Hello INMO, USAMO and all national olympiads awardees,
try these problems and post solutions , specially for IMO 2016:
Find all functions which satisfy the equation:
for all real numbers such that .
Show that if is a prime and , then
Given two circles that intersect at and , prove that there exist four points with the following property.
For any circle tangent to the two given circles, we let and be the points of the tangency an and the intersections of with the line . Then each of the lines passes through one of these four points.
Find all integral solutions to following equations:
Let , , be positive real numbers. Prove that
Let be the in-center of triangle . It is known that for every point , one can find the points and such that is the centroid of the triangle . Prove that is an equilateral triangle.