[/quote]
,
and let 
and 
be its orthocenter and circumcenter, respectively. Let 
be the circle 
.
and the circle of radius 
cross 
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 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 
.
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
The circle inscribed in triangle 
touches 
at point 
. The perpendicular from point on 
intersects the bisector of angle 
at point 
. Prove that segment 
is equal to the radius of the circle inscribed in triangle .
(Oleksii Karliuchenko)
touches 
at point 
. The perpendicular from point on 
intersects the bisector of angle 
at point 
. Prove that segment 
is equal to the radius of the circle inscribed in triangle .(Oleksii Karliuchenko)
This post has been edited 2 times. Last edited by parmenides51, Mar 26, 2024, 2:27 PM
on 
is equal to the radius of the circle inscribed in triangle 
and 
which yields 
is isosceles so 
.
