# Concluding 2014 Fall

Thanks 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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# Solution: 2014-21 Duality

Let $$\mathcal F$$ be a non-empty collection of subsets of a finite set $$U$$. Let $$D(\mathcal F)$$  be the collection of subsets of $$U$$ that are subsets of an odd number of members of $$\mathcal F$$. Prove that $$D(D(\mathcal F))=\mathcal F$$.

The best solution was submitted by Jimin Park (박지민), 전산학과 2012학번. Congratulations!

Here is his solution of Problem 2014-21.

Alternative solutions were submitted by 박민재 (수리과학과 2011학번, +3), 채석주 (수리과학과 2013학번, +3), 이병학 (수리과학과 2013학번, +3), 정경훈 (서울대 컴퓨터공학과 2006학번, +3), 조현우 (경남과학고등학교 3학년, +3), 김경석 (경기과학고등학교 3학년, +3).

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# Solution: 2014-18 Rank

Let $$A$$ and $$B$$ be $$n\times n$$ real matrices for an odd integer $$n$$. Prove that if both $$A+A^T$$ and $$B+B^T$$ are invertible, then $$AB\neq 0$$.

The best solution was submitted by Jimin Park (박지민, 전산학과 2012학번). Congratulations!

Here is his solution of problem 2014-18.

Alternative solutions were submitted by 채석주 (2013학번, +3), 정성진 (2013학번, +3), 장기정 (2014학번, +3), 박민재 (2011학번, +3), 김경석 (경기과학고등학교 3학년, +3).

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Find all field automorphisms of the field of real numbers $$\mathbb{R}$$. (A field automorphism of a field $$F$$ is a bijective map $$\sigma : F \to F$$ that preserves all of $$F$$’s algebraic properties.)