2023 SSMO Accuracy Round Problems/Problem 8

Problem

There is a quadrilateral $ABCD$ inscribed in a circle $\omega$ with center $O$ . In quadrilateral $ABCD$ , diagonal $AC$ is a diameter of the circle, $\angle BAC = 30^\circ,$ and $\angle DAC = 15^\circ.$ Let $E$ be the base of the altitude from $O$ onto side $BA$ . Let $F$ be the base of the altitude from $E$ onto $BO$ . Given that $EF = 3,$ and that the product of the lengths of the diagonals of $ABCD$ is $a\sqrt{b},$ for some squarefree $b,$ find $a+b.$

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2023 SSMO Accuracy Round Problems/Problem 8

Problem

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