HKJSMS newsletter jan 2012

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Hong Kong Joint School Mathematics Society jsms.hk@gmail.com| www.hkjsms.org HKJSMS Newsletter 2nd Issue January 2012

From the Editor Dear readers, entering 2012 and the Year of Dragon, the HKJSMS newsletter is taking a great leap forward. Thanks to the president for supporting my ideas of introducing new “corners” in this publication and to teacher advisors and former committee members for their contributions and guidance. Please enjoy reading. Yeung Hon Wah

Academic Corner – P1 Experienced Contestants Corner – P4 Member Schools Corner – P5 Event Focus Corner – P6 Mathematical and Problem Corner – P7

Academic Corner Interview with Dr. K Y Li (李健賢), Associate Professor, Department of Mathematics, Hong Kong University of Science and Technology, and teacher advisor to HKJSMS By Yeung Hon Wah, HW: Some students are wondering whether they should take mathematics as a major in university. Can you give us some idea about the career path of mathematics majors and what mathematics can do for us? KY:

Mathematics is all about deductive reasoning and applicability. Mathematics is everywhere in our daily lives. For example, we need to calculate how much we need to pay for goods we buy. In weather forecast, the analysts working in observatory need to attain certain level of mathematical skills in order to understand what will happen from the data generated by their very sophisticated instruments and tools. As simple as making a phone call, it actually involves signaling which calls for numerous calculations and digitized codes in transmission. For electronic communication, error correction is very important to ensure accuracy. All these are parts of our daily lives and in this IT era, many technical decisions require a good sense of mathematics. When I was in secondary school, I loved mathematics simply because I was good at it. However, I did not really know many practical applications. Some students may be good in one area, say algebra, but not strong in others. So it is too early to say whether a high-school student, who scores high in mathematics, is capable of coping with different streams of mathematics in university. Having said that, I think the practicality and application side of mathematics would inspire students to pursue further. The more he or she explores, the more passionate he or she will be. In considering mathematics as a major at the university level, a student should focus more on his or her

personal interest and strength. For careers, it depends on what the students can select. HW: I understand that the Department of Mathematics of the HKUST offers several tracks of mathematics courses. Could you tell us more about them? KY: There are two tracks of pure mathematics. One track provides regular scope which many other universities offer. The advanced track is the other, which is for students who plan to pursue postgraduate degrees to become professional mathematicians. There is also an applied mathematics track that includes statistics and financial. Students taking this track usually are pursuing careers in the finance related fields. We also have a computer science track. Those who like programming and IT related jobs may select this track. There is also the mathematics and economics

program. Some of those who selected this program go on to PhD programs in Economics or MSc in Finance. They will most likely work in the finance related fields later.

KY:

HW: Among the many tracks, which track do students like the most? KY:

Actually their interests are quite diverse. In recent years, a lot of students like taking the computer science and finance tracks. However, the pure mathematics track remains to be the most popular as almost up to 50% of the students of our department choose pure mathematics. They choose pure mathematics mainly because they are good at mathematics while they have less knowledge about computer science, statistics and other disciplines. To play safe, they choose pure mathematics. After the first year, some students may get interested in other areas and switch tracks. Students may apply to switch tracks or other programs in UST provided the corresponding department or school accepts them based on their academic performances.

HW: Some people say that people good at mathematics are likely good at physics. Why is that? KY:

HW: I used to think that fewer students will choose pure mathematics because it is rather difficult and not so practical comparing to the other tracks. KY:

Whether the subject is difficult, it is all about how much effort an individual has put into his or her time management and method of study. Students are not advised to rely on memorization or study solely on their own because such practice may not be suitable in dealing with university studies. Students are advised to form study groups. As the depth and breadth of the subjects increase year after year, it is crucial for students to understand the basic concepts thoroughly and do more exercises to increase their breadth in the first year. Mentoring is very important. So students should take initiatives to approach professors, teaching assistants and tutors for help on a frequent basis. The more you interact with the teachers, the clearer your understanding should be. Then you will find it easier to approach assignments such as doing analysis, solving problems and writing proofs.

HW: I would like to ask if some students switch to other tracks because they find pure mathematics very difficult. KY:

No, it is not very common actually. Like all other tracks, pure mathematics track offers students with different courses, such as algebra and number theory. Most likely they perform well in more than one area and will focus on the areas they like the most. Some of them want to study more advanced topics like topology. Students tend to study in groups for mutual support. However, some students decide to switch track for career consideration but not because of the difficulties of the subject. For example, a pure mathematics student will switch to the computer science track because of the job opportunities available later.

