For a (simple) graph \(G\), let \(o(G)\) be the number of odd-sized sets of pairwise non-adjacent vertices and let \(e(G)\) be the number of even-sized sets of pairwise non-adjacent vertices. Prove that if we can delete \(k\) vertices from \(G\) to destroy every cycle, then \[ | o(G)-e(G)|\le 2^{k}.\]

The best solution was submitted by Minjae Park (박민재, 수리과학과 2011학번). Congratulations!

Here is his solution.

An alternative solution was submitted by 김경석 (+3, 경기과학고 3학년). One incorrect solution was received (BHJ).

Prove or disprove that for all positive integers \(m\) and \(n\), \[ f(m,n)=\frac{2^{3(m+n)-\frac12} }{{\pi}} \int_0^{\pi/2} \sin^{ 2n – \frac12 }\theta \cdot \cos^{2m+\frac12}\theta \, d\theta\]  is an integer.

The best solution was submitted by 김경석 (경기과학고등학교 3학년). Congratulations!

Here is his solution.

Alternative solutions were submitted by 이병학 (2013학번, +2), 박훈민 (2013학번, +2), 배형진 (공항중학교 3학년, +2). One incorrect solution was submitted (LSC).