HMMT 二月 2007 · 冲刺赛 · 第 14 题

英文解析

14. [ 9 ] We are given some similar triangles. Their areas are 1 , 3 , 5 . . . , and 49 . If the smallest trianglehas 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 perimetersquared to area, or, equivalently, the same ratio of perimeter to the square root of area. Because thelatter ratio is 4 for the smallest triangle, it is 4 for all the triangles, and thus their perimeters are
    4 · 1 , 4 · 3 , 4 · 5 , . . . , 4 · 49, and the sum of these numbers is [4(1 + 3 + 5 + · · · + 49) = 4(25 ) = 2500 .2