Hong Kong Joint School Mathematics Society jsms.hk@gmail.com| www.hkjsms.org HKJSMS Newsletter 3rd Issue March 2012
From the Editor Since the January newsletter was published, we have been receiving positive and encouraging feedbacks from both teachers and fellow students. Member schools are very active in contributing to the March issue. The theme of this issue is Geometry and the feature interview is about the opportunities and challenges faced by mathematically gifted students. We hope you will find it interesting and insightful. Yeung Hon Wah
Academic Corner – P1 Event Focus Corner – P4 Experienced Contestants Corner – P5 Member Schools Corner – P7 Mathematical and Problem Corner – P9
Academic Corner Interview with Professor Benny Y C Hon Department of Mathematics of the City University of Hong Kong By Yeung Hon Wah
Professor Benny Y. C. Hon is at present a professor in the Mathematics Department of the City University of Hong Kong. His major research interests include numerical computation for solving various types of partial differential equations related to storm surge predication and option price evaluation. He is particularly keen on promoting gifted education in Hong Kong. After organizing the first summer mathcamp for gifted children in 2001 and the first summer project on the mathematics of arts in 2005, he is now serving as an adviser for the Gifted Education Section of the Education and Manpower Bureau and the Hong Kong Association for Parents of Gifted Children and an honourable mentor for the Hong Kong Academy for Gifted Education.
The Opportunities and Challenges faced by mathematically gifted students
In addition to his research on mathematics, Professor Hon has been involved in researches on gifted students in recent years. Through this interview, we would like to seek his points of view towards the opportunities and challenges faced by mathematically gifted students HW: I understand that your primary research interest is Mathematics. However, you have been doing researches on mathematically gifted students recently. Is that right? YC: I am responsible for teaching and research activities relating to mathematics. Giftedness and gifted education will be the focus of educational psychologists. There is a team of academics within the university keen on doing research on mathematically gifted students. I have been involved in such research activities, giving talks and sharing with various parties and authorities, in Hong Kong and the Mainland, such as the Hong Kong Education and Manpower Bureau (Gifted Section), the Hong Kong Academy for Gifted Education, the University of Science and Technology of China, and the Chengdu Normal University in the recent years. I also do sharing with teachers and parents of primary and secondary schools. One of my major focuses is how to define, identify, nurture the mathematically and scientifically gifted children.
HW: Why are you so interested in this area? YC: Well, I would say my elder son Jeffrey inspired me. Since he was a toddler, I noticed his strong observation power and talent in mathematics. It became more obvious after he had grown older and got into primary school. His learning problem in mathematics class when he was studying at P.4, however, confused me and motivated me to organize the first summer mathcamp for gifted children in 2001. The experience I gained from the mathcamp is invaluable in my current research in gifted education.
Characteristics of mathematically gifted students HW: How would you say about mathematically gifted students? Are they typically quick at doing calculation? YC: People used to think that mathematically gifted kids are calculating prodigy. Actually it is a myth. In 1970s, there was a young boy who had very fast computing power and was named 神童輝. He could calculate faster than a calculator. He frequently performed on TV programmes and became the talk of the town. A few years ago, he was interviewed by some newspaper and was found that his academic performance, including mathematics, was not
satisfactory at high school. This shows that mathematically gifted is not simply about quick at calculation. Gifted in Mathematics is, in fact, a special talent on pattern recognition. Take Carl F. Gauss, "the Prince of Mathematicians”, as an example. He was asked to calculate the sum of 1 – 100 when he was a small boy. He immediately realized the pattern (1+100, 2+99, 3+98 …) and gave the right answer (5050). Mind you, during his time, the 18th Century when a lot of the mathematical discoveries were not available. Little kids might only be taught very simple calculation. Let alone the mathematics theories and calculation skills being taught in the present day. I would say that the ability of pattern recognition is inborn and is the principal talent for a person to be named gifted in mathematics. HW: What subjects will these mathematically-gifted kids excel? YC:
They are good at geometry and algebra. These two fields do not require much tedious computation. Again it is all about pattern recognition. For instance, general students may find abstract algebra dealing with three kinds of objects: groups, rings, and fields difficult to comprehend while mathematically gifted students do find it interesting. The mathematically gifted students can easily observe the pattern and structure out of the mathematics problems and then jump to the answer. They seek out patterns. They are interested in finding the beautiful pattern of the universe which also appeals to physicists. Physicists will take a step further to find out the reasons behind the structure and pattern of natural phenomena such as gravitational force, the structure of atoms and particles, etc. That is also the reason why students excel in physics must have mastered mathematics very well. Think about it, if it was not Isaac Newton sitting under the apple tree but some other kid, he might simply eat the apple that hit his head and not bother to find out the reason why the apple dropped onto the ground instead of flying into the sky.
