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-7f53c2fdbbf49ef66cb22ada451a6b05f41284cdfcdd82245cdda50dac80509f | What is the sum of the greatest common factor of 3 and 6 and the least common multiple of 3 and 6? | [
"9"
] | [
"What is the sum of the greatest common factor of 3 and 6 and the least common multiple of 3 and 6?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8c58c42e52ee61f27e3ce302dc95777c2c004d48f4f492344959c85ba2ba8084 | What is the sum of the infinite series $1 + 3 + 9 + 27 + 81 + 243 + \ldots$? | [
"\\infty"
] | [
"What is the sum of the infinite series $1 + 3 + 9 + 27 + 81 + 243 + \\ldots$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-252be93af277995950e44ad45f9df907f2167f496da81ad250557e07ad0fd10e | What is the sum of the infinite series $\frac{2}{3}, \frac{4}{9}, \frac{6}{27}, \frac{8}{81}, \ldots$? | [
"$\\frac{3}{2}$"
] | [
"What is the sum of the infinite series $\\frac{2}{3}, \\frac{4}{9}, \\frac{6}{27}, \\frac{8}{81}, \\ldots$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-9d0616581e44ad8421903add47e3439489adfc5f65eca9e33fd59c1aed4b1909 | What is the sum of the infinite series $\frac{3}{2}+\frac{5}{4}+\frac{7}{8}+\frac{9}{16}+\frac{11}{32}+\ldots$? | [
"5"
] | [
"What is the sum of the infinite series $\\frac{3}{2}+\\frac{5}{4}+\\frac{7}{8}+\\frac{9}{16}+\\frac{11}{32}+\\ldots$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c640f5caed60e916ab13d16a797ea9aa227b4a3c38dd5522a12474612d186320 | What is the sum of the infinite series \(1 + \frac{2}{3} + \frac{6}{3^2} + \frac{10}{3^3} + \frac{14}{3^4} + \cdots\)? | [
"3"
] | [
"What is the sum of the infinite series \\(1 + \\frac{2}{3} + \\frac{6}{3^2} + \\frac{10}{3^3} + \\frac{14}{3^4} + \\cdots\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-cfc23f89d7c84c6ceff4a9532f7b3f40e4430bb4829d5f15a3b9b6a5945d7063 | What is the sum of the infinite series: \(\frac{1}{1} - \frac{3}{3} + \frac{5}{9} - \frac{7}{27} + \frac{9}{81} - \frac{11}{243} + \cdots\)? | [
"\\(\\frac{3}{8}\\)"
] | [
"What is the sum of the infinite series: \\(\\frac{1}{1} - \\frac{3}{3} + \\frac{5}{9} - \\frac{7}{27} + \\frac{9}{81} - \\frac{11}{243} + \\cdots\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-a1ae05bdac220db016485d09a360f49f476dc1bda62420dd81f70730112e3dfe | What is the sum of the integer solutions of the equation $(x^2 + 13x + 21)^{(x^2-6x+8)} = 1$? | [
"\\(-5\\)"
] | [
"What is the sum of the integer solutions of the equation $(x^2 + 13x + 21)^{(x^2-6x+8)} = 1$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-484c698d58539170d5c2a57a49615af9a7b7021c44b62153ae5c597d108a73ca | What is the sum of the integers from $-30$ to $50$, inclusive? | [
"810"
] | [
"What is the sum of the integers from $-30$ to $50$, inclusive?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c911253928102d657e06677a21ab8f04b084232315384486124b6f7516681d76 | What is the sum of the integers that are both greater than 3 and less than 12? | [
"60"
] | [
"What is the sum of the integers that are both greater than 3 and less than 12?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6018f33dda9896c615a8c31eba16f2665cf66f1565a9fefc5e13cd72bae1656c | What is the sum of the internal angles of a triangle? | [
"180"
] | [
"What is the sum of the internal angles of a triangle?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-f2ea39a33daa52720023d926498061e565b67435850eb4c928ff4e03106fde74 | What is the sum of the irrational roots of the equation $4(x^{2}+3x+1)^{2} -5(x^{2}+3x+1)-x=2$? | [
"-2.5"
] | [
"What is the sum of the irrational roots of the equation $4(x^{2}+3x+1)^{2} -5(x^{2}+3x+1)-x=2$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 25
} | 47 | 0 | 0 |
omr-ws-64ebf4e56a8e4488be3f459e954daf0b3c7e361a69078fd8b8bb6100250c32e1 | What is the sum of the largest and smallest prime factors of 990? | [
"13"
] | [
"What is the sum of the largest and smallest prime factors of 990?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-77dcbb7d957d1303686a16af07d90688e52bf16de79c140c86ef6794ae542cad | What is the sum of the last 3 digits in the expansion of $7^{2013}$? | [
"11"
] | [
"What is the sum of the last 3 digits in the expansion of $7^{2013}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a0b2ca39f2356d6bfe15f95a69bc7ba1e972d5f5fe9f932b5633fb2b20fd9c4e | What is the sum of the last five digits of the quantity 8 * 240 * 25 * (1/2) * 50? | [
"0"
] | [
"What is the sum of the last five digits of the quantity 8 * 240 * 25 * (1/2) * 50?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8c18542798ef29497dc4d75c5bbcf4fe8228e5fbee48ffb6c2868a07b0847d2c | What is the sum of the last two digits in the decimal representation of $9^{2004}$? | [
"61"
] | [
"What is the sum of the last two digits in the decimal representation of $9^{2004}$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 6,
