Problem

Source: RMO KV 2024 Q6

Tags: combinatorics



Let $X$ be a set of $11$ integers. Prove that one can find a nonempty subset $\{a_1, a_2, \cdots , a_k \}$ of $X$ such that $3$ divides $k$ and $9$ divides the sum $\sum_{i=1}^{k} 4^i a_i$.