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-cc07d5ba0f50afa7077f81e2be479b24261a507de275f37b9b01af13496b3c93 | Calculate $\phi(n!)$ for all positive integers $n$. | [
"n! \\prod_{\\substack{p \\leq n \\\\ p \\text{ prime}}} \\left(1 - \\frac{1}{p}\\right)"
] | [
"Calculate $\\phi(n!)$ for all positive integers $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-eea020403a297dbe66b826ed62938ac9d80d6dc6480fd8faa83e879d3e7d5cba | Calculate $\pi^{4}+\pi^{5}$ mentally. | [
"400"
] | [
"Calculate $\\pi^{4}+\\pi^{5}$ mentally."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3906b619a6697d82889c055654b9dcc3771c4b7ee185fe0e01c62cbde663a7d7 | Calculate $\prod_{i=1}^{17}\cos\left(\frac{i\pi}{36}\right)$ without using a calculator. The expected solution is $\frac{3\sqrt{2}}{2^{17}}$. | [
"\\(\\frac{3\\sqrt{2}}{2^{17}}\\)"
] | [
"Calculate $\\prod_{i=1}^{17}\\cos\\left(\\frac{i\\pi}{36}\\right)$ without using a calculator. The expected solution is $\\frac{3\\sqrt{2}}{2^{17}}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-94bd719f9d00b8d88e6687ce8cba0e8a4a561deb0a0c16a8d95823c08bb9418f | Calculate $\prod_{x=1}^{89} \sin x$. | [
"$\\frac{3\\sqrt{10}}{2^{89}}$"
] | [
"Calculate $\\prod_{x=1}^{89} \\sin x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 18
} | 66 | 0 | 0 |
omr-ws-1cf16b4d2d53b7238a7f5e022c9af723981581f45aa50a1819d8c86bde88eed2 | Calculate $\sin\left(\frac{13}{2}\right)$. | [
"0.2151"
] | [
"Calculate $\\sin\\left(\\frac{13}{2}\\right)$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-1e00bd81aa693284791145e593b66c3610eb04690d8a6329b1d717ce704e2902 | Calculate $\sin{15^\circ}$ and $\sin{75^\circ}.$ | [
"\\(\\sin 15^\\circ = \\frac{\\sqrt{6} - \\sqrt{2}}{4}\\) and \\(\\sin 75^\\circ = \\frac{\\sqrt{6} + \\sqrt{2}}{4}\\)"
] | [
"Calculate $\\sin{15^\\circ}$ and $\\sin{75^\\circ}.$"
] | [
"aops_c4_high_school_math"
] | [
"tir"
] | {
"tir": 27
} | 27 | 0 | 0 |
omr-ws-51e979e70e70a45b439feb3c501783764ae3df3eb88c7e141a713bbe39b34c68 | Calculate $\sqrt[3]{2 + 2i}$. | [
"\\(\\frac{1+\\sqrt{3}}{2} + i\\frac{\\sqrt{3}-1}{2}\\)"
] | [
"Calculate $\\sqrt[3]{2 + 2i}$."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 21,
"genselect": 5,
"tir": 14
} | 40 | 0 | 0 |
omr-ws-add6bf6010d644c698709f2235131679692197b59d2c200e807a7193ed0587c4 | Calculate $\sqrt[4]{\sqrt{3} + 3i}$. | [
"\\(\\sqrt[8]{12} \\left(\\frac{1+\\sqrt{3}}{2\\sqrt{2}} + i\\frac{\\sqrt{3}-1}{2\\sqrt{2}}\\right)\\)"
] | [
"Calculate $\\sqrt[4]{\\sqrt{3} + 3i}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 35,
"genselect": 3,
"tir": 6
} | 44 | 0 | 0 |
omr-ws-9693f6a0cdcc370e8b82cdfb48784a678142e6fc2991bc138deacbc84cf95db0 | Calculate $\sqrt{(1998)(1996)(1994)(1992)+16}$. Is the method $2000^2 - 2 \times 5 \times 2000 + 5^2 - 5$ justified? Why? | [
"\\(1995^2 - 5\\)"
] | [
"Calculate $\\sqrt{(1998)(1996)(1994)(1992)+16}$. Is the method $2000^2 - 2 \\times 5 \\times 2000 + 5^2 - 5$ justified? Why?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-6299db4b8df4e94a3ebfaddc6a58b0879232c9c625e7d4644ec046f1fb0b8dd5 | Calculate $\sqrt{10p} \cdot \sqrt{5p^2} \cdot \sqrt{6p^4}$ . Express your answer in simplest radical form in terms of $p$. Note: When entering a square root with more than one character, you must use parentheses or brackets. For example, you should enter $\sqrt{14}$ as "sqrt(14)" or "sqrt{14}". | [
"10p^3 \\sqrt{3p}"
] | [
"Calculate $\\sqrt{10p} \\cdot \\sqrt{5p^2} \\cdot \\sqrt{6p^4}$ . Express your answer in simplest radical form in terms of $p$.\n\nNote: When entering a square root with more than one character, you must use parentheses or brackets. For example, you should enter $\\sqrt{14}$ as \"sqrt(14)\" or \"sqrt{14}\"."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6f07297c0d356af343f01d81cd6f48e12912f64d4f08c3c5d87f8c6178da3f9f | Calculate $\sqrt{2+\sqrt{3}}\cdot \sqrt{2+\sqrt{2+\sqrt{3}}}\cdot \sqrt{2+\sqrt{2+\sqrt{2+\sqrt{3}}}}+\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{3}}}}$ | [
"1"
] | [
"Calculate $\\sqrt{2+\\sqrt{3}}\\cdot \\sqrt{2+\\sqrt{2+\\sqrt{3}}}\\cdot \\sqrt{2+\\sqrt{2+\\sqrt{2+\\sqrt{3}}}}+\\sqrt{2-\\sqrt{2+\\sqrt{2+\\sqrt{3}}}}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f3679763c091ff2566c4fbc04e2ece03dfa510857e26dacaf02a6f62e17bca61 | Calculate $\sqrt{2014+\sqrt{4029}}-\sqrt{2014-\sqrt{4029}}$. | [
"\\sqrt{2}"
] | [
