MathDB
Romania District Olympiad 2008 - Grade XI

Source:

April 10, 2011
real analysisreal analysis unsolved

Problem Statement

Let (xn)n≥1(x_n)_{n\ge 1} and (yn)n≥1(y_n)_{n\ge 1} a sequence of positive real numbers, such that:
xn+1≥xn+yn2, yn+1≥xn2+yn22, (∀)n∈N∗x_{n+1}\ge \frac{x_n+y_n}{2},\ y_{n+1}\ge \sqrt{\frac{x_n^2+y_n^2}{2}},\ (\forall)n\in \mathbb{N}^*
a) Prove that the sequences (xn+yn)n≥1(x_n+y_n)_{n\ge 1} and (xnyn)n≥1(x_ny_n)_{n\ge 1} have limit.
b) Prove that the sequences (xn)n≥1(x_n)_{n\ge 1} and (yn)n≥1(y_n)_{n\ge 1} have limit and that their limits are equal.
Back to ProblemsView on AoPS