834ae0517282eea20276e91c7703f119bcbde021 | What is the sum of the consecutive combinations given by the formula: $$\sum_{i=m}^{n} \binom{k}{i},$$ where $m \le n$, and $k \ge n$? | [
"\\sum_{i=m}^{n} \\binom{k}{i}"
] | [
"What is the sum of the consecutive combinations given by the formula:\n$$\\sum_{i=m}^{n} \\binom{k}{i},$$ where $m \\le n$, and $k \\ge n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 2
} | 12 | 0 | 0 |
omr-ws-d65c18ea38a5c6390a5b1f5b5e103ceac0d523df99a88cdf6a842cbbc881322f | What is the sum of the coordinates of the midpoint of the segment with endpoints $(6, 12)$ and $(0, -6)$? | [
"6"
] | [
"What is the sum of the coordinates of the midpoint of the segment with endpoints $(6, 12)$ and $(0, -6)$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-26104cbb2ab8eb03256513995fc838fcbc58523cb56037bded528189433ea510 | What is the sum of the coordinates of the midpoint of the segment with endpoints $(8, 5)$ and $(2, -1)$? | [
"7"
] | [
"What is the sum of the coordinates of the midpoint of the segment with endpoints $(8, 5)$ and $(2, -1)$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ac8c53cb2cc3e91f9ca28d98b52d81883a96dc10160110f8337bc70d36e4a147 | What is the sum of the cubes of the distinct roots of the polynomial \(3x^4 + 2x^3 - 14x^2 + 3x + 4 = 0\)? | [
"\\(\\frac{-341}{27}\\)"
] | [
"What is the sum of the cubes of the distinct roots of the polynomial \\(3x^4 + 2x^3 - 14x^2 + 3x + 4 = 0\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 28
} | 50 | 0 | 0 |
omr-ws-86c3f90ece869d32153ddf1f1baf7a411fe4457ff71dba8d29676744e54c25dd | What is the sum of the cubes of the first n terms of an arithmetic progression (A.P.)? | [
"\\( na^3 + 1.5a^2dn(n-1) + 0.5ad^2n(n-1)(2n-1) + 0.25d^3n^2(n-1)^2 \\)"
] | [
"What is the sum of the cubes of the first n terms of an arithmetic progression (A.P.)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 17,
"tir": 14
} | 31 | 0 | 0 |
omr-ws-e995c4baafb7e9df36724de81bc94ab4f225d42396398ebae2f0dd37bf834060 | What is the sum of the cubes of the roots of the equation $2x^2\sqrt{6} + \frac{x}{i} - \sqrt{15}=0$? | [
"\\(\\frac{-i\\sqrt{6}+36i\\sqrt{15}}{288}\\)"
] | [
"What is the sum of the cubes of the roots of the equation $2x^2\\sqrt{6} + \\frac{x}{i} - \\sqrt{15}=0$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 6,
"tir": 13
} | 30 | 0 | 0 |
omr-ws-ae8e1ef72c587a0f1b7e3d83276b16edc868ed218e665a8230868dfec67bcc92 | What is the sum of the cubes of the roots of the equation \(x^4 - 2x^3 + x^2 - 5x + 1 = 0\)? | [
"17"
] | [
"What is the sum of the cubes of the roots of the equation \\(x^4 - 2x^3 + x^2 - 5x + 1 = 0\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-41700e24ef22912fec79cd73ed2aec8ffc02cd02d4c3a0ccc29fce73013c308b | What is the sum of the different prime factors of $150280?$ | [
"37"
] | [
"What is the sum of the different prime factors of $150280?$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3f9ef91b599456c2038496f690796cee4be621ad9c8526415f5525c178f54f9f | What is the sum of the digits of \(10^{1998} - 1998\)? | [
"17956"
] | [
"What is the sum of the digits of \\(10^{1998} - 1998\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 18
} | 41 | 0 | 0 |
omr-ws-b9f1928db3207f9d7ed27f098a811006336a11b1bd566f0d92011cf3f8e6af11 | What is the sum of the digits of the base $7$ representation of $777_{10}$? | [
"9"
] | [
"What is the sum of the digits of the base $7$ representation of $777_{10}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d750fd7748e2b65b712baba477311ea072cba63d5623f9c6855935fd0bb741c2 | What is the sum of the digits of the base-2 expression for $222_{10}$? | [
"6"
] | [
"What is the sum of the digits of the base-2 expression for $222_{10}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-957a4889239348bbdf9e69aca775e67be249686704d7bb226d74289bb4e8a795 | What is the sum of the digits of the cube root of the number formed by the digit 9 repeated 2021 times, followed by the digit 0 repeated 2021 times, and finally the digit 9 repeated 2022 times? | [
