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