PUMaC 2012 · 几何（A 组） · 第 3 题

PUMaC 2012 — Geometry (Division A) — Problem 3

[ 4 ] Six ants are placed on the vertices of a regular hexagon with an area of 12. At each point in time, each ant looks at the next ant in the hexagon (in counterclockwise order), and measures the distance, s , to the next ant. Each ant then proceeds towards the next ant at a speed of s units per year. After T years, the ants’ new positions are the vertices of a new hexagon 100 with an area of 4. T is of the form a ln b , where b is square-free. Find a + b .

解析

[ 4 ] Six ants are placed on the vertices of a regular hexagon with an area of 12. At each point in time, each ant looks at the next ant in the hexagon (in counterclockwise order), and measures 1 the distance, s , to the next ant. Each ant then proceeds towards the next ant at a speed of s units per year. After T years, the ants’ new positions are the vertices of a new hexagon 100 with an area of 4. T is of the form a ln b , where b is square-free. Find a + b . Solution: At each moment in time, the velocity in the inward radial direction is s cos 60, which is s/ 2. The distance from the ant to the center is s . Thus this is a continuous compound 1 interest problem with the interest rate r = − . The side length at any point in time, as 200 √ 1 − t 200 a function of the original side length D , is De . Set this equal to D/ 3, and we have t = 100 ln 3. a + b = 100 + 3 = 103