62nd Ukrainian National Mathematical Olympiad

The positive integers $x$, $y$ satisfy the conditions: $$\{\sqrt{x^2+2y}\}>\frac{2}{3},\{\sqrt{y^2+2x}\}>\frac{2}{3}.$$ Prove that $x=y$. Here, $\{a\}\in [0;1)$ denotes the fractional part of the number $a$, that is, there exists an integer $n$ for which the equality $a=n+\{a\}$ holds. For example, $\{3.14\}=0.14$.