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*Edit* Hand positioning doesn't really matter as being wider than shoulder width usually becomes more difficult because there is less chest involvement. I have never seen on anyone campus or at LDAC getting corrected for hand positioning. I use the wide arm approach to get more use out of back and less on my triceps. According to the regs your body needs to be in a generally straight line and your arms should lock out in the up position and 90 degrees on the down approach. Bad push ups either involve the person bowing their back, doing the "worm", not coming all the way up, not going low enough or sagging.
You want to be good at pushups? Lift. I never do pushups and I max them every time. You need to shock your muscle fibers so a good blend of incline/reg push up super sets and weight lifting will easily allow you to max the test even in a LDAC environment. I lost 7 at LDAC because I stopped lifting a month prior to drop weight for the run.
doing the worm...
Actually the reason why they sag is that they don't do enough core exercises. Situps aren't even half of the equation and aren't the thing that keep you from sagging. Exercises where your upper torso is suspended in mid-air (horizontal off of bench or inverted while hanging) will do more for your core than any situp.