# Solution: 2021-08 Self-antipodal sets on the sphere

Prove or disprove that if C is any nonempty connected, closed, self-antipodal (ie., invariant under the antipodal map) set on $$S^2$$, then it equals the zero locus of an odd, smooth function $$f:S^2 -> \mathbb{R}$$.

The best solution was submitted by 신준형 (수리과학과 2015학번, +4). Congratulations!

Here is his solution of problem 2021-08.

Another solution was submitted by 고성훈 (수리과학과 2018학번, +2).

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# Solution: 2017-12 Invertible matrices

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

The best solution was submitted by Shin, Joonhyung (신준형, 수리과학과 2015학번). Congratulations!

Here is his solution of the problem 2017-12.

Alternative solutions were submitted by 국윤범 (수리과학과 2015학번, +3), 김동률 (수리과학과 2015학번, +3), 김태균 (수리과학과 2016학번, +3), 위성군 (수리과학과 2015학번, +3), 유찬진 (수리과학과 2015학번, +3), 윤준기 (전기및전자공학부 2014학번, +3), 이본우 (2017학번, +3), 이준협 (하나고등학교, +3), 이태영 (수리과학과 2013학번, +3), 이형진 (청주대 수학교육과 2011학번, +3), 임성혁 (수리과학과 2016학번, +3), 장기정 (수리과학과 2014학번, +3), 조태혁 (수리과학과 2014학번, +3), 최대범 (수리과학과 2016학번, +3), 최인혁 (물리학과 2015학번, +3), Huy Tung Nguyen (수리과학과 2016학번, +3), Saba Dzmanashvili (2017학번, +3).

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# Concluding 2016 Fall

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

1st prize (Gold): Shin, Joonhyung (신준형, 수리과학과 2015학번)
2nd prize (Silver): Jang, Kijoung (장기정, 수리과학과 2014학번).
2nd prize (Silver): Kim, Taegyun (김태균, 수리과학과 2014학번).
2nd prize (Silver): Kook, Yun Bum (국윤범, 수리과학과 2015학번).
3rd prize (Bronze): Lee, Sangmin (이상민, 수리과학과 2014학번).
3rd prize (Bronze): Lee, Jongwon (이종원, 수리과학과 2014학번).

신준형 (수리과학과 2015학번) 32, 장기정 (수리과학과 2014학번) 31, 김태균 (2016학번) 30, 국윤범 (수리과학과 2015학번) 29, 이상민 (수리과학과 2014학번) 19, 이종원 (수리과학과 2014학번) 19, 최대범 (2016학번) 16, 윤준기 (전기및전자공학부 2014학번) 14, 최인혁 (물리학과 2015학번) 13, 채지석 (2016학번) 12, 김재현 (2016학번) 11, 이정환 (수리과학과 2015학번) 9, Ivan Adrian Koswara (전산학부 2013학번) 6, 강한필 (2016학번) 6, 위성군 (수리과학과 2015학번) 6, 김기택 (수리과학과 2015학번) 6, 박기연 (2016학번) 5, 한준호 (수리과학과 2015학번) 5, 조준영 (수리과학과 2012학번) 3, 박현준 (물리학과 2014학번) 3, 오동우 (2015학번) 3, 유찬진 (수리과학과 2015학번) 3, 임성혁 (2016학번) 3, Muhammaadfiruz Hasanov (2014학번) 3, 정의현 (수리과학과 2015학번) 2, 박진호 (물리학과 2015학번) 2, 정성진 (수리과학과 2013학번) 2.

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# Solution: 2016-20 Finding a subspace

Let $$V_1,V_2,\ldots$$ be countably many $$k$$-dimensional subspaces of $$\mathbb{R}^n$$. Prove that there exists an $$(n-k)$$-dimensional subspace $$W$$ of $$\mathbb{R}^n$$ such that $$\dim V_i\cap W=0$$ for all $$i$$.

