HMMT 二月 2004 · 冲刺赛 · 第 29 题

英文解析

29. A regular dodecahedron is projected orthogonally onto a plane, and its image is ann -sided polygon. What is the smallest possible value of n ?
    Solution: 6
    We can achieve 6 by projecting onto a plane perpendicular to an edge of the dodecaheron. Indeed, if we imagine viewing the dodecahedron in such a direction, then 4 ofthe faces are projected to line segments (namely, the two faces adjacent to the edgeand the two opposite faces), and of the remaining 8 faces, 4 appear on the front of thedodecahedron and the other 4 are on the back. Thus, the dodecahedron appears asshown.
    To see that we cannot do better, note that, by central symmetry, the number of edgesof the projection must be even. So we just need to show that the answer cannot be 4.
    But if the projection had 4 sides, one of the vertices would give a projection formingan acute angle, which is not possible. So 6 is the answer.