CSMC 2022 Part A - Question 5, CEMC UWaterloo

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Difficulty: 6

This problem is tagged with csmc, csmc22, highschool.

(Canadian Senior Mathematics Contest 2022, Part A, Question 5, CEMC - UWaterloo)

A circle has centre $O$ and diameter $AB = 2 \sqrt{19}$. Points $C$ and $D$ are on the upper half of the circle. A line is drawn through $C$ and $D$, as shown. Points $P$ and $Q$ are on the line so that $AP$ and $BQ$ are both perpendicular to $PQ$. $QB$ intersects the circle at $R$. If $CP = DQ = 1$ and $2AP = BQ$, what is the length of $AP$?

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