2010 2011issue3

HKJSMS Newsletter - Feb 2011

香港聯校數學學會

Hong Kong Joint School Mathematics Society

jsms.hk@gmail.com| http://www.hkjsms.org

Newsletter - Feb 2011

Interschool Mathematics Contest The Interschool Mathematics Contest (ISMC) was successfully held in Maryknoll Convent School on 12th February, 2011. 19 schools, 34 teams and over 150 participants participated in the competition.

The results are as follows: Junior Event: Champion First runner-up Second runner-up

Lau Chun Ting Hon Pun Yat Lui Ching Yin

St Paul’s Co-educational College Queen’s College Queen’s College

Senior Event: Champion First runner-up Second runner-up

Lum Kai Chun Chan Long Tin Chan Yin Hong

Queen’s College Diocesan Boys’ School St Paul’s Co-educational College

Group Event: Champion First runner-up Second runner-up

Queen’s College Diocesan Boys’ School La Salle College

Congratulations to the winners.

If you would like to see the papers and the solutions, or have a look into our gallery, you can visit our website. There are more events coming in 2011, we hope to see you joining our events in the future.

In this issue of the newsletter, we are introducing the concept of cryptography in a mathematical way.

There is also an article about studying mathematics as a lifelong plan. Trevor Yu, bronze medallist in IMO 2010, has also written an article about his special journey for the 51st IMO held in Kazakhstan.

Enjoy reading the articles.

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HKJSMS Newsletter - Feb 2011

Cryptography Lee Pak Tao Cryptography, the practice of hiding information, is highly related to mathematics and computer science. Simply speaking, when we are given a message, we can encrypt it, that is, to convert it to another form, so as to protect it from theft or alteration. Decryption, the conversion of the encrypted message back to the original message, is the opposite effect. Cryptography is used when someone, let’s call her Alice, wants to send a message to her friend, Bob, without letting others know the content. Alice can encrypt the original message (known as plaintext) to an unintelligible form (known as ciphertext) using an encryption key, and the ciphertext can be transformed back to the plaintext by a decryption key. Therefore, if no one but Bob holds the decryption key, Alice’s aim will be achieved.

Above lists some of the jargons in cryptography, but don’t be discouraged by the seemingly difficult procedures to encode a message. I am going to state the simplest form of cryptography which almost everyone can understand. For the encryption of an English message, we can set A=1, B=2, … , Z=26.

The message JSMS is converted to 10191319, since 10 represents J, 19 represents S, and so on.

However, even an amateur cryptographer would be able to break this simple code. We can alter this method a bit to make it more complicated, by assigning random numbers to letters. But still, this method can’t be used if the message is very secret and can’t be revealed, since computer can decode it very quickly.

A more complex way of encoding and decoding is called the public key method. Suppose you know two very large primes p and q. Let their product be m. The number of integers relatively prime to m is ( ) ( ) ( )( equal to ( ) ). We choose a number k that is relatively prime to m.

Now, the values of m and k are publicized while p and q are kept secret.

Firstly, we convert the message into a string of digits by the simple method in last paragraph. The list ( ) ( ) ( ) by . Next we compute of numbers is successive squaring, and a new list of numbers is formed.

For the decoding part, we have to solve the congruence ( ). If you are familiar with number theory, there is an algorithm * to solve the congruence and find all the s if you can find ( ), which can be computed easily if you know p and q. Therefore, if we are given sufficiently large primes p and q, their product m would be so large that even fast-processing computer needs a lot of time to factor it and decode the message. This encryption is a good one since for those who do not hold the decryption key, they cannot learn the content of the message in a short period of time. The public key method is so called because of the publicized m and k, and this method is a very commonly used one nowadays.

There are many other methods, with some requiring advanced mathematics knowledge, in cryptography. If you find this topic your cup of tea, I suggest you learn more about programming, an indispensable part in modern cryptography. Hope you all are having a great time dealing with the art of cryptography!

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HKJSMS Newsletter - Feb 2011 Remarks: *: To compute

roots modulo m:

Let a, k and m be given integers satisfying

(

)

and

(

( ))

First, compute ( ). Then, find positive integers s and t such that ( ) by successive squaring. The value obtained gives the solution x.

