Evaluate the following sum

\(\displaystyle\sum_{m=1}^\infty \sum_{\substack{n\ge 1\\ (m,n)=1}} \frac{x^{m-1}y^{n-1}}{1-x^m y^n}\)

when |x|, |y|<1.

(We write (m,n) to denote the g.c.d of m and n.)

The best solution was submitted by Hojin Kim (김호진, 2009학번). Congratulations!

Here is his Solution of Problem 2010-3.

Alternative solutions were submitted by 정성구 (수리과학과 2007학번, +3), 임재원 (2009학번, +3), Prach Siriviriyakul (2009학번, +3), 서기원 (2009학번, +3), 김치헌 (수리과학과 2006학번, +2).

The problem had a slight problem when xy=0; It is necessary to assume 0⁰=1.

Let A=(aij) be an n×n matrix of complex numbers such that \(\displaystyle\sum_{j=1}^n |a_{ij}|<1\) for each i. Prove that I-A is nonsingular.

The best solution was submitted by  Sung-Min Kwon (권성민), 2009학번. Congratulations!

Here is his Solution of Problem 2010-2.

Alternative solutions were submitted by 라준현 (수리과학과 2008학번, +3), 서기원 (2009학번, +3), 임재원 (2009학번, +3), 정성구 (수리과학과 2007학번, +3).

Prove that finitely many squares on the plane with total area at least 3 can cover the unit square.

The best solution was submitted by Jeong, Seong-Gu (정성구), 수리과학과 2007학번. Congratulations!

Here is his Solution of Problem 2010-1.

Alternative solutions were submitted by 임재원 & 서기원 (2009학번, +3 -> +2, +2 each) and 권용찬 (2009학번, +2; almost correct). Thank you for participation.