# Year Book 2010

## 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 2010

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 2010 Hang Lung Mathematics Awards comprises of the following members:

 Chair: Professor Shing-Tung Yau Harvard University Professor Chong-Qing Cheng Nanjing University Professor Shiu-yuen Cheng The Hong Kong University of Science and Technology Professor John H. Coates University of Cambridge Professor Jean-Marc Fontaine Paris-Sud 11 University Professor Ka-Sing Lau The Chinese University of Hong Kong Professor Eduard Looijenga Universiteit Utrecht Netherland Professor Ngai-Ming Mok The University of Hong Kong Professor Duong H. Phong Columbia University Professor Hyam Rubinstein University of Melbourne Professor Gilbert Strang Massachusetts Institute of Technology Professor Ngo Viet Trung Institute of Mathematics, Vietnam Professor Tom Yau-Heng Wan The Chinese University of Hong Kong Professor Chin-Lung Wang National University of Taiwan

## Screening Panel 2010

The members of the Screening Panel of the 2010 Hang Lung Mathematics Awards are:

 Chair: Professor Tom Yau-Heng Wan The Chinese University of Hong Kong Professor Wing Sum Cheung The University of Hong Kong Professor Conan Nai Chung Leung The Chinese University of Hong Kong Dr. Chi-Hin Lau The Chinese University of Hong Kong

## Steering Committee 2010

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 2010 Hang Lung Mathematics Awards comprises of the following members:

 Chair: Professor Sir James A. Mirrlees 1996 Nobel Laureate in Economics Master, Morningside College, CUHK Professor Thomas Kwok-Keung Au The Chinese University of Hong Kong Professor Tony Chan The Hong Kong University of Science and Technology Professor Shiu-yuen Cheng The Hong Kong University of Science and Technology Professor Wing-Sum Cheung The University of Hong Kong Professor Ka-Sing Lau Chairman, Mathematics Department, CUHK Professor Lo Yang Chinese Academy of Sciences Dr. Stephen Tommis Executive Director, HK Academy for Gifted Education Mr. Siu-Leung Ma CEO, Fung Kai Public Schools Ms Carolina Yip Hang Lung Properties Limited Mr. Chee-Tim Yip Principal, Pui Ching Middle School

## Executive Committee 2010

The members of the Executive Committee of the 2010 Hang Lung Mathematics Awards are:

 Chair: Professor Thomas Kwok-Keung Au The Chinese University of Hong Kong Dr. Ka-Luen Cheung The Hong Kong Institute of Education Dr. Leung-Fu Cheung The Chinese University of Hong Kong Dr. Charles Chun-Che Li The Chinese University of Hong Kong Secretariat: Ms. Mavis Kit-Ying Chan Ms. Serena Wing-Hang Yip The Chinese University of Hong Kong The Chinese University of Hong Kong

## WINNERS of the 2010 Hang Lung Mathematics Awards

### GOLD

 Topic: Expressibility of Cosines as Sum of Basis Team Members: Kwok Wing TSOI, Ching WONG Teacher: Mr. Yan Ching CHAN School: Po Leung Kuk Centenary Li Shiu Chung Memorial College Abstract: The central issue we are investigating is based on a problem from The Hong Kong (China) Mathematical Olympiad. It is basically about whether a cosine ratio is expressible as sum of rational numbers to powers of reciprocals of primes. In our project, we give the generalization of this problem by using some tricks in Elementary Number Theory and Galois Theory.

### SILVER

 Topic: Curve Optimization Problem Team Members: Ping Ngai CHUNG Teacher: Ms. Mee Lin LUK School: La Salle College Abstract: In this project, we shall introduce a new quantity associated with any given shape on the plane: “optimal curve”, which is defined as the shortest curve such that its convex hull fully covers a given shape S. Here curve can involve straight lines or union of straight lines. We shall investigate on some properties of this kind of curve and also prove a theorem that among shapes with a given fixed length of perimeter, the circle has the maximal optimal curve. Moreover, we will introduce an algorithm to find the shortest curve with convex hull equals a given shape in polynomial time.

