2024 SSMO Accuracy Round Problems/Problem 2

Problem

Equilateral triangle $N$ is inscribed within circle $O$ . A smaller equilateral triangle $P$ is inscribed within $N$ such that the vertices of $P$ lie on the midpoints of $N$ . The ratio of the areas between $O$ and $P$ can be expressed as $\frac{a\pi\sqrt{b}}{c},$ for relatively prime positive integers $a,c$ and squarefree $b.$ Find $a+b+c$ .

Solution

WLOG, let the radius of $O$ be 1. Then, the side length of $N$ is $\sqrt{3},$ meaning the side length of $P$ is $\frac{\sqrt{3}}{2}.$ Thus, the answer is $\frac{\pi}{\left(\frac{\sqrt{3}}{2}\right)^2\cdot\left(\frac{\sqrt{3}}{4}\right)} = \frac{\pi}{\frac{3\sqrt{3}}{16}} = \frac{16\sqrt{3}\pi}{9}\implies 16+3+9 = \boxed{28}.$

~SMO_Team

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2024 SSMO Accuracy Round Problems/Problem 2

Problem

Solution