"tir": 10
} | 30 | 0 | 0 |
omr-ws-9023317af3e91b3f30190ff567c70144d132bdbbdcf666a654464f38999b7673 | What is the sum of the last two digits of $8^{25} + 12^{25}?$ | [
"0"
] | [
"What is the sum of the last two digits of $8^{25} + 12^{25}?$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b35b2e4fbdc5a38351828623f79ba322ca9b9f7dbc2b11633bcb00de6e41d9dc | What is the sum of the last two digits of this portion of the Fibonacci Factorial Series: $1!+1!+2!+3!+5!+8!+13!+21!+34!+55!+89!$? | [
"5"
] | [
"What is the sum of the last two digits of this portion of the Fibonacci Factorial Series: $1!+1!+2!+3!+5!+8!+13!+21!+34!+55!+89!$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0dbe53a87976c2b51ab8be3e6b1f2f634eb8d2acd60e83a595b5176808304bd0 | What is the sum of the least and greatest positive integers with exactly one repeated digit? | [
"99876543221"
] | [
"What is the sum of the least and greatest positive integers with exactly one repeated digit?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 8,
"tir": 3
} | 20 | 0 | 0 |
omr-ws-fb156a1786dce742dcb1d0c872026724376786b8f87453cd98754b3d4b57d66a | What is the sum of the measures of the angles \( A \) and \( C \) in a quadrilateral \( ABCD \) drawn on checkered paper? | [
"135"
] | [
"What is the sum of the measures of the angles \\( A \\) and \\( C \\) in a quadrilateral \\( ABCD \\) drawn on checkered paper?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9bac99bca9fd3715bb08ea9616592bbace8dae50954c7c16b378f83a7cb45243 | What is the sum of the medians of all nonempty subsets of $\{1, 2, \ldots, 9\}$? | [
"2555"
] | [
"What is the sum of the medians of all nonempty subsets of $\\{1, 2, \\ldots, 9\\}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-8e0f77cd68bc687445d039d655789444eb8538ee25f559e71afe042612e94916 | What is the sum of the moduli of the coefficients in the expansion of $(2a - 3b + 5c + 6d - 11e + 3f + 2g)^{10}$? | [
"\\(4^{10}\\)"
] | [
"What is the sum of the moduli of the coefficients in the expansion of $(2a - 3b + 5c + 6d - 11e + 3f + 2g)^{10}$?"
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5dfe3d27eafc04e5e15f01ccb6c884d0b2fb6808f60170029a8b366c5b8a38a5 | What is the sum of the multiples of seven between 0 and 1000? | [
"71071"
] | [
"What is the sum of the multiples of seven between 0 and 1000?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-f8aba99a175bb6687e7359165f14dba320f2129b4a360f2edf9823308e94a035 | What is the sum of the numbers $10111_{-2}$ and $11010_{-2}$ in base $(-2)$? | [
"\\(1101001_{(-2)}\\)"
] | [
"What is the sum of the numbers $10111_{-2}$ and $11010_{-2}$ in base $(-2)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 10
} | 25 | 0 | 0 |
omr-ws-4548980038b6db7426f42ed34b0d9a8262b5cac80c241e2717d8a817a92fa4e8 | What is the sum of the numbers in row \( n \) of Pascal's triangle? | [
"2^n"
] | [
"What is the sum of the numbers in row \\( n \\) of Pascal's triangle?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0588cd0fb3e9bbf0a119241b56e7243ac09b3d1a542df9062d14fe4767c4055a | What is the sum of the numbers in the $19^{th}$ row of the given triangular arrangement? A) $133829$ B) $136819$ C) $135809$ D) $134829$ E) None of these | [
"None of these\n\nExplanation: The forum discussions indicate that the sum of the numbers in the nth row is \\( n^3 \\). For the 19th row, the sum would be \\( 19^3 \\). However, Post 6 points out that none of the provided options (A, B, C, D) are correct since they do not match \\( 19^3 \\). Therefore, the correct... | [
"What is the sum of the numbers in the $19^{th}$ row of the given triangular arrangement?\nA) $133829$\nB) $136819$\nC) $135809$\nD) $134829$\nE) None of these"
] | [
"c4_high_school_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-b149e2f7a509f020f7445bf63a76cf2a7f021e7dd6d073e094e6871ec83ea5ea | What is the sum of the numbers less than $1000$ that are relatively prime to $1000$? | [
"200000"
] | [
"What is the sum of the numbers less than $1000$ that are relatively prime to $1000$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-144830188c0c9aacbfe6d6e3d7197c8f4948f8da5d39abdd02552f435b05d1f5 | What is the sum of the numerators in the partial fraction decomposition of \(\dfrac{11x^2 - 25x + 1}{x^3 - x^2 - 8x + 12}\)? | [
"10"
] | [
"What is the sum of the numerators in the partial fraction decomposition of \\(\\dfrac{11x^2 - 25x + 1}{x^3 - x^2 - 8x + 12}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 6
} | 22 | 0 | 0 |
omr-ws-ca65ef99849de4013a0de772aed6918611e7d983ecd215219414168043b145bb | What is the sum of the numerical coefficients in the expansion of the binomial $(a+b)^8$? | [
"64"
] | [
