Mediterranean Mathematics Olympiad 2003

Prove that the equation $$x^2 + y^2 + z^2 = x + y + z + 1$$ has no rational solutions. In a triangle $ABC$ with $BC = CA + \frac 12 AB$, p...

  1. Prove that the equation $$x^2 + y^2 + z^2 = x + y + z + 1$$ has no rational solutions.
  2. In a triangle $ABC$ with $BC = CA + \frac 12 AB$, point $P$ is given on side $AB$ such that $BP : PA = 1 : 3$. Prove that $\angle CAP = 2 \angle CPA.$
  3. Let $a, b, c$ be non-negative numbers with $a+b+c = 3$. Prove the inequality \[\frac{a}{b^2+1}+\frac{b}{c^2+1}+\frac{c}{a^2+1} \geq \frac 32.\]
  4. Consider a system of infinitely many spheres made of metal, with centers at points $(a, b, c) \in \mathbb Z^3$. We say that the system is stable if the temperature of each sphere equals the average temperature of the six closest spheres. Assuming that all spheres in a stable system have temperatures between $0^\circ C$ and $1^\circ C$, prove that all the spheres have the same temperature.

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Balkan,1,Bosonia,1,Brazil,1,Bulgary,1,Canada,2,Caucasus,2,CentroAmerican,1,CGMO,1,China,5,Cono Sur,2,France,1,Germany,2,Greece,2,IberoAmerican,1,IMO,1,India,2,Indonedia,1,International,42,Italy,1,Itan,1,Japan,1,JBMO,2,Kazakhstan,1,Korea,2,Macedonia,1,Mediterrane,7,Mediterranean,3,MEMO,1,Mexico,1,Miklós Schweitzer,1,Moldova,1,National,30,Olympic Revenge,1,Paenza,1,Paraguayan,1,Rusia,1,TST,9,Turkey,1,Tuymaada,19,Undergraduate,1,Zhautykov,1,
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MATHEMATICAL OLYMPIAD PROBLEMS: Mediterranean Mathematics Olympiad 2003
Mediterranean Mathematics Olympiad 2003
MATHEMATICAL OLYMPIAD PROBLEMS
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