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).