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-96e354db781c47bd671567c43a0a1b456f5c7769e6c73e3c8839821c69fb3e74 | What is the coefficient of $x^2$ when the expression $(1+x)(1+2x)(1+4x) \ldots (1+2^n x)$ is expanded? | [
"\\(\\frac{1}{3}(2^{n+1}-1)(2^{n+1}-2)\\)"
] | [
"What is the coefficient of $x^2$ when the expression $(1+x)(1+2x)(1+4x) \\ldots (1+2^n x)$ is expanded?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 15
} | 62 | 0 | 0 |
omr-ws-39d549c327cb0a39d702881fc850e5d2fbed5db0ac90bbc037a7eac9c5225252 | What is the coefficient of $x^2y^6$ in the expansion of $\left(\frac{3}{5}x-\frac{y}{2}\right)^8$? Express your answer as a common fraction. | [
"\\frac{63}{400}"
] | [
"What is the coefficient of $x^2y^6$ in the expansion of $\\left(\\frac{3}{5}x-\\frac{y}{2}\\right)^8$? Express your answer as a common fraction."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5c6d25ebc4312eb39b194b2d9d047088700a7fe09a562a1dfc054e648d37ef0a | What is the coefficient of $x^3$ in the expansion of $$(x+2\sqrt3)^7?$$ | [
"5040"
] | [
"What is the coefficient of $x^3$ in the expansion of $$(x+2\\sqrt3)^7?$$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-78f9b735c35f3d49f658efdce037e31f850b8ec2abad4f5374c45675041cbb65 | What is the coefficient of $x^3$ when $$x^4-3x^3 + 5x^2-6x + 1$$is multiplied by $$2x^3 - 3x^2 + 4x + 7$$and the like terms are combined? | [
"19"
] | [
"What is the coefficient of $x^3$ when $$x^4-3x^3 + 5x^2-6x + 1$$is multiplied by $$2x^3 - 3x^2 + 4x + 7$$and the like terms are combined?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-41dfa0af731296d8e51642eefc395b1393a65c6e7c6872639650ffdbb56bbf3d | What is the coefficient of $x^3y^{13}$ in $\left(\frac{1}{2}x + y\right)^{17}$? | [
"0"
] | [
"What is the coefficient of $x^3y^{13}$ in $\\left(\\frac{1}{2}x + y\\right)^{17}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-013d8aeabe64603d3e19ab9e0fc29838f6946efe03dbea15d0f5a9f9dc3208f9 | What is the coefficient of $x^4$ in the expansion of $(1-2x^2)^5$? | [
"40"
] | [
"What is the coefficient of $x^4$ in the expansion of $(1-2x^2)^5$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-49164657e426d60cd409ae62167ae9df1cd97997da1f8e12929b8cafe5027e85 | What is the coefficient of $x^4$ in the expansion of $(1-x^5)(1-x^4)(1-x^3)(1-x^2)(1-x)^{-4}$? | [
"20"
] | [
"What is the coefficient of $x^4$ in the expansion of $(1-x^5)(1-x^4)(1-x^3)(1-x^2)(1-x)^{-4}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-6d99657724bc3576ceed7a5c5dc4dd08f7b66b78a56f6544705a4c034a5213c9 | What is the coefficient of $x^4$ in the expansion of $(5x^6 + 10x^5 + 2x^4 + 19x^3 + 2x^2 + 1)^{10000}$? | [
"200000000"
] | [
"What is the coefficient of $x^4$ in the expansion of $(5x^6 + 10x^5 + 2x^4 + 19x^3 + 2x^2 + 1)^{10000}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-d8a058bc73bb584cae29e9dfff1447b24251ed89141223195e8d37a3cfb43c1a | What is the coefficient of $x^4$ in the expansion of $(5x^6+10x^5+2x^4+19x^3+2x^2+1)^{10000}$? | [
"\\(2 \\times 10^8\\)"
] | [
"What is the coefficient of $x^4$ in the expansion of $(5x^6+10x^5+2x^4+19x^3+2x^2+1)^{10000}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-859a65deb24f563a9dd0fab59ab7034d005e74dc1fab71f9bf6ae3af31f4f602 | What is the coefficient of $x^4$ in the expansion of $(5x^{6} + 10x^{5} + 2x^{4} + 19x^{3} + 2x^{2} + 1)^{10000}$? | [
"200000000"
] | [
"What is the coefficient of $x^4$ in the expansion of $(5x^{6} + 10x^{5} + 2x^{4} + 19x^{3} + 2x^{2} + 1)^{10000}$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-5c1f29204d9c5f3c2eb39557084e49cc5167511287f5865687debc3849f54560 | What is the coefficient of $x^5$ in the expansion of $(1 + x + x^2)^9$? | [
"882"
] | [
"What is the coefficient of $x^5$ in the expansion of $(1 + x + x^2)^9$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6ff8868d248f591b9a09bc859aee7f69c59accad97ad51fa801885244a23c29b | What is the coefficient of $x^5$ in the expansion of $(2x+3)^7$? | [
"6048"
] | [
"What is the coefficient of $x^5$ in the expansion of $(2x+3)^7$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9ff1b33a60a4e31f3c1047f99948fc1a67b733fa319b617440f14808c19bef11 | What is the coefficient of $x^6y^8z^2w$ in the expansion of $(2x+2y+6z+11w)^{17}$? | [
"39746658631680"
] | [
"What is the coefficient of $x^6y^8z^2w$ in the expansion of $(2x+2y+6z+11w)^{17}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-968288f7e03b51d5869739fc2ffd67234a6996d14419a0933ea93e9e04231bbc | What is the coefficient of $x^8$ in the expansion of $(x-1)^9$? | [
"-9"
] | [
"What is the coefficient of $x^8$ in the expansion of $(x-1)^9$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9ca22f2b508a7dd872388fc52aa82f35e1bc160650d612d7a3615d3ce167d68c | What is the coefficient of $x^8y^7$ in $(1 + x^2 - 3xy + y^2)^{17}$? | [
"0"
] | [
"What is the coefficient of $x^8y^7$ in $(1 + x^2 - 3xy + y^2)^{17}$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8a5ec6c0b736719884a550c5df823011476432ac6a63743508c523c523dd5ddf | What is the coefficient of $x^9$ in the expansion of $(1+x)(1+x^2)(1+x^3)\cdots(1+x^{100})$? | [
"8"
] | [
"What is the coefficient of $x^9$ in the expansion of $(1+x)(1+x^2)(1+x^3)\\cdots(1+x^{100})$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 12
} | 28 | 0 | 0 |
omr-ws-9452873065cd060a223482f8ace927b206bed11c4b06a0f8d5b51badb3a0b815 | What is the coefficient of $x^j$ in the expansion of $(1 + x + \cdots + x^n)^n$? | [
"\\sum_{k=0}^{\\left\\lfloor \\frac{j}{n+1} \\right\\rfloor} (-1)^k \\binom{n}{k} \\binom{j - k(n+1) + n - 1}{n - 1}"
] | [
"What is the coefficient of $x^j$ in the expansion of $(1 + x + \\cdots + x^n)^n$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b6b16d91f6023f7cc17bad7912aab4d715d68713512a50078124a043b189729c | What is the coefficient of $x^j$ in the expansion of $(1 + x + x^2 + \cdots + x^n)^n$? | [
"\\(\\binom{n+j-1}{j}\\)"
] | [
"What is the coefficient of $x^j$ in the expansion of $(1 + x + x^2 + \\cdots + x^n)^n$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-25fdda6422effc58621a654ca19aaab8a75b2afe240cc59790a7c27692095ead | What is the coefficient of $x^n$ in the expansion of $(1 - x - x^2 - x^3 - x^4 - x^5 - x^6)^{-1}$? | [
"\\(a_n = a_{n-1} + a_{n-2} + a_{n-3} + a_{n-4} + a_{n-5} + a_{n-6}\\) with \\(a_0 = 1\\) and \\(a_n = 0\\) for \\(n < 0\\)."
