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-cf102c13bb1d8d13b963cf4bfddc780f25d8cdfaf6a9077f1129c9459cf89c4d | $\frac{1}{\sqrt [4]{2}+\sqrt [4]{4}+\sqrt [4]{8}+2}$. | [
"$\\frac{2-\\sqrt [4]{8}}{2}$"
] | [
"$\\frac{1}{\\sqrt [4]{2}+\\sqrt [4]{4}+\\sqrt [4]{8}+2}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-de28c7aea199fea75bb1f7a9ca6c78150d8dc6b999532665c8db6d77ba11e17d | $\frac{1}{\sqrt [4]{5}-\sqrt [4]{2}}$. | [
"$\\frac{\\left(\\sqrt [4]{5}+\\sqrt [4]{2}\\right)\\left(\\sqrt 5+\\sqrt 2\\right)}{3}$"
] | [
"$\\frac{1}{\\sqrt [4]{5}-\\sqrt [4]{2}}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f9134a271d7b6e5d43adfb745d4693cbe1d52cca3f480cab2dce6c8c6eb1217e | $\frac{1}{\sqrt {14}+\sqrt {21}+\sqrt {15}+\sqrt {10}}$. | [
"$\\frac{\\sqrt {21}-\\sqrt {15}-\\sqrt {14}+\\sqrt {10}}{2}$"
] | [
"$\\frac{1}{\\sqrt {14}+\\sqrt {21}+\\sqrt {15}+\\sqrt {10}}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e06d50d5914e232ed65360afab2ceda70316a6e779c58b9f83b1532b946f8bd4 | $\frac{1}{\sqrt {2}+\sqrt [3]{3}}$. | [
"$-\\sqrt {\\sqrt 2+\\sqrt [3]{2}}\\left(\\sqrt 2-\\sqrt [3]{3}\\right) \\left(4+2\\sqrt [3]{9}+3\\sqrt [3]{3}\\right)$"
] | [
"$\\frac{1}{\\sqrt {2}+\\sqrt [3]{3}}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d72f7f36f73d579fd9c58cc258294fd144ee803710c4e38fc0a034d25eb444f7 | $\frac{1}{a(a-b)(a-c)}+\frac{1}{b(b-a)(b-c)} +\frac{1}{c(c-a)(c-b)}$ | [
"$\\frac{1}{abc}$"
] | [
"$\\frac{1}{a(a-b)(a-c)}+\\frac{1}{b(b-a)(b-c)} +\\frac{1}{c(c-a)(c-b)}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a6c90633e0dfa41cb02997809a72055312c33fa6c63d786111d7bef0d3f8e789 | $\frac{215\frac{9}{16}-208\frac{3}{4}+\frac{1}{2}}{0.0001:0.005}$ | [
"$365\\frac{5}{8}$"
] | [
"$\\frac{215\\frac{9}{16}-208\\frac{3}{4}+\\frac{1}{2}}{0.0001:0.005}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-88b68fe1206a8d3bfaa1a1632181344f5f98636d12fb72d3c67dce41a3e530e2 | $\frac{2\frac{5}{8}-\frac{2}{3}\cdot 2\frac{5}{14}} {\left(3\frac{1}{12}+4.375\right):19\frac{8}{9}}$ | [
"$2\\frac{17}{21}$"
] | [
"$\\frac{2\\frac{5}{8}-\\frac{2}{3}\\cdot 2\\frac{5}{14}} {\\left(3\\frac{1}{12}+4.375\\right):19\\frac{8}{9}}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-13997a5c7d9479cd770a802689fa380a434786161a2c8d805fa9ac11827224df | $\frac{2a^2(b+c)^{2n}-\frac{1}{2}}{an^2-a^3-2a^2-a}: \frac{2a(b+c)^n-1}{a^2c-a(nc-c)}$ | [
"$-\\frac{[2a(b+c)^n+1]c}{2(n+a+1)}$"
] | [
"$\\frac{2a^2(b+c)^{2n}-\\frac{1}{2}}{an^2-a^3-2a^2-a}: \\frac{2a(b+c)^n-1}{a^2c-a(nc-c)}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-725df1529e0bc393bfad90ea66c992a32c84b85d863915f4a58a806912c4b1ba | $\frac{2x^2+2x+3}{x+2}=\frac{3x^2+2x-1}{x+2}$ and $2x^2+2x+3=3x^2+2x-1$. | [
"No"
] | [
"$\\frac{2x^2+2x+3}{x+2}=\\frac{3x^2+2x-1}{x+2}$ and $2x^2+2x+3=3x^2+2x-1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-bd385c4034569329178fe36ac24e4d320be8b9aaeb1091f3c81ddcb3883b0a7a | $\frac{2x^2+2x+3}{x+3}=\frac{3x^2+2x-1}{x+3}$ and $2x^2+2x+3=3x^2+2x-1$. | [
"Yes"
] | [
"$\\frac{2x^2+2x+3}{x+3}=\\frac{3x^2+2x-1}{x+3}$ and $2x^2+2x+3=3x^2+2x-1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5aeb03525e3a3e1c0bdff0515a22e3da9378fb39b51790412aa2f7278f055c4f | $\frac{2}{3}\left[\frac{1}{1+\left(\frac{2x+1}{\sqrt 3}\right)^2} +\frac{1}{1+\left(\frac{2x-1}{\sqrt 3}\right)^2}\right]$ | [
"$\\frac{x^2+1}{x^4+x^2+1}$"
] | [
"$\\frac{2}{3}\\left[\\frac{1}{1+\\left(\\frac{2x+1}{\\sqrt 3}\\right)^2} +\\frac{1}{1+\\left(\\frac{2x-1}{\\sqrt 3}\\right)^2}\\right]$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e0e279443c303b321fc26b491e8957d062a7307f0bfbd01362403d1f50a67cd8 | $\frac{2}{5}$ of a product has been sold at a 30% margin, and $\frac{3}{5}$ of the product has been sold at a 20% margin. Find the total margin of the product. | [
"24\\%"
] | [
"$\\frac{2}{5}$ of a product has been sold at a 30% margin, and $\\frac{3}{5}$ of the product has been sold at a 20% margin. Find the total margin of the product."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 26
} | 42 | 0 | 0 |
omr-ws-e28898807af55c57acf14de24bd3fa797ea3f68b8643fd8b1501fd6582683599 | $\frac{3:\frac{2}{5}-0.09:\left(0.15:2\frac{1}{2}\right)} {0.32_6+0.03-(5.3-3.88)+0.67}$ | [
"5"
] | [
"$\\frac{3:\\frac{2}{5}-0.09:\\left(0.15:2\\frac{1}{2}\\right)} {0.32_6+0.03-(5.3-3.88)+0.67}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2540a5cfc19ce764d26e086795d03ef92e33906c3fcaccc97b79d1ff8a4e953d | $\frac{4.5:\left[47.375-\left(26\frac{1}{3}-18\cdot 0.75\right) \cdot 2.4:0.88\right]}{17.81:1.37-23\frac{2}{3}:1\frac{5}{6}}$ | [
