UA-UMO 2023 Problem 83

Positive integers $x, y$ satisfy the following conditions: $$\{\sqrt{x^2 + 2y}\}> \frac{2}{3}; \hspace{10mm} \{\sqrt{y^2 + 2x}\}> \frac{2}{3}$$ Show that $x = y$ .

Here $\{x\}$ denotes the fractional part of $x$ . For example, $\{3.14\} = 0.14$ .

Proposed by Anton Trygub