] | [
"What is the coefficient of $x^n$ in the expansion of $(1 - x - x^2 - x^3 - x^4 - x^5 - x^6)^{-1}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 9,
"tir": 18
} | 27 | 0 | 0 |
omr-ws-f083f46526e4cb47bbdf366378de79313ddf46dd073ecc643d95722b53b76bf7 | What is the coefficient of $x^n$ in the power series form of $\sqrt[3]{1-2x}$? | [
"\\(\\left(\\frac{2}{3}\\right)^n \\cdot \\frac{-1 \\cdot 2 \\cdot 5 \\cdots (3n-4)}{n!}\\) for \\( n > 0 \\) and \\( 1 \\) for \\( n = 0 \\)."
] | [
"What is the coefficient of $x^n$ in the power series form of $\\sqrt[3]{1-2x}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 2,
"tir": 7
} | 18 | 0 | 0 |
omr-ws-0e9fa3bbd75484c8900be95b4721b5cad033eca2b113b95e8788b0032e4939ef | What is the coefficient of $x^{11}$ in the expansion of $(1 + x^3 + x^5)^{24}$? | [
"6072"
] | [
"What is the coefficient of $x^{11}$ in the expansion of $(1 + x^3 + x^5)^{24}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-6631271fe4e82b2c182b202b79468225117f477e5366ef65b42bec457044d989 | What is the coefficient of $x^{11}$ in the expansion of $(1-x)(1+x^2)(1-x^3)(1+x^4) \dots (1-x^9)(1+x^{10})$? | [
"-11"
] | [
"What is the coefficient of $x^{11}$ in the expansion of $(1-x)(1+x^2)(1-x^3)(1+x^4) \\dots (1-x^9)(1+x^{10})$?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c49c2c83036a9c9d32d5ca5a982f60e112a454de9284dc2de93fc46b4256a4a0 | What is the coefficient of $x^{11}$ in the expansion of $(x+{}^{25}C_{1}) (x+{}^{25}C_{2}) (x+{}^{25}C_{3})\cdots(x+{}^{25}C_{12})$? | [
"\\( 2^{24} - 1 \\)"
] | [
"What is the coefficient of $x^{11}$ in the expansion of $(x+{}^{25}C_{1}) (x+{}^{25}C_{2}) (x+{}^{25}C_{3})\\cdots(x+{}^{25}C_{12})$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-d974337c59faaea617d640c37027c0314e6e031fd0103aa008b98dd2cb426c5e | What is the coefficient of $x^{13}$ in the expansion of $(1 + x^3 + x^5)^{24}$? | [
"218592"
] | [
"What is the coefficient of $x^{13}$ in the expansion of $(1 + x^3 + x^5)^{24}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c746e7ac4c1099a91b31ee497465b6ecee3b4e0807756ae2833e3c05a47defc8 | What is the coefficient of $x^{15}$ in the expansion of $(1-x^6)^{10} \cdot (1-x)^{-10}$? | [
"\\(\\binom{24}{15} - 10 \\binom{18}{9} + 45 \\binom{12}{3}\\)"
] | [
"What is the coefficient of $x^{15}$ in the expansion of $(1-x^6)^{10} \\cdot (1-x)^{-10}$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b3fe07843e07a81430bb839f71dd35e6006a9d2b8ff2e359acacf5a36e2d7172 | What is the coefficient of $x^{17}$ in the expansion of $\ln(1 + x + x^2)$? | [
"\\dfrac{1}{17}"
] | [
"What is the coefficient of $x^{17}$ in the expansion of $\\ln(1 + x + x^2)$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-14b42dd16389b2d0059e305b9db568ba3f171b05f7c0ddd0fa6a586cc57f7440 | What is the coefficient of $x^{2009}$ in the expansion of $(x+1)(x+2)(x+3) \cdots (x+2011)$? | [
"2045041554425"
] | [
"What is the coefficient of $x^{2009}$ in the expansion of $(x+1)(x+2)(x+3) \\cdots (x+2011)$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-cd322aed365a8482c15072ebe81ddd062ad77c3291802cf8d33e6d6d52be97fa | What is the coefficient of $x^{2010}$ in the expansion of $(x+1)(x+2)(x+3) \cdots (x+2011)$? | [
"2023066"
] | [
"What is the coefficient of $x^{2010}$ in the expansion of $(x+1)(x+2)(x+3) \\cdots (x+2011)$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 32
} | 48 | 0 | 0 |
omr-ws-1b680739c8d1f7b70f8f0f448ae91bc4fa9e1bdeb05c834c500d1ea0c573385a | What is the coefficient of $x^{2011}$ in the expansion of $(x+1)^7 (x^2+1)^4 (x^4+1)^5 (x^8+1) (x^{16}+1) (x^{32}+1) (x^{64}+1) (x^{128}+1) (x^{256}+1) (x^{512}+1) (x^{1024}+1)$? | [
"\\( 2^{13} \\)"
] | [
"What is the coefficient of $x^{2011}$ in the expansion of $(x+1)^7 (x^2+1)^4 (x^4+1)^5 (x^8+1) (x^{16}+1) (x^{32}+1) (x^{64}+1) (x^{128}+1) (x^{256}+1) (x^{512}+1) (x^{1024}+1)$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 2,
"tir": 7
} | 9 | 0 | 0 |
omr-ws-cebfd6b2511a7772431d834fbd5fef258822d1011f2c4792f2ff61b63bcecdd5 | What is the coefficient of $x^{24}$ in the expansion of $(1 + x^2)^{12}(1 + x^{12})(1 + x^{24})$? | [
"926"
] | [
