HMMT 十一月 2022 · 冲刺赛 · 第 18 题

HMMT November 2022 — Guts Round — Problem 18

[10] A regular tetrahedron has a square shadow of area 16 when projected onto a flat surface (light is shone perpendicular onto the plane). Compute the sidelength of the regular tetrahedron. (For example, the shadow of a sphere with radius 1 onto a flat surface is a disk of radius 1.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT November 2022, November 12, 2022 — GUTS ROUND Organization Team Team ID#

解析

[10] A regular tetrahedron has a square shadow of area 16 when projected onto a flat surface (light is shone perpendicular onto the plane). Compute the sidelength of the regular tetrahedron. (For example, the shadow of a sphere with radius 1 onto a flat surface is a disk of radius 1.) Proposed by: Albert Wang √ Answer: 4 2 Solution: Imagine the shadow of the skeleton of the tetrahedron (i.e. make the entire tetrahedron translucent except for the edges). The diagonals of the square shadow must correspond to a pair of opposite edges of the tetrahedron. Both of these edges must be parallel to the plane – if they weren’t, then edges corresponding to the four sides of the square would have to have different lengths. Thus, √ the length of a diagonal of the square (namely, 4 2) must be the same as the edge length of the tetrahedron.