They can spend many hours to read maps. They can give the answers to problems of the number of chicken and rabbits in the same cage (雞兔同籠) but would not be able to tell how they get it. They do not have any idea of linear equation at this age but they can do these because of their pattern recognition power. It is their instinct to find connections among things. At primary 5 or higher, they may be complained by mathematics teachers for not being attentive in classes. In fact, they are bored and may wonder why their classmates spend so much time on problems which appear to be very straight forward to them. In fact, they read a lot of books about mathematics and learn by themselves. That is another reason why they find it so boring in mathematics class. In general, they are happy, naïve and simple. They can be disheveled as clothing is not their priority. Their major focus is only mathematics while they ignore other areas. Dating in high school is relatively rare among these students. Their language abilities are usually weak. Their social and communication skills are weak too. They are regarded as freak and sometimes “high IQ low EQ” type of people. In some cases, they may even appear to be autistic. They are very focused on mathematics while ignoring a lot of things surrounding them.
HW: Why are they good at algebra and geometry?
There may be some extreme cases like Game Theorist, John F Nash, Nobel Prize winner. He was diagnosed with paranoid schizophrenia. He felt that delusional thinking came to him when he was totally immersed into his work.
YC: Mathematics is the study of space, structure and quantity which are included in algebra and geometry. It is not about computation in arithmetic. Students, who are strong at mathematics, must be good at either algebra or geometry or both.
Their memorization abilities are exceptionally high. They are strong at pattern recognition and, at the same time, weak at creative arts. If a person is creative, he will not be bound by any kind of patterns or rules. Therefore an artistic person is usually weak in mathematics.
HW: How do mathematically gifted students behave? Do any indicators suggest their giftedness?
HW: You just mentioned an artistic person is usually weak at mathematics. I know some students, who are good at mathematics, like classical music and play instruments such as the piano and violin very well. YC: Music, in particular classical music, has no conflict with the talent in mathematics. In fact, both of them require good
YC: Their mathematical giftedness can be noticed when they are in kindergartens. At their ages of 4-5 or even younger, most of them like reading maps to find patterns of routes.
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memory and pattern recognition! One of my research collaborators, who is now a mathematics professor at the Tokyo University, always declares that mathematics and music are the center of his life. The artistic talents that will be in conflict with mathematics require good memory and highly creative mind. These artistic persons can be found from distinguished writers, painters, and actors (although I don‟t know Stephen Zhou 周星馳, I am pretty sure that his mathematics won‟t be as good as his creative performance) who are talented in creating, performing, and demonstrating the arts. Mathematicians and musicians do not create but find and appreciate the patterns of the nature. In my recent summer project on the Mathematics of Arts, I had demonstrated in the class how mathematics is related to music.
Assessment of mathematically gifted students HW: People always refer bright students as talented or gifted. Is there any difference between the definitions of „talented‟ and „gifted‟? YC: Firstly let‟s look at what giftedness means. Since the intelligence test was developed in early 1900s, people have been referring high Intelligence Quotient (IQ) scores of over 130 as gifted. IQ test is effective in assessing the logical and analytical abilities of an individual and can basically be divided into verbal part and technical part. Later on, people recognized IQ test overlooks the linguistic, artistic and musical abilities. Different school of thoughts and assessment tools for screening gifted children were developed. Anyway, the most popular view of giftedness is the “three-ring concept” 三環理論. The three rings are the important clusters of human traits: above average ability, creativity and task commitment. All these assessment methods can tell whether the person is all-round-above-average but not the true mathematically gifted. My recent research in gifted education reveals that if a person is regarded as gifted or talented, he must have very good memorization ability. A mathematically gifted person can be defined to be very good in memory and excellent in pattern recognition.
This test can identify the true mathematically gifted student. I have assessed many junior high school students in Hong Kong and mainland China. HW: Why do you need to assess their spatial ability? YC:
Spatial ability reflects their ability in Geometry. It can be assessed by giving questions on maze games and paper folding. A student with high spatial ability can solve a geometric problem in a short time. You can imagine when they approach a geometry problem, the whole picture of the geometry will come to their mind. The answer, to them, can be very obvious. General students may search for formula, use tools including rulers and protractors to solve the problem.
