Difference between revisions of "2023 WSMO Accuracy Round Problems/Problem 1"
(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 11:35, 13 September 2025
Problem
Let 
Find the value of 
Solution
We have
~pinkpig