"What is the coefficient of $x^{24}$ in the expansion of $(1 + x^2)^{12}(1 + x^{12})(1 + x^{24})$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 11
} | 26 | 0 | 0 |
omr-ws-336b6219b9cab6ba114a38fcf825fd6e0075f56e3df40eeb2cfb2a0b24c10009 | What is the coefficient of $x^{296}$ in the expansion of $(x^3 + \sqrt{2}x^2 + \sqrt[3]{3}x + 1)^{100}$? | [
"15684900 + 970200 \\sqrt[3]{3} + 4950 \\sqrt[3]{9} + 9900 \\sqrt{2}"
] | [
"What is the coefficient of $x^{296}$ in the expansion of $(x^3 + \\sqrt{2}x^2 + \\sqrt[3]{3}x + 1)^{100}$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-899fb4a4862a15707cfac40a6ae9206bd42b9fdbbe87f350e5c41762d743edca | What is the coefficient of $x^{33}$ in the generating function $$\frac{x(x^7-1)}{x-1}?$$ | [
"0"
] | [
"What is the coefficient of $x^{33}$ in the generating function \n$$\\frac{x(x^7-1)}{x-1}?$$"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-193f49266dfb04af1c9db9d09e6228fe2f6886d3adc7fb12945fbbd76c27ce2f | What is the coefficient of $x^{43}$ in the product $(x-2)(x-5)\cdots(x-131)$? | [
"\\(-2926\\)"
] | [
"What is the coefficient of $x^{43}$ in the product $(x-2)(x-5)\\cdots(x-131)$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-9f85a9da7b3d764e7397ad36d7306c7fcb53e9da595108d549d6885f62ddcebd | What is the coefficient of $x^{50}$ in the expansion of $(1+2x+3x^2+4x^3+\ldots+101x^{100})(1+x+x^2+\ldots+x^{29})$? | [
"1065"
] | [
"What is the coefficient of $x^{50}$ in the expansion of $(1+2x+3x^2+4x^3+\\ldots+101x^{100})(1+x+x^2+\\ldots+x^{29})$?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b32a39d13c5657c28dec5e8b06e5dc2bc38167dd72b405fd45987e510bb4a3f2 | What is the coefficient of \( x^2 \) in the expansion of \( (1 - x)(1 + 2x)(1 - 3x)(1 + 4x) \cdots (1 + 14x)(1 + 15x) \)? | [
"-588"
] | [
"What is the coefficient of \\( x^2 \\) in the expansion of \\( (1 - x)(1 + 2x)(1 - 3x)(1 + 4x) \\cdots (1 + 14x)(1 + 15x) \\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"tir"
] | {
"cot": 3,
"tir": 18
} | 21 | 0 | 0 |
omr-ws-0a8f32a88220945d9c5dddfecd98efe875aed6a9463a8db2ec6d37459d53b532 | What is the coefficient of \( x^2 \) when \( (...(((x-2)^2-2)^2-2)^2)...\minus{}2)^2 \) is expanded and like terms are combined? | [
"\\(\\frac{4^{n-1}(4^n - 1)}{3}\\)"
] | [
"What is the coefficient of \\( x^2 \\) when \\( (...(((x-2)^2-2)^2-2)^2)...\\minus{}2)^2 \\) is expanded and like terms are combined?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"tir"
] | {
"tir": 9
} | 9 | 0 | 0 |
omr-ws-bd48b3a9b32a983871453f08e1affb93b70b1d2e3bfc8bb846c1f2789468060c | What is the coefficient of \( x^2 \) when the expression \( (1+x)(1+2x)(1+4x) \cdots (1+2^nx) \) is expanded, where \( n = 0, 1, 2, \ldots \)? | [
"\\( 2(2^n - 1)^2 \\)"
] | [
"What is the coefficient of \\( x^2 \\) when the expression \\( (1+x)(1+2x)(1+4x) \\cdots (1+2^nx) \\) is expanded, where \\( n = 0, 1, 2, \\ldots \\)?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 2
} | 2 | 0 | 0 |
omr-ws-a48be69ae2344c622c8eee2989d615081fd95f109003756dab1e0e9b90c748b5 | What is the coefficient of \( x^2 \) when the expression \[ \left ( \cdots \left (\left (\left ( x - 2 \right )^2 - 2 \right )^2 - 2 \right )^2 - \cdots - 2 \right )^2 \] is expanded and like terms are combined? | [
"\\( \\frac{1}{3}4^{n-1}(4^n - 1) \\)"
] | [
"What is the coefficient of \\( x^2 \\) when the expression \n\n\\[ \\left ( \\cdots \\left (\\left (\\left ( x - 2 \\right )^2 - 2 \\right )^2 - 2 \\right )^2 - \\cdots - 2 \\right )^2 \\]\n\nis expanded and like terms are combined?"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 6
} | 6 | 0 | 0 |
omr-ws-e8dda903dba97ac799d24502b3a8335dcb8009c7a06730ec1cff2b3ffa21c72c | What is the coefficient of \( x^5 y^2 z^2 \) in the expansion of \( (x + y + 2z)^9 \)? | [
"756"
] | [
"What is the coefficient of \\( x^5 y^2 z^2 \\) in the expansion of \\( (x + y + 2z)^9 \\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0f395010d178e042c0cf6765d2d88ee8df75e8c9170c9d680c320b82d323a7fb | What is the coefficient of \( x^k \) in the expansion of \( \left( x + \left( r_1 r_2 \cdots r_n \right)^{1/n} \right)^n \)? | [
"\\(\\binom{n}{k} \\left( r_1 r_2 \\cdots r_n \\right)^{(n-k)/n}\\)"
] | [
