Let \(A, B\) be \(N \times N\) symmetric matrices with eigenvalues \(\lambda_1^A \leq \lambda_2^A \leq \cdots \leq \lambda_N^A\) and \(\lambda_1^B \leq \lambda_2^B \leq \cdots \leq \lambda_N^B\). Prove that
\[ \sum_{i=1}^N |\lambda_i^A – \lambda_i^B|^2 \leq Tr (A-B)^2 \]

The best solution was submitted by 라준현, 08학번. Congratulations!

Alternative solutions were submitted by 김호진(09학번, +3), 서기원(09학번, +3), 곽걸담(11학번, +3), 김정민(12학번, +2), 홍혁표(13학번, +2). Thank you for your participation.

Consider all non-empty subsets \(S_1,S_2,\ldots,S_{2^n-1}\) of \(\{1,2,3,\ldots,n\}\). Let \(A=(a_{ij})\) be a \((2^n-1)\times(2^n-1)\) matrix such that \[a_{ij}=\begin{cases}1 & \text{if }S_i\cap S_j\ne \emptyset,\\0&\text{otherwise.}\end{cases}\] What is \(\lvert\det A\rvert\)?

The best solution was submitted by Kim, Taeho (김태호), 수리과학과 2011학번. Congratulations!

Here is his Solution of Problem 2012-24.

Alternative solutions were submitted by 이명재 (2012학번, +3), 임현진 (물리학과 2010학번, +3), 정종헌 (2012학번, +2),  어수강 (서울대학교 수리과학부 석사과정, +3).

Prove that for each positive integer \(n\), there exist \(n\) real numbers \(x_1,x_2,\ldots,x_n\) such that \[\sum_{j=1}^n \frac{x_j}{1-4(i-j)^2}=1 \text{ for all }i=1,2,\ldots,n\] and \[\sum_{j=1}^n x_j=\binom{n+1}{2}.\]

The best solution was submitted by Taehyun Eom (엄태현), 2012학번. Congratulations!

Here is his Solution of Problem 2012-23.

Alternative solutions were submitted by 박민재 (2011학번, +3, Solution), 김태호 (수리과학과 2011학번, +2), 이명재 (2012학번, +2).

Compute \(\int_0^1 \frac{x^k-1}{\log x}dx\).

The best solution was submitted by Myeongjae Lee (이명재), 2012학번. Congratulations!

Here is his Solution of Problem 2012-22.

Alternative solutions were submitted by 박민재 (2011학번, +3), 서기원 (수리과학과 2009학번, +2), 김태호 (수리과학과 2011학번, +2), 임현진 (물리학과 2010학번, +2), 조위지 (Stanford Univ. 물리학과 박사과정, +3), 박훈민 (대전과학고 2학년, +3).