Year Book 2018

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

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 2018 Hang Lung Mathematics Awards comprises of the following members:

Chair: Professor Zhouping Xin 2004 Morningside Medal of Mathematics Gold Medalist
The Chinese University of Hong Kong
Professor Tony F. Chan, JP King Abdullah University of Science and Technology
Professor Shiu Yuen Cheng Tsinghua University
Professor Lawrence C. Evans University of California, Berkeley
Professor David Gabai Princeton University
Professor Brendan Hasset Brown University
Professor Bong Lian Brandeis University
Professor Rafe Mazzeo Stanford University
Professor Ngaiming Mok The University of Hong Kong
Professor Viet Trung Ngo Vietnam Academy of Science and Technology
Professor Raman Parimala Emory University
Professor Hyam Rubinstein University of Melbourne
Professor Tom Yau Heng Wan The Chinese University of Hong Kong
Professor Po Lam Yung The Chinese University of Hong Kong
Professor Jun Zou The Chinese University of Hong Kong

Screening Panel

The members of the Screening Panel of the 2018 Hang Lung Mathematics Awards are:

Chair: Professor Tom Yau Heng Wan The Chinese University of Hong Kong
Dr. Ping Shun Chan The Chinese University of Hong Kong
Dr. Man Chuen Cheng The Chinese University of Hong Kong
Dr. Charles Chun Che Li The Chinese University of Hong Kong
Dr. Mark Jingjing Xiao The Chinese University of Hong Kong
Professor Po Lam Yung The Chinese University of Hong Kong

Steering Committee

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 2018 Hang Lung Mathematics Awards comprises of the following members:

Chair: Professor Shiu Yuen Cheng 2007 Chern Prize Recipient
Tsinghua University
Professor Thomas Kwok Keung Au The Chinese University of Hong Kong
Professor Kwok Wai Chan The Chinese University of Hong Kong
Professor Raymond Chan The Chinese University of Hong Kong
Professor Tony F. Chan, JP King Abdullah University of Science and Technology
Professor Wing Sum Cheung The University of Hong Kong
Mr. Siu Leung Ma, BBS, MH Fung Kai Public School
Professor Ngaiming Mok The University of Hong Kong
Professor Tai Kai Ng Hong Kong Academy of Gifted Education
Professor Yang Wang The Hong Kong University of Science and Technology
Ms. Susan Wong Hang Lung Properties Limited
Professor Zhouping Xin The Chinese University of Hong Kong
Dr. Chee Tim Yip Princeton (Shenzhen) International School

Executive Committee

The members of the Executive Committee of the 2018 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
Professor Kwok Wai Chan The Chinese University of Hong Kong
Professor Ka Luen Cheung The Education University of Hong Kong
Dr. Chi Hin Lau The Chinese University of Hong Kong
Secretariat: Ms. Aggie So Ching Law*
Ms. Judy Wing Lam Chu
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. Law participated up to October 2017.

Winners of the 2018 Hang Lung Mathematics Awards

GOLD 

Topic On the Trapezoidal Peg Problem among Convex Curves
Team Member Zhiyuan Bai
Teacher Mr. Pui Keung Law
School La Salle College
Abstract The Trapezoidal Peg Problem, as one of the generalizations of the famous Square Peg Problem, asks when a prescribed trapezoid can be inscribed in a given Jordan curve. We investigated a possible approach towards the problem by first weakening the similarity condition, in which we have shown that for any trapezoid, some classes of convex curves can actually inscribe, up to two kinds of weaker forms of similarity, infinitely many trapezoids. Our main theorem further analyzed the properties of one of these infinite family of trapezoids, and showed that any given trapezoid can be uniquely inscribed in any strictly convex C1 curve, which we named ‘oval’, up to only translation and a kind of transformation, which we called ‘stretching’, but without rotation, and the resulting trapezoid moves continuously when the given trapezoid rotates. Through this, we consequently obtained a necessary and sufficient condition for an oval to inscribe an arbitrary trapezoid up to similarity, which could be taken as an answer to the problem among ovals. Some other variations are also discussed.

SILVER

Topic Containing Geometric Objects with Random Inscribed Triangles in a Circle
Team Member Tsz Hin Chan
Teacher Mr. Ho Fung Lee
School Pui Ching Middle School
Abstract In this paper, we aim to investigate the probability of an inscribed triangle in a given circle containing certain geometric objects. Our paper is motivated by a Putnam problem in 1992. We study three generalizations in containing: (i) an arbitrary point, (ii) an arbitrary line segment which lies on a diameter, and (iii) a concentric circle. For the case of an arbitrary point, a closed form expressed by the Spence’s function is obtained. For the case of an arbitrary line segment, we use numerical approximations to calculate the probability, namely the trapezoidal rule and the Monte Carlo integration. For the case of a concentric circle, we successfully find an explicit formula that depends on the radius of the concentric circle.

