Let f(x) be a continuous function on I=[a,b], and let g(x) be a differentiable function on I. Let g(a)=0 and c≠0 a constant. Prove that if

|g(x) f(x)+c g′(x)|≤|g(x)| for all x∈I,

then g(x)=0 for all x∈I.

The best solution was submitted by Seungkyun Park (박승균), 수리과학과 2008학번. Congratulations!

Here is his Solution of Problem 2011-18.

Alternative solutions were submitted by 김범수 (수리과학과 2010학번, +3), 장경석 (2011학번, +3),  김태호 (2011학번, +2), 김재훈 (EEWS대학원, +2).

Let M=(m_i,j)_1≤i,j≤n be an n×n matrix such that m_i,j=i(i+1)(i+2)…(i+j-2). (Note that m_1,1=1.) What is the determinant of M?

The best solution was submitted by Seungkyun Park (박승균), 수리과학과 2008학번. Congratulations!

Here is his Solution of Problem 2011-12.

Alternative solutions were submitted by 조상흠 (수리과학과 2010학번, +3), 장경석 (2011학번, +3), 김원중 (2011학번, +3), 박민재 (2011학번, +3),   서기원 (수리과학과 2009학번, +3), 김범수 (2010학번, +3), 어수강 (서울대학교, +3),  조위지 (Stanford Univ. 물리학과 박사과정, +3).

Let A and B be n×n matrices over the real field R. Prove that if A+B is invertible, then A(A+B)^-1B=B(A+B)^-1A.

The best solution was submitted by SeungKyun Park (박승균), 2008학번. Congratulations!

Here is his Solution of Problem 2009-19.

Alternative solutions were submitted by 옥성민 (수리과학과 2003학번, +3), 노호성 (물리학과 2008학번, +3), 송지용 (수리과학과 2006학번, +3), 김현 (2008학번, +3), 정성구 (수리과학과 2007학번, +3), 이재송 (전산학과 2005학번, +3), 정지수 (수리과학과 2007학번, +3), 김호진 (2009학번, +3), 최석웅 (수리과학과 2006학번, +3), 김환문 (물리학과 2008학번, +3),  류종하 (서울대학교 전기과 2008학번). One incorrect solution was received. Thank you for the participation.