"18189"
] | [
"What is the sum of the digits of the cube root of the number formed by the digit 9 repeated 2021 times, followed by the digit 0 repeated 2021 times, and finally the digit 9 repeated 2022 times?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 2
} | 4 | 0 | 0 |
omr-ws-12d524d6935fa6b86de41befdb86156294373027c884295d4fa9636ee2de5525 | What is the sum of the digits of the decimal representation of $2^{2005} \times 5^{2007} \times 3$? | [
"12"
] | [
"What is the sum of the digits of the decimal representation of $2^{2005} \\times 5^{2007} \\times 3$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c740ce66d5e24ab22ab5e515e773d413e111689bd43acdfed7f8491c87134136 | What is the sum of the digits of the first $2010$ positive integers? | [
"28068"
] | [
"What is the sum of the digits of the first $2010$ positive integers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 16,
"genselect": 8,
"tir": 26
} | 50 | 0 | 0 |
omr-ws-b15751ea4c43a463a808aca33de42b4e1352151c8752f9ccfcf3faca7b281fdc | What is the sum of the digits of the first \( 2010 \) positive integers? | [
"28068"
] | [
"What is the sum of the digits of the first \\( 2010 \\) positive integers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 27
} | 49 | 0 | 0 |
omr-ws-38490e8243e19bf8f8d860b274593c570f6bcd9264c5ebfa2dc7aa5a767450e3 | What is the sum of the digits of the fourth smallest triangular number that is also a perfect square? | [
"18"
] | [
"What is the sum of the digits of the fourth smallest triangular number that is also a perfect square?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7a3e2f419f75c25ebf4b1323e74bb3212f64b03dcd56b51934d593c3160b5696 | What is the sum of the digits of the integer $10^{2004} - 2004$? | [
"18031"
] | [
"What is the sum of the digits of the integer $10^{2004} - 2004$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-48a158cc427764b11dd0a4e8b893eb7dbb397a2e944dc6e528437821c4a54876 | What is the sum of the digits of the integer equal to \(777777^2 - 222223^2\)? | [
"74"
] | [
"What is the sum of the digits of the integer equal to \\(777777^2 - 222223^2\\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4b7921bbd925b24e9b7b2d742ad272293c82fd70db14d15fd07a86fded5c346e | What is the sum of the digits of the least positive integer \( n > 1 \) such that \( n^3 - n \) is divisible by 1001? | [
"14"
] | [
"What is the sum of the digits of the least positive integer \\( n > 1 \\) such that \\( n^3 - n \\) is divisible by 1001?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 23
} | 55 | 0 | 0 |
omr-ws-a39236a626b0b86dfa740f8cda34ae91a760a7d9b1de28968636fd0c48645cf9 | What is the sum of the digits of the number $123456789101112...10701071$? | [
"1677"
] | [
"What is the sum of the digits of the number $123456789101112...10701071$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-41ee5a54b296557fc2325d32b394a9566f9bd9465a3de7be46d59960869c7c95 | What is the sum of the digits of the number $\sqrt{2004 \times 2002 \times 1998 \times 1996 + 36}$? | [
"48"
] | [
"What is the sum of the digits of the number $\\sqrt{2004 \\times 2002 \\times 1998 \\times 1996 + 36}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-a4267657c2de79d5fa8e584c6db040dc426b01d8029775b168125acb2cb57149 | What is the sum of the digits of the number 201920182017...00020001? | [
"28140"
] | [
"What is the sum of the digits of the number 201920182017...00020001?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 3
} | 13 | 0 | 0 |
omr-ws-47c4eafa8bb4ca3d8a4f10d7b75dd7fed19d45105847208772eb4959a23ad8d5 | What is the sum of the digits of the number formed by multiplying 100 "1"s by 100 "1"s? For example, \(1 \times 1 = 1\) and the sum of the digits is 1. | [
"10000"
] | [
"What is the sum of the digits of the number formed by multiplying 100 \"1\"s by 100 \"1\"s? For example, \\(1 \\times 1 = 1\\) and the sum of the digits is 1."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-919c315350f02c06cba658231f2c89df1c7b4699c30cfe50444a9cf25518b00a | What is the sum of the digits of the number formed by writing all the numbers from 1 to 1,000,000 consecutively? | [
