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-267ba1d1bdcf46941ee4a68e9b72047df1fc60c0d9e1b4236366993e61d1aff5 | $\sin x+\sin 2x+\sin 3x+\sin 4x=0$ | [
"$x=72^\\circ n$; $x=180^\\circ(2n+1)$; $x=90^\\circ(2n+1)$"
] | [
"$\\sin x+\\sin 2x+\\sin 3x+\\sin 4x=0$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-169d3ab9f2f048152e4ca727785c1dcb3a0004277c7b9e6dfe595c734c2c8eb0 | $\sin x+\sin 2x+\sin 3x=0$. | [
"$\\frac{\\pi }{2}k;\\ \\pm \\frac{2\\pi }{3}+2\\pi n$"
] | [
"$\\sin x+\\sin 2x+\\sin 3x=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c1b7224dcc5d900bb3183b471aceb64b56e0c1c5fe6c6ec928206ac2d137a777 | $\sin x+\sin 2x+\sin 3x=\cos x+\cos 2x+\cos 3x$ | [
"$x=\\frac{2\\pi }{3}(3n\\pm 1);\\ x=\\frac{\\pi }{8}(4n+1)$"
] | [
"$\\sin x+\\sin 2x+\\sin 3x=\\cos x+\\cos 2x+\\cos 3x$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-84ac4658bbfc5fe73c0a44553e9b634f1269591460839694a07939f338715d95 | $\sin x+\sin 3x+\sin x+\cos 3x=0$. | [
"$\\frac{\\pi }{2}+\\pi k;\\ -\\frac{\\pi }{8}+\\frac{\\pi }{2}n$"
] | [
"$\\sin x+\\sin 3x+\\sin x+\\cos 3x=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-daf5b703ae9c18f62cd7f53b65b6db5a3b676cdd0f8daffaba76f06474c0d752 | $\sin x-\cos x-4\cos^2 x\sin x=4\sin^3 x$ | [
"$x=\\pi n-\\arctan \\frac{1}{3}$"
] | [
"$\\sin x-\\cos x-4\\cos^2 x\\sin x=4\\sin^3 x$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ba910c94c88ab053b0622109f3cc8952aa1e796ff58345212f383530a65b1fa6 | $\sin x=-2/5$ $\cot x=-3/\sqrt 5$. | [
"No solution"
] | [
"$\\sin x=-2/5$ $\\cot x=-3/\\sqrt 5$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7f1eebd0c67d69576f37393bfd57b71e3de29cc91024d7438ed35bc3155002b7 | $\sin x=\frac{3}{4}\cos x$. | [
"$\\arctan\\frac{3}{4}+\\pi k$"
] | [
"$\\sin x=\\frac{3}{4}\\cos x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c39b83d37230d5ff271819dbf31d36f6df4189a399ee630748f1f134b7d709e4 | $\sin x\cos x-6\sin x+6\cos x+6=0$. | [
"$\\frac{\\pi }{2}+2\\pi k;\\ \\pi +2\\pi n$"
] | [
"$\\sin x\\cos x-6\\sin x+6\\cos x+6=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-743250d5414442b596e52156d067329547ddb0d84f6922416e23117419ef4af5 | $\sin x\sin 2x \sin 3x=\frac{1}{4}\sin 4x$ | [
"$x=\\frac{\\pi n}{2};\\ x=\\frac{\\pi }{8}(2n+1)$"
] | [
"$\\sin x\\sin 2x \\sin 3x=\\frac{1}{4}\\sin 4x$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-90fef0de6aa83a81c04f249ecf01f82c23b47a7bde993e7baa61ddb3971f3fff | $\sin x\sin 7x=\sin 3x\sin 5x$ | [
"$x=\\frac{\\pi n}{4}$"
] | [
"$\\sin x\\sin 7x=\\sin 3x\\sin 5x$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1cce5a28c1a0f9e0255d5bb6aae9c82423349643cd8607cd2ad35242a0d29b35 | $\sin(\pi -x)+\cot\left(\frac{3\pi }{2}+x\right)=\sec(-x)-\cos(2\pi -x)$ | [
"$x=180^\\circ n$"
] | [
"$\\sin(\\pi -x)+\\cot\\left(\\frac{3\\pi }{2}+x\\right)=\\sec(-x)-\\cos(2\\pi -x)$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-81db4a32ac8362a1ce94927d08df6a83162777dfc74448bec5b256eba6d6e9eb | $\sin(\pi -x)+\cot\left(\frac{\pi }{2}-x\right)=\frac{\sec x-\cos x}{2\sin x}$ | [
"$x=\\frac{2\\pi }{3}(3n\\pm 1)$"
] | [
"$\\sin(\\pi -x)+\\cot\\left(\\frac{\\pi }{2}-x\\right)=\\frac{\\sec x-\\cos x}{2\\sin x}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-68ba570f716e55cb539918e126239102d333086cb603c96845cd3422ffd9acde | $\sin(x-60^\circ)=\cos(x+30^\circ)$ | [
"$x=60^\\circ(3n+1)$"
] | [
"$\\sin(x-60^\\circ)=\\cos(x+30^\\circ)$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-40b4705b7b7a937f1c91b233ba05381175d911e732b4dcea8e032eb4d70ac0dc | $\sin\left(x+\frac{\pi }{6}\right)+\cos\left(x+\frac{\pi }{3}\right) =1+\cos 2x$. | [
"$\\frac{\\pi }{2}+\\pi k;\\ \\pm \\frac{\\pi }{3}+2\\pi n$"
] | [
"$\\sin\\left(x+\\frac{\\pi }{6}\\right)+\\cos\\left(x+\\frac{\\pi }{3}\\right) =1+\\cos 2x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4dc6a0594890d1c89595aa08bb6b1c1a5fb6de9a459d152c84c34538e05351e3 | $\sin^2 x(\tan x+1)=3\sin x(\cos x-\sin x)+3$ | [
"$x=\\frac{\\pi }{4}(4n-1);\\ x=\\frac{\\pi }{3}(3n\\pm 1)$"
] | [
"$\\sin^2 x(\\tan x+1)=3\\sin x(\\cos x-\\sin x)+3$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ba90cad8928fa21db90d2f2d17730a3c0ad18259d44f56520fce08087aa9e8dc | $\sin^2 x+3\cos^2 x-2\sin 2x=0$. | [
"$\\frac{\\pi }{4}+\\pi k;\\ \\arctan 3+\\pi n$"
] | [
"$\\sin^2 x+3\\cos^2 x-2\\sin 2x=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ca66dafc562e5712a0ecc77ef862cdaef6bfe94884000ebd09b1ae914a5e8913 | $\sin^2 x+\frac{3}{2}\cos^2 x=\frac{5}{2}\sin x\cos x$ | [
"$x=\\frac{\\pi }{4}(4n+1);\\ x=\\pi n+\\arctan\\frac{3}{2}$"
] | [
"$\\sin^2 x+\\frac{3}{2}\\cos^2 x=\\frac{5}{2}\\sin x\\cos x$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d0d9fd78d6b09c50c0dac739afb8e95b779fa1679166cc54aeb7018d416aa9db | $\sin^3 x\cos 3x+\sin 3x\cos^3 x=0.375$ | [
"$x=\\frac{\\pi n}{4}+(-1)^n \\frac{\\pi }{24}$"
] | [
"$\\sin^3 x\\cos 3x+\\sin 3x\\cos^3 x=0.375$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-42b88c2596cbd369c597dda609e4633a60e9eb4a384acbffd4d3a0bca709897e | $\sin^4 \frac{x}{3}+\cos^4 \frac{x}{3}=\frac{5}{8}$ | [
"$x=\\displaystyle\\frac{\\pi }{2}(3n\\pm 1)$"
] | [
"$\\sin^4 \\frac{x}{3}+\\cos^4 \\frac{x}{3}=\\frac{5}{8}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6cf130cbf7d88c080e53467b9dd02d329fa6947162cb6a47852ff0ad377c359b | $\sin^4 x+\sin^4 \left(x+\frac{\pi }{4}\right)=\frac{1}{4}$ {\bf 897a.} $\sin^4 x+\sin^4 \left(x+\frac{\pi }{4}\right)+\sin^4 \left(x-\frac{\pi }{4}\right) =\frac{9}{8}$ | [