HW: Other than being a teacher or accountant, what will be the choices available to students studying mathematics?

We have students working at banks as mortgage managers. Some are working as programmers for finance, insurance, securities or even logistic companies to design programs for information security, business transactions, data administration and access security. In fact, people studying mathematics and computer science have a higher employability. For instance, as simply as making a transaction at a bank, the system has to compute and executes tons of lines of codes into different accounts. All these jobs require people with mathematics, computer and financial knowledge. At hospitals and pharmaceutical companies, studying drug effectiveness and analyzing trends of disease occurrences require statistics expertise. Our tracks increase the employability of our students.

Physics and mathematics have the same origin. Mathematics knowledge is fundamental to calculating, analyzing and studying the phenomena of nature. Isaac Newton and Carl Gauss were both mathematicians and physicists. In fact, there are some overlapping between mathematics and physics. However, when comes to experiments, it will be another story. Mathematicians may not find them so interesting.

HW: Some people suggest that people good at mathematics are relatively weak at languages such as English and Chinese. Do you agree? KY:

People like mathematics may like reading mathematics related books and tend not to spend so much time on socializing or writing while focusing more on developing and practicing skills on solving mathematics problems. The language being used in these books is rather simple and straightforward. On the other hand, there is always a finite solution to every mathematics problems. As for learning language, students are expected to express their ideas effectively. However, students may feel less confident as there is no definite answer to assignments of language courses. Gradually they lose interest and motive in such subjects.

HW: I understand that you studied in the USA. Comparing to the students of the USA, are Hong Kong students’ ability in mathematics higher? KY: Yes. Hong Kong has been following the British system even after the reunification to China. Initially the Hong Kong government did not budget for many university quotas like what we have today. The secondary curriculum was to prepare students to work after high school. Therefore the breadth and depth of all subjects, especially mathematics, were expected to cover as much as possible so as to get the students well-equipped for working. On the other hand, universities at that time were prepared to train the most talented. Due to these historical reasons, Hong Kong

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students have been learning more. That’s why they perform better in mathematics than students of other countries. Today, most mathematics programs offered by Hong Kong universities cover more and are taught much faster than those in the USA. If you compare assignments of both, you will find assignments from Hong Kong universities are more difficult. In my opinion, the American secondary education system does not work satisfactorily. It is not surprising to find many undergraduate students, except those from the East and West coasts, still have problem in doing simple arithmetic. American university students, except those from the top ten like Harvard, Stanford, Berkeley, MIT, etc., do not perform so well in mathematics as Hong Kong students at universities on the average. This trend is also noticed among postgraduate students. HW: Most HKJSMS members like reading books on mathematics. However, we have difficulties in finding books right for our level. Do you have any advice to us? KY:

You may search on the websites of mathematics societies of other countries. The Mathematics Association of America (MAA), who publishes mathematics books for university students, is good at presenting difficult concepts in an interesting and reader-friendly manner. Students may also come to the UST library which is the only Hong Kong university library open to public. When you do the search, you will find, even on the same topic, many different books written by different authors. You may sample these books to find one suitable for you.

When I was a university student, the mathematics teachers gave me a list of reference books written by world-renowned mathematicians. However, they were too difficult for me to comprehend. So I started with reading some simple ones first. As soon as I had acquired the fundamental concepts, I could then read the more advanced books. I would advise the freshmen to read more about the topics they are going to study. Reading and practicing problem-solving skills are crucial to acquiring mathematics concepts. HW: Talking about writing mathematic proofs, do students find it very difficult and boring? KY:

In the later stage of mathematic programs, students will learn to interpret and explain certain phenomena or do pattern reasoning. For instance, when we encounter quintic (5th degree) polynomial equations, we do not have any formula to solve them. To study them, we have to be good at inductive and deductive reasoning. Analytic approach is more desirable in this case. In learning pure mathematics, you will encounter more problems about proving. Therefore logical thinking is vital to it. Writing mathematical proofs is the most challenging and interesting part. Nowadays, there are still many mathematical hypotheses need to be proved. “All even integers greater than two are sums of two prime numbers” is one such example.

