HMMT 二月 2017 · 冲刺赛 · 第 10 题

英文解析

10. [ 7 ] Let ABC be a triangle in the plane with AB = 13 , BC = 14 , AC = 15 . Let M denote the smallestnn n n 1
    possible value of ( AP + BP + CP ) over all points P in the plane. Find lim M .nn →∞ n
    Proposed by: Yang Liu
    Answer: 65/8
    Let R denote the circumradius of triangle ABC. As ABC is an acute triangle, it isn’t hard to checkthat for any point P , we have either AP ≥ R, BP ≥ R , or CP ≥ R. Also, note that if we choosen n n n
    P = O (the circumcenter) then ( AP + BP + CP ) = 3 · R . Therefore, we have the inequality
    1 1 1
    n n n nn n n
    R ≤ min ( AP + BP + CP ) ≤ (3 R ) = R · 3 .
    P ∈ R2
    Taking n → ∞ yields
    R ≤ lim M ≤ Rnn →∞
    n 651
    (as lim 3 = 1), so the answer is R = .
    n →∞ 8