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

By Gabor J. Szekely

Some of the most powerful how you can stimulate scholars to get pleasure from highbrow efforts is the clinical festival. In 1894 the Hungarian Mathematical and actual Society brought a mathematical pageant for top college scholars. The luck of highschool competitions led the Mathematical Society to discovered a school point contest, named after Miklós Schweitzer. the issues of the Schweitzer Contests are proposed and chosen through the main renowned Hungarian mathematicians. This publication collects the issues posed within the contests among 1962 and 1991 which diversity from algebra, combinatorics, thought of capabilities, geometry, degree thought, quantity thought, operator idea, chance thought, topology, to set idea. the second one half includes the suggestions. The Schweitzer festival is without doubt one of the most original on the earth. The adventure indicates that this festival is helping to spot learn skills. This selection of difficulties and ideas in different fields in arithmetic can function a consultant for lots of undergraduates and younger mathematicians. the massive number of examine point difficulties can be of curiosity for extra mature mathematicians and historians of arithmetic to boot.

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

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Assume that R, a recursive, binary relation on N (the set of natural numbers), orders N into type w. Show that if f (n) is the nth element of this order, then f is not necessarily recursive. 6) [L. Posa] 2. Let G be an infinite graph such that for any countably infinite vertex set A there is a vertex p joined to infinitely many elements of A. Show that G has a countably infinite vertex set A such that G contains uncountably infinitely many vertices p joined to infinitely many elements of A. 10) [P.

18) [V. T. Sos, L. Szekely] 2. Let I be an ideal of the ring R and f a nonidentity permutation of the set {1, 2, ... , k} for some k. Suppose that for every 0 a E R, aI # 0 and Ia # 0 hold; furthermore, for any elements x1i x2, ... xkf holds. Prove that R is commutative. 35) [R. Wiegandt] 3. Let f : R -+ R be a twice differentiable, 27r-periodic even function. Prove that if 1 fr(X) + f(x) f (x + 37r/2) holds for every x, then f is it/2-periodic. 36) [Z. Szabo, J. Terjeki] 4. For which cardinalities n do antimetric spaces of cardinality ,c exist?

Halasz] 8. Let ao = 0, a1, ... , ak and bo = 0, b1, ... , bk be arbitrary real numbers. ,k, p ) (-1) = ai, p(i) (1) = bi, (1) and max Ipn(X)I (2) n2 where the constant c depends only on the numbers ai, bi. (ii) Prove that, in general, (2) cannot be replaced by the relation lim n2 max IN(x)I = 0. 42) [J. Szabados] 9. Consider a latticelike packing of translates of a convex region K. Let t be the area of the fundamental parallelogram of the lattice defining the packing, and let tm;n(K) denote the minimal value of t taken for all latticelike packings.

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