2020 CIME I Problems/Problem 4

Problem 4

There exists a unique positive real number $x$ satisfying $x=\sqrt{x^2+\frac{1}{x^2}} - \sqrt{x^2-\frac{1}{x^2}}.$ Given that $x$ can be written in the form $x=2^\frac{m}{n} \cdot 3^\frac{-p}{q}$ for integers $m, n, p, q$ with $\gcd(m, n) = \gcd(p, q) = 1$ , find $m+n+p+q$ .

Solution

We simply use the best technique of easy bash.

$x^2 = 2x^2 - 2\sqrt{x^4-\frac{1}{x^4}}$

$x^4 = 4x^4-\frac{4}{x^4}$

$x^8 = \frac{4}{3}$

$x = 2^{\frac{1}{4}}3^\frac{-1}{8}$

The answer is then $14$ .

2020 CIME I (Problems • Answer Key • Resources)
Preceded by Problem 3	Followed by Problem 5
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2020 CIME I Problems/Problem 4

Problem 4

Solution

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