Good luck with your midterm exam next week. Next problem will be posted on April ~~1st~~ 8th.

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Good luck with your midterm exam next week. Next problem will be posted on April ~~1st~~ 8th.

Let \(a_1\le a_2\le \cdots \le a_k\) and \(b_1\le b_2\le \cdots \le b_l\) be sequences of positive integers at most M. Prove that if \[ \sum_{i=1}^{k} a_i^n = \sum_{j=1}^l b_j^n\] for all \(1\le n\le M\), then \(k=l\) and \(a_i=b_i\) for all \(1\le i\le k\).

The best solution was submitted by Cho, Yonghwa (조용화), 수리과학과 석사과정 2010학번.

Here is his Solution of Problem 2011-6.

Alternative solutions were submitted by 김지원 (2010학번, +3), 이재석 (수리과학과 2007학번, +3), 구도완 (해운대고등학교 3학년, +3). One incorrect solution was submitted.

Let \(a_1\le a_2\le \cdots \le a_k\) and \(b_1\le b_2\le \cdots \le b_l\) be sequences of positive integers at most M. Prove that if \[ \sum_{i=1}^{k} a_i^n = \sum_{j=1}^l b_j^n\] for all \(1\le n\le M\), then \(k=l\) and \(a_i=b_i\) for all \(1\le i\le k\).

Find all linear functions

fon the set of n×n matrices such thatf(XY)=f(YX) for every pair of n×n matrices X and Y.

Added: The value f(X) is a scalar.

The best solution was submitted by Jesek Lee (이재석), 수리과학과 2007학번. Congratulations!

Here is his Solution of Problem 2011-5.

Alternative solutions were submitted by 강동엽 (전산학과 2009학번, +3), 박민재 (2011학번, +3), 서기원 (수리과학과 2009학번, +3), 조용화 (수리과학과 석사과정 2010학번, +3), 김지원 (2010학번, +3), 어수강 (홍익대학교 수학교육학과 2004학번, +3), 변범부 (경남대학교 수학교육과 2005학번, +3). One incorrect solution was submitted.

Find all linear functions *f* on the set of n×n matrices such that *f*(XY)=*f*(YX) for every pair of n×n matrices X and Y.

Added: The value f(X) is a scalar.

Let

n>2. Letf(x) be a degree-npolynomial with real coefficients. Iff(x) hasndistinct real zeros r_{1}<r_{2}<…<r_{n}, then Rolle’s theorem implies that the largest real zeroqoff´(x) is between r_{n-1}and r_{n}. Prove thatq>(r_{n-1}+r_{n})/2.

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

Here is his Solution of Problem 2011-4.

Alternative solutions were submitted by 박민재 (2011학번, +3), 강동엽 (전산학과 2009학번, +3), 김태호 (2011학번, +3), 김지원 (2010학번, +3), 이재석 (수리과학과 2007학번, +3), 김현수 (한국과학영재학교 3학년, +3), 구도완 (해운대고등학교 3학년, +3).

Let *n*>2. Let *f *(*x*) be a degree-*n* polynomial with real coefficients. If *f *(*x*) has *n* distinct real zeros r_{1}<r_{2}<…<r_{n}, then Rolle’s theorem implies that the largest real zero *q* of *f´*(x) is between r_{n-1} and r_{n}. Prove that *q*>(r_{n-1}+r_{n})/2.

Let us write \([n]=\{1,2,\ldots,n\}\). Let \(a_n\) be the number of all functions \(f:[n]\to [n]\) such that \(f([n])=[k]\) for some positive integer \(k\). Prove that \[a_n=\sum_{k=0}^{\infty} \frac{k^n}{2^{k+1}}.\]

The best solution was submitted by Kang, Dongyub (강동엽), 전산학과 2009학번. Congratulations!

Here is his Solution of Problem 2011-3.

Alternative solutions were submitted by 서기원 (수리과학과 2009학번, +3), 박민재 (2011학번, +3), 김치헌 (수리과학과 2006학번, +2), 이동민 (수리과학과 2009학번, +2), 구도완 (해운대고등학교 3학년, +2).

P.S. A common mistake is to assume that \(\sum_{i}\sum_{j}\) can be swapped without showing that a sequence converges absolutely.