MathNet 2009-part-04 — 175 problems (EN–VI) Question 3

Let $k$ be a positive integer greater than $1$. Prove: for every non-negative integer $m$ there exist $k$ positive integers $n_1, n_2, \dots, n_k$, such that $$ n_1^2 + n_2^2 + \dots + n_k^2 = 5^{m+k}. $$