Let \(A\) be an \(n\times n\) matrix with complex entries. Prove that if \(A^2=A^*\), then \[\operatorname{rank}(A+A^*)=\operatorname{rank}(A).\] (Here, \(A^*\) is the conjugate transpose of \(A\).)

The best solution was submitted by Kim, Kee Taek (김기택, 2015학번). Congratulations!

Here is his solution of problem 2015-12.

Alternative solutions were submitted by 고경훈 (2015학번, +3), 김기현 (수리과학과 2012학번, +3), 박지민 (전산학과 대학원생, +3), 엄태현 (수리과학과 2012학번, +3), 이수철 (수리과학과 2012학번, +3), 이종원 (수리과학과 2014학번, +3), 진우영 (수리과학과 2012학번, +3), 함도규 (2015학번, +3).

Does \(\frac{1}{n \sin n}\) converge as \(n\) goes to infinity?

The best solution was submitted by Lee, Jongwon (이종원, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2015-11.

Alternative solutions were submitted by 고경훈 (2015학번, +3), 김기현 (수리과학과 2012학번, +3), 신준형 (2015학번, +3), 엄태현 (수리과학과 2012학번, +3), 오동우 (2015학번, +3), 이수철 (수리과학과 2012학번, +3), 진우영 (수리과학과 2012학번, +3), 함도규 (2015학번, +3), 이상민 (수리과학과 2014학번, +2), 이영민 (수리과학과 2012학번, +2). One incorrect solution (KDR) was submitted.

Let \(w_1,w_2,\ldots,w_n\) be positive real numbers such that \( \sum_{i=1}^n w_i=1\). Prove that if \(x_1,x_2,\ldots,x_n\in [0,\pi]\), then \[ \sin \left(\prod_{i=1}^n x_i^{w_i} \right) \ge \prod_{i=1}^n (\sin x_i)^{w_i}.\]

The best solution was submitted by Lee, Young Min (이영민, 수리과학과 2012학번). Congratulations!

Here is his solution of problem 2015-10.

Other (but mostly identical) solutions were submitted by 고경훈 (2015학번, +3), 김기현 (수리과학과 2012학번, +3), 엄태현 (수리과학과 2012학번, +3), 오동우 (2015학번, +3), 이수철 (수리과학과 2012학번, +3), 이종원 (수리과학과 2014학번, +3), 진우영 (수리과학과 2012학번, +3), 함도규 (2015학번, +3).

Let \(n\ge 1\) and \(a_0,a_1,a_2,\ldots,a_{n}\) be non-negative integers. Prove that if \[ N=\frac{a_0^2+a_1^2+a_2^2+\cdots+a_{n}^2}{1+a_0a_1a_2\cdots a_{n}}\] is an integer, then \(N\) is the sum of \(n\) squares of integers.

The best solution was submitted by Lee, Jongwon (이종원, 수리과학과 2014학번). Congratulations!

Here is his solution of problem 2015-9.

Alternative solutions were submitted by 김기현 (수리과학과 2012학번, +3), 엄태현 (수리과학과 2012학번, +3), 이수철 (수리과학과 2012학번, +3), 진우영 (수리과학과 2012학번, +3), 함도규 (2015학번, +3), 윤지훈 (2012학번, +2). One incorrect solution was submitted (YSC).

Does there exist a subset \(A\) of \(\mathbb{R}^2\) such that \( \lvert A\cap L\rvert=2\) for every straight line \(L\)?

The best solution was submitted by Lee, Su Cheol (이수철, 수리과학과 2012학번). Congratulations!

Here is his solution of problem 2015-08.

Alternative solutions were submitted by 김기현 (수리과학과 2012학번, +3), 김동률 (2015학번, +3), 엄태현 (수리과학과 2012학번, +3), 이종원 (수리과학과 2014학번, +3), 진우영 (수리과학과 2012학번, +3), 박훈민 (수리과학과 2013학번, +2), 오동우 (2015학번, +2).