# Solution: 2015-3 Limit

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).

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# 2015-4 An inequality on positive semidefinite matrices

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.)

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# Solution: 2015-2 Monochromatic triangle

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).

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# 2015-3 Limit

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 $\lim_{n\to \infty} a_n (6n)^{1/6} = 1.$

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# Solution: 2015-1 Equal sums

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).

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# 2015-2 Monochromatic triangle

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.

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