"Calculate $\\sqrt{2014+\\sqrt{4029}}-\\sqrt{2014-\\sqrt{4029}}$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-2d5ad17387254739c1c29a14a1989b1633939c66618437c34cc1578f76e9c347 | Calculate $\sqrt{2}$ in the ring of 7-adic integers $\mathbb{Z}_7$ and write the answer as an infinitely left-hand sequence of the digits $0, 1, \ldots, 6$. | [
"$6213\\ldots$"
] | [
"Calculate $\\sqrt{2}$ in the ring of 7-adic integers $\\mathbb{Z}_7$ and write the answer as an infinitely left-hand sequence of the digits $0, 1, \\ldots, 6$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 2,
"genselect": 3,
"tir": 8
} | 13 | 0 | 0 |
omr-ws-2de4ccf089820e78933f553f91625abc741ab692f0c5f54ee8139cdd13fdaf53 | Calculate $\sqrt{30p} \cdot \sqrt{5p} \cdot \sqrt{6p}$ . Express your answer in simplest radical form in terms of $p$. | [
"30p \\sqrt{p}"
] | [
"Calculate $\\sqrt{30p} \\cdot \\sqrt{5p} \\cdot \\sqrt{6p}$ . Express your answer in simplest radical form in terms of $p$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-69a82e4e84fb642fd0bc1041732df5e3a4ba717a297d9d10c2e4502ce9ecd1d4 | Calculate $\sqrt{5.444...4}$ (the decimal point is followed by 100 fours) with an approximation to: a) $10^{-100}$, b) $10^{-200}$ | [
"\\(\\frac{7}{3}\\)"
] | [
"Calculate $\\sqrt{5.444...4}$ (the decimal point is followed by 100 fours) with an approximation to:\na) $10^{-100}$,\nb) $10^{-200}$"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 8
} | 8 | 0 | 0 |
omr-ws-a1185864b4e966bfd2480fcca4857b2a358d846905d8bde2e7a77dd7a4ef9367 | Calculate $\sqrt{6-\sqrt{42}}$. | [
"i \\sqrt{\\sqrt{42} - 6}"
] | [
"Calculate $\\sqrt{6-\\sqrt{42}}$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5fdc7845f73e1baebc7f5a5357e59187a61a8c2d643f4dbb7185547a09b1adb0 | Calculate $\sqrt{75x} \cdot \sqrt{2x} \cdot \sqrt{14x}$ . Express your answer in simplest radical form in terms of $x$. | [
"10x \\sqrt{21x}"
] | [
"Calculate $\\sqrt{75x} \\cdot \\sqrt{2x} \\cdot \\sqrt{14x}$ . Express your answer in simplest radical form in terms of $x$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fa9316fb8301bb1085a046c4ae12fc440760c7590c2704d518a0b48f84f42197 | Calculate $\sqrt{\frac{4417^3 - 3000^3}{109} + 13(13251000)}$ without a calculator. | [
"\\( 7417\\sqrt{13} \\)"
] | [
"Calculate $\\sqrt{\\frac{4417^3 - 3000^3}{109} + 13(13251000)}$ without a calculator."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-2fadc64d5997f476ee164b2f3c8776067bf412ac384af8973791b3dc8819e6a4 | Calculate $\sqrt{\sqrt[3]{0.000064}}$. Express your answer as a decimal to the nearest tenth. | [
"0.2"
] | [
"Calculate $\\sqrt{\\sqrt[3]{0.000064}}$. Express your answer as a decimal to the nearest tenth."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-ac93f8b630ce193eaaf74dc6313be3b15a7929a198b332e7ade29df60d2d8726 | Calculate $\sqrt{\sqrt{5} - \sqrt{3 - \sqrt{29 - 12\sqrt{5}}}}$. | [
"1"
] | [
"Calculate $\\sqrt{\\sqrt{5} - \\sqrt{3 - \\sqrt{29 - 12\\sqrt{5}}}}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-93ac878944c6cef5c39f58939ed7bb3cfc5380570f14a88d4eeb5ce194d201b1 | Calculate $\sum \limits_{i=0}^{2^{2019}} 3^{3^i} \mod 2^{100}$. | [
"3"
] | [
"Calculate $\\sum \\limits_{i=0}^{2^{2019}} 3^{3^i} \\mod 2^{100}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 31
} | 31 | 0 | 0 |
omr-ws-e211b2312fb9676917794db450659db9bc2674d35ca0d2f0b5d889a3f174d5e5 | Calculate $\sum \limits_{i=0}^{3^{2019}} 3^{3^i} \mod 2^{100}$. | [
"1199996907049704462478368150180"
] | [
"Calculate $\\sum \\limits_{i=0}^{3^{2019}} 3^{3^i} \\mod 2^{100}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ac911d4d093d9d7f19573f5c32ffc6ea13e73df2df5ad4742a6082203821a0a5 | Calculate $\sum \limits_{i=1}^{2^{220}} ((i \mod (2^{127}-1)) \mod (2^{89}-1))$. | [
"\\(2^{92} (2^{216} - 3 \\times 2^{127} + 2^{89} + 2^{76} - 2^{38} - 1) + 152\\)"
] | [
"Calculate $\\sum \\limits_{i=1}^{2^{220}} ((i \\mod (2^{127}-1)) \\mod (2^{89}-1))$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 1,
"tir": 4
} | 6 | 0 | 0 |
omr-ws-ea302bb192581729f81c1166e2c30d5ea62ec2ecb45a2d8b1ddf74022a7e83c2 | Calculate $\sum\limits_{n=1,3,5,\ldots}^{+\infty }{\frac{\sin n\alpha }{n^3}}$ for $\alpha \in \left( 0,\frac{\pi }{2} \right)$. | [
"\\(\\frac{\\pi^2}{8}\\,\\alpha-\\frac{\\pi}8\\,\\alpha^2\\)"
] | [
"Calculate $\\sum\\limits_{n=1,3,5,\\ldots}^{+\\infty }{\\frac{\\sin n\\alpha }{n^3}}$ for $\\alpha \\in \\left( 0,\\frac{\\pi }{2} \\right)$."
] | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 9
} | 25 | 0 | 0 |
omr-ws-477dcef3af7d200aa6d4bfa19fe27a1cbe8b1620bbfedc14577e159d9fa10ce6 | Calculate $\sum_{0<i , j\le n}\frac{i^j}{\max(i^j,j^i)+\min(i!,j!)}$. | [
"n"
] | [
"Calculate $\\sum_{0<i , j\\le n}\\frac{i^j}{\\max(i^j,j^i)+\\min(i!,j!)}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 7
} | 11 | 0 | 0 |
omr-ws-567ccf5e39a2b647b391dbde0fd897380f1d02a04fbef7efc6e69385a49d7719 | Calculate $\sum_{0<i\le j\le n}\frac{i}{i+j}$. | [
"\\frac{n(n+1)}{4}"
] | [
"Calculate $\\sum_{0<i\\le j\\le n}\\frac{i}{i+j}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 3
} | 3 | 0 | 0 |
omr-ws-c55a1ece288338c4496ef008ddd6a8c2699f3043bd88eb84c58d6854f9fd7378 | Calculate $\sum_{i=0}^{2015} \left\lfloor \frac{2^i}{25} \right\rfloor$. | [
"\\dfrac{2^{2016} - 25186}{25}"
] | [
"Calculate $\\sum_{i=0}^{2015} \\left\\lfloor \\frac{2^i}{25} \\right\\rfloor$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 4
} | 18 | 0 | 0 |
omr-ws-9f3b6135db07e0bcfc4b13ac0ac52b992c44976bcd7594271d8e4f281a8d0cc9 | Calculate $\sum_{i=0}^{3}\Delta _{3}^{i}$ where the determinant $\Delta _{n}^{i}$ is given by: \[ \Delta _{n}^{i}=\begin{vmatrix} 1 &1 &... &1 \\ a_{1}&a_{2} &... &a_{n} \\ ... & & & \\ a_{1}^{i-1}&a_{2}^{i-1} &... &a_{n}^{i-1} \\ a_{1}^{i+1}&a_{2}^{i+1} &... &a_{n}^{i+1} \\ ...& & & \\ a_{1}^{n}& a_{2}^{n} &... & a_{n... | [
"\\((1+a_1)(1+a_2)(1+a_3)(a_1-a_2)(a_2-a_3)(a_3-a_1)\\)"
] | [
"Calculate $\\sum_{i=0}^{3}\\Delta _{3}^{i}$ where the determinant $\\Delta _{n}^{i}$ is given by:\n\\[\n\\Delta _{n}^{i}=\\begin{vmatrix}\n1 &1 &... &1 \\\\ \n a_{1}&a_{2} &... &a_{n} \\\\ \n... & & & \\\\ \n a_{1}^{i-1}&a_{2}^{i-1} &... &a_{n}^{i-1} \\\\ \n a_{1}^{i+1}&a_{2}^{i+1} &... &a_{n}^{i+1} \\\\... | [
"aops_c7_college_math",
"c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 8
} | 22 | 0 | 0 |
omr-ws-8cbe21ea102c6ed95434875e1504da694140a8a45ebf9d38182c7b1f49e7c6f2 | Calculate $\sum_{k=0}^{16}\cos^{2}\left(\frac{2k\pi}{17}\right)$. | [
"\\(\\frac{17}{2}\\)"
] | [
"Calculate $\\sum_{k=0}^{16}\\cos^{2}\\left(\\frac{2k\\pi}{17}\\right)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-883a592cee6163e1d4c2ab77ee2f0b2a432e885c5cb51d18a6425b1d3c2086c5 | Calculate $\sum_{k=0}^{2n}(-1)^{k}{ 4n-2k \choose 2n-k}{2k \choose k}$. | [
"$\\binom{2n}{n}4^{n}$"
] | [
"Calculate $\\sum_{k=0}^{2n}(-1)^{k}{ 4n-2k \\choose 2n-k}{2k \\choose k}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 46,
"tir": 23
} | 69 | 0 | 0 |
omr-ws-ac50f8a4a9a65dcf75c98f3017326472936dd032b5b0a2582f545d609e05c437 | Calculate $\sum_{k=0}^{n} \sin(k)$. | [
" \\dfrac{ \\sin\\left( \\dfrac{n}{2} \\right) \\sin\\left( \\dfrac{n + 1}{2} \\right) }{ \\sin\\left( \\dfrac{1}{2} \\right) } "
] | [
"Calculate $\\sum_{k=0}^{n} \\sin(k)$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 4
} | 19 | 0 | 0 |
omr-ws-9960e3c4fc92906db5c7e055b95ef351e8d5dbc24e2958cecc2ae87fc6aeedd2 | Calculate $\sum_{k=1}^{2001}\left\lfloor\frac{765\cdot k}{2002}\right\rfloor$. | [
"764502"
] | [
"Calculate $\\sum_{k=1}^{2001}\\left\\lfloor\\frac{765\\cdot k}{2002}\\right\\rfloor$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-69ae4c6a1e67841dc3481d9d1c3138052a3e4c86abb87926239075c869284b99 | Calculate $\sum_{k=1}^{2013} \frac{k+2}{k(k+1)2^k}$. | [
"\\(1 - \\frac{1}{2014 \\cdot 2^{2013}}\\)"
] | [
"Calculate $\\sum_{k=1}^{2013} \\frac{k+2}{k(k+1)2^k}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 8,
"tir": 23
} | 46 | 0 | 0 |
omr-ws-9ba39656126cc0d9bf78824e6d87258c1ca32cc72cdd4403819c6047de3fcb66 | Calculate $\sum_{k=1}^{n}a_k^2$ given that $\sum_{k=1}^{n}a_k = 3n^2 + 4n + 2$ for $n = 1, 2, 3, \ldots$. | [
"\\(12n^3 + 24n^2 + 13n + 32\\)"
] | [
"Calculate $\\sum_{k=1}^{n}a_k^2$ given that $\\sum_{k=1}^{n}a_k = 3n^2 + 4n + 2$ for $n = 1, 2, 3, \\ldots$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 18
} | 34 | 0 | 0 |
omr-ws-9b840d535722fc17acb310de84c1be61179a4b2cf87363c19e8fb04fd9f8e1c6 | Calculate $\sum_{n=0}^{502}\left[\frac{305n}{503}\right]$. | [
"76304"
] | [
"Calculate $\\sum_{n=0}^{502}\\left[\\frac{305n}{503}\\right]$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 28
} | 44 | 0 | 0 |
omr-ws-983d1403c034433e0c9fcacce933407f2be222e0bcbd59879725b2b7d34444c4 | Calculate $\sum_{n=1}^{+\infty} \dfrac{\sqrt{n}}{n(n+1)}$. | [
"\\frac{\\pi}{2}"
] | [
"Calculate $\\sum_{n=1}^{+\\infty} \\dfrac{\\sqrt{n}}{n(n+1)}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7192f3f1a2cb0b73909d835f0aa1f059dc64f33d508d838ef3267b4b10b12974 | Calculate $\sum_{n=1}^{1997}\lfloor\sqrt{n}\rfloor$. | [
"58542"
] | [
"Calculate $\\sum_{n=1}^{1997}\\lfloor\\sqrt{n}\\rfloor$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-f22f6a5fd0f40323f96cf6554a7bb203d06b21d1a636d4e64a1721a080a6a489 | Calculate $\sum_{n=1}^{\infty} \frac{d(n)}{n^2}$. | [
"\\(\\frac{\\pi^4}{36}\\)"
] | [
"Calculate $\\sum_{n=1}^{\\infty} \\frac{d(n)}{n^2}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-70f12e862594493ed8297d80f7ddf349644a6f856509deef3d9485f4de7371ea | Calculate $\sum_{n=1}^{\infty} \frac{d(n)}{n}$. | [
"\\infty"
] | [
"Calculate $\\sum_{n=1}^{\\infty} \\frac{d(n)}{n}$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-8be6abc37edcd801145b2651844c497732578023b6cb33953f349cf83f9ddbc5 | Calculate $\sum_{n=1}^{\infty} \tan^{-1} \dfrac{1}{n^2}$. | [
"\\(\\tan^{-1}\\dfrac{\\tan\\dfrac{\\pi}{\\sqrt{2}}-\\tanh\\dfrac{\\pi}{\\sqrt{2}}}{\\tan\\dfrac{\\pi}{\\sqrt{2}}+\\tanh\\dfrac{\\pi}{\\sqrt{2}}}\\)"
] | [
"Calculate $\\sum_{n=1}^{\\infty} \\tan^{-1} \\dfrac{1}{n^2}$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-363ae0c3338ed7b4b7fd49b9815abaaf7784ccef169f6696e02a0209626fc28d | Calculate $\sum_{n=1}^{\infty} \tan{\frac{\pi}{3^n}}$ | [
"no_valid_answer_found"
] | [
"Calculate $\\sum_{n=1}^{\\infty} \\tan{\\frac{\\pi}{3^n}}$"
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-75c155fdccd4971f5f8682ce95ddefdf1e4a6e75756390681940d3a6f12d0e2b | Calculate $\sum_{n=1}^{\infty}\frac{1}{F_{n}}$, where $F_{n}$ is the n-th Fibonacci number. | [
"$\\psi \\approx 3.359885666243$"
] | [
"Calculate $\\sum_{n=1}^{\\infty}\\frac{1}{F_{n}}$, where $F_{n}$ is the n-th Fibonacci number."