"What is the coefficient of \\( x^k \\) in the expansion of \\( \\left( x + \\left( r_1 r_2 \\cdots r_n \\right)^{1/n} \\right)^n \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 15
} | 31 | 0 | 0 |
omr-ws-c44528e105dfeb46e2aec4923158739f3c61acd02543969cfd20361a50c4c4b9 | What is the coefficient of \( x^n \) in the expression \( \frac{(1+x)^{2m+1}}{(2+x)^{n+1}} \)? | [
"\\frac{1}{2^{2n+1}} \\sum_{k=0}^n (-1)^{n-k} \\binom{2m+1}{k} \\binom{2n -k}{n} 2^k"
] | [
"What is the coefficient of \\( x^n \\) in the expression \\( \\frac{(1+x)^{2m+1}}{(2+x)^{n+1}} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 5
} | 7 | 0 | 0 |
omr-ws-30ec000f3097a76b118ac3ed49f1de43c4881e1679ab529136a2256e17ff30a0 | What is the coefficient of \( x^n \) in the polynomial \( \prod_{i=0}^n \left( \sum_{k=0}^i \frac{x^k}{k!} \right) \)? | [
"\\sum_{k=0}^n \\frac{1}{k!}"
] | [
"What is the coefficient of \\( x^n \\) in the polynomial \\( \\prod_{i=0}^n \\left( \\sum_{k=0}^i \\frac{x^k}{k!} \\right) \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a785be81e087e34f4b51ae9a8228aa0d8cf9249ef2c19f0f839fe74cda199c04 | What is the coefficient of \( x^{14} \) when \( x^{1951} - 1 \) is divided by \( x^4 + x^3 + 2x^2 + x + 1 \)? | [
"-1"
] | [
"What is the coefficient of \\( x^{14} \\) when \\( x^{1951} - 1 \\) is divided by \\( x^4 + x^3 + 2x^2 + x + 1 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 5,
"tir": 1
} | 16 | 0 | 0 |
omr-ws-3b641b11a952492f91c429d868ee855d7603dd6d1d56068ac75c4c846a5e6490 | What is the coefficient of \( x^{15} \) in the product \( (1-x)(1-2x)(1-2^2x)(1-2^3x) \cdots (1-2^{15}x) \)? | [
"\\( 2^{105} - 2^{121} \\)"
] | [
"What is the coefficient of \\( x^{15} \\) in the product \\( (1-x)(1-2x)(1-2^2x)(1-2^3x) \\cdots (1-2^{15}x) \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 43,
"genselect": 4,
"tir": 18
} | 65 | 0 | 0 |
omr-ws-3966959356e5708e0fe7057b0b48616c44ba082ab5849808f74edaf95ada773a | What is the coefficient of \( x^{16} \) in the polynomial \( (x-1)(x-2)(x-3) \cdots (x-18) \)? | [
"13566"
] | [
"What is the coefficient of \\( x^{16} \\) in the polynomial \\( (x-1)(x-2)(x-3) \\cdots (x-18) \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 15
} | 30 | 0 | 0 |
omr-ws-b4b43923f453c68ea1eb2e68be7e86560c0bb4a671557cbf7332599032b35733 | What is the coefficient of \( x^{2009} \) in the expansion of \(\left[(1+x)(1+2x^3)(1+4x^9)(1+8x^{27})(1+16x^{81})(1+32x^{243})(1+64x^{729})\right]^2\)? | [
"2147483648"
] | [
"What is the coefficient of \\( x^{2009} \\) in the expansion of \\(\\left[(1+x)(1+2x^3)(1+4x^9)(1+8x^{27})(1+16x^{81})(1+32x^{243})(1+64x^{729})\\right]^2\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 20
} | 35 | 0 | 0 |
omr-ws-b048ad6d64449a7afee2f6bd94f195bf84a310734bb9c3ae217ef0089a77ad6e | What is the coefficient of \( x^{2012} \) in the expansion of \( (x+1)(x^2+2)(x^4+4) \cdots (x^{1024}+1024) \)? | [
"64"
] | [
"What is the coefficient of \\( x^{2012} \\) in the expansion of \\( (x+1)(x^2+2)(x^4+4) \\cdots (x^{1024}+1024) \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 21
} | 37 | 0 | 0 |
omr-ws-37667b11cc2f4db648babc766b89035d1ebdf6ec718bf6ee9b8fcb1fec840395 | What is the coefficient of \( x^{25} \) in the polynomial \( (x-1)(x-2)(x-3) \cdots (x-50) \)? | [
"-1759975170237895559117007667844300114185022044875"
] | [
"What is the coefficient of \\( x^{25} \\) in the polynomial \\( (x-1)(x-2)(x-3) \\cdots (x-50) \\)?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 1,
"tir": 18
} | 19 | 0 | 0 |
omr-ws-909a6132daa1ef6995ca477a5e5cb05457b8b7f5342c03dab66f19905238b619 | What is the coefficient of \( x^{33} \) in the expansion of \( [(1+x)(1+3x^3)(1+9x^9)(1+27x^{27})]^2 \)? | [
"486"
] | [
"What is the coefficient of \\( x^{33} \\) in the expansion of \\( [(1+x)(1+3x^3)(1+9x^9)(1+27x^{27})]^2 \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-9ae52b9b9bd6185beb4838a40c39e3554f40a9603c696172fd15bb29a055a508 | What is the coefficient of \( x^{52} \) in the expansion of \( (1 + x + x^2 + x^3 + \cdots + x^{12})^5 \)? | [
"495"
] | [
"What is the coefficient of \\( x^{52} \\) in the expansion of \\( (1 + x + x^2 + x^3 + \\cdots + x^{12})^5 \\)?"
