Olympiad problems

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1985 Traian Lălescu, 1.3

Let $ a,b,c $ denote the lengths of a right triangle ($ a $ being the hypothenuse) that satisfy the equality $ a=2\sqrt{bc} . $ Find the angles of this triangle.

2000 Junior Balkan Team Selection Tests - Romania, 4

Tags: geometry , numerical geometry

Two identical squares havind a side length of $ 5\text{cm} $ are each divided separately into $ 5 $ regions through intersection with some lines. Show that we can color the regions of the first square with five colors and the regions of the second with the same five colors such that the sum of the areas of the resultant regions that have the same colors at superpositioning the two squares is at least $ 5\text{cm}^2. $

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