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Baby Circle

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?

Ab Surd Ity

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

Absurdity Again

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.