# Contests in Higher Mathematics: Miklós Schweitzer by Gabor J. Szekely By Gabor J. Szekely

The most potent how one can stimulate scholars to get pleasure from highbrow efforts is the medical pageant. In 1894 the Hungarian Mathematical and actual Society brought a mathematical festival for top university scholars. The luck of highschool competitions led the Mathematical Society to came across a school point contest, named after Miklós Schweitzer. the issues of the Schweitzer Contests are proposed and chosen through the main widespread Hungarian mathematicians. This e-book collects the issues posed within the contests among 1962 and 1991 which variety from algebra, combinatorics, conception of features, geometry, degree idea, quantity thought, operator conception, chance thought, topology, to set idea. the second one half includes the recommendations. The Schweitzer pageant is among the most original on the planet. The event exhibits that this festival is helping to spot learn skills. This choice of difficulties and suggestions in numerous fields in arithmetic can function a advisor for plenty of undergraduates and younger mathematicians. the massive number of learn point difficulties could be of curiosity for extra mature mathematicians and historians of arithmetic to boot.

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Extra info for Contests in Higher Mathematics: Miklós Schweitzer Competitions, 1962-1991 (Problem Books in Mathematics)

Example text

Joo, L. Szekely] 6. For which Lebesgue-measurable subsets E of the real line does a positive constant c exist for which sup -00

N > 5, 1 K(x) _ , 3- 2, n > 5. ), Construct a subsequence Ink} of natural numbers such that if X1, X2, ... are independent, identically distributed random variables with distribution function F, then for all real numbers x x li 00 P n j=1, Xj < 7rx } = 1 + 1 arctan x. 27) [S. Csorgo] 1988 1. Define a partial order on all functions f : R -+ R by the relation f -< g if f (x) < g(x) for all x E R. Show that this partially ordered set contains a totally ordered subset of size greater than but that the latter subset cannot be well-ordered.

Totik] 10. Consider the equation f'(x) = f (x + 1). Prove that (a) each solution f : [0, oc) --* (0, oo) has an exponential order of growth, that is, there exist numbers a > 0, b > 0 satisfying I f (x) I < a ebx, x>0; (b) there are solutions f : [0, oo) -p (-oo, oo) of nonexponential order of growth. 57) [T. Krisztin] 11. Does there exist a bounded linear operator T on a Hilbert space H such that 00 00 n Tn(H) = {0} but n Tn(H)- n_1 n-1 {0}, where - denotes closure? 9) [L. Kerchy] 12. Let X1, X2, ...