52nd Putnam 1991

Problem A4

Can we find an (infinite) sequence of disks in the Euclidean plane such that: (1) their centers have no (finite) limit point in the plane; (2) the total area of the disks is finite; and (3) every line in the plane intersects at least one of the disks?



Answer: yes.


The fact that the centers have no limit point is a red herring - after all putting them close together would hardly help to ensure that the disks blocked all lines. The easiest strategy is evidently to find a continuous line of overlapping disks. That evidently requires that if their radii are ri, then ∑ ri diverges, but the sum of their areas is finite, so ∑ ri2 must converge. That is easy to achieve. For example, take ri = 1/i.

That leaves a little tidying up. We need the line to be infinite in both directions, and we need another line of disks to catch lines parallel to the first line of disks. But multiplying by four leaves convergence and divergence unaffected.

So, define Cn = ∑1n 1/i, Rn = 1/n. Define disk D4n to have center (Cn, 0) and radius Rn. That gives us a line of overlapping disks of finite total area, covering the positive x-axis. Similarly, define D4n+1 to have center (-Cn, 0) and radius Rn; they cover the negative x-axis. Define D4n+2 to have center (0, Cn), radius Rn, and D4n+3 to have center (0, -Cn), radius Rn; they cover the y-axis. The complete set D1, D2, D3, ... have the required properties.



52nd Putnam 1991

© John Scholes
21 Sep 1999