HW: We always think that giving short solution is good to save time and won’t bore anyone. Is that right? KY: A short solution is good. However, it takes people more time to come up with a short solution. In 1994 Andrew Wiles won awards for successfully proofing Fermat’s Last Theorem. Actually he used seven years to develop many new techniques to explain his 200 page proof. Some people may pursue a shorter proof, but it may take them decades. In learning different subjects, your time spent will depend on the subject areas. Take number theory as an example, you may find modulo arithmetic easy to learn. However, some parts of algebraic number theory may take you a decade to learn all the basics because its level of abstraction has accumulated to a great depth over centuries. Basically before you study the work of great mathematicians, you need to look into their previous research works first. As a mathematician with religious faith, I truly appreciate the beauty of mathematics and the laws of nature God has designed with love for us. HW: Could you give us some advice on how to learn mathematics better? KY:

Reading is a good option. However, practice is more important. I would suggest students to participate in mathematics competitions to get more exposure to different types of problems and to interact with people. Going through the solutions to the problems is very important. By doing this, you will improve your thinking and analytic skills.

HW: Last but not least, what is your expectation from HKJSMS? KY:

I think the joint school competition and day camp are very meaningful events. The newsletter is very impressive. It also serves as a platform for members to share and express ideas. I think the society may also interview renowned mathematicians or invite them to give talks on certain topics. If possible, the society may post the videos on the JSMS Facebook and website. That will be very meaningful. Try your best to learn from the experience of the elder generations such as the Pui Ching mathematics training team members of the 1950-1960s when a number of world famous mathematicians were incubated. I wish HKJSMS the best in 2012.

HW: Thank you.

2012 3

Experienced Contestants Corner Interview with Mr. Ching Tak Wing (程垡永), gold medalist of IMO 2009, 2010 and 2011 By Jerry Fong Ching Tak Wing is one of the former IMO Hong Kong team members. He is now studying Mathematics at the University of Hong Kong. In this academic year, he becomes a trainer for the IMO training . JF: You are very good at Mathematics. You have been representing Hong Kong in international competitions. How would you compare the mathematics we are learning at school and Mathematics Olympiad? TW: Mathematics taught at schools is quite different from Mathematics Olympiad (MO). I would say that school-based Mathematics emphasizes on knowledge while MO emphasizes on skills. For Mathematics in schools, the target audience is the students who know little or even nothing about the Mathematics knowledge. What the teachers have to do is to tell them 'what it is'. Hence, the students are not encouraged to think a lot during lessons. They only need to learn the definitions and do some problems in practising the usage of the things they have learnt. However, for MO, the students are already talented in mathematics. They need not to learn too much extra knowledge, possibly except some more advanced ones. But MO lessons encourage the students to think more deeply through doing various types of problems. This helps them improve their thinking skills. More often, there are a lot of ways in solving a particular problem. This even encourages the thinking and creativity of students. Also, that's the reason why I love MO a lot, though I'm not saying that I don't like the Mathematics taught at school! JF: I know that you are conducting training for Moers. Have you got some unforgettable experience and inspiration of being a trainer? TW: Why do I become a trainer for IMO training? One of the reasons is that I hope to pass on my knowledge to the younger students. After years of being a trainee and receiving knowledge from the teachers, I think that it's time for me to contribute to HK students. Indeed, some of the elders also asked me to do so. Although I'm not sure how I could get good results in competitions, I think that sharing my own experience would be quite valuable to some others. As for the other reason, it is because I really love MO very much. I would feel sad if I could no longer be in contact with MO anymore. So, being a trainer can prevent me from that, and I can always recall my fantastic MO life through the lessons and tests. JF: When was your first encounter with Mathematics Olympiad? How did you find it? TW: I started my MO life since primary 5. At that time, I was chosen as a representative to participate in a local competition held by the HKMOS. And then I had a chance to attend its

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would dream to attend the IMO in the life as it is the most prestigious worldwide mathematics competition Such chances and experiences are far more fruitful than the competition itself.

