IZhO 2024 Problem 3

Positive integer $d$ is not a perfect square. For each positive integer $n$, let $s(n)$ denote the number of digits 1 among the first $n$ digits in the binary representation of $\sqrt{d}$ (including the digits before the point). Prove that there exists an integer $A$ such that $s(n) > \sqrt{2n} - 2$ for all integers $n \geq A$.