1997 IMO 第 1 题

$)^{\mathrm{IMO}} \mathrm{An}$ infinite square grid is colored in the chessboard pattern. For any pair of positive integers $m, n$ consider a right-angled triangle whose vertices are grid points and whose legs, of lengths $m$ and $n$, run along the lines of the grid. Let $S_{b}$ be the total area of the black part of the triangle and $S_{w}$ the total area of its white part. Define the function $f(m, n)=\left|S_{b}-S_{w}\right|$. (a) Calculate $f(m, n)$ for all $m, n$ that have the same parity. (b) Prove that $f(m, n) \leq \frac{1}{2} \max (m, n)$. (c) Show that $f(m, n)$ is not bounded from above.

$)^{\mathrm{IMO}} \mathrm{An}$无限方形网格在棋盘图案中着色。对于任何一对正整数 $m, n$，请考虑一个直角三角形，其顶点是网格点，其长度为 $m$ 和 $n$ 的边沿网格线延伸。令 $S_{b}$ 为三角形黑色部分的总面积，$S_{w}$ 为其白色部分的总面积。定义函数 $f(m, n)=\left|S_{b}-S_{w}\right|$。 (a) 计算具有相同奇偶校验的所有 $m, n$ 的 $f(m, n)$。 (b) 证明 $f(m, n) \leq \frac{1}{2} \max (m, n)$。 (c) 证明 $f(m, n)$ 不受上方限制。