Question 184
Find the smallest real number $C$ for which the following condition is true: for any different positive integers $x, y$ the inequality holds
$$
\min\{\{\sqrt{x^2 + 2y}\}, \{\sqrt{y^2 + 2x}\}\} \lt C.
$$
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, $\{3.14\} = 0.14$.
$$
\min\{\{\sqrt{x^2 + 2y}\}, \{\sqrt{y^2 + 2x}\}\} \lt C.
$$
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, $\{3.14\} = 0.14$.