Question 145
In square $ABCD$, $P$ lies on the ray $AD$ past $D$ and lines $PC$ and $AB$ meet at $Q$. Point $X$ is the foot of the perpendicular from $B$ to $DQ$, and the circumcircle of triangle $APX$ meets line $AB$ again at $Y$. Suppose that $DP = \frac{16}{3}$ and $BQ = 27$. The length of $BY$ can be written in the form $p/q$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.