51st Ukrainian National Mathematical Olympiad, 4th Round

Answer: All non-negative reals and negative integers.

Consider 3 cases.

1) $x\ge 0$. Then $|x|=x$, $[x]\ge 0$, and so $|[x]|=[x]=[|x|]$, which means that any non-negative $x$ is a solution of our equation.

2) $x$ is a negative integer. Then $[x]=x$, $[-x]=-x$, $|x|=-x$, and so $|[x]|=[-x]=(-x=|x|=[x]|)$, proving that these values of $x$ are also solutions.

3) $x$ is negative, but not an integer. In this case $|x|=-x$, $|[x]|<-x$. On the other hand, $[x]<x<0$, $|[x]|>-x$, and the equality is not satisfied.