MathDB

3

Part of 1989 Irish Math Olympiad

Problems(2)

Maximum Value of Function in Range

Source: 1989 IrMO Paper 1 Problem 3

9/29/2017
A function ff is defined on the natural numbers N\mathbb{N} and satisfies the following rules:
(a) f(1)=1f(1)=1;
(b) f(2n)=f(n)f(2n)=f(n) and f(2n+1)=f(2n)+1f(2n+1)=f(2n)+1 for all n∈Nn\in \mathbb{N}.
Calculate the maximum value mm of the set {f(n):n∈N,1≤n≤1989}\{f(n):n\in \mathbb{N}, 1\le n\le 1989\}, and determine the number of natural numbers nn, with 1≤n≤19891\le n\le 1989, that satisfy the equation f(n)=mf(n)=m.
functionalgebra
Geometry Triangle inequality

Source: Irish MO

5/10/2012
Suppose P is a point in the interior of a triangle ABC, that x; y; z are the distances from P to A; B; C, respectively, and that p; q; r are the per- pendicular distances from P to the sides BC; CA; AB, respectively. Prove that xyz \geq 8pqr; with equality implying that the triangle ABC is equilateral.
geometryinequalitiestrigonometry