2006 IMO Shortlist A3

The sequence $c_{0}, c_{1}, \ldots, c_{n}, \ldots$ is defined by $c_{0}=1, c_{1}=0$ and $c_{n+2}=c_{n+1}+c_{n}$ for $n \geq 0$. Consider the set $S$ of ordered pairs $(x, y)$ for which there is a finite set $J$ of positive integers such that $x=\sum_{j \in J} c_{j}, y=\sum_{j \in J} c_{j-1}$. Prove that there exist real numbers $\alpha, \beta$ and $m, M$ with the following property: An ordered pair of nonnegative integers $(x, y)$ satisfies the inequality $$m<\alpha x+\beta y<M$$ if and only if $(x, y) \in S$. N. B. A sum over the elements of the empty set is assumed to be 0 . (Russia)

对于 $n \geq 0$，序列 $c_{0}, c_{1}, \ldots, c_{n}, \ldots$ 由 $c_{0}=1, c_{1}=0$ 和 $c_{n+2}=c_{n+1}+c_{n}$ 定义。考虑有序对 $(x, y)$ 的集合 $S$，其中存在正整数的有限集合 $J$，使得 $x=\sum_{j \in J} c_{j}, y=\sum_{j \in J} c_{j-1}$。证明存在实数 $\alpha, \beta$ 和 $m, M$，且具有以下性质：有序非负整数对 $(x, y)$ 满足不等式 $$m<\alpha x+\beta y<M$$ 当且仅当 $(x, y) \in S$。注意：假设空集元素的总和为 0 。 (俄罗斯)