"What is the sum of the numerical coefficients in the expansion of the binomial $(a+b)^8$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2f1f491b8e2156da83f73752e51699daaf7767270eb82997efd387cbbad9dfe3 | What is the sum of the odd integers from 11 through 39, inclusive? | [
"375"
] | [
"What is the sum of the odd integers from 11 through 39, inclusive?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1a9e13a0284a80092be19e4571202afff775ab224ef3735e5ee26551a83f183b | What is the sum of the positive divisors of \( N \) if the prime factorization of \( N \) is \( (p_1)^{a_1}(p_2)^{a_2} \cdots (p_k)^{a_k} \)? | [
"\\prod_{i=1}^{k} \\frac{p_i^{a_i + 1} - 1}{p_i - 1}"
] | [
"What is the sum of the positive divisors of \\( N \\) if the prime factorization of \\( N \\) is \\( (p_1)^{a_1}(p_2)^{a_2} \\cdots (p_k)^{a_k} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5323d65301097c1c5aeb989fd3e99e6da013c8e6c69cba034988d83b9852d045 | What is the sum of the positive even divisors of \(3600\)? | [
"12090"
] | [
"What is the sum of the positive even divisors of \\(3600\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 13
} | 28 | 0 | 0 |
omr-ws-efd3172d74763b7b74b559c9ff61f6016ad06def0d6bd5d171e9aa99c20ee4fc | What is the sum of the positive factors of 24? | [
"60"
] | [
"What is the sum of the positive factors of 24?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ae2f57da1f703e1c618aeee3d35b80bc6af0a39e500b18fdb4d8aa6f001d8fee | What is the sum of the positive factors of 48? | [
"124"
] | [
"What is the sum of the positive factors of 48?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-94c3533659a2f1e0d58cce581d71a2e8919e048f308130c7c119d79b7643c03a | What is the sum of the positive integer divisors of 23? | [
"24"
] | [
"What is the sum of the positive integer divisors of 23?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-93d5582b5a71060397a107c98cf2b9a7ffb16f772b50d5356efc6fc22f84d166 | What is the sum of the positive odd divisors of $60$? | [
"24"
] | [
"What is the sum of the positive odd divisors of $60$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-10a1c0800524fdf7fe57bc1a3a7d4547e668eb2c2e4b6d7639e3615df8cfe999 | What is the sum of the positive whole number divisors of 210? | [
"576"
] | [
"What is the sum of the positive whole number divisors of 210?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ea84d791f94816badba85a55bf543144e3c24229bb0aa720f5c258e8e31620d4 | What is the sum of the prime factors of 91? | [
"20"
] | [
"What is the sum of the prime factors of 91?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1d75604d9c17a600bef1fed698098f485c541d31370632e5f85dd88a706c5c7a | What is the sum of the prime numbers between 10 and 20? | [
"60"
] | [
"What is the sum of the prime numbers between 10 and 20?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-22a171ce4ad14b517a034f12276e30f06db3eac2c8bc7c215ef4b42a606aebf2 | What is the sum of the prime numbers less than 10? | [
"17"
] | [
"What is the sum of the prime numbers less than 10?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-98fcde528bb463573d4834b382c363ba45217ce92d90604ea38b1940f9c31726 | What is the sum of the products between all numbers from the set \(\{2, 4, 6, \ldots, 2000\}\) and the set \(\left\{\frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots, \frac{1}{2001}\right\}\)? The sum can be expressed as: \[ \left(\frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \cdots + \frac{1}{2001}\right) \times (2 + 4 + 6 + ... | [
"\\(1001000 \\cdot \\left[H(2001) - 1\\right]\\)"
] | [
"What is the sum of the products between all numbers from the set \\(\\{2, 4, 6, \\ldots, 2000\\}\\) and the set \\(\\left\\{\\frac{1}{2}, \\frac{1}{3}, \\frac{1}{4}, \\ldots, \\frac{1}{2001}\\right\\}\\)?\n\nThe sum can be expressed as:\n\\[\n\\left(\\frac{1}{2} + \\frac{1}{3} + \\frac{1}{4} + \\cdots + \\frac{1}{... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 10
} | 25 | 0 | 0 |
omr-ws-f79ace7ac501fa5a9b67a172625e669fd467cd8dd913d539407f7be7a77ba83d | What is the sum of the proper divisors of 256? | [
"255"
] | [
"What is the sum of the proper divisors of 256?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-140f2b83c37ea2cb43ec65e59faff2f32f78a5c76f43f86823fc953a5c1cd5e7 | What is the sum of the rational terms in the expansion of $(3^\frac{1}{4} + 4^\frac{1}{3})^{12}$? | [
"283"
] | [
"What is the sum of the rational terms in the expansion of $(3^\\frac{1}{4} + 4^\\frac{1}{3})^{12}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-641d0af7590acb6f1814637a22b30ae3d960541b31aea395f0394dc6b2bdb89f | What is the sum of the rational terms in the expansion of $\left(\sqrt{2} + 3\sqrt{3} + 6\sqrt{5}\right)^{10}$? | [
"2768697856"
] | [
