Age of Worms has a contest like this in the opening chapter. The 1st-level players can compete against a level 5 rogue. The contest is in 3 rounds, where each contestant has to hit a target with a dagger. Each round they have to hit a smaller ring on the target to advance. The AC of each ring is 10, 15 and 20. In 3.5e I am sure the to hit bonuses for the level 5 rogue make hitting that center ring fairly easy. The level 1 PCs generally have a +5 to hit. This was in April, so we were still using play test rules. The rogue would now be a CR5 rogue probably. She would have a +7 to attack. That is almost the same. I handled it in that case by giving her advantage because she is "supposed to be better." Now I probably would play it more like a dart game where the outer ring to inner rings were 1 to 3 points and let each contestant throw 5 daggers. That would average things out while still giving the PCs a reasonable chance of winning some gold.
For an archery competition, I would suggest having each contestant fire 10 shots (or 5 or whatever). Have the player roll 10d20 and count up the hits, or do what I would do with points for different rings.
Each type of competition is probably best handled differently. Using multiple shots or contested rolls is probably best when possible. For archery, that is just a lot of shots. For boxing, you could just use hp. For sumo, it is a series of contested Athletics and Acrobatics checks tracking the contestants as they go back and forth across the mat.
For an archery competition, I would suggest having each contestant fire 10 shots (or 5 or whatever). Have the player roll 10d20 and count up the hits, or do what I would do with points for different rings.
Each type of competition is probably best handled differently. Using multiple shots or contested rolls is probably best when possible. For archery, that is just a lot of shots. For boxing, you could just use hp. For sumo, it is a series of contested Athletics and Acrobatics checks tracking the contestants as they go back and forth across the mat.