"$x=\\pi n\\pm \\frac{1}{2}\\arccos \\left(-1+\\frac{\\sqrt 6}{2}\\right)$"
] | [
"$\\sin^4 x+\\sin^4 \\left(x+\\frac{\\pi }{4}\\right)=\\frac{1}{4}$ {\\bf 897a.} $\\sin^4 x+\\sin^4 \\left(x+\\frac{\\pi }{4}\\right)+\\sin^4 \\left(x-\\frac{\\pi }{4}\\right) =\\frac{9}{8}$ "
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9f3e853b895793b9d73a452a1ad2bebb9e05550b0fb4bf283d07a7b6a6467393 | $\sqrt 2\cos 2x=\cos x+\sin x$. | [
"$\\frac{\\pi }{12}+\\frac{2}{3}\\pi k;\\ -\\frac{\\pi }{4}+2\\pi n$"
] | [
"$\\sqrt 2\\cos 2x=\\cos x+\\sin x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-21974169f85fa3f03ff693c478ff32607977102ebd1f6a9fd9929d7d65e6a8e3 | $\sqrt 2\sin^2 x+\cos x=0$. | [
"$\\pm \\frac{3\\pi }{4}+2\\pi k$"
] | [
"$\\sqrt 2\\sin^2 x+\\cos x=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cf67f24167ee0ac9f319da40a855b118feff43c1925d65d630d8ab1559b498bb | $\sqrt 3\sin 2x+\cos 2x=\sqrt 2$. | [
"$\\frac{\\pi }{6}\\pm \\frac{\\pi }{8}+\\pi k$"
] | [
"$\\sqrt 3\\sin 2x+\\cos 2x=\\sqrt 2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-77ba0d96d1d3377ccdb1fdd221786615009d27eb1161d189dbe0c751e30db77d | $\sqrt 3\sin x-\sqrt {2\sin ^2 x-\sin 2x+3\cos ^2 x}=0$. | [
"$\\frac{\\pi }{4}+2\\pi k;\\ -\\arctan 3+\\pi (2n+1)$"
] | [
"$\\sqrt 3\\sin x-\\sqrt {2\\sin ^2 x-\\sin 2x+3\\cos ^2 x}=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b4000635a01f36c7f3d34a85ee8489372fa9d76c26f2c1fa23e80a6063f12fcd | $\sqrt [6]{8x\left(7+4\sqrt 3\right)}\sqrt [3]{2\sqrt {6x}-4\sqrt {2x}}$ | [
"$-2\\sqrt [3]{x}$. "
] | [
"$\\sqrt [6]{8x\\left(7+4\\sqrt 3\\right)}\\sqrt [3]{2\\sqrt {6x}-4\\sqrt {2x}}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7870b4a6d3d79d2c25a3df6ec13bdc8d3358da16fc5f5ecf73fcddb28cc502f5 | $\sqrt [x-y]{x+y}=2\sqrt 3$ \ $(x+y)2^{y-x}=3$. | [
"$x=7,\\ y=5$"
] | [
"$\\sqrt [x-y]{x+y}=2\\sqrt 3$ \\ $(x+y)2^{y-x}=3$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f6fe24a7ce2d9c73486473d3dd2ee677e525fb742325ea5dd6d53f738b62ccc5 | $\sqrt [x-y]{x+y}=\frac{\sqrt {52-2x}}{\sqrt [4]{x-y}}$ $\frac{3}{2}\log _8(x-y)-\log _{1/\sqrt 2}(x-y)=5$. | [
"$x=20,\\ y=16$"
] | [
"$\\sqrt [x-y]{x+y}=\\frac{\\sqrt {52-2x}}{\\sqrt [4]{x-y}}$ $\\frac{3}{2}\\log _8(x-y)-\\log _{1/\\sqrt 2}(x-y)=5$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3170549d8c4e219277baaf4fa087a0082237a1044f2e002985b8a322d6ba6853 | $\sqrt x+1=\sqrt {2x}+1$ and $\left(\sqrt x+2\right)^2=\left(\sqrt {2x}+1\right)^2$. | [
"Yes"
] | [
"$\\sqrt x+1=\\sqrt {2x}+1$ and $\\left(\\sqrt x+2\\right)^2=\\left(\\sqrt {2x}+1\\right)^2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e86d2cc5c7d8ef3c070f7edd0c34a002ff976d5e7cd0741fadc87a4d1c0be24d | $\sqrt {-3\sin 5x-\cos ^2 x-3}+\sin x=1$. | [
"$\\emptyset $"
] | [
"$\\sqrt {-3\\sin 5x-\\cos ^2 x-3}+\\sin x=1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5ba96fe4cc16269cc5d24b33847da94fad4a6e08025ebe6e6663816585de7448 | $\sqrt {1-2\tan x}-\sqrt {1+2\cot x}=2$. | [
"$-\\frac{3\\pi }{8}+\\pi k$"
] | [
"$\\sqrt {1-2\\tan x}-\\sqrt {1+2\\cot x}=2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0c3e0ed186038a4b9db6edae094f7b4e7037556098a2be1fbcf289ad887054f4 | $\sqrt {2+4\cos x}=\frac{1}{2}+3\cos x$. | [
"$\\pm \\frac{\\pi }{3}+2\\pi k$"
] | [
"$\\sqrt {2+4\\cos x}=\\frac{1}{2}+3\\cos x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c31e04f6425bb64ab25cca690bada72c6d9907e000769eccf0bc20422d096573 | $\sqrt {3+2\tan x-\tan ^2 x}=\frac{1+3\tan x}{2}$. | [
"$\\frac{\\pi }{4}+\\pi k$"
] | [
"$\\sqrt {3+2\\tan x-\\tan ^2 x}=\\frac{1+3\\tan x}{2}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2b011295da39bca6471139027203f8475634ab11fbcd4697cb8d2e17cd67b82f | $\sqrt {5-2\sin x}=6\sin x-1$. | [
"$(-1)^k \\frac{\\pi }{6}+\\pi k$"
] | [
"$\\sqrt {5-2\\sin x}=6\\sin x-1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3cbc788615f9c8c5a1633daf5f30acaa606ead59711666741706ebf12a3ef27e | $\sqrt {\cos 2x}+\sqrt {1+\sin 2x}=2\sqrt {\sin x+\cos x}$. | [
"$-\\frac{\\pi }{4}+\\pi k;\\ 2\\pi n$"
] | [
"$\\sqrt {\\cos 2x}+\\sqrt {1+\\sin 2x}=2\\sqrt {\\sin x+\\cos x}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a917fb9039302caf293573bf6f882b1370c9cdf345fb4c46783e668999943b63 | $\sqrt {\cos ^2 x+\frac{1}{2}}+\sqrt {\sin ^2 x+\frac{1}{2}}=2$. | [
"$\\pm \\frac{\\pi }{4}+2\\pi k$"
] | [
"$\\sqrt {\\cos ^2 x+\\frac{1}{2}}+\\sqrt {\\sin ^2 x+\\frac{1}{2}}=2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d260ac801047ffcf8ef64b0fa198038a6a514f1d0e11e8c3bb33569b69ebd17a | $\sqrt {\frac{(1+a)\sqrt [3]{1+a}}{3a}}\cdot \sqrt [3]{\sqrt 3}{9+18a^{-1}+9a^{-2}}$ | [
"$\\frac{1}{3}\\sqrt [6]{a}$"
] | [
"$\\sqrt {\\frac{(1+a)\\sqrt [3]{1+a}}{3a}}\\cdot \\sqrt [3]{\\sqrt 3}{9+18a^{-1}+9a^{-2}}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6eb8905eab745225dcd52dab984b9795407f518863b8a641692f69c3e62aea77 | $\sqrt {a(2^x-2)+1}=1-2^x$. | [
"$x=\\log _2 a$ for $0<a\\le 1$, there are no solutions for the remaining values of $a$"
] | [
"$\\sqrt {a(2^x-2)+1}=1-2^x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0a4af17c323206cb866707bcf3baa9fe9383ea28d0e03816c425c03c04440e92 | $\sqrt {x+y}+\sqrt {2x+4y}=\sqrt 2+4$ \ $\sqrt {x+2y}-\sqrt {2x+2y}=2\sqrt 2-2$. | [
"$x=-4,\\ y=6$"
] | [
