Home > Affine Geometry, Geometry, Olympiad > Traian Lalescu contest 2010 Problem 2

Traian Lalescu contest 2010 Problem 2


Consider the hyperboloid of one sheet (\mathcal{H}) \displaystyle \ \frac{x^2}{a^2}+\frac{y^2}{b^2}-\frac{z^2}{c^2}=1, in the system of coordinates Oxyz. Suppose there exist points M,N,P \in \mathcal{H} such that the vectors \overrightarrow{OM}, \overrightarrow{ON},\overrightarrow{OP} are mutually orthogonal.

Prove that \displaystyle \frac{1}{a^2}+\frac{1}{b^2} > \frac{1}{c^2}.

Traian Lalescu student contest 2009

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