BRONZE

Topic On the Divisibility of Catalan Numbers
Team Member Tsz Chung Li
Teacher Dr. Kit Wing Yu
School United Christian College
Abstract In this paper, we propound an efficacious method to derive the p-adic valuation of the Catalan number by analyzing the properties of the coefficients in the base p-expansion of n. We unearth a new connection between those coefficients and the p-adic valuation of the Catalan number. In fact, we have discovered that the highest power of p dividing the Catalan number is relevant to the number of digits greater than or equal to half of p + 1, the nature of distribution of digits equal to half of p – 1, and the frequency of carries when 1 is added to n. Meanwhile, we remark that the method we apply is more natural than the current way used by Alter and Kubota, which is quite artificial. Applications, examples of our new formula, and details about Catalan numbers are also included in this paper.

HONORABLE MENTIONS

(Arranged by school name and research topic in alphabetical order)

Topic Doing Indefinite Integrals without Integration
Team Member Chun Szeto
Teacher Mr. Alexander Kin Chit O
School G.T. (Ellen Yeung) College
Abstract Residue Theorem has been frequently used to tackle certain complicated definite integrals. However, it is never applied to indefinite integrals. Therefore, in this report, Residue Theorem and some small tricks are applied to find antiderivatives.

There are mainly three interesting results:

  • Antiderivatives can be found without integration: antiderivatives can be represented by residues, while calculation of residues requires no knowledge of integration.
  • A universal functional form of antiderivatives can be obtained: the antiderivative obtained by this method has a functional form that converges wherever it should converge. The functional form has the largest possible region of convergence on the complex plane.
  • As a weak tool for analytic continuation: since the universal functional form of an antiderivative is obtained, differentiating yields a universal functional form of the integrand.
Topic A Generalization of the Gauss Sum
Team Member Ho Leung Fong
Teacher Mr. Hing Pan Fong
School Hoi Ping Chamber of Commerce Secondary School
Abstract This essay will analyze a function that is a generalization of the Gauss sum. The function happens to be closely related to the cycle index of the symmetric group, which will also be analyzed. Some properties of the Gauss sum will be generalized. A number-theoretic inequality is also obtained.
Topic A Markov Model of the Busy Footbridge Problem
Team Members Lok Kan Yuen, Omega Nok To Tong, Ethan Lok Kan Tsang
Teacher Mr. Ho Fung Lee
School Pui Ching Middle School
Abstract The central problem we are investigating is based on a problem from the 2018 Singapore International Mathematics Challenge. It is about a mathematical model of the probabilities that the people on a footbridge from two sides meet. In our paper, we generalize the contest problem in various cases. We develop a Markov model, and then formulate a transition matrix to solve the generalized problem. Also, we define an expansion rule of the transition matrices to reduce the time complexity to compute. Furthermore, we propose a new topic on the expected number of collisions. We tackle the problem by performing Jordan decomposition. Lastly, we optimize the method of finding eigenvalues by observing the recursive relationships in transition matrices.
Topic Investigation on Mordell’s Equation
Team Member Tin Wai Lau
Teacher Mr. Ho Fung Lee
School Pui Ching Middle School
Abstract This paper aims to investigate the integral solutions of the Mordell’s Equation y2 = x3 + k for a particular class of integers k. We employ some classical approaches, i.e. factorization in number fields and quadratic reciprocity. When k = p2 for certain primes p, we can determine the set of solutions. The equation for two other classes of integers k are also solved in this paper.
Topic Old and New Generalizations of Classical Triangle Centres to Tetrahedra
Team Members Trevor Kai Hei Cheung, Hon Ching Ko
Teacher Mr. Pak Leong Cheung
School St. Paul’s Co-educational College
Abstract The classical triangle centres, namely centroid, circumcentre, incentre, excentre, or thocentre, and Monge point, will be generalized to tetrahedra in a unified approach as points of concurrence of special lines. Our line characterization approach will also enable us to create new tetrahedron centres lying on the Euler lines, which will be a family with nice geometry including Monge point and the twelve-point centre.

Two tetrahedron centres generalizing orthocentre of triangles from new perspectives will be constructed through introducing antimedial tetrahedra, tangential tetrahedra, and a new kind of orthic tetrahedra. The first one, defined as the circumcentre of the antimedial tetrahedron of a tetrahedron, will be proven to lie on the Euler line. The second one, defined as the incentre or a suitable excentre of the new orthic tetrahedron of a tetrahedron, will be discovered to be collinear with its circumcentre and twenty-fifth Kimberling centre X25. Surprisingly, these two differently motivated geometric generalizations turn out to have analogous algebraic representations.

