# Concluding 2014 Fall

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

• 1st prize (Gold): Park, Minjae (박민재) – 수리과학과 2011학번
• 2nd prize (Silver): Chae, Seok Joo (채석주) – 수리과학과 2013학번
• 3rd prize (Bronze): Lee, Byeonghak (이병학) – 수리과학과 2013학번
• 4th prize: Park, Jimin (박지민) – 전산학과 2012학번
• 5th prize: Park, Hun Min (박훈민) – 수리과학과 2013학번

박민재 (2011학번) 30
채석주 (2013학번) 22
이병학 (2013학번) 20
박지민 (2012학번) 19
박훈민 (2013학번) 15
장기정 (2014학번) 14
허원영 (2014학번) 4
정성진 (2013학번) 3
김태겸 (2013학번) 3
윤준기 (2014학번) 3

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# Solution: 2014-24 Random points on a sphere

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).

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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 $\frac{1}{n}\sum_{i=1}^n \frac{1}{f'(x_i)}=1.$