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).

Let \( A\) be a set of \(n\ge 2\) odd integers. Prove that there exist two distinct subsets \(X\), \(Y\) of \(A\) such that \[ \sum_{x\in X} x\equiv\sum_{y\in Y}y \pmod{2^n}.\]

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

Here is his solution of problem 2015-1.

Alternative solutions were submitted by 고경훈 (2015학번, +3), 김경석 (2015학번, +3), 김기현 (2012학번, +3), 김동철 (2013학번, +3), 배형진 (마포고 1학년, +2), 어수강 (서울대 수리과학부 대학원생, +3), 엄태현 (2012학번, +3), 오동우 (2015학번, +3), 유찬진 (2015학번, +3), 윤성철 (홍익대 수학교육과, +3), 이명재 (수리과학과 2012학번, +3), 이병학 (2013학번, +3), 이상민 (수리과학과 2014학번, +3), 이수철 (2012학번, +3), 이시우 (POSTECH 수학과 2013학번, +3), 이영민 (2012학번, +3), 장기정 (수리과학과 2014학번, +3), 정성진 (2013학번, +3), 진우영 (수리과학과 2012학번, +3), 최두성 (수리과학과 2011학번, +3), 최인혁 (2015학번, +3), Muhammadfiruz Hassnov (2014학번, +3).