[/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 
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 Adventure10, Mango247
Ana and Beto play against each other. Initially, Ana chooses a non-negative integer 
and announces it to Beto. Next Beto writes a succession of 
numbers, 
of them equal to 
and of them equal to 
. Once this is done, Ana must split the succession into several blocks of consecutive terms (each term belonging to exactly one block), and calculate the sum of the numbers of each block. Finally, add the squares of the calculated numbers. If this sum is equal to , Ana wins. If not, Beto wins. Determine all values of for which Ana can ensure victory, no matter how Beto plays.
and announces it to Beto. Next Beto writes a succession of 
numbers, 
of them equal to 
and of them equal to 
. Once this is done, Ana must split the succession into several blocks of consecutive terms (each term belonging to exactly one block), and calculate the sum of the numbers of each block. Finally, add the squares of the calculated numbers. If this sum is equal to , Ana wins. If not, Beto wins. Determine all values of for which Ana can ensure victory, no matter how Beto plays.
.
,
.
to 
,
.
or 
blocks, such that the sum in each block is either 
blocks all summing to either 
There exists 
consecutive blocks, 
and 
that sum to 
are of the same sign, which implies that
impossible as
If 
,
and combine them into one block. Otherwise, take 
and combine them into one block. We have found a new sequence of 
