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Hong Kong Joint School Mathematics Society jsms.hk@gmail.com| www.hkjsms.org HKJSMS Newsletter 4th Issue July 2012

From the Editor July marks the end of our school term but turns a new page for students especially for public examinations candidates. It is also a busy month for the editorial team. To conclude this year, the team has invited more friends to contribute. The feature interview is on the role of mathematics in preparing Hong Kong students to compete globally and the theme of this issue is Mathematics Crises. Please enjoy reading. Yeung Hon Wah Yeung Hon Wah

Academic Corner – P1 Experienced Contestants Corner – P3 Event Focus Corner – P5 Member Schools Corner – P7 Mathematical and Problem Corner – P12

Academic Corner Interview with Dr. Koopa Koo Committee Member of International Mathematical Olympiad (IMO) Hong Kong By Yeung Hon Wah

Dr Koopa Koo obtained his PhD in mathematics at the University of Washington, USA. International Mathematical Olympiad Committee.

He has served as a committee member of the Hong Kong

He is a leading tutor in the mathematics tutorial industry in Hong Kong.

The Role of Core Mathematics in Preparing Hong Kong Students to Compete Globally Since

the reunification of Hong Kong with China, Hong Kong has been undergoing some major education reforms, including the mother-tongue teaching policy being implemented in over 310 secondary schools. The lastest reform is the 334 New Secondary System Academic Structure. With the new structure, two major public examinations, the Hong Kong Certificate of Education Examination (HKCEE) and Hong Kong Advanced Level Examination (HKAL), have been eliminated. A new examination, Hong Kong Diploma of Secondary Education Examination (HKDSE), has been introduced, and Mathematics is one of the core subjects. Some people criticized that designating mathematics as a basic requirement for local university admission is unfair to a lot of school candidates, especially to those who are from the arts stream. Some people even say that such requirement may strangle some students from the opportunities of further studies.

Dr. Koopa Koo shares with us his experience and his views towards the latest changes and his teaching experience. HW: I understand you have been very busy, these few days, with your students who are HKDSE candidates since the result announcement of the first HKDSE was made on July 20. It was probably regarded as the most nerve-racking day for a lot of students. Wasn’t it?

KK: You can expect a lot of students being anxious about their results. Afterall, this year is the first HKDSE. As their tutor, it is very important for me to give them support and advice at this critical moment. HW: Some people think that mathematics should not be made a core subject in secondary school, especially senior high school, because most of the areas, such as geometry, being covered are irrelevant to the future studies or careers of some students. KK: To answer your question, let’s first take a look at what Mathematics is all about. Learning mathematics is a way to train students’ analytical, logical and critical thinking. Mathematics education provides students with a wide range of knowledge and skills to help them develop an understanding of the physical world. With such knowledge and skills, students are able to describe, to illustrate, to interpret, to predict and to explain phenomena of the world. Unless a society or a nation does not need any kind of technological or scientific advancement or even wants to lead a very primitive life, there is a need for people to learn mathematics which is a fundamental way for training people to think critically. Carl F. Gauss, one of the three greatest mathematicians of all time, described mathematics


as ‘the Queen of Science’. Literally, it means that mathematics is the ruler of science. If you take a step back and look at the global university standards, almost all developed countries’ university requires mathematics as a basic requirement, because they also see mathematics education as a fundamental and effective way to build a logical and critical mind for the students. If Hong Kong does not make Mathematics as a core subject, then the Hong Kong students are not even competitive globally. Therefore, in order to help Hong Kong students to develop logical and critical thinking skills, and to stay globally competitive, mathematics should be a core subject in secondary education. HW: Some people say that mathematical competency is in-born and cannot be acquired. Therefore, if mathematics is a basic requirement of a university place, it will strangle the opportunities of future studies for some students who do not possess such competency. What do you think? KK: I really don’t think mathematical competency is in-born. It has been taught and trained to any person. The key, and it would be the same for any subject, is to learn from a knowledgeable and good teacher. Similarly, Chinese and English are core subjects. Every year, many students failed to get in universities because of failing language subjects. Failing language subjects does not mean they are illiterate. They may be just weak at presenting ideas. Can we say that having language as core subjects is unfair to students who are good at mathematics but weak in languages? The reason why there are basic requirements for a university place is to ensure the candidates possess the fundamental skillsets for their future studies. Such requirements do not mean to deprive anyone of the opportunities. HW: Now that HKDSE has replaced HKAL and HKCEE and has made Mathematics as core subject while HKAL did not. Some students and parents find it very difficult to adapt to the change. What is the advice to them? KK: I won’t say it is a change. Mathematics was a core subject in HKCEE. Since the HKCEE and HKAL are being replaced by one examination, Mathematics is therefore a core subject in DSE. Actually, it is a tradition in Hong Kong to have Chinese, English and Mathematics as core subjects which are basic requirements of university admission. HW: Is the depth and breadth of the current syllabus set at a right level for students in general?

KK: I would say it is acceptable. The syllabus of NSS Core Mathematics covers a much wider and deeper scope than that of HKCEE. However, when comparing to USA, and some major countries or cities in Asia like China, Japan, Korea and Taiwan, as far as mathematics is concerned, the math syllabus in Hong Kong is considerably easier and this is likely to affect the competitiveness of Hong Kong students in the long run. HW: So much about the education system, may I ask questions about you? Why do you love mathematics?

