### Absurdity Again

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

### Ball Bearings

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

### Incircles

The incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles have radii 1 and 2 units respectively. What about triangles with an inradius of 3, 4 or 5 or ...?

# Look Before You Leap

##### Stage: 5 Challenge Level:

If $a+b+c=4$,
$ab+bc+ca=6$
and $abc=3$,

what are the values of:
${1\over a}+ {1\over b }+ {1\over c}$ (think of fractions),

${1\over ab}+ {1\over bc }+ {1\over ca}$

and $a^2 +b^2 + c^2$?

In the diagram the coloured squares have sides of length $a$, $b$ and $c$. Use the areas in the diagram to write down a formula for the expansion of $(a + b + c)^2$ and explain your method.

Using your expansion of $(a + b + c)^2$ to help you, expand $(a + b + c)^3$. Can you explain each term of the expansion using a diagram of a cube where each face has been cut up in a similar way to the square above?