MathDB

2

Part of 2009 Pan African

Problems(2)

If DEF is equilateral then AD, BE and CR are concurrent

Source: Pan African Olympiad 2009

10/1/2011
Point PP lies inside a triangle ABCABC. Let D,ED,E and FF be reflections of the point PP in the lines BC,CABC,CA and ABAB, respectively. Prove that if the triangle DEFDEF is equilateral, then the lines AD,BEAD,BE and CFCF intersect in a common point.
geometrygeometric transformationreflectiongeometry proposed
find all f such that f(x^2-y^2)=f(x)f(y)

Source: Pan African Olympiad 2009

10/1/2011
Find all functions f:N0→N0f:\mathbb{N}_0\to\mathbb{N}_0 for which f(0)=0f(0)=0 and f(x2−y2)=f(x)f(y)f(x^2-y^2)=f(x)f(y) for all x,y∈N0x,y\in\mathbb{N}_0 with x>yx>y.
functioninductionnumber theory proposednumber theory