[/quote]
,
and 
for 
.
be a positive integer. Suppose that for every positive integer 
there exists a positive integer 
such that 
.
satisfying following conditions.
for all 
.
with 
,
holds.
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 
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 
Y by
Let 
and 
be two non-empty subsets of 
with 
and 
. Let 
be the product of all elements of and let 
be the product of all elements of . Find the minimum possible value of sum 
.
PS. It is a variation of JBMO Shortlist 2019 A3
and 
be two non-empty subsets of 
with 
and 
. Let 
be the product of all elements of and let 
be the product of all elements of . Find the minimum possible value of sum 
.PS. It is a variation of JBMO Shortlist 2019 A3
This post has been edited 2 times. Last edited by parmenides51, Nov 14, 2020, 8:17 PM
for which 
is minimal. W.l.o.g. 
.
.
,
.
do not divide 
.
,
and 
.
is the minimal value of 
.
