Question 5
Points $A_1,A_2,\ldots,A_{100}$ are on a line, in this order, such that $A_1A_2=\frac1{1\times2}$ cm, $A_2A_3=\frac1{2\times3}$ cm, $A_3A_4=\frac1{3\times4}$ cm, ..., $A_{99}A_{100}=\frac1{99\times100}$ cm. If the segment $A_mA_n$, where $1\leq m<n\leq100$, has a length of $\frac1{15}$ cm, then what is the largest possible value of $m+n$ that satisfies the given conditions?