courses and learn mathematics in a different way. Before that, I had never heard of MO, so I didn't have much feeling about it. But years later, of courses, my interest towards mathematics grew gradually. And that motivated me to play MO all through my secondary school life JF: You have been the gold medalist of IMO for three years. What is your opinion towards IMO? TW: Hmm... In my opinion, participating in IMO is just an additional reward for those who are comparatively luckier and more skillful in MO. MO training is already a wonderful experience for everyone. But, of course, any one would dream to attend the IMO in the life, as it is the most prestigious worldwide mathematics competition for secondary students. Another gain from it is to have a trip to somewhere in the world free of charge! This could probably be the one and only one chance in life to visit that place. For comparison, such chances and experiences are far more fruitful than the competition itself. JF: Have you got any ideas about your future career? Would that be related to mathematics or others? TW: Actually, I haven't decided yet for that. Though I think that doing researches or being a teacher (mathematics, of course,) could suit my style, that still needs careful considerations. JF: To get good results in competitions or exams, you need to spend a lot of time on revising and practicing. Could you tell us how you prepare for exams and competitions? TW: I would not consider myself as a hardworking person. Unlike many others, I seldom work especially hard before a competition or an exam. But that doesn't mean that I do not need practices! As Mathematics is my interest, I consider it not as a subject in school, but something that I can play with. Whenever I have spare time and nothing to do, I will think of

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mathematics. I always think of various mathematical problems and try to solve them. No matter I could solve them or not, I did enjoy it a lot. So, in reality, I am always practising mathematics. And I think that a gradual improvement in the thinking skills through this way should be the key to success, but not a rush practice before the exam. JF: Have you got any learning tips for mathematics? What do you think about the ways that HK students used in learning Mathematics? Could you comment on it? TW: Oh, in fact, I have specified my ways of learning Mathematics in previous question. But I agree that this really needs strong interest towards mathematics. For the students

who are studying Mathematics only for examinations (that's the truth, though), I would like to emphasize on practising Mathematics. Unlike other subjects, only reciting the definitions and some examples in mathematics books are definitely not enough. And that's why most students find it difficult in studying Mathematics. They have to put a lot of in practising in order to solve any problems with similar technique that they've come across. Also, I strongly discourage people to rely on the solutions a lot. To have improvements in Mathematics, as well as in thinking skills, one always needs to explore themselves. It's no use reading the solutions whenever the problems 'seem' to be difficult. JF: Thanks very much for your time and your valuable sharing.

Member Schools Corner Mak Hugo Wai Leung St. Joseph’s College

(Former Executive Committee Member of HKJSMS)

Fascinating mathematical activities JOINED BY SJC STUDENTS

When I was the president of mathematics society of SJC, I engaged myself in organizing various educational activities and teaching training sessions of mathematics team in my school. In November 2010, on behalf of school, I had invited Dr. Koopa Koo Tak Lun(1) to be speaker of a number theory talk in hall, and it attracted around 200 students to participate. Though the topics involved such as q-th power Lemma and Dirichlet's Theorem were quite advanced to most students, mathematics lovers were still willing to devote their time in listening and studying related books. Personally, I also explained the topics in a more detailed way afterwards when it was my turn to teach training sessions. I enjoy opportunities to educate young ones of my school, and to provide enormous guidance for them to learn more about number theory. Then, in February 2011, our school joined the Inter-school Mathematics Contest organized by HKJSMS, which was taking place in Maryknoll Convent School (Secondary Section). I was glad to be one of the question setters of the contest, and I strongly encourage students from my school to participate in public mathematics activities to explore more and develop friendships with outstanding mathematics learners from other schools. I believe learning mathematics should not only emphasize in achieving marvelous results, but we should also appreciate its intrinsic aesthetics and view beauty as central to mathematics. Therefore, joining different kinds of mathematics activities are certainly beneficial to all of us.

Afterwards, in May, our school organized the “Joint School Mathematics Meet” in HKUST(2) with several grand girl schools in HK. The activity included a games session, a mathematics contest and treasure hunt. There were 16 participants from SJC, and they all enjoyed the activity, especially the sharing of mathematics knowledge with others. I believe that these activities will be more successful if we could invite a professor for giving a talk next time, and I am sure students can learn a lot from such fascinating activities. That’s why I devote my time in organizing related activities throughout the year. In a nutshell, I really hope that the mathematics society of my school can continue to provide training programs for talented students, and try to build up a good communication network with other mathematical organizations such as HKJSMS, thus organizing more educational and attractive mathematics activities in the foreseeable future. January 2012

Note: (1) Dr. Koo earned his PhD degree in Mathematics in University of Washington, and is currently the coach of Hong Kong International Mathematical Olympiad (IMO) Team. (2) Thanks to the help of Dr. Kin Li for providing venue and support.

