Transcription of Shortlisted Problems with Solutions
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Shortlisted Problems with Solutions
Shortlisted Problems with Solutions53rdInternational Mathematical OlympiadMar del Plata, Argentina 2012 Note of ConfidentialityThe Shortlisted Problems should be keptstrictly confidential until IMO 2013 Contributing CountriesThe Organizing Committee and the Problem Selection Committee of IMO2012 thank thefollowing 40 countries for contributing 136 problem proposals:Australia, Austria, Belarus, Belgium, Bulgaria, Canada, Cyprus,Czech Republic, Denmark, Estonia, Finland, France, Germany,Greece, Hong Kong, India, Iran, Ireland, Israel, Japan,Kazakhstan, Luxembourg, Malaysia, Montenegro, Netherlands,Norway, Pakistan, Romania, Russia, Serbia, Slovakia, Slovenia,South Africa, South Korea, Sweden, Thailand, Ukraine,United Kingdom, United States of America, UzbekistanProblem Selection CommitteeMart n Avenda noCarlos di FioreG eza K osSvetoslav all the functionsf:Z Zsuch thatf(a)2+f(b)2+f(c)2= 2f(a)f(b) + 2f(b)f(c) + 2f(c)f(a)for all integersa,b,csatisfyinga+b+c= the sets of integers and rationals ) Does there ex
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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