By Gábor J. Székely (auth.), Gábor J. Székely (eds.)

One of the best how one can stimulate scholars to take pleasure in highbrow efforts is the clinical pageant. In 1894 the Hungarian Mathematical and actual Society brought a mathematical pageant for top university scholars. The good fortune of highschool competitions led the Mathematical Society to chanced on a school point contest, named after Miklós Schweitzer. the issues of the Schweitzer Contests are proposed and chosen through the main favourite Hungarian mathematicians. This publication collects the issues posed within the contests among 1962 and 1991 which diversity from algebra, combinatorics, conception of services, geometry, degree idea, quantity thought, operator conception, likelihood conception, topology, to set idea. the second one half comprises the recommendations. The Schweitzer festival is likely one of the most original on the planet. The adventure indicates that this festival is helping to spot study abilities. This selection of difficulties and strategies in numerous fields in arithmetic can function a consultant for lots of undergraduates and younger mathematicians. the massive number of study point difficulties could be of curiosity for extra mature mathematicians and historians of arithmetic as well.

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Ii) To any neighborhood U of 0 there is a neighborhood V of 0 such that V + (- V) ~ U, a suitable choice being V = U. (iii) If a is contained in some neighborhood U of 0, then there exists a neighborhood V of 0 such that a + V ~ U. Again the choice V = U will do. (iv) The intersection of any two neighborhoods of 0 contains a neighborhood of 0, because for any two neighborhoods, one of them contains the other. We now come to the proof of completeness. •. , 'fJn,· .. is a Cauchy sequence, then it converges to some element of H.

Show that I = sup{gj : j E J, gj E cp} = inf {h m : m E M, h m E cp} implies I E CP. Does this statement remain true if (1) is replaced with the following condition? (2) k: heX) -+ JR is continuous on the closure of heX) in the product topology. 20) [A. Csaszar] 5. Let us draw a circular disc of radius r around every integer point in the plane different from the origin. Let Er be the union of these discs, and denote by dr the length of the longest segment starting from the origin and not intersecting E r .

45) [Y. Totik] 9. Show that there exists a constant Ck such that for any finite subset V of the k-dimensional unit sphere there is a connected graph G such that the set of vertices of G coincides with V, the edges of G are straight line 42 1. PROBLEMS OF THE CONTESTS segments, and the sum of the kth powers of the lengths of the edges is less than Ck. 41) [V. Totik] 10. Let F be a probability distribution function symmetric with respect to the origin such that F(x) = 1 - x- 1 K(x) for x ~ 5, where K(x) = {~ 3- • 2n!