"4"
] | [
"$\\frac{4.5:\\left[47.375-\\left(26\\frac{1}{3}-18\\cdot 0.75\\right) \\cdot 2.4:0.88\\right]}{17.81:1.37-23\\frac{2}{3}:1\\frac{5}{6}}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2dc571563330d25fbd871902747cd9f25f127b0623258ab12428c86cfddcc258 | $\frac{7}{4}\cos \frac{x}{4}=\cos ^3 \frac{x}{4}+\sin \frac{x}{2}$. | [
"$2\\pi +4\\pi k;\\ (-1)^n \\frac{2\\pi }{3}+4\\pi n$"
] | [
"$\\frac{7}{4}\\cos \\frac{x}{4}=\\cos ^3 \\frac{x}{4}+\\sin \\frac{x}{2}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-110132c00dce4a61d55b9b3d6eb80c5558c60d58b9ca1b2f5c4591595030fd45 | $\frac{\cos x}{1+\cos 2x}=0$. | [
"$\\emptyset $"
] | [
"$\\frac{\\cos x}{1+\\cos 2x}=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ccfcc352a0cb01a3782883d0a99b4287d8afe2e4cea9c473feff6b08edf31a67 | $\frac{\left(1.75:\frac{2}{3}-1.75\cdot 1\frac{1}{8}\right) :\frac{7}{12}}{\left(\frac{17}{80}-0.0325\right):400} :(6.79:0.7+0.3)$ | [
"250"
] | [
"$\\frac{\\left(1.75:\\frac{2}{3}-1.75\\cdot 1\\frac{1}{8}\\right) :\\frac{7}{12}}{\\left(\\frac{17}{80}-0.0325\\right):400} :(6.79:0.7+0.3)$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3f6bf05aa3e736dc47db1a0d45eb80331a112063294eb2b560eb283e595f5fb1 | $\frac{\left(49\frac{5}{24}-46\frac{7}{20}\right)\cdot 2\frac{1}{3}+0.6}{0.2}$ | [
"$36\\frac{25}{72}$"
] | [
"$\\frac{\\left(49\\frac{5}{24}-46\\frac{7}{20}\\right)\\cdot 2\\frac{1}{3}+0.6}{0.2}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1f885c8bdff3047cd494788088803279fd0947f9d74d3837a0d9adac60f74ee1 | $\frac{\left(68\frac{7}{30}-66\frac{5}{18}\right):6\frac{1}{9} +\left(\frac{7}{40}+\frac{3}{32}\right)\cdot 4.5}{0.04}$ | [
"$38\\frac{15}{64}$"
] | [
"$\\frac{\\left(68\\frac{7}{30}-66\\frac{5}{18}\\right):6\\frac{1}{9} +\\left(\\frac{7}{40}+\\frac{3}{32}\\right)\\cdot 4.5}{0.04}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-596b68a6de4280e41e5e359da681cca36e101406141f16084a61134a1418313e | $\frac{\left(6\frac{3}{5}-3\frac{3}{14}\right)\cdot 5\frac{5}{6}} {(21-1.25):2.5}$ | [
"2.5"
] | [
"$\\frac{\\left(6\\frac{3}{5}-3\\frac{3}{14}\\right)\\cdot 5\\frac{5}{6}} {(21-1.25):2.5}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-535afe7a49bde6ed6dea7caa63768de8e8dfeeefb25bb05026ca012ff8cefb96 | $\frac{\left(85\frac{7}{30}-83\frac{5}{18}\right):2\frac{2}{3}}{0.04}$ | [
"$18\\frac{1}{3}$"
] | [
"$\\frac{\\left(85\\frac{7}{30}-83\\frac{5}{18}\\right):2\\frac{2}{3}}{0.04}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-67d608a7d1d78d0cd4d4890a0fec66843a8e1ddd84fdb44985ed77c2ffa97cf5 | $\frac{\left[\left(40\frac{7}{30}-38\frac{5}{12}\right):10.9 +\left(\frac{7}{8}-\frac{7}{30}\right)\cdot 1\frac{9}{11}\right] \cdot 4.2}{0.008}$ | [
"700"
] | [
"$\\frac{\\left[\\left(40\\frac{7}{30}-38\\frac{5}{12}\\right):10.9 +\\left(\\frac{7}{8}-\\frac{7}{30}\\right)\\cdot 1\\frac{9}{11}\\right] \\cdot 4.2}{0.008}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fd319cecfa3d4f1098ce28f902542ec14f61b55e2bf3f640c86064191287254b | $\frac{\left[\left(6.2:0.31-\frac{5}{6}\cdot 0.9\right)\cdot 0.2 +0.15\right]:0.02}{\left(2+1\frac{4}{11}\cdot 0.22:0.1\right) \cdot \frac{1}{3}}$ | [
"1320"
] | [
"$\\frac{\\left[\\left(6.2:0.31-\\frac{5}{6}\\cdot 0.9\\right)\\cdot 0.2 +0.15\\right]:0.02}{\\left(2+1\\frac{4}{11}\\cdot 0.22:0.1\\right) \\cdot \\frac{1}{3}}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3bd4f33be16f3c19e627edcba7b745a80912e2121a452cf75a314cbeac7474d8 | $\frac{\sin x+\cos x}{\cos 2x}=0$. | [
"$\\emptyset $"
] | [
"$\\frac{\\sin x+\\cos x}{\\cos 2x}=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5862de090d78c89e3111dd582e43918ebcb005f99e5e0be68b58bd3e6a6717c8 | $\frac{\sin(60^\circ+x)+\sin(60^\circ-x)}{2} =\frac{\tan x}{(1+\tan^2 x)^2}+\frac{\cot x}{(1+\cot^2 x)^2}$ | [
"$x=180^\\circ n+(-1)^n 60^\\circ$"
] | [
"$\\frac{\\sin(60^\\circ+x)+\\sin(60^\\circ-x)}{2} =\\frac{\\tan x}{(1+\\tan^2 x)^2}+\\frac{\\cot x}{(1+\\cot^2 x)^2}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-882da47f04a0a1c9bf000b6bc8ee6ef30777016dbee6844933b87f151a05c530 | $\frac{\sqrt 2-\cos \alpha -\sin\alpha }{\sin \alpha -\cos \alpha }$ | [
"$\\tan\\frac{\\alpha -45^\\circ}{2}$"
] | [
"$\\frac{\\sqrt 2-\\cos \\alpha -\\sin\\alpha }{\\sin \\alpha -\\cos \\alpha }$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-81f7b66586c8c01a0b704c9525c64670c0a779081e6a87232315798ddc03d916 | $\frac{\sqrt {5-2\sqrt 6}\left(5+2\sqrt 6\right)\left(49-20\sqrt 6\right)} {\sqrt {27}-3\sqrt {18}+3\sqrt {12}-\sqrt {8}}=1.$ | [
