# 2010-4 Power and gcd

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

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# Solution: 2010-3 Sum

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.

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# Solution: 2010-2 Nonsingular matrix

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).

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# 2010-3 Sum

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.)

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# 2010-2 Nonsingular matrix

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.

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# Solution: 2010-1 Covering the unit square by squares

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.

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# 2010-1 Covering the unit square by squares

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.

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# Concluding Fall 2009

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
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