The first issue of *Parabola* appeared in July 1964.

Michael A. B. Deakin

One of the most significant advances in the entire history of mathematics was the development of Calculus in the latter years of the 17th Century.

David Angell

Fermat’s Last Theorem is one of the most famous problems in mathematics. Its origin can be traced back to the work of the Greek mathematician Diophantus

Gerardo Sozio

Trigonometry, as it is taught in high school using the trigonometric ratios, has an interesting history.

N. P. Singh

In a recent article in *Parabola* (Vol.37, No.3, 2001, 13–16), Anard Kumar proved that the dynamical system defined by summing the squares of the digits of a positive integer repeatedly leads to a fixed point solution.

**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.

**Q1161.** Find all values of x (real number) satisfying (x−1)/(2004) + (x−3)/(2002) + (x−5)/(2000) + ⋯ + (x−2003)/2 = ⋯

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.

J. Guest

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.

Garth Lien

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.

Michael A. B. Deakin

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

x + 2y + z = 1

−2x + λy − 2z = -2

2x + 6y + 2λz = 3

**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.

This issue of Parabola celebrates some of the mathematics of George and Esther Szekeres.

George Szekeres

In the October 1964 issue of *Parabola*, the article on the Four Colour Problem called your attention to the existence of numerous unsolved mathematical problems which can be stated in quite simple non-technical terms.

George Szekeres

The theory of combinatorial configurations abounds in unsolved problems, some of which can be stated in simple non-technical terms. One of the most famous among these is the following problem due to the French mathematician, J. Hadamard.

Catherine Greenhill

How many people must attend a party before you are sure that you can find either three people who all know each other, or three people who do not know each other? This is a question in an area called Ramsey Theory.

Catherine Greenhill

George Szekeres made many contributions to various areas of mathematics. In combinatorics, one of his most important contributions was to ask a question which we still don’t know how to answer.

Yi Liu

We need counting in our daily life. Collecting cash from the supermarket, checking the bill at a restaurant, counting the number of place settings at a dinner party.... This sort of counting is pretty easy because we can count one by one.

Michael Deakin

In my last column, I showed the way in which the solution of cubic equations led to the introduction of complex numbers. Here I will concentrate on the solution of cubic equations themselves.

**Q1191.** In the triangle ABC,M is the midpoint of BC. Points X and AB and Y on AC are such that XY ‖ BC. Show that BY and CX intersect at a point P on AM.

**Q1181.** Consider the following set of linear equations x + 2y + z = 1

−2x + λy − 2z = -2

2x + 6y + 2λz = 3