MathDB

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Part of 2014 Ukraine National Mathematical Olympiad

Problems(2)

Equality of fractional parts

Source: Ukrainian National Math Olympiad 4th round

10/20/2014
Suppose that for real x,y,z,tx,y,z,t the following equalities hold:{x+y+z}={y+z+t}={z+t+x}={t+x+y}=1/4\{x+y+z\}=\{y+z+t\}=\{z+t+x\}=\{t+x+y\}=1/4. Find all possible values of {x+y+z+t}\{x+y+z+t\}.(Here{x}=x−[x]\{x\}=x-[x])
algebra unsolvedalgebra
Trigonometric inequality regarding minimums

Source: Ukrainian National Math Olympiad 4th round

10/20/2014
Find the values of xx such that the following inequality holds: min⁡{sin⁡x,cos⁡x}<min⁡{1−sin⁡x,1−cos⁡x}\min\{\sin x,\cos x\}<\min\{1-\sin x,1-\cos x\}
inequalitiestrigonometryalgebra unsolvedalgebra