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 1 of 6: Reduce to one triangle

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ABCD=\triangle ABD\cup\triangle BCD

Detailed analysis

The parallelogram is the union of congruent triangles $ABD$ and $BCD$ . By symmetry, the four unit circles cover the parallelogram exactly when the three unit circles centered at $A,B,D$ cover $\triangle ABD$ .