Hello i was looking through problems and found this one which i thought was quite challenging and i still haven't managed to find a solution. anyway here is the problem.
Given $541$ points in the interior of a circle of unit radius, show that there must be a subset of $10$ points whose diameter (the maximum distance between any pair of points) is less than $\frac{\sqrt{2}}{4}$.
Hint: Superimpose a $\frac14$ unit by $\frac14$ unit grid over the circle with one of the grid points falling on the center of the circle. You should be able to show that the circle is covered by $60$ grid squares.
Then as suggested by JeanMarie in a comment, apply the Pigeonhole Principle. https://en.wikipedia.org/wiki/Pigeonhole_principle