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HMMT 十一月 2012 · 冲刺赛 · 第 21 题

HMMT November 2012 — Guts Round — Problem 21

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

  1. [ 11 ] ABC is a triangle with AB = 7, BC = 10, and CA = 13. Point D lies on segment BC such that
    DC = 2 BD . Point E lies on segment AD such that AE = 4 ED . Point F lies on segment AC suchthat F C = 5 AF . Find the area of △ EF C .
    HMMT NOVEMBER 2012, 10 NOVEMBER 2012 — GUTS ROUND

英文原题

[ 11 ] ABC is a triangle with AB = 7, BC = 10, and CA = 13. Point D lies on segment BC such that
DC = 2 BD . Point E lies on segment AD such that AE = 4 ED . Point F lies on segment AC such
that F C = 5 AF . Find the area of △ EF C .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
HMMT NOVEMBER 2012, 10 NOVEMBER 2012 — GUTS ROUND

解析

英文解析

  1. [ 11 ]
    √
    80 3
    Answer: By shared bases, we know that
    ( ) ( ) ( ) ( ) ( ) ( )9
    5 5 4 5 4 2
    [ EF C ] = [ AEC ] = [ ADC ] = [ ABC ] .
    6 6 5 6 5 3
    √
    √ √
    80 3
    By Heron’s formula, we find that [ ABC ] = (15)(8)(2)(5) = 20 3, so [ AEC ] =
    Guts Round 9