The opportunities and challenges faced by mathematically gifted students HW: I was told that Chinese students are very strong at mathematics but weak at language. Is that true? YC: The nature is fair. Everyone is born with certain talents. However, if one is gifted at one area, he probably puts most of his effort and time in that particular area. He will not have much time spared for other areas. As I mentioned earlier, mathematically gifted students are good at pattern recognition. Chinese characters, such as 一、二、三、木、 林、森, were created in pattern-form while English words, such as “one, two, three, tree, wood, forest”, were not. Therefore, Chinese mathematically gifted children may have difficulty in learning English in their early childhood as the English words make no pattern sense to them. In fact, this is true also for some subjects such as liberal studies and biology. HW: What is the implication then? YC:
The situation can be disastrous. Since they usually do not perform well in non-patterned subjects, the compulsory requirement on general education subjects such as English language and Liberal Studies (LS) in the new 3-3-4 curriculum will eliminate their chances of being admitted into universities.
HW: How can you tell a person is gifted in Mathematics? YC: I have designed a test method to assess the true mathematically gifted students. Different from IQ test which assesses the students‟ verbal and technical abilities, this test consists of three parts: The first part is a questionnaire for the parents to answer. The questions are to find out the behavior of the kids because the “pattern-recognition” ability of mathematically gifted children will have a great influence to their childhood. The second part is a set of questions to test the pattern recognition and spatial ability of the kids. The third part is another set of questions to assess the memorization ability of the kids.
For example, if there is a LS question about the implications of Shenzhou 5 Spacecraft (神州 5 號), they may immediately associate the question with aerodynamics 空氣動力學 instead of economic issues. HW: But they can still demonstrate their ability in Mathematics, can‟t they? YC: With the world trend of fair education, the current curriculum has simplified the syllabus of both core mathematics and even extended modules. The scope of geometry and algebra being taught in secondary schools in present day is rather limited compared to what I had in my secondary school education in Hong Kong. To some extent, geometry has disappeared from Hong Kong secondary
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What should be done to help them excel? school curriculum. Hong Kongâ€&#x;s school curriculum caters to general students and is rather rigid. Mathematically gifted students have no platform to demonstrate their talent in mathematics. On the opposite, these mathematically gifted students may not even obtain a 5** grade in mathematics at the public examination. HW: What about students who get straights As including mathematics in public examinations? YC: I am sure those who get 10 As are not mathematically gifted. They are gifted in multi-talents. Under the current examination requirements, students with good memory can obtain high grade in mathematics by memorizing solutions of past examination papers. For mathematically gifted students, they may easily come up with the correct answers but do not take the required steps to reach the solution required in the examination. In another word, their competitive edge becomes their drawback under the current system. To the 10 As students, the curriculum might have misled them too. I have learned that some 10 As students, who excelled at general, additional and pure mathematics, chose mathematics as their major in bachelor degree study in universities such as Princeton. Just in the first year when the real mathematics subject such as Algebra was taught, they found it difficult to comprehend. Later on they regretted because they found it was far beyond their ability. They ended up with switching to another field.
HW: It seems they are facing a lot of challenges. be done by the government to help them?
What should
YC: I hope that the education policy and school curriculum should acknowledge their special needs and should be reviewed to allow some flexibility to them. The existing curriculum forces many mathematically gifted students either losing chance to be admitted into universities or leaving Hong Kong to further their mathematical studies in some other countries. HW: What can schools and teachers do to help? YC:
Sometimes these students can be annoying especially they appear not attentive in class and even disturb others. Some teachers may even think these students are autistic, hyperactive or ADHD and refer them to psychologists. There are cases which students being misdiagnosed and given unnecessary treatments. Consequently, their development in mathematics is hindered. I think teachers may pay more effort to understand the reason behind their behavior instead of jumping to conclusion too soon. Schools may consider various means to help them explore their potential.
HW: What about parents? YC: Parents have to acknowledge the characteristics of their kids and address their needs early. HW: Thank you very much for your time for this interview. YC: Thanks.
Event Focus Corner Inter School Mathematics Contest (ISMC), the most important annual event of JSMS, will be held on May 5 (Saturday). This year, interesting post competition events will also be organized. To encourage more students to participate, special discount will be offered to both member and non-member schools sending more than one team.
Deadline for application is April 16.
March 14 (Ď€ Day) was one of the most important day for maths lovers. Our members celebrated the day with their friends, teachers and family members by sharing this pie at 1:59 pm.
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Experienced Contestants Corner Interview with Mr. Bobby Poon Head of Information Services & Technology, St. Paul‟s College By Jerry Fong
Mr. Bobby Poon represented Hong Kong to participate in IMO in 1994, 1995 and 1996 and received one silver award and two bronze awards. He earned an MPhil in Mathematics at the University of Hong Kong. Since 1999, he has been an IMO trainer for the Hong Kong Team. He was also the Deputy Leader of the Hong Kong Team in 2005, 2006 and 2011. He is now the mathematics teacher and the Head of Information Services & Technology at St. Paul’s College.
