# 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?

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

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# Solution: 2014-22 Limit

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).

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# 2014-23 Differentiable function

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

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# Solution: 2014-21 Duality

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).

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# 2014-22 Limit

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  $$\lim_{n\to\infty}f_n(x)$$.

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# 2014-21 Duality

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

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# Solution: 2014-20 Abelian group

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).

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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.