"$\\sqrt {x+y}+\\sqrt {2x+4y}=\\sqrt 2+4$ \\ $\\sqrt {x+2y}-\\sqrt {2x+2y}=2\\sqrt 2-2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b65ca55d17dea64f66463431732d07a04961157c8d3511b375f59cc880fa245d | $\sqrt {|x|+1}-\sqrt {|x|}=a$. | [
"$x_{1,2}=\\pm (1-a^2)^2/4a^2$ for $0<a<1$, $x=0$ for $a=1$, and there are no solutions for the remaining values of $a$"
] | [
"$\\sqrt {|x|+1}-\\sqrt {|x|}=a$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9bdf678c2284c16ce8454eb399bf61c44e2e48ffca4c6d9e6fcca7d56a5c7a84 | $\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}=$ | [
"1"
] | [
"$\\sqrt[3]{2+\\sqrt{5}}+\\sqrt[3]{2-\\sqrt{5}}=$"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-5e878163cd7bce203db78813d36619b60a40c6ba51610234ac13bc4898a96691 | $\sqrt{53+20\sqrt{7}}$ can be written in the form $a+b\sqrt{c}$, where $a,$ $b,$ and $c$ are integers and $c$ has no factors which is a perfect square of any positive integer other than 1. Find $a+b+c$. | [
"14"
] | [
"$\\sqrt{53+20\\sqrt{7}}$ can be written in the form $a+b\\sqrt{c}$, where $a,$ $b,$ and $c$ are integers and $c$ has no factors which is a perfect square of any positive integer other than 1. Find $a+b+c$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fc7a416fc12392b6a8fe6338883ce1099c63f17d7277bcd2f4134955d44164f3 | $\tan ^2 2x+2\sqrt 3\tan 2x+3=-\cot^2 \left(4y-\frac{\pi }{6}\right)$. | [
"$\\left(-\\frac{\\pi }{6}+\\frac{\\pi }{2}k;\\frac{\\pi }{6}+\\frac{\\pi }{4}n\\right)$"
] | [
"$\\tan ^2 2x+2\\sqrt 3\\tan 2x+3=-\\cot^2 \\left(4y-\\frac{\\pi }{6}\\right)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-81d2116ea994e2b66540bf6d27f3b4d4e5d7fcf82ea5c5c8a658e5c58b2ecfd5 | $\tan x+\cot x-\cos 4x=3$. | [
"$\\frac{\\pi }{4}+\\pi k;\\ (-1)^n \\frac{1}{2}\\arcsin \\frac{\\sqrt 5-1}{2} +\\frac{\\pi }{2}n$"
] | [
"$\\tan x+\\cot x-\\cos 4x=3$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7da9325126eb2bc2e2bbf6e2dcd989961adc8198ea08d05d081f3a9ccfb2135f | $\tan x+\frac{1}{9}\cot x=\sqrt {\frac{1}{\cos ^2 x}-1}-1$. | [
"$-\\arctan \\frac{1}{6}+\\pi k;\\ -\\arctan \\frac{1}{3}+\\pi n$"
] | [
"$\\tan x+\\frac{1}{9}\\cot x=\\sqrt {\\frac{1}{\\cos ^2 x}-1}-1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-836b85073b9dcf795a1a248b04d0d3d11c4396b504befa350fb50fc22c9252b4 | $\tan x+\tan 2x=\tan 3x$ | [
"$x=\\frac{\\pi n}{3}$"
] | [
"$\\tan x+\\tan 2x=\\tan 3x$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-0e133ad15460638cbb7ffe6a963993fc6cdfb2b127b219424203827aa635c5fe | $\tan x-1+\sin x(1-\tan x)+\frac{1}{1+\tan^2 x}$ | [
"$4\\tan x\\cos^2 \\frac{x}{2}\\sin^2 \\left(45^\\circ-\\frac{x}{2}\\right)$"
] | [
"$\\tan x-1+\\sin x(1-\\tan x)+\\frac{1}{1+\\tan^2 x}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f192c0e290b91762dc652a44c024678b288d9f4e650915885f5981118c4ac17d | $\tan\left(\alpha +\frac{\pi }{4}\right)+\tan\left(\alpha -\frac{\pi }{4}\right)$ | [
"$2\\tan 2\\alpha $"
] | [
"$\\tan\\left(\\alpha +\\frac{\\pi }{4}\\right)+\\tan\\left(\\alpha -\\frac{\\pi }{4}\\right)$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4704bd2321b2079785cd62d701fe7aecbc40433ec87a1f72d9352a14db4adbab | $\tan\left(\frac{\pi }{4}+x\right)+\tan x-2=0$ | [
"$x=\\pi n+\\arctan \\left(2\\pm \\sqrt 3\\right)$"
] | [
"$\\tan\\left(\\frac{\\pi }{4}+x\\right)+\\tan x-2=0$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b136d7c868b7598b52f72178f81e39c888902bf16fd1cc22f5d218eebdd7b583 | $\tan^2 \pi (x+y)+\cot^2 \pi (x+y)=\sqrt {\frac{2x}{x^2+1}}+1$. | [
"$x=1$, $y=(2k-3)/4$"
] | [
"$\\tan^2 \\pi (x+y)+\\cot^2 \\pi (x+y)=\\sqrt {\\frac{2x}{x^2+1}}+1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6e6296b6b22df25a95ac866568ea3d498d51fcd4be22fb05c4155deda8c84cd9 | $\tan^2 x+2\tan x(\sin y+\cos y)+2=0$. | [
"$x_1=-\\arctan \\sqrt 2+k\\pi $, $y_1=(\\pi /4)+2n\\pi $; $x_2=\\arctan\\sqrt 2+k\\pi $, $y_2=(5\\pi /4)+2n\\pi $. Reduce the equation to the form $[\\tan x+(\\sin y+\\cos y)]^2+(1-\\sin 2y)=0$"
] | [
"$\\tan^2 x+2\\tan x(\\sin y+\\cos y)+2=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d13c24af680fedb9f551bc4bfcf3fa525c2e6a5ebd6c5b6afefa087d5d419e2c | $\tan^3 x+\tan^2 x-3\tan x=3$. | [
"$\\pm \\frac{\\pi }{3}+\\pi k;\\ -\\frac{\\pi }{4}+\\pi k$"
] | [
"$\\tan^3 x+\\tan^2 x-3\\tan x=3$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ca2a49fb599312f522ecb3d955ded0ecc300984a75ce0cc9860632fabd66a4f9 | $\textbf{Juan's Old Stamping Grounds}$ Juan organizes the stamps in his collection by country and by the decade in which they were issued. The prices he paid for them at a stamp shop were: Brazil and France, $6$ cents each, Peru $4$ cents each, and Spain $5$ cents each. (Brazil and Peru are South American countries and... | [
"\\$ 1.06"
] | [
"$\\textbf{Juan's Old Stamping Grounds}$\n\nJuan organizes the stamps in his collection by country and by the decade in which they were issued. The prices he paid for them at a stamp shop were: Brazil and France, $6$ cents each, Peru $4$ cents each, and Spain $5$ cents each. (Brazil and Peru are South American coun... | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-d8ecfdc46b3fbdd9d4bbdc9ea41a805bbf8ad085ad7ce4e0a1eed3794b6054c3 | $\triangle ABC$ has all of its vertices on the parabola $y=x^{2}$. The slopes of $AB$ and $BC$ are $10$ and $-9$, respectively. If the $x$-coordinate of the triangle's centroid is $1$, find the area of $\triangle ABC$. | [
"665"
] | [
