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

(This is the last problem of 2014. Thank you!)

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

(This is the last problem of 2014. Thank you!)

For a nonnegative real number \(x\), let \[ f_n(x)=\frac{\prod_{k=1}^{n-1} ((x+k)(x+k+1))}{ (n!)^2}\] for a positive integer \(n\). Determine \(\lim_{n\to\infty}f_n(x)\).

The best solution was submitted by Hun-Min Park (박훈민), 수리과학과 2013학번. Congratulations!

Here is his solution of problem 2014-22.

Alternative solutions were submitted by 박민재 (수리과학과 2011학번, +3, his solution), 박지민 (전산학과 2012학번, +3), 이병학 (수리과학과 2013학번, +3), 채석주 (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 \[ \frac{1}{n}\sum_{i=1}^n \frac{1}{f'(x_i)}=1.\]

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).

For a nonnegative real number \(x\), let \[ f_n(x)=\frac{\prod_{k=1}^{n-1} ((x+k)(x+k+1))}{ (n!)^2}\] for a positive integer \(n\). Determine \(\lim_{n\to\infty}f_n(x)\).

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 \(D(D(\mathcal F))=\mathcal F\).

Let \(G\) be a group such that it has no element of order \(2\) and \[ (ab)^2=(ba)^2\] for all \(a,b\in G\). Prove that \(G\) is abelian.

The best solution was submitted by Chae, Seok Joo (채석주), 수리과학과 2013학번. Congratulations!

Here is his solution of problem 2014-20.

Alternative solutions were submitted by 박민재 (수리과학과 2011학번, +3), 박지민 (수리과학과 2012학번, +3), 장기정 (2014학번, +3), 박훈민 (수리과학과 2013학번, +3), 이병학 (수리과학과 2013학번, +3), 한미진 (순천향대학교 2014학번, +3), 한대진 (인천신현여중 교사, +3), 김경석 (경기과학고 3학년, +3), 진형준 (인천대학교 수학과 2014학번, +3), 장일승 (인천대학교 수학과, +3), 조현우 (경남과학고 3학년, +2).

Let \(G\) be a group such that it has no element of order \(2\) and \[ (ab)^2=(ba)^2\] for all \(a,b\in G\). Prove that \(G\) is abelian.

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