# Solution: 2013-14 Nilpotent matrix

Let $$A, B$$ are $$N \times N$$ complex matrices satisfying $$rank(AB – BA) = 1$$. Prove that $$(AB – BA)^2 = 0$$.

The best solution was submitted by 김호진. Congratulations!

Similar solutions were submitted by 강동엽(+3), 김범수(+3), 김홍규(+3), 박민재(+3), 박지민(+3), 박훈민(+3), 안가람(+3), 어수강(+3), 엄문용(+3), 유찬진(+3), 이성회(+3), 이시우(+3), 이주호(+3), 장경석(+3), 전한솔(+3), 정동욱(+3), 정성진(+3), 정종헌(+3), 정우석(+3), 진우영(+3), Fardad Pouran(+3). Thank you for your participation.

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# Concluding Spring 2013

The top 5 participants of the semester are:

• 1st: 라준현 (08학번): 38 points
• 2nd: 서기원 (09학번): 29 points
• T-3rd: 김호진 (09학번): 25 points
• T-3rd: 황성호 (13학번): 25 points
• 5th: 김범수 (10학번): 19 points

Hearty congratulations to the prize winners! The prize ceremony will be held on Jun. 19 (Wed.) at 2PM.

We thank all of the participants for the nice solutions and your intereset you showed for POW. We hope to see you next semester with even better problems.

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# Solution: 2013-10 Mean and variance of random variable

Let random variables $$\{ X_r : r \geq 1 \}$$ be independent and uniformly distributed on $$[0, 1]$$. Let $$0 < x < 1$$ and define a random variable $N = \min \{ n \geq 1 : X_1 + X_2 + \cdots + X_n > x \}.$
Find the mean and variance of $$N$$.

The best solution was submitted by 김호진, 09학번. Congratulations!

Similar solutions were also submitted by 라준현(08학번, +3), 서기원(09학번, +3), 김범수(10학번, +3), 황성호(13학번, +3), 어수강(서울대, +3), 이시우(POSTECH, +3), Fardad Pouran(Sharif University of Tech, Iran, +3), 양지훈(10학번, +2), 이정민(서울대, +2). Thank you for your participation.

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# Solution: 2013-02 Functional equation

Let $$\mathbb{Z}^+$$ be the set of positive integers. Suppose that $$f : \mathbb{Z}^+ \to \mathbb{Z}^+$$ satisfies the following conditions.

i) $$f(f(x)) = 5x$$.

ii) If $$m \geq n$$, then $$f(m) \geq f(n)$$.

iii) $$f(1) \neq 2$$.

Find $$f(256)$$.

The best solution was submitted by 김호진, 09학번. Congratulations!

Similar solutions were also submitted by 황성호(13학번, +3), 양지훈(10학번, +3), 홍혁표(13학번, +3), 김준(13학번, +3), 서기원(09학번, +3), 이주호(12학번, +3), 박훈민(13학번, +3), 송유신(10학번, +3), 임현진(10학번, +3), 라준현(08학번, +3), 김정민(12학번, +3), 박지민(12학번, +3), 김태호(11학번, +3), 김범수(10학번, +3), 전한솔(고려대 13학번, +3), 어수강(서울대 석사과정, +3), 이시우(POSTECH 13학번, +3), 정우석(서강대 11학번, +3), 윤성철(홍익대 09학번, +3), 김재호(하나고, +3), 이정준(08학번, +2). Thank you for your participation.

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# Solution: 2010-3 Sum

Evaluate the following sum

$$\displaystyle\sum_{m=1}^\infty \sum_{\substack{n\ge 1\\ (m,n)=1}} \frac{x^{m-1}y^{n-1}}{1-x^m y^n}$$

when |x|, |y|<1.

(We write (m,n) to denote the g.c.d of m and n.)

The best solution was submitted by Hojin Kim (김호진, 2009학번). Congratulations!

Here is his Solution of Problem 2010-3.

Alternative solutions were submitted by 정성구 (수리과학과 2007학번, +3), 임재원 (2009학번, +3), Prach Siriviriyakul (2009학번, +3), 서기원 (2009학번, +3), 김치헌 (수리과학과 2006학번, +2).

The problem had a slight problem when xy=0; It is necessary to assume 00=1.

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# Solution: 2009-10 x and cos x

Find all real-valued continuous function f on the reals such that f(x)=f(cos x) for every real number x.

The best solution was submitted by Hojin Kim (김호진), 2009학번. Congratulations!

Here is his Solution of Problem 2009-10.

Alternative solutions were submitted by 백형렬(수리과학과 2003학번, +3), 김치헌(수리과학과 2006학번, +3), 이재송(전산학과 2006학번, +3), 권상훈(수리과학과 2006학번, +3), 조용화(수리과학과 2006학번, +2), 김일희 & 오성진 (Princeton Univ., Graduate students).

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