] | [
"aops_c6_high_school_olympiads",
"c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 14,
"genselect": 8,
"tir": 15
} | 37 | 0 | 0 |
omr-ws-828a91a6f7cf5938876ba462abbd4158592faba58da7b773da3a2af60ca5a9c6 | Calculate $\sum_{n=1}^{\infty}\frac{1}{n^2+\frac{n}{2}}$. | [
"\\(4 - 2\\ln{2}\\)"
] | [
"Calculate $\\sum_{n=1}^{\\infty}\\frac{1}{n^2+\\frac{n}{2}}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-473cfa1de9114c686c1f5d5003234f311e706942de1cb372c7d1c5ebca852a4e | Calculate $\sum_{n=1}^{\infty}\frac{1}{n^2+a^2}$ using the Fourier series of $\cosh(ax)$, where $a$ is a non-integer real number. | [
"\\(\\frac{1}{2}\\left(\\frac{\\pi}{a\\sinh(a\\pi)} - \\frac{1}{a^2}\\right)\\)"
] | [
"Calculate $\\sum_{n=1}^{\\infty}\\frac{1}{n^2+a^2}$ using the Fourier series of $\\cosh(ax)$, where $a$ is a non-integer real number."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 3,
"tir": 7
} | 21 | 0 | 0 |
omr-ws-f6f11a8d0e83bbc169ef81e0dd37128c6942f6ced878269d006bc2f821807062 | Calculate $\sum_{n=1}^{\infty}\left(\frac{1}{n^2+a^2}\right)^2$ using the Fourier series of $\cosh(ax)$, where $a$ is a non-integer real number. | [
"\\frac{\\pi}{4a^3} \\coth(\\pi a) + \\frac{\\pi^2}{4a^2 \\sinh^2(\\pi a)} - \\frac{1}{2a^4}"
] | [
"Calculate $\\sum_{n=1}^{\\infty}\\left(\\frac{1}{n^2+a^2}\\right)^2$ using the Fourier series of $\\cosh(ax)$, where $a$ is a non-integer real number."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 5
} | 17 | 0 | 0 |
omr-ws-3a258c7c0b408d24fcf5a9ee9bc3dc863efa34c3465dbf6fb3b246e57fbd0c9d | Calculate $\sum_{x=0}^{\infty} {{4+x} \choose 4} \left( \frac12 \right)^x$. | [
"32"
] | [
"Calculate $\\sum_{x=0}^{\\infty} {{4+x} \\choose 4} \\left( \\frac12 \\right)^x$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-5d79e5966c24c52a8e530ba47dbfbdc4de693e1821379b88868e0a38ba5e9888 | Calculate $\tan x + \tan 2x + \tan 3x + \cdots + \tan nx$. | [
"\\frac{\\sin \\frac{n(n+1)x}{2}}{\\prod_{k=1}^n \\cos kx}"
] | [
"Calculate $\\tan x + \\tan 2x + \\tan 3x + \\cdots + \\tan nx$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 3,
"genselect": 2
} | 5 | 0 | 0 |
omr-ws-7966c99c037d00c99e951309b80df65ebe511844552cefca7c15a6227b4e3e74 | Calculate $\tan\frac{\alpha }{2}$ if $\frac{\sin 3\alpha }{\sin \alpha }=\frac{11}{25}$. | [
"$-2,\\ -\\frac{1}{2},\\ \\frac{1}{2},\\ 2$"
] | [
"Calculate $\\tan\\frac{\\alpha }{2}$ if $\\frac{\\sin 3\\alpha }{\\sin \\alpha }=\\frac{11}{25}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0c5d7a3c0add5bc45b576487ddc2ddc4a265646d346de8ce9322b97321bc793f | Calculate $\tan\frac{\alpha }{2}+\tan\frac{\beta }{2}$ if $\sin \alpha +\sin \beta =a,\ \cos \alpha +\cos \beta =b$. | [
"$\\frac{4a}{a^2+b^2+2b}$"
] | [
"Calculate $\\tan\\frac{\\alpha }{2}+\\tan\\frac{\\beta }{2}$ if $\\sin \\alpha +\\sin \\beta =a,\\ \\cos \\alpha +\\cos \\beta =b$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8abca844b492d42a756b779f0c12ed66f5ccd341904a0e5997fed61a2e5781d3 | Calculate $\text{card}(\{a \in [0;3^{2000}] \cap \mathbb N : s_3(a)=2024 \}) \mod 10^6$, where $s_3(a)$ is the sum of all digits of $a$ in base $3$. | [
"0"
] | [
"Calculate $\\text{card}(\\{a \\in [0;3^{2000}] \\cap \\mathbb N : s_3(a)=2024 \\}) \\mod 10^6$, where $s_3(a)$ is the sum of all digits of $a$ in base $3$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 9,
"genselect": 2
} | 11 | 0 | 0 |
omr-ws-d4dcde9198add370917b6316e167701bfa81f884fa53cf37e0644992da13329c | Calculate $\underset{n\to \infty }{\mathop{\lim }}\,n{{I}_{n}}\,.$ | [
"\\(\\frac{1}{2}\\)"
] | [
"Calculate $\\underset{n\\to \\infty }{\\mathop{\\lim }}\\,n{{I}_{n}}\\,.$"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 6,
"genselect": 6
} | 12 | 0 | 0 |
omr-ws-efdb9bbd46d50ef4f1655e8c0cbb4df89d45207227d67ac607e47ea64b097103 | Calculate $\underset{x \to 0^-}{\lim}\frac{1}{x}+\sqrt{\frac{1}{x^2}+\frac{1}{x}+1}$. | [
"\\(-\\frac{1}{2}\\)"
] | [
"Calculate $\\underset{x \\to 0^-}{\\lim}\\frac{1}{x}+\\sqrt{\\frac{1}{x^2}+\\frac{1}{x}+1}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-8da385b48277ac616ab987e608f12d04aac9a795a20257192e48c31936d2050b | Calculate $\underset{x \to 0}{\lim}\frac{\sin^m(x)}{x^n}$. | [
"\n\\begin{cases}\n0 & \\text{if } m > n, \\\\\n1 & \\text{if } m = n, \\\\\n\\infty & \\text{if } m < n.\n\\end{cases}\n"
] | [
"Calculate $\\underset{x \\to 0}{\\lim}\\frac{\\sin^m(x)}{x^n}$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-8de50765af0a87faf705bf171cec54395dd4926713f9372c9fb169b1a165f167 | Calculate $\underset{x \to 1}{\lim }\frac{1-x^n}{(1-x)^n}$. | [
"\\( 0 \\) if \\( n > 1 \\) or \\( n = 0 \\), \\( 1 \\) if \\( n = 1 \\)."
