2007 IMO Shortlist C1

Let $n>1$ be an integer. Find all sequences $a_{1}, a_{2}, \ldots, a_{n^{2}+n}$ satisfying the following conditions: (a) $a_{i} \in\{0,1\}$ for all $1 \leq i \leq n^{2}+n$; (b) $a_{i+1}+a_{i+2}+\ldots+a_{i+n}<a_{i+n+1}+a_{i+n+2}+\ldots+a_{i+2 n}$ for all $0 \leq i \leq n^{2}-n$. (Serbia) Answer. Such a sequence is unique. It can be defined as follows: $$a_{u+v n}=\left\{\begin{array}{ll} 0, & u+v \leq n, \\ 1, & u+v \geq n+1 \end{array} \quad \text { for all } 1 \leq u \leq n \text { and } 0 \leq v \leq n\right.$$ The terms can be arranged into blocks of length $n$ as $$(\underbrace{0 \ldots 0}_{n})(\underbrace{0 \ldots 0}_{n-1} 1)(\underbrace{0 \ldots 0}_{n-2} 11) \ldots(\underbrace{0 \ldots 0}_{n-v} \underbrace{1 \ldots 1}_{v}) \ldots(0 \underbrace{1 \ldots 1}_{n-1})(\underbrace{1 \ldots 1}_{n}) .$$

令 $n>1$ 为整数。查找满足以下条件的所有序列 $a_{1}, a_{2}, \ldots, a_{n^{2}+n}$： (a) $a_{i} \in\{0,1\}$ 对于所有 $1 \leq i \leq n^{2}+n$； (b) $a_{i+1}+a_{i+2}+\ldots+a_{i+n}<a_{i+n+1}+a_{i+n+2}+\ldots+a_{i+2 n}$ 对于所有 $0 \leq i \leq n^{2}-n$。 (塞尔维亚)回答。这样的顺序是独一无二的。它可以定义如下： a_{u+v n}=\left\{\begin{array}{ll} 0, & u+v \leq n, \\ 1, & u+v \geq n+1 \end{array} \quad \text { for all } 1 \leq u \leq n \text { 和 } 0 \leq v \leq n\right。 这些项可以排列成长度为 $n$ 的块，如下所示 $$(\underbrace{0 \ldots 0}_{n})(\underbrace{0 \ldots 0}_{n-1} 1)(\underbrace{0 \ldots 0}_{n-2} 11) \ldots(\underbrace{0 \ldots 0}_{n-v} \underbrace{1 \ldots 1}_{v}) \ldots(0 \underbrace{1 \ldots 1}_{n-1})(\underbrace{1 \ldots 1}_{n}) 。$$