### BRONZE

 Topic: Orchard Visibility Problem Team Members: Trevor Chak Yin CHEUNG, Yin To CHUI Teacher: Mr. Kwok Kei CHANG School: Buddhist Sin Tak College Abstract: In this paper, we discuss the generalization of the orchard visibility problem – from that of grid shapes to that of the shapes of the trees. We will even take a look at the problem of the visibility problem on a sphere surface and 3-D space.

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

 Topic: A Study of Infectious Diseases by Mathematical Models Team Members: On Ping CHUNG, Winson Che Shing LI, Hon Kei LAI, Wing Yan SHIAO, Sung Him WONG Teacher: Mr. Wing Kwong WONG School: Hong Kong Chinese Women’s Club College Abstract: Diseases are devastating. The SARS in 2003 and the swine influenza in 2009 sparked myriad of questions in our minds. Our major concern is the spread of germs. Throughout the entire project, we investigate disease-related issues and try to study the impacts of a disease by mathematical modeling. We first start with the simplest model followed by more complicated ones. We focus on different factors that affect the spread of diseases. Diagrams are included in each chapter to see how the values of different groups vary. Then we come up with possible ways to prevent epidemics. Altering the models by adding more conditions, we find one that fits the real life situation – the SEIRS model. The situation in Hong Kong (Swine Influenza from April 2009 to April 2010 in Hong Kong) is simulated by putting the data into the model and our goal is fulfilled. Topic: Dividing a Circle with the Least Curve Team Members: Chung Yin CHAN, Rennie LEE Teacher: Mr. Ka Wo LEUNG School: Hong Kong True Light College Abstract: In this project we planned to study the division of a circle with the shortest curve. In a party, we often divide a circular cake into equal and unequal parts. Suppose that bacteria grow on the exposed surface area of a cake. In order to keep the cake hygienic, we should divide the cake with the shortest cut. We investigated this problem by using a simple mathematical model: dividing a circle into equal or unequal areas with the shortest curve. The first possible solution was the radius method. It meant that we used radii to divide a circle into parts. But, were there any ways to divide a circle with a curve shorter than that of the radius method? The results included: Radius method is the solution of the problem for $$n$$ = 2, 3 and equal division. Radius method is not a solution of the problem for $$n$$ = 4 and equal division. Orthogonal circular arc is the solution of the problem for $$n$$ = 2 and unequal division. We found a necessary condition of the problem for $$n$$ = 3 and unequal division by a “Y-shaped” curve. Topic: Magic Squares of Squares Team Members: Pak Hin LI Teacher: Mr. Chi Ming CHAN School: P.L.K. Vicwood K.T. Chong Sixth Form College Abstract: In this report, we want to know whether there is a magic square whose entries are distinct perfect squares. Firstly, we analyze the basic properties of a magic square and find that the magic sum of a magic square is equal to 3 times of the central entry and the 9 entries of a magic square contain 8 arithmetic progressions. Secondly, we focus on our main target, magic square of squares. Investigating the properties of the prime factors of those 9 entries, we find that if the greatest common divisor of all entries is equal to 1, the prime factors of central entry are of the form $$p \equiv 1$$ (mod 4), the central entry must not be a square of a prime number and the common prime factors of any two adjacent entries (if exist) are not of the form $$p \equiv 3$$ (mod 4). Thirdly, we find that this problem is equivalent to a system of Diophantine equations with ten variables. We provide a construction method of the solution to these partial equations: $$a^2 + b^2 = c^2 + d^2 = e^2 + f^2 = g^2 + h^2 = 2M^2$$ , where these nine perfect squares are distinct. Finally, based on the theorems obtained, we find that given a positive integer $$N$$, there exists a positive integer $$M$$ such that it has $$N$$ essentially different representations of a sum of two perfect squares. Topic: Spherical Fagnano’s Problem and its Extensions Team Members: Ho Kwan SUEN, Tin Yau CHAN, Tsz Shan MA, Pak Hay CHAN Teacher: Mr. Cheuk Yin AU School: Pui Ching Middle School Abstract: In a given acute triangle, the inscribed triangle with minimum perimeter is the orthic triangle. This problem was proposed and solved using calculus by Fagnano in 1775. Now we wonder, will the result remain unchanged when the problem is discussed on a sphere? In this paper, we will first try to find the answer of the “spherical Fagnano’s problem”. Based on our results in spherical triangle cases, we will go further to generalize the problem to quadrilateral and n-sided spherical polygon in spherical geometry. Topic: The Erdős-Szekeres Conjecture (“Happy End Problem”) Team Members: Ho Ming WONG, Man Han LEUNG, Wing Yee WONG, Hon Ka HUI, Tin Chak MAK Teacher: Mr. Chi Keung LAI School: Shatin Pui Ying College Abstract: The survey [1] conducted by W. Morris and V. Soltan mentioned that in 1935 Erd˝os-Szekeres proved that for any integer $$n \geq 3$$, there exists a smallest positive integer $$g(n)$$ points in general position in the plane containing n points that are the vertices of a convex n-gon. [See reviewer’s comment (3)] They also conjectured that $$g(n) = 2n−2 + 1$$ for any integer $$n \geq 3$$. The conjecture is far from being solved for decades though many mathematicians had tried their very best on it. This paper is to investigate the Erd˝os-Szekeres conjecture by studying the greatest positive integer $$f(n)$$ points in general position in the plane which contains no convex n-gons. We successfully prove the cases when $$n = 4$$, 5 i.e. $$f(4) = 4$$ and $$f(5) = 8$$. For $$n = 6$$, we arrive at the conclusion that $$f(6) \geq 16$$ by creating an example of 16 points containing no convex hexagons. Moreover, we excitedly find an elegant proof for this example that one more point added to it will certainly give birth to a convex hexagon.

