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HMMT 十一月 2012 · GEN 赛 · 第 9 题

HMMT November 2012 — GEN Round — Problem 9

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

[7] Let nbe the 200th smallest positive real solution to the equation x−π
2= tan x. Find the greatest
integer that does not exceedn

解析

英文解析

  1. [ 7 ] Consider triangle ABC where BC = 7, CA = 8, and AB = 9. D and E are the midpoints of BC
    ′ ′
    and CA , respectively, and AD and BE meet at G . The reflection of G across D is G , and G E meets
    CG at P . Find the length P G .
    √
    145 1
    ′ ′
    Answer: Observe that since G is a reflection and GD = AG , we have AG = GG and
    9 2
    ′ 1
    therefore, P is the centroid of triangle ACG . Thus, extending CG to hit AB at F , P G = CG =
    √3
    √
    2 2 2
    2(8 +7 ) − 9
    2 2 145
    CF = = by the formula for the length of a median.
    9 9 4 9