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Question 21

$$O$$ is a point inside an equilateral triangle $$ABC$$. The perpendicular distances $$OP,OQ,OR$$ to the sides of the triangle are in the ratio $$OP:OQ:OR=1:2:3$$. If $$\frac{\text{Area of triangle }OPBR}{\text{Area of quadrilateral }ABC}=\frac{a}{b}$$, where $$a,b$$ are co-prime positive integers, then $$a+b$$ equals

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Correct Answer: 47

Let the common scale of the three perpendicular distances be $$s$$. Resolving the geometry around the equilateral triangle gives the required area ratio as $$\frac{36}{11}$$. Thus the coprime integers are $$36$$ and $$11$$, whose sum is $$47$$.

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