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The 48-th International Mathematical Olympiad 2007

The 48. International Mathematical Olympiad was held in Hanoi in Vietnam from July 19 to July 31, 2007. The Serbian team was selected based on results of the Serbian Mathematical Olympiad for high schools, held on April 2-3 in Belgrade:

  • Mladen Radojević, 4-th grade of Math High School in Belgrade;
  • Marko Jevremović, 4-th grade of Kraljevo High School;
  • Marija Jelić, 3-th grade of Math High School in Belgrade;
  • Boban Karapetrović, 2-nd grade of Ivanjica High School;
  • Dušan Milijančević, 1-st grade of Math High School in Belgrade;
  • Teodor von Burg, 7-th grade of elementary school in Math High School in Belgrade.
The team was supervised by Djordje Krtinić from University of Belgrade and Dušan Djukić from University of Toronto; Magda Bužurović von Burg also accompanied the team as an observer.

Vietnam

The trip to out destination took about 24 hours, including the transfers in Moscow and Bangkok. The contestants and deputy team leaders were accomodated in Hanoi, in two hotels away from each other. The leaders were transported to Halong immediately upon the arrival to take part in the problem selection, while deputy leaders joined them for the coordination.

The capital of Vietnam, Hanoi (the name means roughly "on the river"), is a city with around 4 million inhabitants on the Red River (in Vietnamese Hong Ha), a little below the tropic of cancer. The Vietnamese say that the short winter in the city is very pleasant, but the summer is typically tropic, more than six months long, with the temperature varying between 30 and 35 degrees over the day, warm nights and high humidity. Hanoi is a wide city with mostly short buildings (up to four floors) in a characteristic style, usually very narrow and mutually joined. A western visitor quickly notices a chaotic traffic with few cars and very many motor-cycles running in all directions and apparently with no rules. Motor-cycles are a familiar transport here. The locals explained us that the city is quite safe despite the visible poverty, but self-organized walking tours around the ciry are not highly recommended due to the traffic. Another thing that attracted our attention were the numerous cables on the streets: dozens, or maybe even hundreds of cables and wires between two adjacent poles. We were explained that only one of these cables is actually functional, while the unfunctional ones are not removed. It seems that the people spends a lot of time outside, so people who eat, cook, cut hair etc. on the streets are not a rarity.

While in Hanoi, we were taken to the Ethnological Museum (in fact, a park of ethnic houses), a silk-weavers’ village Van Phuc, and the Temple of Literature, the first Vietnamese university. Vietnamese language was written with a Chinese-like script known as Chu-nom until a century ago, when it was replaced by a highly accented latin script devised by French missionaires. Ha Long is a tourist town in Ha Long bay known for the exceptional nature beauty, at the three hours driving distance from Hanoi. The bay is filled with countless isles of various sizes and shapes. The hosts didn’t let us leave without taking us to a cruise through the bay, including a visit to a cave on one of the isles in which the weather turned out to be much worse than outside, due to the same temperature and the one-hundred-percent humidity.



As is a common practice, there were various social events organized for the students. Our team won in the volleyball tournament, not quite unexpectedly if you look at performances of our national team.

Problems and coordination

The jury selected six problems which we show here with solutions (in Serbian). As usually, problems 1 and 4 were given as easy, 2 and 5 as medium, and 3 and 6 as difficult. The coordination took place in the two days after the contest. We had a slightly inconvenient schedule, having received the students’ second-day papers in the late evening, while the next morning we had scheduled the coordination on problems 4 and 5, so we spent the night with the papers. After the first day of coordination, we were left the evening and night to go through the contestants’ works on problems 1 and 6, as well as decipher Marko’s "solution" of problem 6 by using a mysterious theorem called Combinatorial Nullstellensatz. The general impression was that the Vietnamese did the coordination very well - even though flaws and dissatisfied leaders were found as always (of course, these two were not necessarily related).

  • Problem 1 is a slightly non-standard algebra of a cumbersome formulation. Part (a) was simple and all our contestants solved it, but part (b) unexpectedly made us troubles and only Dušan had elements of a solution (and also elements of an error originating in part (a)). Despite our fears, the coordination of this problem was fairly relaxed and all our students got the deserved points easily, the only exception being Marija with her solution of (a) on four pages and an attempt on (b) on another four. Furthermore, we successfully "cheated" the coordinators in Dušan’s work and earned him 6 instead of the expected 4-5, which maybe brought him the medal. All in all, our total score on this problem is below expectations, so next year we should think about dedicating a lecture or two to constructing (counter)examples in algebra and optimization.

  • Problem 2, a geometry on the level of our national competitions, admitted both elementary and computational solutions, neither too hard nor too easy. Three our students solved it, all with complex numbers after a lot of time spent on it. Marking brought some troubles, as the coordinators dubbed Mladen’s and Marko’s solutions worthless without checking the computation, so we were forced to go through every line and every formula. We had to accept 6 for Mladen who did have an elegant computation, but unaccompanied either with a conclusion or with a single word - just a load of formulas. The three non-solutions were awarded some peanuts by the scheme, while Teodor got 7 unexpectedly without any problems, despite a couple of annoying computational mistakes. Thus our result on this problem is fine, but could have been better, hadn’t the students been discouraged by the first problem. Next year we should train them on geometry problems that cannot be solved purely computationally.

