WINNERS 2008 
Gold 
Topic:  Isoareal and Isoperimetric Deformation of Curves 
Members:  Kwok Chung Li, Chi Fai Ng 
Teacher:  Mr. Wing Kay Chang 
School:  Shatin Tsung Tsin Secondary School 
Abstract:  In the report, we want to answer the following question: how to deform a curve such that the rate of change of perimeter is maximum while the area and the total kinetic energy are fixed? First we work on isosceles triangle as a trial. Then we study smooth simple closed curve and obtain information about the velocity of each point of the curve and its relation to the curvature. We also consider the applications of the results and the velocity for the dual isoperimetric problem.

Silver 
Topic:  Sufficient Condition of WeightBalance Tree 
Members:  Chi Yeung Lam, Yin Tat Lee 
Teacher:  Mr. Chun Kit Ho 
School:  The Methodist Church Hong Kong Wesley College 
Abstract:  Huffman's coding provides a method to generate a weightbalance tree, but it is not generating progressively. In other words, we cannot have meaningful output if we terminate the algorithm halfway in order to save time. For this purpose, we want to design an alternative algorithm, therefore this paper aims at finding out a sufficient condition of being a weightbalance tree. In this paper, we have found out the sufficient condition. Besides, as the solution of building a weightbalance tree can be applied to solving other problems, we abstract the problem and discuss it in the manner of graph theory. The applications are also covered.

Bronze 
Topic:  Fermat Point Extension ¡V Locus, Location, and Local Use 
Members:  Fung Ming Ng, Chi Chung Wan, Wai Kwun Kung, Ka Chun Hong 
Teacher:  Mr. Yiu Kwong Lau 
School:  Sheng Kung Hui Tsang Shiu Tim Secondary School

Abstract:  Published in 1659, the solutions of Fermat Point problem help people find out the point at which the sum of distances to 3 fixed points in the plane is minimized. In this paper, we are going to further discuss when the number of fixed points is greater than 3, the relationship between the fixed points and the point minimizing the sum of distances to more than three given points. Also, we would like to find out if there exists a way such that the location of point minimizing the sum of distances to more than three given points can be determined just by compass and ruler, or approximated by mathematical methods.

4 Honorable Mentions 
(arranged in alphabetical order of school name) 
Topic:  A Cursory Disproof of Euler's Conjecture Concerning GraecoLatin Squares by means of Construction 
Members:  Jun Hou Fung 
Teacher:  Mr. Jonathan Hamilton 
School:  Canadian International School of Hong Kong

Abstract:  In this report, our team has explored a mathematical structure commonly known as GraecoLatin squares. Although we do give a broad scope of this field, we are particularly focused on one aspect: Euler's Conjecture. According to this conjecture, there are certain types of GraecoLatin squares that do not exist. In this report, we disprove this conjecture by demonstrating a means to construct an infinite number of these socalled nonexistent squares. This branch of mathematics is highly related to group theory, combinatorics, and transversal design; therefore, we will also provide a brief overview of these topics throughout this report.

Topic:  Equidecomposition Problem 
Members:  Cheuk Ting Li 
Teacher:  Miss Mee Lin Luk 
School:  La Salle College 
Abstract:  The equidecomposition problem is to divide a shape into pieces, and then use the pieces to form another shape. In this project, we are going to investigate the conditions under which a given shape can be broken down and combined into another specified shape. The classical problem on the equidecomposability of polygons has already been solved by mathematicians. We start by presenting the proof of the classical problem, which is the keystone of this research. Then the problem is generalized to weighted shapes, shape with curves, etc. Some interesting new results are obtained.

Topic:  3n+1 Conjecture 
Members:  Shun Yip 
Teacher:  Mr. Chi Keung Lai 
School:  Shatin Pui Ying College 
Abstract:  The aim of our project is to investigate the 3n + 1 conjecture. It is very hard to give a general path for each natural number to arrive at 1. So we investigate its negation i.e. there exists a natural number k with no path to 1. There are two possibilities: either k takes a path which is a cycle to itself after n steps or its path is increasing indefinitely. These two possibilities lead us to study prenumbers of any odd natural number and the number of peaks of paths. In the project, several interesting results were obtained by studying backward paths, number of peaks and cycles or forward paths.

Topic:  Geometric Construction  Area Trisection of a Circle 
Members:  Shun On Hui, Kin Ho Lo, Kai Ming To, Maureen Tsz Yan Ho , Wai Hang Ng 
Teacher:  Mr. Wai Hung Ho 
School:  Tsuen Wan Public Ho Chuen Yiu Memorial College 
Abstract:  When dividing a cake of circle shape into equal parts, it is quite easy to divide it from the centre. However, if we need to divide it from its edge, how can we accomplish this task accurately?
This report aims to find a method to divide the area of a circle into 3 equal parts with two straight lines by Euclidean construction, i.e. the construction with compass and straightedge only. However, we were aware that it is difficult, if not impossible, to find the exact method of construction. Therefore, we try to find some methods to divide the area of circle approximately into three equal parts. In this report, we have three analytic approaches: by Lagrange Interpolating Polynomial, by infinite series of sine function and by method of bisection. Then, we will discuss three methods of construction, which are: inscribing a regular polygon with a large number of sides, inscribing a regular polygon with a small number of sides and bisecting the slope. At last, we will give a comparison of these three methods


