CZ-SK-MO 2020 Problem 2

The triangle $ABC$ is given. Inside its sides $AB$ and $AC$ , the points $X$ and $Y$ are respectively selected Let $Z$ be the intersection of the lines $BY$ and $CX$ . Prove the inequality $$[BZX] + [CZY]> 2 [XY Z]$$ , where $[DEF]$ denotes the content of the triangle $DEF$ .

(David Hruska, Josef Tkadlec)