PUMaC 2013 · 几何（A 组） · 第 4 题

PUMaC 2013 — Geometry (Division A) — Problem 4

[ 4 ] Draw an equilateral triangle with center O . Rotate the equilateral triangle 30 , 60 , 90 with respect to O so there would be four congruent equilateral triangles on each other. Look at the diagram. If the smallest triangle has area 1, the area of the original equilateral triangle √ could be expressed as p + q r where p, q, r are positive integers and r is not divisible by a square greater than 1. Find p + q + r .

解析

[ 4 ] Draw an equilateral triangle with center O . Rotate the equilateral triangle 30 , 60 , 90 with respect to O so there would be four congruent equilateral triangles on each other. Look at the diagram. If the smallest triangle has area 1, the area of the original equilateral triangle p could be expressed as p + q r where p, q, r are positive integers and r is not divisible by a square greater than 1. Find p + q + r . p 1 Solution of triangle is similar to the smallest triangle with a ratio of 3 + 2 3. Answer is 3 p 63 + 36 3 hence 102 .