MathDB

3

Part of 2000 Turkey Team Selection Test

Problems(2)

Sequences (x_k, y_k) and expressions P(x)=x+1,Q(x)=x^2+1

Source: Turkey TST 2000 P3

3/12/2011
Let P(x)=x+1P(x)=x+1 and Q(x)=x2+1.Q(x)=x^2+1. Consider all sequences ⟨(xk,yk)⟩k∈N\langle(x_k,y_k)\rangle_{k\in\mathbb{N}} such that (x1,y1)=(1,3)(x_1,y_1)=(1,3) and (xk+1,yk+1)(x_{k+1},y_{k+1}) is either (P(xk),Q(yk))(P(x_k), Q(y_k)) or (Q(xk),P(yk))(Q(x_k),P(y_k)) for each k.k. We say that a positive integer nn is nice if xn=ynx_n=y_n holds in at least one of these sequences. Find all nice numbers.
number theory unsolvednumber theory
f:R→R; f(x+y)=f(x)+f(y)+1, Show that there exists g:R→R

Source: Turkey TST 2000 P6

3/12/2011
Suppose f:R→Rf:\mathbb{R} \to \mathbb{R} is a function such that ∣f(x+y)−f(x)−f(y)∣≤1   for all  x,y∈R.|f(x+y)-f(x)-f(y)|\le 1\ \ \ \text{for all} \ \ x, y \in\mathbb R. Prove that there is a function g:R→Rg:\mathbb{R}\to\mathbb{R} such that ∣f(x)−g(x)∣≤1|f(x)-g(x)|\le 1 and g(x+y)=g(x)+g(y)g(x+y)=g(x)+g(y) for all x,y∈R.x,y \in\mathbb R.
functionlimitinequalitiesalgebra