# Concluding 2010 Spring

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

1st prize:  Jeong, Seong-Gu (정성구) – 수리과학과 2007학번

2nd prize: Kim, Chiheon (김치헌) – 수리과학과 2006학번

3rd prize: Suh, Gee Won (서기원) – 2009학번

Congratulations!

In addition to those three people, I have selected two students. They received 1 movie ticket each.

Lim, Jae Won (임재원) – 2009학번

Kim, Ho Jin (김호진) – 2009학번 정성구 (2007학번) 30 pts
김치헌 (2006학번) 20 pts
서기원 (2009학번) 19 pts
임재원 (2009학번) 12 pts
김호진 (2009학번) 7 pts
최홍석 (2006학번) 6 pts
라준현 (2008학번) 6 pts
Prach Siriviriyakul (2009학번) 6 pts
강동엽 (2009학번) 5 pts
권성민 (2009학번) 4 pts
권용찬 (2009학번) 2 pts
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# Thanks for participating POW; We will resume on Sep. 2010.

Problem 2010-11 was the last problem of this semester. Good luck to your final exam! We wish you to come back in the fall semester. We will start in the first week of September.

Next week, we will have a small ceremony to award winners.

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# Solution: 2010-11 Integral Equation

Let z be a real number. Find all solutions of the following integral equation: $$f(x)=e^x+z \int_0^1 e^{x-y} f(y)\,dy$$ for 0≤x≤1.

The best solution was submitted by Gee Won Suh (서기원), 2009학번. Congratulations!

Here is his Solution of Problem 2010-11.

Alternative solutions were submitted by 최홍석 (화학과 2006학번, +3), 정성구 (수리과학과 2007학번, +3).

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# Solution: 2010-10 Metric space of matrices

Let  Mn×n be the space of real n×n matrices, regarded as a metric space with the distance function

$$\displaystyle d(A,B)=\sum_{i,j} |a_{ij}-b_{ij}|$$

for A=(aij) and B=(bij).
Prove that $$\{A\in M_{n\times n}: A^m=0 \text{ for some positive integer }m\}$$ is a closed set.

The best solution was submitted by Gee Won Suh (서기원), 2009학번. Congratulations!

Here is his Solution of Problem 2010-10.

Alternative solutions were submitted by 정성구 (수리과학과 2007학번, +3), 김치헌 (수리과학과 2006학번, +3), 강동엽 (2009학번, +2).

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Let M>0 be a real number. Prove that there exists N so that if n>N, then all the roots of $$f_n(z)=1+\frac{1}{z}+\frac1{{2!}z^2}+\cdots+\frac{1}{n!z^n}$$ are in the disk |z|<M on the complex plane.