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.

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.

Let n>2. Let f (x) be a degree-n polynomial with real coefficients. If f (x) has n distinct real zeros r₁<r₂<…<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.

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 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.

Evaluate the sum \[ \sum_{n=1}^{\infty} \frac{n \sin n}{1+n^2}. \]

The best solution was submitted by Minjae Park (박민재), 2011학번. Congratulations!

Here is his Solution of Problem 2011-1.

A solution for an incorrect version of the problem was submitted by 구도완 (해운대고등학교 3학년).

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).

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.

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).