Question

考虑所有三角形 $OPQ$ ，其中 $O$ 是原点， $P$ 和 $Q$ 是平面上互不相同的点，且坐标为非负整数 $(x, y)$ ，满足 $41x + y = 2009$ 。求这些不同三角形中，面积为正整数的个数。

Consider the set of all triangles $OPQ$ where $O$ is the origin and $P$ and $Q$ are distinct points in the plane with nonnegative integer coordinates $(x, y)$ such that $41x + y = 2009.$ Find the number of such distinct triangles whose area is a positive integer.

答案

答案：600

知识点：格点鞋带公式奇偶性

难度评级：2840

解答：

直线上非负整数坐标的点为 $P_i = (i,\, 2009 - 41i)$ ，其中 $i = 0, 1, \ldots, 49$ ，共五十个点。若 $P = P_i$ ， $Q = P_j$ ，用鞋带公式得 $\begin{aligned} [OPQ] &= \small \frac{1}{2}\left|\,i(2009 - 41j) - j(2009 - 41i)\,\right| \\ &= \frac{2009}{2}\,|i - j|. \end{aligned}$

对互不相同的点，面积自动为正；又因为 $2009$ 是奇数，它是整数当且仅当 $i - j$ 为偶数，也就是 $i$ 和 $j$ 同奇偶。偶数编号有 $25$ 个，奇数编号也有 $25$ 个，所以三角形个数为 $\binom{25}{2} + \binom{25}{2} = 300 + 300 = 600.$

The points on the line with nonnegative integer coordinates are $P_i = (i,\, 2009 - 41i)$ for $i = 0, 1, \ldots, 49$ — fifty points in all. For $P = P_i$ and $Q = P_j,$ the shoelace formula gives $\begin{aligned} [OPQ] &= \small \frac{1}{2}\left|\,i(2009 - 41j) - j(2009 - 41i)\,\right| \\ &= \frac{2009}{2}\,|i - j|. \end{aligned}$

This is automatically positive for distinct points, and since $2009$ is odd, it is an integer exactly when $i - j$ is even, that is, when $i$ and $j$ have the same parity. There are $25$ even and $25$ odd indices, so the number of triangles is $\binom{25}{2} + \binom{25}{2} = 300 + 300 = 600.$

2009 AIME I 第 11 题

答案：600

难度评级：2840

解答：

其他年份的第 11 题