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Equality of fractional parts

Source: Ukrainian National Math Olympiad 4th round

October 20, 2014
algebra unsolvedalgebra

Problem Statement

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}=xāˆ’[x]\{x\}=x-[x])
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