Help with this Putname problem: Is it possible to find an infinite sequence of closed disks $D_1,D_2,...$ in the plane with centers $c_1,c_2,...$ such that
a) the $c_1$ have no limit point in the finite plane.
b) the sum of the areas of the $D_i$ is finite and
c) every line in the plane intersects at least one of the $D_i$?
Put a unit disk at the origin. Now put disks with centers along each of the four coordinate axes. Put a disk of radius $\frac 12$ tangent to the unit disk, so there are four with centers $(\pm \frac 32,0)$ and $(0,\pm \frac 32)$. Put disks of radius $\frac 13$ tangent to the disks of radius $\frac 12$ and in general put disks of radius $\frac 1{n+1}$ tangent to the disks of radius $\frac 1n$. The radii form the harmonic series, which is unbounded, so we cover all of the coordinate axes. Every line in the plane therefore hits at least one disk. The sum of the areas is $$\pi\left(4\sum_{i=1}^\infty \frac 1{i^2}-3\right)=\pi\left(\frac {4\pi^2}6-3\right)$$ which is nicely finite. The first three stages of the construction are shown below.