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).

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).

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.)

The best solution was submitted by 박지민. Congratulations!

Similar solutions are submitted by 고진용(+3), 김호진(+3), 박경호(+3), 박민재(+3), 박훈민(+3), 어수강(+3), 전한솔(+3), 정성진(+3), 진우영(+3), 한대진(+3). Thank you for your participation.