JF: You took part in IMO competitions at high school and now you are both a mathematics teacher and a MO trainer. Would you mind sharing with us your first encounter with MO and why you took the career path as a mathematics teacher and MO trainer? BP: When I was young, I liked to learn mathematics by myself through watching ETV. I remember I learnt solving quadratic equation in primary 5. In F.2 I took part in an inter-class competition and was able to win over some F.3 classes. The Chairman of the Mathematics Society in my school then brought me to IMO training secretly (it was held in BUHK). He also gave me some MO material which was very precious at that time as there was no internet, and book stores in HK did not have Olympiad Mathematics books. I finally got to IMO training officially in F.4 and was able to represent Hong Kong in F.5, 6 and 7. At my time, no IMO students continued in mathematics in University. They usually studied engineering, medicine, accounts or actuarial science. I was admitted to the Faculty of Architecture in HKU, after my parents‟ rejection of letting me take Mathematics. I graduated in 1999 and at that time the financial situation in HK was so bad that even architectural firms were closing down. So I was able to convince my parents that taking mathematics and being a teacher was not a bad choice. I then took a year in mathematics undergrad and moved onto MPhil in mathematics in HKU. CJ (Mr. Cesar Jose Alaban) has been giving advices throughout as a teacher and a friend. He also encouraged me to talk a few lessons in IMO. So I took the job and since then I have been around the IMO training. JF: You have been training Hong Kong MOers for quite some time. How are MOers different from general students?
BP: I would say that there is no big difference between MOers and general students. MOers are just being interested and strong in mathematics. As usual, some of them are naughty, some are quite talkative and some are very quiet. However, I notice that some of the MOers are probably too focused on reading and doing mathematics that they ignore some social life. The problem is not very serious.
Mr. Bobby Poon has been an IMO trainer for the Hong Kong Team and was the Deputy Leader of the Hong Kong Team in 2005, 2006 and 2011. “My advice is to think more when learning and solving geometric problem. It is different from other areas of Maths where use of formulas and numerical calculation is more important …”
JF: Universities in China take MO competition results into consideration when it comes to enrolment. Do you think universities here should do the same? BP: Yes. But the problem is not just about University. I once had a student who represented Hong Kong in IMO and IPhO, but because of getting Bs in HKCEE, he was not allowed to take mathematics and physics in F.6!! (You know his school has good HKCEE results.)
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JF: Most of your training sessions are about Geometry. you so interested in Geometry?
Why are
visual intuition and logical reasoning in order to be successful in solving geometric problem.
BP: In F.2, I had a very good mathematics teacher who inspired me in Geometry and encouraged me to read a lot. I then spent some years borrowing geometry books from the public library for studying. Later on, during my MO training, I was also being fascinated by the trainer, Mr. Tsang Yuk Sing (now the president of LEGCO).
My advice, as usual, is to think more when learning and solving geometric problem. It is different from other areas of mathematics where use of formulas and numerical calculation is more important.
I enjoy solving Geometric Problems as the questions are clearly visible and the arguments are usually indirect. However, once the explanation is understood, it becomes so clear that the reason why we have such a geometric phenomenon. I like to use the nine-point circle as an example. For the usually proof, checking the nine points are on the same circle are just boring. Once you understand that it is actually homothety of the circumcentre from the orthocentre, everything is now explained clearly. For me, studying Geometry is just like understanding the world. The physical world is created under certain rules and it is for us to explore the relationships between them. JF: Since 2008, HKMO competition has added a separate paper on “Geometry Construction”. So HKMO is giving more focus on Geometry. Is that right? BP: It is actually the Education and Manpower Bureau of Hong Kong (EDB) who pays more emphasis on Geometry. We have found that proving in Geometry is good for our student as a good mental training. However, it is always hard to have geometric proofs in ordinary tests and examinations. That‟s why we are finding alternative ways to enhance this topic. Geometric Construction is part of this Geometry movement. JF: It seems that Core Mathematics does not cover much about geometry. Would you suggest general students should be taught more about Geometry? BP: Yes and No. I would say that the topics covered in Geometry are very rich indeed in our Core Mathematics. What is lacking is a good strategy for students to talk and explore more on that. As it is very difficult to put a geometric proof in public examinations, students and teachers find it less important. Of course, we should not be driven by these examinations. So I encourage more projects, exploration in Geometry to give students more chance to play around with Geometry. JF: We notice that a lot of students find Geometry very difficult. Do you have any advice to them? BP: Learning Geometry is actually a mental training. There is no short cut to train our mind. We gain experience in solving each Geometric problem. We have to combine both
JF: How would you compare the MOers of the present day and those in the past? BP: No much difference. JF: It seems that more students especially primary students are taking part into MO competitions in Hong Kong. Why is that? BP: This is a social issue. Parents of primary students want extra-curricular activities and certificate for the kids so that their sons and daughters may become more competitive. The syllabus in primary mathematics is too simple to the top students, so they want to learn more about mathematics. However, learning only general mathematics has no sense of achievement to them. So learning MO Mathematics with competitions give them achievement and certificates. JF: Thank you very much for your time and your sharing. BP: Thank you.
