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 exist a partition ofZinto three non-empty subsetsA, B, Csuch that the setsA+B,B+C,C+Aare disjoint?

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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