Question

平面上有 $40$ 条直线，其中任意 $2$ 条都不平行。已知有 $3$ 个点恰好有 $3$ 条直线相交，有 $4$ 个点恰好有 $4$ 条直线相交，有 $5$ 个点恰好有 $5$ 条直线相交，有 $6$ 个点恰好有 $6$ 条直线相交，并且没有超过 $6$ 条直线相交的点。求恰好有 $2$ 条直线相交的点的个数。

A plane contains $40$ lines, no $2$ of which are parallel. Suppose that there are $3$ points where exactly $3$ lines intersect, $4$ points where exactly $4$ lines intersect, $5$ points where exactly $5$ lines intersect, $6$ points where exactly $6$ lines intersect, and no points where more than $6$ lines intersect. Find the number of points where exactly $2$ lines intersect.

答案

答案：607

知识点：交点计数组合双重计数

难度评级：2090

解答：

因为这 $40$ 条直线中任意两条都不平行，所以每两条直线相交，共有 $\binom{40}{2} = 780$ 对直线，并且每一对直线恰好对应一个交点。一个恰好有 $k$ 条直线相交的点，恰好对应这些直线对中的 $\binom{k}{2}$ 对。

已给出的交点共对应 $3\binom{3}{2} + 4\binom{4}{2}$ $+ 5\binom{5}{2} + 6\binom{6}{2}$ $= 9 + 24 + 50 + 90 = 173$ 对直线。剩下的每一对直线都在一个恰好有 $2$ 条直线相交的点相交，每对给出一个这样的点，所以共有 $780 - 173 = 607$ 个。

Since no two of the $40$ lines are parallel, every two lines cross, giving $\binom{40}{2} = 780$ pairs of lines, and each pair meets at exactly one point. A point where exactly $k$ lines meet accounts for exactly $\binom{k}{2}$ of these pairs.

The given points account for $3\binom{3}{2} + 4\binom{4}{2}$ $+ 5\binom{5}{2} + 6\binom{6}{2}$ $= 9 + 24 + 50 + 90 = 173$ pairs of lines. Each remaining pair meets at a point where exactly $2$ lines intersect, one point per pair, so there are $780 - 173 = 607$ such points.

2023 AIME I 第 3 题

答案：607

难度评级：2090

解答：

其他年份的第 3 题