# Midterm Exam Break

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

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# Solution: 2011-6 Equal sums

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.

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# 2011-6 Equal sums

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 $$1\le i\le k$$.

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# Solution: 2011-5 Linear function on matrices

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.

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.

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# 2011-5 Linear function on matrices

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.

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# Solution: 2011-4 A polynomial with distinct real zeros

Let n>2. Let f (x) be a degree-n polynomial with real coefficients. If f (x) has n distinct real zeros r1<r2<…<rn, then Rolle’s theorem implies that the largest real zero q of (x) is between rn-1 and rn. Prove that q>(rn-1+rn)/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).

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# 2011-4 A polynomial with distinct real zeros

Let n>2. Let f (x) be a degree-n polynomial with real coefficients. If f (x) has n distinct real zeros r1<r2<…<rn, then Rolle’s theorem implies that the largest real zero q of (x) is between rn-1 and rn. Prove that q>(rn-1+rn)/2.

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# Solution: 2011-3 Counting functions

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.

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