] | [
"Calculate $\\underset{x \\to 1}{\\lim }\\frac{1-x^n}{(1-x)^n}$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4c879ff2f7e955a288780c4ce1a077b92481c38c3e056b08b4875aa60cc60bee | Calculate $\underset{x \to 1}{\lim }\frac{x^2+mx+1}{x-1}$ where $m \in \mathbb{R}$. | [
"0 (if \\( m = -2 \\), otherwise the limit does not exist)"
] | [
"Calculate $\\underset{x \\to 1}{\\lim }\\frac{x^2+mx+1}{x-1}$ where $m \\in \\mathbb{R}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 8,
"tir": 24
} | 37 | 0 | 0 |
omr-ws-bde357cfda6f15434f2ac3c54db8ab4a9ce83a89e26bf1c1454733f669227dc9 | Calculate $\underset{x \to 1}{\lim}\frac{(1-x)^n}{1-x^n}$. | [
"$\\infty$ for $n > 1$, $1$ for $n = 1$, and $0$ for $n = 0$."
] | [
"Calculate $\\underset{x \\to 1}{\\lim}\\frac{(1-x)^n}{1-x^n}$."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-325ad13973c4fe77f3a2a8d3f82b60c9abde51c13ccef5b7b4d838d6027b1208 | Calculate $\underset{x \to 3^+}{\lim }\frac{\sqrt{x}-\sqrt{3}+\sqrt{x-3}}{\sqrt{x^2-9}}$. | [
"$\\frac{\\sqrt{6}}{6}$"
] | [
"Calculate $\\underset{x \\to 3^+}{\\lim }\\frac{\\sqrt{x}-\\sqrt{3}+\\sqrt{x-3}}{\\sqrt{x^2-9}}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 27
} | 43 | 0 | 0 |
omr-ws-17d71b5fb795e0355a9dd3c51b8d727c6162bbb5725137c95febee5c0756f3b4 | Calculate $\underset{x \to \frac{3}{2}}{\lim}\frac{x^2([x]+2)^2}{-3[x+1]}$ where $[ \; ]$ denotes the integer part function. | [
"$-\\frac{27}{8}$"
] | [
"Calculate $\\underset{x \\to \\frac{3}{2}}{\\lim}\\frac{x^2([x]+2)^2}{-3[x+1]}$ where $[ \\; ]$ denotes the integer part function."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 15,
"tir": 27
} | 42 | 0 | 0 |
omr-ws-7faf7354e81248f750d2cae2008759e05e545513c8bea0e53da0c11096e776cb | Calculate $\underset{x \to \frac{\pi}{2}}{\lim}\frac{(1-\sin x)(1-\sin x^2) \dots (1-\sin^k x)}{\cos^{2k} x}$ where $k \in \mathbb{N}$. | [
"$\\frac{k!}{2^k}$"
] | [
"Calculate $\\underset{x \\to \\frac{\\pi}{2}}{\\lim}\\frac{(1-\\sin x)(1-\\sin x^2) \\dots (1-\\sin^k x)}{\\cos^{2k} x}$ where $k \\in \\mathbb{N}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 15,
"genselect": 7,
"tir": 8
} | 30 | 0 | 0 |
omr-ws-9edd0007b420c2ac8daedf24071d41962bd3a079de2ea651eb9011fd52b3d502 | Calculate $\{3^{\sqrt{2}}\}$ to 6 decimal places in base 10 using a calculator. | [
"0.728804"
] | [
"Calculate $\\{3^{\\sqrt{2}}\\}$ to 6 decimal places in base 10 using a calculator."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 8,
"genselect": 7,
"tir": 28
} | 43 | 0 | 0 |
omr-ws-6805fe1903554b2521d36dbaf6d6ea8a60516e899a17574a703c53bed51c3ab4 | Calculate $a(m,n)$, the number of ordered $m$-coverings of the set $A=\{a_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n\}$. | [
"\\( (2^m - 1)^{2n} \\)"
] | [
"Calculate $a(m,n)$, the number of ordered $m$-coverings of the set $A=\\{a_1,a_2,\\cdots,a_n,b_1,b_2,\\cdots,b_n\\}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 47,
"tir": 11
} | 58 | 0 | 0 |
omr-ws-909792dd692fba31fafc315efe3c2b622b445ddd8123f1f6e13d0f1141861f32 | Calculate $arctan(1-\sqrt2) + arctan(1+\sqrt2)$. | [
"\\(\\frac{\\pi}{4}\\)"
] | [
"Calculate $arctan(1-\\sqrt2) + arctan(1+\\sqrt2)$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-bbd2e8415582f64e819ab985377e2008355498a9b983c18642502a9f02ef6f74 | Calculate $b(m,n)$, the number of ordered $m$-coverings of the set $A=\{a_1,a_2,\cdots,a_n,b_1,b_2,\cdots,b_n\}$ without pairs. | [
"\\( b(m,n) = \\left(3^m - 2^{m+1} + 1\\right)^n \\)"
] | [
"Calculate $b(m,n)$, the number of ordered $m$-coverings of the set $A=\\{a_1,a_2,\\cdots,a_n,b_1,b_2,\\cdots,b_n\\}$ without pairs."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 42,
"genselect": 8,
"tir": 8
} | 58 | 0 | 0 |
omr-ws-a2d5fc133ea7d8f0007fafd0a726e6cdc6465f498925420f79d7ce79c4bfee65 | Calculate $f(-4) + f(-1) - f(4)$ given that $f(-3) = f(3)$, $f(0) = \frac{1}{3}$, and $(x+2)(x-1)f'(x) = 1$ for $x \neq -2, 1$. | [
"\\(\\frac{1+\\ln 2}{3}\\)"
] | [
"Calculate $f(-4) + f(-1) - f(4)$ given that $f(-3) = f(3)$, $f(0) = \\frac{1}{3}$, and $(x+2)(x-1)f'(x) = 1$ for $x \\neq -2, 1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 46,
"tir": 29
} | 75 | 0 | 0 |
