I did something like this in my last campaign. The ranger had to hit a fuse with a flaming arrow to start an explosion that would destroy a drow city. I just back-figured it. I wanted the chance to succeed very low, so that the players would have to work together to grant him enough bonuses to make the chance a little better than 50%.
To make your contest more exciting, start at extreme range (our ranger had the Far Shot feat) to get the range penalties. Maybe toward the end of the competition, when its just the hero and his rival, it starts to rain and a wind whips up (-4 for severe wind and/or rain). Maybe its late in the day and the light begins to fade or the smoke from the nearby hog roast drifts pass the archery field (-2 concealment). In our case, the target was so far away, I made the ranger make a Spot check just to see it.
You can throw in any number of circumstance penalties to even the score. Maybe the archer's rival sabotaged his bow somehow, or the bow string is too wet or too dry. Maybe the crowd is heckling the archer and he has to make a Concentration check or take a penalty to the shot. This should go the other way too. Encourage the player to do all sorts of preparation to increase his chances (like a Craft check to tweak his bow or arrows for the conditions, or a Concentration check to "get into the zone").