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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 $\frac{\sqrt{2}}{4}$.

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  • $\begingroup$ This type of problem is usually solved using pigeonhole principle. Do you know it ? See an example of application in (math.stackexchange.com/q/2054210) $\endgroup$ – Jean Marie May 9 '17 at 14:09
  • $\begingroup$ i've not seen this before thank you $\endgroup$ – user395952 May 9 '17 at 15:10
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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

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