1

Part of 1980 Vietnam National Olympiad

Problems(2)

Sum of sines is greater than 1.

Source: Vietnam MO 1980 P1

Let \alpha_{1}, \alpha_{2}, \cdots , \alpha_{n} be numbers in the interval [0, 2\pi] such that the number \displaystyle\sum_{i=1}^n (1 + \cos \alpha_{i}) is an odd integer. Prove that \displaystyle\sum_{i=1}^n \sin \alpha_i \ge 1

trigonometryvectorabsolute valueinequalities unsolvedinequalities

Ratio of projections of a tetrahedron on two planes.

Source:

Prove that for any tetrahedron in space, it is possible to find two perpendicular planes such that ratio between the projections of the tetrahedron on the two planes lies in the interval

[\frac{1}{\sqrt{2}}, \sqrt{2}].

ratiogeometry3D geometrytetrahedron