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!)
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).
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 \[\sum_{j=1}^n x_j=\binom{n+1}{2}.\]
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).
Compute \(\int_0^1 \frac{x^k-1}{\log x}dx\).
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 \(n\) 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 \(D_n\).
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 \(A\).