This problem – #15 from the 2006 AMC 10A – is a hard problem:
Problem 15 from the 2006 AMC 10A
“Odell and Kershaw run for 30 minutes on a circular track. Odell runs clockwise at 250 m/min and uses the inner lane with a radius of 50 meters. Kershaw runs counterclockwise at 300 m/min and uses the outer lane with a radius of 60 meters, starting on the same radial line as Odell. How many times after the start do they pass each other?”
I mean **hard**.
My number one struggle in teaching my kids is identifying ahead of time what problems are going to be relatively easy and which ones will be hard. I constantly miss on both sides, and I probably guessed incorrectly on this one by as much as I ever have.
But, spending 20 minutes talking through it helped me get a better feel for why this problem gave him trouble, and also a better feel for the how step by step process you need to go through to solve it. When I read the problem, that step by step process was essentially invisible to me, unfortunately.
Mike's Math Page 
Do you think a square or triangular track shape would have made it easier for him to see the steps?
The \pi certainly was one of the steps to climb up on this problem, but only one.
I showed him how to approximate \pi as 22/7 to make approximate calculations a little easier. That was actually a fun part of the morning and something that I’d not give a lot of thought to previously.
I found this the other day. From Maclaurin’s Treatise on Algebra:
This is 300 years ago!