Let \(A\) be an unbounded subset of the set \(\mathbb R\) of the real numbers. Let \(T\) be the set of all real numbers \(t\) such that \(\{tx-\lfloor tx\rfloor : x\in A\}\) is dense in \([0,1]\). Is \(T\) dense in \(\mathbb R\)?

The best solution was submitted by Kim, Kihyun (김기현, 수리과학과 2012학번). Congratulations!

Here is his solution of problem 2015-6.

Alternative solutions were submitted by 엄태현 (수리과학과 2012학번, +3), 이명재 (수리과학과 2012학번, +3), 이수철 (수리과학과 2012학번, +3), 이종원 (수리과학과 2014학번, +3), 정성진 (수리과학과 2013학번, +3), 최인혁 (2015학번, +3), 진우영 (수리과학과 2012학번, +3), 배형진 (마포고 1학년, +2). One incorrect solution was submitted (KDR).

Determine all \(n\times n\) matrices A such that \( \operatorname{tr}(AXY)=\operatorname{tr}(AYX)\) for all \(n\times n\) matrices \(X\) and \(Y\).

The best solution was submitted by Choi, Doo Seong (최두성, 수리과학과 2011학번). Congratulations!

Here is his solution of problem 2015-5.

Alternative solutions were submitted by 고경훈 (2015학번, +3), 김기현 (수리과학과 2012학번, +3), 김경석 (2015학번, +3), 엄태현 (수리과학과 2012학번, +3), 오동우 (2015학번, +3), 유찬진 (2015학번, +3), 이수철 (수리과학과 2012학번, +3), 이명재 (수리과학과 2012학번, +3), 이영민 (수리과학과 2012학번, +3), 장기정 (수리과학과 2014학번, +3), 정성진 (수리과학과 2013학번, +3), 진우영 (수리과학과 2012학번, +3), 최인혁 (2015학번, +3), 함도규 (2015학번, +3), 홍혁표 (수리과학과 2013학번, +3), 박성혁 (수리과학과 2014학번, +2), 이종원 (수리과학과 2014학번, +2), 전한솔 (고려대, +3), 어수강 (서울대 수리과학부 대학원생, +3).

Let \( M=\begin{pmatrix} A & B \\ B^*& C \end{pmatrix}\) be a positive semidefinite Hermian matrix. Prove that \[ \operatorname{rank} M \le \operatorname{rank} A +\operatorname{rank} C.\] (Here, \(A\), \(B\), \(C\) are matrices.)

The best solution was submitted by 엄태현 (수리과학과 2012학번). Congratulations!

Here is his solution of problem 2015-04.

Alternative solutions were submitted by 고경훈 (2015학번, +3), 김경석 (2015학번, +3), 김기현 (수리과학과 2012학번, +3), 박성혁 (수리과학과 2014학번, +3), 오동우 (2015학번, +3), 이명재 (수리과학과 2012학번, +3), 이수철 (수리과학과 2012학번, +3, solution), 이종원 (수리과학과 2014학번, +3, solution), 장기정 (수리과학과 2014학번, +3), 정성진 (수리과학과 2013학번, +3), 진우영 (수리과학과 2012학번, +3), 최인혁 (2015학번, +3).

Let \(\{a_n\}\) be a sequence of non-negative reals such that \( \lim_{n\to \infty} a_n \sum_{i=1}^n a_i^5=1\). Prove that \[ \lim_{n\to \infty} a_n  (6n)^{1/6} = 1.\]

The best solution was submitted by 고경훈 (2015학번). Congratulations!

Here is his solution of Problem 2015-3.

Alternative solutions were submitted by 이종원 (수리과학과 2014학번, +3), 김경석 (2015학번, +3), 김기현 (수리과학과 2012학번, +3), 엄태현 (수리과학과 2012학번, +3), 이명재 (수리과학과 2012학번, +3), 정성진 (수리과학과 2013학번, +3), 진우영 (수리과학과 2012학번, +3), 최인혁 (2015학번, +3), 이수철 (수리과학과 2012학번, +3), 국윤범 (2015학번, +3), 박지현 (경상고등학교 2학년, +3). One incorrect solution was submitted (SKB).

Let \(T\) be a triangle. Prove that if every point of a plane is colored by Red, Blue, or Green, then there is a triangle similar to \(T\) such that all vertices of this triangle have the same color.

The best solution was submitted by 박훈민 (수리과학과 2013학번). Congratulations!

Here is his solution of problem 2015-2.

Alternative solutions were submitted by 국윤범/고경훈 (2015학번, +3 jointly / +2 each), 김경석 (2015학번, +3), 김기현 (수리과학과 2012학번, +3), 엄태현 (수리과학과 2012학번), 오동우 (2015학번, +3), 이명재 (수리과학과 2012학번, +3), 이수철 (2012학번, +2), 이영민 (수리과학과 2012학번, +3), 이종원 (수리과학과 2014학번, +3), 장기정 (수리과학과 2014학번, +3), 정성진 (수리과학과 2013학번, +3), 진우영 (수리과학과 2012학번, +3), 최인혁 (2015학번, +3). There was 1 incorrect solution (SML).