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

ISBN-10: 1461207339

ISBN-13: 9781461207337

ISBN-10: 1461268869

ISBN-13: 9781461268864

One of the simplest how one can stimulate scholars to take pleasure in highbrow efforts is the medical pageant. In 1894 the Hungarian Mathematical and actual Society brought a mathematical pageant for top tuition scholars. The good fortune of highschool competitions led the Mathematical Society to chanced on a faculty point contest, named after Miklós Schweitzer. the issues of the Schweitzer Contests are proposed and chosen by means of the main widespread Hungarian mathematicians. This e-book collects the issues posed within the contests among 1962 and 1991 which variety from algebra, combinatorics, thought of services, geometry, degree concept, quantity conception, operator conception, chance conception, topology, to set concept. the second one half comprises the suggestions. The Schweitzer pageant is likely one of the most unusual on the planet. The adventure exhibits that this festival is helping to spot study skills. This choice of difficulties and recommendations in numerous fields in arithmetic can function a consultant for plenty of undergraduates and younger mathematicians. the big number of study point difficulties can be of curiosity for extra mature mathematicians and historians of arithmetic as well.

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**Extra info for Contests in Higher Mathematics: Miklós Schweitzer Competitions 1962–1991**

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20) (Y. Totik] L fk(Z) is ab00 9. Let us assume that the series of holomorphic functions k=l solutely convergent for all Z E C. Let H S;; C be the set of those points where the above sum function is not regular. Prove that H is nowhere dense but not necessarily countable. 21) [L. Kerchy] 10. Prove that if ai (i = 1,2,3,4) are positive constants, a2 - a4 > 2, and alaS - a2 > 2, then the solution (x(t), yet)) of the system of differential equations :i; = a1 - a2x + asxy, if = a4x - Y - asxy (x,y E JR) with the initial conditions x(O) = 0, yeO) ~ al is such that the function x(t) has exactly one strict local maximum on the interval [0,00).

RESULTS OF THE CONTESTS 1988 1. 2. 3. H. 1989 1. 2. 3. H. 1990 1. 2. 3. H. 1991 1. 53 Geza Kos, Zoltan SzabO Laszl6 Erdos Istvan Sigray, Jeno Torocsik Sandor Kovacs, Akos Magyar, Andras BenczUr, Tamas Keleti, Miklos Mocsy Laszl6 Erdos Sandor Kovacs Andras Benczur, Gyorgy Birkas, Andras Biro, Gabor Drasny, Geza K6s, Akos Magyar, Laszlo Majoros, Mikl6s M6csy, Tibor SzabO, Zoltan Szabo Andras BenczUr Gabor Drasny Andras Biro Tamas Hausel, Geza Makay Andras Biro 2. - 3. Tamas Fleiner, Geza Kos H. Matyas Domokos, Gabor Hajdu, Gergely Harcos, Tamas Keleti, Vu HaVan 3.

Show that there is no graph H E 1{ such that every graph in 1{ is a subgraph of H. 7) [F. Galvin] 3. In a lattice, connect the elements a/\b and avb by an edge whenever a and b are incomparable. Prove that in the obtained graph every connected component is a sublattice. 31) [M. Ajtai] 4. Let T E SL(n, Z), let G be a nonsingular n x n matrix with integer elements, and put S = G-ITG. Prove that there is a natural number k such that Sk E SL(n, Z). 14) [Gy. Szekeres] 5. Let G be a transitive subgroup of the symmetric group S25 different from S25 and A 25 .

### Contests in Higher Mathematics: Miklós Schweitzer Competitions 1962–1991 by Gábor J. Székely (auth.), Gábor J. Székely (eds.)

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