Let $ \mathcal{M} $ a set of $ 3n\ge 3 $ planar points such that the maximum distance between two of these points is $ 1 $. Prove that: a) among any four points,there are two aparted by a distance at most $ \frac{1}{\sqrt{2}} . $ b) for $ n=2 $ and any $ \epsilon >0, $ it is possible that $ 12 $ or $ 15 $ of the distances between points from $ \mathcal{M} $ lie in the interval $ (1-\epsilon , 1]; $ but any $ 13 $ of the distances can´t be found all in the interval $ \left(\frac{1}{\sqrt 2} ,1\right]. $ c) there exists a circle of diameter $ \sqrt{6} $ that contains $ \mathcal{M} . $ d) some two points of $ \mathcal{M} $ are on a distance not exceeding $ \frac{4}{3\sqrt n-\sqrt 3} . $