Question 1
$ABCD$ is a parallelogram; $AB=a$, $AD=1$, $\alpha$ is the size of $\angle DAB$, and the three angles of the triangle $ABD$ are acute. Prove that the four circles $K_A,K_B,K_C,K_D$, each of radius $1$, whose centers are the vertices $A,B,C,D$, cover the parallelogram if and only if $a\leq\cos\alpha+\sqrt3\sin\alpha$.