B1 Alice and Bob run in the clockwise direction around a circular track, each running at a constant speed. Alice can complete a lap in $t$ seconds, and Bob can complete a lap in 60 seconds. They start at diametrically-opposite points.
When they meet for the first time, Alice has completed exactly 30 laps. Determine all possible values of $t$.

Solution: The answer is $t=59$ or $t=61$.

Since Alice ran exactly 30 laps, Bob meets Alice at where Alice started. Since Bob started diametrically across from Alice, Bob ran $n+\frac{1}{2}$ laps for some positive integer $n$. Since Alice and Bob meet only the first time they meet, the number of laps that Alice ran and the number of laps Bob ran cannot differ by more than 1. Therefore, Bob ran either 29.5 laps or 30.5 laps.

Note that Alice and Bob ran for the same amount of time and the number of seconds each person ran is the number of laps he/she ran times the number of seconds it takes he/she to complete a lap.

If Bob ran 29.5 laps, then $30 t=29.5 \times 60$. Hence, $t=29.5 \times 2=59$.

If Bob ran 30.5 laps, then similarly, $30 t=30.5 \times 60$. Hence, $t=30.5 \times 2=61$.

Therefore, $t=59$ or $t=61$.