KK: It was mathematics competition which triggered my love for mathematics. Problems in mathematics competition are interesting and challenging. My experience to tackle such problems with ingenious applications of simple math theories have been rewarding and amazing. HW: Why did you choose to teach secondary students? KK: Learning is a lifelong career and it is advantageous to build an analytical and logical mind at an early stage. Teaching students, who are still in the stage of building up their knowledge foundation, is very meaningful. At the secondary level, I am not only transferring knowledge to them, but also fostering young people and helping them to find their directions. Most importantly, I can be exposed to more students in tutorial schools. This means I can inspire more people. HW: How would you compare Hong Kong students and US students? KK: Hong Kong students are too shy to ask and answer questions. On one hand, it is about their mentality, they are afraid of asking silly questions or being seen as showing off. (In fact, no question is silly.) On the other hand, it is a matter of our culture. Therefore, students and teachers in Hong Kong do not interact as much as those in the USA, where students proactively ask questions and voice their opinions. HW: Thanks very much for your sharing KK: Thank you.

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Experienced Contestants Corner .

Interview with Mr. Kenneth Hung Silver Medalist of IMO 2009 and Bronze Medalist of IMO 2010. By Jerry Fong

Kenneth Hung is a former IMO Hong Kong Team member. He was the president of the Hong Kong Joint School Mathematics Society in 2009. He is now studying mathematics at California Institute of Technology (Caltech). His dream job is a teaching professor.

JF: Kenneth, you are a very devoted math lover. Can I say that? I understand that you represented Hong Kong twice in the IMO with brilliant results and you are now studying Mathematics in California. Why are you so interested in mathematics and how do you get yourself trained? Could you tell us about your story? KH: I like the simplicity of Math. The goal of science is to find out fundamental laws, to find out the pattern between different things, to find out a set of universal laws that applies to everything in the world. And Math is probably the field that achieves these the most. Everything goes down to the basic definition that we have. Anything that you can find in Math stems out of the few basic definitions. So when I first got into high school, in Form 1, I was lucky enough to be selected to represent DBS in IMO Preliminary contest. That probably was the time when my life changed. Before that, Math was something I did casually. But the training that I received, after the contest, in Hong Kong Academy of Gifted Education (HKAGE) was much more organized, so everything speeded up at that time. The rest is, probably just a compulsion to succeed, just like what it takes an athlete to win a competition. It's a compulsion to become one of those 6 people on the stage (IMO team) one day. JF: Why did you choose to study in the USA? Is there any difference between studying Mathematics in California and that in Hong Kong? KH: Frankly, there is a huge difference. I feel Hong Kong lacks the respect that it should have for mathematicians. Let me give an anecdote here. I remember when I received my HKCEE (that turned into the HKDSE nowadays) result, my family was overjoyed. My mom called some of the relatives to spread the good news. Naturally, they asked my mom what I am planning to do in college, "Is Kenneth going to be a doctor, a lawyer or an actuary?" They almost had a heart attack when my mom said I wanted to do math. They spend one week just trying to convince my mom to stop me from "acting foolishly". No one really take Math as a serious subject.

On the other hand, here in California, the percentage of people getting a Math degree, whether for doing finance, accounting, programming or research in the future, is a lot higher. Doing Math is not an anomaly here anymore. It's considered as a … It turned into sort of a compulsive thing for me to grab some old math problems and start working on it whenever I am free. So I had a hard time stopping myself from "going insane"…. There were lots of voices inside my brain that told me, "Kenneth, you must succeed!" …

good basis for further studies or work. It's considered as something that prepares you to think logically. People hold higher respect for people doing Math in the States. JF: How would you compare the Hong Kong students’ ability in Mathematics to that of students in the USA? KH: Hong Kong students definitely had a better start than the US students. For example, one of the Putnam fellow (Putnam competition is an intercollegiate Math competition in North America) in Caltech, Sam Elder, was not exposed to math competitions at all until high school. But compared to Hong Kong students, he is a lot more dedicated to math and thus catches up with other people, exceeded other people, and even manage to come first five in the aforementioned competition. On the other hand, I remember that there were a lot of students who did Math with me ever since I was in Primary 5. A lot of them were talented, and most of them were actually more talented that I am. However, some of them just left, some of them just didn't put enough time in it until they faded away.

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So I would say Hong Kong students are more talented in the early stage, but they never have the dedication compared to US students. JF: Let’s get back to IMO. How did you prepare for every IMO competitions? KH: So for IMO, I had a year long preparation for it. With a bold assumption that I will make the team again, I decided to start preparing for it ever since I got back from IMO 2009 in Bremen. At school, my teacher was lenient enough to let me "mind my own math" instead of paying attention in class. Since I was taking Pure Math and Applied Math and Physics in high school, having those periods freed up for MO training helped a lot. It gets me an extra 3 hours per day for math / relaxing. So when I do the training, usually I just go through past IMO problems which are all available on MathLinks or official websites. Then later on I ran out of problems after than 90's, so I started on Balkan MO. And later, APMO and BJMO were useful resources as well. My idea was, training problems usually make the purpose and the solution too easy. Let's say you have a combinatorics book, on the chapter of "generating function". Obviously you have to use generating functions, which actually reduces the difficulties of the problem by a lot. By working on actual contest problems, you get to get more field experience. Also, it's a more enjoyable process as well, as most competition committees select problems based on elegance as well. Working on past competition problems allows you to appreciate the beauty of math. JF: Each year, in Hong Kong, there are thousands of participants sitting for Maths Olympiad competitions ranging from kindergarten level to HKMO and the International Mathematics Olympiad Selection Contest? How would you perceive such phenomenon? What do you think about the competitions for junior levels or even kindergarten levels? Would that be too much for little kids? KH: In my opinion, if the kids choose to do math themselves, just naturally, certainly that means they are into it. But the fact that a lot of parents are forcing little kids to take classes, this and that, is not quite healthy. It's the same reasoning for any other extracurricular activities: An activity is fun only when you enjoy it, not when you are forced to do it. Indeed, this trend can, in the end, cause the children to lose interest in math, even though they might like math and be good at math otherwise. JF: Throughout your ‘career’, so to speak, in IMO, have you ever faced any difficulties, how would you perceive and counteract with it?

