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On Fractional Parts of Square Roots

Source: Ukrainian Mathematical Olympiad 2023. Day 1, Problem 8.3

April 5, 2023
inequalitiesalgebrafractional part

Problem Statement

Positive integers x,yx, y satisfy the following conditions:
{x2+2y}>23;{y2+2x}>23\{\sqrt{x^2 + 2y}\}> \frac{2}{3}; \hspace{10mm} \{\sqrt{y^2 + 2x}\}> \frac{2}{3}
Show that x=yx = y.
Here {x}\{x\} denotes the fractional part of xx. For example, {3.14}=0.14\{3.14\} = 0.14.
Proposed by Anton Trygub
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