HMMT February

Geometry Difficulty 4.9 AIME Find the answer

Is it possible for the projection of the set of points $(x, y, z)$ with $0 \leq x, y, z \leq 1$ onto some two-dimensional plane to be a simple convex pentagon?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

It is not possible. Consider $P$ , the projection of $\left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right)$ onto the plane. Since for any point (x, y, z) $in the cube,$ (1-x, 1-y, 1-z) is also in the cube, and the midpoint of their projections will be the projection of their midpoint, which is P, the projection of the cube onto this plane will be a centrally symmetric region around P$, and thus cannot be a pentagon.

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