AMC 10 Weekly Practice Round 2
Complete problem set with solutions and individual problem pages
On a circular track, flags are placed at equal intervals. Person and person start at the same time from the same flag, running in the same direction. When they both return to the starting point together again, has run laps and has run laps. Excluding the starting flag position, how many times does overtake exactly at a flag position?
- A.
- B.
- C.
- D.
- E.
Let the distance between two adjacent flags be , so the circumference of the track is .
Since , let and .
For to overtake , must run more laps than .
Then:
Thus, the time taken for to overtake is:
In this time, covers a distance of:
For to overtake exactly at a flag position, the distance covered must be an integer, which requires to be even.
Thus: (the maximum is extra laps).
However, when , both runners return to the starting point.
Therefore, overtakes exactly at a flag position times.
