# 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
GD Star Rating

# Solution: 2009-14 New notion on the convexity

Let C be a continuous and self-avoiding curve on the plane. A curve from A to B is called a C-segment if it connects points A and B, and is similar to C. A set S of points one the plane is called C-convex if for any two distinct points P and Q in S, all the points on the C-segment from P to Q is contained in S.

Prove that a bounded C-convex set with at least two points exists if and only if C is a line segment PQ.

(We say that two curves are similar if one is obtainable from the other by rotating, magnifying and translating.)

The best solution was submitted by Jae-song Lee (이재송), 전산학과 2005학번. Congratulations!

Here is his Solution of Problem 2009-14.

Alternative solutions were submitted by 최범준 (수리과학과 2007학번, +3), 정성구 (수리과학과 2007학번, +3). One incorrect solution was received.

GD Star Rating

# Solution: 2009-9 min or max

Suppose that * is an associative and commutative binary operation on the set of rational numbers such that

1. 0*0=0
2. (a+c)*(b+c)=(a*b)+c for all rational numbers a,b,c.

Prove that either

1. a*b=max(a,b) for all rational numbers a,b, or
2. a*b=min(a,b) for all rational number a,b.

The best solution was submitted by Jaesong Lee (이재송), 전산학과 2005학번. Congratulations!

Check his Solution of Problem 2009-9.

Alternative solutions were submitted by 김치헌 (수리과학과 2006학번, +3), 권상훈 (수리과학과 2006학번, +3), 양해훈 (수리과학과 2008학번, +3), 백형렬 (수리과학과 2003학번, +2), 김일희 & 오성진 (Princeton Univ., Graduate Student). One incorrect solution was submitted (0 point) and one (incorrect) solution was submitted but later withdrawn.

GD Star Rating
Let $$a_0=a$$ and $$a_{n+1}=a_n (a_n^2-3)$$. Find all real values $$a$$ such that the sequence $$\{a_n\}$$ converges.