"27000001"
] | [
"What is the sum of the digits of the number formed by writing all the numbers from 1 to 1,000,000 consecutively?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-f5f02ca1f31d622b87921aeb91bdbf079eccbf907f1edd06e058b3edd952e868 | What is the sum of the digits of the positive integer $n$ such that the number $\frac{1! \cdot 2! \cdot 3! \cdots 2019! \cdot 2020!}{n!}$ is a perfect square? | [
"1010"
] | [
"What is the sum of the digits of the positive integer $n$ such that the number $\\frac{1! \\cdot 2! \\cdot 3! \\cdots 2019! \\cdot 2020!}{n!}$ is a perfect square?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-76512b773bfa7476988b3cee577550c7ad2c4730989beae85f5da11a764c7cc3 | What is the sum of the digits of the product $9 \times 99 \times 999 \times 9999 \times \cdots \times 99\ldots9$ (where the last term has $2^n$ nines)? | [
"9 \\times 2^n"
] | [
"What is the sum of the digits of the product $9 \\times 99 \\times 999 \\times 9999 \\times \\cdots \\times 99\\ldots9$ (where the last term has $2^n$ nines)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-4424a9cc148561ada81a0a7254f913907d6c2977722fdaaf35fe1cd44d2a7c74 | What is the sum of the digits of the product of 2002 and the number consisting of 2000 repeated ones (111...111)? | [
"7988"
] | [
"What is the sum of the digits of the product of 2002 and the number consisting of 2000 repeated ones (111...111)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ae1aa62a2f6c355a39e65ef47a4800bba60c1b2d5632b6905055f46a5a23db50 | What is the sum of the digits of the product of a number with 2023 digits, all of which are 6, and a number with 2023 digits, all of which are 9? | [
"18207"
] | [
"What is the sum of the digits of the product of a number with 2023 digits, all of which are 6, and a number with 2023 digits, all of which are 9?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-02b477bd80f7b9bfde21576b35d8a936a2a5ef000890b25b7d14f0696fe119c8 | What is the sum of the digits of the resulting expression $\sqrt{\frac{144^4 + 25^4 + 13^8}{2}}$? | [
"22"
] | [
"What is the sum of the digits of the resulting expression $\\sqrt{\\frac{144^4 + 25^4 + 13^8}{2}}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-4c999c107b16435a89b30ed236e112d1edce9a7b6006de4d2a30e1995e2d53a6 | What is the sum of the digits of the smallest prime that can be written as a sum of 5 distinct primes? | [
"7"
] | [
"What is the sum of the digits of the smallest prime that can be written as a sum of 5 distinct primes?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-01a916de7db94886ca17441ed3a31ad789c7eabed5a6e8464660303dda7ebc25 | What is the sum of the digits of the square of the number $n$, where $n$ is a number consisting of $1990$ digits of $6$? | [
"17910"
] | [
"What is the sum of the digits of the square of the number $n$, where $n$ is a number consisting of $1990$ digits of $6$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 19
} | 34 | 0 | 0 |
omr-ws-93f58b09b2da95ce5ba8d75c2831b440724643d49c7e1c98597e192f1fe313c9 | What is the sum of the digits of the sum S, where S is the sum of all five-digit palindromes of the form $abcba$ with $a$ being non-zero? | [
"18"
] | [
"What is the sum of the digits of the sum S, where S is the sum of all five-digit palindromes of the form $abcba$ with $a$ being non-zero?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 14
} | 30 | 0 | 0 |
omr-ws-a8034e2751b3d232064545e3a5f255aa810962a554a6a7335b22a9690aefb908 | What is the sum of the digits of the sum of the digits of the sum of the digits of \(4444^{4444}\)? | [
"7"
] | [
"What is the sum of the digits of the sum of the digits of the sum of the digits of \\(4444^{4444}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 16
} | 48 | 0 | 0 |
omr-ws-0416a81e93fee3838b544e0329ccbdf377922f4076d98a08a3c669f5495af783 | What is the sum of the distinct prime factors of 315? | [
"15"
] | [
