100 Discs


Suppose there are 100 points in plane, O_1,...,O_{100} such that if we construct the disks C_i,\ i=1..100 of radius 1 centered at O_i, than for any triplet i,j,k there exists a line which intersects C_i,C_j and C_k. Prove that if we construct the disks D_i,\ i=1..100 centered at O_i and of radius 2 then there exists a line which intersects each D_i.

Steps: 1.Prove that any triangle O_iO_jO_k has one of its heights at most equal to 2.
2. Pick the segment O_iO_j with the greatest length. Then the height of the other vertex O_k,\ k\neq i,j is the smallest height in triangle O_iO_jO_k, and therefore is smaller than 2. This means that the disk D_k intersects O_iO_j. Therefore O_iO_j 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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