jmc

Let the two mathematicians be $M_1$ and $M_2$. Consider plotting the times that they are on break on a coordinate plane with one axis being the time $M_1$ arrives and the second axis being the time $M_2$ arrives (in minutes past 9 a.m.). The two mathematicians meet each other when $|M_1-M_2|\le m$. Also because the mathematicians arrive between 9 and 10, $0\le M_1,M_2\le 60$. Therefore, $60\times 60$ square represents the possible arrival times of the mathematicians, while the shaded region represents the arrival times where they meet.It's easier to compute the area of the unshaded region over the area of the total region, which is the probability that the mathematicians do not meet: $\frac{(60-m{)}^2}{60^2}=.6$ $(60-m{)}^2=36\cdot 60$ $60-m=12\sqrt{15}$ $\Rightarrow m=60-12\sqrt{15}$ So the answer is $60+12+15=\boxed{87}$.