The best solution was submitted by Shin, Joonhyung (신준형, 수리과학과 2015학번). Congratulations!

Here is his solution of problem 2016-20.

Alternative solutions were submitted by 김태균 (2016학번, +3), 국윤범 (수리과학과 2015학번, +3), 장기정 (수리과학과 2014학번, +3, alternative solution). One incorrect solution was submitted.

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# Solution: 2016-18 Partitions with equal sums

Suppose that we have a list of $$2n+1$$ integers such that whenever we remove any one of them, the remaining can be partitioned into two lists of $$n$$ integers with the same sum. Prove that all $$2n+1$$ integers are equal.

The best solution was submitted by Joonhyung Shin (신준형, 수리과학과 2015학번). Congratulations!

Here is his solution of problem 2016-18.

Alternative solutions were submitted by 국윤범 (수리과학과 2015학번, +3), 장기정 (수리과학과 2014학번, +3), 이종원 (수리과학과 2014학번, +3, solution), 김태균 (2016학번, +3), 윤준기 (전기및전자공학부 2014학번, +3), 김재현 (2016학번, +3), 채지석 (2016학번, +3), 강한필 (2016학번, +3), Ivan Adrian Koswara (전산학부 2013학번, +3), 김기현 (수리과학과 대학원생, +3). One incorrect solution was received.

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# Concluding 2015 Fall

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

1st prize (Gold): Lee, Jongwon (이종원), 수리과학과 2014학번.
2nd prize (Silver): Park, Sunghyuk (박성혁), 수리과학과 2014학번.
3rd prize (Bronze): Shin, Joonhyung (신준형), 2015학번.
3rd prize (Bronze): Jang, Kijoung (장기정), 수리과학과 2014학번.
3rd prize (Bronze): Choi, Inhyeok (최인혁), 2015학번.

이종원 (수리과학과 2014학번) 37점, 박성혁 (수리과학과 2014학번) 36점, 신준형 (2015학번) 33점, 장기정 (수리과학과 2014학번) 32점, 최인혁 (2015학번) 32점, 이영민 (수리과학과 2012학번) 18점, 박훈민 (수리과학과 2013학번) 17점, 김동률 (2015학번) 10점, 이상민 (수리과학과 2014학번) 8점, 김재준 (2014학번) 6점, 이정환 (2015학번) 6점, 오동우 (2015학번) 5점, 유찬진 (2015학번) 5점, 함도규 (2015학번) 5점, 이신영 (물리학과 2012학번) 4점, 김경석 (2015학번) 3점, 김기택 (2015학번) 3점, 김희주 (2015학번) 2점, 이호일 (수리과학과 2013학번) 2점,  이경훈 (수리과학과 2014학번) 1점.

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# Solution: 2015-20 Dense function

Prove or disprove the following statement:
There exists a function $$f : \mathbb{R} \to \mathbb{R}$$ such that
(1) $$f \equiv 0$$ almost everywhere, and
(2) for any nonempty open interval $$I$$, $$f(I) = \mathbb{R}$$.

The best solution was submitted by Joonhyung Shin (신준형, 2015학번). Congratulations!

Here is his solution of problem 2015-20.

Alternative solutions were submitted by 박성혁 (수리과학과 2014학번, +3, his solution), 이영민 (수리과학과 2012학번, +3), 이종원 (수리과학과 2014학번, +3, his solution), 장기정 (수리과학과 2014학번, +3, his solution), 최인혁 (2015학번, +3), 김동률 (2015학번, +2), 이신영 (물리학과 2012학번, +2),  송교범 (서대전고등학교 2학년, +3).

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What is the determinant of the $$n\times n$$ matrix $$A_n=(a_{ij})$$ where $a_{ij}=\begin{cases} 1 ,&\text{if } i=j, \\ x, &\text{if }|i-j|=1, \\ 0, &\text{otherwise,}\end{cases}$ for a real number $$x$$?