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How efficiently you can schedule matches between N teams where each team has to play once against other team. 

Given n distinct points prove that for all k,0 divide points into n-k and k points by drawing a line segment.... Proof given on a site There are n points, so max n(n-1)/2 different directions of the line segments joining two points among these n points. Now draw any line which has direction different than above at most n(n-1)/2 values. By choosing a different direction you are making sure that no two points can lie on the line. Now you can 'drag' the line as much as you want and get as many points on it's one of the sides as you want.