11. On a circular track, 2015 flags are evenly spaced. A and B start from a flag at the same time and in the same direction. When A and B return to the starting point at the same time, A has run 23 laps, and B has run 13 laps. Excluding the starting flag position, how many times does A catch up to B exactly at a flag position?
【Answer】 5 times
【Explanation】
Let the distance between each pair of flags be 1, i.e., the circumference of the track is 2015. Since $v_{甲}: v_{乙}=23: 13$, let $v_{\text {甲 }}=23 x, v_{乙}=13 x$. For 甲 to catch up with 乙, 甲 needs to run $n$ more laps than 乙, $(23 x-13 x) t=2015 n, 10 x \times t=2015 n$, which means the time 甲 takes to catch up with 乙 is $t=\frac{403 n}{2 x}$, and the distance 甲 runs when catching up with 乙 is $23 x \times \frac{403 n}{2 x}=\frac{23 \times 403}{2} n$; to catch up exactly at the flag position, the distance run must be an integer, so $n$ must be even, thus $n=2,4,6,8,10$ (a maximum of 10 more laps); in summary, 甲 catches up with 乙 exactly at the flag position 5 times.