Question 179
The positive integers $x$, $y$ satisfy the conditions:
$$
\{\sqrt{x^2 + 2y}\} \gt \frac{2}{3}, \{\sqrt{y^2 + 2x}\} \gt \frac{2}{3}.
$$
Prove that $x = y$.
Here, $\{a\} \in [0; 1)$ denotes the fractional part of the number $a$, that is, there exists an integer $n$ for which the equality $a = n + \{a\}$ holds. For example, $\{3.14\} = 0.14$.
$$
\{\sqrt{x^2 + 2y}\} \gt \frac{2}{3}, \{\sqrt{y^2 + 2x}\} \gt \frac{2}{3}.
$$
Prove that $x = y$.
Here, $\{a\} \in [0; 1)$ denotes the fractional part of the number $a$, that is, there exists an integer $n$ for which the equality $a = n + \{a\}$ holds. For example, $\{3.14\} = 0.14$.