# Problem 180: Rational zeros of a function of three variables
For any integer n, consider the three functions f1,n(x,y,z) = xn+1 +
yn+1 − zn+1f2,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.