HMMT 二月 2007 · 冲刺赛 · 第 14 题

HMMT February 2007 — Guts Round — Problem 14

[ 9 ] We are given some similar triangles. Their areas are 1 , 3 , 5 . . . , and 49 . If the smallest triangle has a perimeter of 4, what is the sum of all the triangles’ perimeters?

解析

[ 9 ] We are given some similar triangles. Their areas are 1 , 3 , 5 . . . , and 49 . If the smallest triangle has a perimeter of 4, what is the sum of all the triangles’ perimeters? Answer: 2500 . Because the triangles are all similar, they all have the same ratio of perimeter squared to area, or, equivalently, the same ratio of perimeter to the square root of area. Because the latter ratio is 4 for the smallest triangle, it is 4 for all the triangles, and thus their perimeters are 2 4 · 1 , 4 · 3 , 4 · 5 , . . . , 4 · 49, and the sum of these numbers is [4(1 + 3 + 5 + · · · + 49) = 4(25 ) = 2500 .