MathDB

2

Part of 1990 Vietnam Team Selection Test

Problems(2)

Inequality in a tetrahedron

Source: Vietnam TST 1990, Problem 5

7/29/2008
Given a tetrahedron such that product of the opposite edges is 1 1. Let the angle between the opposite edges be α \alpha, β \beta, γ \gamma, and circumradii of four faces be R1 R_1, R2 R_2, R3 R_3, R4 R_4. Prove that \sin^2\alpha \plus{} \sin^2\beta \plus{} \sin^2\gamma\ge\frac {1}{\sqrt {R_1R_2R_3R_4}}
inequalitiesgeometry3D geometrytetrahedrontrigonometrygeometry unsolved
Find limit

Source: Vietnam TST 1990, Problem 2

7/29/2008
Let be given four positive real numbers a a, b b, A A, B B. Consider a sequence of real numbers x1 x_1, x2 x_2, x3 x_3, … \ldots is given by x_1 \equal{} a, x_2 \equal{} b and x_{n \plus{} 1} \equal{} A\sqrt [3]{x_n^2} \plus{} B\sqrt [3]{x_{n \minus{} 1}^2} ( n \equal{} 2, 3, 4, \ldots). Prove that there exist limit \lim_{n\to \plus{} \propto}x_n and find this limit.
limitalgebra unsolvedalgebra