"True"
] | [
"$\\frac{\\sqrt {5-2\\sqrt 6}\\left(5+2\\sqrt 6\\right)\\left(49-20\\sqrt 6\\right)} {\\sqrt {27}-3\\sqrt {18}+3\\sqrt {12}-\\sqrt {8}}=1.$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-48e469684807a48dfcca91202daa9c4d4ac22732678563e9e9769c1b728ff21b | $\frac{\sqrt {\frac{1+a}{1-a}}+\sqrt {\frac{1-a}{1+a}}} {\sqrt {\frac{1+a}{1-a}}-\sqrt {\frac{1-a}{1+a}}}-\frac{1}{a}$ | [
"0"
] | [
"$\\frac{\\sqrt {\\frac{1+a}{1-a}}+\\sqrt {\\frac{1-a}{1+a}}} {\\sqrt {\\frac{1+a}{1-a}}-\\sqrt {\\frac{1-a}{1+a}}}-\\frac{1}{a}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8eaf1b09b96f7ce5789b298d4d305593485c035e7d11b1bedd8df4111f7220c9 | $\frac{\sqrt {\sqrt 5+\sqrt 3}}{\sqrt {\sqrt 5-\sqrt 3}}$. | [
"$\\frac{\\sqrt 2\\left(\\sqrt 3+\\sqrt 5\\right)}{2}$"
] | [
"$\\frac{\\sqrt {\\sqrt 5+\\sqrt 3}}{\\sqrt {\\sqrt 5-\\sqrt 3}}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7a205dac469f3e645e83a0d6359ee09acee1bf27997e26f5b4d70f3edd6f04ee | $\frac{\tan 2x\tan x}{\tan 2x-\tan x}-2\sin(45^\circ+x)\sin(45^\circ-x)=0$ | [
"$x=90^\\circ n+22^\\circ 30'$"
] | [
"$\\frac{\\tan 2x\\tan x}{\\tan 2x-\\tan x}-2\\sin(45^\\circ+x)\\sin(45^\\circ-x)=0$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1e5d20885e13c229de8cb47870b956ea56174e8cecc4e643b5df40e73aa175f7 | $\frac{a^2-1}{n^2+an}\cdot \left(\frac{1}{1-\frac{1}{n}}-1\right) \cdot \frac{a-an^3-n^4+n}{1-a^2}$ | [
"$\\frac{n^2+n+1}{n}$"
] | [
"$\\frac{a^2-1}{n^2+an}\\cdot \\left(\\frac{1}{1-\\frac{1}{n}}-1\\right) \\cdot \\frac{a-an^3-n^4+n}{1-a^2}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-58ceb4f9f20d2f0d10225a287c18fbd86131e6b4c7442f67501fdd46e65cfef2 | $\frac{a}{2}\sqrt [4]{(a+1)(a^2-1)(1+2a+a^2)} \cdot \left(\frac{a^2+3a+2}{\sqrt {a-1}}\right)^{-1}$ | [
"$\\frac{a}{2}\\cdot \\frac{\\sqrt [4]{(a-1)^3}}{a+2}$"
] | [
"$\\frac{a}{2}\\sqrt [4]{(a+1)(a^2-1)(1+2a+a^2)} \\cdot \\left(\\frac{a^2+3a+2}{\\sqrt {a-1}}\\right)^{-1}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2ec7e8666eac03f8869841c9dc93f6dc0018888a3d2ba2fa0992642eb322af9c | $\frac{b}{a-b}\sqrt [3]{(a^2-2ab+b^2)(a^2-b^2)(a+b)} \cdot \frac{a^3-b^3}{\sqrt [3]{(a+b)^2}}$ | [
"$b(a^3-b^3)$. "
] | [
"$\\frac{b}{a-b}\\sqrt [3]{(a^2-2ab+b^2)(a^2-b^2)(a+b)} \\cdot \\frac{a^3-b^3}{\\sqrt [3]{(a+b)^2}}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a89d016bdbe31bc29baad2c67784ad646527d011efe3e16d1ce7bf7216f6094b | $\frac{n(n + 1)}{2}$ distinct numbers are arranged at random into $n$ rows, with the first row having $1$ number, the second row having $2$ numbers, and so on. Find the probability that the largest number in each row is smaller than the largest number in each row with more numbers. | [
"\\(\\frac{2^n}{(n+1)!}\\)"
] | [
"$\\frac{n(n + 1)}{2}$ distinct numbers are arranged at random into $n$ rows, with the first row having $1$ number, the second row having $2$ numbers, and so on. Find the probability that the largest number in each row is smaller than the largest number in each row with more numbers."
] | [
"aops_c6_high_school_olympiads"
] | [
"cot",
"tir"
] | {
"cot": 21,
"tir": 12
} | 33 | 0 | 0 |
omr-ws-931fde1214a0e88505c6310fe177a8782713678a5e79321efc3d33efcdf8a1f3 | $\frac{|\cot xy|}{\cos ^2 xy}=\log _{1/8}(9y^2-18y+10)+2$. | [
"$x=(2k+1)\\pi /4$, $y=1$"
] | [
"$\\frac{|\\cot xy|}{\\cos ^2 xy}=\\log _{1/8}(9y^2-18y+10)+2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b3f0b06aae05a4d45567f1f28f0e7b9d2768857e2e28d02fae85d1c749ef4353 | $\left(10:2\frac{2}{3}+7.5:10\right):\left(\frac{3}{40} -\frac{7}{30}\cdot 0.25+\frac{157}{360}\right)$ | [
"$2\\frac{3}{80}$"
] | [
"$\\left(10:2\\frac{2}{3}+7.5:10\\right):\\left(\\frac{3}{40} -\\frac{7}{30}\\cdot 0.25+\\frac{157}{360}\\right)$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-72436e56fa3c460ac582acdf0003ad96a5ec2dd75503eaf229c16745dae53540 | $\left(2\sin ^4 \frac{x}{2}-1\right)\frac{1}{\cos ^4 \frac{x}{2}}=2$. | [
"$\\pm \\frac{2\\pi }{3}+2\\pi k$"
] | [
"$\\left(2\\sin ^4 \\frac{x}{2}-1\\right)\\frac{1}{\\cos ^4 \\frac{x}{2}}=2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6b829b0a52fd776079dac6972e559fa6e0a4bac3cf920959448cc23473b74f39 | $\left(\cos ^2 x+\frac{1}{\cos ^2 x}\right)(1+\tan ^2 2y)(3+\sin 3z)=4$. | [
"$\\left(\\pi k;\\frac{\\pi }{2}n;-\\frac{\\pi }{6}+\\frac{2\\pi }{3}m\\right)$"
] | [
