Question

考虑顶点为 $(10, 45)$ ， $(10, 114)$ ， $(28, 153)$ ，和 $(28, 84)$ 的平行四边形。一条经过原点的直线把这个图形分成两个全等多边形。该直线的斜率为 $\frac{m}{n}$ ，其中 $m$ 和 $n$ 是互质的正整数。求 $m + n$ 。

Consider the parallelogram with vertices $(10, 45),$ $(10, 114),$ $(28, 153),$ and $(28, 84).$ A line through the origin cuts this figure into two congruent polygons. The slope of the line is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$

答案

答案：118

知识点：平行四边形中点斜率对称性

难度评级：1790

解答：

平行四边形关于其中心作 $180^\circ$ 旋转对称，所以任何经过中心的直线都会把它分成两个在该旋转下互相对应的部分，因此这两个部分全等。中心是某条对角线的中点： $\begin{aligned} &\left(\frac{10 + 28}{2}, \frac{45 + 153}{2}\right) \\ &= (19, 99). \end{aligned}$

经过原点和 $(19, 99)$ 的直线斜率为 $\frac{99}{19}$ ，且 $\gcd(99, 19) = 1$ ，所以 $m + n = 99 + 19 = 118$ 。

Congruent pieces have equal area. For any fixed direction, there is only one line in that direction that bisects the area of a convex figure; because a parallelogram is centrally symmetric, that line passes through its center. Thus the required line through the origin must also pass through the center. Conversely, the $180^\circ$ rotation about the center swaps the two pieces cut by such a line, so they are congruent. The center is the midpoint of a diagonal: $\begin{aligned} &\left(\frac{10 + 28}{2}, \frac{45 + 153}{2}\right) \\ &= (19, 99). \end{aligned}$

The line through the origin and $(19, 99)$ has slope $\frac{99}{19},$ and $\gcd(99, 19) = 1,$ so $m + n = 99 + 19 = 118.$

1999 AIME 第 2 题

答案：118

难度评级：1790

解答：

其他年份的第 2 题