HMMT 十一月 2021 · 冲刺赛 · 第 10 题

HMMT November 2021 — Guts Round — Problem 10

[8] Squares ABCD and DEF G have side lengths 1 and , respectively, where E is on CD and points 3 m A , D , G lie on a line in that order. Line CF meets line AG at X . The length AX can be written as , n where m, n are positive integers and gcd( m, n ) = 1. Find 100 m + n . n

解析

[8] Squares ABCD and DEF G have side lengths 1 and , respectively, where E is on CD and points 3 A , D , G lie on a line in that order. Line CF meets line AG at X . The length AX can be written as m , where m, n are positive integers and gcd( m, n ) = 1. Find 100 m + n . n Proposed by: David Vulakh Answer: 302 Solution: C F B E X G D A There are a variety of solutions involving similar triangles. One fast way to solve the problem without hunting for many geometric relationships is to notice that, if one continues to add squares inscribed 1 between AX and XC as shown in the diagram above, each square has side length equal to of the 3 1 1 1 3 length of the previous square. Then AX = 1 + + + + · · · = . Note that this construction can 3 9 27 2 be used to geometrically prove the formula for infinite geometric sums! n