Let $s(k)$ denotes sum of digits of positive integer $k$. Prove that there are infinitely many positive integers $n$, which are not divisible by $10$ and satisfies:
$$s(n^2) < s(n) - 5$$
We will prove that $4\underbrace{9...9}_{x \; \text{times }}$ works for all positive integers $x$.
Note that:
$$(4\underbrace{9...9}_{x \; \text{times }})^2=(5\underbrace{0..0}_{x \; \text{times}}-1)^2=25\underbrace{0..0}_{2x \; \text{times}}-1\underbrace{0..}_{x+1 \; \text{times}}+1 = 24\underbrace{9..9}_{x-1 \; \text{times}}\underbrace{0..0}_{x \; \text{times}}1$$Therefore we need to check if:
$$2 +4+9(x-1)+1 <4+9x-5 \implies 7-9+9x<9x-1 \implies 0<1$$We are done.