IMO, so I felt really compelled to do ultra-über-good in IMO. It turned into sort of a compulsive thing for me to grab some old math problems and start working on it whenever I am free. So I had a hard time stopping myself from "going insane”. Then in senior year, things were going alright. I would say the biggest challenge came in the very last month. There was only one chance left. I had to get it right. I had to get a better medal. There were lots of voices inside my brain that told me, "Kenneth, you must succeed!" Ultimately, I ended up one point below the silver medal cutting score, so I ended up with a bronze medal. I was far away from home in Kazakhstan dealing with this bad news alone, while my teammates were overjoyed with their medals. Dealing with the failure was probably the biggest challenge I've faced and the biggest lesson I have taken. JF: What is you dream job? Does the dream job relate to your interest in mathematics ? KH: My dream job is probably being a teaching professor. I enjoy seeing bright minds learning math, and I personally enjoy helping a lot of my fellow students at Caltech with problem sets. I could have chosen research professor, but to be honest, the academia nowadays has evolved into a "publish or perish" dead loop, which is not quite favorable for a lot of mathematics research that are on the theoretical side. Unless you are in the very top notch, getting grants and funding is getting harder and harder. JF: In your opinion, are there enough opportunities for most of the mathematical-gifted children to receive suitable training in mathematics? KH: In California, and I dare say, for the entire US, they have a lot of things called "math camps". Such math camps are usually partly just a summer camp, and partly math training. It's nowhere as rigorous as the training that people would receive in HK, but in a more entertaining way. This inspired a lot of the best mathematicians in my school indeed. Afterwards, when they do national and regional selections, they may enroll in the US summer camp for IMO team selection. However, one interesting point here is that, Americans don't really label them as "gifted" vs. "non-gifted". Generally, they just give them tests and select. I would say that over all, the training that they receive is a lot less than what it is in Hong Kong. However, the American training focuses more on inspiring youngsters to do the math themselves, and making them interested in solving problems.

KH: I have faced a lot of difficulties. In Form 5, it was mainly handling the HKCEE exam and getting decent training for the team that was troubling. Sometimes it really feels like even 30 hours a day would not be enough.

JF: Would you like to give some comments or advice to students who aim at representing Hong Kong to participate in international competitions?

Then later on, after HKCEE was over, it was mainly handling stress. I don't have too many chances of going to big events like

KH: My advice is, "go for it". There is a saying, "Shoot for the moon. Even if you miss, you'll land among the stars". If you try

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hard doing math, even if you can't make it to the team, it would train your logical thinking and make you more prepared for other career options as well. For example, a law degree would require a lot of logical thinking. Indeed, I have a lot of MO friends doing a law degree right now, and their math background definitely helped them. In short, you don't lose anything by trying, but you would certainly miss out on the beauty and the excitement of math if you don't try. JF:Do you think these days, MOers have enough commitments and enthusiasm in study advanced mathematics?

KH: People seem to commit less in mathematics nowadays compared to MOers from my age. However, I think this is not a bad trend. Doing MO is a commitment, but it doesn't mean that it's the only thing in your life. As long as you enjoy doing, who cares if you actually did it well or not? Who cares if you spend a lot of time or a little bit of time doing it? If the commitment turned into something constraining, then the person probably have the wrong mentality about doing MO. MO should be just a hobby, that you enjoy and spend time as much as you wish on.

. 

The interview was conducted through Facebook, in mid July, as Kenneth was in California

Event Focus Corner Inter School Mathematics Contest 2012 The Inter-School Mathematics Contest 2012 was successfully held on May 5 at Diocesan Boys’ School. 246 contestants, represented 150% participation of last year, joined the event. Number of awards was three times of the previous year. 8 trophies and 58 medals were given out to the following outstanding contestants: JUNIOR INDIVIDUAL: Champion: Poon Wing Hei Levi (Ying Wa College) 1st Runner Up: Yu Hoi Wai (La Salle College) 2nd Runner UP: Chan Tsz Yu (La Salle College) MERIT - JUNIOR INDIVIDUAL Choi Wang Hei (Ying Wa College) Hung Yui Chi (Queen Elizabeth School) Frankie Lam Shun Leung (Diocesan Boys’ School) Gerry Lam Chun Hei (Diocesan Boys’ School) Lam Chung Yin (Ying Wa College) Hui Shing Yau Justin (NLSI Lui Kwok Pat Fong College) Cheung Ka Long (La Salle College) Paul Ng Pak Lun (Diocesan Boys’ School) Lam Ho (St. Joseph College) Leung Kei Wai (Wah Yan College Hong Kong) SENIOR INDIVIDUAL: Champion: Chong Hip Kuen (Queen’s College) 1st Runner Up: Tse Shun Chung (La Salle College) 2nd Runner Up: Chan Chung Hong Nigel (St. Paul’s Co-educational College) SENIOR INDIVIDUAL MERIT Lo Wai Kit (Diocesan Boys’ School) Wong Hing Shing (La Salle College) Yeung Yat Kwan (St. Paul’s Co-educational College) Fung Kin Yiu (King’s College)

The JSMS would like to express our heartfelt thanks to the members of the organizing committee and volunteers for their kind assistance and support. Last but not least, we would like to thank all participating schools and contestants for their unfailing support.

