# Concluding Spring 2009

Thanks all for participating POW actively. Here’s the list of winners:

1st prize:  Kim, Chiheon (김치헌) – 수리과학과 2006학번

2nd prize: Baik, Hyung Ryul (백형렬) – 수리과학과 2003학번

3rd prize: Lee, Jaesong (이재송) – 전산학과 2005학번

Congratulations!

In addition to those three people, I have selected two students. They received 1 movie ticket each.

Kwon, Sang Hoon (권상훈) – 수리과학과 2006학번

Cho, Kangjin (조강진) – 2009학번 김치헌 (2006학번) 33pts
백형렬 (2003학번) 30pts
이재송 (2005학번) 22pts
권상훈 (2006학번) 21pts
조강진 (2009학번) 17pts
김호진 (2009학번) 11pts
조용화 (2006학번) 6pts
김린기 (2003학번) 5pts
박승균 (2008학번) 3pts
양해훈 (2008학번) 3pts
류연식 (2008학번) 2pts
이병찬 (2007학번) 2pts
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# Solution: 2009-13 Distances between points in [0,1]^2

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.

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# Solution: 2009-12 Colorful sum

Suppose that we color integers 1, 2, 3, …, n with three colors so that each color is given to more than n/4 integers. Prove that there exist x, y, z such that x+y=z and x,y,z have distinct colors.

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

Here is his Solution of Problem 2009-12.

Alternative solutions were submitted by 백형렬 (수리과학과 2003학번, +3), 조강진 (2009학번, +2), 권상훈 (수리과학과 2006학번, +2).

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# Solution: 2009-11 Circles and lines

Does there exist a set of circles on the plane such that every line intersects at least one but at most 100 of them?

The best solution was submitted by Hyung Ryul Baik (백형렬), 수리과학과 2003학번. Congratulations!

Here is his Solution of Problem 2009-11.

There were 2 other incorrect solutions submitted.

Reference: L. Yang, J. Zhang, and W. Zhang, On number of circles intersected by a line, J. Combin. Theory Ser. A, 98 (2002), pp. 395–405.

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# Concluding 2009 Spring

Problem 2009-13 was the last problem of the spring 2009. I’ll soon write collected solutions for problems 2009-11,12,13. (Sorry for the delays because I was away.) Next week we will have a prize ceremony.

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# 2009-13 Distances between points in [0,1]^2

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 $$r_1^2+r_2^2+\cdots+r_n^2\le 4$$.

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