"What is the sum of the distinct prime factors of 315?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b53741b746a89102f07df392f930dfa6693991c6fff7b46613b76d8824feacbb | What is the sum of the distinct real roots of the polynomial $x^5 - 2x^2 - 9x - 6$? | [
"1"
] | [
"What is the sum of the distinct real roots of the polynomial $x^5 - 2x^2 - 9x - 6$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-90f70b1dd53709378a36a2e553e06d3b1b9de0d78a6a392865405f2d4c64393c | What is the sum of the distinct real roots of the polynomial \(x^5 + x^4 - 4x^3 - 7x^2 - 7x - 2\)? | [
"0"
] | [
"What is the sum of the distinct real roots of the polynomial \\(x^5 + x^4 - 4x^3 - 7x^2 - 7x - 2\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 4
} | 20 | 0 | 0 |
omr-ws-a1a6aa6e705c6a0db29a5179db752fe5e61973d8fc265029080c22165a6e4a3e | What is the sum of the distinct values that the expression $x\lfloor x - \frac{x}{10} \rfloor$ takes for $1 \le x \le 100$, $x$ a positive integer? | [
"302325"
] | [
"What is the sum of the distinct values that the expression $x\\lfloor x - \\frac{x}{10} \\rfloor$ takes for $1 \\le x \\le 100$, $x$ a positive integer?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 6
} | 21 | 0 | 0 |
omr-ws-dfd581fde553717db8c8f140efb0d9a276e69e83edfa0f37076d82028e8ceb47 | What is the sum of the divisors of $N = 3^{16} - 1$ that lie between 75 and 85 inclusive? | [
"247"
] | [
"What is the sum of the divisors of $N = 3^{16} - 1$ that lie between 75 and 85 inclusive?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-77156cb6d62275fb2b42077d7899fcfdb7bced976b807f7d3f2f7b446a6c80cd | What is the sum of the eight terms in the arithmetic sequence $-2, 3, \dots, 33$? | [
"124"
] | [
"What is the sum of the eight terms in the arithmetic sequence $-2, 3, \\dots, 33$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6f3c0ec69dc634f9e44d26fe8f8d9c89bb0a10ab1396c059b6c5d3448b468176 | What is the sum of the elements of $S$ if $S$ is the set of all values of $(-1)^{n-k}\binom{n}{k}2^{k}$ such that $n$ and $k$ are nonnegative integers such that $2n-k=100$? | [
"101"
] | [
"What is the sum of the elements of $S$ if $S$ is the set of all values of $(-1)^{n-k}\\binom{n}{k}2^{k}$ such that $n$ and $k$ are nonnegative integers such that $2n-k=100$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 6
} | 20 | 0 | 0 |
omr-ws-b81d6ab9816362333dea941ea933fed6a55222d099e258968249e37c9e7dbb6b | What is the sum of the elements of $S$ if $S$ is the set of all values of $(-1)^{n-k}\binom{n}{k}2^{k}$ such that $n$ and $k$ are nonnegative integers such that $2n-k=500$? | [
"501"
] | [
"What is the sum of the elements of $S$ if $S$ is the set of all values of $(-1)^{n-k}\\binom{n}{k}2^{k}$ such that $n$ and $k$ are nonnegative integers such that $2n-k=500$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 13
} | 13 | 0 | 0 |
omr-ws-52a262ea9690f4dea10f529d1ff974c1aa2d404ad2727c987f690ea5cd60c7c7 | What is the sum of the even, positive integers less than 62? | [
"930"
] | [
"What is the sum of the even, positive integers less than 62?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-24423b4d34641c9b4b45a3d5e6ff59a192daa2403922669025d8b2987c53c43a | What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides $12!$? | [
"8"
] | [
"What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides $12!$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-2c8f4ba20b33bf31fe2707600c3879f7eff7978962311c9fc1846ab622de2076 | What is the sum of the expression $f(x) = 1 + 2x + 3x^2 + \ldots + nx^{n-1}$? | [
"\\( f(x) = \\begin{cases} \n\\frac{n(n+1)}{2} & \\text{if } x = 1 \\\\\n\\frac{1 - (n+1)x^n + nx^{n+1}}{(1-x)^2} & \\text{if } x \\neq 1 \n\\end{cases} \\)"
] | [
"What is the sum of the expression $f(x) = 1 + 2x + 3x^2 + \\ldots + nx^{n-1}$?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 15
} | 15 | 0 | 0 |
omr-ws-3ef3dac6e0ce2333e95f02d9acf772cdb8c40b03c147dfbbf69f6307cc1c0d8a | What is the sum of the factorials from 1! to n!, i.e., \(1! + 2! + 3! + \ldots + n!\)? | [
"\\sum_{k=1}^{n} k!"