"$\\left(\\cos ^2 x+\\frac{1}{\\cos ^2 x}\\right)(1+\\tan ^2 2y)(3+\\sin 3z)=4$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-47fa2a04dfdcac47277192aba01f8bcbbed1d712dc1b1b8162c5bd73e44388b4 | $\left(\frac{1+x}{\sqrt x}\right)^2+2a\frac{1+x}{\sqrt x}+1=0$. | [
"$x_{1,2}=\\left(\\sqrt {a^2-1}-a\\pm \\sqrt {2a^2-5-2a\\sqrt {a^2-1}}\\right)^2$ for $a<-5/4$; $x=1$ for $a=-5/4$; no solution for $a>-5/4$. Find the roots of the quadratic $f(y)=y^2+2ay+1$ that satisfy the condition $y\\ge 2$. If $D=a^2-1<0$, then there are no real roots at all, and so it must be true that $a^2-1... | [
"$\\left(\\frac{1+x}{\\sqrt x}\\right)^2+2a\\frac{1+x}{\\sqrt x}+1=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9b8becf3528e3e171af69e0b224546d57195de89c69c8f022bf2028f5462e03f | $\left(\frac{1}{\sqrt a-\sqrt {a-b}}+\frac{1}{\sqrt a+\sqrt {a+b}}\right)" \left(1+\sqrt {\frac{a+b}{a-b}}\right)$ | [
"$\\frac{\\sqrt {a-b}}{b}$"
] | [
"$\\left(\\frac{1}{\\sqrt a-\\sqrt {a-b}}+\\frac{1}{\\sqrt a+\\sqrt {a+b}}\\right)\" \\left(1+\\sqrt {\\frac{a+b}{a-b}}\\right)$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-cbd1d9a1dacdec9443ea4f41284c45fc221170a54b762d346581a7cb69dcc048 | $\left(\frac{1}{b-\sqrt a}+\frac{1}{b+\sqrt a}\right): \frac{\sqrt [-2]{\frac{1}{9}a^{-2}b^{-1}}}{a^{-2}-a^{-1}b^{-2}}$ | [
"$\\frac{2}{3}$"
] | [
"$\\left(\\frac{1}{b-\\sqrt a}+\\frac{1}{b+\\sqrt a}\\right): \\frac{\\sqrt [-2]{\\frac{1}{9}a^{-2}b^{-1}}}{a^{-2}-a^{-1}b^{-2}}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8cd2a88f556a349a0e32a801a4d4d37f88c7acab0306bbe6fb81e8e6752648d4 | $\left(\frac{2a+10}{3a-1}+\frac{130-a}{1-3a}+\frac{30}{a}-3\right) \cdot \frac{3a^3+8a^2-3a}{1-\frac{1}{4}a^2}$ | [
"$\\frac{12(2a+5)(a+3)}{a-2}$"
] | [
"$\\left(\\frac{2a+10}{3a-1}+\\frac{130-a}{1-3a}+\\frac{30}{a}-3\\right) \\cdot \\frac{3a^3+8a^2-3a}{1-\\frac{1}{4}a^2}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fa0fa7eeedc05a09b497eec185cd6ab7782f57f267a6ffcbcedc30f3581b0431 | $\left(\frac{3}{\sqrt [3]{64}\!-\!\sqrt [3]{25}} +\frac{\sqrt [3]{40}}{\sqrt [3]{8}+\sqrt [3]{5}} -\frac{10}{\sqrt [3]{25}}\right)^{-2} \left(13\!-\!4\sqrt [3]{5}-2\sqrt [3]{25}\right)+\sqrt [3]{25}=4$. | [
"True"
] | [
"$\\left(\\frac{3}{\\sqrt [3]{64}\\!-\\!\\sqrt [3]{25}} +\\frac{\\sqrt [3]{40}}{\\sqrt [3]{8}+\\sqrt [3]{5}} -\\frac{10}{\\sqrt [3]{25}}\\right)^{-2} \\left(13\\!-\\!4\\sqrt [3]{5}-2\\sqrt [3]{25}\\right)+\\sqrt [3]{25}=4$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b004cc7b6a5d4d66dacf8494d43dffa5c61c79e13066cce5bf9310a87c4b43fd | $\left(\frac{6+4\sqrt 2}{\sqrt 2+\sqrt {6+4\sqrt {2}}} +\frac{6-4\sqrt {2}}{\sqrt {2}-\sqrt {6-4\sqrt {2}}}\right)^2=8$. | [
"True"
] | [
"$\\left(\\frac{6+4\\sqrt 2}{\\sqrt 2+\\sqrt {6+4\\sqrt {2}}} +\\frac{6-4\\sqrt {2}}{\\sqrt {2}-\\sqrt {6-4\\sqrt {2}}}\\right)^2=8$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-da2c2e979ea48fc1938b9d787755a31bbeeb9b8381153a29f8256f0d682b2a6a | $\left(\frac{x-y}{2y-x}-\frac{x^2+y^2+y-2}{x^2-xy-2y^2}\right): \frac{4x^4+4x^2y+y^2-4}{x^2+y+xy+x}$ | [
"$\\frac{x+1}{(2y-x)(2x^2+y+2)}$"
] | [
"$\\left(\\frac{x-y}{2y-x}-\\frac{x^2+y^2+y-2}{x^2-xy-2y^2}\\right): \\frac{4x^4+4x^2y+y^2-4}{x^2+y+xy+x}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-65d24d063aae4d4f95bf6f2ee1f960334a42d6980a6b14ffa88dfad1efe18624 | $\left(\sin 2x+\sqrt 3\cos 2x\right)^2-5=\cos \left(\frac{\pi }{6}-2x\right)$. | [
"$\\frac{7\\pi }{12}+\\pi k$"
] | [
"$\\left(\\sin 2x+\\sqrt 3\\cos 2x\\right)^2-5=\\cos \\left(\\frac{\\pi }{6}-2x\\right)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e9b1d4f99e3dcb4e7cb930fcd2b3b81441df7bc55ff6ed0d7dab7cf3511ef3ad | $\left(\sin ^2 x+\frac{1}{\sin ^2 x}\right) +\left(\cos ^2 x+\frac{1}{\cos ^2 x}\right)^2=12+\frac{1}{2}\sin y$. | [
"$\\left(\\frac{\\pi }{4}+2\\pi k;\\frac{\\pi }{2}+2\\pi n\\right)$"
] | [
"$\\left(\\sin ^2 x+\\frac{1}{\\sin ^2 x}\\right) +\\left(\\cos ^2 x+\\frac{1}{\\cos ^2 x}\\right)^2=12+\\frac{1}{2}\\sin y$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-fbde76eca0715d3981970284121528ea5ec44708a7ed1bff14f121192541b9a4 | $\left(\sin x+\sqrt 3\cos x\right)\sin 3x=2$. | [
"$\\frac{\\pi }{6}+\\pi k$"
] | [
"$\\left(\\sin x+\\sqrt 3\\cos x\\right)\\sin 3x=2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6495e99e856c53f69fd4bc0b2a443c720133aacf5d92013d3bca27f86a733bc1 | $\left(\sqrt 5+\sqrt 2\right)^{20}$. % | [
"$314 925\\cdot 10^5$"
] | [
"$\\left(\\sqrt 5+\\sqrt 2\\right)^{20}$. % "
