GULF 2015 Problem 4

a) We have a geometric sequence of $3$ terms. If the sum of these terms is $26$ , and their sum of squares is $364$ , find the terms of the sequence.

b) Suppose that $a,b,c,u,v,w$ are positive real numbers , and each of $a,b,c$ and $u,v,w$ are geometric sequences. Suppose also that $a+u,b+v,c+w$ are an arithmetic sequence. Prove that $a=b=c$ and $u=v=w$ c) Let $a,b,c,d$ be real numbers (not all zero), and let $f(x,y,z)$ be the polynomial in three variables defined by $$f(x,y,z) = axyz + b(xy + yz + zx) + c(x+y+z) + d$$ .Prove that $f(x,y,z)$ is reducible if and only if $a,b,c,d$ is a geometric sequence.