Putnam 1967 B2

Let $0\leq p,r\leq 1$ and consider the identities $$a) (px+(1-p)y)^{2}=a x^2 +bxy +c y^2, \;\;\; b) (px+(1-p)y)(rx+(1-r)y) =\alpha x^2 + \beta xy + \gamma y^2.$$ Show that $$a) \max(a,b,c) \geq \frac{4}{9}, \;\;\; b) \max( \alpha, \beta , \gamma) \geq \frac{4}{9}.$$