Let \( A, B \) are \( n \times n \) Hermitian matrices and \( p, q \in [1, \infty] \) with \( \frac{1}{p} + \frac{1}{q} = 1 \). Prove that
\[
| Tr (AB) | \leq \| A \|_{S^p} \| B \|_{S^q}.
\]
(Here, \(\| A \|_{S^p} \) is the \(p\)-Schatten norm of \( A \), defined by
\[
\| A \|_{S^p} = \left( \sum_{i=1}^n |\lambda_i|^p \right)^{1/p},
\]
where \( \lambda_1, \lambda_2, \dots, \lambda_n \) are the eigenvalues of \( A \).)

The best solution was submitted by Park, Sunghyuk (박성혁, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2015-24.

Alternative solutions were submitted by 신준형 (2015학번, +3), 이정환 (2015학번, +3), 이종원 (수리과학과 2014학번, +3), 장기정 (수리과학과 2014학번, +3), 김동률 (2015학번, +2), 최인혁 (2015학번, +2).

Let \(f:[0,1)\to[0,1)\)  be a function such that \[ f(x)=\begin{cases} 2x,&\text{if }0\le 2x\lt 1,\\ 2x-1, & \text{if } 1\le 2x\lt 2.\end{cases}\] Find all \(x\) such that \[ f(f(f(f(f(f(f(x)))))))=x.\]

The best solution was submitted by Park, Sunghyuk (박성혁, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2015-23.

Alternative solutions were submitted by 김동률 (2015학번, +3), 신준형 (2015학번, +3), 유찬진 (2015학번, +3), 이상민 (수리과학과 2014학번, +3), 이종원 (수리과학과 2014학번, +3, his solution), 장기정 (수리과학과 2014학번, +3), 정호진 (동북고등학교 2학년, +3), 최인혁 (2015학번, +3), Daulet Kurmantayev (?, +3), 최동준 (포항공대 수학과 2013학번, +2).