Well archery competition is quite easy to pull. The target itself is small immobile object, therefore having AC of 6 (5+1). The next circle is tiny with AC 7 (5+2). And the inner circle diminutive AC 9 (5+4). The target is then placed at a distance of 300, 600 or 900 feet. Calculate the necessary range-increment penalties depending on the weapon. Hitting the target will earn a point, hitting the next circle will earn 2 points and hitting the inner circle 4 points. Calculate the points at every round and drop half of the contestants with lowest score. Move the target further away every few rounds. To make it harder you can use moving targets for +5 AC.
So with a basic longbow the effective AC of immobile target at 300 feet would be 12/13/15, at 600 18/19/21 and at 900 24/25/27.