# Solution: 2020-22 Regular simplex

Let $$S$$ be the unit sphere in $$\mathbb{R}^n$$, centered at the origin, and $$P_1 P_2 \dots P_{n+1}$$ a regular simplex inscribed in $$S$$. Prove that for a point $$P$$ inside $$S$$,
$\sum_{i=1}^{n+1} (PP_i)^4$
depends only on the distance $$OP$$ (and $$n$$).

The best solution was submitted by 이준호 (수리과학과 2016학번, +4). Congratulations!

Here is his solution of problem 2020-22.

Other solutions was submitted by 고성훈 (수리과학과 2018학번, +3), 채지석 (수리과학과 2016학번, +3).

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# Solution: 2020-15 The number of cycles of fixed lengths in random permutations

Let $$m_0=n$$. For each $$i\geq 0$$, choose a number $$x_i$$ in $$\{1,\dots, m_i\}$$ uniformly at random and let $$m_{i+1}= m_i – x_i$$. This gives a random vector $$\mathbf{x}=(x_1,x_2, \dots)$$. For each $$1\leq k\leq n$$, let $$X_k$$ be the number of occurrences of $$k$$ in the vector $$\mathbf{x}$$.

For each $$1\leq k\leq n$$, let $$Y_k$$ be the number of cycles of length $$k$$ in a permutation of $$\{1,\dots, n\}$$ chosen uniformly at random. Prove that $$X_k$$ and $$Y_k$$ have the same distribution.

The best solution was submitted by 이준호 (수리과학과 2016학번). Congratulations!

Here is his solution of problem 2020-15.

GD Star Rating # Concluding 2016 Spring

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

1st prize (Gold): Kook, Yun Bum (국윤범, 수리과학과 2015학번)
2nd prize (Silver): Jang, Kijoung (장기정, 수리과학과 2014학번).
3rd prize (Bronze): Lee, Sangmin (이상민, 수리과학과 2014학번)
3rd prize (Bronze): Lee, Jongwon (이종원, 수리과학과 2014학번).
3rd prize (Bronze): Lee, Junho (이준호, 2016학번).

국윤범 (수리과학과 2015학번), 장기정 (수리과학과 2014학번), 이상민 (수리과학과 2014학번), 이종원 (수리과학과 2014학번), 이준호 (2016학번), 강한필 (2016학번), 유찬진 (수리과학과 2015학번), 윤준기 (전기및전자공학부 2014학번), Muhammaadfiruz Hasanov (2014학번), 김동규 (수리과학과 2015학번), 최백규 (2016학번), 김기택 (수리과학과 2015학번), 조태혁 (수리과학과 2014학번), 김동률 (수리과학과 2015학번), 김태균 (2016학번), 박기연 (2016학번), 최대범 (2016학번), 이정환 (수리과학과 2015학번), 김강식 (포항공대 수학과 2013학번), 김동하 (기계공학과 2014학번), 김재현 (2016학번), 이태영 (2013학번), 장창환 (기계공학과 2015학번), 정성진 (수리과학과 2013학번), 최인혁 (물리학과 2015학번), 김홍규 (수리과학과 2011학번), 노희광 (화학과 2014학번), 안현수 (2016학번), 홍혁표 (수리과학과 2013학번).

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