[/quote]
,
and let 
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 
.
be the antipode of 
is chosen on the internal angle bisector of 
such that 
.
,
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 Adventure10, Mango247
Let 
be a triangle with orthocenter 
and let 
be the feet of the altitudes lying on sides 
respectively. Let 
be the intersection of the lines 
and 
. Prove that the line 
is perpendicular to 
-median of 
.
Since this is in the chapter of harmonic divisions of the book, looking for a solution along those lines. What I could find were:
1.
is harmonic
2.
3.
, i.e. 
is a harmonic quadrilateral.
4.
where
is the midpoint of , 
, 
, 
be a triangle with orthocenter 
and let 
be the feet of the altitudes lying on sides 
respectively. Let 
be the intersection of the lines 
and 
. Prove that the line 
is perpendicular to 
-median of 
.Since this is in the chapter of harmonic divisions of the book, looking for a solution along those lines. What I could find were:
1.
is harmonic2.
3.
, i.e. 
is a harmonic quadrilateral.4.
where
is the midpoint of , 
, 
, 
is the 
are concyclic.
are concyclic. Particularly, this gives that 
is perpendicular to the 
concur (Follows easily from Radical Axes Theorem)
and 
are tangent to 
.
bisects 
This gives that 
is cyclic, thus proving Property 1. And, 
This, in turn, gives that 
is cyclic, proving Property 3. As for Property 4, just apply Radical Axes theorem on 
and 
.
concur, say at 
,
,
.
,
and 
are cyclic. Since 
is cyclic, 
