INMO 2005 Problem 3

Let $p, q, r$ be positive real numbers, not all equal, such that some two of the equations

$$p x^{2}+2 q x+r=0, q x^{2}+2 r x+p=0, r x^{2}+2 p x+q=0$$

have a common root, say $\alpha$. Prove that

(a) $\alpha$ is real and negative; and

(b) the third equation has non-real roots.