Break for the midterm exam

Published on March 12, 2010 in annoncement. 0 Comments

KAIST POW will take a break for the midterm exam. Good luck to all students!

Next problem will be posted on March 26th.

Solution: 2010-5 Dependence over Q

Published on March 12, 2010 in solution. 1 Comment Tags: .

Find the set of all rational numbers x such that 1, cos πx, and sin πx are linearly dependent over rational field Q.

The best solution was submitted by Chiheon Kim(김치헌), 수리과학과 2006학번. Congratulations!

Here is his Solution of Problem 2010-5.

An alternative solution was submitted by 정성구 (수리과학과 2007학번, +3), 임재원 (2009학번, +2). One incorrect solution was submitted.

2010-5 Dependence over Q

Published on March 5, 2010 in problem. 0 Comments Tags: , , .

Find the set of all rational numbers x such that 1, cos πx, and sin πx are linearly dependent over rational field Q.

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Rating: +1 (from 3 votes)

Solution: 2010-4 Power and gcd

Published on March 5, 2010 in solution. 0 Comments Tags: .

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

The best solution was submitted by Chiheon Kim(김치헌), 수리과학과 2006학번. Congratulations!

Here is his Solution of Problem 2010-4.

Alternative solutions were submitted by 정성구 (수리과학과 2007학번, +3), Prach Siriviriyakul (2009학번, +3), 라준현 (수리과학과 2008학번, +3), 서기원 (2009학번, +3), 강동엽 (2009학번, +3), 임재원 (2009학번, +2).

2010-4 Power and gcd

Published on February 26, 2010 in problem. 0 Comments Tags: .

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

Published on February 26, 2010 in solution. 0 Comments Tags: .

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.

Solution: 2010-2 Nonsingular matrix

Published on February 20, 2010 in solution. 0 Comments Tags: .

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).

2010-3 Sum

Published on February 19, 2010 in problem. 0 Comments Tags: .

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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Rating: 4.7/5 (3 votes cast)
VN:F [1.8.4_1055]
Rating: +2 (from 2 votes)

2010-2 Nonsingular matrix

Published on February 12, 2010 in problem. 2 Comments Tags: .

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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Rating: +1 (from 5 votes)

Solution: 2010-1 Covering the unit square by squares

Published on February 12, 2010 in solution. 0 Comments Tags: .

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.