# Concluding 2010 Fall

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

1st prize: Kim, Chiheon (김치헌) – 수리과학과 2006학번

2nd prize: Park, Minjae (박민재) – 한국과학영재학교 (KAIST 2011학번 입학예정)

3rd prize: Jeong, Jinmyeong (정진명) – 수리과학과 2007학번.

Congratulations!

In addition to these three people, I selected one more student to receive 2 movie tickets.

Jeong, Seong-Gu (정성구) – 수리과학과 2007학번. 김치헌 (2006학번) 28 pts
박민재 (KSA) 25 pts
정진명 (2007학번) 19 pts
정성구 (2007학번) 16 pts
서기원 (2009학번) 9 pts
심규석 (2007학번) 9 pts
권용찬 (2009학번) 3 pts
정유중 (2006학번) 3 pts
진우영 (KSA) 3 pts
서영우 (2010학번) 2 pts
오상국 (2007학번) 2 pts
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# Solution: 2010-21 Limit

Let $$a_1=0$$, $$a_{2n+1}=a_{2n}=n-a_n$$. Prove that there exists k such that $$\lvert a_k- \frac{k}{3}\rvert > 2010$$ and yet $$\lim_{n\to \infty} \frac{a_n}{n}=\frac13$$.

The best solution was submitted by Chiheon Kim (김치헌), 수리과학과 2006학번. Congratulations!

Here is his Solution of Problem 2010-21.

Alternative solutions were submitted by 한대진 (신현여중 교사, +2), 이승훈 (연세대학교 경제학과 06학번, +2).

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# Solution:2010-19 Fixed Points

Suppose that $$V$$ is a vector space of dimension $$n>0$$ over a field of characterstic $$p\neq 0$$. Let $$A: V\to V$$ be an affine transformation. Prove that there exist $$u\in V$$ and $$1\le k\le np$$ such that $A^k u = u.$

The best solution was submitted by Chiheon Kim (김치헌), 수리과학과 2006학번. Congratulations!

Here is his Solution of Problem 2010-19.

An alternative solution was submitted by 박민재 (KSA-한국과학영재학교, +3).

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# Solution: 2010-18 Limit of a differentiable function

Let f be a differentiable function. Prove that if $$\lim_{x\to\infty} (f(x)+f'(x))=1$$, then $$\lim_{x\to\infty} f(x)=1$$.

The best solution was submitted by Chiheon Kim (김치헌), 수리과학과 2006학번. Congratulations!

Here is his Solution of Problem 2010-18.

Alternative solutions were submitted by 정성구 (수리과학과 2007학번, +3), 서기원 (수리과학과 2009학번, +3), 심규석 (수리과학과 2007학번, +3), 진우영 (KSA-한국과학영재학교, +3), 박민재 (KSA-한국과학영재학교, +2), 한대진 (?, +2), 문정원 (성균관대학교 수학교육과, +2).

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# Solution: 2010-15 Characteristic Polynomial

Let A, B be 2n×2n skew-symmetric matrices and let f be the characteristic polynomial of AB. Prove that the multiplicity of each root of f is at least 2.

The best solution was submitted by Chiheon Kim (김치헌), 수리과학과 2006학번. Congratulations!

Here is his Solution of Problem 2010-15.

An alternative solution was submitted by 정진명 (수리과학과 2007학번, +2).

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# Solution: 2010-6 Identity on Binomial Coefficients

Prove that $$\displaystyle \sum_{m=0}^n \sum_{i=0}^m \binom{n}{m} \binom{m}{i}^3=\sum_{m=0}^n \binom{2m}{m} \binom{n}{m}^2$$.

The best solution was submitted by Chiheon Kim(김치헌), 수리과학과 2006학번. Congratulations!

Here is his Solution of Problem 2010-6.

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# Solution: 2010-5 Dependence over Q

Find the set of all rational numbers x such that 1, cos πx, and sin πx are linearly dependent over rational field Q.

The best solution was submitted by Chiheon Kim(김치헌), 수리과학과 2006학번. Congratulations!

Here is his Solution of Problem 2010-5.

An alternative solution was submitted by 정성구 (수리과학과 2007학번, +3), 임재원 (2009학번, +2). One incorrect solution was submitted.

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# Solution: 2010-4 Power and gcd

Let n, k be positive integers. Prove that $$\sum_{i=1}^n k^{\gcd(i,n)}$$ is divisible by n.

The best solution was submitted by Chiheon Kim(김치헌), 수리과학과 2006학번. Congratulations!

Here is his Solution of Problem 2010-4.

Alternative solutions were submitted by 정성구 (수리과학과 2007학번, +3), Prach Siriviriyakul (2009학번, +3), 라준현 (수리과학과 2008학번, +3), 서기원 (2009학번, +3), 강동엽 (2009학번, +3), 임재원 (2009학번, +2).

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# Solution: 2009-13 Distances between points in [0,1]^2

Let $$P_1,P_2,\ldots,P_n$$ be n points in {(x,y): 0<x<1, 0<y<1} (n>1). Let $$r_i=\min_{j\neq i} d(P_i,P_j)$$ where d(x,y) means the distance between two points x and y. Prove that $$r_1^2+r_2^2+\cdots+r_n^2\le 4$$.

The best solution was submitted by Chiheon Kim (김치헌), 수리과학과 2006학번. Congratulations!

Here is his Solution of Problem 2009-13.

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