HMMT 十一月 2013 · THM 赛 · 第 3 题

HMMT November 2013 — THM Round — Problem 3

[ 5 ] Let ABC be a triangle with AB = 5, BC = 4, and CA = 3. Initially, there is an ant at each vertex. The ants start walking at a rate of 1 unit per second, in the direction A → B → C → A (so − − → the ant starting at A moves along ray AB , etc.). For a positive real number t less than 3, let A ( t ) be the area of the triangle whose vertices are the positions of the ants after t seconds have elapsed. For what positive real number t less than 3 is A ( t ) minimized?

解析

[ 5 ] Let ABC be a triangle with AB = 5, BC = 4, and CA = 3. Initially, there is an ant at each vertex. The ants start walking at a rate of 1 unit per second, in the direction A → B → C → A (so − − → the ant starting at A moves along ray AB , etc.). For a positive real number t less than 3, let A ( t ) be the area of the triangle whose vertices are the positions of the ants after t seconds have elapsed. For what positive real number t less than 3 is A ( t ) minimized? 47 1 Answer: We instead maximize the area of the remaining triangles. This area (using xy sin θ ) 24 2 1 3 1 4 1 1 47 2 is ( t )(5 − t ) + ( t )(3 − t ) + ( t )(4 − t )1 = ( − 12 t + 47 t ), which has a maximum at t = ∈ (0 , 3). 2 5 2 5 2 10 24