XXIX International Mathematical Olympiad

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XXIX
International Mathematical Olympiad

Candidate of Physical and Mathematical Sciences
V. V. VAVILOV,
Candidate of Physical and Mathematical Sciences
A. A. FOMIN

The XXIX International Mathematical Olympid was held from July 9 to 21, 1988 in Australia in Sydney and Canberra and was part of the state program dedicated to celebrating the 200th anniversary of the founding of the Union of Australia.

Of course, we were all really looking forward to this Olympiad and prepared for it diligently throughout the year. Our country was represented by Sergei Berlov, Sergei Ivanov, Nikolai Filonov, Yuri Khokhlov (all four from Leningrad), Dmitry Ivanov (Dolgoprudny, Moscow region), Dmitry Tulyakov (Zhdanov). Sergei and Dmitry Ivanov now participated in the tenth grade and have a real chance of being included in the USSR team at the XXX Olympiad, which is in Germany in 1989; In addition, for Sergei Ivanov, the Olympid in Australia was already the second International Olympid, and in both he added substitute results. The leaders of the team’s training camp and the team at the Olympics itself were the authors of this article. Members of the editorial board of the journal «Kvant» provided great assistance in preparing the team.

At the opening ceremony, Union Member of Parliament for Employment, Education and Training John Downis said: “Participation in this important international event shows what can be achieved when countries around the world come together to provide an enabling environment for the education of young people… By participating in this competition you are affirming the core principles that underpin education systems around the world: the principle of striving for excellence in all we do; the principle of cooperation both in the classroom and in all areas of working life; and the simple, but so important, principle of hard work…"

A record number of countries took part in the XXIX IMO - 58, of which 50 countries were represented by teams of schoolchildren, and 8 countries were represented only by observed ones. The Olympiad was held at a very high scientific and organizational level. It was successfully facilitated by extensive preparatory work, led by Prime Minister of Australia Robert Hawke, Minister of Employment, Education and Training John Downis, Chairman of the Academy of Sciences David Curtis, Director of the Canber College of Higher Education Roger Scott, Chairman of the Legal Public Mathematics Teacher Patrick Costello, Director of IBM Australia Brain Finn and many others.

According to the established tradition, international Olympids are held in two rounds, each of which offers three tasks. The round will take 4.5 hours of work time, and each task (regardless of its difficulty) is worth 7 points.

Mission XXIX MMO
1 (Luxembourg). Two circles with a common center are given on the plane, the radii of which are equal to r and R, r<R. Let P be a fixed point on a small circle and B a variable point on a larger circle. Taking BP intersects a second time the great circle at point C. Perpendicular l to line BC passing

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Through the point $P$, the small circle intersects for the second time at the point $A$ (in the case when $l$ is a tangent to the circle, we assume that $A=P$).

1. Find the set of values ​​of the expression $BC^2+CA^2+AB^2$.

2. Find the set of points that are the midpoints of the segments $AB$.

2 (Czechoslovakia). Let $n$ be a natural number and $A_1, A_2, \dots, A_{2n+1}$ be subsets of some set $B$. Suppose that $a)$ each set $A_i(i=1, 2, \dots, 2n+1)$ contains exactly $2n$ elements;

$b)$ each set $A_i \cap A_j(1 \leq i < j \leq 2n+1)$ contains exactly one element;

$c)$ any element of the set $B$ takes at least two from the set $A_i(i=1, 2, \dots, 2n+1)$.

For what values ​​of $n$ can each element of the set $B$ be assigned one of the numbers 0 or 1 so that each of the set $A_1, A_2, \dots, A_{2n+1}$ contains equal to $n$ elements corresponding to the number zero?

3 (England). The function $f$ is defined on the set of positive integers and satisfies the following conditions:

$$f(1)=1, f(3)=3, f(2n)=f(n),$$
$$f(4n+1)=2f(2n+1)-f(n),$$
$$f(4n+3)=3f(2n+1)-2f(n).$$

Find the number of all such values ​​$n$, $1 \leq n \leq 1988$ for which $f(n)=n$.

