Let \( H \) be an \( N \times N \) real symmetric matrix. Suppose that \( |H_{kk}| < 1 \) for \( 1 \leq k \leq N \). Prove that, if \( |H_{ij}| > 4 \) for some \( i, j \), then the largest eigenvalue of \( H \) is larger than \( 3 \).

The best solution was submitted by 김범수, 10학번. Congratulations!

Other solutions were submitted by 라준현(08학번, +3), 서기원(09학번, +3), 김호진(09학번, +3), 김범수(10학번, +3), 박훈민(13학번, +3), 노수현(13학번, +2). Thank you for your participation.

Let \( t \) be a positive real number and \( m \) be a positive integer. Show that if both \( \cosh \, mt \) and \( \cosh \, (m+1)t \) are rational then \( \cosh \, t \) is also rational.

The best solution was submitted by 홍혁표, 13학번. Congratulations!

Other solutions were submitted by 라준현(08학번, +3), 서기원(09학번, +3), 김호진(09학번, +3), 김범수(10학번, +3), 박지민(12학번, +3), 김정민(12학번, +2), 양지훈(10학번, +2), 황성호(13학번, +2). Thank you for your participation.

Let \( \mathbb{Z}^+ \) be the set of positive integers. Suppose that \( f : \mathbb{Z}^+ \to \mathbb{Z}^+ \) satisfies the following conditions.

i) \( f(f(x)) = 5x \).

ii) If \( m \geq n \), then \( f(m) \geq f(n) \).

iii) \( f(1) \neq 2 \).

Find \( f(256) \).

The best solution was submitted by 김호진, 09학번. Congratulations!

Similar solutions were also submitted by 황성호(13학번, +3), 양지훈(10학번, +3), 홍혁표(13학번, +3), 김준(13학번, +3), 서기원(09학번, +3), 이주호(12학번, +3), 박훈민(13학번, +3), 송유신(10학번, +3), 임현진(10학번, +3), 라준현(08학번, +3), 김정민(12학번, +3), 박지민(12학번, +3), 김태호(11학번, +3), 김범수(10학번, +3), 전한솔(고려대 13학번, +3), 어수강(서울대 석사과정, +3), 이시우(POSTECH 13학번, +3), 정우석(서강대 11학번, +3), 윤성철(홍익대 09학번, +3), 김재호(하나고, +3), 이정준(08학번, +2). Thank you for your participation.

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).