Difference between revisions of "2020 CIME I Problems/Problem 4"
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==Solution== | ==Solution== | ||
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| + | We simply use the best technique of easy bash. | ||
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| + | <math>x^2 = 2x^2 - 2\sqrt{x^4-\frac{1}{x^4}}</math> | ||
| + | |||
| + | <math>x^4 = 4x^4-\frac{4}{x^4}</math> | ||
| + | |||
| + | <math>x^8 = \frac{4}{3}</math> | ||
| + | |||
| + | <math>x = 2^{\frac{1}{4}}3^\frac{-1}{8}</math> | ||
| + | |||
| + | The answer is then 14. | ||
{{solution}} | {{solution}} | ||
Revision as of 15:49, 31 August 2020
Problem 4
There exists a unique positive real number 
satisfying ![\[x=\sqrt{x^2+\frac{1}{x^2}} - \sqrt{x^2-\frac{1}{x^2}}.\]](http://latex.artofproblemsolving.com/c/e/5/ce549c814c450bcb941089630021b4b0dd539a02.png)
Given that can be written in the form 
for integers 
with 
, find 
.
Solution
We simply use the best technique of easy bash.
The answer is then 14. This problem needs a solution. If you have a solution for it, please help us out by adding it.
| 2020 CIME I (Problems • Answer Key • Resources) | ||
| Preceded by Problem 3 |
Followed by Problem 5 | |
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| All CIME Problems and Solutions | ||
The problems on this page are copyrighted by the MAC's Christmas Mathematics Competitions. 