4 (Ireland). Prove that the set of solutions to the inequality

$$\sum_{k=1}^{70} \frac{k}{x-k} \geq \frac{5}{4}$$

is the union of disjoint intervals whose sum of lengths is 1988.

5 (Greece). Let $AD$ be the height in right triangle $ABC, <A=90^\circ$. The prima line passing through the centers of the circles inscribed in triangles $ABD$ and $ACD$ intersects the sides $AB$ and $AC$, respectively, at points $K$ and $L$. Prove that

$$S_{ABC} \geq 2S_{AKL}.$$

6 (Germany). Let $a$ and $b$ be positive integers such that $a^2+b^2$ is divisible by $ab+1$ without remainder. Prove that $\frac{a^2+b^2}{ab+1}$ is the square of an integer.

Let’s briefly talk about how tasks for optimization are selected, how the MMO jury is formed and how it works. Each participating country is given the opportunity to propose no more than 6 problems with complete solutions, which, if possible, should cover different sections of the school mathematics course and have varying degrees of difficulty. From all the problems submitted to the competition, a special commission of the country carrying out the optimization selects, as a rule, 20-30 problems for further development.

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further discussion at the jury meetings. The MMO jury consists of the scientific directors of all participating countries, as well as a jury and alternate appointed by the organizations. All issues at jury meetings are resolved by a simple majority of votes. Major discussions within the jury will arise when assessing the complexity of the tasks. The most typical example here is the problem of stereometry - most countries do not want to include such problems in the final list of Olympiad problems.

Of the 275 students who participated in the MMO, approximately 100,000 students were awarded gold, silver or bronze medals (approximately 1:2:3 ratio). In the USSR team, N. Filonov (42), S. Ivanov (41), D. Tulyakov (37) and D. Ivanov (36) were awarded gold medals, and Y. Khokhlov (31) and S. Berlov (30) were awarded silver medals; The number of points scored by participants (out of 42 possible) is indicated in parentheses.

In the unofficial (as well as at sports Olympiads) team competition, the USSR team took first place (217 points out of 252 possible). This is followed by: China (201), Romania (201), Germany (174), Vietnam (166), USA (153), East Germany (145), Bulgaria (144), France (128), Canada (124). Let us especially note that the teams of Bulgaria and the GDR competed with incomplete teams - 5 schoolchildren each. The Australian team scored 110 points, one of the participants, Terence Tao, won a gold medal (34 points) and one, Jeffrey Bailey (20 points), won a silver medal. Terence Tao is competing for the third time in the national team - last year in Cuba he won a silver medal, and in 1986 in Warsaw - a bronze medal. And the most important thing is that during this year’s Olympics he turned only 13 years old and during lunch all the MMO participants gave him a loud shout, thereby expressing their admiration and respect for him.

Mathematical tests took place over two days, and all the remaining days of the stay were devoted to getting to know the country, its attractions, culture and art, nature, simple and well-organized individual sports events and mutual communication between members of various teams. One can talk endlessly about extraneous experiences from the excursion, about acquaintances that arose during the Olympic period, the inverted place and the creation of the Southern Cross, left-hand traffic, a kangaroo, a koala - a small bear cub the size of a pillow, a cockatoo, flying squirrels, etc., etc….

We will end with stories from the speech of the Prime Minister of Australia, Roberta Hawke, at the very solemn and moving closing ceremony of the XXIX IMO: “In areas such as debate and the arts, there is much opportunity for the encouragement and early discovery of talent, but in the intellectual field this is less common. The real Olympics represents just such an opportunity, and in a context where the pursuit of excellence is extremely important not only for specific individuals, but also for countries as a whole. Encouraging excellence in mathematics can only promote industrial and economic development, which in turn can lead to a better life for the people of all our countries…

As important as the mathematical accomplishment itself is, I believe that the cultural exchange that took place here is equally important. I hope that you will continue to maintain these forged connections after you leave Canberra, that you will use your talents for the benefit of your homeland and the whole world, as well as for personal satisfaction and fulfillment of human life…

Join these words and wish the members of the Soviet team great creative success, good health, great hard work and determination in their future lives.