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-019085b57b155ae30b4f967bf27b9de52f0d45c8aade08c02997e0fb90d33913 | $\left(\sqrt x-2\right)^2=\left(\sqrt {2x}+1\right)^2$ and $x-4\sqrt {x}+4=2x+2\sqrt {2x}+1$. | [
"Yes"
] | [
"$\\left(\\sqrt x-2\\right)^2=\\left(\\sqrt {2x}+1\\right)^2$ and $x-4\\sqrt {x}+4=2x+2\\sqrt {2x}+1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6b6a265af1e77ceb836038c97dd224e0760f6954b4e890961470b0afbfc025c5 | $\left[\frac{2+ba^{-1}}{a+2b}-6b(4b^2-a^2)^{-1}\right]: \left(2a^n b+3a^{n+1}-\frac{6a^{n+2}}{2a-b}\right)^{-1}$ | [
"$a^{n^{-1}}b$"
] | [
"$\\left[\\frac{2+ba^{-1}}{a+2b}-6b(4b^2-a^2)^{-1}\\right]: \\left(2a^n b+3a^{n+1}-\\frac{6a^{n+2}}{2a-b}\\right)^{-1}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-8fac718edda296771f2b283f7fc928ec8ff01d83c2d5bf8740be5e69d16f9f1f | $\left[\frac{3(x+2)}{2(x^3+x^2+x+1)} +\frac{2x^2-x-10}{2(x^3-x^2+x-1)}\right]$ $:\left[\frac{5}{x^2+1}+\frac{3}{2(x+1)}-\frac{3}{2(x-1)}\right]$ | [
"$\\frac{x+2}{2}$"
] | [
"$\\left[\\frac{3(x+2)}{2(x^3+x^2+x+1)} +\\frac{2x^2-x-10}{2(x^3-x^2+x-1)}\\right]$ $:\\left[\\frac{5}{x^2+1}+\\frac{3}{2(x+1)}-\\frac{3}{2(x-1)}\\right]$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c9ee99e54e5c130d6a9683f9368836866117409a93ee06a81845ca5df9b6f77f | $\left[\frac{\left(2.4+1\frac{5}{7}\right)\cdot 4.375} {\frac{2}{3}-\frac{1}{6}} -\frac{\left(2.75-1\frac{5}{6}\right)\cdot 21} {8\frac{3}{20}-0.45}\right]:\frac{67}{200}$ | [
"100"
] | [
"$\\left[\\frac{\\left(2.4+1\\frac{5}{7}\\right)\\cdot 4.375} {\\frac{2}{3}-\\frac{1}{6}} -\\frac{\\left(2.75-1\\frac{5}{6}\\right)\\cdot 21} {8\\frac{3}{20}-0.45}\\right]:\\frac{67}{200}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-efd563174f3c20152d936f3c5ce11800c379b71cedd34ffd6278d1a4f89821f0 | $\left[\frac{\left(6-4\frac{1}{2}\right):0.03} {\left(3\frac{1}{20}-2.65\right)\cdot 4+\frac{2}{5}} -\frac{\left(0.3-\frac{3}{20}\right)\cdot 1\frac{1}{2}} {\left(1.88+2\frac{3}{25}\right)\cdot \frac{1}{80}}\right] :2\frac{1}{20}$ | [
"10"
] | [
"$\\left[\\frac{\\left(6-4\\frac{1}{2}\\right):0.03} {\\left(3\\frac{1}{20}-2.65\\right)\\cdot 4+\\frac{2}{5}} -\\frac{\\left(0.3-\\frac{3}{20}\\right)\\cdot 1\\frac{1}{2}} {\\left(1.88+2\\frac{3}{25}\\right)\\cdot \\frac{1}{80}}\\right] :2\\frac{1}{20}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d4015c94a72835c3258fc6c234dca69aeee2b31e614127e51425a48f37065fc7 | $\left[\frac{a-1}{a^2-2a+1}+\frac{2(a-1)}{a^2-4} -\frac{4(a+1)}{a^2+a-2}+\frac{a}{a^2-3a+2}\right]$ $\times \frac{36a^3-144a-36a^2+144}{a^3+27}$ | [
"$\\frac{72}{a^2-3a+9}$"
] | [
"$\\left[\\frac{a-1}{a^2-2a+1}+\\frac{2(a-1)}{a^2-4} -\\frac{4(a+1)}{a^2+a-2}+\\frac{a}{a^2-3a+2}\\right]$ $\\times \\frac{36a^3-144a-36a^2+144}{a^3+27}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6338350d88daa7aeeb0d960ba436c8b3253b26d02f9f0f0b49ff69740bb57f7f | $\left\{\begin{array}{lll} \cos x<0\medskip \\ \sin \frac{3}{5}x>0. \end{array}\right.$ | [
"$\\frac{\\pi }{2}+10\\pi k<x<\\frac{3\\pi }{2}+10\\pi k$; $\\frac{10\\pi }{3}+10\\pi k<x<\\frac{7\\pi }{2}+10\\pi k$; $\\frac{9\\pi }{2}+10\\pi k<x<5\\pi +10\\pi k$; $\\frac{20\\pi }{3}+10\\pi k<x<\\frac{25\\pi }{3}+10\\pi k$"
] | [
"$\\left\\{\\begin{array}{lll} \\cos x<0\\medskip \\\\ \\sin \\frac{3}{5}x>0. \\end{array}\\right.$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-28e0966704e1ba0c56f418234b6a5cb65025faf06f0f417f72343914f1c2be3f | $\left\{\begin{array}{lll} \sin \frac{x}{2}<\frac{1}{2}\medskip \\ \cos 2x>-\frac{1}{2}. \end{array}\right.$ | [
"$-\\frac{\\pi }{3}+2\\pi k<x<\\frac{\\pi }{3}+2\\pi k$; $\\frac{8\\pi }{3}+4\\pi k<x<\\frac{10\\pi }{3}+4\\pi k$"
] | [
"$\\left\\{\\begin{array}{lll} \\sin \\frac{x}{2}<\\frac{1}{2}\\medskip \\\\ \\cos 2x>-\\frac{1}{2}. \\end{array}\\right.$ "
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ac566eed805b0e381b30ca78dac7f2543ff41d68db3dbec87ad5094a84119585 | $\log _2 \left(\cos ^2 xy+\frac{1}{\cos ^2 xy}\right) =\frac{1}{y^2-2y+2}$. | [
"$x=k\\pi $, $y=1$"
] | [
"$\\log _2 \\left(\\cos ^2 xy+\\frac{1}{\\cos ^2 xy}\\right) =\\frac{1}{y^2-2y+2}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-6651c5835e9bba54bbb8c79ea6fe1bce180c5a236d0ab0df9d819297328ec870 | $\log _2 x\log _4 y+\log _4 z=2$ \ $\log _3 y+\log _9z+\log _9 x=2$ \ $\log _4 z+\log _{16}x+\log _{16}y=2$. | [
"$x=2/3,\\ y=27/8,\\ z=32/3$"
] | [