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 24
} | 38 | 0 | 0 |
omr-ws-1987ad15c3605e3709f79e0c6d14cdc2685903069b707c8446a7ca70dc5c886e | What is the coefficient of \( x^{999} \) in the polynomial \( p(x) = (x-1)(x-\frac{1}{2})(x-\frac{1}{4}) \cdots (x-\frac{1}{2^{999}}) \)? | [
"\\(\\frac{1}{2^{999}} - 2\\)"
] | [
"What is the coefficient of \\( x^{999} \\) in the polynomial \\( p(x) = (x-1)(x-\\frac{1}{2})(x-\\frac{1}{4}) \\cdots (x-\\frac{1}{2^{999}}) \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 17
} | 33 | 0 | 0 |
omr-ws-b57962da3d8559a73faa3ee049e7740f15a0c1c9a731b2811cf1ded3cb4ed415 | What is the coefficient of \( x^{n - 1} \) in \( (x + 1)^n + (x + 1)^{n + 2} + \cdots + (x + 1)^{n + 2m} \)? | [
"\\binom{n + 2m + 2}{n + 1} - \\binom{n}{n + 1}"
] | [
"What is the coefficient of \\( x^{n - 1} \\) in \\( (x + 1)^n + (x + 1)^{n + 2} + \\cdots + (x + 1)^{n + 2m} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0d7fb0eeaa89f33a56905de49b97ba495221c23e3677d98255ae604b5d2440af | What is the coefficient of \( x^{n-3} \) in the power series expansion of \( \frac{2}{x+1} - \frac{x \ln x}{x+1} \)? | [
"2(-1)^{n-3}"
] | [
"What is the coefficient of \\( x^{n-3} \\) in the power series expansion of \\( \\frac{2}{x+1} - \\frac{x \\ln x}{x+1} \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-8b6b52fb66ca07b84b61f073837d9f993450e02e0e4e3ba70e47d8803639f5ef | What is the coefficient of \(a^n b^n\) in the expansion of \([(1+a)(1+b)(a+b)]^n\)? | [
"\\(\\sum_{k=0}^n \\binom{n}{k}^3\\)"
] | [
"What is the coefficient of \\(a^n b^n\\) in the expansion of \\([(1+a)(1+b)(a+b)]^n\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 31,
"tir": 22
} | 53 | 0 | 0 |
omr-ws-8d61d2061ea1416a1f123b8423e8aa89b3cfc1674f82806acd2f4e0838f24278 | What is the coefficient of \(x^2\) in the expansion of \((1 - x)(1 + 2x)(1 - 3x)(1 + 4x) \cdots (1 - 15x)\)? | [
"-588"
] | [
"What is the coefficient of \\(x^2\\) in the expansion of \\((1 - x)(1 + 2x)(1 - 3x)(1 + 4x) \\cdots (1 - 15x)\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e9eb29b179510d56a46ea9c2022561bef889581520f27e86c304196cbdf8ca7e | What is the coefficient of \(x^2\) in the expression \( (((( \cdots ((((x - 2)^2 - 2)^2 - 2)^2 - 2)^2 \cdots - 2)^2 \)? | [
"\\(\\frac{4^{2n-1}-4^{n-1}}{3}\\)"
] | [
"What is the coefficient of \\(x^2\\) in the expression \\( (((( \\cdots ((((x - 2)^2 - 2)^2 - 2)^2 - 2)^2 \\cdots - 2)^2 \\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 2,
"tir": 9
} | 12 | 0 | 0 |
omr-ws-f35aa70835ce7b99043c52cdafc1b710b97b9f867223d375236782511e035f11 | What is the coefficient of \(x^3 y^4 z^6\) in \((x + y + z)^{13}\)? | [
"60060"
] | [
"What is the coefficient of \\(x^3 y^4 z^6\\) in \\((x + y + z)^{13}\\)?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5010f468af8a36bfb4b9b19180273af43ec28b0c674cdfd09ea7b8fc5ab0645b | What is the coefficient of \(x^3\) in the expansion of \((1 + x^2 + x^3 + x^4 + x^5)^6\)? | [
"21"
] | [
"What is the coefficient of \\(x^3\\) in the expansion of \\((1 + x^2 + x^3 + x^4 + x^5)^6\\)?"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a18c27eab75552b13032bed1f89da1ab972fc2ec60a9c973abfb2b3d2aa243db | What is the coefficient of \(x^{13}\) after expanding and combining all like terms in the expression \((1 + x^3 + x^5)^{10}\)? | [
"360"
] | [
"What is the coefficient of \\(x^{13}\\) after expanding and combining all like terms in the expression \\((1 + x^3 + x^5)^{10}\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-8d6946072303904fae1e69233ac6c9f3ab77d1d1e37307a85a3c474ce20f46c6 | What is the coefficient of the $x^5$ term in the simplified expansion of $(x + \sqrt{x} + \sqrt[3]{x})^{10}$? | [
"2101"
] | [
"What is the coefficient of the $x^5$ term in the simplified expansion of $(x + \\sqrt{x} + \\sqrt[3]{x})^{10}$?"
] | [
"aops_c4_high_school_math",
"c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-042ccf69d3b5bf26df2df8d7997c8ca1758da1e19ce368ff97cb636617a03047 | What is the coefficient of the \(x^3\) term in the expansion of \((x^2 + x - 4)^7\)? | [
"-34048"
] | [
"What is the coefficient of the \\(x^3\\) term in the expansion of \\((x^2 + x - 4)^7\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ccca68e5fe45200c8c4214f0b7610e092f8902a74a862bf8ff0ae32ec640150b | What is the coefficient of the term \( x^9 \) when \( (1 + x)(1 + x^2)(1 + x^3)(1 + x^4)(1 + x^5) \) is expanded? | [
"3"
] | [
"What is the coefficient of the term \\( x^9 \\) when \\( (1 + x)(1 + x^2)(1 + x^3)(1 + x^4)(1 + x^5) \\) is expanded?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e4da330c47d01220130f5ed4237e57d1a8650dacf330c8fc40795dd8c28db468 | What is the common ratio of the infinite series given by \[ 1 + \frac{1}{3} + \frac{2}{3} + \frac{2}{9} + \frac{4}{9} + \frac{4}{27} + \frac{8}{27} + \frac{8}{81} + \dotsb \]? | [
"\\(\\frac{2}{3}\\)"
] | [
"What is the common ratio of the infinite series given by\n\n\\[ 1 + \\frac{1}{3} + \\frac{2}{3} + \\frac{2}{9} + \\frac{4}{9} + \\frac{4}{27} + \\frac{8}{27} + \\frac{8}{81} + \\dotsb \\]?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-a4700811c723942c403bde353e9cefa7489312e79039de1d6bc2694c84a3cf5b | What is the common tangent line for the functions \( f(x) = \log_a x \) and \( g(x) = a^x \) where \( a > 0 \) and \( a \ne 1 \)? | [
"\\( y = x \\) for \\( a = e^{e^{-1}} \\)"
] | [
"What is the common tangent line for the functions \\( f(x) = \\log_a x \\) and \\( g(x) = a^x \\) where \\( a > 0 \\) and \\( a \\ne 1 \\)?"
