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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.
be reals with 
and 
. Prove that
is the incentre and point 
is defined such that 
and 
is the perpendicular bisector of 
and 
Prove that
Let and 
Prove that
Let and 
Prove that
Let and 
Prove that
people seated around a circular table, and 
cookies
be one of these people. Find the least such that no matter cookies are distributed initially, there is a strategy to pass receives at least one cookie.
be the midpoint of 
, then 
. Suppose, without loss of generality, that 
, we know that 
is a perpendicular bisector of 
, if 
then 
. through 
intersects and 
at 
and 
, respectively, so 
. We see that 
so 
(*). Since 
and 
we have 
, that is, is the reflection of wrt 
then 
is isosceles with 
, then by (*) we have that 
is the centroid of so 
and since 
we have that 
. Let's apply the bisector theorem in 
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
.
be non-negative reals such that 
.
and 
.
.