If 650 points in a circle of radius 16, prove that some 10 must lie in a ring of inner radius 2 and outer radius 3.
The area of any such ring is $5\pi$ and the area covered by the union of all rings is $19^2\pi = 361\pi$. On average, each ring will contain $650 * 5/361 \approx 9.002$ points. So there must be at least one ring with more than 9 points lying within.
Here I am using the Law of Large Numbers (http://en.wikipedia.org/wiki/Law_of_large_numbers). If I repeatedly pick rings randomly from the infinite set of rings whose centers lie within the circle, the average number of points in each ring must approach 9.002.