2021 WSMO Accuracy Round Problems/Problem 9

Problem

Let $x=1+\frac{5}{2+\frac{3}{2+\frac{3}{2+\ldots}}}.$ If $\sqrt{x+\sqrt{x+\sqrt{x+\ldots}}}$ can be written as $\frac{a+\sqrt{b}}{c},$ where $b$ is not divisible by the square of any prime, find $a+b+c.$

Solution

Let $y = 2 + \frac{3}{2 + \frac{3}{2+...}}$ : $\Rightarrow y = 2 + \frac{3}{y}$ $\Rightarrow y^2 - 2y - 3 = 0$ $\Rightarrow y = 3$

Thus, $x = 1 + \frac{5}{y} = 1 + \frac{5}{3} = \frac{8}{3}$ . Now let $z = \sqrt{x+\sqrt{x+\sqrt{x+\ldots}}}$ $\Rightarrow z = \sqrt{\frac{8}{3}+\sqrt{\frac{8}{3}+\sqrt{\frac{8}{3}+\ldots}}}$ $\Rightarrow z = \sqrt{\frac{8}{3} + z}$ $\Rightarrow 3z^2 - 3z - 8 = 0$ $\Rightarrow z = \frac{3 + \sqrt{105}}{6}$

Our answer is thus $3 + 105 + 6 = \boxed{114}$ .

~BigKahuna227

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2021 WSMO Accuracy Round Problems/Problem 9

Problem

Solution