HMMT-Nov 2022 guts, Problem 4

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

Solution: Solution: $W X Y Z$ is a kite with diagonals $X Z$ and $W Y$, which have lengths $2 \sqrt{3}-2$ and 2 , so the area is $2 \sqrt{3}-2=\sqrt{12}-2$. $\fbox{10}$.

Answer: 10