Question 15
Let $\overline{MN}$ be a diameter of a circle with diameter $1.$ Let $A$ and $B$ be points on one of the semicircular arcs determined by $\overline{MN}$ such that $A$ is the midpoint of the semicircle and $MB = \frac{3}{5}.$ Point $C$ lies on the other semicircular arc. Let $d$ be the length of the line segment whose endpoints are the intersections of diameter $\overline{MN}$ with the chords $\overline{AC}$ and $\overline{BC}.$ The largest possible value of $d$ can be written in the form $r - s\sqrt{t},$ where $r,$ $s,$ and $t$ are positive integers and $t$ is not divisible by the square of any prime. Find $r + s + t.$