Problem

Source: Ukrainian Mathematical Olympiad 2023. Day 1, Problem 9.4

Tags: algebra, fractional part



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