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?

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

The Root of the Problem

Age 14 to 18 Challenge Level:

You may wish to start by exploring using a spreadsheet.
Think about other ways to write $\frac{1}{\sqrt{n}+\sqrt{n+1}}$, particularly ways that have no surds in the denominator.