Let X be a finite non-empty set. Suppose that there is a function \(f:X\to X\) such that \( f^{20120407}(x)=x\) for all \(x\in X\). Prove that the number of elements x in X such that \(f(x)\neq x\) is divisible by 20120407.

The best solution was submitted by Myeongjae Lee (이명재), 2012학번. Congratulations!

Here is his Solution of Problem 2012-8.

Alternative solutions were submitted by Phan Kieu My (전산학과 2009학번, +3), 김태호 (수리과학과 2011학번, +3), 박민재 (2011학번, +3), 서기원 (수리과학과 2009학번, +3), 천용 (전남대 의예과 2011학번, +3), 어수강 (서울대학교 석사과정, +3), 정우석 (서강대학교 2011학번, +3), 박훈민 (대전과학고 2학년, +3). There were 2 incorrect solutions (S. B., S. H.).