# Year Book 2016

## Organization

The Scientific Committee and the Steering Committee are the two principal committees of the Hang Lung Mathematics Awards. The Scientific Committee, comprising world-renowned mathematicians, is the academic and adjudicating body of the Awards. The Steering Committee, comprising mathematicians and representatives from different sectors of society, serves as the advisory body.

## Scientific Committee 2016

The Scientific Committee upholds the academic standard and integrity of Hang Lung Mathematics Awards. Its members actively participate in the evaluation of all project reports, determination of the teams that will be invited to the oral defense, and adjudication at the oral defense.

The Screening Panel under the Scientific Committee handles the initial review of each report, supervises the second and final round of the review process, and liaises with all referees and members of the Scientific Committee regarding the project reports.

The Scientific Committee for the 2016 Hang Lung Mathematics Awards comprises of the following members:

 Chair: Professor Shing Tung Yau 1982 Fields Medalist Harvard University The Chinese University of Hong Kong Professor Raymond Hon Fu Chan* The Chinese University of Hong Kong Professor Shiu Yuen Cheng Tsinghua University Professor Dejun Feng The Chinese University of Hong Kong Professor Wee Teck Gan National University of Singapore Professor Wei Ping Li The Hong Kong University of Science and Technology Professor Bong Lian Brandeis University Professor Chang Shou Lin Taiwan University Professor Ngai Ming Mok The University of Hong Kong Professor Ye Tian Chinese Academy of Sciences Professor Tom Yau Heng Wan The Chinese University of Hong Kong Professor Michael Zieve University of Michigan

*Note: Professor Chan was unable to join the Oral Defense and will be represented by Professor Dejun Feng.

## Screening Panel 2016

The members of the Screening Panel of the 2016 Hang Lung Mathematics Awards are:

 Chair: Professor Tom Yau Heng Wan The Chinese University of Hong Kong Dr. Man Chuen Cheng The Chinese University of Hong Kong Dr. Chi Hin Lau The Chinese University of Hong Kong Professor Conan Nai Chung Leung The Chinese University of Hong Kong Dr. Charles Chun Che Li The Chinese University of Hong Kong

## Steering Committee 2016

The Steering Committee serves as an advisory body to Hang Lung Mathematics Awards, and comprises of mathematicians and representatives from different sectors of society, including leading educators and heads of mathematics departments in major Hong Kong universities. It also enlists some members from other Hang Lung Mathematics Awards committees to provide an overall oversight.

The Executive Committee, which reports to the Steering Committee, is responsible for the operation and administration of the competition, as well as managing the Resource Center and acting as the Secretariat for the Awards.

The Steering Committee for the 2016 Hang Lung Mathematics Awards comprises of the following members:

 Chair: Professor Sir James A. Mirrlees 1996 Nobel Laureate in Economics The Chinese University of Hong Kong Professor Thomas Kwok Keung Au The Chinese University of Hong Kong Professor Tony F. Chan The Hong Kong University of Science and Technology Professor Shiu Yuen Cheng Tsinghua University Professor Wing Sum Cheung The University of Hong Kong Professor Ka Sing Lau The Chinese University of Hong Kong Mr. Siu Leung Ma Fung Kai Public School Ms. Michelle Sau Man Mak Hang Lung Properties Limited Professor Tai Kai Ng The Hong Kong University of Science and Technology Professor Zhouping Xin The Chinese University of Hong Kong Dr. Chee Tim Yip Pui Ching Middle School

## Executive Committee 2016

The members of the Executive Committee of the 2016 Hang Lung Mathematics Awards are:

 Chair: Professor Thomas Kwok Keung Au The Chinese University of Hong Kong Dr. Kai Leung Chan The Chinese University of Hong Kong Dr. Ka Luen Cheung The Education University of Hong Kong Dr. Leung Fu Cheung The Chinese University of Hong Kong Secretariat: Ms. Aggie So Ching Law Ms. Konnie Wan Yu Pak* Ms. Serena Wing Hang Yip The Chinese University of Hong Kong The Chinese University of Hong Kong The Chinese University of Hong Kong

*Note: Ms. Pak participated up to August 2015

## Winners of the 2016 Hang Lung Mathematics Awards

### GOLD

 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 (see [3]). Throughout the years, many mathematicians have contributed to the estimation of the sum $$\sum_{n \leq N} \left\{\alpha n\right\}$$ , where α is an irrational number. In Section 2, 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 3, we continue to make use of the sum in order to study the distribution of quadratic residues and ‘relatively prime numbers’ modulo integers.

### SILVER

 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 $$P_1,P_2,P_3,\dotsc$$ be a sequence of points in $$R^d$$ such that for every $$i \geq d + 2$$ the points $$P_i-1,P_i-2,P_i-3,\dotsc,P_i-d-1$$ are distinct, lie on a unique sphere, and further, Pi 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

 Topic A Geometric Approach to the Second Non-trivial Case of the Erdös-Szekeres Conjecture Team Members Wai Chung Cheng Teacher Ms. Dora Po Ki Yeung School Diocesan Girls’ School Abstract The 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.

### (ARRANGED IN ALPHABETICAL ORDER OF SCHOOL NAME)

 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 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. Topic On Hilbert Functions and Positive-definite 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 $$a^2 + 2 = (a – l)^2 + b^2$$, 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 Littlewoods 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.

### 2016 Award Ceremony Video

Young talents were recognized by a Gold, a Silver, a Bronze and five Honorable Mentions.
They received the trophies and certificates from world class scholars.
Many guests shared their joy and honor.

### Presentation of Souvenirs

##### Mr. Ronnie C. ChanChairman, Hang Lung Properties Limited

2016 DEFENSE MEETING VIDEO

Please click on the “Playlist” menu below to select the team’s video you want to watch.

## Finalist Teams Selected for the Oral Defense at the 2016 Hang Lung Mathematics Awards

### (arranged by school name in alphabetical order)

 Congruences of Solutions of the Pell’s Equation Baptist Lui Ming Choi Secondary School On the Summation of Fractional Parts and its Application Bishop Hall Jubilee School A Geometric Approach to the Second Non-trivial Case of the Erdös-Szekeres Conjecture Diocesan Girls’ School 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 G.T. (Ellen Yeung) College On Family of Triangles – from Medians to Concurrent Lines and Angle Bisectors Hong Kong Chinese Women’s Club College The Generalized Tower of Hanoi Problem La Salle College Are Gray Code and Gros Sequence the Solution of Chinese Ring? Maryknoll Fathers’ School On Hilbert Functions and Positive-definite Quadratic Forms P.L.K. Centenary Li Shiu Chung Memorial College On the Iterated Circumcentres Conjecture and its Variants Pui Ching Middle School Rational Distance in Rational-sided Triangles Pui Ching Middle School Voting Power Queen Elizabeth School Triples of Sums of Two Squares Wong Shiu Chi Secondary School