Jason Tsang (St. Joseph College) Wong Kin Cheung (La Salle College) Ho Wui Hang (Wah Yan College Hong Kong) Chow Tsz Yin (Carmel Pak U Secondary School) Chan Ki Fung (HKSYCIA Wong Tai Shan Memorial College) Ryan Lau Ho Hon (Diocesan Boys’ School) GROUP EVENT Champion: La Salle College, Team 1 1st Runner Up: La Salle College, Team 2 2nd Runner Up: Diocesan Boys' School, Team 2 GROUP EVENT MERIT Diocesan Boys’ School, Team 1 Carmel Pak U Secondary School Queen Elizabeth School St. Paul’s Co-educational College, Team 1 Wah Yan College Hong Kong, Team 1

Special Thanks to Diocesan Boys’ School for Sponsoring Venue for this Event.

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53rd International Mathematical Olympiad (IMO) (27 July 2012, Hong Kong) A team of six student members of the Hong Kong Academy for Gifted Education (HKAGE) representing Hong Kong at the 53rd International Mathematical Olympiad (IMO) in Argentina finished the contest with a flourish, winning three silver medals, one bronze medal and two honourable mentions for Hong Kong. The Silver Medalists are Secondary Six student Li Yau Wing from Ying Wa College, Secondary Six student Loo Andy from St. Paul’s Co-educational College and Secondary Seven student Chow Chi Hong from The Bishop Hall Jubilee School. The Bronze Medalist is Secondary Four student Wong Sze Nga from Diocesan Girls’ School. She has been the first female team-mate since 2002. Secondary Six student Kung Man Kit from S.K.H. Lam Woo Memorial Secondary School and Secondary Three student Lau Chun Ting from St. Paul’s Co-educational College are awarded Honourable Mentions. Students with outstanding performances in the IMO Preliminary Selection Contest – Hong Kong 2011 were selected to take part in a series of Mathematics courses organised by the HKAGE and the IMOHKC before taking part in the IMO 2012..

Source: International Mathematical Olympiad: HKAGE Student Members Proudly Returned with Six Awards HKAGE Students’, Press Release, HKAGE, July 2012

Upcoming Events

Shoot for the Moon HKJSMS proudly presents the Day Camp 2012. Besides city tracing, there will be an exciting survival game known as "Math Survivor". Dr. Koopa Koo, Committee Member of IMO Hong Kong will also give his talk on the second day. You should never miss it! Day 1(1/8 Wed) : Day 2(6/8 Mon):

Day 3(10/8 Fri): Venue: Time: Fee:

City Tracing Math Survivor (Part I) & “Your Strategy for HKDSE” Talk – by Dr. Koopa Koo Math Survivor (Part II) Lecture Hall, Diocesan Boys’ School 09:00-18:00 $50 for three days

Online Registration: www.hkjsms.org

全港青少年數學挑戰賽 2012 主辦單位: 全港青少年數學挑戰賽 2012 委員會 日期:8 月 12 日(星期日) 對象:2011-2012 學年中、小學生 地點:新生命教育協會呂郭碧鳳中學

查詢:電郵至 logtin@live.hk

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Member Schools Corner Au Chi Chun Nelson St. Francis Xavier’s College

A New Experience to Me In April, my friend Jeff told me that there would be a Mathematics competition, ISMC. I was curious about joining Mathematics competitions. Therefore, I joined the competition with a few of my friends to see what I could learn from the competition. On the day of the competition, when I received the question paper, I found that the questions were too hard for me. There were different kinds of questions that I had never seen. I ended up spending very long time on solving one of the thousand questions. I could only try and try until I succeeded within the competition’s time.

Participants are doing their best to achieve the prizes! process, we will succeed at the end.

Although I lost in the competition, I learnt a lot. First of all, I understood that there were different kinds of Mathematics questions in the world instead of only those in our textbooks. I would explore them more on Mathematics and get a deeper knowledge in Mathematics. Second, I learned that when facing challenges, we may need to try many times before reaching the goal. If we don’t give up in the

Finally, I also learned some problem solving skills, such as trial-and-error, in the competition, which is also useful in my daily learning. To conclude, it is a right choice to join this competition and hope I can join the competition next year. I hope I can also join the afternoon games section next year as this year I missed it, but I have already felt the joy of Mathematics in the competition. Participants are doing their best to achieve the prizes!

Chong Hip Kuen, Champion of Individual Event (Senior), ISMC 2012 Queen’s College

A Joyful Moment Inter-School Mathematical Contest, which is held on 5th May, impressed me much. Although I had other things to do that day, I determined to participate in this meaningful competition. I knew it was organized by some secondary mathematical talents. They put a lot of efforts in it. So I should respect them and join this competition. Also, it can promote friendship among students of different schools, and arouse students’ interest in mathematics. Every contest, to me, is a challenge. It is not important whether we obtain a higher or lower score in the mental challenge. I think it is more important to obtain experiences from every contest. I learn more about the skills and tactics from the contests because the problems are tricky and require careful thinking before getting the correct answers. I was very happy when I got the championship in this contest. However, I got far more than a champion. I got knowledge. I got experience. I am inspired by

the spirit of competition.

this

Mathematics is the base of all sciences and analysing skills. It is important for us to learn mathematics better to build foundations of knowledge which allow us to get a even My very joyful moment higher achievement. I would recommend this meaningful competition to all secondary students to enhance their interest of mathematical study.

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Sam Yam, Vice President of HKJSMS, 2011-2012 La Salle College

Simple Logic? Or not?