"$\\triangle ABC$ has all of its vertices on the parabola $y=x^{2}$. The slopes of $AB$ and $BC$ are $10$ and $-9$, respectively. If the $x$-coordinate of the triangle's centroid is $1$, find the area of $\\triangle ABC$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-fb385c498961bdad848513c919013dc5bf6929336bc5091437477e7a38b343fd | $\triangle ABC$ has all of its vertices on the parabola $y=x^{2}$. The slopes of $AB$ and $BC$ are $10$ and $-9$, respectively. If the x-coordinate of the triangle's centroid is $1$, find the area of $\triangle ABC$. | [
"665"
] | [
"$\\triangle ABC$ has all of its vertices on the parabola $y=x^{2}$. The slopes of $AB$ and $BC$ are $10$ and $-9$, respectively. If the x-coordinate of the triangle's centroid is $1$, find the area of $\\triangle ABC$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-4b34aac7be90e83d2a3f9b4d1e986e96cd9d6950f90d0986729a7f3bb6c2f73e | $\triangle ABC$ has side lengths $AB=7$, $BC=8$, and $AC=10$. A point $D$ is on $AC$ such that $AD \cong BC$, and $BD$ intersects the circumcircle of $\triangle ABC$ at $E$. Then the length of $AE$ can be expressed as $\frac{a}{\sqrt{b}}$, where $a$ and $b$ are positive integers. Compute $a+b$. | [
"69"
] | [
"$\\triangle ABC$ has side lengths $AB=7$, $BC=8$, and $AC=10$. A point $D$ is on $AC$ such that $AD \\cong BC$, and $BD$ intersects the circumcircle of $\\triangle ABC$ at $E$. Then the length of $AE$ can be expressed as $\\frac{a}{\\sqrt{b}}$, where $a$ and $b$ are positive integers. Compute $a+b$."
] | [
"aops_c4_high_school_math"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f8bb467790847cbdcc0f340a0a43a8e98a8ad4d9fcd274111d0a7de5ed380d65 | $\triangle ABC$ is a triangle such that $AB=5$, $AC=3$, and $BC=4$. Points $D$ and $E$ are on side $AB$ such that $AD=2$ and $BE=2$. Find the area of $\triangle DEC$. | [
"\\dfrac{6}{5}"
] | [
"$\\triangle ABC$ is a triangle such that $AB=5$, $AC=3$, and $BC=4$. Points $D$ and $E$ are on side $AB$ such that $AD=2$ and $BE=2$. Find the area of $\\triangle DEC$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-648bee7601379a139f1374643d84520cd32862cfcc2918768fa1a1f5b76488f7 | $\triangle ABC$ is a triangle with $AB = 5$, $BC = 6$, and $CA = 7$. Squares are drawn on each side. Find the area of hexagon $DEFGHI$. | [
"\\(24\\sqrt{6} + 110\\)"
] | [
"$\\triangle ABC$ is a triangle with $AB = 5$, $BC = 6$, and $CA = 7$. Squares are drawn on each side. Find the area of hexagon $DEFGHI$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 3,
"tir": 2
} | 8 | 0 | 0 |
omr-ws-3a30f810aa428aa12dffa293c40fd500fe57ec6bdd66e992d7a26d6c7a0150f6 | $\triangle ABC$ is an equilateral paper triangle with sides that each measure 5. $\triangle DEF$ is an equilateral triangle drawn in the interior of $\triangle ABC$ with sides that measure 2 so that the two triangles share the same centroid, $\overline {EF} \parallel \overline {BC}$, and $E$ is closer to $B$ than to $C... | [
"144"
] | [
"$\\triangle ABC$ is an equilateral paper triangle with sides that each measure 5. $\\triangle DEF$ is an equilateral triangle drawn in the interior of $\\triangle ABC$ with sides that measure 2 so that the two triangles share the same centroid, $\\overline {EF} \\parallel \\overline {BC}$, and $E$ is closer to $B$... | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 6
} | 18 | 0 | 0 |
omr-ws-8d7be1458e4c56a88ce362c364affddb7b74d54a93a54da497d73776d73329e1 | $\triangle ABC$ is an equilateral triangle with side length $3$. Point $D$ lies on the segment $BC$ such that $BD = 1$ and $E$ lies on $AC$ such that $AE = AD$. Compute the area of $\triangle ADE$. | [
"\\(\\frac{\\sqrt{21}}{2}\\)"
] | [
"$\\triangle ABC$ is an equilateral triangle with side length $3$. Point $D$ lies on the segment $BC$ such that $BD = 1$ and $E$ lies on $AC$ such that $AE = AD$. Compute the area of $\\triangle ADE$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 30
} | 46 | 0 | 0 |
omr-ws-7cb002accf7d83b9fa7dc26b419188c4bf48a62c385f54bc6ae253cc437121aa | $\triangle ABC$ is inscribed in circle $\omega$ with $AB = 5$, $BC = 7$, $AC = 3$. The bisector of $\angle A$ meets side $BC$ at $D$ and circle $\omega$ at a second point $E$. Let $\gamma$ be the circle with diameter $DE$. Circles $\omega$ and $\gamma$ meet at $E$ and at a second point $F$. Then $AF^{2} = \frac{m}{n}$ ... | [
"919"
] | [
"$\\triangle ABC$ is inscribed in circle $\\omega$ with $AB = 5$, $BC = 7$, $AC = 3$. The bisector of $\\angle A$ meets side $BC$ at $D$ and circle $\\omega$ at a second point $E$. Let $\\gamma$ be the circle with diameter $DE$. Circles $\\omega$ and $\\gamma$ meet at $E$ and at a second point $F$. Then $AF^{2} = \... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-c184684688dab638a0bf97ef2d390ed57a58a2d15619dd52d269eb0579f18d1f | $\triangle ABC$ satisfies $AB = 3$, $BC = 4$, $CA = 5$ and is circumscribed in circle $\omega$ with center $O$. The tangents to $\omega$ at $A, B, C$ are lines $\ell_A$, $\ell_B$, $\ell_C$ respectively. Let $P_1 = \ell_A \cap \ell_B$ and $P_2 = \ell_B \cap \ell_C$ ($\cap$ refers to the intersection between two lines). ... | [
"\\dfrac{3}{2}"
] | [
"$\\triangle ABC$ satisfies $AB = 3$, $BC = 4$, $CA = 5$ and is circumscribed in circle $\\omega$ with center $O$. The tangents to $\\omega$ at $A, B, C$ are lines $\\ell_A$, $\\ell_B$, $\\ell_C$ respectively. Let $P_1 = \\ell_A \\cap \\ell_B$ and $P_2 = \\ell_B \\cap \\ell_C$ ($\\cap$ refers to the intersection be... | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 12,
