Question

方程 $9x + 223y = 2007$ 的图像画在方格纸上，每个小正方形在两个方向上都表示一个单位。有多少个 $1$ 乘 $1$ 的方格纸小正方形，其内部完全位于该图像下方且完全位于第一象限内？

The graph of the equation $9x + 223y = 2007$ is drawn on graph paper with each square representing one unit in each direction. How many of the $1$ by $1$ graph paper squares have interiors lying entirely below the graph and entirely in the first quadrant?

答案

答案：888

知识点：格点最大公约数对称性

难度评级：2390

解答：

直线与坐标轴交于 $(223, 0)$ 和 $(0, 9)$ ，所以所有符合条件的方格都在 $223 \times 9$ 的矩形内，该矩形含有 $223 \cdot 9 = 2007$ 个单位方格。因为 $\gcd(9, 223) = 1$ ，线段不经过内部格点；它穿过 $222$ 条内部竖直网格线和 $8$ 条内部水平网格线，每次穿线都进入一个新方格，所以它经过 $223 + 9 - 1 = 231$ 个方格的内部。

其余 $2007 - 231 = 1776$ 个方格完全位于线段上方或下方。线段的中点 $\left(\frac{223}{2}, \frac{9}{2}\right)$ 是矩形中心，绕它旋转 $180^\circ$ 会交换这两组方格。因此恰有一半，即 $888$ ，位于图像下方。

The line meets the axes at $(223, 0)$ and $(0, 9),$ so all qualifying squares lie inside the $223 \times 9$ rectangle, which contains $223 \cdot 9 = 2007$ unit squares. Because $\gcd(9, 223) = 1,$ the segment passes through no interior lattice point; it crosses $222$ interior vertical lines and $8$ interior horizontal lines, entering a new square at each crossing, so it passes through the interiors of $223 + 9 - 1 = 231$ squares.

The other $2007 - 231 = 1776$ squares lie entirely above or entirely below the segment. The segment's midpoint $\left(\frac{223}{2}, \frac{9}{2}\right)$ is the center of the rectangle, so rotating by $180^\circ$ about it swaps the two groups. Hence exactly half of them, $888,$ lie below the graph.

2007 AIME II 第 5 题

答案：888

难度评级：2390

解答：

其他年份的第 5 题