### 2010 Awards Ceremony Video

The 2010 Hang Lung Mathematics Awards winners were announced and recognized on December 18, 2010. Eight awards were announced: a Gold Award, a Silver Award, a Bronze Award, and five Honorable Mentions.

Winning students, teachers, and schools were recognized on stage, and received crystal trophies and certificates from world renowned scholars.

### Closing Remarks

##### Dr. Gerald L. ChanNon-Executive Director, Hang Lung Group

2010 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 2010 Hang Lung Mathematics Awards

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

 The Erdos-Szekeres Conjecture Shatin Pui Ying College Dividing a circle with the least curve Hong Kong True Light College Orchard Visibility Problem Buddhist Sin Tak College A Study of Infectious Diseases by Mathematical Models Hong Kong Chinese Women’s Club College The Centres of Tetrahedra Baptist Lui Ming Choi Secondary School Investigation on Mastermind and its generalization Munsang College (HK Island) Expressibility of Cosines as Sum of Basis Po Leung Kuk Centenary Li Shiu Chung Memorial College Introduction and Applications of Fuzzy Homotopy and Fuzzy Deformation Retraction Tang Shiu Kin Victoria Government Secondary School Zero-knowledge mutual authenticaion in the two-party password-authenticated key exchange setting: future goal or mission impossible? Carmel Secondary School Spherical Fagnano’s Problem and Its Extensions Pui Ching Middle School Starting from Combinatorial Geometry St. Joseph’s College Curve Optimization Problem La Salle College A new method to improve the ranking system for students studying the NSS: Using the calculation of eigenvector to find the weights of different subjects in NSS Carmel Holy Word Secondary School Magic squares of squares P.L.K. Vicwood K.T.Chong Sixth Form College n-Puzzle: An Innovation Sha Tin Government Secondary School