2012/13 AHSMC Part 1 Problem 3

Problem 3 of 16

A circle of diameter $r$ is drawn inside a circle of diameter $R$ . For which of the following pairs $(r, R)$ is the area of the smaller circle closest to half the area of the larger circle?

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Solution

We want $\bigl(\dfrac{r}{R}\bigr)^2$ to be close to $\dfrac{1}{2}$ . We have $\bigl(\dfrac{1}{3}\bigr)^2 < \bigl(\dfrac{2}{4}\bigr)^2 < \bigl(\dfrac{3}{5}\bigr)^2 < \bigl(\dfrac{4}{6}\bigr)^2 = \dfrac{4}{9}$ and $\bigl(\dfrac{5}{7}\bigr)^2 = \dfrac{25}{49}$ . Since $\dfrac{25}{49} - \dfrac{1}{2} = \dfrac{1}{98} < \dfrac{1}{18} = \dfrac{1}{2} - \dfrac{4}{9}$ , the answer is (E).

University of Alberta

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