Upcoming competitions 第二十九屆香港數學競賽（2011 / 2012）HKMO 決賽: 2012 年 4 月 21 日（星期六）
Inter-School Mathematics Contest 2012 Date: Venue: Address: Time:
5th May 2012 (Saturday) Diocesan Boys' School, School Hall 131, Argyle Street, Mong Kok, Kowloon, 09:00 – 17:00
第十一屆培正數學邀請賽 決賽: 2012 年 5 月 12 日（星期六） IMO Preliminary Selection Contest Hong Kong 2012 Deadline for application: Apr 19, 2012 Competition Date: May 26th 2012 (Sat)
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.
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Member Schools Corner Angus Chung (Former Executive Committee Member of HKJSMS) King‟s College
How is the Hong Kong IMO Team Selected?
When you read the newspaper in July, you may find news about Hong Kong participates in the International Mathematical Olympiad (IMO). Every year, a team of 6 secondary students is chosen to represent Hong Kong for the competition. How are the team members selected? Actually, every team member as well as reserved member has to achieve good results in 5 tests. The first and the easiest one, IMO Preliminary Selection Contest (Hong Kong), is held in late May. It is a 3-hour contest with 20 questions which require answers only. Those who perform well in the contest are given the opportunity to participate in the training of IMO Hong Kong team. The training lasts for 8 months, from July to February. Every Saturday afternoon (or some weekdays in the summer vacation) trainers from Hong Kong IMO Committee teach the trainees some advanced Mathematical knowledge, such as Number Theory or Combinatorics. There are 3 phases of trainings, each with 2 different levels. The basic level introduces some elementary concepts to students, while the advanced level focuses on some sophisticated and complex techniques. During the training, 4 tests are given. All the tests, which consist of a few questions that require a complete proof, last for 3 hours with the exception of the last one, which takes 4 hours. The later tests are generally more difficult than the earlier ones, so they count more when the participants are finally ranked.
Test 1, which is held in late August, contains 6 questions, while Test 2 and the Hong Kong (China) Mathematics Olympiad (CHKMO, the third test) contain 4 questions each. Test 2 is held in late October, and CHKMO is held in early December. Apart from being a selection test, CHKMO is also a chance for students to meet the China and Singapore team, as both countries join the test too. In Dec 2011, a joint sharing and a gathering are organized for students from the three regions. The last test, Asian Pacific Mathematics Olympiad (APMO), is a regional 5-question contest involving countries from Asian Pacific Region and also the USA. Every year, APMO is held in the afternoon of the second Monday of March for participating countries in the North and South Americas, and in the morning of the second Tuesday of March for participating countries on the Western Pacific and in Asia, including Hong Kong. In each region, at most 7 students can be given a gold, silver or bronze award. Therefore, it is surely a great honor for one to obtain an award in APMO. After all the tests are passed, the students are ranked by the Hong Kong IMO Committee. 6 team members and 6 reserved members will be chosen and they will attend some further training in June. After that, the team members will participate in the International Mathematical Olympiad. This year, all the trainings and tests have been over. Let‟s hope that the Hong Kong IMO team will achieve great results in Argentina.
Alice Yuen St. Francis‟ Canossian College
Game Theory Fun Day On 14th-15th March, a fun day concerning Game Theory was held in our school. It was opened for all students. Nerds, geeks or simply passersby were all welcome. Through the classic example, Prisoner Dilemma, students were given introduction to the basic terminologies in Game Theory and how strategies were formed. Getting introduced how Game Theory works, participants were then required to tackle various classic problems and real life dilemmas. And since 14th March was Pi day, tens of handmade HAPPY PI DAY files were distributed to students as celebration. It was hoped that students‟ interests in mathematics can be aroused.
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To Yick Hin Wah Yan College Kowloon
Sudokus and Rubik’s Cube Competitions To enhance the interest of our members in the logical puzzle, The latest competition has been scheduled for March 30 (Friday).
