Volume 43
, Issue 2

2007

Each year Parabola celebrates the winners of the annual UNSW School of Mathematics Competition and each year there are standout performances from many Sydney Schools including James Ruse Agricultural High School and Sydney Boys High.

If you have already begun studying complex numbers at school, you have probably been taught that it makes no sense to say that one complex number is less than another. However, there are various plausible ways in which we might attempt to do just that. Is it really true that none of them works?

Numerical integration enables approximations to be found for $\int^b_a f(x) dx$ where the integral for $f(x)$ cannot be written in terms of elementary functions.  Integration is the process of measuring the `signed area' between the curve $y=f(x)$ and the $x$ axis in between the end points $x=a$ and $x=b.$

Many of you reading this article will be aware of the problems the world faces over energy supply: can we rely on fossil fuels (oil and coal), or should we look again at nuclear power? By nuclear power we usually mean fission, the break up of heavy atoms (uranium) to lighter ones.

I could kick myself! I have to begin this column by confessing to a stupid mistake. Here is what happened.  I was surfing the net when I came upon a website that held great interest for me.

Problem 1. You are given nine square tiles, with sides of lengths $1, 4, 7, 8, 9, 10, 14,15$ and $18$ units, respectively. They can be used to tile a rectangle without gaps or overlaps. Find the lengths of the sides of the rectangle, and show how to arrange the tiles.

Competition Winners – Senior Division

Q1241 Show that Simpson's Elementary Rule
$$\int^b_a f (x) dx \approx \left(\frac{b-a}{6}\right)\left( f(a) + 4f\left(\frac{a+b}{2}\right) + f(b)\right)$$
is an exact equality for the quadratic function
$$f(x) = Ax^2 + Bx + C.$$

Q1231 Given $a>0$, prove that
$$\underbrace{\sqrt{a+\sqrt{a+\cdots+\sqrt{a}}}}_{n \text{ times}} < \frac{1+\sqrt{4a+1}}{2}.$$