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.

Sorry man, can u reduce ur answer by 1. Because it is a circle, not a sequence.

To find out the maximum number of runners that can be arranged in a circle such that the product of the numbers on their bibs is less than 1000, we need to consider the constraints on the product.

The maximum value of a single bib number is 100 since there are 100 runners with distinct bibs labeled from 1 to 100.

We want to find the maximum number of runners that satisfy the condition where the product of any two neighboring bib numbers is less than 1000.

Let's analyze the possible values for the neighboring bib numbers.

The maximum possible value of a single bib number is 100. In order for the product of two neighboring bib numbers to be less than 1000, we need to find the maximum value of the second bib number with respect to the first bib number.

If the first bib number is 1, then the maximum value for the second bib number is 999. If the first bib number is 2, then the maximum value for the second bib number is 499, and so on. We can find this pattern by dividing 1000 by the first bib number and subtracting 1.

Now, we can start from the maximum bib number (100) and work our way down to find the first bib number. For each bib number, we find the maximum value for its neighboring bib number using the pattern mentioned above. If the neighboring bib number is still within the range of 1 to 100, we keep decreasing the first bib number. Otherwise, we stop.

Let's calculate this step-by-step:

- For bib number 100, the maximum possible neighboring bib number is 999/100 - 1 = 8.99, which is not within the range 1 to 100.

- For bib number 99, the maximum possible neighboring bib number is 999/99 - 1 = 9.09, which is not within the range 1 to 100.

- For bib number 98, the maximum possible neighboring bib number is 999/98 - 1 = 10.18, which is not within the range 1 to 100.

- Continuing this process, we find that for bib number 11, the maximum possible neighboring bib number is 999/11 - 1 = 87. However, the neighboring bib number 87 is within the range of 1 to 100.

Therefore, the maximum number of runners that can be arranged in a circle such that the product of the numbers on their bibs is less than 1000 is 87.