Sudoku, and to provide an event to let the Sudoku addicts compete against each other, our Club organized three Sudoku Competitions this year, and was rewarded with the enthusiastic responses of the members. The members were required to complete two Sudokus within a limited time, and the first three contestants who completed the Sudokus in the fastest time and were completely correct were the winners of these competitions. They were awarded with abundant prizes. In these competitions, the contestants focused on every number on the Sudoku and tried their very best to be the fastest to solve the puzzle correctly. Apart from the Sudoku Competitions, our Club organized a Rubik‟s Cube Competition this year. We received active participation from our members. Similar to the Sudoku Competition, the first three contestants who solved their Rubik‟s Cubes in the fastest time became the winners of this competition and were awarded prizes
Sam Yam La Salle College
Tag
You must have played tag (捉) when you are small.
One of the children needed to count to ten, and then try to catch the other children in the game. Do you still remember what would you do, while you know your opponent is faster than you, and he/she is trying to catch you? Yes, you will hide behind some obstacle, and when your opponents try to catch you from the left path, you run away at right, and if he tries to catch you from right, you will run to the left. These obstacles usually delay the catcher, and buy you some precious time to escape. Recently, I have come across an interesting question about tag.
Mickey Mouse is playing tag with Sun Wukong (孫悟空). Mickey hides in a circle “protected area”, which Sun Wukong cannot enter the region. However, Mickey can‟t stay forever in the region, he needs to find a way out. Sun Wukong, of course is faster than Mickey. The problem comes: Will Mickey be able to escape, when Sun Wukong is 2 times, 3 times, 4 times or even 5 times faster than Mickey? If Mickey escapes, what will his path be like?
Our dear readers, try it out!
Reference Source: Problem of the week 5, La Salle College Mathematics Society, 07-08.
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Mathematical & Problem Corner Brief introduction to Geometry You all might have struggles when learning Geometry in
a)
the past. However, in the past, mathematicians were so smart that over 3000 years, they developed much on it. This article will include some introduction of a few mathematicians who were very good at geometry.
Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.
b)
When a straight line set up on a straight line makes the adjacent angles equal to one another, each of the equal angles is right, and the straight line standing on the other is called a perpendicular to that on which it stands.
1. Pythagoras (570 BC – 495 BC) He was a famous Greek philosopher and mathematician. He was so enthusiastic in learning that he had travelled to a lot of places in order to learn more when he was young. In his life, he had a lot of contributions in Mathematics, such as in the number theory and the famous Pythagoras Theorem.
(Fig.1) A graphical representation of the proof of the Pythagoras Theorem.
Although these statements may seem silly to some of you if you have learnt geometry before, they were very important to mathematicians investigating the world of geometry after Euclid. Also, he was the first mathematician to summarize these definitions, in a famous book called “Element”. He also proposed a lot of propositions of requiring people drawing only using straight edge (Ruler) and compasses. An example is shown below, which he demonstrates how to construct an icosahedron and comprehend it in a sphere. (It is called Proposition 16 in the “Elements” Book XIII).
Source: http://www.mathsisfun.com/definitions/pythagoras-theorem.html
He also contributed a lot in music by linking music with Mathematics, such as the string length for producing different sounds in a musical instrument (Example: violin). More interesting is that, he discovered that due to his interest in listening to blacksmiths hammering. He thought that the different sounds were beautiful so he wanted to discover how the sounds were produced. He had been famous for some time and he had a lot of followers in Greece. They are called “The Pythagoreans”. However, it was believed that at around 500 BC an uprising was held against the Pythagoreans and Pythagoras was being killed. However, his discoveries contributed a lot in different areas of science.
Fig.2 Source: http://aleph0.clarku.edu/~djoyce/java/elements/bookXIII/propXIII16.html
He was very famous at that time. Many students followed him to learn. In addition to mathematics, Euclid was also famous for energetic in doing scientific investigations in his life. However, his contributions in geometry were much more important than that as nowadays, people investigate geometry based on his principles! This shows how great Euclid‟s work is! 3. Leonhard Euler (AD 1707 – AD 1783)
2. Euclid (Year: Unknown, probably between 300 BC and 200 BC) He was a Greek mathematician. He was so important in the development of geometry so he was often referred as the “Father of Geometry”. Many geometrical definitions were defined by him. Here are a few examples:
Leonhard Euler was one of the most famous mathematicians in the last few centuries. He was well known for investigating much in different branches of mathematics. His contributions to geometry will be discussed in this article. He had proposed a lot of theories in coordinate geometry, such as relating complex numbers to trigonometric relations, with discovering the exponential function for complex
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numbers. He related them and leaded to the discovery of the Euler‟s formula, which means, for any real number a, the relation eia = cos a + i sin a holds in any exponential function for complex numbers.
