HMMT 二月 2011 · ALGGEO 赛 · 第 2 题

HMMT February 2011 — ALGGEO Round — Problem 2

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 EF are parallel and divide triangle ABC into three parts of equal area, determine the number of possible integral values for BC .

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 EF are parallel and divide triangle ABC into three parts of equal area, determine the number of possible integral values for BC . Answer: 13 C 14 F E D 7 A B 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 height is 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 triangle inequalities are satisfied: d + 7 > 14 7 + 14 > d Hence 7 < d < 21 and there are 13 possible integral values for BC . Algebra & Geometry Individual Test