MathDB

1

Part of 1990 Vietnam National Olympiad

Problems(2)

Vietnam NMO 1990_1

Source:

10/26/2008
The sequence (xn) (x_n), n∈N∗ n\in\mathbb{N}^* is defined by ∣x1∣<1 |x_1|<1, and for all n≥1 n \ge 1, x_{n\plus{}1} \equal{}\frac{\minus{}x_n \plus{}\sqrt{3\minus{}3x_n^2}}{2} (a) Find the necessary and sufficient condition for x1 x_1 so that each xn>0 x_n > 0. (b) Is this sequence periodic? And why?
trigonometryalgebra unsolvedalgebra
Vietnam NMO 1990_4

Source:

10/26/2008
A triangle ABC ABC is given in the plane. Let M M be a point inside the triangle and A′ A', B′ B', C′ C' be its projections on the sides BC BC, CA CA, AB AB, respectively. Find the locus of M M for which MA \cdot MA' \equal{} MB \cdot MB' \equal{} MC \cdot MC'.
geometrycircumcircleanalytic geometryconicshyperbolaperpendicular bisectorgeometry proposed