Let f(n) be the largest integer k such that n! is divisible by \(n^k\). Prove that \[ \lim_{n\to \infty} \frac{(\log n)\cdot \max_{2\le i\le n} f(i)}{n \log\log n}=1.\]

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

Here is his Solution of Problem 2011-7.

Alternative solutions were submitted by 양해훈 (수리과학과 2008학번, +3), 이재석 (수리과학과 2007학번, +2).

Prove that for every skew-symmetric matrix A, there are symmetric matrices B and C such that A=BC-CB.

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

Here is his Solution of Problem 2011-11.

Alternative solutions were submitted by 강동엽 (전산학과 2009학번, +3), 서기원 (수리과학과 2009학번, +3), 어수강 (홍익대 수학교육과, +3, Alternative Solution of Problem 2011-11).

Let \(t_1,t_2,\ldots,t_n\) be positive integers. Let \(p(x_1,x_2,\dots,x_n)\) be a polynomial with n variables such that \(\deg(p)\le t_1+t_2+\cdots+t_n\). Prove that \(\left(\frac{\partial}{\partial x_1}\right)^{t_1} \left(\frac{\partial}{\partial x_2}\right)^{t_2}\cdots \left(\frac{\partial}{\partial x_n}\right)^{t_n} p\) is equal to \[\sum_{a_1=0}^{t_1} \sum_{a_2=0}^{t_2}\cdots \sum_{a_n=0}^{t_n} (-1)^{t_1+t_2+\cdots+t_n+a_1+a_2+\cdots+a_n}\left( \prod_{i=1}^n \binom{t_i}{a_i} \right)p(a_1,a_2,\ldots,a_n).\]

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

Here is his Solution of Problem 2011-10

An alternative solution was submitted by 박민재 (2011학번, +3).

Prove that there is a constant c>1 such that if  \(n>c^k\) for positive integers n and k, then the number of distinct prime factors of \(n \choose k\) is at least k.

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

Here is his Solution of Problem 2011-9.

An alternative solution was submitted by 어수강 (홍익대 수학교육과 2004학번, +3).

Let f be a continuous function on [0,1]. Prove that \[ \lim_{n\to \infty}\int_0^1 \cdots \int_0^1 f(\sqrt[n]{x_1 x_2 \cdots x_n } ) dx_1 dx_2 \cdots dx_n = f(1/e).\]

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

Here is his Solution of Problem 2011-8.

Alternative solutions were submitted by 김치헌 (수리과학과 2006학번, +3), 어수강 (홍익대학교 수학교육과 2004학번, +3).

(Here is a Solution by Chiheon Kim for Problem 2011-8.)

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.