(Fig. 3) Euler relates his formula with geometry
(Fig.4) The “Königsberg bridge problem”
Source: www.ie.u-ryukyu.ac.jp/~wada/design07/spec_e.html
He discovered the graph theory at the same time too. There was an interesting story of how he discovered this. One day he arrived a city called Königsberg, Prussia, which consisted 2 main islands connected by 7 bridges. He wanted to investigate whether there was a path which each bridge could be crossed once, and the challenger could return to the starting point. Later he proved that it was not possible. This question was named as “Königsberg bridge problem” by the people living there. Later he proposed the graph theory based on this idea.
Euler had also done a lot of investigations in Physics, algebra too. He was one of the most successful mathematicians in the recent centuries. Conclusion In addition to the mathematicians mentioned in this article, there are also a lot of other mathematicians who are enthusiastic in investigating geometry. If you are interested in geometry, perhaps you can be one of them in the future.
Article contributed by Jeff Siu
Tackling Geometry Problems in Mathematical Olympiad A) Introduction
In
attending Mathematical Olympiad competitions, we usually come across three types of geometry problems, namely Transformational Geometry, Deductive Geometry (do proofs), and doing Geometry evaluation. Some people even try to use coordinate geometry techniques to provide a complete proof of a pure geometry problem. Actually, some geometry problems in MO competition can be further established and applied in daily lives, such as using trigonometric and spherical trigonometry in the development of astronomy, and making use of “rigid” geometrical shapes (such as lines and spheres) to find out the densest packing of spheres of equal size in space, i.e. the Kepler Conjecture.
Figure 1. Some of the applications of geometry (Kepler Conjecture, Astronomy and Tranformation)
B) Basic Useful theorems and formulae in tackling Olympiad Geometry Questions There are multitude geometry formulae that can be applied in Math Olympiad, such as the Power Chord Theorem, Ceva‟s Theorem, Stewart‟s Theorem, or even Brianchon‟s
Theorem, Pascal‟s Theorem and Desargues‟ Theorem, which are useful tools in helping us finding a way to a complete solution. Here are details of some more important theorems.
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(I)
C)
Ceva‟s Theorem Let ABC be a triangle and D, E and F be points on lines BC, CA and AB respectively, if AD, BE and CF are concurrent, we can have AF BD CE 1 FB DC EA
Tackling real competition questions
As mentioned before, we mainly focus on doing proofs in real mathematical Olympiad competition. Here are some interesting questions in my question bank. Hope all of you can get some insights from them. (The solution provided may not be the best way, but it should be the easiest way that most people can think of when encountering such problems) Example 1.
Figure 2.1 Ceva’s Theorem
(II)
Menelaus„ Theorem Let ABC be a triangle and D, E and F be points on lines BC,
(2012 Japan Mathematical Olympiad Finals Problem 4)
Question : Given 2 triangles PAB and PCD such that PA = PB, PC = PD, P, A, C and B, P, D are collinear in this order respectively. The circle S1 passing through A, C intersects with circle S2 passing through B, D at distinct points X and Y. Show that the circumcentre of the triangle PXY is the midpoint of the centres of S1 and S2.
CA and AB respectively, if D, E and F are collinear, we can Solution: Let O1 and O2 be the circumcentres of S1 and S2 respectively, and M be midpoint of O1O2. Then, let MO1 = MO2 = p, and r1 and r2 be the radii of S1 and S2.By Power chord theorem,
have AF BD CE 1 FB DC EA
Figure 2.2 Menelaus‘ Theorem
PA PC PO1 r1 , PB PD r2 PO2 2
2
2
2
……….(1) Note that XO12 XO2 2 YO12 YO2 2 r12 r2 2 , Therefore, PO1 PO2 r1 r2 2
2
2
2
So, by Stewart‟s Theorem,
(III) Pascal‟s Theorem
2 PM 2 2 p 2 PO1 PO2 2
Given cyclic points A, B, C, D, E, F and let
r1 r2 2
CD AF R, BC EF Q, AB DE P
2
2
2 XM 2 p 2 2YM 2 2 p 2 2
.
Then P, Q and R are collinear. (Note: Degenerate case take place when 2 of the cyclic points coincide, they determine the tangent line at the point)
Figure 2.3 Pascal’s Theorem
(IV) Brianchon‟s Theorem Given points A, B, C, D, E, F such that AB, BC, CD, DE, EF, FA are tangents to a fixed circle. Then AD, BE and CF are concurrent. (Note: Degenerate case takes place when 2 of the points coincide, they determine a tangent line if they lie on the circle) Figure 2.4 Brianchon’s Theorem
Hence XM = YM = PM, so M is the circumcentre of triangle PXY.
(QED)
(Analysis: This is a rather straightforward Olympiad Geometry question, in which using Power Chord and Stewart’s Theorem can directly tackle the question completely.)