omr-ws-e981971e87664e2588dc2486c704be8a84681b8889ee10c48b1146fdfa5e7609 | Calculate $f(3)$, given that $f(x) = x^3 + f'(-1)x^2 + f''(1)x + f'(-1)f(-1)$. | [
"198"
] | [
"Calculate $f(3)$, given that $f(x) = x^3 + f'(-1)x^2 + f''(1)x + f'(-1)f(-1)$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 14,
"tir": 9
} | 23 | 0 | 0 |
omr-ws-1722d9b96c30be1d9c7456fedbd3b6197f588ea6bb9d57d06329e549fd929e46 | Calculate $f(m, n)$ for all $m, n$ that have the same parity, where $f(m, n) = |S_{b} - S_{w}|$, $S_{b}$ is the total area of the black part of a right-angled triangle with legs of lengths $m$ and $n$ on a chessboard-patterned grid, and $S_{w}$ is the total area of the white part of the triangle. | [
"\\(\\frac{1}{2}\\)"
] | [
"Calculate $f(m, n)$ for all $m, n$ that have the same parity, where $f(m, n) = |S_{b} - S_{w}|$, $S_{b}$ is the total area of the black part of a right-angled triangle with legs of lengths $m$ and $n$ on a chessboard-patterned grid, and $S_{w}$ is the total area of the white part of the triangle."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 16,
"genselect": 3
} | 19 | 0 | 0 |
omr-ws-3d166cb4bb0505593f945e578acae51046154380aed5327a8ab7b459361a7d7f | Calculate $f(m, n)$ for all numbers $m, n$ that have the same parity, where $f(m, n) = \left|S_{b} - S_{w}\right|$ and $S_{b}$ and $S_{w}$ are the total areas of the black and white parts, respectively, of a right-angled triangle with legs of lengths $m$ and $n$ on an infinite square grid colored in a chessboard patter... | [
"0"
] | [
"Calculate $f(m, n)$ for all numbers $m, n$ that have the same parity, where $f(m, n) = \\left|S_{b} - S_{w}\\right|$ and $S_{b}$ and $S_{w}$ are the total areas of the black and white parts, respectively, of a right-angled triangle with legs of lengths $m$ and $n$ on an infinite square grid colored in a chessboard... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 5
} | 9 | 0 | 0 |
omr-ws-bc2961a8b6a750ac505f2da7410deb2643d17958d328f1e6883e6e7d3147c2c6 | Calculate $f^{(50)}(1)$ for the function $f(x) = e^{3x}\sin(3x+2)$. | [
"\\( e^3 (3\\sqrt{2})^{50} \\sin(5) \\)"
] | [
"Calculate $f^{(50)}(1)$ for the function $f(x) = e^{3x}\\sin(3x+2)$."
] | [
"aops_c7_college_math"
] | [
"tir"
] | {
"tir": 1
} | 1 | 0 | 0 |
omr-ws-813eefb72d1786915822be2a0a7ddce91281a19c29c93bbad4160115fcb301d2 | Calculate $f^{(n)}(0)$, where $f(x) = \arcsin^2(x)$. | [
" \\begin{cases} \n0 & \\text{if } n \\text{ is odd}, \\\\ \n2^{n-1} \\left( \\left( \\dfrac{n}{2} - 1 \\right)! \\right)^2 & \\text{if } n \\text{ is even} \n\\end{cases} "
] | [
"Calculate $f^{(n)}(0)$, where $f(x) = \\arcsin^2(x)$."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2789c8681258e4e60374a944ab6b93e896c32cff289207462ee141bca7a77702 | Calculate $f^{2004}(2005)$ for $f(x) = \frac{1}{1-x}$. | [
"2005"
] | [
"Calculate $f^{2004}(2005)$ for $f(x) = \\frac{1}{1-x}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 4,
"genselect": 6
} | 10 | 0 | 0 |
omr-ws-f5c749f07b596b202b6a023815f32670820a72c12f5334201d5d9e7a689cbb02 | Calculate $ord_{3^n}(2)$ for any $n \in \mathbb{N}$. | [
"\\(2 \\cdot 3^{n-1}\\)"
] | [
"Calculate $ord_{3^n}(2)$ for any $n \\in \\mathbb{N}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-895166b4e78dfd9afff03d5ba275251e6eeed4a7e54fc288f03fd7c193810a95 | Calculate $p_3$, where $p_n$ is the number of precious triplets $(A, B, C)$ of sets such that $A \cup B \cup C = A_n$ and $A \cap B \cap C$ has exactly two elements, with $A_n = \{1, 2, \ldots, n\}$. | [
"18"
] | [
"Calculate $p_3$, where $p_n$ is the number of precious triplets $(A, B, C)$ of sets such that $A \\cup B \\cup C = A_n$ and $A \\cap B \\cap C$ has exactly two elements, with $A_n = \\{1, 2, \\ldots, n\\}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 48
} | 48 | 0 | 0 |
omr-ws-aabb32496462e8ffbce2c48ffc0a399adda19b828795fcb9e3b1d23184bc8e1a | Calculate $q_{r}=P(R=r)$, where $R$ is the number of trials required for one of the objects to be sampled twice, and $2 \le r \le n+1$. | [
"\\( q_r = \\frac{(n)_{r-1} (r-1)}{n^r} \\)"
] | [
"Calculate $q_{r}=P(R=r)$, where $R$ is the number of trials required for one of the objects to be sampled twice, and $2 \\le r \\le n+1$."
