1992 AMC 12 第 22 题

Question

在

x

轴正半轴

\mathrm{X}^+

上选取十个点，在

y

轴正半轴

\mathrm{Y}^+

上选取五个点。连接

\mathrm{X}^+

上每个选定点与

\mathrm{Y}^+

上每个选定点，共画出五十条线段。位于第一象限内部的这些线段交点数最多可能是多少？

Ten points are selected on the positive

x

-axis,

\mathrm{X}^+,

and five points are selected on the positive

y

-axis,

\mathrm{Y}^+.

The fifty segments connecting the ten selected points on

\mathrm{X}^+

to the five selected points on

\mathrm{Y}^+

are drawn. What is the maximum possible number of points of intersection of these fifty segments that could lie in the interior of the first quadrant?

250

450

500

1250

2500

答案：B

知识点：combinatorial geometry intersections binomial coefficient

难度评级：2110

小提示：

一个内部交点由每条坐标轴上各选两个点确定

An interior crossing is determined by choosing two points from each axis

大提示：

对每条坐标轴上选定的两个点，四条连接线段中恰有一对相交

For each selected pair on each axis, exactly one pair of the four connecting segments crosses

解答：

从这

10

个

x

轴上的点中选两个，再从这

5

个

y

轴上的点中选两个。在四条连接线段中，端点次序相反的两条恰好相交一次。可以选取这些点，使得没有三条线段在同一个内部点相交，因此最大值为

\binom{10}{2}\binom{5}{2}=45\cdot10=450\text{。}

因此正确答案是 B。

Choose two of the

10

points on the

x

-axis and two of the

5

points on the

y

-axis. Among the four connecting segments, the two with reversed endpoint order cross exactly once. The points can be chosen so no three segments meet at one interior point, so the maximum is

\binom{10}{2}\binom{5}{2}=45\cdot10=450.

Thus the correct answer is B.

1992 AMC 12 第 22 题

答案：B

难度评级：2110

小提示：

大提示：

解答：

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