Mediterranean Mathematics Olympiad 1999

Do there exist a circle and an infinite set of points on it such that the distance between any two of the points is rational? A plane figu...

  1. Do there exist a circle and an infinite set of points on it such that the distance between any two of the points is rational?
  2. A plane figure of area $A > n$ is given, where $n$ is a positive integer. Prove that this figure can be placed onto a Cartesian plane so that it covers at least $n+1$ points with integer coordinates.
  3. Let $a,b,c\not= 0$ and $x,y,z\in\mathbb{R}^+$ such that $x+y+z=3$. Prove that \[\frac{3}{2}\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}}\geq\frac{x}{1+a^2}+\frac{y}{1+b^2}+\frac{z}{1+c^2}\]
  4. In triangle $\triangle ABC$ we have $BC=a$, $CA=b$, $AB=c$ and $\angle B=4\angle A$ Show that \[ab^2c^3=(b^2-a^2-ac)((a^2-b^2)^2-a^2c^2)\]

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Balkan,1,Bosonia,1,Brazil,1,Bulgary,1,Canada,1,CentroAmerican,1,CGMO,1,China,5,Cono Sur,2,France,1,Germany,2,Greece,2,IberoAmerican,1,IMO,1,India,2,Indonedia,1,International,40,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,29,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 1999
Mediterranean Mathematics Olympiad 1999
MATHEMATICAL OLYMPIAD PROBLEMS
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