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Yeung Hon Wah Diocesan Boys’ School

The Beauty of Geometry

At times, in reading a piece of work, whether it is a poem or lyric, we may interpret with our personal, impressionistic, or emotional apprehension. However, personal appreciation can lead to subjective and vague judgment. So when you are reading this poem ”Mission to the Frontier” by Wang Wei, a famous poet of Tang Dynasty, especially the 5th and the 6th lines about the lone column of smoke in the desert and the setting sun over the river, what will come to your mind?

Nature Those who Lover

Geometry

Lover

Love natu And litera

大漠孤煙直

使至塞上 王維 單車欲問邊， 征蓬出漢塞， 大漠孤煙直， 蕭關逢候騎，

長河落日圓

屬國過居延。 歸雁入胡天。 長河落日圓。 都護在燕然。

Event Focus Corner Chinese Mathematical Olympiad, one of the most well-known mathematical Olympiad in the world, is held on two consecutive days of January every year. Top students who perform well in the competition will be selected as members of the national team, and will be trained for the International Mathematical Olympiad held in July. This year, the 27th CMO has been successfully held on 7th and 8th January in Xi’an. A total of 244 mathematically talented students from 34 teams took part in the competition. Competitors are asked to solve 3 challenging problems in 4.5 hours on each day. This year, our Hong Kong representative team has achieved satisfying results. Eight student represented Hong Kong to participate in the China Mathematical Olympiad 2012 (CMO 2012) The team had an outstanding competition winning two silver and six bronze medals. The youngest member in the Hong Kong Team is Lau Chun Ting (14 years old). One of the awardees, Jeffrey Hui of La Salle College, is a HKJSMS executive committee member.

Upcoming competitions 第十一屆培正數學邀請賽 初賽: 2012 年 2 月 4 日（星期六） 決賽: 2012 年 5 月 12 日（星期六） 全港青年學藝比賽 第十四屆香港青少年數學精英選拔賽 日期: 2012 年 2 月 11 日(星期六) 第二十九屆香港數學競賽（2011 / 2012）HKMO 初賽: 2012 年 2 月 25 日（星期六）

Source: HKAGE Students’ Outstanding Achievement at the China Mathematical Olympiad 2012, Snap Shot, HKAGE, Jan 2012

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Mathematical & Problem Corner Brief introduction to algebraic development History of algebraic development

At first, ancient Babylonians wrote all algebra expressions in full sentences. Example:

using the concepts in algebra. Also, it has helped a lot in the development of abstract algebra, which is all about mathematical structures. Because of that, the formula for mastering our favourite toy “Rubik’s cube” is deduced. Cubic functions

“2y + 4 =5” is written as “2 of that term plus 4 is equal to five”. Such style of expression remained for a long stage until the BC 1500s. People started feeling annoying as the algebraic expressions started becoming more complicated. Therefore they improved the system to a new stage, which is called the syncopated algebraic stage. After the syncopated algebraic stage, which is a stage of expressing algebra in sentences and some forms of simple symbols the algebraic system improved and it is the current system we are using today.

1.

Format: f(x) = ax3 + bx2 + cx + d , where a is a non 0 number.

2.

A cubic equation has a format of 0 = ax3 + bx2 + cx + d Solving methods: a. By using the factor theorem, we can factorize the equation and find out the roots, just like factor method which is used to solve quadratic equations. b. By graphical method Like quadratic equations, we can find the nature of roots of cubic equations by a formula below:

3.

18abcd – 4b3d + b2c2 – 4ac3 - 27a2d2 = S

The algebraic system we use today is called symbolic algebra, in which expressions are all in a form of symbols. It makes the whole thing more reader-friendly. From observing the development of mathematics in Greek, we must thank for the invention of those symbols as they facilitate the development of algebra in the later years. The Greeks produced some of the greatest mathematicians in history, but their work was mainly accomplished in geometry. They made very little progress in algebra, because they did not have appropriate symbols. Contributions of algebraic development

There are mainly two contributions, to science and mathematics. Just at the beginning of the development of symbolic algebra, quadratic equations are simplified into symbols (Such as x2 + rx = t) and become more important in the investigation of science. For example, in Physics, Galileo, a famous scientist has found out the relationship between displacement, time and initial velocity with the help of the development of quadratic equation. (i.e. s= ut + 0.5at2). The algebraic system development has helped with the development of other branches of mathematics also. For example, in geometry mathematicians can solve unknowns in a figure by

If S > 0, the equation has 3 non-equal real roots. If S = 0, the equation has 3 equal real roots. If S < 0, the equation has 1 real root, 2 roots in a form of complex number. 4.

