The end of February is twenty weeks away. With that much time until the competition, you should periodize your training into a general preparation, specific training, and competition lead-up. Doing just push-ups might work too, but you will achieve better results with a more balanced approach.
General preparation - 6 weeks
For the general preparation phase, you should consider your cardiovascular ability, your level of strength, and your joint mobility. If any of these are pathological, it would make sense to deal with that issue. If you're average in each of these, then the general approach I'd use is to do some cardio and strength training each week. This program, or any of the others on that page, would be fine. I'd make room at the beginning and end of each workout to do at least a few sets of push-ups.
So in this stage, in order to get yourself generally ready for physical competition, you would do something like this, three times a week:
- Warm up with a quick jog or stationary bike, plus a set or two of push-ups to near-failure
- Bench press, 3 sets of 12, heavy, with a spotter or dumbbells
- Some kind of squat (air, goblet, barbell, dumbbell, kettlebell), two sets of five
- Pull-ups, three sets to failure (with good form)
- Deadlifts, one set of five to ten, heavy
- Short cardio finisher (unless you want to do this, or a longer cardio session, on non-lifting days), such as two or three 100m sprints or 5 minutes of a barbell or kettlebell complex
- Two sets of push-ups to failure
I would do yoga, particularly Down Dog and Sun Salutations peppered with push-ups, every day or at least on off days. Be careful with jumping in too quickly to this training, however, lest you get injured. Take it easy the first week, a little less easy the second and third weeks, then push yourself hard in the fourth week and beyond.
Specific preparation - 6 weeks
After six weeks of getting yourself generally fit and strong, it's time to narrow your training towards push-ups while still developing qualities like whole-body strength and endurance. I would focus on short-to-medium cardio efforts (which broadly match the timing of 200 pushups) and upper-body strength (which is closely related to your push-up goal). Of course I would continue with the push-ups. So, six weeks of:
- Warm up with a quick jog or stationary bike
- Five sets of 25+ push-ups each, not to failure, alternated with front and side planks, push-up variations (wide, narrow, staggered), uneven push-ups using a ball or stack of books on one side, and so on
- Pull-ups, three sets of one-half your maximum (or if that's less than 3, then 5, using negatives), to balance the pressing work
- Bench press, 3 heavy sets of 10--make sure you use dumbbells or have a spotter
- Either squat or deadlift, light
- Supplemental pressing work: dips, three sets to failure
Of course, I would be trying to increase the weight on the bar each week for the bench press. On off days, I would go for a jog and then do at least two hundred total push-ups. The five sets should be totaling around or over two hundred.
Competition preparation - eight weeks
In the final lead-up to the test, I would keep doing one general workout per week up until four weeks out: bench press, dips, a little squatting/deadlifting/pull-upping. At least once a week, I'd test myself: two sets of push-ups to failure, of at least fifty each, shooting for 110 or 120. All my other workouts would be specific to the competition, and I'd have at least three of them. Something like one of the following:
- Four or five hundred total push-ups
- Pyramids with short rest times, starting at 10 and adding one each set until you get to 20
- A set of push-ups, then run a quarter mile (or do burpees, or otherwise make yourself exhausted), then another set of push-ups without stopping
- Working with diamond push-ups and wide push-ups
You should aim to be exceeding your goal by the end of January, with four weeks to spare. At that point I'd drop the general workouts entirely and just keep working on long sets of push-ups. The last week before the competition I'd barely work out at all, just rest.
Other approaches
The plan I outlined above is rough and should be taken as a template, not a specialized plan. It is meant to show the broad strokes of a plan: get strong and fit while doing push-ups, then get really good at pushing motions, then get awesome at push-ups. You could match those broad strokes with other approaches, too. For instance:
- Four times a week, run for a mile then do a circuit of push-ups, pull-ups, and air squats. Each month increase the number of push-ups you do, but keep everything else constant. Start doing extra sets of clap push-ups, one-arm push-ups, super-slow and super-fast push-ups. In the third month, run a shorter distance and do fewer pull-ups and squats. In the last month, just warm up and do push-ups exactly like you will for the competition.
...or:
- Spend a month training yoga daily at a studio, to get your shoulders and back limber. Do push-ups at home at least three times a week. Then spend two months with dumbbells, doing high-rep bench presses, rows, and squats, finishing each workout with four sets of push-ups to failure, using a different push-up variation in each one. In the last month, stop lifting and just do huge numbers of regular push-ups every weekday.
...or:
- Join a gym and do a whole-body circuit on the machines, five days a week. Finish each workout with five sets of push-ups and a short jog. After two months, cut the number of exercises in the circuit in half and double the number of push-ups. In the last month, just warm up and do every kind of push-up you can think of for ninety minutes.
The point is, you have plenty of time to get your entire body ready. You don't want to overtrain or get injured. So yes, absolutely do a lot of push-ups. But don't neglect other methods of getting strong, like dips and bench press, and don't forget to keep your shoulders, back, and chest healthy with pulling exercises and overhead work.
Perspective
Remember, this competition is dumb. In particular, the part about "run until you puke" tells me that either someone's going to drop out before December, or everyone involved is a little juvenile. Go ahead and win it, but do it by making yourself awesome instead of valuing foolish grandstanding.
Really?
A skeptic might justifiably ask, "why do anything other than push-ups to prepare for a push-up contest?" The justification comes from experiments in sport science, which have found that general physical preparation will provide a greater and more effective training stimulus than just the target activity itself. Without putting too much credence in the exercises/sets/reps of this specific proposal, it's a well understood athletic principle that just doing the specific thing that will be tested is rarely the optimal approach for non-elite athletes. When there's time, a periodized approach of general -> specific -> competition prep produces best results.
You'll see this in many sports. For instance, in running (except for extreme-distance runs like ultra-marathons), a little strength and speed work early in the season is better than just running. The same is found in football, powerlifting, judo, and so on. Squats and deadlifts (programmed with less intensity, priority, and volume than the multiple forms of pressing work) produce an enormous training effect in the novice. They develop core strength and the ability to exert maximum force that will stave off injury and--yes--improve one's ability to do large numbers of push-ups.
