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2023 SSMO Accuracy Round Problems/Problem 8

Revision as of 22:21, 15 December 2023 by Pinkpig (talk | contribs) (Created page with "==Problem== There is a quadrilateral <math>ABCD</math> inscribed in a circle <math>\omega</math> with center <math>O</math>. In quadrilateral <math>ABCD</math>, diagonal <math...")
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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 $a+b.$

Solution

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