. ( )

. Compute

Acknowledgement: A Friendly Introduction to Number Theory by Joseph H. Silverman

An everlasting plan – studying mathematics Mak Hugo Wai Leung Nowadays, when people are asked about feelings towards mathematics, most of them would just believe that studying mathematics is useful to capture “A” or “A*” in public examinations, while some talented students also think of capturing gold medals in mathematics competitions. But is studying mathematics that monotonous and stupid? For people who have extreme passion and tremendous enthusiasm in it will never think of the above, but they will try their best in tackling more mathematical problems or even striving hard to make contributions to the world in the field of mathematics. Let me share the experience of a boy to you.

When the boy was enjoying his wonderful Christmas holiday at home at the age of seven, his father played a game with him, distributing the chopsticks into sets of 3, 5 and 7, and then gave the boy each remainder and required him to find out the total number of chopsticks his father got. The boy thought for the whole afternoon and could eventually find out the answer, but could not provide a complete explanation, so he felt very interested and amazed. His father used that game to teach him Chinese Remainder Theorem and the concept of modular arithmetic. He felt puzzled at first, and could hardly understand any of the theories, because he was just a Primary 2 student at that time. However, he began to know that learning mathematics does not only limit to books, but can also be applied into daily lives and learnt from experience, so he started developing interest towards mathematics.

Since primary school, the boy was selected to be in the school team, representing the school for various highlighted mathematics competition, such as The Hong Kong Mathematical High Achievers Selection Contest, HKMO and even IMO Preliminary Selection Contest. Sometimes he did win satisfactory awards in those competitions and his parents appreciated his effort in mathematics, but that was not the only thing he wanted. He also treasures the opportunities to educate the young ones in his school and to provide enormous help for the internal mathematics training programs. No matter how poor the students do, he still has patience to teach them, and he always has the passion to discuss mathematics with them. This was because he learns that it is not that important if students really can’t solve those competition questions. A true mathematics lover will not only focus on these stuffs, but will devote his entire time and effort in reading mathematics books, doing mathematical research and playing with numbers instead. Therefore, he does not only concentrate on achieving extraordinary good results in the mathematics competitions he joins, but also on learning from the competitions and establish more interest towards mathematics. Moreover, he had also competed with Form 4 students in the Chong Gene Hang Mathematics Competition since he was in Form 2, and with the help of his teammates, he did not make his school disgraced and captured prizes in three consecutive years.

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HKJSMS Newsletter - Feb 2011 When the boy was in Form 3, he was nominated to join the Mathematics and Leadership Streams of the Exceptionally Gifted Students Scheme. After a series of interviews and examinations, he successfully got the opportunities to study university mathematics courses at an earlier stage. During the lessons, he has not only learnt more about mathematics, but has also developed a positive attitude towards life. He always wanted a utopian life like mathematics, in which everything could be solved in a nice and direct way. But he soon understood that was impossible, and he starts to observe that beauty is an important foundation of mathematics, in which people can easily experience it, no matter in the sublime beauty and elegance of a “book proof”, or in the surprising beauty of unexpected relationships, such as Wiles’ proof of Fermat’s Last Theorem, with the connection between modular forms and elliptical equations.

After having a much larger picture of mathematics, he started thinking about doing related research, which led him to a higher stage in mathematical field.

The boy then joined “Hang Lung Mathematics Awards 2010” when he was in Form 5. He was confused when choosing the topic. He didn’t know whether choosing Geometry or Number Theory was more suitable. After reading a mathematical paper about Circle Packing, he made his final decision. The topic was “Starting from Combinatorial Geometry”, on which the boy had written a 28-page mathematical paper. The paper included some famous theorems, conjectures and definitions in combinatorial geometry such as Sylvester-Gallai theorem, Dirac & Motzkin’s Conjecture, Covering, Packing and Ramsey Theorem. Then, he carried out a research and study related to Ramsey’s Theorem, mainly showing that for any triangle ABC, one can use three colours to paint every point in a plane, such that the triangle formed by any three vertexes having same colour will NOT be congruent to ABC. He had considered two different methods, by filling in the plane with parallel bars of different colours and considering a hexagonal plane respectively. He at first thought he succeeded in showing the requirement (by using the hexagonal approach) and handed in the report to Hang Lung Corporation happily in late August 2010. He received a letter from HLMA in December 2010, notifying the boy that he had entered the final 15 teams of the competition and was invited to the Oral Defense Section for competing scholarships and prizes. He read his own report again, but was depressed very soon as he eventually found out that the rod or line was not fixed. What he proposed became unacceptable. However, there were only 3 days left, while he needed to prepare for his internal examination at the same time. His brain got stuck and couldn’t think of ways to remedy, what he could do was to accept failure. Although he tried hard to ask and discuss with his teachers and professors, the result remained the same. Finally, he couldn’t get scholarships in the award presentation ceremony held in the Hong Kong Convention and Exhibition Centre. He had set a target for himself, that is to win the Gold Prize in the HLMA 2 years later, and make more contributions to the mathematical field.

