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IMO2018 Shortlisted Problems with Solutions

ShortlistedProblems(with Solutions )59thInternational Mathematical OlympiadCluj-Napoca Romania, 3 14 July 2018 NoteofCon dentialityTheShortlisthastobekeptstri tly on dentialuntilthe on lusionofthefollowingInternationalMathema ti tionCommitteeofIMO2018thankthefollowing4 9 ountriesfor ontributing168problemproposals:Armenia,A ustralia,Austria,Azerbaijan,Belarus,Belg ium,BosniaandHerzegovina,Brazil,Bulgaria ,Canada,China,Croatia,Cyprus,Cze hRepubli ,Denmark,Estonia,Germany,Gree e,HongKong,I eland,India,Indonesia,Iran,Ireland,Israe l,Japan,Kosovo,Luxembourg,Mexi o,Moldova,Mongolia,Netherlands,Ni aragua,Poland,Russia,Serbia,Singapore,Sl ovakia,Slovenia,SouthAfri a,SouthKorea,Switzerland,Taiwan,Tanzania ,Thailand,Turkey,Ukraine,UnitedKingdom, tionCommitteeCalinPopes u,RaduGologan,MarianAndrona he,MihailBaluna,Ni olaeBeli,IlyaBogdanov,PavelKozhevnikov,G zaK s, tionsf:Q 0 Q 0satisfyingf`x2fpyq2 fpxq2fpyqforallx, yPQ 0.

pro 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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Transcription of IMO2018 Shortlisted Problems with Solutions

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Shortlisted Problems with Solutions

www.imo-official.org

Prove that the quadrilateral BXCY is cyclic. G5. Let ABC be a triangle with ∠BCA = 90 , and let C 0 be the foot of the altitude from C. Choose a point X in the interior of the segment CC 0, and let K,L be the points on the segments AX,BX for which BK = BC and AL = AC respectively. Denote by M the intersection of AL and BK. Show that MK = ML. G6.

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52 Mathematical Olympiad - imo-official.org

www.imo-official.org

Mathematical 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

  Mathematical, Olympiad, Mathematical olympiads

Shortlisted Problems with Solutions

www.imo-official.org

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 ...

  With, Solutions, Problem, Shortlisted, Shortlisted problems with solutions

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