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A small circle fits between two touching circles so that all three circles touch each other and have a common tangent? What is the exact radius of the smallest circle?

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Ab Surd Ity

Find the value of sqrt(2+sqrt3)-sqrt(2-sqrt3)and then of cuberoot(2+sqrt5)+cuberoot(2-sqrt5).

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Absurdity Again

What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?

Fibonacci Fashion

Age 16 to 18 Challenge Level:

$F_n={1\over\sqrt5}(\alpha^n-\beta^n)$ where $\alpha$ and $\beta$ are solutions of the quadratic equation $x^2-x-1=0$ and $\alpha > \beta$ is the explicit formula for the $n$th Fibonacci number as given in this question.

The Golden Ratio is one of the roots of the quadratic equation and this explains the many connections between Fibonacci numbers and the Golden Ratio.

This problem complements the material in the article The Golden Ratio, Fibonacci Numbers and Continued Fractions
For a sequence of, mainly more elementary, problems on these topics see
Golden Mathematics

See this article if you want to know more about the method of proof by mathematical induction.