How do you train to max pushups on an APFT?

Quite a bit of decent training advice. I'm going to suggest something very simple that worked for me, although I am short and stocky. Try bringing your elbows in and doing pushups Ranger Style. It made all the difference for me. YMMV, but it's a simple fix that simply isolates different muscles.

Eat meat, work out at the gym doing youre entire body, do pull ups, swimming

I was told by my drill to do 20 rest for a sec then 20 more rest and then 20 more rest and then in the final secs knock out what you can

Do push ups early in the morning and late at night.

It's your own body! Push it up!!

Unfortunately us tall lanky guys have long arms and it takes us longer to complete one push up than say someone with short arms so speed as well as conditioning is something you need to work on. I would suggest resistance training with your push ups but do them as fast and as efficiently as you can.

I would hit the weights in the evening for strength and do some diamond push ups to help strengthen your triceps, do your push-ups in sets you can only get stronger, it got to a point where I was hitting 75 for one minute and 114 for two minutes

Way back in the eighties...I'm a tall lanky too, it's the travel time to break the plane that gets us. Try using a wide arm position, it shortens the distance significantly.

What form do you use? 90° or tricep-ups? Try doing push-ups solely relying on your triceps. They are a more solid muscle and don't pain very easily. I easily bumped my push-ups at least by 20 by pulling my elbows in to my sides and using my triceps to lift my body weight. Give it a shot and good luck.

I use the Tabata method. I spent my deployments doing a lot of heavy weightlifting and found that I could benchpress well over my bodyweight; however I found that during the APFT, I would get extremely tight.

I've attached an article which explains a similar predicament. The article is written by a former Physical Education instructor at West Point. The author found a way to train for a bodybuilding competition and train for the APFT simultaneously.

Tabata is 8 sets of an exercise. I use it for sit-ups and push-ups. You do 20 seconds of work and 10 seconds of rest for 8 sets. I do this twice a week and consistently max push-ups and sit-ups.

Here is the link to the article: http://startingstrength.com/articles/army_weak_long.pdf

It's a good read. I have handed this out to countless Soldiers and have had success stories from those who applied it.

Best of luck.