A person walks one lap along a circle at a speed $$v$$. Thereafter, he runs one lap along the boundary of the largest square that can be inscribed in the circle at a speed $$3v$$. The ratio of the time he walks to the time he runs is

Let circle radius $$= r$$. Largest inscribed square has diagonal $$2r$$, so side $$r\sqrt 2$$, perimeter $$4r\sqrt 2$$.

Walking time: $$\dfrac{2\pi r}{v}$$. Running time: $$\dfrac{4r\sqrt 2}{3v}$$.

Ratio $$= \dfrac{2\pi r/v}{4r\sqrt 2/(3v)} = \dfrac{2\pi \cdot 3}{4\sqrt 2} = \dfrac{3\pi}{2\sqrt 2}$$.

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