### Circles Ad Infinitum

A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?

### Climbing Powers

$2\wedge 3\wedge 4$ could be $(2^3)^4$ or $2^{(3^4)}$. Does it make any difference? For both definitions, which is bigger: $r\wedge r\wedge r\wedge r\dots$ where the powers of $r$ go on for ever, or $(r^r)^r$, where $r$ is $\sqrt{2}$?

A quadrilateral changes shape with the edge lengths constant. Show the scalar product of the diagonals is constant. If the diagonals are perpendicular in one position are they always perpendicular?

# Unit Interval

##### Stage: 5 Short Challenge Level:

Take any two numbers between $0$ and $1$. Prove that the sum of the numbers is always less than one plus their product. That is, if $0< x< 1$ and $0< y< 1$ then prove $$x+y< 1+xy$$
Did you know ... ?

Pure inequalities such as this one are often used in the analysis of far more difficult mathematics problems: whilst the inequalities might be simple to prove in themselves, they can be surprisingly useful as tools.