## The 48-th International Mathematical Olympiad 2007The 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 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. 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.
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?"
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:
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:
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:
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:
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