IR-TST 2020 Problem 5

Given $k \in \mathbb{Z}$ prove that there exist infinite pairs of distinct natural numbers such that \begin{align*} n+s(2n)=m+s(2m) kn+s(n^2)=km+s(m^2). \end{align*} ( $s(n)$ denotes the sum of digits of $n$ .)

Proposed by Mohammadamin Sharifi