Let n, k be positive integers. Prove that \(\sum_{i=1}^n k^{\gcd(i,n)}\) is divisible by n.

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Let n, k be positive integers. Prove that \(\sum_{i=1}^n k^{\gcd(i,n)}\) is divisible by n.

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 0^{0}=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).

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.)

Let A=(a_{ij}) 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.

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.

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

각각의 정사각형의 면적을 다 더했을 때 3 이상이 되는 유한개의 정사각형들이 있을 때, 이 정사각형들로 면적이 1인 단위정사각형을 완전히 덮을 수 있음을 증명하세요.

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

1st prize: Jeong, Seong-Gu (정성구) – 수리과학과 2007학번

(shared) 2nd prize: Ok, Seong min (옥성민) – 수리과학과 2003학번

(shared) 2nd prize: Lee, Jaesong (이재송) – 전산학과 2005학번

Congratulations! (We have two students sharing 2nd prizes.) *POW for 2010 Spring will start on Feb. 5th.*

정성구 (2007학번) 35 pts

이재송 (2005학번) 10 pts

옥성민 (2003학번) 10 pts

김호진 (2009학번) 5 pts

양해훈 (2008학번) 4 pts

류연식 (2008학번) 4 pts

박승균 (2008학번) 4 pts

Prach Siriviriyakul (2009학번) 3 pts

노호성 (2008학번) 3 pts

김현 (2008학번) 3 pts

김환문 (2008학번) 3 pts

최범준 (2007학번) 3 pts

정지수 (2007학번) 3 pts

심규석 (2007학번) 3 pts

김치헌 (2006학번) 3 pts

송지용 (2006학번) 3 pts

최석웅 (2006학번) 3 pts