jmc

From the Pythagorean Theorem, $A{H}^2+C{H}^2=1994^2$, and $(1995-AH{)}^2+C{H}^2=1993^2$. Subtracting those two equations yields $A{H}^2-(1995-AH{)}^2=3987$. After simplification, we see that $2\ast 1995AH-1995^2=3987$, or $AH=\frac{1995}{2}+\frac{3987}{2\ast 1995}$. Note that $AH+BH=1995$. Therefore we have that $BH=\frac{1995}{2}-\frac{3987}{2\ast 1995}$. Therefore $AH-BH=\frac{3987}{1995}$. Now note that $RS=|HR-HS|$, $RH=\frac{AH+CH-AC}{2}$, and $HS=\frac{CH+BH-BC}{2}$. Therefore we have $RS=\left|\frac{AH+CH-AC-CH-BH+BC}{2}\right|=\frac{|AH-BH-1994+1993|}{2}$. Plugging in $AH-BH$ and simplifying, we have $RS=\frac{1992}{1995\ast 2}=\frac{332}{665}\to 332+665=\boxed{997}$.