25 darts land inside a regular-hexagonal dart board with edge length the square root of 3. Show that at least five of the darts land inside the same unit circle.
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Proof by Snail_Erato: Pigeon hole principle. The circle is precisely the circumcircle of each of the 6 triangles in which we can dived the hexagon. if none had more than four darts it could not make more than 24 darts: Q.E.D!