HMMT February, 2016

Geometry Difficulty 4.6 AIME Prove it United States

Problem:

Dodecagon $Q W A R T Z S P H I N X$ has all side lengths equal to $2$ , is not self-intersecting (in particular, the twelve vertices are all distinct), and moreover each interior angle is either $90^{\circ}$ or $270^{\circ}$ . What are all possible values of the area of $\triangle S I X$ ?

Solution

Solution:

The dodecagon has to be a "plus shape" of area $20$ , then just try the three non-congruent possibilities.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.