"$\\log _2 x\\log _4 y+\\log _4 z=2$ \\ $\\log _3 y+\\log _9z+\\log _9 x=2$ \\ $\\log _4 z+\\log _{16}x+\\log _{16}y=2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f21b0ab99fc4acb3cd20c0cbe937e9af67f92e5d513dd6f84f8082f91ed57fff | $\log _2^2 x+(x-1)\log _2 x=6-2x$. | [
"$x_1=1/4$, $x_2=2$. Reduce the equation to the form \\ $(\\log _2 x+2)(\\log _2 x+x-3)=0$. The equation $\\log _2 x=3-x$ has a \\ unique root, $x=2$; for $x>2$ the left member is greater than 1, \\ the right member is less than 1; for (admissible) $x<2$, vice versa"
] | [
"$\\log _2^2 x+(x-1)\\log _2 x=6-2x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-26072e60b5b8857cd884b3e217c08f68be37eccc07809576701d6ade3de8f50a | $\log _3 |\pi x|+\log _{|\pi x|}3=\frac{2}{\sin ^2 (x+y)-2\sin (x+y)+2}$. | [
"$x=\\pm 3/\\pi $, $y=[(4k+1)\\pi /2]\\mp (3/\\pi )$"
] | [
"$\\log _3 |\\pi x|+\\log _{|\\pi x|}3=\\frac{2}{\\sin ^2 (x+y)-2\\sin (x+y)+2}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dfc855f9a99a4554343d6e75f1a0e2676c774d94bf894cec4e78f25f4635b7eb | $\log _9 (x^2+1)-\log _3(y-2)=0$ $\log _2 (x^2-2y^2+10y-7)=2$. | [
"$x_1=\\sqrt 3,\\ y_1=4$; $x_2=-\\sqrt 3,\\ y_2=4$"
] | [
"$\\log _9 (x^2+1)-\\log _3(y-2)=0$ $\\log _2 (x^2-2y^2+10y-7)=2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-195ec3f616a85210a3af1ff7c03835c358acb84bacc48497f9be26713973691f | $\log _9 (x^2+2)+\log _{81}(y^2+9)=2$ \ $2\log _4 (x+y)-\log _2(x-y)=0$. | [
"$x=5,\\ y=0$"
] | [
"$\\log _9 (x^2+2)+\\log _{81}(y^2+9)=2$ \\ $2\\log _4 (x+y)-\\log _2(x-y)=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-b583be195eb5ceb8d0346f32cbe1280bd5e6dc79c984264773830f2f1a97870d | $\log _{100}x^2=\log _{\sqrt x}10\left(\log _{10}10a -\left|\log _{10}\frac{x}{a}\right|\right)$. | [
"$x_1=10^{1-\\sqrt 3}$, $x_2=10^{-1+\\sqrt {4\\log _{10}a+3}}$ for $10^{1-\\sqrt 3}\\le a\\le 10^{1+\\sqrt 3}$; $x_{1,2}=10^{1\\pm \\sqrt 3}$ for $a>10^{1+\\sqrt 3}$; there are no solutions for the remaining values of $a$"
] | [
"$\\log _{100}x^2=\\log _{\\sqrt x}10\\left(\\log _{10}10a -\\left|\\log _{10}\\frac{x}{a}\\right|\\right)$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-2c7dcdd77cba79fc4c1d4d7bcee52fe3e6855c48675df1a26d7f382de6541d3b | $\log _{\sqrt x}a\cdot \left|\log _a \frac{x}{2}\right| =\log _{a^2}2\cdot \log _{\sqrt x}a^2-\log _a \sqrt x$. | [
"$x=a^4$ for $1<a\\le \\sqrt [4]{2}$; $x=a^{\\sqrt {8\\log _a 2+4}-2}$ for $a>\\sqrt [4]{2}$; there are no solutions for the remaining values of $a$"
] | [
"$\\log _{\\sqrt x}a\\cdot \\left|\\log _a \\frac{x}{2}\\right| =\\log _{a^2}2\\cdot \\log _{\\sqrt x}a^2-\\log _a \\sqrt x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4b693c342b24fac44f291174e586df883964b858984ba808e9addc29c371805f | $\log _{\sqrt x}a^2\cdot \left|\log _{a^2}\frac{x}{2}\right|+\log _a x =\log _{a^2} 4\cdot \log _{\sqrt x}a$. | [
"$x=a^2$ for $1<a\\le \\sqrt 2$, $x=a^{-1+\\sqrt {4+\\log _a 2}}$ for $a>\\sqrt 2$; no solutions for the remaining values of $a$"
] | [
"$\\log _{\\sqrt x}a^2\\cdot \\left|\\log _{a^2}\\frac{x}{2}\\right|+\\log _a x =\\log _{a^2} 4\\cdot \\log _{\\sqrt x}a$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5b767cc2e43df57e4822dce3288409afc30b9b612fe9193daff5d187d807c86d | $\omega_1$ and $\omega_2$ are concentric circles with center $O$, such that the radius of $\omega_1$ is $1$ and the radius of $\omega_2$ is $2$. Points $A$ and $B$ are on $\omega_1$ and points $C$ and $D$ are on $\omega_2$ such that $ABCD$ is the smallest square possible. If $\cos (\angle AOB)=\frac{\sqrt{a}-b}{c}$ in ... | [
"714"
] | [
"$\\omega_1$ and $\\omega_2$ are concentric circles with center $O$, such that the radius of $\\omega_1$ is $1$ and the radius of $\\omega_2$ is $2$. Points $A$ and $B$ are on $\\omega_1$ and points $C$ and $D$ are on $\\omega_2$ such that $ABCD$ is the smallest square possible. If $\\cos (\\angle AOB)=\\frac{\\sqr... | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 8
} | 24 | 0 | 0 |
omr-ws-b37d9e88912e9549445da00a1355eae5e9a6d2baf97466a36cae8054f4807cce | $\overline{AB}$ and $\overline{ED}$ are parallel tangents of the circle $O$ at points $A$ and $E$, respectively, cutting a third tangent $\overline{BCD}$ at points $B$ and $D$, respectively. If $AB = 8$ and $ED = 18$, what is the radius of the circle? | [
"12"
] | [
"$\\overline{AB}$ and $\\overline{ED}$ are parallel tangents of the circle $O$ at points $A$ and $E$, respectively, cutting a third tangent $\\overline{BCD}$ at points $B$ and $D$, respectively. If $AB = 8$ and $ED = 18$, what is the radius of the circle?"
