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.

Let \(P_1,P_2,\ldots,P_n\) be n points in {(x,y): 0<x<1, 0<y<1} (n>1). Let \(r_i=\min_{j\neq i} d(P_i,P_j)\) where d(x,y) means the distance between two points x and y. Prove that \(r_1^2+r_2^2+\cdots+r_n^2\le 4\).

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

Here is his Solution of Problem 2009-13.

The best solution was submitted by Hojin Kim (김호진), 2009학번. Congratulations!

Here is his Solution of Problem 2009-10.

Alternative solutions were submitted by 백형렬(수리과학과 2003학번, +3), 김치헌(수리과학과 2006학번, +3), 이재송(전산학과 2006학번, +3), 권상훈(수리과학과 2006학번, +3), 조용화(수리과학과 2006학번, +2), 김일희 & 오성진 (Princeton Univ., Graduate students).

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 Hojin Kim (김호진), 2009학번. Congratulations!

Here is his Solution of Problem 2009-8.

There were 6 other solutions submitted by KAIST undergraduates; 조강진 (2009학번), 이재송 (전산학과 2005학번), 백형렬 (수리과학과 2003학번), 조용화 (수리과학과 2006학번), 김치헌 (수리과학과 2006학번), 권상훈 (수리과학과 2006학번). All will receive 3 points each. In addition, there were 3 other correct solutions submitted; 김성윤 (Mathematics, MIT, Undergraduate Class of ’09), 김일희 (PACM, Princeton Univ., Graduate Student), 정준혁 (Mathematics, Princeton Univ., Graduate Student).

Let n>1 be an integer and let x>1 be a real number. Prove that if
\(\sqrt[n]{x+\sqrt{x^2-1}}+\sqrt[n]{x-\sqrt{x^2-1}}\)
is a rational number, then x is rational.

The best solution was submitted by Sungyoon Kim (김성윤) (Mathematics, MIT, Class of ’09). Congratulations! (Though, he is not eligible for earning points and taking prizes.)

Here is his Solution of Problem 2009-7.

There were 5 other solutions submitted: 김호진 (2009학번), 백형렬 (수리과학과 2003학번), 이재송 (전산학과 2005학번), 조강진 (2009학번), 박승균 (수리과학과 2008학번). All will receive 3 points each.

Prove that each positive integer x can be written as a sum of at most 53 integers to the fourth power. In other words, for every positive integer x, there exist \(y_1,y_2,\ldots,y_k\) with \(k\le 53\) such that \(x=\sum_{i=1}^k y_i^4\).

The best solution was submitted by SangHoon Kwon (권상훈), 수리과학과 2006학번.

Here is his Solution of Problem 2009-6.

There were 4 other submitted solutions: 백형렬, 이재송, 조강진, 김치헌 (+3).

If we choose n points on a circle randomly and independently with uniform distribution, what is the probability that the center of the circle is contained in the interior of the convex hull of these n points?

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

There were 2 incorrect solutions submitted.

Click here for his Solution of Problem 2009-5.