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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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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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 \[ \sum_{x\in X} x\equiv\sum_{y\in Y}y \pmod{2^n}.\]

Remark (added March 3): n is an integer greater than or equal to 2 and A is a set of n odd integers.

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Concluding 2014 Fall

pow2014fall-624Thanks all for participating POW actively. Here’s the list of winners:

  • 1st prize (Gold): Park, Minjae (박민재) – 수리과학과 2011학번
  • 2nd prize (Silver): Chae, Seok Joo (채석주) – 수리과학과 2013학번
  • 3rd prize (Bronze): Lee, Byeonghak (이병학) – 수리과학과 2013학번
  • 4th prize: Park, Jimin (박지민) – 전산학과 2012학번
  • 5th prize: Park, Hun Min (박훈민) – 수리과학과 2013학번

박민재 (2011학번) 30
채석주 (2013학번) 22
이병학 (2013학번) 20
박지민 (2012학번) 19
박훈민 (2013학번) 15
장기정 (2014학번) 14
허원영 (2014학번) 4
정성진 (2013학번) 3
김태겸 (2013학번) 3
윤준기 (2014학번) 3

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