## Rational zeros of a function of three variables.

### Problem 180

Published on Saturday, 2nd February 2008, 09:00 am; Solved by 827

For any integer n, consider the three functions

f1,n(x,y,z) = xn+1 + yn+1 zn+1
f2,n(x,y,z) = (xy + yz + zx)*(xn-1 + yn-1 zn-1)
f3,n(x,y,z) = xyz*(xn-2 + yn-2 zn-2)

and their combination

fn(x,y,z) = f1,n(x,y,z) + f2,n(x,y,z) f3,n(x,y,z)

We call (x,y,z) a golden triple of order k if x, y, and z are all rational numbers of the form a / b with
0 a b k and there is (at least) one integer n, so that fn(x,y,z) = 0.

Let s(x,y,z) = x + y + z.
Let t = u / v be the sum of all distinct s(x,y,z) for all golden triples (x,y,z) of order 35.
All the s(x,y,z) and t must be in reduced form.

Find u + v.