Question

有三个半径均为 $3$ 的圆，其圆心分别为 $(14,92)$ 、 $(17,76)$ 和 $(19,84)$ 。一条经过 $(17,76)$ 的直线，将三个圆各自在直线一侧的部分面积相加，所得总面积恰好等于另一侧各部分的总面积。求这条直线斜率的绝对值。

Three circles, each of radius $3,$ are drawn with centers at $(14,92),$ $(17,76),$ and $(19,84).$ A line passing through $(17,76)$ is such that the total area of the parts of the three circles to one side of the line is equal to the total area of the parts of the three circles to the other side of it. What is the absolute value of the slope of this line?

答案

答案：24

知识点：圆面积对称性斜率

难度评级：2410

小提示：

这条直线已经平分了以 $(17,76)$ 为圆心的圆

The line already bisects the circle centered at $(17,76)$

大提示：

对于另外两个相等的圆，它们的圆心到直线的有向距离必须互为相反数

For the other two equal circles, their signed distances from the line must be opposites

解答：

对于半径为 $3$ 的圆，设 $G(d)$ 表示圆心到直线的有向距离为 $d$ 时，直线两侧面积的有向差。函数 $G$ 是奇函数，在 $-3<d<3$ 上严格递增；只有当直线不再与圆相交时，它才取常值。以 $(17,76)$ 为圆心的圆贡献为零，因此这条直线必须将另外两个圆心分隔在两侧。

设这两个圆心的中点为 $M=(\frac{33}{2},88)$ 。在所有经过 $(17,76)$ 并将它们分隔开的直线中，当两个圆心到直线的距离相等时，这两个距离中的较小者最大；此时直线经过 $M$ 。即使在这种情形下，该距离也只有 $\frac{56}{\sqrt{577}}<3$ ，所以直线必与两个圆中的至少一个相交。要使两者的面积贡献平衡，直线就必须与两个圆都相交。由 $G$ 的严格递增性，两个有向距离必须互为相反数，因此所求直线经过 $M$ 。其斜率为 $\frac{88-76}{\frac{33}{2}-17}=-24\text{。}$ 所以斜率的绝对值为 $24$ 。

For a circle of radius $3,$ let $G(d)$ be the signed difference between the areas on the two sides of a line when the center’s signed distance from the line is $d.$ The function $G$ is odd and strictly increasing for $-3<d<3,$ and is constant only after the line no longer cuts the circle. The circle centered at $(17,76)$ contributes zero, so the line must separate the other two centers.

Let $M=(\frac{33}{2},88)$ be the midpoint of those two centers. Among the lines through $(17,76)$ that separate them, the smaller of their two distances to the line is largest when the distances are equal, namely for the line through $M.$ Even then, the distance is $\frac{56}{\sqrt{577}}<3,$ so the line must cut at least one of the two circles. Balance then forces it to cut both. The strict increase of $G$ therefore forces the two signed distances to be opposites, so the required line passes through $M.$ Its slope is $\frac{88-76}{\frac{33}{2}-17}=-24.$ Its absolute value is $24.$

1984 AIME 第 6 题

答案：24

难度评级：2410

小提示：

大提示：

解答：

其他年份的第 6 题