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?

Bina-ring

Stage: 5 Challenge Level:

There is more than one way to prove this result.

On way is to define $(1 + \sqrt2)^n = A_n + B_n\sqrt2$ and then find and use $A_{n+1} + B_{n+1}\sqrt 2$.