Shortlisted Problems with Solutions
Shortlisted Problems with Solutions53rdInternational Mathematical OlympiadMar del Plata, Argentina 2012Note of ConfidentialityThe Shortlisted Problems should be keptstrictly confidential until IMO 2013Contributing 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 exist a partition ofZinto three non-empty subsetsA, B, Csuch that the setsA+B,B+C,C+Aare disjoint?
G´eza Ko´s Svetoslav Savchev. 4 Algebra A1. Find all the functions f : Z→ Zsuch that ... intersect ℓ at the points X,Y,Z different from P. Prove that the circumcircles of the triangles AXP,BYP and CZP have a common point different from P or are mutually tangent at P. 7 Number Theory N1.
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