More on the European Union Contest for Young Scientists

In Delta 5/1995 we wrote about the Contest for Young Scientists, organised by the European Commission since 1989 and covering all domains of science, in the following terms:

The contest consists in the competition of papers which have been prepared by the participants. Most appreciated are those which contain a complete solution of an interesting research problem. Obviously, not all papers may be considered of real significance, but the winning ones correspond (at least) to the level of a good Master's thesis in Poland. Here is a serious challenge for prospective Polish participants.

After two years it seems that the challenge had been treated with due interest. Polish participants proved they can do very well among numerous prize-hungry contest finalists. Thanks to the possibilities offered to Poland after its association with the European Community, our representatives appeared in the finals twice (Newcastle, September 1995, and Helsinki, September 1996), each time carrying off prizes. Before presenting the details, one more quotation from our 1995 article:

The contest is highly esteemed in the European Community: the prizes of the European Contest are appreciated more than laurels won at International Olympiads. The prestige of the contest stems from the high level of the winning papers on the one hand, and on the other - from the acknowledged competence and authority of the Jury members. Last but not least, the prizes are important: 5000 ECU for each of the three first prizes, 3000 ECU for each of the three second prizes and 1500 ECU for each of the six third prizes. You may look for exchange rates in any newspaper and compute the values in national currency.

In Newcastle Poland was represented by three papers (in biology, physics and mathematics) and the mathematicians from Warsaw, Marcin Kowalczyk and Marcin Sawicki, were awarded a third prize. In Force of a set you will find an extensive account of their work, which introduces a common generalisation of notions such as Euler's characteristic, cardinality, measure and probability.

In the Helsinki finals both Polish contributions were considered prize-worthy. This gave Poland an excellent standing among prize-winners, second only to Germany, ex-aequo with the United Kingdom. The success of the mathematicians Maciej Kurowski and Tomasz Osman and of the palaeontologist Radoslaw Skibinski was made known to the general public in Poland both in newspapers and on TV on Monday, September 30, 1996.

Radoslaw Skibinski discovered remains of hitherto unknown species of oligocene fish in the region of Rudawka Rymanowska (in the Beskid Niski mountains in the south of Poland) and reconstructed their anatomic structure and mode of life.

Tomasz Osman and Maciej Kurowski presented an extended version of the paper for which T. Osman was awarded a gold medal at the Polish School Mathematical Contest (co-organised by Delta) in 1995. Here follows a not very precise resumé of the main result of their work.

If a polynomial W1 of n real variables is indecomposable (i.e. is not the product of two polynomials of lower degree) and assumes both positive and negative values, while the polynomial W2 becomes zero wherever this happens to W1, then W1 is a divisor of W2, i.e., there is a polynomial D of n real variables such that for all x1,…,xn,

W2(x1,…,xn)=D(x1,…,xn) W1(x1,…,xn).

The key step in the proof consists in the description of the position the set Z of all zeros of W1 occupies within the space R^n. The authors prove that the projection of Z onto some (n-1)-dimensional subspace formed by all but one coordinate axis in R^n contains an (n-1)-dimensional open sphere. The details may be found below, in Common solution sets of real polynomials.

The repeated success of young Polish mathematicians in the finals of the European Union Contest seems to suggest the following humble conjecture: the gold medal in the Polish School Mathematical Contest, supported by self-confidence, a perfectly elaborated English version of the paper and a tiny bit of luck, is sufficient to win at least a third prize in the EU Contest. We'll see whether this conjecture is corroborated during the finals of the 9th European Union Contest for Young Scinetists in Milan, between the 9th and 14th of September 1997. This year's gold medalist in the Polish competition was Michal Stukow from Gdansk You can read his paper in this issue of WWW Delta (A short history of the proof of an interesting theorem); let's just mention here that his main result is, as in the case of Osman and Kurowski, an interesting and hitherto unknown generalisation of an 18th-century result "with a name".

M. Stukow generalised to the tetragon the so-called Euler's formula, which claims that in any triangle the radius R of the circumscribed circle, the radius r of the inscribed circle and the distance d between the centres of these circles are related by the equality

It turns out that if an (otherwise arbitrary) tetragon has the property that a circle can be circumscribed on it and a circle can be inscribed in it, then the distance d between the centres of these circles satisfies the following formula:

(where R and r are the radii of the circumscribed and the inscribed circle, respectively).

The Editors of Delta have no hope of winning laurels in the EU Contest, be it only for its being restricted to youngsters between 15 and 21 years old. If you are of the appropriate age and have ideas, knowledge and good will, you can verify our conjecture experimentally. It suffices to take part in both contests, first in the Polish and then in the European.

Polish participants can find detailed information on the conditions of participation in the European Union Contest for Young Scientists in the office of the Krajowy Fundusz na Rzecz Dzieci (Polish Children's Fund), ul. Chocimska 14, 00-791 Warsaw, Poland. Good luck!