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.

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 2009-4. (This proof can be slightly improved in the second half.)

Alternative solutions were submitted by 김치헌 (+2), 백형렬 (+3).