### 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

##### Stage: 4 and 5 Challenge Level:

Correct solutions were recieved from Charlene from Brunei, Kiang from Singapore, Andre from Bucharest and Jing of Madras College. Well done to all of you. Charlene's solution is given below. Not as hard as it at first looks! The moral is not to be put off by appearances.

The numerator and denominator of the terms can be multiplied to give a more convenient value as follows:

\begin{eqnarray}&&\frac{1 \times(\sqrt{1} - \sqrt{2})}{(\sqrt{1} + \sqrt{2})(\sqrt{1} - \sqrt{2})} + \frac{1 \times (\sqrt{2} - \sqrt{3})}{(\sqrt{2} + \sqrt{3})(\sqrt{2} - \sqrt{3})} + \dots + \frac{1 \times (\sqrt{99} - \sqrt{100})}{(\sqrt{99} + \sqrt{100})(\sqrt{99} - \sqrt{100})}\\ &=& \frac{(\sqrt{1} - \sqrt{2})}{-1} + \frac{(\sqrt{2} - \sqrt{3})}{-1} + \dots + \frac{(\sqrt{99} - \sqrt{100})}{-1}\\ &=& (-\sqrt{1} + \sqrt{2}) + (-\sqrt{2} + \sqrt{3}) + \dots + (-\sqrt{99} + \sqrt{100}) \\ &=& -\sqrt{1} + \sqrt{100}\\ &=& -1 + 10 \\ &=& 9 \end{eqnarray}