

Year Book
Winners 2016
Gold Award
Topic  On the Summation of Fractional Parts and its Application 
Team Members  Sun Kai Leung 
Teacher  Mr. Yiu Chung Leung 
School  Bishop Hall Jubilee School 
Abstract  The summation of fractional parts is an old topic in number theory since the time of G.H. Hardy and J.E. Littlewood. In section 1, we estimate the fractional part sum of certain nonlinear functions, which can be applied to refine an existing bound of the discrepancy. In section 2, we continue to make use of the sum in order to study the distribution of quadratic residues and 'relatively prime numbers' modulo integers. 
Silver Award
Topic  On the Iterated Circumcentres Conjecture and its Variants 
Team Members  Tsz Fung Yu, Tsz Chun Wong, Janice Ling 
Teacher  Mr. Ho Fung Lee 
School  Pui Ching Middle School 
Abstract  We study the Iterated Circumcentres Conjecture proposed by Goddyn in 2007: Let P_1, P_2, P_3, ... be a sequence of points in the ddimension space such that for every i>=d+2, the points P_(i1), P_(i2), ..., P_(id1) are distinct, lie on a unique sphere, and further, P_i is the center of this sphere. If this sequence is periodic, then its period must be 2d+4.
We focus on cases of d=2 and d=3 and obtain partial results on the conjecture. We also study the sequence and prove its geometrical properties. Furthermore, we propose and look into several variants of the conjecture, namely the Skipped Iterated Circumcentres Conjecture and the Spherical Iterated Circumcentres Conjecture. 
Bronze Award
Topic  A Geometric Approach to the Second Nontrivial Case of the ErdösSzekeres Conjecture 
Team Members  Wai Chung Cheng 
Teacher  Ms. Dora Po Ki Yeung 
School  Diocesan Girls’ School 
Abstract  The ErdösSzekeres conjecture, developed from the famous HappyEnding Problem, hypothesizes on the number of points in general position needed on a plane to guarantee the existence of a convex ngon. The research conducted aims to examine geometric characteristics of different constructions of points in general position, organized by number of points forming the convex hull of the set. This paper has explored the case of pentagons, reestablishing the previously proven result of the case using a geometrical approach in contrast to the combinatorial approaches generally adopted when exploring this problem. This paper also proves that the lower bound to the conjecture is not sharp under certain circumstances, an aspect never explored in the past. 
Honorable Mentions (arranged by school name in alphabetical order)
Topic  Congruences of Solutions of the Pell's Equation 
Team Members  Man Yi Kwok 
Teacher  Mr. Kim Fung Lee 
School  Baptist Lui Ming Choi Secondary School 
Abstract  In this research, we are interested in how the solutions of the famous Pell's equation look like. It is well known that the solutions of the Pell's equation are generated by the fundamental solution of the equation, which could be represented by a set of recursive equations. Therefore, we would like to explore the characteristics of such recurrence sequences and tell the relationship between the cycle length of the congruence modulo a number and divisibility of the terms. 

Topic  A Synthetic Approach on Studying the Mysterious Right Kite and its Applications on Cryptography in related to Poincaré Disk Model in the Views of Euclid Geometry 
Team Members  Chit Yuen Lam, Christy Sze Wai Kok, King Chun Chan, Hin Tung Chung 
Teacher  Mr. Tat Cheong Wong 
School  G.T. (Ellen Yeung) College 
Abstract  In this study, it gives a synthetic approach to the quadrilateral "Kite" and right kite. It mainly based on the definitions, postulates (axioms), propositions (theorem and constructions) from the Euclid's Elements, which is known as one of the most successful and influential mathematical textbook attributed to the ancient Greek mathematician Euclid in Alexandria, Ptolemaic Egypt, c. 300 BC.
Linked with the definitions of "Right Kite" and the lines which are that meet the boundary of a said circle orthogonally described in the Poincaré Disk Model, we attempt to combine it in a mathematical task namely "Cryptography". The application of Poincaré Disk Model will be acted as a bridge to form a single key for encryption and decryption. Even the single common trick we use, it leads to infinite possibilities by experiencing various and distinct mathematical skills in cryptography.
Last but not least, we would like to dedicate to the publish of Euclid's Element and the discovery of Euclid's Geometry so that we can admire the Beauty of Mathematics. Our ultimate goal is to lay the new insight into some of the most enjoyable and fascinating aspects of geometry regarding to the most unaware quadrilateral, Kite. 

Topic  The Generalized Tower of Hanoi Problem 
Team Members  Hoi Wai Yu 
Teacher  Ms. Mee Lin Luk 
School  La Salle College 
Abstract  In this paper, we look into a generalized version of the wellknown Tower of Hanoi problem. We will investigate the shortest methods of traversing between any two valid configurations of discs in the standard problem, as well as in some variants. 

Topic  On Hilbert Functions and Positivedefinite Quadratic Forms 
Team Members  Chak Him Au 
Teacher  Mr. Yan Ching Chan 
School  P.L.K. Centenary Li Shiu Chung Memorial College 
Abstract  In this project, we give an explicit construction of positive definite quadratic forms of arbitrary dimension by using a family of real analytic functions whose coefficients in their Taylor expansions are strictly positive. We also prove a variant result that allows the construction if the number of positive coefficients has a positive upper density. 

Topic  Triples of Sums of Two Squares 
Team Members  Kin Ip Mong, Chun Ming Lai, Siu Hong Mak 
Teacher  Mr. Chun Yu Kwong 
School  Wong Shiu Chi Secondary School 
Abstract  In 1903, an anonymous reader submitted a question to Mathematical Questions in The Educational Times
Find all consecutive triples of sums of two squares. J.E. Littlewood later posed a question on whether in general there exist infinitely many triples n, n+h, n+k that are simultaneously sums of two squares? By solving the equation a^2+2=(al)^2+b^2, we give all consecutive triples of sums of two squares such that the first number is a perfect square. This method is generalised to solve Littlewood's problem for the case when h is a perfect square.
We also prove that there are infinitely many pairs of consecutive triples of sums of two squares such that the first numbers of the two triples differ by 8. 