"genselect": 8
} | 20 | 0 | 0 |
omr-ws-ad6908d3f538a0b4c82241b938bd5844b764e4b27f52d0790ea40df04306c1d5 | $\triangle DEF$ is inscribed inside $\triangle ABC$ such that $D,E,F$ lie on $BC, AC, AB$, respectively. The circumcircles of $\triangle DEC, \triangle BFD, \triangle AFE$ have centers $O_1,O_2,O_3$, respectively. Also, $AB = 23, BC = 25, AC=24$, and $\stackrel{\frown}{BF} = \stackrel{\frown}{EC},\ \stackrel{\frown}{AF... | [
"14"
] | [
"$\\triangle DEF$ is inscribed inside $\\triangle ABC$ such that $D,E,F$ lie on $BC, AC, AB$, respectively. The circumcircles of $\\triangle DEC, \\triangle BFD, \\triangle AFE$ have centers $O_1,O_2,O_3$, respectively. Also, $AB = 23, BC = 25, AC=24$, and $\\stackrel{\\frown}{BF} = \\stackrel{\\frown}{EC},\\ \\sta... | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 5
} | 12 | 0 | 0 |
omr-ws-089c34f87668e27e9ff23121dbd3ab6ac6fba17d3e4ab200d3ad3a542d044e4a | $\zeta_1, \zeta_2,$ and $\zeta_3$ are complex numbers such that \[\zeta_1+\zeta_2+\zeta_3=1\]\[\zeta_1^2+\zeta_2^2+\zeta_3^2=3\]\[\zeta_1^3+\zeta_2^3+\zeta_3^3=7\] Compute $\zeta_1^{7} + \zeta_2^{7} + \zeta_3^{7}$. | [
"71"
] | [
"$\\zeta_1, \\zeta_2,$ and $\\zeta_3$ are complex numbers such that\n\\[\\zeta_1+\\zeta_2+\\zeta_3=1\\]\\[\\zeta_1^2+\\zeta_2^2+\\zeta_3^2=3\\]\\[\\zeta_1^3+\\zeta_2^3+\\zeta_3^3=7\\]\nCompute $\\zeta_1^{7} + \\zeta_2^{7} + \\zeta_3^{7}$.\n"
] | [
"MATH_training_set"
] | [
"cot",
"genselect"
] | {
"cot": 15,
"genselect": 8
} | 23 | 0 | 0 |
omr-ws-8add6858ec452f42b3d1367849f92527fc3416f657bedfc2d6a30902e4a2d9cd | $a$ and $b$ are nonnegative real numbers such that $\sin(ax + b) = \sin(29x)$ for all integers $x$. Find the smallest possible value of $a$. | [
"10\\pi - 29"
] | [
"$a$ and $b$ are nonnegative real numbers such that $\\sin(ax + b) = \\sin(29x)$ for all integers $x$. Find the smallest possible value of $a$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 7,
"genselect": 7
} | 14 | 0 | 0 |
omr-ws-fb8b4de0e1aa5779c10ba52d104be98d0a0e04dc6c409a1f141ef956afb36446 | $a$ and $b$ are positive integers so that $20a+12b$ and $20b-12a$ are both powers of $2$, but $a+b$ is not. Find the minimum possible value of $a + b$. | [
"482"
] | [
"$a$ and $b$ are positive integers so that $20a+12b$ and $20b-12a$ are both powers of $2$, but $a+b$ is not. Find the minimum possible value of $a + b$."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-14c6be6b2a94cfbc19ee2f53ac0439222baebe2f82e938b0eefceb0c28428bf8 | $a$ and $b$ are relatively prime positive integers such that each of them has at least 3 divisors. What is the smallest possible value of $a + b$? | [
"13"
] | [
"$a$ and $b$ are relatively prime positive integers such that each of them has at least 3 divisors. What is the smallest possible value of $a + b$?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-3f724c2c49e047fc01b0287b8e2d34bfa678b714c7b0b6e680daa24b2bb9deae | $a$, $b$, and $c$ are integers such that $$0 < ab^2 + a^2c + bc^2 - ac^2 - b^2c - a2b \ge 2020$$. Find the maximum possible value of $a - b$. | [
"46"
] | [
"$a$, $b$, and $c$ are integers such that $$0 < ab^2 + a^2c + bc^2 - ac^2 - b^2c - a2b \\ge 2020$$. Find the maximum possible value of $a - b$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-28b9a538db8cb427ea404d5544035bc5e7ff3428ef8053d91554bcf67c43b871 | $a, b, c$ are distinct real roots of $x^3 - 3x + 1 = 0$. What is $a^8 + b^8 + c^8$? | [
"186"
] | [
"$a, b, c$ are distinct real roots of $x^3 - 3x + 1 = 0$. What is $a^8 + b^8 + c^8$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-c78d2c341041b067b802fcc3e27b80e678dd1105d85acfb745e96f671acca81d | $a, b, c, d$ are positive numbers with product 1. What is the minimum of $a^2+b^2+c^2+d^2+ab+ac+ad+bc+bd+cd$? | [
"10"
] | [
"$a, b, c, d$ are positive numbers with product 1. What is the minimum of $a^2+b^2+c^2+d^2+ab+ac+ad+bc+bd+cd$?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-4468569a96f33ba887d184c3052d6999d0321a23c203115485fa39f3298872bd | $a\%$ of some quantity of goods is sold at a profit of $p\%$ and $b\%$ of the rest of the goods is sold at a profit of $q\%$. What profit is made on selling the remaining goods, if the total profit is $r\%$? | [
"$\\displaystyle\\frac{r-\\displaystyle\\frac{ap}{100}-\\displaystyle\\frac{bq}{100}\\left(1-\\displaystyle\\frac{a}{100}\\right)} {\\left(1-\\displaystyle\\frac{a}{100}\\right)\\left(1-\\displaystyle\\frac{b}{100}\\right)}\\%$"
] | [
"$a\\%$ of some quantity of goods is sold at a profit of $p\\%$ and $b\\%$ of the rest of the goods is sold at a profit of $q\\%$. What profit is made on selling the remaining goods, if the total profit is $r\\%$?"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d6bc52ca0216efa0f7e8b68ebdb377b226bd296951324e338afff21855c2d7a0 | $ab\sqrt [n]{a^{1-n}b^{-n}-a^{-n}b^{1-n}} \cdot \sqrt [n]{(a-b)^{-1}}$ | [
"1"
] | [
"$ab\\sqrt [n]{a^{1-n}b^{-n}-a^{-n}b^{1-n}} \\cdot \\sqrt [n]{(a-b)^{-1}}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-44f11a17f996e2c41acbca102ae84dbfa3484e45c22c19719ac03cb636455a01 | $f (x) = x + 3$ and $g(x) = x^2 -6$, what is the value of $f (g(2))$? | [
"1"
] | [
