Problem 1

Let

ABCD

be a parallelogram with

AB=a

AD=1

, and

\angle BAD=\alpha

. If

\triangle ABD

is acute, prove that the four circles of radius

1

centered at

A,B,C,D

cover the parallelogram if and only if

a\le\cos\alpha+\sqrt3\sin\alpha

Step 2 of 6: Use the circumcenter

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R=OD\le1

Detailed analysis

Let $O$ be the circumcenter of the acute triangle $ABD$ , and $R=OD$ . The standard medial-triangle argument shows that the three unit disks cover $\triangle ABD$ if and only if $O$ lies in the unit disk centered at $D$ , namely $R\le1$ .