Question 41
Let $f(x) = x^{2} - r_{2} x + r_{3}$ for all real numbers $x$, where $r_{2}$ and $r_{3}$ are some real numbers. Define a sequence $\{g_{n}\}$ for all nonnegative integers $n$ by $g_{0} = 0$ and $g_{n+1} = f(g_{n})$. Assume that $\{g_{n}\}$ satisfies the following three conditions:
(i) $g_{2i} \lt g_{2i+1}$ and $g_{2i+1} \gt g_{2i+2}$ for all $0 \leq i \leq 2011$;
(ii) there exists a positive integer $j$ such that $g_{i+1} \gt g_{i}$ for all $i \gt j$;
(iii) $\{g_{n}\}$ is unbounded.
If $A$ is the greatest number such that $A \leq |r_{2}|$ for any function $f$ satisfying these properties, find $A$.
(i) $g_{2i} \lt g_{2i+1}$ and $g_{2i+1} \gt g_{2i+2}$ for all $0 \leq i \leq 2011$;
(ii) there exists a positive integer $j$ such that $g_{i+1} \gt g_{i}$ for all $i \gt j$;
(iii) $\{g_{n}\}$ is unbounded.
If $A$ is the greatest number such that $A \leq |r_{2}|$ for any function $f$ satisfying these properties, find $A$.