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Download 50th IMO - 50 years of International Mathematical Olympiads by Hans-Dietrich Gronau, Hanns-Heinrich Langmann, Dierk PDF

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By Hans-Dietrich Gronau, Hanns-Heinrich Langmann, Dierk Schleicher

In July 2009 Germany hosted the fiftieth foreign Mathematical Olympiad (IMO). For the first actual time the variety of partaking international locations passed a hundred, with 104 nations from all continents. Celebrating the fiftieth anniversary of the IMO offers an awesome chance to seem again over the last 5 a long time and to study its improvement to develop into a global occasion. This booklet is a file concerning the fiftieth IMO in addition to the IMO heritage. loads of info approximately all of the 50 IMOs are incorporated. We record the main profitable contestants, the result of the 50 Olympiads and the 112 nations that experience ever taken half. it truly is remarkable to work out that some of the world’s prime learn mathematicians have been one of the such a lot winning IMO members of their early life. Six of them gave shows at a distinct party: Bollobás, Gowers, Lovász, Smirnov, Tao and Yoccoz. This booklet is aimed toward scholars within the IMO age staff and all those that have curiosity during this world wide best pageant for high school students.

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1 National Teams ALB Leader Deputy Contestants ALG Leader Deputy Contestants ARG Leader Deputy Contestants ARM Leader Deputy Contestants Albania Fatmir Hoxha Artur Baxhaku Andi Reçi, Tedi Aliaj, Ornela Xhelili, Arlind Gjoka, Niko Kaso, Ridgers Mema Algeria Abed-Seddik Bouchoucha Ali Atia Lamia Attouche, Hacen Zelaci, Oussama Guessab, Kacem Hariz, Argentina Patricia Fauring Flora Gutierrez Germán Stefanich, Alfredo Umfurer, Iván Sadofschi, Federico Cogorno, Ariel Zylber, Miguel Maurizio Armenia Nairi Sedrakyan Koryun Arakelyan Anna Srapionyan, Nerses Srapionyan, Vahagn Kirakosyan, Vahagn Aslanyan, Hayk Saribekyan, Vanik Tadevosyan 5 Participants 49 ALB ALG ARG ARM 50 AUS 5 Participants Australia Leader Deputy Contestants Observer A AUT Leader Deputy Contestants AZE Leader Deputy Contestants BGD Leader Deputy Contestants Observer C Angelo Di Pasquale Ivan Guo Aaron Wan Yau Chong, Andrew Elvey Price, Stacey Wing Chee Law, Alfred Liang, Dana Ma, Sampson Wong Hans Lausch, Ian Roberts Austria Robert Geretschläger Walther Janous Felix Dräxler, Adrian Fuchs, Pfannerer, Valerie Roitner Johannes Hafner, Clemens Müllner, Stephan Azerbaijan Fuad Garayev Jafar Jafarov Ilgar Ramazanli, Elvin Aliyev, Zulfu Aslanli, Altun Shukurlu, Subhan Rustamli, Emil Jafarli Bangladesh Mahbub Alam Majumdar A A Munir Hasan Samin Riasat, Nazia Naser Chowdhury, Tarik Adnan Moon, Haque Muhammad Ishfaq, Kazi Hasan Zubaer, Pranon Rahman Khan Jesmin Akter, Md.

Gauss himself revealed that among his primary inspirations for these theoretical geometric investigations were astronomy and (theoretical) surveying. For an oriented smooth surface S in R3 , the scalar curvature that is now known as the Gaussian curvature is defined as follows: for each point x ∈ S, choose a unit vector n(x) perpendicular to S at x. This defines n(x) uniquely up to sign, and orientability of S means that one can define n(x) continuously on all of S. Now n(x) is a point on the unit sphere S2 ⊂ R3 ; this defines a smooth map G : S → S2 , x → n(x).

On a sphere of radius r, say centered at the origin the Gauss map is G(x) = x/r, so the curvature is constant and equal to 1/r2 . On 3 Carl Friedrich Gauss, Curvature, and the Cover Art of the 50th IMO 25 domains where the surface has a saddle, the Gauss map changes orientation, and thus the Gaussian curvature is negative. Note that this definition of the Gaussian curvature depends on how S is embedded in R3 . One of Gauss’ key results is his “theorema egregium” (remarkable theorem) that states that this curvature depends only on the geometry of the surface S itself (all necessary measurements can be performed on S), but it does not depend on any embedding of S into an ambient space such as R3 : Gaussian curvature is an intrinsic quantity of any surface.

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