Let’s do a test first. In front of you, there are 4 cards: “A”, “B”, “1” and “2”. If now a person tells you that “For every card that has a vowel on a side, there is an odd number on the other side.” Now in order to test whether this statement is true for all 4 cards, which card(s) are necessary to flip over? According to a research, more than 90 percent of university students will answer “A”. But only about 20% of students will answer “2”, which is also necessary to prove the statement right. Moreover, about 20% of the students say “1”, which is not a correct answer. The question is not that challenging, but why would we make such mistakes? Now, let’s do another question. You are a cop, there are 4 guys in front of you. Guy A: 16 years old, Guy B: 20 years old, Guy C: Drinking Coca-cola. Guy and D: Drinking Beer.

Realize that the first and second questions involve similar logic, but why does the first one require you to think thoroughly to get the answer, while for the other on you can tell by instinct?

By law, it is illegal to drink alcoholic drinks, under 18 years old, in public places. Which guy do you need to check to ensure they obey the law?

Studies show that, when people read the first question, they use their brain zone that are related to mathematics to answer it. But when they read the second one, they use their brain zone which is in charge of social righteous.

100% of the students know the answer: Guy A & Guy D. So next time when you see some logic problems, try to put the situation in reality, you may find it turns simple.

Priscilla Chan Diocesan Girls’ School

Simple Arithmetic I have two interesting mathematical questions for sharing:

2. Is three minus one equals two?

1. Is one plus one equals two?

No, necessarily.

No, necessarily.

For example, if there are three birds resting on a branch of a tree. A hunter fired at one bird and shot it. How many birds left on the branch?

1+1 ≠ 2?

For example, a man and a woman, after getting married, can give birth to one baby, two babies, or more. Therefore, one plus one may equal to two or more. It all depends on the situation.

If you ask a kindergarten child, the answer will probably be two. But, as you grow up, you probably know that there will not be any birds left after they heard the shooting sound. The answer is obviously zero, which means three minus one equals zero. Do you agree?

3 -1 ≠ 2? In the above cases, you may argue that those are not mathematics but IQ questions. Yes, the answers are quite tricky, and in most cases, the normal answer should apply.

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Gerry Lam Diocesan Boys’ School

Thankful to the ISMC Organising Committee Being one of the members of DBS math team, I went to school early in the morning to get ready for the contest. Similar to the two international math competitions I joined before in Bali and Beijing, the ISMC comprised a group event and an individual event. While ISMC was not a large-scale contest, it was still very exciting as with many math talents coming from other schools, the ISMC was keen and we could know the results later on the same day. With the start of ISMC at 9:00 am, we first had our group event. When our group got the paper, the group leader quickly had the first few questions properly assigned to each member for solving. After finishing these questions, we only had little time left for discussing those much more difficult ones. We could not complete the paper eventually and lost a couple of marks. The level of difficulty of the questions was a bit more difficult than that of those international competitions I had. We then took a short break before the individual event started. The two hours given for the contest flew so soon but I could luckily finish most of the questions. Following the end of the individual and group events, there were some fun mini math games for us to play. This helped us to relieve our anxiety after the competition and provided good chance for us to cultivate friendship with contestants from other schools who were also passionate about math. If we were to form teams with contestants from different schools in playing the math games, it would have been a special experience to us. Besides, if I were asked to introduce a math game to the ISMC, I would suggest the KenKen Math Puzzle (or called 聰明格).

It is to be played like Sudoku but with arithmetic questions added for filling the digits into the grids. As swiftness in completing the puzzle is a crucial factor to win, it would be an exciting experience to the contestants on the spot. I was satisfied for attaining the Merit result in the Individual (junior) event. Following the events, there were some fun mini math games for us to play. This helped us to relieve our anxiety.

Then here came the high tide of the ISMC. All of us were anxious when was satisfied for the Iresults were announced. I was satisfied for attaining the Merit result attaining the Merit in the Individual (junior) event. However, our group was a bit result in the Individual disappointed as we did not do quite well in the Group event and only got (junior) event. the fourth place in overall. We could have got a higher position if we had our group not However, lost marks for being careless in doing some of the questions which was a bit disappointed were not too difficult. as we did not do quite in the Group event All well in all, I found the ISMC a joyful experience. We must be thankful to and only got the fourth the members of the organising committee as the 2012 ISMC would not be place in overall.

held successfully without their dedicated efforts. I look forward to joining the 2013 ISMC and hope to meet you and other new contestants in the contest!

Rachel Fung St. Francis’ Canossian College

What a Great Event! After participating in the Inter-School Mathematics Contest 2012, I found that I have acquired lots of skills. For example, I have learnt how to solve problems in the ways that I have never thought of. I have also learnt more than we learn in schools, like different kinds of formula. I regret that I did not join the competition in group event because it would help competitors to develop not just academic skills, but also interpersonal and communication skills. I would definitely join it next year!

Keep Quiet! Individual Event is going to start

Listen! Here are the Instructions for Treasure Hunt

Thanks to the HKJSMS.

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Jeffrey Hui, Math Co-ordinator of HKJSMS La Salle College