"$f (x) = x + 3$ and $g(x) = x^2 -6$, what is the value of $f (g(2))$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f3875a7010bc2e0d06579dff2f50450ec9e5e8ddf7c65109722beca7b7a2e3a3 | $f$ is a bijective function from the set $\{0, 1, 2, ..., 11\}$ to $\{0, 1, 2, ..., 11\}$, with the property that whenever $a$ divides $b$, $f(a)$ divides $f(b)$. How many such $f$ are there? | [
"2"
] | [
"$f$ is a bijective function from the set $\\{0, 1, 2, ..., 11\\}$ to $\\{0, 1, 2, ..., 11\\}$, with the property that whenever $a$ divides $b$, $f(a)$ divides $f(b)$. How many such $f$ are there?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 14,
"genselect": 7
} | 21 | 0 | 0 |
omr-ws-2a3b113a853fe3e4d9ab73cff1893ddbfee4a16cfb6930ba1e1087c5e7068b0c | $f(x)$ is a monic polynomial such that $f(0)=4$ and $f(1)=10$. If $f(x)$ has degree $2$, what is $f(x)$? Express your answer in the form $ax^2+bx+c$, where $a$, $b$, and $c$ are real numbers. | [
"x^2+5x+4"
] | [
"$f(x)$ is a monic polynomial such that $f(0)=4$ and $f(1)=10$. If $f(x)$ has degree $2$, what is $f(x)$? Express your answer in the form $ax^2+bx+c$, where $a$, $b$, and $c$ are real numbers."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-f9a5dba9f8f30fed8472a582249679b7e925818d01305629637f8aaeaa011cd0 | $k$ smaller boxes are put in a box. Then in each of the smaller boxes either $k$ still smaller boxes are put or no boxes and so on. Determine the number of empty boxes if it is known that there are $m$ filled boxes. | [
"m(k-1)+1"
] | [
"$k$ smaller boxes are put in a box. Then in each of the smaller boxes either $k$ still smaller boxes are put or no boxes and so on. Determine the number of empty boxes if it is known that there are $m$ filled boxes. "
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ae35aacc7ffe2cb57658e0dd1e229d74a4fa606c7cfa6b17f376429eeaf94c39 | $k, a_2, a_3$ and $k, b_2, b_3$ are both nonconstant geometric sequences with different common ratios. We have $$a_3-b_3=3(a_2-b_2).$$Find the sum of the common ratios of the two sequences. | [
"3"
] | [
"$k, a_2, a_3$ and $k, b_2, b_3$ are both nonconstant geometric sequences with different common ratios. We have $$a_3-b_3=3(a_2-b_2).$$Find the sum of the common ratios of the two sequences."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-282ab0a81cb8e3187e252cb80bae020420ae57e9b7bff65bee2d38a2c94086df | $k_1, k_2, k_3$ are three circles. $k_2$ and $k_3$ touch externally at $P$, $k_3$ and $k_1$ touch externally at $Q$, and $k_1$ and $k_2$ touch externally at $R$. The line $PQ$ meets $k_1$ again at $S$, the line $PR$ meets $k_1$ again at $T$. The line $RS$ meets $k_2$ again at $U$, and the line $QT$ meets $k_3$ again at... | [
"2"
] | [
"$k_1, k_2, k_3$ are three circles. $k_2$ and $k_3$ touch externally at $P$, $k_3$ and $k_1$ touch externally at $Q$, and $k_1$ and $k_2$ touch externally at $R$. The line $PQ$ meets $k_1$ again at $S$, the line $PR$ meets $k_1$ again at $T$. The line $RS$ meets $k_2$ again at $U$, and the line $QT$ meets $k_3$ aga... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-59b68dc7469882e684a021c3ce4327dc47428c74388a708ba05208cd8bcb72f9 | $m$ points are taken on one of two given parallel lines and $n$ points on the other. Join with line segments each of the $m$ points on the former line to each of the $n$ points on the latter. What is the number of points of intersection of the segments if it is known that there are no points in which three or more segm... | [
"\\frac{m(m-1)n(n-1)}{4}"
] | [
"$m$ points are taken on one of two given parallel lines and $n$ points on the other. Join with line segments each of the $m$ points on the former line to each of the $n$ points on the latter. What is the number of points of intersection of the segments if it is known that there are no points in which three or more... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-46ea9158ed520ba41bf108969bcbdff264342757f7f2aac8d6ddaf88b475d576 | $n - 1$ planes are drawn parallel to one face of a regular tetrahedron so that they divide the altitude to that face into $n$ equal segments, and the same is done for the other three faces. Into how many regions do these planes divide the interior of the tetrahedron? | [
"\\dfrac{n(n + 1)(n + 2)(n + 3)}{24}"
] | [
"$n - 1$ planes are drawn parallel to one face of a regular tetrahedron so that they divide the altitude to that face into $n$ equal segments, and the same is done for the other three faces. Into how many regions do these planes divide the interior of the tetrahedron?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 8,
"genselect": 7
} | 15 | 0 | 0 |
omr-ws-0b566b3e3ebd7ba8d75d891f02a4a45eabf2737784dd7d5e6cfdc7cdb660905a | $n$ balls are placed independently and uniformly at random into $n$ boxes. One box is selected at random and found to contain $b$ balls. Let $e_n$ be the expected value of $b^4$. Find $$\lim_{n \to \infty} e_n.$$ | [
"15"
] | [
"$n$ balls are placed independently and uniformly at random into $n$ boxes. One box is selected at random and found to contain $b$ balls. Let $e_n$ be the expected value of $b^4$. Find\n$$\\lim_{n \\to \\infty} e_n.$$"
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-04bcacfd04fd8d258db527ef71502f52e1687a20c92879f268a9c767a11d7979 | $n$ coins are simultaneously flipped. The probability that at most one of them shows tails is $\frac{3}{16}$. Find $n$. | [
"5"
] | [
"$n$ coins are simultaneously flipped. The probability that at most one of them shows tails is $\\frac{3}{16}$. Find $n$."
