MathDB

4

Part of 2013 Greece Team Selection Test

Problems(2)

Counting Regions

Source: 2013 Greek TST,Pr.4

5/24/2016
Given are nn different concentric circles on the plane.Inside the disk with the smallest radius (strictly inside it),we consider two distinct points A,BA,B.We consider kk distinct lines passing through AA and mm distinct lines passing through BB.There is no line passing through both AA and BB and all the lines passing through kk intersect with all the lines passing through BB.The intersections do not lie on some of the circles.Determine the maximum and the minimum number of regions formed by the lines and the circles and are inside the circles.
combinatoricscountingcircleslines
Tiling with trapezoids in an equilateral grid

Source: Greek 2nd TST 2013-pr4

5/25/2016
Let nn be a positive integer. An equilateral triangle with side nn will be denoted by TnT_n and is divided in n2n^2 unit equilateral triangles with sides parallel to the initial, forming a grid. We will call "trapezoid" the trapezoid which is formed by three equilateral triangles (one base is equal to one and the other is equal to two). Let also mm be a positive integer with m<nm<n and suppose that TnT_n and TmT_m can be tiled with "trapezoids". Prove that, if from TnT_n we remove a TmT_m with the same orientation, then the rest can be tiled with "trapezoids".
trapezoidcombinatoricsTiling