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AIME 2023 I · 第 3 题

AIME 2023 I — Problem 3

专题
Contest Math
难度
L4
来源
AIME

题目详情

Problem

A plane contains 4040 lines, no 22 of which are parallel. Suppose that there are 33 points where exactly 33 lines intersect, 44 points where exactly 44 lines intersect, 55 points where exactly 55 lines intersect, 66 points where exactly 66 lines intersect, and no points where more than 66 lines intersect. Find the number of points where exactly 22 lines intersect.

解析

Solution

In this solution, let n\boldsymbol{n}-line points be the points where exactly nn lines intersect. We wish to find the number of 22-line points.

There are (402)=780\binom{40}{2}=780 pairs of lines. Among them:

  • The 33-line points account for 3⋅(32)=93\cdot\binom32=9 pairs of lines.

  • The 44-line points account for 4⋅(42)=244\cdot\binom42=24 pairs of lines.

  • The 55-line points account for 5⋅(52)=505\cdot\binom52=50 pairs of lines.

  • The 66-line points account for 6⋅(62)=906\cdot\binom62=90 pairs of lines.

It follows that the 22-line points account for 780−9−24−50−90=607780-9-24-50-90=\boxed{607} pairs of lines, where each pair intersect at a single point.

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

~MRENTHUSIASM

Video Solution by TheBeautyofMath

https://youtu.be/3fC11X0LwV8

~IceMatrix