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A diamond (a parallelogram with equal sides) is given, and its sides are 2 cm long. The sharp angels are 60 degrees. If there are nine points inside the diamond, prove that there must be two of them so that the distance between them is at most 1 cm.

Ideas where to start from?

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    Nine points and the pigeonhole principle suggests trying to find 8 "pigeonholes" for nine points.2012-06-09
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    Your diamond is a two-dimensional figure, I think?2012-06-09
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    I believe that “a parallelogram with equal sides” is usually called a “rhombus”.2014-01-16

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Hint: Split the diamond into eight congruent equilateral triangles. The triangles are your pigeon holes.

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    The diamond may be a regular tetrahedron of side $2$.2012-06-09
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    @André, that possibility did not occur to me. Well spotted! But in that case there would be no need to specify the acute angle? I'm sticking to the interpretation: a diamond = a parallelogram with equal sides, but let's wait for the OP to comment.2012-06-09
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    @André, thanks for the thought, but it is two-dimensional diamond.Therefore, this hint will do the job.2012-06-10