Question 13
Exercise 4. Trapezoid $ABCD$ has $AB\parallel CD$, $BC=CD=43$, and $AD\perp BD$.
Let $O$ be the intersection of the diagonals $AC$ and $BD$, and let $P$ be the midpoint of $BD$. Given that $OP=11$, the length of $AD$ can be written in the form $m\sqrt n$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. What is $m+n$?
**Editorial input convention:** Use the usual convex trapezoid with vertices A,B,C,D in cyclic order. Enter the requested integer m+n.
Let $O$ be the intersection of the diagonals $AC$ and $BD$, and let $P$ be the midpoint of $BD$. Given that $OP=11$, the length of $AD$ can be written in the form $m\sqrt n$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. What is $m+n$?
**Editorial input convention:** Use the usual convex trapezoid with vertices A,B,C,D in cyclic order. Enter the requested integer m+n.