AoPS Academy
and 
Find the value of 
such that 
for all 
+1 w
be an even integer and let 
be real numbers satisfying 
.
be a positive integer. Ana and Banana play a game. Banana thinks of a function 
and a prime number 
. He tells Ana that 
is nonconstant, 
, and 
for all integers 
. Ana's goal is to determine the value of . She writes down integers 
. After seeing this list, Banana writes down 
in order. Ana wins if she can determine the value of from this information. Find the smallest value of for which Ana has a winning strategy.
Let 
be positive reals such that 
.Prove the inequality 
of positive integers is called central if for every positive integer , the arithmetic mean of the first 
terms of the sequence is equal to .
of positive integers such that for every central sequence 
there are infinitely many positive integers with 
.
be a set of 
points in the plane, no three of which are collinear. Initially these points are connected with segments so that each point in is the endpoint of exactly two segments. Then, at each step, one may choose two segments 
and 
sharing a common interior point and replace them by the segments 
and 
if none of them is present at this moment. Prove that it is impossible to perform 
or more such moves.
is the center of the inscribed circle and holds 
. Prove that 
.
be a primes ,
be positive integers such that 
and 
.
, where 
is a prime number. Show that if is an even number, then the polynomial ![\[-1+\prod_{k=0}^{n-1} P\left(X^{p^k}\right)\]](http://latex.artofproblemsolving.com/5/d/f/5df80f885e63852e4f53a6145dafd18d63dc3526.png)
is divisible by 
.
and 
. 
is even, we have: 
is odd, we have: 
. 
Prove that
,
,
for 