PUMaC 2013 · 几何（B 组） · 第 3 题

PUMaC 2013 — Geometry (Division B) — Problem 3

[ 4 ] Consider all planes through the center of a 2 × 2 × 2 cube that create cross sections that are regular polygons. The sum of the cross sections for each of these planes can be written in √ the form a b + c , where b is a square-free positive integer. Find abc .

解析

[ 4 ] Consider all planes through the center of a 2 ⇥ 2 ⇥ 2 cube that create cross sections that are regular polygons. The sum of the cross sections for each of these planes can be written in p the form a b + c , where b is a square-free positive integer. Find abc . Solution There are two possible regular polygons: a square and a regular hexagon. Square: There are 3 planes that create squares of area 4. Hexagon: Hexagons are formed by connecting midpoints of the cube. There are 4 planes that p create hexagons of area 3 3. p The total area will be 12 3 + 12. Then, abc = 432 .