2021 CMO 第 3 题

Let $m>1$ be an integer. Find the smallest positive integer $n$ , such that for any integers $a_1,a_2,\ldots ,a_n; b_1,b_2,\ldots ,b_n$ there exists integers $x_1,x_2,\ldots ,x_n$ satisfying the following two conditions:

i) There exists $i\in \{1,2,\ldots ,n\}$ such that $x_i$ and $m$ are coprime

ii) $\sum^n_{i=1} a_ix_i \equiv \sum^n_{i=1} b_ix_i \equiv 0 \pmod m$

令 $m>1$ 为整数。找到最小的正整数 $n$ ，使得对于任何整数 $a_1,a_2,\ldots ,a_n; b_1,b_2,\ldots ,b_n$ 存在整数 $x_1,x_2,\ldots ,x_n$ 满足以下两个条件：

i) 存在 $i\in \{1,2,\ldots ,n\}$ 使得 $x_i$ 和 $m$ 互质

ii) $\sum^n_{i=1} a_ix_i \equiv \sum^n_{i=1} b_ix_i \equiv 0 \pmod m$