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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?

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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:

Parts (2) and (3) of this problem use the results of previous parts.

In part (4) try some small values of $n$, look for a pattern and make a conjecture about the result you suspect might always be true, then prove your conjecture.