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-2535ac3889adbdf55ae1bdd4377df0f5a77e9c732ed3fc1c93b5f1b5a0ec62fa | $n$ cups are arranged in a circle. A coin is hidden under one of the cups. Each move consists of choosing 4 cups and examining whether the coin is under one of them. After each move, the cups are turned over and the coin moves to one of the two neighboring positions. What is the minimal number of moves that guarantees ... | [
"$\\left\\lfloor \\frac{n}{3} \\right\\rfloor$"
] | [
"$n$ cups are arranged in a circle. A coin is hidden under one of the cups. Each move consists of choosing 4 cups and examining whether the coin is under one of them. After each move, the cups are turned over and the coin moves to one of the two neighboring positions. What is the minimal number of moves that guaran... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e92c28da1c6844899d94f365f6b7816b88cca19ebc103c6f25d3daf56033b4d1 | $n$ dwarfs are crossing a river with a boat. The boat can only carry $n$ pounds. The dwarfs' weights are $1, 2, 3, \ldots, n$ pounds. When going to the other side, a dwarf driving the boat is not needed, but one is needed when coming back. Each dwarf can only drive the boat once. For which values of $n$ can they succes... | [
"n \\text{ is odd}"
] | [
"$n$ dwarfs are crossing a river with a boat. The boat can only carry $n$ pounds. The dwarfs' weights are $1, 2, 3, \\ldots, n$ pounds. When going to the other side, a dwarf driving the boat is not needed, but one is needed when coming back. Each dwarf can only drive the boat once. For which values of $n$ can they ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 13,
"genselect": 5
} | 18 | 0 | 0 |
omr-ws-93073baeac7c858515d230e21a67f74987b85fc097f140cc6519c40e77bd6a83 | $n$ equal circles are inscribed in a regular $n$-gon with side $a$ so that each circle is tangent to two adjacent sides of the polygon and to two other circles. Find the area of the star-shaped figure formed in the center of the polygon. | [
"\\[\nS=\\frac{a^2}{4}\\cdot \\frac{n\\cot\\frac{\\pi }{n}-(n-2)\\frac{\\pi }{2}} {\\left(1+\\tan\\frac{\\pi }{n}\\right)^2}\n\\]"
] | [
"$n$ equal circles are inscribed in a regular $n$-gon with side $a$ so that each circle is tangent to two adjacent sides of the polygon and to two other circles. Find the area of the star-shaped figure formed in the center of the polygon."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d50793915315e2184a15986935df7ebe02ca470f7e9dc8d5d3e4d62440ccaa8d | $n$ integers are arranged along a circle such that each number is equal to the absolute value of the difference of the two numbers following it in the clockwise direction. If the sum of all numbers is $278$, how many different values can $n$ take? | [
"2"
] | [
"$n$ integers are arranged along a circle such that each number is equal to the absolute value of the difference of the two numbers following it in the clockwise direction. If the sum of all numbers is $278$, how many different values can $n$ take?"
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 44,
"genselect": 8
} | 52 | 0 | 0 |
omr-ws-6049808a03da33cc257b06233f053d219b9f5739aafb269c559881c119e497d0 | $n$ lattice points are marked on a lattice grid where $n$ is a positive integer, and the distances between each possible pair of different points are listed. If it is known that all numbers listed are different, find the minimum possible sum of those numbers. | [
"\\frac{n(n-1)(n+1)}{6}"
] | [
"$n$ lattice points are marked on a lattice grid where $n$ is a positive integer, and the distances between each possible pair of different points are listed. If it is known that all numbers listed are different, find the minimum possible sum of those numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 2
} | 2 | 0 | 0 |
omr-ws-c33f1ff95ba5e4a6d9dc185d9e3c2bff1b800090018dc51a571cb06d2c5792b1 | $n$ paper circles of radius $1$ are laid on a plane so that their borders pass through one point, and this point is inside the entire area covered by the circles. This area forms a polygon with curved sides. Find the perimeter of this polygon. | [
"\\( 2\\pi \\)"
] | [
"$n$ paper circles of radius $1$ are laid on a plane so that their borders pass through one point, and this point is inside the entire area covered by the circles. This area forms a polygon with curved sides. Find the perimeter of this polygon."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 3,
"genselect": 3,
"tir": 2
} | 8 | 0 | 0 |
omr-ws-ccf4f36d7e9588a33175e177257c767412fa0e1f3b65957daca70a6c3fbaa77a | $n$ people, $Q_{1}, Q_{2}, ..., Q_{n}$, sit in order around a circular table with $Q_{1}$ sitting to $Q_{n}$'s right. Each person makes the statement "either the people to my left and right are both lying or both are telling the truth". If at least one person is lying, find the smallest possible value of $n$ such that ... | [
"3"
] | [
"$n$ people, $Q_{1}, Q_{2}, ..., Q_{n}$, sit in order around a circular table with $Q_{1}$ sitting to $Q_{n}$'s right. Each person makes the statement \"either the people to my left and right are both lying or both are telling the truth\". If at least one person is lying, find the smallest possible value of $n$ suc... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a587036a3e0715f5e50f1c467283f452ddaf20adf7f723159a24d5b3fc781baf | $n$ people, $Q_{1}, Q_{2}, ..., Q_{n}$, sit in order around a circular table with $Q_{1}$ sitting to $Q_{n}$'s right. Each person makes the statement "the person on my left is lying and the person on my right is telling the truth". If $n$ is the smallest number of people for which this scenario is possible, find $n$. | [
"2"
] | [
"$n$ people, $Q_{1}, Q_{2}, ..., Q_{n}$, sit in order around a circular table with $Q_{1}$ sitting to $Q_{n}$'s right. Each person makes the statement \"the person on my left is lying and the person on my right is telling the truth\". If $n$ is the smallest number of people for which this scenario is possible, find... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect"
] | {
"cot": 5,
"genselect": 7
} | 12 | 0 | 0 |
omr-ws-648d0a2ccf04d9cadbfc130f000d87a3dbbd08f6bc7c2f86a092ceecfe354913 | $n$ persons sit around a round table. The number of persons having the same gender as the person to their right is equal to the number of persons not having the same gender as the person to their right. If $n$ is the smallest number for which this arrangement is possible, find $n$. | [
"4"
] | [