Brief history about ancestors who developed methods to solve cubic equations: The ancient Babylonians were the first ones who knew how to solve cubic equations. At that time, they usually did this by doubling the cube. Until the 7th century, China was the second country which developed methods to solve cubic equations, but just suitable for equations in a form of x3 + px2 + qx = N. In the 11th century, a Persian poet mathematician who is called Omar Khayyam, successfully found that a cubic equation actually can have more than 1 solution. It was a huge success. Moreover, in the 12th century, another Persian Mathematician found that a cubic equation actually can have a solution of less than 0. Also, he developed concepts of derivative functions and found out how to draw the graphs of each cubic function. This totally speeded up the development of solving cubic equations as the graphs of them can be drawn.

Article contributed by Jeff Siu

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Question:

Let (x1, y1), (x2, y2) and (x3, y3) be three different real solutions to the system of equations x3 – 5xy2 = 21 and y3 – 5x2y = 28. �1

Find the value of (11 –

đ?‘Ś1

) (11 –

đ?‘Ľ2 đ?‘Ś2

) (11 –

đ?‘Ľ3 đ?‘Ś3

Thus, we get x3 – (Îą + β + Îł) x2y + (ιβ + βγ + γι)xy2 – ιβγy3 =0 Multiply it by 4, we get 4x3 – 4(Îą + β + Îł) x2y + 4(ιβ + βγ + γι)xy2 – 4ιβγy3 = 0 --- (2) By comparing the coefficients in (1) and (2), we get

).

[IMO Preliminary Selection Contest 2011 Q.20]

Îą+β+Îł= (11 –

đ?‘Ľ1 đ?‘Ś1

−15 4

4

đ?‘Ľ2

) (11 –

đ?‘Ś2

−15

= 1331 – 121( Solution: Let

đ?‘Ś1

= 1331 +

= Îą,

đ?‘Ľ2 đ?‘Ś2

= β and

đ?‘Ľ3

2

đ?‘Ľ3 đ?‘Ś3

)

3

đ?‘Ś3

1815 4

4

3

) + 11(-5) – ( )

– 55 –

4

3 4

= 1729

=Îł

4(x – 5xy ) – 3(y – 5x y) = 4  21 – 3  28 = 0 4x3 + 15x2y – 20xy2 – 3y3 = 0 --- (1) 3

) (11 –

= (11 – Îą) (11 – β) (11 – Îł) = 1331 – 121(Îą + β + Îł) + 11(ιβ + βγ + γι) – ιβγ

Answer: 1729

đ?‘Ľ1

3

, ιβ + βγ + γι = -5, ιβγ = .

Solution provided by Jeffrey Hui

2

âˆľ x1 = Îąy1, x2 =βy2, x3 = Îły3 ∴ (x1 – Îąy1) (x2 – βy2) (x3 – Îły3) = 0 âˆľ (x1, y1), (x2, y2) and (x3, y3) are three different real solutions to x and y. ∴ (x - Îąy) (x – βy) (x – Îły) = 0

[This method uses Vieta’s formulas (é&#x;‹é ”厚ç?†) to solve this problem. It may not be the most efficient and convenient, but it can solve many algebraic problems involve their roots especially in high power polynomial. You can think another method to solve this problem.]

Acknowledgement

Need your contributions:

The publication team would like to thank the following persons for their contributions and guidance:

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. The deadline for submission to our next issue is March 30, 2012.

Dr. K Y Li Dr. Koopa Koo Mr. Ching Tak Wing Mr. Hugo Mak Mr. Lawrence Au

ć‹œ ĺżƒćƒłäş‹ćˆ? éž?錏粞缞

çĽ?ĺ?„ä˝?č€ 帍 čŽ˜čŽ˜ĺ­¸ĺ­?

ĺš´

éž?嚴大ĺ?‰

學漭進歼

躍éŤ”ĺ Ľĺşˇ

科科弪ĺ† éŚ™ć¸Żč Żć Ąć•¸ĺ­¸ ĺ­¸ćœƒäť?äşşéž 躏

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