USA & International Mathematical Olympiads 2003 by Titu Andreescu

By Titu Andreescu

The Mathematical Olympiad examinations, overlaying the united states Mathematical Olympiad (USAMO) and the foreign Mathematical Olympiad (IMO), were released every year on account that 1976 by means of the MAA American arithmetic Competitions. this is often the fourth quantity in that sequence released by way of the MAA in its challenge ebook sequence. The IMO is a global arithmetic festival for prime university scholars that occurs every year in a special state. scholars from around the globe perform this pageant. The USAMO and the workforce choice attempt are the final phases of the method that result in the choice of the group representing america within the IMO. difficulties and strategies from either one of those competitions for the yr 2003 are incorporated during this quantity. those Olympiad variety checks encompass a number of hard essay-type difficulties. even supposing an accurate and whole way to an Olympiad challenge usually calls for deep research and cautious argument, the issues require not more than an excellent historical past in highschool arithmetic coupled with a dose of mathematical ingenuity. There are precious tricks supplied for every of the issues. those tricks frequently support lead the scholar to an answer of the matter. whole suggestions to every of the issues is additionally incorporated, and lots of of the issues are provided including a set of exceptional suggestions constructed via the exam committees, contestants and specialists, in the course of or after the competition. for every challenge with a number of strategies, a few universal the most important effects are offered in the beginning of those ideas.

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By Titu Andreescu

The Mathematical Olympiad examinations, overlaying the united states Mathematical Olympiad (USAMO) and the foreign Mathematical Olympiad (IMO), were released every year on account that 1976 by means of the MAA American arithmetic Competitions. this is often the fourth quantity in that sequence released by way of the MAA in its challenge ebook sequence. The IMO is a global arithmetic festival for prime university scholars that occurs every year in a special state. scholars from around the globe perform this pageant. The USAMO and the workforce choice attempt are the final phases of the method that result in the choice of the group representing america within the IMO. difficulties and strategies from either one of those competitions for the yr 2003 are incorporated during this quantity. those Olympiad variety checks encompass a number of hard essay-type difficulties. even supposing an accurate and whole way to an Olympiad challenge usually calls for deep research and cautious argument, the issues require not more than an excellent historical past in highschool arithmetic coupled with a dose of mathematical ingenuity. There are precious tricks supplied for every of the issues. those tricks frequently support lead the scholar to an answer of the matter. whole suggestions to every of the issues is additionally incorporated, and lots of of the issues are provided including a set of exceptional suggestions constructed via the exam committees, contestants and specialists, in the course of or after the competition. for every challenge with a number of strategies, a few universal the most important effects are offered in the beginning of those ideas.

