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).

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).

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.

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.

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).

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 \(x\)?

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

Here is his soluton of problem 2015-18.

Alternative solutions were submitted by 김동률 (2015학번, +3), 박지민 (전산학부 석사 2015학번, +3), 최인혁 (2015학번, +3), 박성혁 (수리과학과 2014학번, +2), 박훈민 (수리과학과 2013학번, +2), 이상민 (수리과학과 2014학번, +2), 이영민 (수리과학과 2012학번, +2), 이종원 (수리과학과 2014학번, +2), 이호일 (수리과학과 2013학번, +2), 장기정 (수리과학과 2014학번, +2), 함도규 (2015학번, +2).