Suppose that $V$ is a vector space of dimension $n>0$ over a field of characterstic $p\neq 0$. Let $A: V\to V$ be an affine transformation. Prove that there exist $u\in V$ and $1\le k\le np$ such that $$A^k u = u.$$

The best solution was submitted by Chiheon Kim (김치헌), 수리과학과 2006학번. Congratulations!

Here is his Solution of Problem 2010-19.

An alternative solution was submitted by 박민재 (KSA-한국과학영재학교, +3).