MathLabs

Problem 1

Let ABCDABCD be a parallelogram with AB=aAB=a, AD=1AD=1, and ∠BAD=α\angle BAD=\alpha. If △ABD\triangle ABD is acute, prove that the four circles of radius 11 centered at A,B,C,DA,B,C,D cover the parallelogram if and only if a≤cos⁡α+3sin⁡αa\le\cos\alpha+\sqrt3\sin\alpha.
Step 5 of 6: Solve the quadratic
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cos⁡α−3sin⁡α≤a≤cos⁡α+3sin⁡α\cos\alpha-\sqrt3\sin\alpha\le a\le\cos\alpha+\sqrt3\sin\alpha
Detailed analysis

The roots of the quadratic are cos⁡α±3sin⁡α\cos\alpha\pm\sqrt3\sin\alpha, so the inequality holds between them. If a≤1a\le1, coverage is immediate because the whole triangle lies in the unit disk centered at AA, and the upper bound is greater than 11 for acute α\alpha. If a>1a>1, the lower bound is at most 11, so it is automatic. Thus only a≤cos⁡α+3sin⁡αa\le\cos\alpha+\sqrt3\sin\alpha remains.