## 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.