Instructions
Put a Pokemon at the head of your party that has a lot of health points and moves that can easily incapacitate other Pokemon. Something like sleep powder, stun spore or hypnosis is ideal. Don't put a Pokemon that is weak against bugs in the lead as they are all you will be facing in the contest.
Travel to the entrance of the National Park on the side of Route 35 and speak to the man in the red suit. This should be done on one of the days when the contest is open: Tuesday, Thursday or Saturday.
Enter the park after you receive your 20 sports balls from the man in red. You have 20 balls and 20 minutes to catch the best bug Pokemon in order to win.
Quickly walk into any tall grass and move around to attract the Pokemon. Use one of your moves that does no damage but can stun or paralyze your opponents first. Do not attack the bugs directly, as they are worth a lot more points depending on how much health they have when you capture them. For maximum points, you should do no damage to them.
Throw your sports balls at them as soon as they are stunned, asleep or paralyzed. It may take a few tries, but eventually you will catch your bug. If you have sports balls left, keep walking around until you encounter another bug. If it is better than the one you already have, catch it and replace your previous bug. You can keep one bug Pokemon for entry in the contest, so make sure it's the one with the highest level and most health.
Walk to the exit once you have found a bug you are confident will win. The contest will automatically end if your time is over or you run out of balls. Prizes will then be handed out and if you have a rare Pokemon with a level higher than 15 and full health, the chances are good of you winning.