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* 1 Henri Cartan Les travaux de Koszul, I (Lie algebra cohomology)
* 2 Claude Chabauty Le théorème de Minkowski-Hlawka (Minkowski-Hlawka theorem)
* three Claude Chevalley L'hypothèse de Riemann pour les corps de fonctions algébriques de caractéristique p, I, d'après Weil (local zeta-function)
* four Roger Godement Groupe complexe unimodulaire, I : Les représentations unitaires irréductibles du groupe complexe unimodulaire, d'après Gelfand et Neumark (representation conception of the complicated distinct linear group)
* five Léo Kaloujnine Sur los angeles constitution de p-groupes de Sylow des groupes symétriques finis et de quelques généralisations infinies de ces groupes (Sylow theorems, symmetric teams, countless crew theory)
* 6. Pierre Samuel l. a. théorie des correspondances birationnelles selon Zariski (birational geometry)
* 7 Jean Braconnier Sur les suites de composition d'un groupe et los angeles journey des groupes d'automorphismes d'un groupe fini, d'après H. Wielandt (finite groups)
* eight Henri Cartan, Les travaux de Koszul, II (see 1)
* nine Claude Chevalley, L'hypothèse de Riemann pour les groupes de fonctions algébriques de caractéristique p, II,, d'après Weil (see 3)
* 10 Luc Gauthier, Théorie des correspondances birationnelles selon Zariski (see 6)
* eleven Laurent Schwartz, Sur un mémoire de Petrowsky : "Über das Cauchysche challenge für ein procedure linearer partieller Differentialgleichungen im gebiete nichtanalytischen Funktionen" (partial differential equations)
* 12 Henri Cartan, Les travaux de Koszul, III (see 1)
* thirteen Roger Godement, Groupe complexe unimodulaire, II : l. a. transformation de Fourier dans le groupe complexe unimodulaire à deux variables, d'après Gelfand et Neumark (see 4)
* 14 Marc Krasner, Les travaux récents de R. Brauer en théorie des groupes (finite groups)
* 15 Laurent Schwartz, Sur un deuxième mémoire de Petrowsky : "Über das Cauchysche challenge für process von partiellen Differentialgleichungen" (see 11)
* sixteen André Weil Théorèmes fondamentaux de los angeles théorie des fonctions thêta, d'après des mémoires de Poincaré et Frobenius (theta functions)
* 17 André Blanchard, Groupes algébriques et équations différentielles linéaires, d'après E. Kolchin (differential Galois theory)
* 18 Jean Dieudonné, Géométrie des espaces algébriques homogènes, d'après W. L. Chow (algebraic geometry)
* 19 Roger Godement, Sommes keeps d'espaces de Hilbert, I (functional research, direct integrals)
* 20 Charles Pisot, Démonstration élémentaire du théorème des nombres premiers, d'après Selberg et Erdös (prime quantity theorem)
* 21 Georges Reeb, Propriétés des trajectoires de certains systèmes dynamiques (dynamical systems)
* 22 Pierre Samuel, Anneaux locaux ; advent à l. a. géométrie algébrique (local rings)
* 23 Marie-Hélène Schwartz, Compte-rendu de travaux de M. Heins sur diverses majorations de l. a. croissance des fonctions analytiques et sous-harmoniques (complex research, subharmonic functions)
* 24 Charles Ehresmann, Les connexions infinitésimales dans un espace fibré différentiable (connections on fiber bundles)
* 25 Roger Godement, Sommes maintains d'espaces de Hilbert, II (see 19)
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* 27 Jean-Pierre Serre, Extensions de groupes localement compacts, d'après Iwasawa et Gleason (locally compact groups)
* 28 René Thom, Les géodésiques dans les variétés à courbure négative, d'après Hopf (geodesics)
* 29 Armand Borel, Groupes localement compacts, d'après Iwasawa et Gleason (see 27)
* 30 Jacques Dixmier, Facteurs : class, size, hint (von Neumann algebras)
* 31 Jean-Louis Koszul, Algèbres de Jordan (Jordan algebras)
* 32 Laurent Schwartz, Sur un mémoire de ok. Kodaira : "Harmonic fields in riemannian manifolds (generalized strength theory)", II (see 26)
* 33 Armand Borel, Sous-groupes compacts maximaux des groupes de Lie, d'après Cartan, Iwasawa et Mostow (maximal compact subgroups)
* 34 Henri Cartan, Espaces fibrés analytiques complexes (analytic geometry, fiber bundles)
* 35 Charles Ehresmann, Sur les variétés presque complexes (almost-complex manifolds)
* 36 Samuel Eilenberg, Exposition des théories de Morse et Lusternick-Schnirelmann (Morse thought, Lyusternik-Schnirelmann category)
* 37 Luc Gauthier, Quelques variétés usuelles en géométrie algébrique (algebraic geometry)
* 38 Jean-Louis Koszul, Cohomologie des espaces fibrés différentiables et connexions (Chern-Weil theory)
* 39 Jean Delsarte, Nombre de recommendations des équations polynomiales sur un corps fini, d'après A. Weil (Weil conjectures)
* forty Jacques Dixmier, Anneaux d'opérateurs et représentations des groupes (operator algebras, illustration theory)
* forty-one Roger Godement, Théorie des caractères dans les groupes unimodulaires (unimodular groups)
* forty two Pierre Samuel, Théorie du corps de sessions neighborhood selon G. P. Hochschild (local category box theory)
* forty three Laurent Schwartz, Les théorèmes de Whitney sur les fonctions différentiables (singularity theory)
* forty four Jean-Pierre Serre, Groupes d'homotopie (homotopy groups)
* forty five Armand Borel, Cohomologie des espaces homogènes (cohomology of homogeneous areas of Lie groups)
* forty six Samuel Eilenberg, Foncteurs de modules et leurs satellites, d'après Cartan et Eilenberg (homological algebra)
* forty seven Marc Krasner, Généralisations non-abéliennes de l. a. théorie locale des corps de periods (local fields)
* forty eight Jean Leray, los angeles résolution des problèmes de Cauchy et de Dirichlet au moyen du calcul symbolique et des projections orthogonales et obliques (Dirichlet difficulties and Cauchy difficulties for partial differential equations, symbolic calculus)
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Extra resources for USA & International Mathematical Olympiads 2003

Sample text

A a 0 a 0 0 0→a 0→0 0 a a 0 a 0 0 31 Formal Solutions Third Solution. (By Tiankai Liu) In the beginning, because A + B + C + D + E + F is odd, either A + C + E or B + D + F is odd; assume without loss of generality it is the former. Perform the following steps repeatedly. a. In this case we assume that A, C, E are all nonzero. Suppose without loss of generality that A ≥ C ≥ E. Perform the sequence of moves A B C (A − C) C D→A (C − E) F E (A − E) E → (C − E) (A − C) C (C − E), (A − E) (A − C) which decreases the sum of the numbers in positions A, C, E while keeping that sum odd.

We will show that from any position in which the sum is odd, it is possible to reach the all-zero position. Our strategy alternates between two steps: 27 Formal Solutions (a) from a position with odd sum, move to a position with exactly one odd number; (b) from a position with exactly one odd number, move to a position with odd sum and strictly smaller maximum, or to the all-zero position. Note that no move will ever increase the maximum, so this strategy is guaranteed to terminate, because each step of type (b) decreases the maximum by at least one, and it can only terminate at the all-zero position.

Observe that from positions of the form 0 1 1 0 1 1 (mod 2) or rotations it is impossible to reach the all-zero position, because a move at any vertex leaves the same value modulo 2. Dividing out the greatest common divisor of the six original numbers does not affect whether we can reach the all-zero position, so we may assume that the numbers in the original position are not all even. Then by a more complete analysis in step (a), one can show from any position not of the above form, it is possible to reach a position with exactly one odd number, and thus the all-zero position.

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