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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 a convex quadrilateral, and write 
; 
; 
; 
; and 
. Assuming that 
, 
, and 
, prove that![\[(DB+BC)^2=AD^2+AC^2\]](http://latex.artofproblemsolving.com/f/b/c/fbc6a5e2822ce61505195c5342157e1bf53ca902.png)
be a convex quadrilateral, and write ; ; ; ; and . Assuming that , , and , prove that
. The conclusion follows.
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
.