A clear definition of tetrahedron cent res, as a generalization of triangle centres to tetrahedra, will be coined to set up a framework for studying the analogies between geometries of triangles and tetrahedra.

The fundamental properties of tetrahedron centres will be studied.

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2018 Award Ceremony Video

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

Welcome Remarks

Mr. Weber Lo
Chief Executive Officer, Hang Lung Properties Limited

Professor Zhouping Xin
Chairman, 2018 Hang Lung Mathematics Awards Scientific Committee
Professor Shiu-Yuen Cheng
Chairman, 2018 Hang Lung Mathematics Awards Steering Committee

Opening Remarks

The Hon Mrs. Carrie Lam, GBM, GBS
The Chief Executive
Hong Kong Special Administrative Region

Dialogue: My Mathematical Journey to Neuroscience

Awards Announcement and Presentation

Gold Award
La Salle College

Silver Award
Pui Ching Middle School

Bronze Award
United Christian College

Honorable Mention
G.T. (Ellen Yeung) College

Honorable Mention
Hoi Ping Chamber of Commerce Secondary School

Honorable Mention
Pui Ching Middle School

Honorable Mention
Pui Ching Middle School

Honorable Mention
St. Paul’s Co-educational College

Presentation of Souvenirs

Mr. Weber Lo
Chief Executive Officer, Hang Lung Properties Limited

2018 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 2018 Hang Lung Mathematics Awards

(Arranged by school name and research topic in alphabetical order)

Solving Cubic Pell’s Equation by Bifurcating Continued Fraction  
Team Member: Ka Lam Wong
Teacher: Mr. Yiu Chung Leung
School: Bishop Hall Jubilee School
Heine-Cantor Theorem, Lebesgue’s Number and Compactness
Team Member: Kin Fung Wong
Teacher: Dr. Chi Kwan Leung
School: Cognitio College (Kowloon)
Doing Indefinite Integrals without Integration
Team Member: Chun Szeto
Teacher: Mr. Alexander Kin Chit O
School: G.T. (Ellen Yeung) College
Finding the Expected Number of Random Reversals to Sort a Permutation Using Matrix Equation for Application in Genetics 
Team Members: Man Hon Fan, Kwok Yan Lo
Teacher: Mr. Ho Cheung Lai
School: HKUGA College
A Generalization of the Gauss Sum
Team Member: Ho Leung Fong
Teacher: Mr. Hing Pan Fong
School: Hoi Ping Chamber of Commerce Secondary School
On the Trapezoidal Peg Problem among Convex Curves
Team Member: Zhiyuan Bai
Teacher: Mr. Pui Keung Law
School: La Salle College
A Markov Model of the Busy Footbridge Problem
Team Members: Lok Kan Yuen, Omega Nok To Tong, Ethan Lok Kan Tsang
Teacher: Mr. Ho Fung Lee
School: Pui Ching Middle School
Containing Geometric Objects with Random Inscribed Triangles in a Circle
Team Member: Tsz Hin Chan
Teacher: Mr. Ho Fung Lee
School: Pui Ching Middle School
From Close Match Problem to the Generation of Identities of Binomial Coefficients and Trigonometric Terms
Team Members: Chi Hang Tam, Joshua Pui Sang Cheung, Chi Ki Ngai
Teacher: Mr. Ho Fung Lee
School: Pui Ching Middle School
Investigation on Mordell’s Equation
Team Member: Tin Wai Lau
Teacher: Mr. Ho Fung Lee
School: Pui Ching Middle School
Graphical Approach to the Lonely Runner Conjecture
Team Members: Ka Ho Mok, Sum Kiu Law, Ho Lam Wan, Cheuk Yin Lee
Teacher: Mr. Pun Sin Chan
School: S.K.H. Tsang Shiu Tim Secondary School
Inradius Numbers and the Investigations on the Inradius Number Diophantine Equation
Team Members: Long Kiu O, Hei Long Young
Teacher: Ms. Jasmine Sze Lai Ku
School: St. Joseph’s College
Old and New Generalizations of Classical Triangle Centres to Tetrahedra
Team Members: Trevor Kai Hei Cheung, Hon Ching Ko
Teacher: Mr. Pak Leong Cheung
School: St. Paul’s Co-educational College
On the Divisibility of Catalan Numbers
Team Member: Tsz Chung Li
Teacher: Dr. Kit Wing Yu
School: United Christian College
On Length Preserving Curve Flow to Isoperimetric Inequality
Team Member: Man Hei Law
Teacher: Mr. Ching Ping Lam
School: Wah Yan College, Hong Kong