"What is the sum of the rational terms in the expansion of $\\left(\\sqrt{2} + 3\\sqrt{3} + 6\\sqrt{5}\\right)^{10}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-b61d77f85cb0c9f852f5ddfe6719f25e42e4401cca95b7c4bef0a11b6027ea53 | What is the sum of the real roots of \((2 + (2 + (2 + x)^2)^2)^2 = 2000\)? | [
"-4"
] | [
"What is the sum of the real roots of \\((2 + (2 + (2 + x)^2)^2)^2 = 2000\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 10
} | 26 | 0 | 0 |
omr-ws-4911477535f1c6fbdcd15df71f6db96f114b5d4fbc1b01ddb88a710313043e63 | What is the sum of the real roots of \((x\sqrt{x})^x = x^{x\sqrt{x}}\)? | [
"\\(\\frac{18}{7}\\)"
] | [
"What is the sum of the real roots of \\((x\\sqrt{x})^x = x^{x\\sqrt{x}}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-37b2f2203c74d55eb49a235673f332c962238410ce3942a8ecd2f5e65c54fcfc | What is the sum of the real roots of the cubic polynomial $x^3 - x^2 - x - 2$? | [
"2"
] | [
"What is the sum of the real roots of the cubic polynomial $x^3 - x^2 - x - 2$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1deb44674a6f9fa82fb479a840ebbcd4f618543bb3344ffbc59dd7dce924106b | What is the sum of the real roots of the equation \(x^4 - 8x^3 + 13x^2 - 24x + 9 = 0\)? | [
"7"
] | [
"What is the sum of the real roots of the equation \\(x^4 - 8x^3 + 13x^2 - 24x + 9 = 0\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-1d2211960eb0247c7581802fcec5e6d88344652132029ec459baeda1244fcd4b | What is the sum of the real roots of the equation \[ \left ( x + 1\right )\left ( x + \dfrac 14\right )\left ( x + \dfrac 12\right )\left ( x + \dfrac 34\right )= \dfrac {45}{32}? \] | [
"\\(-\\frac{5}{4}\\)"
] | [
"What is the sum of the real roots of the equation\n\\[\n\\left ( x + 1\\right )\\left ( x + \\dfrac 14\\right )\\left ( x + \\dfrac 12\\right )\\left ( x + \\dfrac 34\\right )= \\dfrac {45}{32}?\n\\]"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-2bf8dbf4050e4be0c04908e712e29886b716bfe07a4b23eebe6884d76f59b1fc | What is the sum of the real solutions of \((x^2 + 4x + 4)^{x+2} = 1\)? Let \(0^0\) be defined as 1. | [
"-6"
] | [
"What is the sum of the real solutions of \\((x^2 + 4x + 4)^{x+2} = 1\\)? Let \\(0^0\\) be defined as 1."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 4
} | 19 | 0 | 0 |
omr-ws-89aadec7ff4742471c81f0b3aaf6c222fb7943d363fbf7cd3da213be3b82d2b8 | What is the sum of the real solutions of the equation \( x^6 - 14x^4 - 40x^3 - 14x^2 + 1 = 0 \)? | [
"5"
] | [
"What is the sum of the real solutions of the equation \\( x^6 - 14x^4 - 40x^3 - 14x^2 + 1 = 0 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-c1a4410682b980dca41906816688e79e3d7db3148ee97b3bead56cf4703e7f3e | What is the sum of the reciprocals of all the factors of 1092? | [
"\\dfrac{112}{39}"
] | [
"What is the sum of the reciprocals of all the factors of 1092?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7e25fccb4c3b38c93e865715662cd7518dad094427fab933cc493c3efd801c08 | What is the sum of the reciprocals of every number in row \( n \) of Pascal's triangle (where the top row is row 0) in terms of \( n \)? | [
"\\(\\frac{n+1}{2^{n+1}} \\sum_{k=1}^{n+1} \\frac{2^k}{k}\\)"
] | [
"What is the sum of the reciprocals of every number in row \\( n \\) of Pascal's triangle (where the top row is row 0) in terms of \\( n \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 8
} | 8 | 0 | 0 |
omr-ws-ecd8d705924e54d840ff24ccf302602d080963bd064f82d584a7bdce694f6f00 | What is the sum of the reciprocals of the 5 solutions to the equation $5x^5 + 4x^4 - 3x^3 + 2x^2 + x - 1 = 0$? | [
"1"
] | [
"What is the sum of the reciprocals of the 5 solutions to the equation $5x^5 + 4x^4 - 3x^3 + 2x^2 + x - 1 = 0$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-25a38c4eb55bdd165381e6907f2d86fabe17ad3be67082b8c9eb2312d08b3833 | What is the sum of the reciprocals of the divisors of 360? | [
"\\(\\frac{1170}{360}\\)",
"\\(\\frac{13}{4}\\)"
] | [
"What is the sum of the reciprocals of the divisors of 360?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 62
} | 94 | 0 | 0 |
omr-ws-f84b42ff0819f80bf5e8401863b75710eb97421326af93727f6196e3d4ae92ef | What is the sum of the reciprocals of the natural-number factors of 6? | [
"2"
] | [
"What is the sum of the reciprocals of the natural-number factors of 6?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-46f4d4318c7e731cd52204d3b94bf4d48663997d8f86b89bf2da4cfe7f97354f | What is the sum of the reciprocals of the roots of the equation $\frac{2003}{2004}x + 1 + \frac{1}{x} = 0$? | [
"-1",
"\\(-1\\)"
] | [
"What is the sum of the reciprocals of the roots of the equation $\\frac{2003}{2004}x + 1 + \\frac{1}{x} = 0$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 30
} | 30 | 0 | 0 |
omr-ws-1003881ecae4d3e82a62916617652f0b1c47a598087d80252cf21d041f56edf2 | What is the sum of the reciprocals of the roots of the equation \(\frac{2003x}{2004} + 1 + \frac{1}{x} = 0\)? | [
"-1."
