Question 92
Three speed skaters have a friendly “race” on a skating oval. They all start from the same point and skate in the same direction, but with different speeds that they maintain throughout the race. The slowest skater does $1$ lap a minute, the fastest one does $3.14$ laps a minute, and the middle one does $L$ laps a minute for some $1 \lt L \lt 3.14$. The race ends at the moment when all three skaters again come together to the same point on the oval (which may differ from the starting point.) Find how many different choices for $L$ are there such that exactly $117$ passings occur before the end of the race. (A passing is defined when one skater passes another one. The beginning and the end of the race when all three skaters are together are not counted as passings.)