Suppose that \(n\) points are chosen randomly on a sphere. What is the probability that all points are on some hemisphere?

The best solution was submitted by 채석주 (수리과학과 2013학번). Congratulations!

Here is his solution of 2014-24.

An alternative solution was submitted by 이병학 (수리과학과 2013학번, +3).

Let \(f:[0,1]\to \mathbb R\) be a differentiable function with \(f(0)=0\), \(f(1)=1\). Prove that for every positive integer \(n\), there exist \(n\) distinct numbers \(x_1,x_2,\ldots,x_n\in(0,1)\) such that \[ \frac{1}{n}\sum_{i=1}^n \frac{1}{f'(x_i)}=1.\]

The best solution was submitted by Heo, Won Yeong (허원영), 2014학번. Congratulations!

Here is his solution of 2014-23.

Alternative solutions were submitted by 김태겸 (전기및전자공학과 2013학번, +3), 박민재 (수리과학과 2011학번, +3), 윤준기 (2014학번, +3), 장기정 (2014학번, +3),  박훈민 (수리과학과 2013학번, +3), 채석주 (수리과학과 2013학번, +3), 박지민 (전산학과 2012학번, +2), 이병학 (수리과학과 2013학번, +3), 어수강 (서울대학교 수리과학부, +3), 김동률 (강원과학고등학교 2학년, +3), 박지현 (경상고등학교 1학년, +3), 진형준 (인천대학교 수학과 2014학번, +3).