Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 5.1 AIME, harder Find the answer

Given that $r$ and $s$ are relatively prime positive integers such that $\frac{r}{s} = \frac{2(\sqrt{2} + \sqrt{10})}{5(\sqrt{3 + \sqrt{5}})}$ , find $r$ and $s$ .

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

Squaring both sides of the given equation yields $\frac{r^{2}}{s^{2}} = \frac{4(12 + 4 \sqrt{5})}{25(3 + \sqrt{5})} = \frac{16(3 + \sqrt{5})}{25(3 + \sqrt{5})} = \frac{16}{25}$ . Because $r$ and $s$ are positive and relatively prime, then by inspection, $r = 4$ and $s = 5$ .

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