2023 WSMO Accuracy Round Problems/Problem 1

Problem

Let $x = \sqrt{69+\sqrt{69+\sqrt{69\dots}}}.$ Find the value of $(2x-1)^2.$

Solution

We have $\begin{align*} x &= \sqrt{69+x}\implies\\ x^2 &= 69+x\implies\\ x^2-x-69 &= 0\implies\\ x &= \frac{1\pm\sqrt{277}}{2}\implies\\ (2x-1)^2 &= \left(2\left(\frac{1\pm\sqrt{277}}{2}\right)-1\right)^2\\ &= \left(\pm\sqrt{277}\right)^2 = \boxed{277}. \end{align*}$

~pinkpig

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2023 WSMO Accuracy Round Problems/Problem 1

Problem

Solution