Subcontests
(3)Iberoamerican Olympiad 2014, Problem 6
Given a set X and a function f:X→X, for each x∈X we define f1(x)=f(x) and, for each j≥1, fj+1(x)=f(fj(x)). We say that a∈X is a fixed point of f if f(a)=a. For each x∈R, let π(x) be the quantity of positive primes lesser or equal to x.Given an positive integer n, we say that f:{1,2,…,n}→{1,2,…,n} is catracha if ff(k)(k)=k, for every k=1,2,…n. Prove that:(a) If f is catracha, f has at least π(n)−π(n)+1 fixed points.
(b) If n≥36, there exists a catracha function f with exactly π(n)−π(n)+1 fixed points. Prove that 4 points lie on a circumference
Let ABC be an acute triangle and H its orthocenter. Let D be the intersection of the altitude from A to BC. Let M and N be the midpoints of BH and CH, respectively. Let the lines DM and DN intersect AB and AC at points X and Y respectively. If P is the intersection of XY with BH and Q the intersection of XY with CH, show that H,P,D,Q lie on a circumference.