"$n$ persons sit around a round table. The number of persons having the same gender as the person to their right is equal to the number of persons not having the same gender as the person to their right. If $n$ is the smallest number for which this arrangement is possible, find $n$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-078a0e351f94e27631b2022102e8dcc9e6adf654d29a2f497801d0515e933822 | $n$ points are chosen randomly and independently on a segment of unit length. Find the expected length of the longest of the $n+1$ resulting segments. | [
"\\(\\frac{H_{n+1}}{n+1}\\)"
] | [
"$n$ points are chosen randomly and independently on a segment of unit length. Find the expected length of the longest of the $n+1$ resulting segments."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 19
} | 35 | 0 | 0 |
omr-ws-6422a0c53b1d03923c7aca1d4a838a84b13f580aefbd5fd6fae14852bf9f960b | $n$ points are connected to each other by a certain number of segments such that from each point, no more than $11$ segments leave. Determine the minimum number of colors needed to paint the points so that no segment connects two points of the same color. | [
"12"
] | [
"$n$ points are connected to each other by a certain number of segments such that from each point, no more than $11$ segments leave. Determine the minimum number of colors needed to paint the points so that no segment connects two points of the same color."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-27489009b18b17beccfe628baf01f02bef408abf7fd413cf187e7b16f6aa0997 | $n$ students, numbered from $1$ to $n$, are sitting next to each other in a class. Initially, the $1$st student has $n$ pieces of paper in one pile. The goal is to distribute the $n$ pieces of paper so that each student gets exactly one piece. The teacher claps once per minute, and for each clap, the students can eithe... | [
"\\( 2(n-1) \\)"
] | [
"$n$ students, numbered from $1$ to $n$, are sitting next to each other in a class. Initially, the $1$st student has $n$ pieces of paper in one pile. The goal is to distribute the $n$ pieces of paper so that each student gets exactly one piece. The teacher claps once per minute, and for each clap, the students can ... | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 11,
"genselect": 3,
"tir": 2
} | 16 | 0 | 0 |
omr-ws-6f6628f9eb8841f3392db76b0becb44d5a8faac5d7a25bdd211ac8cb752f19a5 | $n$ vectors are chosen on the plane. From the origin, all possible sums of one, two, ..., $n$ vectors are considered, and the ends of these sums are marked. What is the smallest $n$ such that among the marked points, both 4 vertices of a square and 3 vertices of a regular triangle can be found? | [
"4"
] | [
"$n$ vectors are chosen on the plane. From the origin, all possible sums of one, two, ..., $n$ vectors are considered, and the ends of these sums are marked. What is the smallest $n$ such that among the marked points, both 4 vertices of a square and 3 vertices of a regular triangle can be found?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 11,
"genselect": 7
} | 18 | 0 | 0 |
omr-ws-74f2f67414e757c20ebccda7ab6c87ada5003271617d7aace67a1ff0bef9348c | $n$ white and $n$ black balls are placed at random on the circumference of a circle. Starting from a certain white ball, number all white balls in a clockwise direction by $1, 2, \dots, n$. Likewise, number all black balls by $1, 2, \dots, n$ in anti-clockwise direction starting from a certain black ball. Let $S$ be th... | [
"2"
] | [
"$n$ white and $n$ black balls are placed at random on the circumference of a circle. Starting from a certain white ball, number all white balls in a clockwise direction by $1, 2, \\dots, n$. Likewise, number all black balls by $1, 2, \\dots, n$ in anti-clockwise direction starting from a certain black ball. Let $S... | [
"aops_c6_high_school_olympiads"
] | [
"cot"
] | {
"cot": 5
} | 5 | 0 | 0 |
omr-ws-3df943b57477f73741847008194685b9bc203bef677c6266623da0c0d3cd105e | $n+1$ points are marked on the plane. No three of them lie on the same straight. Find all $k$ for which it is possible to draw several segments in such a way that for any $n$ points there are exactly $k$ of the drawn segments with both ends at these $n$ points. | [
"k = \\frac{m(n-1)}{2} \\text{ for some integer } m \\text{ with } 0 \\leq m \\leq n"
] | [
"$n+1$ points are marked on the plane. No three of them lie on the same straight. Find all $k$ for which it is possible to draw several segments in such a way that for any $n$ points there are exactly $k$ of the drawn segments with both ends at these $n$ points."
] | [
"aops_c4_high_school_math"
] | [
"cot"
] | {
"cot": 15
} | 15 | 0 | 0 |
omr-ws-5a01de2fa5a71be852d7ddb5b87dbf1f3c61c8f4994a6c799cb61e9485d71425 | $p$ is a prime of the form $4k+3$. Solve in integers the equation $3a^{2}+1=p^{2}$. | [
"(a, p) = (4, 7)"
] | [
"$p$ is a prime of the form $4k+3$. Solve in integers the equation $3a^{2}+1=p^{2}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 1,
"genselect": 3,
"tir": 20
} | 24 | 0 | 0 |
omr-ws-18dca329ec8f42d36c3d24a0da5449014d117c3e2747a43ecee03565a3112a17 | $p$, $q$, and $r$ are positive real numbers such that \[ p^2 + pq + q^2 = 211, \] \[ q^2 + qr + r^2 = 259, \] \[ r^2 + rp + p^2 = 307. \] Compute the value of $pq + qr + rp$. | [
"253"
] | [
"$p$, $q$, and $r$ are positive real numbers such that\n\\[\np^2 + pq + q^2 = 211,\n\\]\n\\[\nq^2 + qr + r^2 = 259,\n\\]\n\\[\nr^2 + rp + p^2 = 307.\n\\]\nCompute the value of $pq + qr + rp$."
] | [
"aops_c5_contests_amp_programs"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-b7aa887f093dd2c0f5b594b29cdcc75a2c21b947f1e49812be23b0979e645775 | $r(x)$ has domain $\{-1,0,1,2\}$ and range $\{0,2,4,6\}$. $s(x)$ has domain $\{1,2,3,4\}$ and is defined by $s(x)=x+1$. What is the sum of all possible values of $s(r(x))$? | [
"8"
] | [
"$r(x)$ has domain $\\{-1,0,1,2\\}$ and range $\\{0,2,4,6\\}$. $s(x)$ has domain $\\{1,2,3,4\\}$ and is defined by $s(x)=x+1$. What is the sum of all possible values of $s(r(x))$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
omr-ws-0ec185cc573a29f158714def45551b8acfc6c4d9002d8b04e4635c491bb61be3 | $x = {1+\frac{\sqrt{2}}{1+\frac{\sqrt{2}}{1+...}}}$. Find $\frac{1}{(x+1)(x-2)}$. When your answer is in the form $\frac{A+\sqrt{B}}{C}$, where $A$, $B$, and $C$ are integers, and $B$ is not divisible by the square of a prime, what is $|A|+|B|+|C|$? | [
"6"
] | [
"$x = {1+\\frac{\\sqrt{2}}{1+\\frac{\\sqrt{2}}{1+...}}}$. Find $\\frac{1}{(x+1)(x-2)}$. When your answer is in the form $\\frac{A+\\sqrt{B}}{C}$, where $A$, $B$, and $C$ are integers, and $B$ is not divisible by the square of a prime, what is $|A|+|B|+|C|$?"
] | [
"MATH_training_set"
] | [
"cot"
] | {
"cot": 16
} | 16 | 0 | 0 |
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