] | [
"What is the sum of the reciprocals of the roots of the equation \\(\\frac{2003x}{2004} + 1 + \\frac{1}{x} = 0\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-4f020b3b5523eca7796d22a18e57925fb143871e09f821ef557098b73e989633 | What is the sum of the reciprocals of the roots of the polynomial \(x^4 - 11x^3 + 28x^2 + 36x - 144\)? | [
"\\(\\frac{1}{4}\\)"
] | [
"What is the sum of the reciprocals of the roots of the polynomial \\(x^4 - 11x^3 + 28x^2 + 36x - 144\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-d46abb621ac65ee941de24a9bbdcf6f63253f3f748cc7b77d8dc9bb49a82a737 | What is the sum of the roots of $\tan(x) = \dfrac{-4 \pm 2\sqrt{7}}{3}$ in the interval $0 < x < 2\pi$? Express your answer in terms of $\pi$. | [
"\\(2 \\tan^{-1} (-8/7) + 4\\pi\\)"
] | [
"What is the sum of the roots of $\\tan(x) = \\dfrac{-4 \\pm 2\\sqrt{7}}{3}$ in the interval $0 < x < 2\\pi$? Express your answer in terms of $\\pi$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-032f2f5e71f60bcc3d4ceb997330dda532a99fbdd7b0190011ea82fde9429d2f | What is the sum of the roots of $x^2 - 4x + 3 = 0$? | [
"4"
] | [
"What is the sum of the roots of $x^2 - 4x + 3 = 0$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-03e94462a279028a18d3c6118fb3f0ae183436a30d7215aaa5f687a35ff262ee | What is the sum of the roots of the equation $(x - 5)^2 = 9$? | [
"10"
] | [
"What is the sum of the roots of the equation $(x - 5)^2 = 9$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-db96544db6928fa5d507a3d31c5b6953a2b23f42edf2be2578e6590efe801103 | What is the sum of the roots of the equation $x^2 - 7[x] + 5 = 0$, where $[x]$ denotes the greatest integer function? | [
"0"
] | [
"What is the sum of the roots of the equation $x^2 - 7[x] + 5 = 0$, where $[x]$ denotes the greatest integer function?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-91fee286808a78427489a5a8d6f24162de04bfeecf4d11205d0c34741c39c037 | What is the sum of the roots of the equation $x^4 + 2x^3 - x + \frac{1}{4}$? | [
"-1"
] | [
"What is the sum of the roots of the equation $x^4 + 2x^3 - x + \\frac{1}{4}$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f8e4812db686fb2103c97dfb2e1b961327358c1160296d4257b00c090b1bdcf2 | What is the sum of the roots of the equation $z^2 = \frac{-3}{\sqrt{5}}|z| + 4i$, where $z$ is a complex number? | [
"0"
] | [
"What is the sum of the roots of the equation $z^2 = \\frac{-3}{\\sqrt{5}}|z| + 4i$, where $z$ is a complex number?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-13f1860f6898ec51e53bc1c064601017b7fb38fed8b370a23e3063e4b382f364 | What is the sum of the roots of the equation \( |x^2 - 3x + 2| + |x^2 + 2x - 3| = 11 \)? | [
"\\dfrac{5\\sqrt{97} - 19}{20}"
] | [
"What is the sum of the roots of the equation \\( |x^2 - 3x + 2| + |x^2 + 2x - 3| = 11 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-31a41edcced30ac23c56c77bf46294860ae939729a465a053edcb15c40f16ae4 | What is the sum of the roots of the equation \((2 - x)^{2012} - x^{2012} = 0\)? | [
"4024"
] | [
"What is the sum of the roots of the equation \\((2 - x)^{2012} - x^{2012} = 0\\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-00926e8c3454733ff5f5b29d5ab288b55f74798660b7a8ebda2e66c1d82acd12 | What is the sum of the roots of the polynomial $x^5 + 10x^4 + 40x^3 + 80x^2 + 80x + 32$? | [
"-10"