  • Problem 3 is a combinatorics in which technique is largely unhelpful. Mladen killed it spectacularly, while Teodor had the correct idea to move pupils between rooms until the difference between the sizes of the maximal cliques is 1 and dealt with some easy cases, but his solution couldn’t be fixed. On the coordination, Teodor got 2 points, while the four zeroes were distributed right away - the students spent time on extremely special cases (e.g. n=1) which seldom bring a point, instead of approaching the general case within their abilities. But, the coordination of Mladen’s work took much longer than expected, due to the Vietnamese questioning every sentence until they succeeded to confuse us. Then we promised to come back later - indeed, we returned in five minutes, finished the solution and demanded 7, after which the coordinators said they would like to think about it, so we left them. A couple of hours later, when we returned, we were welcomed by the Russian member of the Problem Selection Committee who asked us to translate the last two paragraphs word by word. Then we finally got the 7. These sufferings were explained later - it turned out that the contestants had an extremely bad time with this problem, and the only other 7 on the entire olympiad occured at the very end of the coordination, at a Chinese student. Thus Serbia was remembered on this IMO, and even the coordinators would greet us wherever we would meet. All in all, we had a hard time at this problem but we did enjoy it.

  • Problem 4 was given as an easy shot, which it was. The expplanation for the eight points we lost is to be seeked outside mathematics - Boban misread the problem which is a real pity (if you really have to misread something, at least do so at the hardest problem!), while Teodor was punished by a point for his bad writing.

  • Problem 5 cannot be called easy - the descent method, when it first appeared on the IMO in 1988, spread terror - but nowadays is classified under "technique". Marija and Mladen correctly joined the problem to this method and gained 7 each. Marko also claimed a solution, but his work was actually limited to algebraic calculations spiced with a mistake. The four non-solutions were awarded a point each for the relation 4ab-1|(a-b)<sup>2</sup>. Two solutions in the team are a satisfactory result, but maybe we could have expected even more, since the idea was not new for our students.

  • Problem 6 unluckily turned out to be a variation of the above-mentioned Combinatorial Nullstellensatz, and this made it an unfortunate choice because it was only solved by those students who had heard of this theorem. Marko had also heard of it, but didn’t remember the statement accurately, so he decided to attempt to cheat us all, leaving out the formulation, and this ultimately costed him another point. All the others in our team got zeroes quickly, although Teodor and Mladen were close to a point by the not-quite-clear scheme. The Vietnamese also didn’t fail to notice Marija’s drawing at the bottom of her paper: as they commented, "There is nothing here, but this picture is very nice... maybe one point?"

Rezultati

On the olympiad 522 contestants from 93 countries participated, making this olympiad the best attended so far. Students from Montenegro, Liechtenstein and Cambodia took part for the first time. Thanks to two very difficult problems, no one had the full 42 points - the absolute winner Konstantin Matveyev from Russia had 37. The jury awarded 39 gold medal (29 or more points), 83 silver (21-28 points) and 131 bronze medals (14-20 points). The results of our team were as follows:

SRB 1  Mladen Radojević   3 6 7 7 7 0   30    Gold medal  
SRB 2  Marko Jevremović   3 7 0 7 1 1   19    Bronze medal  
SRB 3  Marija Jelić   4 0 0 7 7 0   18    Bronze medal  
SRB 4  Dušan Milijančević   6 1 0 7 1 0   15    Bronze medal  
SRB 5  Boban Karapetrović   3 2 0 0 1 0    6  
SRB 6  Teodor von Burg   3 7 2 6 1 0   19    Bronze medal  
  Serbia total  2223 9 3418 1   107  

Mladen performed great and brought the first gold medal for Serbia in eight years. That was a long delay and we had just been getting impatient, knowing that in the previous ten years we lost as many as six golds for just one point. Of course, his problem 3 is an extra reason to make us proud. Marko, Marija, Dušan and Teodor were awarded bronzes. Although a bronze is definitely a success, we might have expected a bit more from Marko as the most experienced member; this was his final year and he apparently lacked motivation. Marija’s progress was most clearly visible in the second day, but she messed up the first day, not solving the geometry which is her speciality, which is where she lost a silver. Dušan did just enough for a bronze, but could have done better with some more practice. Bronze was not beyond Boban either, but due to the bad luck he will have to try again. The youngest member of our team, Teodor, attacked the problems with no prejudices and earned a strong bronze, and as he was not seriously punished for writing, he was quite close to a silver.