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 20
} | 36 | 0 | 0 |
omr-ws-ab2d030011fe8441542bed1cfdc3f12f1585e2bb37a1801d707f2d95217ac38f | Calculate $sin ^{3}x + cos ^{3}x$ if $sinx + cosx = 1$. | [
"1"
] | [
"Calculate $sin ^{3}x + cos ^{3}x$ if $sinx + cosx = 1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0cf4575849726bb08cdf55c21bf5e335ff4df63ee4c204a222e85498b334451b | Calculate $tg20^\circ - tg40^\circ + tg80^\circ$. | [
"\\(3\\sqrt{3}\\)"
] | [
"Calculate $tg20^\\circ - tg40^\\circ + tg80^\\circ$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 29
} | 45 | 0 | 0 |
omr-ws-f3c6bda1fe14aa06ed7e789299d97cda5da65a1d8750ff0644b0e2c85911f6ee | Calculate $u_{2^{2020}} \mod (2^{89}-1)$ with $u_0=5$ and the recurrence relation $u_{n+1}=u_n(u_n^2+3)$. | [
"5"
] | [
"Calculate $u_{2^{2020}} \\mod (2^{89}-1)$ with $u_0=5$ and the recurrence relation $u_{n+1}=u_n(u_n^2+3)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 8
} | 15 | 0 | 0 |
omr-ws-8a8e75174c8da73a5ed14bd08bdac7d41561cf85a34272f25998457d9c59e3cc | Calculate $u_{2^{2020}} \mod (2^{89}-1)$ with $u_0=5$ and the recurrence relation $u_{n+1}=u_n(u_n^2-3)$. | [
"5"
] | [
"Calculate $u_{2^{2020}} \\mod (2^{89}-1)$ with $u_0=5$ and the recurrence relation $u_{n+1}=u_n(u_n^2-3)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-3747eacd9a98b5fb423bb303d9b4a8d900b6cb5a6264d1a18ccac76c0c841189 | Calculate $u_{2^{2023}} \mod (2^{89}-1)$ for the sequence defined by $u_{n+2}=u_n+u_{n+1}+3\times u_n\times u_{n+1}$ with initial conditions $u_0=1$ and $u_1=2$. | [
"0"
] | [
"Calculate $u_{2^{2023}} \\mod (2^{89}-1)$ for the sequence defined by $u_{n+2}=u_n+u_{n+1}+3\\times u_n\\times u_{n+1}$ with initial conditions $u_0=1$ and $u_1=2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 9
} | 9 | 0 | 0 |
omr-ws-0a23359e42d63a0c05e6dbca1f2cca803b895f960b9806796874518800d5c923 | Calculate $u_{2^{2^{2023}}} \mod (2^{89}-1)$ for the sequence defined by $u_{n+2}=u_n+u_{n+1}+3\times u_n\times u_{n+1}$ with initial conditions $u_0=1$ and $u_1=2$. | [
"0"
] | [
"Calculate $u_{2^{2^{2023}}} \\mod (2^{89}-1)$ for the sequence defined by $u_{n+2}=u_n+u_{n+1}+3\\times u_n\\times u_{n+1}$ with initial conditions $u_0=1$ and $u_1=2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 7
} | 7 | 0 | 0 |
omr-ws-ffc44e6d013825f4e452eeee4f7204c75a73a38e5059522109e1830a1b9bfc08 | Calculate $x^2 + y^2 + t^2 + z^2$ given the equations: \[ xz - 2yt = 3 \] \[ xt + yz = 1 \] | [
"11"
] | [
"Calculate $x^2 + y^2 + t^2 + z^2$ given the equations:\n\\[\nxz - 2yt = 3\n\\]\n\\[\nxt + yz = 1\n\\]"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 8
} | 13 | 0 | 0 |
omr-ws-d8f94d2a775e01983ea8573ef3d2f05a1f1983c08e57367979458c7441559a80 | Calculate $x^2$ if $3\tan^{-1}(2x) + 2\tan^{-1}(3x) = \pi$. | [
"\\( \\frac{67 - 5\\sqrt{145}}{72} \\)"
] | [
"Calculate $x^2$ if $3\\tan^{-1}(2x) + 2\\tan^{-1}(3x) = \\pi$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 48,
"tir": 29
} | 77 | 0 | 0 |
omr-ws-9c7fd50f8c48c6b669783cc21db25631dfd51087b022987aaffe3132531d5064 | Calculate $x^2$ if $3\tan^{-1}(3x) + 2\tan^{-1}(2x) = \pi$. | [
"\\dfrac{57 - 5\\sqrt{105}}{72}"
] | [
"Calculate $x^2$ if $3\\tan^{-1}(3x) + 2\\tan^{-1}(2x) = \\pi$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-313423d24f144f2a217a80004412e14d3ab35d1ee6bae7ae64bfa9e9debf7d7d | Calculate $x^2$ if $4\tan^{-1}x + 6\tan^{-1}(3x) = \pi$. | [
"\\( \\frac{23 - 8\\sqrt{7}}{81} \\)"
] | [
"Calculate $x^2$ if $4\\tan^{-1}x + 6\\tan^{-1}(3x) = \\pi$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 39,
"tir": 8
} | 47 | 0 | 0 |
omr-ws-4308dec28c453e31b38b4bc62490ee6e1f005a5b16324fa3ebb980aa6e372c89 | Calculate $x_{1}, x_{2}, x_{3}$ from the iteration $x_{n+1} = \frac{e^{-x_{n}}}{x_{n} +1}$ when $x_{0} = 0.45$ and hence find an approximation to the root $X$ of the equation $x^{2} + x = e^{-x}$. | [
"0.4416"
] | [
"Calculate $x_{1}, x_{2}, x_{3}$ from the iteration $x_{n+1} = \\frac{e^{-x_{n}}}{x_{n} +1}$ when $x_{0} = 0.45$ and hence find an approximation to the root $X$ of the equation $x^{2} + x = e^{-x}$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 7
} | 15 | 0 | 0 |
omr-ws-79b91346b7105259c2aa80dd2a4235288b6c88a4da0cfc6be80890e95a1f4433 | Calculate ${\zeta }'\left ( 3 \right )$. | [
"-0.198126"
] | [
"Calculate ${\\zeta }'\\left ( 3 \\right )$."
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 5,
"genselect": 5,
"tir": 25
} | 35 | 0 | 0 |
omr-ws-dee44122b8f700b0bafd2abe15193cd139f2d7fbe8ff6f121b761fb4d2592211 | Calculate $~$ $\frac{1}{2\pi}\int_{0}^{2\pi}(2\cos 2\theta)^{2n}d\theta$ | [
"\\(\\frac{(2n)!}{(n!)^2}\\)"
] | [
"Calculate $~$ $\\frac{1}{2\\pi}\\int_{0}^{2\\pi}(2\\cos 2\\theta)^{2n}d\\theta$"
] | [
"aops_c7_college_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 7
} | 23 | 0 | 0 |
omr-ws-5df020e84c87559efb8d6c01e4f19a3c110994de041744b265f2ed95a069057d | Calculate (after investigating the convergence) the integral \[\int\limits_0^1 \frac{x^a - x^b}{\ln x} \, \mathrm{d}x.\] | [
"\\(\\ln\\left(\\frac{a+1}{b+1}\\right)\\)"
] | [
"Calculate (after investigating the convergence) the integral \\[\\int\\limits_0^1 \\frac{x^a - x^b}{\\ln x} \\, \\mathrm{d}x.\\]"
] | [
"aops_c7_college_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 12,
"genselect": 5,
"tir": 29
} | 46 | 0 | 0 |
omr-ws-f197348705b777e8f6ab3ac2123a0ec8f38ced581c80cdf0833bb0682098b1a9 | Calculate (simplify) \( F(2) - F\left(\frac{1}{2}\right) \) where \( F(x) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1} x^{2n-1}}{(2n-1)^2} \). | [
"\\dfrac{\\pi \\ln 2}{2}"
] | [
"Calculate (simplify) \\( F(2) - F\\left(\\frac{1}{2}\\right) \\) where \\( F(x) = \\sum_{n=1}^{\\infty} \\frac{(-1)^{n-1} x^{2n-1}}{(2n-1)^2} \\)."
