USAMO 2007 Problem 2

[zefuri]

A square grid on the Euclidean plane consists of all points [unparseable or potentially dangerous latex formula], where [unparseable or potentially dangerous latex formula] and [unparseable or potentially dangerous latex formula] are integers. Is it possible to cover all grid points by an infinite family of discs with non-overlapping interiors if each disc in the family has radius at least [unparseable or potentially dangerous latex formula]?

[Kurt]

No.

(How many points is that worth? :P)

[bradp]

this seems kinda ez.

[stupidityismygam]

[Kurt]

[zefuri]

it's hard to write rigorously

[Kurt]

Hmm, well now I've looked at this problem and I still don't have any ideas on how to approach it. It looks desperately like something that would involve curvature ([unparseable or potentially dangerous latex formula] for circles), but I don't know any applications for curvature outside of vector calculus...and no USAMO problems are supposed to require that.

For people who took USAMO (or think they know how to do it), how did you approach the problem? Yay for # at all things geometry. :D

[zefuri]

It is not possible. The proof is by contradiction. Suppose that such a covering
family F exists. Let D(P, Ï