PUMaC 2017 · 几何（B 组） · 第 5 题

PUMaC 2017 — Geometry (Division B) — Problem 5

A right regular hexagonal prism has bases ABCDEF , A B C D E F and edges AA , BB , ′ ′ ′ ′ CC , DD , EE , F F , each of which is perpendicular to both hexagons. The height of the ′ prism is 5 and the side length of the hexagons is 6. The plane P passes through points A , C , √ and E . The area of the portion of P contained in the prism can be expressed as m n , where n is not divisible by the square of any prime. Find m + n .

解析

The portion of the plane inside the prism is a pentagon, which can be seen as a rectangle and a triangle. The base length of the triangle and rectangle can be calculated by drawing triangle √ AF E , which has base 6 3. The height at which the rectangular portion of the pentagon stops 2 is the height of the prism. Hence, the length of the rectangle, by the Pythagorean theorem, 3 √ 2 106 is . The height of the triangle can also be found by the Pythagorean theorem, through 3 √ 1 106 legs at altitude of BCD and of the height of the prism. This gives a height of . Hence 3 3 ( ( ) ) √ √ √ √ 2 106 106 the area of the pentagon is 6 3 · + / 2 , or 5 318 for a final answer of 323 . 3 3 Problem written by Nathan Bergman ′ ′