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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.
is a point inside a convex quadrilateral 
such that ![\[ OA^2 + OB^2 + OC^2 + OD^2 = 2\mathcal A[ABCD] , \]](http://latex.artofproblemsolving.com/4/0/3/403fd4d612e421609d3c18fe6383b3afde36f3eb.png)
where by ![$\mathcal A[ABCD]$](http://latex.artofproblemsolving.com/a/4/0/a4094948d182803d9c58e16cbde40906dd7eabb2.png)
we have denoted the area of . Prove that is a square and is its center.
.![$OA \cdot OB \geq OA \cdot OB\sin \alpha = 2\left[OAB\right]$](http://latex.artofproblemsolving.com/e/4/1/e413691ef9bba370f9e95aaebe9a057b461060f8.png)
because the 
is on 
.![$OA^2 + OB^2 \geq 4\left[OAB\right]$](http://latex.artofproblemsolving.com/3/c/f/3cf9968b44e0c647fd98ecb9b703c240f1296505.png)
and ![$OA^2 + OB^2 + OC^2 + OD^2 \geq 2\left[ABCD\right]$](http://latex.artofproblemsolving.com/5/4/d/54dd00ee44ffd5436d92020189d59bd29b977673.png)
.
for the first and 
for the second one. is the center of a square.
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
.