] | [
"aops_c4_high_school_math"
] | [
"cot",
"tir"
] | {
"cot": 16,
"tir": 1
} | 17 | 0 | 0 |
omr-ws-28263a2d6a01398f4f681867ec67f28ef2f2a9006417688c6d7b60affefc947a | $\overline{AB}$ is a diameter of a circle. Tangents $\overline{AD}$ and $\overline{BC}$ are drawn so that $\overline{AC}$ and $\overline{BD}$ intersect in a point on the circle. If $\overline{AD}=a$ and $\overline{BD}=b$, $a \not= b$, what is the diameter of the circle? | [
"\\(\\sqrt{ab}\\)"
] | [
"$\\overline{AB}$ is a diameter of a circle. Tangents $\\overline{AD}$ and $\\overline{BC}$ are drawn so that $\\overline{AC}$ and $\\overline{BD}$ intersect in a point on the circle. If $\\overline{AD}=a$ and $\\overline{BD}=b$, $a \\not= b$, what is the diameter of the circle?"
] | [
"aops_c5_contests_amp_programs"
] | [
"cot",
"genselect",
"tir"
] | {
"cot": 6,
"genselect": 1,
"tir": 5
} | 12 | 0 | 0 |
omr-ws-3a7a8f17537f831bf1930add5221ed3fc0fda51ad461932fc97f7be1acc1ca0e | $\overline{AB}$ is a diameter of circle $C_1$. Point $P$ is on $C_1$ such that $AP > BP$. Circle $C_2$ is centered at $P$ with radius $PB$. The extension of $\overline{AP}$ past $P$ meets $C_2$ at $Q$. Circle $C_3$ is centered at $A$ and is externally tangent to $C_2$. Circle $C_4$ passes through $A$, $Q$, and $R$. Cal... | [
""
] | [
"$\\overline{AB}$ is a diameter of circle $C_1$. Point $P$ is on $C_1$ such that $AP > BP$. Circle $C_2$ is centered at $P$ with radius $PB$. The extension of $\\overline{AP}$ past $P$ meets $C_2$ at $Q$. Circle $C_3$ is centered at $A$ and is externally tangent to $C_2$. Circle $C_4$ passes through $A$, $Q$, and $... | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 1 |
omr-ws-dbeb970d4c369c990713064f82898a0b805ebbe0446244766deb84c6206f7b8d | $\overline{BC}$, $\overline{CE}$, and $\overline{AE}$ are tangents to circle $G$ at $B$, $D$, and $F$, respectively. Given $BC=5$, $CE=7$, and $BG=3$, find the perimeter of quadrilateral $BFEC$. | [
"20"
] | [
"$\\overline{BC}$, $\\overline{CE}$, and $\\overline{AE}$ are tangents to circle $G$ at $B$, $D$, and $F$, respectively. Given $BC=5$, $CE=7$, and $BG=3$, find the perimeter of quadrilateral $BFEC$."
] | [
"aops_c4_high_school_math"
] | [
"cot",
"genselect"
] | {
"cot": 10,
"genselect": 5
} | 15 | 0 | 0 |
omr-ws-f65ed376c3fb6d1f3b3042f1479101df83e7315616c7191f10d85fb8aab5258a | $\sin (x+30^\circ)+\cos (x+60^\circ)=1+\cos 2x$ | [
"$x=90^\\circ(2n+1);\\ x=60^\\circ(6n\\pm 1)$"
] | [
"$\\sin (x+30^\\circ)+\\cos (x+60^\\circ)=1+\\cos 2x$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-10d6c03834432a88a5ff1094a1964d45459d4bd4afa7c38b8db01f24b87eb61f | $\sin 18x+\sin 10x+\sin 2x=3+\cos ^2 2x$. | [
"$\\emptyset $"
] | [
"$\\sin 18x+\\sin 10x+\\sin 2x=3+\\cos ^2 2x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-35cf667f639aa900f8fb6299bba98f0ac8d251f093a1259e4eb3bb9de69ef33d | $\sin 2x+\cos 2x=\sin x+\cos x$. | [
"$2\\pi k;\\ \\frac{\\pi }{6}+\\frac{2\\pi }{3}n$"
] | [
"$\\sin 2x+\\cos 2x=\\sin x+\\cos x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-15bd7a15201ba2218df794c057a810e6d4ecac5fafa7b14de04045528cfac812 | $\sin 2x+\tan x=2$. | [
"$\\frac{\\pi }{4}+\\pi k$"
] | [
"$\\sin 2x+\\tan x=2$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-26502e55fe344a75db28244900862b7710134f7a4761ebdfb13a9908e9bab8db | $\sin 3x+\cos 3x=\sqrt 2$ | [
"$x=15^\\circ(8n+1)$"
] | [
"$\\sin 3x+\\cos 3x=\\sqrt 2$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-ad983ea35c8eb403f611a77beb63278c35ab946db00cb2867809d8cb231d157d | $\sin 3x=4\sin x\cos 2x$ | [
"$x=180^\\circ n;\\ x=180^\\circ n\\pm 30^\\circ$"
] | [
"$\\sin 3x=4\\sin x\\cos 2x$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-a38393bb185907f031c72ca235254f3c81ac5d1ed9171b04fd26f4001e3ddb5b | $\sin 3x=\cos 2x$ | [
"$x=\\frac{\\pi }{2}(4n+1);\\ x=\\frac{\\pi }{10}(4n+1)$"
] | [
"$\\sin 3x=\\cos 2x$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4153fb37af9240669fe8bed61e6283a3b5c1270cda0a39e167a0ad5ff399cafc | $\sin 3x=\cos 2x$. | [
"$\\frac{\\pi }{10}+\\frac{2\\pi }{5}k$"
] | [
"$\\sin 3x=\\cos 2x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1a148c5415e11a2a387525c5b16da58eeaa52534c749f4ce112d38395ac91c95 | $\sin 4x\cos x\tan 2x=0$. | [
"$\\frac{\\pi }{2}k$"
] | [
"$\\sin 4x\\cos x\\tan 2x=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1bded0cb46d7fdd221eb7ccf5ebcf7f9bc4ac66fd24360d4deb6d7320d482619 | $\sin 5x+\cos 5x=\sqrt 2\cos 13x$. | [
"$-\\frac{\\pi }{32}+\\frac{\\pi }{4}k;\\ \\frac{\\pi }{72}+\\frac{\\pi }{9}n$"
] | [
"$\\sin 5x+\\cos 5x=\\sqrt 2\\cos 13x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-d06c31856857de7a2b74d9c55b6493e898eb6dd9f889797091f1c7a1d98ae2f5 | $\sin 5x+\sin x+2\sin^2 x=1$ | [
"$x=45^\\circ(2n+1);\\ x=60^\\circ n+(-1)^n 10^\\circ$"
] | [
"$\\sin 5x+\\sin x+2\\sin^2 x=1$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c993610a99763f8fa104f8d8233e33f564b6d0a9b80324f3e83ef814b39882f2 | $\sin 5x+\sin x=2+\cos ^2 x$. | [
