In rectangle $ABCD$, shown below, point $M$ is the midpoint of side $BC$, and point $N$ lies on $CD$ such that $DN:NC$ = $1:4$. Segment $BN$ intersects $AM$ and $AC$ at points $R$ and $S$, respectively. If $NS:SR:RB = x:y:z$, where $x$, $y$, and $z$ are positive integers, what is the minimum possible value of $x + y + z$?
My thought: $NS/SB = 4/5$, then $SB=SR+RB$, assuming $NS=4$, $SR=2$, $RB=3$ would satisfy the ratio, this can't be right. and I can't see a way to get relationship between $SR$ and $RB$; point mass geometry maybe?
Drop from $S$ a perpendicular $SH$ to $DC$: the similitude of triangle $NSH$ with $NBC$, and of triangle $SCH$ with $ADC$ gives: $$ {SN\over NB}={SH\over AD}={CH\over 5DN}={4DN-CH\over4DN}, $$ whence: $$ {CH\over DN}={20\over9} \quad\hbox{and}\quad {SN\over NB}={4\over9}. $$ In a similar way drop from $R$ a perpendicular $RK$ to $BC$. From the similitude of $BRK$ with $BNC$ and of $HRK$ with $HAB$ we get: $$ {RB\over NB}={RK\over NC}={5RK\over4DC}={BK\over2BM}, \quad {RK\over DC}={BM-BK\over BM} $$ whence: $$ {BK\over2BM}={RB\over NB}={5\over14} \quad\hbox{and}\quad {SR\over NB}=1-{SN\over NB}-{RB\over NB}={25\over126}. $$ It follows that $NS:SR:RB=56:25:45$.