1
Part of 2004 Vietnam National Olympiad
Problems(2)
Vietnam NMO 2004_1
Source:
10/26/2008
Solve the system of equations \begin{cases} x^3 \plus{} x(y \minus{} z)^2 \equal{} 2\\ y^3 \plus{} y(z \minus{} x)^2 \equal{} 30\\ z^3 \plus{} z(x \minus{} y)^2 \equal{} 16\end{cases}.
algebrasystem of equationsalgebra proposed
Vietnam NMO 2004_4
Source:
10/26/2008
The sequence (x_n)^{\infty}_{n\equal{}1} is defined by x_1 \equal{} 1 and x_{n\plus{}1} \equal{}\frac{(2 \plus{} \cos 2\alpha)x_n \minus{} \cos^2\alpha}{(2 \minus{} 2 \cos 2\alpha)x_n \plus{} 2 \minus{} \cos 2\alpha}, for all , where is a given real parameter. Find all values of for which the sequence given by y_n \equal{} \sum_{k\equal{}1}^{n}\frac{1}{2x_k\plus{}1} has a finite limit when n \to \plus{}\infty and find that limit.
trigonometrylimitalgebra unsolvedalgebra