UA-UMO 2023 Problem 94

Find the smallest real number $C$ , such that for any positive integers $x \neq y$ holds the following: $$\min(\{\sqrt{x^2 + 2y}\}, \{\sqrt{y^2 + 2x}\})<C$$ Here $\{x\}$ denotes the fractional part of $x$ . For example, $\{3.14\} = 0.14$ .

Proposed by Anton Trygub