] | [
"What is the sum of the roots of the polynomial $x^5 + 10x^4 + 40x^3 + 80x^2 + 80x + 32$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-87840ba5c47529889ff5d0befc4d8bb29f0d4a7c1f1138fef5335c795dd5b1ca | What is the sum of the sequence $8, 2, 2, 1, \frac{1}{2}, \frac{1}{2}, \frac{1}{8}, \frac{1}{4}, \frac{1}{32}, \frac{1}{8}, \ldots$? | [
"\\(\\frac{44}{3}\\)"
] | [
"What is the sum of the sequence $8, 2, 2, 1, \\frac{1}{2}, \\frac{1}{2}, \\frac{1}{8}, \\frac{1}{4}, \\frac{1}{32}, \\frac{1}{8}, \\ldots$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-747ef89cd49ffe54c48cf90cb2e6da31b68815f20ecf783f057deea98b7ff903 | What is the sum of the sequence \(1^4 + 2^4 + \cdots + n^4\)? | [
"\\(\\frac{n(n+1)(2n+1)(3n^2+3n-1)}{30}\\)"
] | [
"What is the sum of the sequence \\(1^4 + 2^4 + \\cdots + n^4\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-23fa06fbf18cca468b3ecdb102c9833ff7a4832ae6dc546b7dc71b8c38af6d98 | What is the sum of the sequence \(1^5 + 2^5 + 3^5 + 4^5 + \cdots + n^5\)? | [
"\\dfrac{n^2 (n + 1)^2 (2n^2 + 2n - 1)}{12}"
] | [
"What is the sum of the sequence \\(1^5 + 2^5 + 3^5 + 4^5 + \\cdots + n^5\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d298c6ae835caf6db44c1421d80d0bd6f84a80f746766137626174e27e5be34c | What is the sum of the series $1 + 2^2 + 3^3 + \cdots + n^n$? | [
"\\sum_{k=1}^n k^k"
] | [
"What is the sum of the series $1 + 2^2 + 3^3 + \\cdots + n^n$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f287d383cbd04954bce9e7dda8efff9dcc7b5a997531fcd4544489721326cedb | What is the sum of the series $1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{n}$? | [
"\\(H_n = \\Psi(n+1) + \\gamma\\)"
] | [
"What is the sum of the series $1 + \\frac{1}{2} + \\frac{1}{3} + \\ldots + \\frac{1}{n}$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 4
} | 5 | 0 | 0 |
omr-ws-3899d0d8bcfeee6a5380f962c8b1accef6c843ce25bcbe027b62ee1411e805c7 | What is the sum of the series $12 + 24 + 48 + 60 + 72 + 84 + \ldots$? | [
"\\infty"
] | [
"What is the sum of the series $12 + 24 + 48 + 60 + 72 + 84 + \\ldots$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-614b12e95710c9dff694eac4855962905079531e27e55a2d8013392c78f0cf90 | What is the sum of the series $\frac{1}{1} + \frac{1 \cdot 2}{1 \cdot 3} + \frac{1 \cdot 2 \cdot 3}{1 \cdot 3 \cdot 5} + \frac{1 \cdot 2 \cdot 3 \cdot 4}{1 \cdot 3 \cdot 5 \cdot 7} + \cdots$? | [
"1 + \\dfrac{\\pi}{2}"
] | [
"What is the sum of the series $\\frac{1}{1} + \\frac{1 \\cdot 2}{1 \\cdot 3} + \\frac{1 \\cdot 2 \\cdot 3}{1 \\cdot 3 \\cdot 5} + \\frac{1 \\cdot 2 \\cdot 3 \\cdot 4}{1 \\cdot 3 \\cdot 5 \\cdot 7} + \\cdots$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-aa29a1f8d89ac2d7c22786365acc7313221f0567de3c4cad212fcda78daf8e2a | What is the sum of the series $\sum_{n=1}^{\infty} \cot^{-1} \left(n^2 + \frac{3}{4}\right)$? | [
"\\(\\frac{\\pi}{2} - \\tan^{-1}\\left(\\frac{1}{2}\\right)\\)"
] | [
"What is the sum of the series $\\sum_{n=1}^{\\infty} \\cot^{-1} \\left(n^2 + \\frac{3}{4}\\right)$?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 3,
"tir": 27
} | 45 | 0 | 0 |
omr-ws-1a9938e06aac1fec10c045ae7d833c3ba0614a28620480a0a604c4a742965565 | What is the sum of the series $x + x^2 + \ldots + x^n$? | [
"\\(\\frac{x^{n+1}-x}{x-1}\\) for \\(x \\neq 1\\) and \\(n\\) for \\(x = 1\\)."
