Solution: 2013-01 Inequality involving eigenvalues and traces

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.

GD Star Rating
loading...

2013-02 Functional equation

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 \( f(256) \).

GD Star Rating
loading...

Math Problem of the Week 2013 will begin at March 8.

The first problem of 2013 spring semester will be posted at March 8. As usual, problems will be posted on every Thursday at noon 3:30PM Friday at 3PM and solutions will be due next Wednesday at noon. Please submit your solution to jioon at kaist.ac.kr or bring it to the department of mathematical sciences (to put it into the mailbox of Prof. Ji Oon Lee).

GD Star Rating
loading...

Concluding 2012 Fall

Thanks all for participating POW actively. Here’s the list of winners:

  • 1st prize: Lee, Myeongjae  (이명재) – 2012학번
  • 2nd prize: Kim, Taeho (김태호) – 수리과학과 2011학번
  • 3rd prize: Park, Minjae (박민재) – 2011학번
  • 4th prize: Suh, Gee Won (서기원) – 수리과학과 2009학번
  • 5th prize: Lim, Hyunjin (임현진) – 물리학과 2010학번

Congratulations! We again have very good prizes this semester – iPad 16GB for the 1st prize, iPad Mini 16GB for the 2nd prize, etc.

2012 Fall POW


이명재 (2012학번) 32
김태호 (2011학번) 30
박민재 (2011학번) 25
서기원 (2009학번) 21
임현진 (2010학번) 17
김주완 (2010학번) 10
조상흠 (2010학번) 8
임정환 (2009학번) 7
김홍규 (2011학번) 5
곽걸담 (2011학번) 5
김지원 (2010학번) 5
이신영 (2012학번) 5
윤영수 (2011학번) 5
엄태현 (2012학번) 4
조준영 (2012학번) 3
박종호 (2009학번) 3
정종헌 (2012학번) 2
장영재 (2011학번) 2
양지훈 (2010학번) 2
최원준 (2009학번) 2
김지홍 (2007학번) 2
GD Star Rating
loading...

Solution: 2012-24 Determinant of a Huge Matrix

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).

 

GD Star Rating
loading...

Solution: 2012-23 A solution

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).

GD Star Rating
loading...

2012-24 Determinant of a Huge Matrix

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 \(\lvert\det A\rvert\)?

(This is the last problem of this semester. Good luck with your final exam!)

GD Star Rating
loading...

Solution: 2012-22 Simple integral

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).

GD Star Rating
loading...

2012-23 A solution

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 \[\sum_{j=1}^n x_j=\binom{n+1}{2}.\]

GD Star Rating
loading...