Being the Chief Examiner of ISMC 5 May 2012 is a very meaningful day in my life because it is the first time for me to be the chief examiner of a mathematics competition, the Inter-school Mathematics Contest (ISMC) 2012. In the previous years, I joined various mathematics competitions in Hong Kong. As soon as I knew about news of any mathematics competitions, I would join it because I enjoy solving mathematics problems. When I was in Form one, it is the first time I joined ISMC. I was very excited because I enjoyed participating mathematics competition. Also, I thought this competition to be very mysterious as I could not find any past question papers of this event. (There was no JSMS official website before 2009.) During the competition, I was surprised when I read the paper because there were many short-questions. And the type of problems were different from other competitions I have joined, especially it required the knowledge of some discoveries of well-known mathematicians, etc. If you only knew how to calculate, you could not get full marks in this contest. Finally, I got a merit in Junior Session. In the next year, I was joyful because I could participate in ISMC again! It was a big surprise for me that calculators are allowed in the contest. Another surprise was that the organizing committees was a group of students, I admired them as they could organise such a big event by themselves. At that time, I had a dream: I would be one of them. This year, I am a Math Coordinator of JSMS. Before I took on this post, I hesitated a bit because I would give up the opportunity of being a contestant in ISMC. However, when I knew that one of the duty of Math Coordinator would be to compose problems for the ISMC, and I might be a chief examiner of ISMC. I was willing to take this post because I’d never host a mathematics competition, and it would be a fun and unforgettable experience! In March and April, we started to search for questions for the ISMC. It was a hard task for me. I could feel the pressure getting heavier day after day. I was so worried in case the problems I found would not be considered by the committee. Another concern was what if the finalized paper became a bad one. Luckily, many current and past JSMS committee members helped me. Finally, ISMC paper was finalized. “Vice president Sam, how about we read out the scores and you’ll help us enter all the numbers on the board, OK?”

I feel happy for you guys, the Group Champion. I also envy you. You know, I won’t be able to participate as a contestant . I will be too old for the ISMC next year.

On that day, after every contestant got seated, I stood on the stage and gave instructions to them. It was a wonderful experience to look down from the stage and see every participant work hard to finish the paper. Although ISMC delayed for 20 minutes due to bad weather, ISMC was held successfully, at least no participants complained. Markers are doing the final checking on the answers sheets of all contestants, helpers are setting up the venue for the afternoon session, the … No lunch break!

You may think, I have gained nothing from it because I was the chief examiner and I don’t have to finish the paper. I can tell you, being the chief examiner; I need to check all the answers and solutions of every problem carefully. Sometimes, I don’t know how to tackle some problems, which are given by my friends. When I read the solutions, I learnt new algorithm and quicker methods to do the problems. I enjoy being the chief examiner. Not only because the participants would know who I am, but also I can feel the enthusiasm of all ISMC participants towards Mathematics.

“Sure!”

10


Mathematical & Problem Corner The Three Mathematics Crises 1. Introduction The existence of the three Mathematics Crisis is because of the proof of some mathematical axioms are not all rounded. These crisis nearly made Mathematics collapse.

2. The First Mathematics Crisis At around BC 500, most Mathematicians believed that only rational number existed in the world, and every number can be expressed by a fraction. The reason why the First Mathematics Crisis happened was the argument of the presence of “Irrational number�. Pythagoras (A famous ancient Greek Mathematician and Physicisian) found out that not every number could be expressed by a fraction after he proposed the Pythagoras Theorem. The formula of the Pythagoras Theorem is a2 + b2 = c2. Even we substitute a rational number into a, b, the value of c may not be rational. For example, 12 + 12 = c2, i.e. c2 = 2. The value of c (√2) is not rational as it could not be expressed by a fraction. That means, there are not only rational numbers existed in the world.

It contradicts with the original assumption (i.e. The H.C.F. of P, Q is 1) √đ?&#x;? is not a rational number.

3. The Second Mathematics Crisis There was a famous paradox in the ancient times. It proposed that everything which existed in the world didn’t actually move. The reason why we could see objects move was the result of illusion. Of course, nowadays we all know that it is totally ridiculous. However, at that time a lot of people believed in that theory. About 600 years ago, some Mathematicians tried to prove the presence of “Instantaneous speed� so they were able to prove that everything in the world could move. Later, Newton introduced Calculus, which was supposed to find the change of an event, such as the instantaneous speed of a moving object. This established the concept of “Moving object�. However, this leaded to the Second Mathematics Crisis.

The mathematicians argued a lot on this issue as the findings of Pythagoras totally contradict of what they thought in the past (i.e. Every number can be expressed by a fraction). Luckily, they finally accepted the truth that rational numbers and irrational numbers both existed on the number line. The First Mathematics Crisis therefore solved. The proof why √2 is not a rational number by using contradiction Assume √2 =

đ?‘ƒ đ?‘„

(P, Q are natural numbers and their H.C.F is 1)

√2 =

2=

đ?‘ƒ đ?‘„

đ?‘ƒ2 đ?‘„2

2Q2 = P2

As P is even, let P = 2M, where M is a natural number. 2Q2 = (2M)2 Q2 = 2M2

Fig. 1 Comics introducing the First Mathematics Crisis

This shows that Q is even. As both Q and P are even, their H.C.F. is not 1

11


As Calculus is based on the concept of “Limit”, which included the concept of “Infinitely small”. That means the value of “Infinitely small“ is very close to 0, but not equal to 0. Some Mathematicians argued that the value of “Infinitely small“ is actually equal to 0. If their argument was true, the contributions of the Mathematicians proving that “Everything can move” in those long years would be all destroyed. It caused the Second Mathematics Crisis. Luckily, a Mathematician, called Augustin Louis Cauchy solved this problem by proposing the system of Mathematical Analysis. It stated a clear concept of “Variable” and “Infinity”. Therefore the concept of Calculus was still correct and the Second Mathematics Crisis was solved.

4. The Third Mathematics Crisis The Third Mathematics Crisis was because of the “Russell’s Paradox”. It is the most famous of the logical or set-theoretical paradoxes. It considers the set of all sets that are not members of themselves. It seems that a set appears to be a member of itself or not a member of itself. That means, a thing seems to be right, but it can be wrong. This could destroy the whole structure of Mathematics as its basis was based on proofing and set theory. (i.e. If the set theory is wrong, all theorems in Mathematics are wrong).

