Question 20
Triangle $ABC$ is an isosceles right triangle with $AB = AC = 3.$ Let $M$ be the midpoint of hypotenuse $\overline{BC}.$ Points $I$ and $E$ lie on sides $\overline{AC}$ and $\overline{AB},$ respectively, so that $AI \gt AE$ and $AIME$ is a cyclic quadrilateral. Given that triangle $EMI$ has area $2,$ the length $CI$ can be written as $\tfrac{a - \sqrt{b}}{c},$ where $a,$ $b,$ and $c$ are positive integers and $b$ is not divisible by the square of any prime. What is the value of $a + b + c?$