There are 100 runners, each given a distinct bib labeled 1 to 100. What is the most number of runners that we could arrange in a circle, such that the product of the numbers on the bibs of any 2 neighboring runners, is less than 1000?

Since 31^2=961<1000, and 32^2=1024>1000 we can have a maximum of 31 pairs of numbers whose product would be less than 1000.
So the maximum number of runners would be 62.