MathDB

3

Part of 1996 Turkey Team Selection Test

Problems(2)

Turkey TST 1996 Problem 3

Source: Turkey TST 1996 Problem 3

9/28/2011
If 0=x1<x2<...<x2n+1=10=x_{1}<x_{2}<...<x_{2n+1}=1 are real numbers with xi+1−xi≤hx_{i+1}-x_{i} \leq h for 1≤i≤2n1 \leq i \leq 2n, show that 1−h2<∑i=1nx2i(x2i+1−x2i−1)≤1+h2\frac{1-h}{2}<\sum_{i=1}^{n}{x_{2i}(x_{2i+1}-x_{2i-1})}\leq \frac{1+h}{2}
inequalities proposedinequalities
Turkey TST 1996 Problem 6, must have the limit 0

Source: Turkey TST 1996 Problem 6

9/28/2011
Determine all ordered pairs of positive real numbers (a,b)(a, b) such that every sequence (xn)(x_{n}) satisfying lim⁡n→∞(axn+1−bxn)=0\lim_{n \rightarrow \infty}{(ax_{n+1} - bx_{n})} = 0 must have lim⁡n→∞xn=0\lim_{n \rightarrow \infty} x_n = 0.
limitalgebra proposedalgebra