2023 WSMO Accuracy Round Problems/Problem 1: Difference between revisions
Created page with "==Problem== Let <math>x = \sqrt{69+\sqrt{69+\sqrt{69\dots}}}.</math> Find the value of <math>(2x-1)^2.</math> ==Solution==" |
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==Solution== | ==Solution== | ||
We have | |||
<cmath>\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*}</cmath> | |||
~pinkpig | |||
Latest revision as of 10:35, 13 September 2025
Problem
Let 
Find the value of 
Solution
We have
~pinkpig
