HMMT 二月 2013 · 冲刺赛 · 第 17 题

英文解析

17. [ 11 ] The lines y = x , y = 2 x , and y = 3 x are the three medians of a triangle with perimeter 1. Findthe 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 bwhich 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,
    c = − b = a.1
    From this, three points can be represented as ( a, a ), ( − 2 a, − 4 a ), and ( a, 3 a ). Using the distance 2
    √ √
    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