[/quote]
,
be an acute triangle with 
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 
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 
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 
.
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 pavel kozlov
Initially, a pair of numbers 
is written on the board. If for some 
and 
one of the pairs 
and 
is written on the board, then you can add the other one. Similarly for 
and 
. Prove that for each pair that appears on the board, its first number will be positive.
is written on the board. If for some 
and 
one of the pairs 
and 
is written on the board, then you can add the other one. Similarly for 
and 
. Prove that for each pair that appears on the board, its first number will be positive.This post has been edited 2 times. Last edited by VicKmath7, Apr 19, 2022, 5:46 PM
the expression 
is strictly positive, where 
is an arbitrary pair that can ever appear on the board. Substituting 
and 
gives us the result we need.
