1.  An integer n is a sum of two squares if there are integers x, y
(possibly 0) with n = x^2+y^2.  If each of the integers a and b is a
sum of two squares, show that their product ab is also the sum of two
squares.  Prove that the number 5^64 - 3^64 is a sum of two squares.

2.  The cubic polynomial p(x)= x^3-4x^2+2 has three distinct real
roots, say a, b, and c.  Find a^4 + b^4 + c^4.

3.  Suppose that each positive integer is colored either red or blue. 
Show that there exist three positive integers x<y<z having the same
color and such that z-y=y-x.

Hi steeprock:

Thanks for the question. I found the answers to all three of your
questions at:

Wisconsin Mathematics Science & Engineering Talent Search
URL: http://www.math.wisc.edu/~talent/T02-5a.pdf

You'll note that your questions are there verbatim. Your first
question is answer #1. Your second question is answer #3. Your third
question is answer #4.


I hope this information helps with your research. 
                       
If you need any clarification of the information I have provided,
please ask using the clarification feature and provide me with
additional details as to what you are looking for. As well, please
allow me to provide you with clarification(s) *before* you rate this
answer.
                       
Thank you.                        
                       
websearcher-ga                        
                       
                       
Search Strategy (on Google):                        
 
product "sum of two squares"
"sum of two squares" 64 3 5
"cubic polynomial" "a^4" "b^4" "c^4"

Easiest $30 I've ever seen :)