"$\\frac{\\pi }{2}+2\\pi k$"
] | [
"$\\sin 5x+\\sin x=2+\\cos ^2 x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-1a2a0243542bcee0a2da4ff7376493f8f398552df72b60db0589a051e44c1f48 | $\sin 5x\cos 3x=\sin 9x\cos 7x$. | [
"$\\frac{\\pi }{4}k;\\ \\frac{\\pi }{24}+\\frac{\\pi }{12}n$"
] | [
"$\\sin 5x\\cos 3x=\\sin 9x\\cos 7x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f7ca103a518baac5f540fe882b780ba67998359cc28a8288096dab6c07fe45d3 | $\sin 6x\cos 2x=\sin 5x\cos 3x-\sin 2x$. | [
"$\\frac{\\pi }{3}k;\\ \\frac{\\pi }{2}+\\pi n$"
] | [
"$\\sin 6x\\cos 2x=\\sin 5x\\cos 3x-\\sin 2x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-f4eee9dcca0a63f6de2b06e257353d3e2d838dbd73cacde3b62c3711f56acadf | $\sin \alpha +\sin \beta +\sin \gamma $ if $\alpha +\beta +\gamma =180^\circ$ | [
"$4\\cos\\frac{\\alpha }{2}\\cos\\frac{\\beta }{2}\\cos\\frac{\\gamma }{2}$"
] | [
"$\\sin \\alpha +\\sin \\beta +\\sin \\gamma $ if $\\alpha +\\beta +\\gamma =180^\\circ$ "
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e43074a81d0d7b888e26cfc774b5a2216034041d07e674ee09eced67f5b7e960 | $\sin \frac{x}{4}+2\cos \frac{x-2\pi }{3}=3$. | [
"$2\\pi +24\\pi k$"
] | [
"$\\sin \\frac{x}{4}+2\\cos \\frac{x-2\\pi }{3}=3$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-84445400b148335c18a17fc1e5ea73192ebfbf9644dd54dfa7016293e41103b5 | $\sin ^2 x-\cos 2x=2-\sin 2x$. | [
"$\\frac{\\pi }{2}+\\pi k;\\ \\arctan \\frac{3}{2}+\\pi n$"
] | [
"$\\sin ^2 x-\\cos 2x=2-\\sin 2x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-c0963b985f29a9fd322ab64e83029afeda63ba9bb9d4452bd2aba4c65c771387 | $\sin ^2 x\cos ^2 x-10\sin x\cos ^3 x+21\cos ^4 x=0$. | [
"$\\frac{\\pi }{2}+\\pi k;\\ \\arctan 7+\\pi n;\\ \\arctan 3+\\pi m$"
] | [
"$\\sin ^2 x\\cos ^2 x-10\\sin x\\cos ^3 x+21\\cos ^4 x=0$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-3c2d0acfc4ec2a6725c936e1502139c621a08956b909cbd55ef577aa8e8581ed | $\sin ^3 x+\cos ^3 x=1$. | [
"$\\frac{\\pi }{2}+2\\pi k;\\ 2\\pi n$"
] | [
"$\\sin ^3 x+\\cos ^3 x=1$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-dea9a3b4d5d3a789fba963cf783a9d2feff1c77346fc610723535d399ece9ce8 | $\sin ^5 x-\cos ^5 x=\frac{1}{\cos x}-\frac{1}{\sin x}$. | [
"$\\frac{\\pi }{4}+\\pi k$"
] | [
"$\\sin ^5 x-\\cos ^5 x=\\frac{1}{\\cos x}-\\frac{1}{\\sin x}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-4bdc4fa72f46bb85531b932cf167db0f18d485e48543dde0483be7394a35080a | $\sin ^6 x+\sin ^4 x+\cos ^6 x+\cos ^4 x+\sin \frac{x}{2}=3$. | [
"$\\pi +4\\pi k$"
] | [
"$\\sin ^6 x+\\sin ^4 x+\\cos ^6 x+\\cos ^4 x+\\sin \\frac{x}{2}=3$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-7b1c364986df0432f0c222cbca774910faa59c73499ad975efc15c3158c58333 | $\sin ^6 x+\sin ^4 x\cos ^2 x=\sin ^2 x\cos ^3 x+\sin x\cos ^5 x$. | [
"$\\pi k;\\ \\frac{\\pi }{4}+\\pi n$"
] | [
"$\\sin ^6 x+\\sin ^4 x\\cos ^2 x=\\sin ^2 x\\cos ^3 x+\\sin x\\cos ^5 x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-e9d9b792fca1682c67feef31e0e515ad039514237b2a7a39d30a2e293213337c | $\sin ^{10}x+\cos ^{10}x=\frac{29}{16}\cos ^4 2x$. | [
"$\\frac{\\pi }{8}+\\frac{\\pi }{4}k$"
] | [
"$\\sin ^{10}x+\\cos ^{10}x=\\frac{29}{16}\\cos ^4 2x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-06cc896402eb328ceb2cc028bfeaa90403120262e31d43aa5d733f1a55780d35 | $\sin ^{10}x+\cos ^{10}x=\frac{29}{64}$. | [
"$\\frac{\\pi }{8}+\\frac{\\pi }{4}k$"
] | [
"$\\sin ^{10}x+\\cos ^{10}x=\\frac{29}{64}$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-5cd7088ffbe9f161fc78a169e0b9c9dc3fc549e3c22e73e96c25b93ad334ef89 | $\sin x+2\cos x=\cos 2x-\sin 2x$. | [
"$-\\frac{\\pi }{2}+\\pi k;\\ \\frac{\\pi }{4}\\pm \\arccos \\frac{1}{2\\sqrt 2}+2\\pi n$"
] | [
"$\\sin x+2\\cos x=\\cos 2x-\\sin 2x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-58855164f1925c6318278b5995af04409cf1cebe2718644481705c51cef8f86b | $\sin x+\cos x=1$ | [
"$x=90^\\circ \\cdot 4n;\\ x=90^\\circ(4n+1)$"
] | [
"$\\sin x+\\cos x=1$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-10743e75b0bc4f524471e55aa100b4dcbb3c6459a13a6b48b350e3d6d97fa683 | $\sin x+\cos x=1+\sin 2x$ | [
"$x=\\frac{\\pi }{4}(4n-1);\\ x=\\frac{\\pi }{2}(4n+1);\\ x=\\frac{\\pi }{2}\\cdot 4n$"
] | [
"$\\sin x+\\cos x=1+\\sin 2x$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-072b73b99321c068114673050260578c6a0dd0fafc3a62f3c2a85edf7206c169 | $\sin x+\cos x=\frac{1}{\sin x}$ | [
"$x_1=\\frac{\\pi }{2}(2n+1)$; $x_2=\\frac{\\pi }{4}(4n+1)$"
] | [
"$\\sin x+\\cos x=\\frac{1}{\\sin x}$"
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
omr-ws-9a48d011af45881fab21b46a1b9f0297cd72608361c0df261c9e475736727fc9 | $\sin x+\cos x=\sqrt 2+\sin ^4 4x$. | [
"$\\frac{\\pi }{4}+2\\pi k$"
] | [
"$\\sin x+\\cos x=\\sqrt 2+\\sin ^4 4x$."
] | [
"aops_c6_high_school_olympiads"
] | [
"additional_problems"
] | {
"additional_problems": 1
} | 1 | 0 | 0 |
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