The easy answer is to purchase the ponytail with the curl you desire.
But, if you're looking to recreate the look - and I'm taking a guess at this - depending on the curl pattern & texture, I honestly think thin, tight braids wrapped around tight spiral curl rods could achieve the look and hold up through a competition. (I might be wrong, but I feel like it's possible.)
The braids would definitely need to go in wet with a setting lotion and a curling mousse. You may need to let the hair to air dry for 24hrs, and take the braids down with the sharp end of a rat-tail comb. (No blow drying. No finger take down. Blow drying and finger take down may cause unwanted frizz.)
Add more curling mousse throughout the pony tail, and finish with ample hair spray and a bit of oil sheen spray with humidity block so it doesn't look crunchy and resists some sweating.