AoPS Academy
and 
be two sequences of integers such that 
and 
for all 
. Suppose that 
divides 
for an infinity of integers . Prove that there exist an integer 
such that 
for all
are coprime. Consider the sequence 
given by 
. Suppose that there exist positive integers 
such that 
is even and 
is an integer. Prove that the numerator in its simplest form of 
is an odd integer greater than 
.
be an integer and let 
be 
distinct subsets of 
. Show that there exists 
such that the n subsets 
are also disjoint.
there are three circles 
each of which is tangent to two sides of the triangle and to its incircle 
. The radii of are 
, and 
. Determine the radius of 
be the triangle and 
be the circle next to the vertex 
where 
Let 
be the triangle’s incenter. Let 
and 
be the projection of 
and 
on 
Then we obviously have the following:
and similarly 
we have:
and summing equation 
over 
we finally get:
and 
and rearranging, we get 
Because 
we have 
such that ![\[f(2x+y) + f(x+f(2y)) = f(x)f(y) - xy\]](http://latex.artofproblemsolving.com/0/6/2/0623b7c328af6d088cfc5dd288afe3e17a94ff29.png)
for all reals 
and 
.
s
.
