1995 IMO 第 2 题

A2 (SWE) Let $a$ and $b$ be nonnegative integers such that $a b \geq c^{2}$, where $c$ is an integer. Prove that there is a number $n$ and integers $x_{1}, x_{2}, \ldots, x_{n}, y_{1}, y_{2}, \ldots, y_{n}$ such that $$\sum_{i=1}^{n} x_{i}^{2}=a, \quad \sum_{i=1}^{n} y_{i}^{2}=b, \quad \text { and } \quad \sum_{i=1}^{n} x_{i} y_{i}=c$$

A2 (SWE) 令 $a$ 和 $b$ 为非负整数，使得 $a b \geq c^{2}$，其中 $c$ 为整数。证明存在数字 $n$ 和整数 $x_{1}, x_{2}, \ldots, x_{n}, y_{1}, y_{2}, \ldots, y_{n}$ 使得 $$\sum_{i=1}^{n} x_{i}^{2}=a, \quad \sum_{i=1}^{n} y_{i}^{2}=b, \quad \text { 和 } \quad \sum_{i=1}^{n} x_{i} y_{i}=c$$