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 10
} | 10 | 0 | 0 |
omr-ws-b24ad8624638944d273bf8523b2ffb315fa4fe76c32cf567fb574d687ed6a2c3 | What is the commutator subgroup of the group of (bounded) invertible linear operators on a complex Hilbert space? The commutator subgroup is defined as the subgroup generated by the commutators $[A,B] = ABA^{-1}B^{-1}$ for invertible linear operators $A$ and $B$. | [
"GL(H)"
] | [
"What is the commutator subgroup of the group of (bounded) invertible linear operators on a complex Hilbert space? The commutator subgroup is defined as the subgroup generated by the commutators $[A,B] = ABA^{-1}B^{-1}$ for invertible linear operators $A$ and $B$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-54944c1b40b4c84f6d851dada053736e0d33acc42fc1acf8b79c07f1460d5c6a | What is the complementary angle of \(-900\) degrees? | [
"990 degrees\n\nThe discussions in the forum indicate that the complementary angle of \\(-900\\) degrees is \\(990\\) degrees. While some posts suggest simplifying the angle to \\(270\\) degrees by subtracting multiples of \\(360\\) degrees, others argue for leaving the angle as \\(990\\) degrees to maintain the co... | [
"What is the complementary angle of \\(-900\\) degrees?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 7,
"tir": 10
} | 27 | 0 | 0 |
omr-ws-48d3063053d57513699f07f11ac9478ed0b3905cf95aeb72cd33648516470c9d | What is the complex conjugate of \(-3 \text{cis}\left(\frac{2\pi}{3}\right)\)? | [
"\\(3 \\text{cis}\\left(\\frac{\\pi}{3}\\right)\\)"
] | [
"What is the complex conjugate of \\(-3 \\text{cis}\\left(\\frac{2\\pi}{3}\\right)\\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-729ad7c99092e4b637315fc76ab14fc37abbb1a7bb667fa89d9d6df83d732695 | What is the complex slope of a line that makes an angle $t$ with the positive x-axis? | [
"\\( e^{it} \\)"
] | [
"What is the complex slope of a line that makes an angle $t$ with the positive x-axis?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 4,
"tir": 3
} | 19 | 0 | 0 |
omr-ws-f2ed497b5a654221a8bb0973e859e7b838f80ac95a4c3ca996f9abac918dd6e2 | What is the complexity of computing a minimum-cost Euclidean matching for 2n points in the plane? The cost of a matching is the total length of the edges in the matching. | [
"P"
] | [
"What is the complexity of computing a minimum-cost Euclidean matching for 2n points in the plane? The cost of a matching is the total length of the edges in the matching."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 3
} | 12 | 0 | 0 |
omr-ws-b7dd675916c020e107031112b9a65e9dd7b57b449a9a7c0c48af033f6942bfab | What is the condition for 4 given sides to form a quadrilateral and 5 given sides to form a pentagon? Is there a general approach for an n-gon? | [
"\\text{Each side must be less than the sum of the others.}"
] | [
"What is the condition for 4 given sides to form a quadrilateral and 5 given sides to form a pentagon? Is there a general approach for an n-gon?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-50ea2c85b7cf286d7b2e244056093f9471aa480f3c0d45655020c5632a044e9a | What is the condition for a common root between the cubic equation $ax^3 + bx^2 + cx + d = 0$ and the quadratic equation $ax^2 + bx + c = 0$? | [
"\\( d = 0 \\)"
] | [
"What is the condition for a common root between the cubic equation $ax^3 + bx^2 + cx + d = 0$ and the quadratic equation $ax^2 + bx + c = 0$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-abc6e9c3c3d5b6e9f05964e68a8f9dd78247bf3ad9c8a62dc0014d5841ec9d96 | What is the condition for a composite number \( N \) to not have any prime factors greater than \( \sqrt{N} \)? | [
"\\( N \\) must be the square of a prime number."
] | [
"What is the condition for a composite number \\( N \\) to not have any prime factors greater than \\( \\sqrt{N} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 1,
"tir": 1
} | 4 | 0 | 0 |
omr-ws-64b4523c2e8731e4eac1224aa189161cce81869e41102e010ebd2f81097fdbef | What is the condition for lines $l_1$, $l_2$, and $l_3$ to be concurrent in the complex plane, given that $l_1$ passes through points $a_1$ and $b_1$, $l_2$ passes through points $a_2$ and $b_2$, and $l_3$ passes through points $a_3$ and $b_3$? | [
"\\begin{vmatrix} a_1 & b_1 & 1 \\\\ a_2 & b_2 & 1 \\\\ a_3 & b_3 & 1 \\end{vmatrix} = 0"
] | [
"What is the condition for lines $l_1$, $l_2$, and $l_3$ to be concurrent in the complex plane, given that $l_1$ passes through points $a_1$ and $b_1$, $l_2$ passes through points $a_2$ and $b_2$, and $l_3$ passes through points $a_3$ and $b_3$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 2,
"genselect": 1
} | 3 | 0 | 0 |
omr-ws-d2480cc931e396f3c44bb33ecefde57d57962d4f73b088de0b348c7223a8a9e9 | What is the condition for real numbers \(a < b\) such that there exists a non-zero polynomial \(f\) with integer coefficients satisfying \(|f(x)| < 1\) for all \(x\) in the interval \((a, b)\)? | [
"\\(b - a < 4\\)"
] | [
"What is the condition for real numbers \\(a < b\\) such that there exists a non-zero polynomial \\(f\\) with integer coefficients satisfying \\(|f(x)| < 1\\) for all \\(x\\) in the interval \\((a, b)\\)?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 13,
"genselect": 4,
"tir": 2
} | 19 | 0 | 0 |
omr-ws-859ac36eb7e5808a01d6515ca0d5dbb14e2e4cdb693b687434b144ef8602bce2 | What is the condition for the concurrency of the lines joining the pairs of complex numbers \(z_1\) and \(z_2\), \(z_3\) and \(z_4\), and \(z_5\) and \(z_6\) in the complex plane? | [
"\\(\\begin{vmatrix}z_2-z_1 & \\bar z_2-\\bar z_1 & z_1\\bar z_2-z_2\\bar z_1 \\\\ z_4-z_3 & \\bar z_4-\\bar z_3 & z_3\\bar z_4-z_4\\bar z_3 \\\\ z_6-z_5 & \\bar z_6-\\bar z_5 & z_5\\bar z_6-z_6\\bar z_5\\end{vmatrix}=0\\)"
] | [
"What is the condition for the concurrency of the lines joining the pairs of complex numbers \\(z_1\\) and \\(z_2\\), \\(z_3\\) and \\(z_4\\), and \\(z_5\\) and \\(z_6\\) in the complex plane?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 9,
"genselect": 5,
"tir": 5
} | 19 | 0 | 0 |
omr-ws-2844655eb3df5530f3eb2b09f421e38f604b65e69b6839a9347390bb3dbcc4b1 | What is the condition for the cubic equation $x^3 - px^2 + qx - r = 0$ to have integer roots, where $p$, $q$, and $r$ are integers? | [
"\\( p = a + b + c \\), \\( q = ab + ac + bc \\), \\( r = abc \\) where \\( a, b, \\) and \\( c \\) are integers."
