[/quote]
,
be an acute triangle with 
and let 
and 
be its orthocenter and circumcenter, respectively. Let 
be the circle 
.
and the circle of radius 
and 
and circle 
are concurrent.
of functions from the set of real numbers to itself that satisfy ![\[g(f(x+y)) = f(x) + (2x + y)g(y)\]](http://latex.artofproblemsolving.com/0/a/3/0a3bc25299a0bcbd3f3602a0e6881d68fce750c5.png)
for all real numbers 
and 
.
such that for all 
be given pairwise coprime positive integers where 
.
be positive integers. We call 
to be a grandson of 
if and only if, for all possible piles of stones whose total mass adds up to 
of primes (greater than 5) such that : 
of positive integers such that:
.
and incircle touching 
at 
.
be the antipode of 
such that 
.
be the midpoint of arc 
,
be the midpoint of 
.
be distinct natural numbers each of which has exactly 
positive integer factors. For instance, 
has exactly 
.
divides any of the 
'
and 
intersect at points 
.
and 
intersects 
,
intersects 
.
,
.
be such that![\[u_{1} = 1; \quad u_{n+1} = \dfrac{u_{1} + u_{2} +...+u_{n}}{n}+n-1 \quad \forall n \in \mathbb{N^{*}}\]](http://latex.artofproblemsolving.com/1/5/e/15e5bf0cc14794e7028dbfe4a8754f6803e850ea.png)
and 
.
.
denote the set of positive real numbers. Find all functions 
such that for each 
,
satisfying 
and primes 
such that 
Y by
Let 
be two non-parallel lines and 
be positive reals. The set of points 
, such that 
is a multiple of 
is called a 
. Let 
be finite set of points, not all collinear. A triangle with vertices in is called 
if no points from lie inside or on the sides of the triangle. Given that all empty triangles have the same area, show that is the intersection of a grid 
and a convex polygon 
.
be two non-parallel lines and 
be positive reals. The set of points 
, such that 
is a multiple of 
is called a 
. Let 
be finite set of points, not all collinear. A triangle with vertices in is called 
if no points from lie inside or on the sides of the triangle. Given that all empty triangles have the same area, show that is the intersection of a grid 
and a convex polygon 
.
