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Fractional Parts of Square Roots, Continued

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

April 5, 2023

algebrafractional part

Problem Statement

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