Category Archives: solution

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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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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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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Solution: 2011-2 Power

Prove that for all positive integers m and n, there is a positive integer k such that \[ (\sqrt{m}+\sqrt{m-1})^n = \sqrt{k}+\sqrt{k-1}.\]

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

Here is his Solution of Problem 2011-2.

Alternative solutions were submitted by 김인환 (2010학번, +3), 박민재 (2011학번, +3), 김지원 (2010학번, +3),강동엽 (전산학과 2009학번, +3), 서기원 (수리과학과 2009학번, +3), 김재훈 (EEWS대학원 2010학번, +3), 김현수 (한국과학영재학교 3학년, +3), 어수강 (홍익대학교 수학교육학과 2004학번, +3).

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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: 2010-20 Monochromatic line

Let X be a finite set of points on the plane such that each point in X is colored with red or blue and there is no line having all points in X. Prove that there is a line L having at least two points of X such that all points in L∩X have the same color.

The best solution was submitted by Minjae Park (박민재), 한국과학영재학교 (KSA). Congratulations!

Here is his Solution of Problem 2010-20.

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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: 2010-17 Two Hermitian Matrices

Let A, B be Hermitian matrices. Prove that tr(A2B2) ≥ tr((AB)2).

The best solution was submitted by Jeong, Jinmyeong (정진명), 수리과학과 2007학번. Congratulations!

Here is his Solution of Problem 2010-17.

Alternative solutions were submitted by 정성구 (수리과학과 2007학번, +3), 김치헌 (수리과학과 2006학번, +3), 박민재 (KSA-한국과학영재학교, +3).

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