MathDB

2

Part of 1986 Vietnam National Olympiad

Problems(2)

Geometry Inequality

Source: Vietnam NMO 1986 Problem 2

2/6/2009
Let R R, r r be respectively the circumradius and inradius of a regular 1986 1986-gonal pyramid. Prove that \frac{R}{r}\ge 1\plus{}\frac{1}{\cos\frac{\pi}{1986}} and find the total area of the surface of the pyramid when the equality occurs.
geometryinequalitiescircumcircleinradiustrigonometry3D geometrypyramid
Find n s.t the inequality holds for all real numbers x_i

Source: Vietnam NMO 1986 Problem 5

2/6/2009
Find all n>1 n > 1 such that the inequality \sum_{i\equal{}1}^nx_i^2\ge x_n\sum_{i\equal{}1}^{n\minus{}1}x_i holds for all real numbers x1 x_1, x2 x_2, … \ldots, xn x_n.
inequalitiesfunctioninequalities unsolved