Shortlisted Problems with Solutions
Let k and n be fixed positive integers. In the liar’s guessing game, Amy chooses integers x and N with 1 ≤ x ≤ N. She tells Ben what N is, but not what x is. Ben may then repeatedly ask Amy whether x ∈ S for arbitrary sets S of integers. Amy will …
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52 Mathematical Olympiad - imo-official.org
www.imo-official.orgMathematical Olympiad Am sterdam 2011 IMO2011 Amsterdam Problem Shortlist with Solutions. 52nd International Mathematical Olympiad 12-24 July 2011 Amsterdam The Netherlands Problem shortlist with solutions. IMPORTANT IMO regulation: these shortlist problems have to be kept strictly confidential
Shortlisted Problems with Solutions
www.imo-official.orgShortlisted 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 ...
IMO2018 Shortlisted Problems with Solutions
www.imo-official.orgpro cess es tak place on an initially y empt b oard. piq If there is at least a pair of equal b umers n on the b oard, e w ho cose h suc and increase one of its compts onen y b a and the other b. piiq If no h suc pair exists, e w write wn do o w t times the b umer n 0. e v Pro that, no matter w ho e w e mak the hoices c in piq, op eration piiq ...
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