HMMT November

Geometry Difficulty 4.7 AIME Find the answer

Let $ABCD$ be a square of side length 2. Let points $X, Y$ , and $Z$ be constructed inside $ABCD$ such that $ABX, BCY$ , and $CDZ$ are equilateral triangles. Let point $W$ be outside $ABCD$ such that triangle $DAW$ is equilateral. Let the area of quadrilateral $WXYZ$ be $a+\sqrt{b}$ , where $a$ and $b$ are integers. Find $a+b$ .

Your answer

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

Solution

$WXYZ$ is a kite with diagonals $XZ$ and $WY$ , which have lengths $2 \sqrt{3}-2$ and 2, so the area is $2 \sqrt{3}-2=\sqrt{12}-2$ .

Want a route through all this instead of an archive? The track puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.

Source: Omni-MATH, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.