] | [
"What is the condition for the cubic equation $x^3 - px^2 + qx - r = 0$ to have integer roots, where $p$, $q$, and $r$ are integers?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 2
} | 18 | 0 | 0 |
omr-ws-0e5053f06c3dfd6d1858f4ead1f45a9ea35aba2fb64e363e779c9c70c76c0b21 | What is the condition for the line $\frac{x}{a} + \frac{y}{b} = 1$ to be tangent to the curve $x^{\frac{2}{3}} + y^{\frac{2}{3}} = 1$ in terms of $a$ and $b$? | [
"a^2 + b^2 = 1"
] | [
"What is the condition for the line $\\frac{x}{a} + \\frac{y}{b} = 1$ to be tangent to the curve $x^{\\frac{2}{3}} + y^{\\frac{2}{3}} = 1$ in terms of $a$ and $b$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 14
} | 14 | 0 | 0 |
omr-ws-dd9ac47c8aebd71685bc143d4654c314c2fc7144657adca8be0b1a55b79c55e1 | What is the condition for the line \( ax + by + c = 0 \) to be the normal to the conic \( xy = c^2 \)? | [
"\\((a^2 - b^2)^2 + ab = 0\\)"
] | [
"What is the condition for the line \\( ax + by + c = 0 \\) to be the normal to the conic \\( xy = c^2 \\)?"
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 8,
"tir": 17
} | 35 | 0 | 0 |
omr-ws-097b8407d44a55394900c78fe79357c0d44a854d2190ee1567798deff9c196ba | What is the condition for the lines joining the pairs of points \((z_1, z_2)\), \((z_3, z_4)\), and \((z_5, z_6)\) on the complex plane to be concurrent? | [
"\n\\begin{vmatrix}\n\\overline{z_1} - \\overline{z_2} & z_2 - z_1 & z_1 \\overline{z_2} - z_2 \\overline{z_1} \\\\\n\\overline{z_3} - \\overline{z_4} & z_4 - z_3 & z_3 \\overline{z_4} - z_4 \\overline{z_3} \\\\\n\\overline{z_5} - \\overline{z_6} & z_6 - z_5 & z_5 \\overline{z_6} - z_6 \\overline{z_5}\n\\end{vmatri... | [
"What is the condition for the lines joining the pairs of points \\((z_1, z_2)\\), \\((z_3, z_4)\\), and \\((z_5, z_6)\\) on the complex plane to be concurrent?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 7
} | 17 | 0 | 0 |
omr-ws-c101096d0777669fd1ad21ead856a4d7e207778a2cc7cf294fb49238850be6ff | What is the condition for the quadratic equation $f(x) = ax^2 + bx + c$ to have at least one negative real root? | [
"\\( b > 0 \\text{ and } b^2 - 4ac \\geq 0 \\)"
] | [
"What is the condition for the quadratic equation $f(x) = ax^2 + bx + c$ to have at least one negative real root?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b49a1dcfe54750f364f8f5186bce150ffdbcb9a73ce437e56cb515d8a1875f06 | What is the condition for the quadratic equation $f(x) = ax^2 + bx + c$ to have at least one positive real root? | [
"\\( ac < 0 \\)"
] | [
"What is the condition for the quadratic equation $f(x) = ax^2 + bx + c$ to have at least one positive real root?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 3
} | 6 | 0 | 0 |
omr-ws-7305ce340ea7578d97936b5b88c7c8d447c0414297838179581d581260d6d6b0 | What is the condition for the quadratic equation $f(x) = ax^2 + bx + c$ to have integral roots? | [
"a \\text{ divides } b \\text{ and } c \\text{, and } b^2 - 4ac \\text{ is a perfect square}"
] | [
"What is the condition for the quadratic equation $f(x) = ax^2 + bx + c$ to have integral roots?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 5
} | 16 | 0 | 0 |
omr-ws-2b29861dd35598354d8bab9120c1ea9ab700350d882e3b10949b96745fe42bad | What is the condition for the series \( \sum_{k=0}^{\infty} {\alpha \choose k} {\beta \choose z-k} \) to converge, and what is the limit if the series converges? | [
"{\\alpha + \\beta \\choose z}"
] | [
"What is the condition for the series \\( \\sum_{k=0}^{\\infty} {\\alpha \\choose k} {\\beta \\choose z-k} \\) to converge, and what is the limit if the series converges?"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 7
} | 22 | 0 | 0 |
omr-ws-6e4e6b8747fa9da498bc640c71587673f91c9767912f7fcad2adc82db305f45e | What is the condition on positive real numbers \(a, b, c\) such that for all \(n \in \mathbb{N}\) and \(n \neq 0\), the following holds: \(\lfloor na \rfloor \cdot \lfloor nb \rfloor = \lfloor n^2 c \rfloor\)? Here, \(\lfloor x \rfloor\) denotes the greatest integer less than or equal to \(x\). | [
"\\(c = ab\\) with \\(a\\) and \\(b\\) being integers."
