# 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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