HMMT 二月 2013 · 冲刺赛 · 第 17 题

HMMT February 2013 — Guts Round — Problem 17

[ 11 ] The lines y = x , y = 2 x , and y = 3 x are the three medians of a triangle with perimeter 1. Find the length of the longest side of the triangle.

解析

[ 11 ] The lines y = x , y = 2 x , and y = 3 x are the three medians of a triangle with perimeter 1. Find the length of the longest side of the triangle. √ 58 √ √ Answer: The three medians of a triangle contain its vertices, so the three vertices of 2+ 34+ 58 the triangle are ( a, a ), ( b, 2 b ) and ( c, 3 c ) for some a , b , and c . Then, the midpoint of ( a, a ) and ( b, 2 b ), a + b a +2 b which is ( , ), must lie along the line y = 3 x . Therefore, 2 2 a + 2 b a + b = 3 · , 2 2 a + 2 b = 3 a + 3 b, − 2 a = b. b + c 2 b +3 c Similarly, the midpoint of ( b, 2 b ) and ( c, 3 c ), which is ( , ), must lie along the line y = x . 2 2 Therefore, 2 b + 3 c b + c = , 2 2 2 b + 3 c = b + c, b = − 2 c, 1 c = − b = a. 2 From this, three points can be represented as ( a, a ), ( − 2 a, − 4 a ), and ( a, 3 a ). Using the distance √ √ formula, the three side lengths of the triangle are 2 | a | , 34 | a | , and 58 | a | . Since the perimeter of the √ 1 58 √ √ √ √ triangle is 1, we find that | a | = and therefore the longest side length is . 2+ 34+ 58 2+ 34+ 58