(Fig. 2) Comics introducing the Third Mathematics Crisis

5. Conclusion However, the founder of the Third Mathematics Crisis, was also the one who solved it. He proved that Russell’s is not true. Therefore the Mathematics theorems in the still true. His proof also leaded to the birth Zermelo-Fraenkel Set Theory, which is very famous.

Russell, Paradox past are of the

From the three Mathematics Crisis, we can observe that overcoming obstacles and difficulties are important for the development of Mathematics. Also, they give warning to Mathematicians that they should not only develop the new branches of Mathematics, but also investigating the basis of Mathematics, such as the set theory. We can also see why the Mathematics proofs have to be strict and all –rounded.

Article contributed by Jeff Siu

Source: comics from: http://www.mathsisfun.com/definitions/pythagoras-theorem.html

Questions

1.

Find the fraction

b a

in the simplest form (where a

2. When expressing

√2

in decimal

and b are both positive integers smaller than 50),

notation, we call the digit before the decimal point

such that the value is the closest to √2.

as r, while the digit after the decimal point as b. Find r2 + b2. (i.e. √2

Please refer to page 15 for solutions

= …r.b…)

Questions Contributed by Jeffrey Hui

12


Solutions to Questions – Newsletter, March 2012, Page 12 Extra question 1

Official Solution: Due to Dr. Tat Wing Leung

Now, map P to the centre of the circle by projection, by noticing symmetry, the three points P, E and O will be

Extend AP, BP, CP and DP to meet the circumcircle of

collinear.

ABCD again at A’, B’, C’ and D’ respectively.

Now 900 =

Extra question 2

1 1 1 PAB  PCB  A' OB  C ' OB  A' OC' 2 2 2

.

Solution: Let E and F be the points on the rays AC and BC

So A’C’ is actually the diameter of the circumcircle.

respectively such that AE = AD and BF = BD, then AD +

Similarly B’D’ is also a diameter of the circumcircle. So

AC = BC + BD, i.e. CE = CF.

A' C'B' D'  O .

Also, let Q be the intersection of the internal angle bisectors of DAC and DBC . Let X, Y and M be the

Now, let B’A’ and C’D’ intersect at Q, applying Pascal’s

midpoints of DE, DF and EF respectively.

Theorem to B, B’, A, C, C’, D on the circumcircle, we see that

BB'CC '  P, B' A  C ' D  Q , and

AC  BD  E

are collinear, next, we apply Pascal’s

Theorem to A, A’, C’, D, D’, B’ on the circumcircle, we see

AA'DD'  P, A' C 'D' B'  O C ' D  B' A  Q are collinear.

that

and

Hence combining the result, O, P, Q and E are collinear.

Lemma 1: M, P and C are collinear First suppose CE = CG, note that AD = AE as well, so

AED  ADE , hence CAD  2AED , implying CAQ  AED , it follows that AQ // ED. But since PA and QA are the interior exterior bisectors of the same angle,

PAQ  90 0 , i.e. PX ' D  90 0 ,

where X’ is the intersection point of PA and ED. Consider triangle ADE is isosceles and AX is perpendicular to ED, so X = X’. Similarly, BP meets FD at its mid-point, so P is the circumcentre of triangle DEF since P is the intersection of the perpendicular bisectors of DE and DF. Hence, we get PM as the perpendicular bisector of EF, so is CM.

As CE = CF, now lemma 1 is claimed, M, P and C are collinear.

Alternative Solution: Due to Mak Hugo Wai Leung Let A’, B’, C’ and D’ be the intersection point of the circle and AP, BP, CP and DP respectively. We can easily deduce that A’B’C’D’ is a rectangle by simple angle tracing. Hence, O will be the intersection point of A’C’ and B’D’.

13


Lemma 2:

ďƒ?BPC  ďƒ?MPY  ďƒ?MFY  ďƒ?EMX  ďƒ?EPX  ďƒ?XPD  ďƒ?APD Notice that FBYM and PMEX are both cyclic quadrilaterals, and MX // DF. Now suppose

ďƒ?APD  ďƒ?BPC , similarly as in lemma

1, we can easily claim that P is the circumcentre of triangle DEF, by angle tracing, it suffices to show that

ďƒ?BPC  ďƒ?APD  ďƒ?MPY . Similar arguments can prove lemma 2, which further implies that M, P and C are collinear.

Combining lemma 1 and lemma 2, C lies on the perpendicular bisector of EF, i.e. CE = CF.

Contents contributed by Mak Hugo Wai Leung, St Joseph College

Solutions to Questions (page 13) Solution to Question 1 (provided by Man Siu Hang, Gordon, CCC Ming Yin College) Think about continued fraction. √2 = 1 1

+

= 1

+√

1

=

+

1 1 2+ 2+â‹Ż

7

, 1

+

1 2+ 2

1

=

1 2+ 1 2+ 2

4 9

, 1

+

=

1

1 2+ 1 2+ 1 2+ 2

Since a, b < 50, the value is the closest to â&#x2C6;&#x161;2 is

99 . 7

đ?&#x;&#x2019;đ?&#x;? . đ?&#x;?đ?&#x;&#x2014;

Solution to Question 2 (provided by Mak Hugo Wai Leung, St. Joseph College) 2â&#x2C6;&#x161; +

= â&#x2C6;&#x161;2 â&#x2C6;&#x161; Let x = 2â&#x2C6;&#x161; is an integer. Note that

2â&#x2C6;&#x161;

2

=

2â&#x2C6;&#x161; â&#x2C6;&#x161; 2 â&#x2C6;&#x161;

=

, by Binomial Theorem, the odd powers of 2â&#x2C6;&#x161;

+ â&#x2C6;&#x161;