I bet all of you should know I am the boy mentioned above. I really hope that people should not only emphasize good grades or marvelous results when learning mathematics, but also explore more and treat mathematics as an everlasting plan in our life journey. So that our life can become more uplifting and elevated. Therefore, I encourage all of you to cherish every opportunity when you are in contact with mathematics. It will not bring any fear to you, but instead become your useful tool and lovely toy.

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HKJSMS Newsletter - Feb 2011

International Mathematical Olympiad 2010, Astana, Kazakhstan Team Leader: Deputy Leader: Observer: Contestants:

Yu Tak Hei Trevor

Dr Lau Yuk Kam Mr CJ Lam Mr Stanley Lee Ching Tak Wing (Derek), Chung Ping Ngai (Brian), Hung Ka Kin (Kenneth), Tam Ka Yu (Gary), Yip Hok Pan (John), Yu Tak Hei (Trevor)

Before the thing The gang arrived at Hong Kong International Airport at 11 am on 4 July. As usual, we ought to take some group photos there, and thanks to the good weather, it favored our parents’ job as cameramen. One thing that I was afraid of, was that the observer of IMO Team 09, another Mr Lee, tried his uttermost efforts in permuting and thereby boosting the tally of photos, but testing our degree of tolerance every time. Luckily, this Mr Lee was so caring for us.

Despite the adjacency of Kazakhstan and China, the whole travelling cost 20 hours. (You can imagine that it was weird when my clear memory reminded me it took only 11 hours to Germany in IMO last year.) Taking a plane wasn’t comfortable. Noise and oscillations made our studies of Olympiad problems uneasy.

5 July. After a long time of flight, we finally arrived at the Astana Airport. At passport control, a woman from the ministry had organised a dedicated IMO queue. We left the airport and travelled across the stunningly flat steppe towards the spires of Astana. We marched to the Duman Hotel, a construction with a stunning 18 storey lobby inside and a giant IMO poster outside – it felt like the competition is really kicking off! We were presented with our colour-coded lanyard. We received a CD/DVD of Kazakh music and cinema, a T-shirt and baseball cap, a booklet detailing the rules of various traditional Kazakh equestrian sports and an official programme. All these were contained within a stylish IMO branded knapsack. We were intrigued by a certain piece in the programme that mentions the Deputy Leaders will be given a ‘Night of Surprises’ one particular evening.

Our accommodation was a “slightly-less-than-world-class” hotel according to my standard, with beautiful configuration inside and a pleasant outlook from my room, where I enjoyed my first night in Kazakhstan with Brian, Derek and Kenneth. Internet service was available, and Brian and Kenneth briskly set up their laptops surfing the net and leaving some status like “Kazakhstan, safe and sound” in the facebook. There was also a piece of lunch ticket, in which we were amazed at our Kazakh names printed. (I am IO TAK XEЍ. ) 6 July. A less-than-perfect night’s sleep, keeping the air conditioning on in a room filled with 6 people is generally desirable, but through some configuration of vents I ended up sleeping in an arctic gale.

We awaked to find no guests in our room. We felt tired and excited. It was the Opening Ceremony day!

Our IMO guide informed us that we need to be on the coach by 09:00. Most of the guides were students at a language school, but ours was just a volunteer from somewhere and proved to be particularly misleading indeed. In this instance she was perhaps too unskilled, as although some other teams like USA arrived on the coaches at the correct time it took like almost an hour for us and a few teams following us to embark.

The Palace of Independence was a truly stunning setting for the ceremony. We entered and exchanged manic waving before taking our seats between Honduras and Hungary. There were many opening speeches and J´ozsef Pelik´an got the biggest round for speaking a sentence in Kazakh.

The acts were many and varied. We were greeted by a myriad of small children running amok onstage, wearing T-shirts bearing the numbers 1 to 10 and various mathematical symbols. There was the traditional dancing and a massed youth dombra (a traditional Kazakh musical instrument) ensemble.

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HKJSMS Newsletter - Feb 2011 We were first treated to a promo video narrated in a Kazakh accent. Then the band comes on and quickly strikes up Mozart’s classic. After a series of speeches and performances, it became our turn.

