37th Iranian Mathematical Olympiad

Let $P_k$ be the set of solutions of $$\{\begin{array}{l}n+S(n)=m+S(m),\\ kn+S(kn)=km+S(km).\end{array}$$ We want to map a single solution $(m_0,n_0)\in P_k$, where $10\nmid m_0,n_0$, to infinite solutions like $(m_1,n_1)$, where $(m_1,n_1)\in P_{k+1}$ or $(m_1,n_1)\in P_{k-1}$ and $10\nmid m_1,n_1$. If $(m_0,n_0)$ be a solution in $P_k$, then we claim that $(m_0+10^{\alpha },n_0+10^{\alpha })$ is a solution in $P_{k+1}$ for every large $\alpha$. It follows by the same easy calculations and noting that $$S(n^2+2\times 10^{\alpha }n+10^{2\alpha })=S(n^2)+S(2n)+1,$$ for all large $\alpha$. And if $(m_0,n_0)$ is a solution in $P_k$ where $10\nmid m_0,n_0$, then we replace them by $(10^{\alpha }-m_0,10^{\alpha }-n_0)$ for all large $\alpha$ to reach an infinite number of solutions in $P_{k-1}$. Therefore, it's enough to find a simple solution. For example we have $(9,12)\in P_0$ and we're done. ■