HMMT 二月 2011 · ALGCALC 赛 · 第 4 题
HMMT February 2011 — ALGCALC Round — Problem 4
题目详情
英文原题
- Let ABC be a triangle such that AB = 7, and let the angle bisector of ∠ BAC intersect line BC at
D . If there exist points E and F on sides AC and BC , respectively, such that lines AD and EFare parallel and divide triangle ABC into three parts of equal area, determine the number of possibleintegral values for BC .
解析
英文解析
- Let ABC be a triangle such that AB = 7, and let the angle bisector of ∠ BAC intersect line BC at
D . If there exist points E and F on sides AC and BC , respectively, such that lines AD and EFare parallel and divide triangle ABC into three parts of equal area, determine the number of possibleintegral values for BC .
Answer: 13
14C
7DEF
A B
Algebra & Calculus Individual Test
Note that such E, F exist if and only if
[ ADC ] = 2 . (1)
[ ADB ]
([ ] denotes area.) Since AD is the angle bisector, and the ratio of areas of triangles with equal heightis the ratio of their bases,
AC DC [ ADC ] = = .
AB DB [ ADB ]
Hence (1) is equivalent to AC = 2 AB = 14. Then BC can be any length d such that the triangleinequalities are satisfied:
d + 7 > 14
7 + 14 > d
Hence 7 < d < 21 and there are 13 possible integral values for BC .