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52 Mathematical Olympiad

52nd International Mathematical Olympiad12 24 July 2011 AmsterdamThe NetherlandsInternationalMathematicalOlym piad Amsterdam 2011 IMO2011 AmsterdamProblem Shortlistwith Solutions52nd InternationalMathematical Olympiad12-24 July 2011 AmsterdamThe NetherlandsProblem shortlistwith solutionsIMPORTANTIMO regulation:these shortlist problems have tobe kept strictly confidentialuntil IMO problem selection committeeBart de Smit (chairman), Ilya Bogdanov, Johan Bosman,Andries Brouwer, Gabriele Dalla Torre, G eza K os,Hendrik Lenstra, Charles Leytem, Ronald van Luijk,Christian Reiher, Eckard Specht, Hans Sterk, Lenny TaelmanThe committee gratefully acknowledges the receipt of 142 problem proposalsby the following 46 countries.

Algebra Problemshortlist 52ndIMO2011 Algebra A1 A1 For any set A = {a 1,a 2,a 3,a 4} of four distinct positive integers with sum sA = a 1+a 2+a 3+a 4, let pA denote the number of pairs (i,j) with 1 ≤ i < j ≤ 4 for which ai +aj divides sA.Among all sets of four distinct positive integers, determine those sets A for which pA is maximal. A2

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Shortlisted Problems with Solutions

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Shortlisted Problems with Solutions

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Shortlisted problems 3 Problems Algebra A1. Let nbe a positive integer and let a 1,...,an´1 be arbitrary real numbers. Define the sequences u 0,...,un and v 0,...,vn inductively by u 0 “ u 1 “ v 0 “ v 1 “ 1, and uk`1 “ uk `akuk´1, vk`1 “ vk `an´kvk´1 for k“ 1,...,n´1. Prove that un “ vn. (France) A2. Prove that in any set of 2000 distinct real numbers there exist two ...

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