Every single national team would come to the stage and say hello to the 51st IMO. Quite a lot of countries managed to somewhat show off and rock on the stage. Australians threw away to make a trick with their typical national figure – koala bears. South Africans played their noisy vuvuzela, trying to note the ongoing World Cup Finals.

After the lunch and a chat with an Australian contestant, a girl who was a post-HongKonger, we embarked on the 3-hour coach journey (the 3-hour one is informed by the organizer, but we later witnessed that it was lengthened to a 7-hour exhausting one). We travelled in a long convoy, which made a halt at periodic intervals to signify that one or other of the coaches has bitten the dust. We were forced to sleep uncomfortably because the only alternative, reading the IMO shortlisted problems, wasn’t practicable.

We had eventually gone to the lakeside retreat at 10 pm. There was a campsite called Baldauren where the competition itself would take place. With the first exam the following morning, everyone was keen to eat dinner and get as early a night as possible. The camp is adamant that we should be given their full welcome, but it would appear that only a certain number of mathematicians can be welcomed at any one time. HongKongers were firstly cheated by their guide together with a staff there. They led us to the 4th floor, where our room was located according to their “instruction”. We went around, finding ourselves in one of the most stupid situation we have ever faced. Next we proceeded to the 2nd floor, walked around and determined that our guide was a liar. After a few minutes, the staff said firmly that our room was on the 1st floor. This was so frustrating that we finally marched on our room on the 3rd floor! (We were so furious, and later declared her the P guide.) The dinner was ridiculous. We were told to go downstairs on 10:30 but the guards firmly turned us away from the canteen. We entered it after an hour of reluctant patience. After having dinner, we eventually slept at about 1:30 am.

7 July. We treasured our 5-hour rest after being rudely awoken at 6:30 am by an impassioned rendition of the IMO hymn. This is the low point of the fortnight. We felt that Baldauren was subjecting us to intense psychological torture designed to disrupt our mathematical function.

The contest And so to the first exam. We were given four and a half hours to complete 3 problems, supposedly varying from simple to difficult in ascending order. The organisation for this seems to be in the hands of a German, Dierk Schleicher, and so runs remarkably smoothly. (It was later revealed that Dierk had, 4 days previously, uncovered some unfortunate oversights of Kazakh exam provision, namely toilet facilities and clocks.) Our exam pack came with 5 handy colour-coded cards. These could be waved to request more paper, a trip to the toilet, more water, for a question to be sent to the jury, and general help.

I completed the first problem in an hour, so my thoughts turned to problem 2. This was on geometry.

Over the next 3 hours, having made no discernable progress, my mind on this problem faded away. The problem concerned an oddly constructed point G. I tried but I found no purchase on this point, and thus neither on the problem. Problem 3 looked beautiful in its symmetry but I only got to address it for a short period, after I eventually lost hope in the geometry.

After the first half of the contest, we gathered our information and our summary was: Derek, Brian and Kenneth got 2 out of 3, whilst Gary, John and I had one. The Q3 number theory wasn’t done by Derek, and we were shocked. Having only 1 on the first day means at least 2 on the second is necessary for my desired silver medal.

Here comes the second half of the contest. The set-up was identical to the first day. Problem 4 was a geometry. Unbelievably weird, I couldn’t believe 5 times of angle tracing will do. It was like fooling myself in the largest International Math Competition to submit this piece of solution in 4 lines with a drawing. Anyway, I moved on to problem 5, an algorithmic combinatorics, much of my preference as

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HKJSMS Newsletter - Feb 2011 I recalled my success in another algorithmic combinatorics in the mock IMO given by CJ. Spirit high, marching to Q5 was, however, the start that things went wrong.

The question involved generating a very large number of coins, 2010 2010 2010, yet it took 30 minutes for me to generate a number over a hundred, and an hour to make one exceeding a thousand. When faced with such a huge quantity in comparison to my offerings, combined with the fact that the number of coins is bounded, I proposed that creating such a number was impossible and tried to prove it. 4 hours later, I had failed to do this and was extremely grouchy, when I came close to a key recursive step which could create a much greater amount of coins, but I just missed it. After brutally defeated, Q6 was a zero for me.

After the contest, we were absolutely amazed at Gary’s Q5 solution with the use of my missed recursive step to a perfect extent. Derek did almost all in Q5 but claimed to have only 3 points there. All of us could do Q4 but Q6.

