Year Book

Winners 2016

Gold Award
TopicOn the Summation of Fractional Parts and its Application
Team MembersSun Kai Leung
TeacherMr. Yiu Chung Leung
SchoolBishop Hall Jubilee School
AbstractThe 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 non-linear 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
TopicOn the Iterated Circumcentres Conjecture and its Variants
Team MembersTsz Fung Yu, Tsz Chun Wong, Janice Ling
TeacherMr. Ho Fung Lee
SchoolPui Ching Middle School
AbstractWe study the Iterated Circumcentres Conjecture proposed by Goddyn in 2007: Let P_1, P_2, P_3, ... be a sequence of points in the d-dimension space such that for every i>=d+2, the points P_(i-1), P_(i-2), ..., P_(i-d-1) 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
TopicA Geometric Approach to the Second Non-trivial Case of the Erdös-Szekeres Conjecture
Team MembersWai Chung Cheng
TeacherMs. Dora Po Ki Yeung
SchoolDiocesan Girls’ School
AbstractThe Erdös-Szekeres conjecture, developed from the famous Happy-Ending Problem, hypothesizes on the number of points in general position needed on a plane to guarantee the existence of a convex n-gon. 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)
TopicCongruences of Solutions of the Pell's Equation
Team MembersMan Yi Kwok
TeacherMr. Kim Fung Lee
SchoolBaptist Lui Ming Choi Secondary School
AbstractIn 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.
 
TopicA 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 MembersChit Yuen Lam, Christy Sze Wai Kok, King Chun Chan, Hin Tung Chung
TeacherMr. Tat Cheong Wong
SchoolG.T. (Ellen Yeung) College
AbstractIn 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.
 
TopicThe Generalized Tower of Hanoi Problem
Team MembersHoi Wai Yu
TeacherMs. Mee Lin Luk
SchoolLa Salle College
AbstractIn this paper, we look into a generalized version of the well-known 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.
 
TopicOn Hilbert Functions and Positive-definite Quadratic Forms
Team MembersChak Him Au
TeacherMr. Yan Ching Chan
SchoolP.L.K. Centenary Li Shiu Chung Memorial College
AbstractIn 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.
 
TopicTriples of Sums of Two Squares
Team MembersKin Ip Mong, Chun Ming Lai, Siu Hong Mak
TeacherMr. Chun Yu Kwong
SchoolWong Shiu Chi Secondary School
AbstractIn 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=(a-l)^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.
 
Hang Lung Mathematics Awards c/o Department of Mathematics
The Chinese University of Hong Kong, Shatin, NT, Hong Kong
exec-hanglung@math.cuhk.edu.hk