Problem

Source: Iran TST1 Day 2 P5

Tags: number theory, sum of digits, Iranian TST



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