] | [
"What is the sum of the series $x + x^2 + \\ldots + x^n$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 21
} | 23 | 0 | 0 |
omr-ws-c12c8dfe7cd00739590c598836f4e2f0cf331cb4ab2574b22c8f61010df42c8f | What is the sum of the series \( \frac{1}{2} + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \frac{1}{11} + \frac{1}{13} + \cdots \), where the denominators are all prime numbers? | [
"\\(\\infty\\)"
] | [
"What is the sum of the series \\( \\frac{1}{2} + \\frac{1}{3} + \\frac{1}{5} + \\frac{1}{7} + \\frac{1}{11} + \\frac{1}{13} + \\cdots \\), where the denominators are all prime numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 23
} | 39 | 0 | 0 |
omr-ws-134761d8cabbdf0a7d9681af474d36f042db5c85dd9e37b53f34c9ab27ebd282 | What is the sum of the series \( \sum_{n=1}^{\infty} \frac{(\tau * u)(n)}{n^s} \)? | [
"\\( \\zeta(s)^3 \\)"
] | [
"What is the sum of the series \\( \\sum_{n=1}^{\\infty} \\frac{(\\tau * u)(n)}{n^s} \\)?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 16
} | 32 | 0 | 0 |
omr-ws-ba8164d7c89369198712809435ab76de3a70a32da5266641bfa5660799dceb9b | What is the sum of the series \(0 + \frac{1}{2} + \frac{2}{3} + \frac{3}{4} + \cdots + \frac{7}{8}\)? | [
"\\(\\frac{1479}{280}\\)"
] | [
"What is the sum of the series \\(0 + \\frac{1}{2} + \\frac{2}{3} + \\frac{3}{4} + \\cdots + \\frac{7}{8}\\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 30
} | 52 | 0 | 0 |
omr-ws-ee032ca028fd2a82147778f00a2d8f7ec0537143977f8216654ebb2177ceaadd | What is the sum of the series \(1 + \frac{1}{3} + \frac{1}{5} + \frac{1}{7} + \cdots + \frac{1}{2n-1}\)? | [
"\\(\\Psi\\left(n + \\frac{1}{2}\\right) + \\frac{\\gamma}{2} + \\ln 2\\)"
] | [
"What is the sum of the series \\(1 + \\frac{1}{3} + \\frac{1}{5} + \\frac{1}{7} + \\cdots + \\frac{1}{2n-1}\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-192b1f046d3b039bc9018407eda0b5de1286fd4e40b101eabdb4edc4ea1fe1a7 | What is the sum of the series \(1, (1+2), (1+2^2), \ldots, (1+2+ \cdots + 2^{n-1})\)? | [
"\\(2^{n+1} - n - 2\\)"
] | [
"What is the sum of the series \\(1, (1+2), (1+2^2), \\ldots, (1+2+ \\cdots + 2^{n-1})\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 8,
"tir": 5
} | 26 | 0 | 0 |
omr-ws-c85dc42b6008a45917084fa2c1eec27a9eb41e887b25018601181164bdcfddcb | What is the sum of the series \(5 + 55 + 555 + 5555 + \ldots\) for any coefficient (in this case 5)? | [
"\\(\\frac{5 \\cdot 10^{n+1} - 45n - 50}{81}\\)"
] | [
"What is the sum of the series \\(5 + 55 + 555 + 5555 + \\ldots\\) for any coefficient (in this case 5)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-4e7d88943f7b7de379b27c99e61901a59b4587a933f4b408495133bff49f3102 | What is the sum of the series \(\frac{1}{3^2 + 1} + \frac{1}{4^2 + 2} + \frac{1}{5^2 + 3} + \cdots\)? | [
"\\dfrac{13}{36}"
] | [
"What is the sum of the series \\(\\frac{1}{3^2 + 1} + \\frac{1}{4^2 + 2} + \\frac{1}{5^2 + 3} + \\cdots\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e1b2547a42d15aa7652aae31289181d2781d6edc529855680a2a1af51bf2aa44 | What is the sum of the seven smallest distinct positive integer multiples of 9? | [
"252"
] | [
"What is the sum of the seven smallest distinct positive integer multiples of 9?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-46d52b11cc8ea64f21cbe29ec7a02f8877d6ca102a1b69ffe1463116da949624 | What is the sum of the six divisors of $2^{48}-1$ that lie between $50$ and $100$? | [
"452"
] | [
"What is the sum of the six divisors of $2^{48}-1$ that lie between $50$ and $100$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 25
} | 41 | 0 | 0 |
omr-ws-3af653a468d6753d2f9b9ab2e87883247e02c5d81381ef840de14be1207ba050 | What is the sum of the six positive integer factors of 18? | [
"39"
] | [
"What is the sum of the six positive integer factors of 18?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a1b676b0a1d580e2098128994c17a60e4038e0215fca15626c91665d93882602 | What is the sum of the sixth powers of the solutions to the equation \(x^6 - 16x^4 + 16x^2 - 1 = 0\)? | [
"6662"
] | [
"What is the sum of the sixth powers of the solutions to the equation \\(x^6 - 16x^4 + 16x^2 - 1 = 0\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 4,
"tir": 3
} | 20 | 0 | 0 |
omr-ws-f2b18815468d258b050e0c3c6eb6d1d8d993f1efed5f6fe737ddf35e488e85e5 | What is the sum of the smallest 100 positive integers not divisible by 2, 3, or 5? | [