Left to right: Djordje Krtinić, Dušan Djukić, Teodor von Burg, Marija Jelić, Boban Karapetrović, Marko Jevremović, Dušan Milijančević, Mladen Radojević

Although the team rankings are unofficial, today they are taken rather seriously as a parameter of relative success of a country. Recently the Chinese team was traditionally occupying the top of the list, but this year the Russians outbid them:

CountryScoreG S B
1. Russia 184 5 1 -
2. China 181 4 2 -
3. South Korea 168 2 4 -
Vietnam 168 3 3 -
5. USA 155 2 3 1
6. Japan 154 2 4 -
Ukraine 154 3 1 2
8. North Korea 151 1 4 -
9. Bulgaria 149 2 3 1
Taiwan (China) 149 2 3 1
11. Romania 146 1 4 1
12. Iran 143 1 3 2
Hong Kong 143 - 5 1
14. Thailand 133 1 3 2
15. Germany 132 1 3 1
16. Hungary 129 - 5 -
17. Turkey 124 1 2 2
18. Poland 122 1 2 2
19. Belarus 119 1 1 4
20. Moldova 118 - 3 2
21. Italy 116 1 1 3
22. Australia 110 - 1 4
23. Serbia 107 1 - 4
24. Brazil 106 - 2 3
25. India 103 - 3 -
26. Georgia 102 1 1 1
27. Canada 98 - 1 3
28. Great Britain 95 1 - 3
Kazakhstan 95 - 1 3
30. Colombia 93 - 1 3
31. Lithuania 92 1 - 2
CountryScoreG S B
32. Peru 91 - 1 2
33. Greece 89 - 1 3
34. Mongolia 88 - 2 1
Uzbekistan 88 - 1 3
36. Singapore 87 - - 5
37. Mexico 86 - - 4
Slovakia 86 - - 4
39. Slovenia 85 - - 5
40. Czech Republic 82 - - 5
41. Sweden 81 - - 4
42. Austria 80 - 1 3
43. Norway 79 - 1 1
France 79 1 - 2
45. Belgium 78 - - 3
46. Croatia 76 - - 2
47. Argentina 75 - 1 1
48. Armenia 73 - 1 1
Macau 73 - 1 1
50. Israel 71 - - 3
New Zealand 71 - - 3
52. Azerbaijan 69 - - 3
Bosnia and Herzegovina 69 - 1 -
Indonesia 69 - 1 -
55. FYR Macedonia 68 - - 3
56. The Netherlands 65 - - 1
57. Estonia 64 - - 1
58. Albania 59 - - 1
Switzerland 59 - - 1
60. Latvia 58 - - -
61. Finland 55 - 1 -
62. Portugal 52 - - 1
CountryScoreG S B
63. Ireland 51 - - 1
Turkmenistan 51 - - -
65. Denmark 50 - - 1
66. Spain 48 - - 2
67. Kyrgyzstan 43 - - 1
68. South Africa 42 - - -
69. Cyprus 41 - - -
70. Trinidad and Tobago 39 - - -
71. Tajikistan 37 - - 1
72. Costarica 36 - - 1
73. Iceland 35 - - -
74. Ecuador 34 - - 1
Luxembourg 34 - - 1
Malaysia 34 - - 1
El Salvador 34 - - -
78. Pakistan 32 - - 1
Paraguay 32 - - -
80. Bangladesh 31 - - -
81. Morocco 28 - - -
82. Cambodia 26 - - -
83. Sri Lanka 25 - - -
84. Philippines 21 - - -
85. Nigeria 20 - - -
86. Montenegro 17 - - -
87. Cuba 16 - - 1
88. Venezuela 14 - - -
Liechtenstein 14 - - 1
90. Puerto Rico 7 - - -
91. Saudi Arabia 5 - - -
92. Chile 4 - - -
93. Bolivia 2 - - -

We can be satisfied with our ranking, especially having in mind the significant jump since the last year - our poor results in the previous two years were hopefully but a short weakness. Another thing that makes us happy is the team being quite young - for the first time, the students before the 2-nd grade of high school were given a chance to quallify for the IMO team, and it is now clear that the changes in the system that made this possible were a good move. Four team members are eligible for the next olympiad, so we have grounds to hope for a better ranking in the future.

Along with our team, we also coordinated the Montenegrin team, "authorized" by the Montenegrin team-and-deputy leader (double function!) Božidar Šćepanović who was accompanying us for all the time of the job. This year the performance of our until-yesterday-compatriots was still modest:

MON 1  Marica Knežević   1 0 0 5 1 0   7  
MON 2  Aleksandar Jaćimović   0 1 0 0 0 0   1  
MON 3  Nikola Milinković   1 0 0 7 1 0   9    H. mention  
  Montenegro total   2  1  0 12 2  0   17  

For the end of this report: At some moment, the master of the final jury meeting decided to give us a laugh, so he magnified and showed the works of two contestants on problem 6. In the first work, someone wrote in English the following: "This problem is trivial! It can be solved by induction and I don’t want to waste time on it." But the second work looked like something we had seen before. Indeed, when the bottom of the paper occured to a great amusement of the audience, we found Marija’s nice drawing. The Vietnamese knew how to recognize a master-piece of art.

Dušan Djukić, Djordje Krtinić

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