Subcontests
(11)2020 Geometry: Estimation
Gunmay picks 6 points uniformly at random in the unit square. If p is the probability that their convex hull is a hexagon, estimate p in the form 0.abcdef where a,b,c,d,e,f are decimal digits. (A convex combination of points x1,x2,…,xn is a point of the form α1x1+α2x2+⋯+αnxn with 0≤αi≤1 for all i and α1+α2+⋯+αn=1. The convex hull of a set of points X is the set of all possible convex combinations of all subsets of X.) 2020 Geometry 5
For every positive integer k, let Tk=(k(k+1),0), and define Hk as the homothety centered at Tk with ratio 21 if k is odd and 32 is k is even. Suppose P=(x,y) is a point such that
(H4∘H3∘H2∘H1)(P)=(20,20). What is x+y?
(A homothety H with nonzero ratio r centered at a point P maps each point X to the point Y on ray PX such that PY=rPX.)