] | [
"What is the condition on positive real numbers \\(a, b, c\\) such that for all \\(n \\in \\mathbb{N}\\) and \\(n \\neq 0\\), the following holds: \\(\\lfloor na \\rfloor \\cdot \\lfloor nb \\rfloor = \\lfloor n^2 c \\rfloor\\)? Here, \\(\\lfloor x \\rfloor\\) denotes the greatest integer less than or equal to \\(x... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 44,
"genselect": 6,
"tir": 10
} | 60 | 0 | 0 |
omr-ws-c0c2eb3d473bb5f9698a293e91619d9e1bf95d7f78f8f6122afcb166548c4a30 | What is the conditional probability that Barbara is dealt four of a kind, given that Angus has been dealt two pairs? | [
"\\dfrac{10}{35673}"
] | [
"What is the conditional probability that Barbara is dealt four of a kind, given that Angus has been dealt two pairs?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-840d891e7e024e81065e76b79d2762d33e5343bee516a19ab717c0d83d94d5fd | What is the conditional probability that Conor is dealt a full house, given that Barbara has been dealt four of a kind? | [
"\\dfrac{2}{1081}"
] | [
"What is the conditional probability that Conor is dealt a full house, given that Barbara has been dealt four of a kind?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-87e0fe527d49831659a16d36fd978862398ae6ba722d5e2af901d554f137aca9 | What is the conjugate of $2+4$? | [
"6"
] | [
"What is the conjugate of $2+4$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 7
} | 16 | 0 | 0 |
omr-ws-85cd1b0fa1632f8b838e93ddad65572807f256b1d908496881faeac056421235 | What is the conjugate of \( \frac{3 - 2i}{12 + 5i} \)? | [
"\\( \\frac{2 + 3i}{13} \\)"
] | [
"What is the conjugate of \\( \\frac{3 - 2i}{12 + 5i} \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 29
} | 44 | 0 | 0 |
omr-ws-b50fd40081457769e25f408dc6137c3d0987f9b8797c6c8e33c0c9267aa7734d | What is the conjugate of the pure imaginary number \( bi \)? | [
"-bi"
] | [
"What is the conjugate of the pure imaginary number \\( bi \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-66ea902ca4045fbedc7221a62d92ea8ce980338f1dfd7834dc1730edd7e374ac | What is the conjugate of the real number \( a \)? | [
"a"
] | [
"What is the conjugate of the real number \\( a \\)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5e7047f76629edc727571ec894e63199bf093d3697b57179b9b088047b6bb416 | What is the constant term in the Laurent polynomial \[ \prod_{1\le i\neq j\le n}\left(1 - \frac{x_i}{x_j}\right)^{k_i} \] where $k_i$'s are given natural numbers? | [
"\\dfrac{(k_1 + k_2 + \\cdots + k_n)!}{k_1! \\, k_2! \\, \\cdots \\, k_n!}"
] | [
"What is the constant term in the Laurent polynomial\n\\[ \\prod_{1\\le i\\neq j\\le n}\\left(1 - \\frac{x_i}{x_j}\\right)^{k_i} \\]\nwhere $k_i$'s are given natural numbers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 7
} | 15 | 0 | 0 |
omr-ws-550359dd76b5dbf86f48423042de47e7b925ae78e375601bd49f6368821acf54 | What is the constant term in the expansion of $(1+x+2x^3)\left(\frac{3x^2}{2}-\frac{1}{3x}\right)^9$? | [
"\\(\\frac{17}{54}\\)"
] | [
"What is the constant term in the expansion of $(1+x+2x^3)\\left(\\frac{3x^2}{2}-\\frac{1}{3x}\\right)^9$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-8fcda020ec3a0a04dde02c8a1482a849a299e1451f81f99b04e052a2d1db5a24 | What is the constant term in the expansion of $(x^4+x+5)(x^5+x^3+15)$? | [
"75"
] | [
"What is the constant term in the expansion of $(x^4+x+5)(x^5+x^3+15)$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-21be4c2657bc9c67d250558e452fa2a35f1aad4e6e2cbdacf339d26cc1643db5 | What is the constant term in the expansion of $\left(\sqrt{x}+\dfrac5x\right)^{9}$? | [
"10500"
] | [
"What is the constant term in the expansion of $\\left(\\sqrt{x}+\\dfrac5x\\right)^{9}$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-73f512a2823337d3bbaeb81222b859f117e236a8db348da9caf93aa079058447 | What is the constant term in the expression $\prod_{1\leq i \neq j\leq n}{(1-z_i/z_j)^{a_i}}$, where $a_1, \ldots, a_n$ are positive integers? | [
"\\dfrac{(a_1 + a_2 + \\cdots + a_n)!}{a_1! \\, a_2! \\, \\cdots \\, a_n!}"
] | [
"What is the constant term in the expression $\\prod_{1\\leq i \\neq j\\leq n}{(1-z_i/z_j)^{a_i}}$, where $a_1, \\ldots, a_n$ are positive integers?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-a881c4ff51df4ec8499c0f0ac6b9a1f4782ec3bb3b410733ffb2aa717cc48553 | What is the constant term of the expansion of $\left(6x+\dfrac{1}{3x}\right)^6$? | [
"160"
] | [
"What is the constant term of the expansion of $\\left(6x+\\dfrac{1}{3x}\\right)^6$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-7b59e40a540b29f9f88f93f0e8989e6c356acafd4dc74cfc2ac0c0257e1ae0f4 | What is the constant that represents the upper bound of the sum $\sin(0) + \sin(1) + \sin(2) + \cdots$? It is observed that the upper bound is around 1.95 and does not appear to be a multiple of $\pi$. | [
"\\(\\frac{1}{2} \\cot \\frac{1}{4}\\) or approximately 1.95815868."
] | [
"What is the constant that represents the upper bound of the sum $\\sin(0) + \\sin(1) + \\sin(2) + \\cdots$? It is observed that the upper bound is around 1.95 and does not appear to be a multiple of $\\pi$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 10,
"genselect": 3,
"tir": 25
} | 38 | 0 | 0 |
omr-ws-9523b74e14381a77d512700eaf428054ce1cb58b1d4940e6c8e4c3df7a5d5630 | What is the correct formula for the number of permutations of \( n \) things, taken \( r \) at a time, when \( m \) specified things always come together? | [
"\\((r-m+1) \\times [(n-m)P(r-m)] \\times m!\\)"
] | [
"What is the correct formula for the number of permutations of \\( n \\) things, taken \\( r \\) at a time, when \\( m \\) specified things always come together?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 13,
"tir": 9
} | 22 | 0 | 0 |
omr-ws-00a49f6303c76c5bdf0f04e3fb44379bb6c536f9970ea959beb461135e533343 | What is the correct formula for the perimeter (or parameter) of a circular sector with radius \( r \) and central angle \( \theta \) (in degrees)? | [
"2r + \\frac{\\pi r \\theta}{180}"
] | [
"What is the correct formula for the perimeter (or parameter) of a circular sector with radius \\( r \\) and central angle \\( \\theta \\) (in degrees)?"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b3ef8ae5cda7de771a2bcf91d77bfecce41ae522278463b35f6e3c8504920775 | What is the correct method to sum the series $\sum_{n=0}^{\infty} \frac{\text{Cat}_n^2}{16^n}$, where $\text{Cat}_n$ is the Catalan number given by $\text{Cat}_n = \frac{(2n)!}{n!(n+1)!}$? | [
"\\(\\frac{16}{\\pi} - 4\\)"
] | [
"What is the correct method to sum the series $\\sum_{n=0}^{\\infty} \\frac{\\text{Cat}_n^2}{16^n}$, where $\\text{Cat}_n$ is the Catalan number given by $\\text{Cat}_n = \\frac{(2n)!}{n!(n+1)!}$?"
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 4
} | 4 | 0 | 0 |
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