### 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?

### Absurdity Again

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

### Strange Rectangle 2

Find the exact values of some trig. ratios from this rectangle in which a cyclic quadrilateral cuts off four right angled triangles.

# Ab Surd Ity

##### Stage: 5 Challenge Level:
Why do this problem?

The problem provides good practice in the manipuation of surds and in algebra (involving the expansions of $(p+q)^2$ and $(p+q)^3$, the difference of two squares and the use of the Remainder Theorem to factorise a cubic equation). If care is taken with the algebra the result comes out in a satisfyingly neat way. The question looks complicated but it turns out to be simple.

Possible approach
Although this is a longer, two part, question, the Hint gives sufficient guidance for this to be set to a class to work on independently.

Key questions

If 'extra' solutions are introduced by squaring or cubing, how do you decide which are the correct solutions?

This is how the problem was used by Peter Thomas, a Sixth Form College teacher:

I was absent at a meeting and set the class work to consolidate topics taught the previous lesson. The work was routine exercises from a textbook (Emanuel and Wood) which I encouraged them to approach selectively (what I called 'bread and butter' with some specific questions as a 'doggy bag' for homework).

Alongside this I set them the four nrich problems as 'cake' with the instruction to tackle at least one.

Ab Surd Ity (this problem)