Let A₁, A₂, A₃, …, A_n be finite sets such that |A_i| is odd for all 1≤i≤n and |A_i∩A_j| is even for all 1≤i<j≤n. Prove that it is possible to pick one element a_i in each set A_i so that a₁, a₂, …,a_n are distinct.

The best solution was submitted by Ilhee Kim (김일희) and Ringi Kim (김린기), Graduate Students, Princeton University.

Here is Solution of Problem 2011-16.

Alternative solutions were submitted by 강동엽 (전산학과 2009학번, +3, solution by 강동엽), 문상혁 & 박상현 (2010학번, +3), 장경석 (2011학번, +3), 이재석 (수리과학과 2007학번, +2). Three incorrect solutions were submitted by B. Kim, J. Lee, Y. Park.

Let n be a positive integer. Let ω=cos(2π/n)+i sin(2π/n). Suppose that A, B are two complex square matrices such that AB=ω BA. Prove that (A+B)ⁿ=Aⁿ+Bⁿ.

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

Here is his Solution of Problem 2011-15.

Alternative solutions were submitted by 박민재 (2011학번, +3, alternative solution), 장경석 (2011학번, +3).

For a positive integer n>1, let f(n) be the largest real number such that for every n×n diagonal matrix M with positive diagonal entries, if tr(M)<f(n), then M-J is invertible. Determine f(n). (The matrix J is the square matrix with all entries 1.)

The best solution was submitted by Kyoungseok Jang(장경석), 2011학번. Congratulations!

Here is his Solution of Problem 2011-14.

Alternative solutions were submitted by 곽영진 (2011학번, +3), 박민재 (2011학번, +3), 라준현 (수리과학과 2008학번, +3), 서기원 (수리과학과 2009학번, +3), 배다슬 (수리과학과 2008학번, +3), 김범수 (수리과학과 2010학번, +3), 어수강 (서울대학교 수리과학부 대학원, +3), 조위지 (Stanford Univ. 물리학과 박사과정, +3).

PS. There were solutions without computing the determinant. Here is a Solution of Problem 2011-14 by 김범수.

Let a₁, a₂, … be a sequence of non-negative real numbers less than or equal to 1. Let \(S_n=\sum_{i=1}^n a_i\) and \(T_n=\sum_{i=1}^n S_i\). Prove or disprove that \(\sum_{n=1}^\infty a_n/T_n\) converges. (Assume a₁>0.)

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

Here is his Solution of Problem 2011-13. (There is a minor mistake in the proof.)

Alternative solutions were submitted by 어수강 (서울대학교 대학원, +2), 백진언 (한국과학영재학교, +2).

Let M=(m_i,j)_1≤i,j≤n be an n×n matrix such that m_i,j=i(i+1)(i+2)…(i+j-2). (Note that m_1,1=1.) What is the determinant of M?

The best solution was submitted by Seungkyun Park (박승균), 수리과학과 2008학번. Congratulations!

Here is his Solution of Problem 2011-12.

Alternative solutions were submitted by 조상흠 (수리과학과 2010학번, +3), 장경석 (2011학번, +3), 김원중 (2011학번, +3), 박민재 (2011학번, +3),   서기원 (수리과학과 2009학번, +3), 김범수 (2010학번, +3), 어수강 (서울대학교, +3),  조위지 (Stanford Univ. 물리학과 박사과정, +3).

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.)