International Mathematical Olympiad 1967 Problem 1

Let $ABCD$ be a parallelogram with side lengths $AB = a$, $AD = 1$, and with $\angle BAD = \alpha$. If $\triangle ABD$ is acute, prove that the four circles of radius 1 with centers $A, B, C, D$ cover the parallelogram if and only if $$a \leq \cos \alpha + \sqrt{3} \sin \alpha.$$