## 100 Discs

Suppose there are points in plane, such that if we construct the disks of radius centered at , than for any triplet there exists a line which intersects and . Prove that if we construct the disks centered at and of radius then there exists a line which intersects each .

**Steps:** **1.**Prove that any triangle has one of its heights at most equal to .

**2.** Pick the segment with the greatest length. Then the height of the other vertex is the smallest height in triangle , and therefore is smaller than . This means that the disk intersects . Therefore is the line we are looking for.

*Note: This problem, with helping subproblems was proposed in the contest for obtaining a math teacher job in Romania, 2010. In my opinion, this is a little tricky, and those who can solve this kind of problem are going to be good teachers. See another of the problems proposed in the contest in this link. *

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Categories: Combinatorics, Geometry, Problem Solving
extremal
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