] | [
"What is the sum of the factorials from 1! to n!, i.e., \\(1! + 2! + 3! + \\ldots + n!\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-31a1e5fc71d5c514464d4e43cf6489b951d38065eff08ed8044ee8c325f714b2 | What is the sum of the finite series $\frac{2}{1*3}+\frac{2}{2*4}+\frac{2}{3*5}+...+\frac{2}{19*21}$? | [
"\\(\\frac{589}{420}\\)"
] | [
"What is the sum of the finite series $\\frac{2}{1*3}+\\frac{2}{2*4}+\\frac{2}{3*5}+...+\\frac{2}{19*21}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-8d65d4d8d0aad8d125f3a257607d709e60501018c2fe70bb4bdd9664b93cf5b4 | What is the sum of the first $31$ integers that can be written as a sum of distinct powers of $3$? | [
"1936"
] | [
"What is the sum of the first $31$ integers that can be written as a sum of distinct powers of $3$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 16,
"genselect": 4,
"tir": 13
} | 33 | 0 | 0 |
omr-ws-7c2b86f6f49e1d3209023bf699bd64783c7754990a28b371992a1046fcc3d97e | What is the sum of the first 10 odd positive integers? | [
"100"
] | [
"What is the sum of the first 10 odd positive integers?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-37908f0b839e02e952acf1127c5dab945b445b9668a99c75d0b38e65697ea9a6 | What is the sum of the first 100 square numbers? | [
"336683"
] | [
"What is the sum of the first 100 square numbers?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-889a2ce37fe9c678341fcef89bad379ea280a95473c3a58529f6dc6d17041b36 | What is the sum of the first 101 prime numbers? | [
"24680"
] | [
"What is the sum of the first 101 prime numbers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 5
} | 20 | 0 | 0 |
omr-ws-a388a0f41d2fc19525260cc1f569d679868a82fb0bd8c4b9023ab0882ebe92b9 | What is the sum of the first 107 terms in Pascal's Triangle? The answer should be expressed as $a^b+c$, where $a$ is a prime positive integer. | [
"2^{14} + 14"
] | [
"What is the sum of the first 107 terms in Pascal's Triangle? The answer should be expressed as $a^b+c$, where $a$ is a prime positive integer."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-01f4b4e5165d4d2a8f7fcd868508acf9a46ca967f8b8901eb81d3c455fef3b86 | What is the sum of the first 200 terms of an infinite geometric series that has a 5th term of 1 and a 25th term of 3486784401? | [
"\\(\\frac{3^{200} - 1}{162}\\)"
] | [
"What is the sum of the first 200 terms of an infinite geometric series that has a 5th term of 1 and a 25th term of 3486784401?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 24
} | 40 | 0 | 0 |
omr-ws-ddba7b37fc7d51b8db25ca14d28a91c8f716c2b075346e869c2e92f67345e553 | What is the sum of the first 21 odd positive integers? | [
"441"
] | [
"What is the sum of the first 21 odd positive integers?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a538265661092504efae95292a6fa0ca2808900107f944dd6f4f7f5040a69bc2 | What is the sum of the first 40 positive integers whose digits consist only of 1s, 2s, and 3s? | [
"7309"
] | [
"What is the sum of the first 40 positive integers whose digits consist only of 1s, 2s, and 3s?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 21
} | 36 | 0 | 0 |
omr-ws-d9f0e4014bd99959c62c3c94081a4e575b90a408666bcdbd3957dabf5177798b | What is the sum of the first 500 numbers in the sequence where the nth number repeats n times? | [
"10544"
] | [
"What is the sum of the first 500 numbers in the sequence where the nth number repeats n times?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-dd9e502e40c0bb38c9041852e1d7a75b8574bc12e568599dd059a0cacb20d45a | What is the sum of the first 800 terms of the sequence 3, 4, 4, 5, 5, 5, 6, 6, 6, 6, ...? | [
