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Harvard-MIT Mathematics Tournament

Geometry Difficulty 5.2 AIME, harder Prove it United States

Problem:
Let $ABCD$ be a convex quadrilateral inscribed in a circle with shortest side $AB$ . The ratio $[BCD]/[ABD]$ is an integer (where $[XYZ]$ denotes the area of triangle $XYZ$ .) If the lengths of $AB$ , $BC$ , $CD$ , and $DA$ are distinct integers no greater than $10$ , find the largest possible value of $AB$ .

Solution

Solution:
Note that
$\frac{[BCD]}{[ABD]} = \frac{\frac{1}{2} BC \cdot CD \cdot \sin C}{\frac{1}{2} DA \cdot AB \cdot \sin A} = \frac{BC \cdot CD}{DA \cdot AB}$
since $\angle A$ and $\angle C$ are supplementary. If $AB \geq 6$ , it is easy to check that no assignment of lengths to the four sides yields an integer ratio, but if $AB = 5$ , we can let $BC = 10$ , $CD = 9$ , and $DA = 6$ for a ratio of $3$ . The maximum value for $AB$ is therefore $5$ .

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.