[/quote]
,
and 
for 
.
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 
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 
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 Mango247
Georg has 
cards with one number written on each card. On one card the integer 
is written, and among the rest of the cards, the integers 
appear, each twice. Georg wants to place the cards in a row in such a way that the -card is in the middle, and for each , the two cards with the number 
have the distance (meaning that there are exactly 
cards between them).
For which
is this possible?
cards with one number written on each card. On one card the integer 
is written, and among the rest of the cards, the integers 
appear, each twice. Georg wants to place the cards in a row in such a way that the -card is in the middle, and for each , the two cards with the number 
have the distance (meaning that there are exactly 
cards between them).For which
is this possible?