] | [
"aops_c7_college_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a5a77bb88e4e52a76ed6ae9de31316303aa2938bf1db5939fb6650ef1c46efec | Calculate 8 divided by $\frac{1}{8}.$ | [
"64"
] | [
"Calculate 8 divided by $\\frac{1}{8}.$"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-6bbb47816430569bc9083f15519bd1d76919b003c515a20b6d921ae102db9af2 | Calculate \( (935.82)^2 \). | [
"875759.0724"
] | [
"Calculate \\( (935.82)^2 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-381982ccd23429daca114b56665f0959033dbebce25d93ed85fff80b324b25cc | Calculate \( (A(x))^n \) for \( n \in \mathbb{N} \), where \( A(x) = \begin{pmatrix} 1-x & 0 & x \\ 0 & 0 & 0 \\ x & 0 & 1-x \end{pmatrix} \). | [
"\\( (A(x))^n = A((-2)^{n-1} \\cdot (x - \\frac{1}{2})^n + \\frac{1}{2}) \\)"
] | [
"Calculate \\( (A(x))^n \\) for \\( n \\in \\mathbb{N} \\), where \\( A(x) = \\begin{pmatrix} \n1-x & 0 & x \\\\ \n0 & 0 & 0 \\\\ \nx & 0 & 1-x \n\\end{pmatrix} \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dd1b53a98043fff2c7daba65f23873cbd34d3b2a4db2dc87c04120a7444f46d1 | Calculate \( 17^{2024} \mod 210 \). | [
"121"
] | [
"Calculate \\( 17^{2024} \\mod 210 \\)."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 5
} | 21 | 0 | 0 |
omr-ws-4247826267637898d3c5444b358ee6cd686e3f6eb6ad497b0a2a3221441c2bdf | Calculate \( 3^{15^{62944331} + 2} \mod 37 \). | [
"28"
] | [
"Calculate \\( 3^{15^{62944331} + 2} \\mod 37 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-fd7e925e6cb10977e95a853b71dfe4aa8da9c6b70ddf675c291a693bc1a660d8 | Calculate \( 628.306 + 6.1325 \times 44.0268 \). | [
"898.300351"
] | [
"Calculate \\( 628.306 + 6.1325 \\times 44.0268 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-a5f1c788dd625fceb31e820855e820f531607565bcfa9a90a0f28f48e99171f6 | Calculate \( 68\% \) of \( 595 \) minus \( 43\% \) of \( 372 \). | [
"244.64"
] | [
"Calculate \\( 68\\% \\) of \\( 595 \\) minus \\( 43\\% \\) of \\( 372 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-9fecf46e1b91a034739eb630428a352cc7da2d9496f375a2251ee9665b22f043 | Calculate \( 92.5\% \) of \( 550 \). | [
"508.75"
] | [
"Calculate \\( 92.5\\% \\) of \\( 550 \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c7930552a33593c9e96c9a6fbdeecfb8a9b409d1c913df419f2d939286f81c5f | Calculate \( A = \tan \frac{\pi}{7} \tan \frac{3\pi}{7} \tan \frac{5\pi}{7} \) without using complex numbers or exotic formulas. | [
"\\(-\\sqrt{7}\\)"
] | [
"Calculate \\( A = \\tan \\frac{\\pi}{7} \\tan \\frac{3\\pi}{7} \\tan \\frac{5\\pi}{7} \\) without using complex numbers or exotic formulas."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 38,
"genselect": 8,
"tir": 28
} | 74 | 0 | 0 |
omr-ws-3cc7e43b16122a3d8f6965c83b161c7ce33df83d7e3c568f789c4c762abc32d5 | Calculate \( A = a_{0} + a_{1} + a_{2} + \cdots + a_{1000} \) for the polynomial \( f(x) = (1 + x + x^2 + \cdots + x^{1000})^{1000} \), where \( f(x) = a_{0} + a_{1}x + a_{2}x^2 + \cdots + a_{10^6}x^{10^6} \). | [
"\\(\\binom{2000}{1000}\\)"
] | [
"Calculate \\( A = a_{0} + a_{1} + a_{2} + \\cdots + a_{1000} \\) for the polynomial \\( f(x) = (1 + x + x^2 + \\cdots + x^{1000})^{1000} \\), where \\( f(x) = a_{0} + a_{1}x + a_{2}x^2 + \\cdots + a_{10^6}x^{10^6} \\)."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 42,
"genselect": 8,
"tir": 13
} | 63 | 0 | 0 |
omr-ws-c83b46528cf935acba8ac38e4b3d559b58984f89158375c9b7aec0d449400f96 | Calculate \( A \) if \[ A = \sqrt[3]{\sqrt[3]{2} + \frac{\sqrt[3]{2} + 1}{3} \cdot \sqrt{\frac{8\sqrt[3]{2} - 1}{3}}} + \sqrt[3]{\sqrt[3]{2} - \frac{\sqrt[3]{2} + 1}{3} \cdot \sqrt{\frac{8\sqrt[3]{2} - 1}{3}}} \] | [
"1"
] | [
"Calculate \\( A \\) if \n\\[ A = \\sqrt[3]{\\sqrt[3]{2} + \\frac{\\sqrt[3]{2} + 1}{3} \\cdot \\sqrt{\\frac{8\\sqrt[3]{2} - 1}{3}}} + \\sqrt[3]{\\sqrt[3]{2} - \\frac{\\sqrt[3]{2} + 1}{3} \\cdot \\sqrt{\\frac{8\\sqrt[3]{2} - 1}{3}}} \\]"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-e1b69ca47f6b07808711b217504313437df61e46b2776b91fea16ee4e2621b4d | Calculate \( A^k \) where \( A = \begin{pmatrix} 1 & 2 & 3 & \cdots & n-1 & n \\ 0 & 1 & 2 & \cdots & n-2 & n-1 \\ 0 & 0 & 1 & \cdots & n-3 & n-2 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & 0 & \cdots & 1 & 2 \\ 0 & 0 & 0 & \cdots & 0 & 1 \end{pmatrix} \). | [
"\\( (I - J)^{-2k} \\)"
] | [
"Calculate \\( A^k \\) where \\( A = \\begin{pmatrix} 1 & 2 & 3 & \\cdots & n-1 & n \\\\ 0 & 1 & 2 & \\cdots & n-2 & n-1 \\\\ 0 & 0 & 1 & \\cdots & n-3 & n-2 \\\\ \\vdots & \\vdots & \\vdots & \\ddots & \\vdots & \\vdots \\\\ 0 & 0 & 0 & \\cdots & 1 & 2 \\\\ 0 & 0 & 0 & \\cdots & 0 & 1 \\end{pmatrix} \\)."
] | [
"aops_c7_college_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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