"22940"
] | [
"What is the sum of the first 800 terms of the sequence 3, 4, 4, 5, 5, 5, 6, 6, 6, 6, ...?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 26
} | 74 | 0 | 0 |
omr-ws-2638f17452afb61a5c73f4a7beff2ceb99938aabaf1214c066f38e4ca51550c3 | What is the sum of the first 999 terms of the sequence 1, 2, 3, 6, 7, 14, 15, 30, 31, 62, 63, ...? A term in an even numbered position is twice the previous term, while a term in an odd numbered position is one more than the previous term. | [
"\\(3(2^{500})-1503\\)"
] | [
"What is the sum of the first 999 terms of the sequence 1, 2, 3, 6, 7, 14, 15, 30, 31, 62, 63, ...? A term in an even numbered position is twice the previous term, while a term in an odd numbered position is one more than the previous term."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3e868685096b9386fd761d65ef04c1bf846ba148048da0a49ad67e1723f7f372 | What is the sum of the first ten positive multiples of $13$? | [
"715"
] | [
"What is the sum of the first ten positive multiples of $13$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-77cf59309ce6812738b419674b3056ca4ef9c115ac724e453a9a3f97ff5ba12d | What is the sum of the first twenty powers of 2 (i.e., $2^1 + 2^2 + 2^3 + \ldots + 2^{20}$)? | [
"2097150"
] | [
"What is the sum of the first twenty powers of 2 (i.e., $2^1 + 2^2 + 2^3 + \\ldots + 2^{20}$)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f5f68724803697d45fc7a8f2b3647b68c48b86b2c638f04e8af679215781476b | What is the sum of the first two positive integers \( n \) such that \( 360 \cdot n \) has exactly 72 positive divisors? | [
"72"
] | [
"What is the sum of the first two positive integers \\( n \\) such that \\( 360 \\cdot n \\) has exactly 72 positive divisors?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ef14517cdadec5a9ab6f7ba9dd9d4268b4e696a84d5824d9987861866535a6af | What is the sum of the following series: \(1 + \frac{1}{4} + \frac{1 \cdot 4}{4 \cdot 8} + \frac{1 \cdot 4 \cdot 7}{4 \cdot 8 \cdot 12} + \cdots\)? | [
"\\(\\sqrt[3]{4}\\)"
] | [
"What is the sum of the following series: \\(1 + \\frac{1}{4} + \\frac{1 \\cdot 4}{4 \\cdot 8} + \\frac{1 \\cdot 4 \\cdot 7}{4 \\cdot 8 \\cdot 12} + \\cdots\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 14
} | 29 | 0 | 0 |
omr-ws-b894360b1fa5dece4b4faf0ccccb6dd69f9edebb238050467425cb2d23d92b98 | What is the sum of the geometric series $\sum_{n=0}^{\infty} ax^n$ where $|x| < 1$? | [
"$\\frac{a}{1-x}$"
] | [
"What is the sum of the geometric series $\\sum_{n=0}^{\\infty} ax^n$ where $|x| < 1$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-979b820c5f1fa65ea382c3761af652ec49e90836b29ca37e207ddd050cf60cb1 | What is the sum of the greatest common divisor of $50$ and $5005$ and the least common multiple of $50$ and $5005$? | [
"50055"
] | [
"What is the sum of the greatest common divisor of $50$ and $5005$ and the least common multiple of $50$ and $5005$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b7b77f5b93d76c07f6097464a8edc4ac74b988d005a6b289f5ac4bf331a4fab3 | What is the sum of the greatest common divisor of 30 and 81 and the least common multiple of 36 and 12? | [
"39"
] | [
"What is the sum of the greatest common divisor of 30 and 81 and the least common multiple of 36 and 12?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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