Volume 41
, Issue 2

2005

Welcome to this issue of Parabola which contains, in addition to regular articles and problems, the list of prizewinners and complete solutions for the UNSW School of Mathematics Competition in 2005.

When I was a young mathematics student, I often wondered whether there was an easy way of checking determinants. By recently studying the checking of contractants I found there is a fairly easy way to accomplish this.

We begin by looking at the definition of an oval, or as it is more formally known, an ellipse. An ellipse is the set of all points $(x, y)$ in the plane such that the sum of the distances from $(x, y)$ to two fixed points is some constant.

In my previous column, I outlined the story of the most recent extension of the number system, so that it expanded to include "infinitesimals", numbers smaller than any of our familiar real numbers, and yet not the same as zero.

Problem 1. On the Island of New Monia, the natives made totem poles out of square-heads and long-heads (which were twice as tall as square-heads). The square-heads were made of mahogany, while the long-heads were made of ebony or sandalwood. The heads were stacked upright.

Prize Winners – Junior Division

First Prize
Vinoth Nandakumar               James Ruse Agricultural High School

Q1181. Consider the following set of linear equations
\begin{eqnarray*}
x+2y+z&=&1\\
-2x+\lambda y-2z&=&-2\\
2x +6y+ 2\lambda z&=&3
\end{eqnarray*}

Q1171. The first digit of a $6$-digit number is $1$. If the $1$ is shifted to the other end, the new number is $3$ times the original number. Find this number.