< , so 9

2â&#x2C6;&#x161;

are cancelled out, so x

< 0.1

Hence, b = 9. x = [51006 + (5)1005(2â&#x2C6;&#x161; + (5)1004(2â&#x2C6;&#x161; 2 + â&#x20AC;Ś + (5)(2â&#x2C6;&#x161; 1005 + (2â&#x2C6;&#x161; (5)1005(2â&#x2C6;&#x161; + (5)1004(2â&#x2C6;&#x161; 2 - â&#x20AC;Ś (5)(2â&#x2C6;&#x161; 1005 + (2â&#x2C6;&#x161; 1006] 1006 1004 2 =2Ă&#x2014;5 +2Ă&#x2014; (5) (2â&#x2C6;&#x161; + â&#x20AC;Ś + 2 Ă&#x2014; (2â&#x2C6;&#x161; 1006 = 10 Ă&#x2014; (â&#x20AC;Ś) + 2 Ă&#x2014; 24503 The last digit of 24503 is 4, so x ends in 8, while r = 8 â&#x20AC;&#x201C; 1 = 7, so r2 + b2 = 130

1006

] + [51006 -

14


Hong Kong Joint School Mathematics Society 2011-2012

Year in Pictures

Facebook Page constructed on Sep 14

Executive Committee Breakfast meeting on Oct 15

1st Newsletter published on Nov 26

2011-2012 Executive Committee and Sub-committee at AGM, Nov 27

President Longtin discussed with guests at AGM

Sharing with Singapore team on Dec 2

2nd Issue published on Jan 30 Pi Day Celebration on Mar 14

Heads-up for the next newsletter and ISMC 3rd Issue published on Mar 30

How many medals are there? 58

ISMC Result is Ready on May 5 2012

Speech before prize presentation at ISMC

Day Camp Preparation Meeting Jul 30

15


Yeung Hon Wah, Publication Officer of HKJSMS Diocesan Boys’ School

Other Learning Experience Time flies. It’s already a year since I took on the post as the publication officer of the Society. Prior to my present capacity, I had little experience of publication and editorial work. Being the chief editor of this newsletter, I really learn a lot.

For interviews, I need my camera, audio recorder, notebook and pen.

Last July, I learned that the Society was recruiting executive committee members. As I wanted to have some other learning experience, I applied for the post and got accepted. The first executive committee meeting was intimidating to me as I found that most of the members were form 5 students who were a lot more experienced in various areas. Luckily, Lawrence Au, the publication officer of the previous year rendered help and gave me a lot of guidance. He actually helped save a lot of my time in the learning curve. I then became more confident to do the job. While I was doing my year plan, I made reference to newsletters and publications of various organizations. President Longtin and the Executive Committee gave me free-hand to do whatever I thought was appropriate. The green light was on – I went ahead with my plan.

Prior to that, I need to do some research and preparations too.

I remembered one day in February, I got a email from my most favorite primary school teacher, Ms. Yu, who read the January Newsletter. She wrote “ 樺樺:….正好告訴我,你已掌握數學 的美妙!繼���你的理想,繼續努力!…” What else could I ask for in addition to such great encouragement. Taking this opportunity, on behalf of the publication team, I would like to thank all people who have given us advices, guidance and support throughout the year.

Acknowledgement As the chief editor, I have successfully reformed the newsletter to incorporate more diversified and substantial contents such as academic corner, school contribution corner and experienced contestant corner. These new corners have helped attract readers of a wider range of parties from teachers, parents and secondary school students. It has also facilitated a two-way communication between readers and the Society. For instance, students from member schools are interested in contributing articles and are free to share with their experiences and opinions. The interviews with academics in town have given us new ideas. No doubt, the reformed newsletter has helped promote the Society beyond the traditional circle of maths lovers. Doing the newsletter has never been a easy job. I have to say it was full of excitement especially when deadline was approaching. From the moment I picked up the phone to contact interviewees, the long journey began. Taking notes during the interviews, typing transcripts, refining the write-ups … all these were not the most ‘fun’ part. Actually, the ‘fun’ part was editing and formatting which were the most time consuming and sometimes frustrating. The team was lucky to get the supports and advices from a lot of people including Augus Chung of King’s College, Hugo Mak of St. Joseph College and Lawrence Au of Diocesan Boys’ School. They are very resourceful and generous. On top of that, we received very positive and encouraging feedbacks from teachers and fellow students. These feedbacks were the momentum to keep us going.

The editorial team would like to thank the following persons for their contributions: Dr. Koopa Koo, Committee Member of IMO Hong Kong Mr. Kenneth Hung, California Institute of Technology Mr. Au Chi Chun Nelson, St Francis Xavier’s College Mr. Chong Hip Kuen, Queen’s College Mr. Sam Yam, La Salle College Miss Priscilla Chan, Diocesan Girls’ School Mr. Gerry Lam, Diocesan Boys’ School Mr. Ryan Lau, Diocesan Boys’ School Miss Rachel Fung, St Francis’ Canossian College Mr. Jeffrey Hui, La Salle College Mr. Angus Chung, King’s College Mr. Mak, Hugo Wai Leung, St Joseph College Mr. Man Siu Hang, Gordon, CCC Ming Yin College

Need your contributions: The HKJSMS welcomes contributions to the upcoming issues of our newsletters from teachers and fellow students of our member schools. Please feel free to send your submission, such as event write-ups, mathematics problem corners or any related items, which you would like to share with our society members, to jsms.hk@gmail.com.

Past Presidents Corner is under construction JSMS would like to get connected to all the past presidents. May past presidents please drop us an e-mail (jsms.hk@gmail.com) at your convenience. Editorial team: Jeff Siu, Jerry Fong, Yeung Hon Wah

16


HKJSMS newsletter jul 2012