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).

Let \(n\) be a fixed positive integer and let \(p\in (0,1)\). Let \(D_n\) be the determinant of a random \(n\times n\) 0-1 matrix whose entries are independent identical random variables, each of which is 1 with the probability \(p\) and 0 with the probability \(1-p\).  Find the expected value and variance of \(D_n\).

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

Here is his Solution of Problem 2012-21.

Alternative solutions were submitted by 박민재 (2011학번, +3), 김태호 (수리과학과 2011학번, +3), 임현진 (물리학과 2010학번, +3), 김지홍 (수리과학과 2007학번, +2), 서기원 (수리과학과 2009학번, +2).

Let \(A=(a_{ij})\) be an \(n\times n\) upper triangular matrix such that \[a_{ij}=\binom{n-i+1}{j-i}\] for all \(i\le j\). Find the inverse matrix of \(A\).

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

Here is his Solution of Problem 2012-20.

Alternative solutions were submitted by 서기원 (수리과학과 2009학번, +3), 이명재 (2012학번, +3), 김태호 (수리과학과 2011학번, +3), 임현진 (물리학과 2010학번, +3), 박훈민 (대전과학고 2학년, +3), 윤성철 (홍익대학교 수학교육학과 2009학번, +3), 어수강 (서울대학교 수리과학부 석사과정, +3).

Let \(a_0=3\) and \(a_{n}=a_{n-1}+\sqrt{a_{n-1}^2+3}\) for all \(n\ge 1\). Determine \[\lim_{n\to\infty}\frac{a_n}{2^n}.\]

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

Here is his Solution of Problem 2012-19.

Alternative solutions were submitted by 박민재 (2011학번, +3), 김태호 (수리과학과 2011학번, +3). Two incorrect solutions were submitted (YSC, KJW).

Let \(r_1,r_2,r_3,\ldots\) be a sequence of all rational numbers in \( (0,1) \) except finitely many numbers. Let \(r_j=0.a_{j,1}a_{j,2}a_{j,3}\cdots\) be a decimal representation of \(r_j\). (For instance, if \(r_1=\frac{1}{3}=0.333333\cdots\), then \(a_{1,k}=3\) for any \(k\).)

Prove that the number \(0.a_{1,1}a_{2,2}a_{3,3}a_{4,4}\cdots\) given by the main diagonal cannot be a rational number.

The best solution was submitted by Kim, Joo Wan (김주완, 수리과학과 2010학번). Congratulations!

Here is his Solution of Problem 2012-18.

Alternative solutions were submitted by 이명재 (2012학번, +3), 김태호 (수리과학과 2011학번, +3), 임현진 (물리학과 2010학번, +3), 박민재 (2011학번, +3), 서기원 (수리과학과 2009학번, +2), 이신영 (2012학번, +2),  윤영수 (2011학번, +2), 박훈민 (대전과학고 2학년, +3), 어수강 (서울대학교 수리과학부 석사과정, +2), 윤성철 (홍익대학교 수학교육학과 2009학번, +2). There were 3 incorrect solutions submitted (JWS, KDR, JSH).

Let \(m\) and \(n\) be odd integers. Determine \[ \sum_{k=1}^\infty \frac{1}{k^2}\tan\frac{k\pi}{m}\tan \frac{k\pi}{n}.\]

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

Here is his Solution of Problem 2012-17.

One incorrect solution was received (PHM).

Prove that if a finite ring has two elements \(x\) and \(y\) such that \(xy^2=y\), then \( yxy=y\).

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

Here is Solution of Problem 2012-16.

Alternative solutions were submitted by 김주완 (수리과학과 2010학번, +3), 김지원 (수리과학과 2010학번, +3), 서기원 (수리과학과 2009학번, +3), 김태호 (수리과학과 2011학번, +3), 임현진 (물리학과 2010학번, +3), 박민재 (2011학번, +3), 조상흠 (수리과학과 2010학번, +3), 정우석 (서강대 수학과 2011학번, +3). One incorrect solution (KHK) was submitted.