Example 2. (International Zhautykov Olympiad 2012 Day 2 Question 2) Question: Equilateral triangles ACB’ and BDC’ are drawn on the diagonals of a convex quadrilateral ABCD so that B and B’ are on the same side of AC, and C and C’ are on the same sides of BD. Find BAD CDA if B’C’ = AB + CD. Solution (In pure geometry way): Construct an equilateral triangle AMD such that M and C are on the same side of AD. Since DAC MAB', DAC MAB' Similarly, as
MDC' ADB, ADB MDC'
Therefore, MB’ = CD and MC’ = AB, so B’C’ = AB + CD = MC’ + MB’, which implies that M B'C ' There are a lot more….You can find and understand the magic of geometry in the Internet at free time.
.
0 0 Hence, BAD CDA C ' MD AMB' 180 AMD 120 (QED)
Hence XM =This YM =isPM, so M is straightforward the circumcentre of triangle (Analysis: another Olympiad PXY. (QED) (Analysis: This is a rather straightforward Olympiad Geometry question, in which using Power Chord and
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Geometry question as it requires competitors to find out the answer only…In normal situation, answer only contains 1 mark, while steps and explanations for claims contain 6 marks, which is far more vital. Furthermore, we can solve the above question by using complex number techniques as follows) 1 3 1 3 Z B ' Z C ' i Z A Z B i Z C Z D Z A Z B Z C Z D 2 2 2 2
Hence,
1 1 3 3 Z 1 i Z A Z B , Z 2 i Z C Z D 2 2 2 2 which are of the same direction, so the sum of required angles is 1200. (QED) After tackling these 2 rather easier problems, there are a lot to go in order to tackle all geometry questions successfully in Mathematical Olympiads. The following is a more complicated geometry shortlist problem for International Mathematical Olympiad 2006. Example 3. G3.
(IMO 2006 Shortlist G3)
Let ABCDE be a convex pentagon such that
, AQ CM DR 1 QC MD AR
which can be further reduced to
CM = MD, which completes the proof. (Analysis: Actually, the relation (**) can be directly deduced after knowing quadrilaterals ABCD and ACDE are similar) After gaining insights from the above three different kinds of geometry problems, I think all of you can understand and apply some basic techniques in tackling related problems….Here are 2 of them that worth trying. Hope you can enjoy the fun of geometry via tackling questions and applying theorems!! Extra Questions: 1. (The 9th Hong Kong (China) Mathematical Olympiad 2006 Question 3) A convex quadrilateral ABCD with AC≠BD is inscribed in a circle with center O. Let E be the intersection of diagonals AC and BD. If P is a point inside ABCD such that ∠PAB+∠PCB+∠PBC+∠PDC=900. Prove that O, P and E are collinear. (Hint : Use Pascal’s Theorem)
BAC CAD DAE and
ABC ACD ADE .The diagonals BD and CE meet at P. Prove that the line AP bisects the side CD. Solution: Let the diagonals AC and BD meet at Q, the diagonals AD and CE meet at R, and the ray AP meet the side CD at M. To prove : CM = MD holds.
2. (Italy ITAMO 2011 Question 4) ABCD is a convex quadrilateral. P is the intersection of external bisectors of ∠DAC and∠DBC. Prove that ∠APD=∠BPC if and only if AD + AC = BC + BD. (Answers will be provided in the next newsletter.)
Article and problems contributed by Mr. Mak Hugo Wai Leung, St.Joseph College Acknowledgement The editorial team would like to thank the following persons for their contributions :
Proof: The idea is to show that Q and R divide AC and AD in the same ratio, i.e.
AQ AR CQ DR
(which is equivalent to QR //
CD) (**) The given angle equalities imply that triangles ABC, ACD and ADE are all similar, so we have .
AB AC AD AC AD AE
it follows
Since
AB AD AC AE
BAD BAC CAD CAD DAE CAE
Need your contributions:
from that the triangles ABD and
ACE are also similar. Their angle bisectors in A are AQ and AR respectively, so AC AQ AD AR
Professor Benny CY Hon, City University of HK Mr. Bobby Poon, St. Paul‟s College Miss Alice Yuen, St. Francis Canossian College Mr. Angus Chung, King‟s College Mr. To Yick Hin, Wah Yan College, Kowloon Mr. Hugo Mak, St. Joseph College Mr. Ryan Lau, Diocesan Boys‟ School Mr. Sam Yam, La Salle College
AB AQ AC AR
As ,
AB AC AC AD
we get ,
which is (**).
Finally, using Ceva‟s Theorem for triangle ACD yields
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. Deadline for submission to our next issue is April 30, 2012. Editorial team: Jeff Siu, Jerry Fong, Yeung Hon Wah
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