Post-exam This is supposedly all about various excursions, but here I had to say beforehand that all the travelling of coaches involved at least 2 hours, and it wasn’t desirable. We didn’t enjoy very much of the planned activities, so my note here wasn’t too detailed.

9 July. We headed to the provincial capital Kokshetau, where a concert was to be given in our honour.

We arrived at the Palace of Culture an hour late, but were nonetheless greeted by an enthusiastic junior brass band. The concert itself was utterly incredible. To summarise, it consisted of a traditional Kazakh orchestra augmented by cellos and double bass, accompanying a number of singers.

All the while, coordination results are filtering through via text. John and I were happy to know a partial point given to us in Q2 and Q5 respectively. North Korea had been disqualified. We were shocked but decided that speculation was pointless.

10 July. We had another excursion, to an archaeological site. The journey was ferociously long (3.5 hours each way). On the outbound trip, the road narrowed to a single lane and then a dirt track. This was exciting! Then, inevitably, we discovered that this was because the coaches had gone the wrong way and needed to turn around. The driver was astonishing in his skill, but the entire experience is terrifying for us as the track is raised some 2 metres above the surrounding countryside.

We approached to a field filled with dried horse manure and shuffle towards another welcoming ceremony, adorned by the worst PA system known to humanity. The archaeological dig itself was wholly fascinating, being a cross-section of a Stone Age burial mound. But it was only a brief delight, as we were soon on the coaches for the long return journey. At Baldauren, we congratulated Derek on his 6 points in Q5, but Kenneth was dissatisfied with a 2-point deduction in the easy Q1. Speculation for bronze medal was 14 to 16, and that for silver was 20 to 22. None of us were secured to grasp our respective expected medal.

11 July. It was a rest day. We stayed in our room and played card games like Hearts and Bridge. That was so wonderful when we turned down the P Guide’s offer of some strange activities. In the evening, all results were out. Originally all of us were doomed by the unreasonable choice of problems and their placement, but it turned out that almost all of us were saved. Derek’s name just appear in the bottom of list of gold medalists, Brian had an obvious silver, and due to Gary’s Godlike performance in Q5, he got a silver as well. The rest of us were titled bronze medalists, but it was frustrating for Kenneth since the deduction in Q1 just drove him away from silver. Hong Kong was ranked the 20th among more than a hundred countries! Anyway, it’s over! And we soon watched the World Cup Finals between Netherlands and Spain.

12 July. Saying goodbye to Baldauren, we drove through Astana, and straighted out the other side

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HKJSMS Newsletter - Feb 2011 again. Our destination (an equestrian centre) would seem to only be 5 minutes from the town. We saw a large crowd of people cowering in the shade provided by the grandstand covers – we assumed these were the team leaders and deputies. We were armed with the description of Kazakh equestrian activities given to us in our original freebee pack. The show was magnificent.

This time we lived in a nicer hotel than the earlier one. The afternoon and the night was wholly playing cards.

13 July. Closing Ceremony Day! Similar to the Opening, 2 of the delegates of all teams were to go up the stage, but altogether this time, and the purpose was to mark an end of the IMO. Derek and I were the two. It was so frustrating and embarrassing that all teams got their national flags waving with pride except Hong Kong because we weren’t given one. Another interesting point was that in the IMO Presentation Ceremony (held in HK before IMO), we were seriously noted that the Flag can never touch the ground. Yet, what we saw was countries firstly gathered their flags on the ground.

How bureaucratic and weird HK officials were! Anyway, afterwards, everybody received the medal and respective presents. Derek got a cool laptop!

After the ceremony we took photos with China team and Taiwan team. We also had a short chat with the experienced Russian Leader of many years’ IMO, Nazar Agakhanov, who would become the Chairman of the IMO Advisory Board starting from next year.

14 July. It was time to say goodbye to Kazakhstan, but it wasn’t easy. Some problems about transfer of our luggage occurred. It was absolutely silly. I remembered clearly the staff in the airport said this for many times, “We can transfer your luggage to Hong Kong. We have an agreement with Hong Kong. It is that we cannot transfer luggage to Hong Kong from Kazakhstan, so we cannot transfer your luggage to Hong Kong.” You fool! Mr Lee got furious, so did CJ. This was the last one of many proofs, that Kazakh is not understandable, and that the coordination ability of them is less than nothing.

Anyway, we left Kazakhstan, headed to Hong Kong through India. The last comment made, by CJ, was, “Kazakhstan was a bad dream.”

Hong Kong Team in the 51th International Mathematical Olympiad

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