"17312"
] | [
"What is the sum of the smallest 100 positive integers not divisible by 2, 3, or 5?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8ae2a71556ad66618a140a513f327d6d6863c60b75c398254f208504f586061f | What is the sum of the solid angles in a tetrahedron? Is it fixed? | [
"\\text{No, the sum of the solid angles in a tetrahedron is not fixed.}"
] | [
"What is the sum of the solid angles in a tetrahedron? Is it fixed?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9183cbd122bf66dd94af73b22aca28ce35d1f394995b1e11cb30aca4e5bc2404 | What is the sum of the solutions of the equation $(3x+5)(2x-9) = 0$? Express your answer as a common fraction. | [
"\\frac{17}{6}"
] | [
"What is the sum of the solutions of the equation $(3x+5)(2x-9) = 0$? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-117a171c86f9fba7e922fc8359fe88f1335b8641d43797421e0c0481f646ecbc | What is the sum of the solutions of the equation \(\sqrt[4]{x} = \frac{12}{7 - \sqrt[4]{x}}\)? | [
"337"
] | [
"What is the sum of the solutions of the equation \\(\\sqrt[4]{x} = \\frac{12}{7 - \\sqrt[4]{x}}\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a7d0624beb8cb19e9f22693afd2a0ca89d61a894316f3ecc2077c1de06cd60f3 | What is the sum of the solutions to $\frac{x+8}{5x+7} = \frac{x+8}{7x+5}$? | [
"-7"
] | [
"What is the sum of the solutions to $\\frac{x+8}{5x+7} = \\frac{x+8}{7x+5}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-e651f23434e91cb961d2c51852807c252751273e154457630bbd6bcb149ee2d3 | What is the sum of the solutions to the equation $$2^x(43-3^x-4^{x+4})=216?$$ | [
"0"
] | [
"What is the sum of the solutions to the equation $$2^x(43-3^x-4^{x+4})=216?$$"
] | [
"aops_c4_high_school_math"
] | [
"genselect"
] | {
"genselect": 8
} | 8 | 0 | 0 |
omr-ws-3d0f322767f35de743d7fec13ebc0a2e97f6f88916ddd7d280272b211d1296a3 | What is the sum of the solutions to the equation \(3^x(43 - 3^x - 3^{x+1}) = 108\)? | [
"3"
] | [
"What is the sum of the solutions to the equation \\(3^x(43 - 3^x - 3^{x+1}) = 108\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e0d7393eff89382bcc6afb15fa1ebfbcd194610460662e26c19b47b2c140bc1c | What is the sum of the squares of the coefficients of $4(x^4 + 3x^2 + 1)$? | [
"176"
] | [
"What is the sum of the squares of the coefficients of $4(x^4 + 3x^2 + 1)$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-510e1cc3a3dcf003618ad745cdd9908877ade5d7e75fbfc0297b7f824e0b8904 | What is the sum of the squares of the lengths of the $\textbf{medians}$ of a triangle whose side lengths are $10,$ $10,$ and $12$? | [
"258"
] | [
"What is the sum of the squares of the lengths of the $\\textbf{medians}$ of a triangle whose side lengths are $10,$ $10,$ and $12$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-3cddee3ec3f788b068b8ae1adea91a17e45a1fb80756dc87bb585841f1897928 | What is the sum of the squares of the real numbers \( x \) for which \( x^2 - 20\lfloor x \rfloor + 19 = 0 \)? | [
"724"
] | [
"What is the sum of the squares of the real numbers \\( x \\) for which \\( x^2 - 20\\lfloor x \\rfloor + 19 = 0 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d5d1ac1218feb3954a4ad5a7b30b413682c2e37a48a1f4b910bf3b92d146aa0c | What is the sum of the squares of the roots of the equation $x^2 - 7 \lfloor x \rfloor + 5 = 0$? | [
"92"
] | [
"What is the sum of the squares of the roots of the equation $x^2 - 7 \\lfloor x \\rfloor + 5 = 0$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 2
} | 17 | 0 | 0 |
omr-ws-1d872ae32eae8d647c83bcaffb1f8019bb1869a2277c7d20170ae5124e63bf0a | What is the sum of the squares of the roots of the equation $x^2 - 7[x] + 5 = 0$, where $x$ is a real number and $[x]$ represents the greatest integer less than or equal to $x$? | [
"39"
] | [
"What is the sum of the squares of the roots of the equation $x^2 - 7[x] + 5 = 0$, where $x$ is a real number and $[x]$ represents the greatest integer less than or equal to $x$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2729548f9d4973940c22c3252e25dc3c18126a6a72ded1efba7e37e7fbfea7cd | What is the sum of the squares of the roots of the equation $x^2 - 7[x] + 5 = 0$? | [
"92"
] | [
"What is the sum of the squares of the roots of the equation $x^2 - 7[x] + 5 = 0$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 32,
"tir": 4
} | 36 | 0 | 0 |
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