IMPORTANT! 3 mile run on PRMC is
not a 3 MILE best effort. It's 1.5 miles run as a
squad in
12.5 mins, then 1.5 mile best effort back the same way, to be done in under
10.5 mins. This changed back in May/April so if you didn't know this shoot your AFCO.
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Jarhead - Sprints are important, they make up the final evolution of Gym Test 2 and are also used during Gym Test 1 and bottom field determination tests on PRMC.
Also, sprints are an excellent way of improving anaerobic fitness. Doing this gives you a massive advantage during Bottom Field, Gym test 2 and the bleep test stage of Gym test 1.
Finally downhill sprints / running helps to develop good form, sprint power and overall speed. Not to mention that focusing on uphill sprints only, can lead to a muscle imbalance in the legs which may affect performance.
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marcus2007 - You should do your cardio work after your resistance training. Cardio can burn the body's glycogen stores and then moves onto fat. Resistance training uses glycogen, but it can't utilise fat for energy. Doing weights / resistance first means that your body will use the glycogen to its fullest potential, and then during your cardio you'll have no choice but to burn fat for energy.
Also during your weights sessions do your heaviest/hardest exercises first. If you exhaust your muscles in the order you've given, your increasing your chance for injury by leaving the heaviest squats to last. Squats are the best total-body exercise so you should do these first to make sure you get as much gain from them as possible.
You should also move your pressups, pullups and situps onto your weights day, making a total-body work out done 3 times a week. Your current schedule does not leave much time for your muscles to recover. If you're muscles don't have sufficiant recovery time, they will not be able to adapt and grow.
Other than that, your weight routine looks pretty simple, and it's mainly compound exercises (which is a good thing). If you have access to the equipment, I'd drop the shouder raises and include dips and bench presses.
Of those 3 weight sessions make two a 'circuit' style and leave the other as the normal '3-sets-of-each-exercise-style-with-rest-in-between-style'.

The two circuit sessions should be ordered: upper body/lower body/upper body etc.. with max 30s rest in between exercises and a 4 min rest between circuits. Keep all gym sessions to a maximum of one hour.
You should also be looking to replace one of your 3 mile runs with a longer run (6 miles is a good target) to build a good endurance base. You can afford to cut back on the volume of running.
Here is, I think, a more sensible plan:
Mon: Gym (circuit style, emphasis on bodyweight exercises).
Tues: 3 mile run.
Wed: Gym (Normal style, emphasis on weights, do staple bodyweight exercises first - presups, pullups, dips).
Thurs: Am - Sprints, Pm - 6 mile steady state run.
Fri: Gym (circuit style, emphasis on bodyweight exercises).
Sat: 3 mile run.
Rest is as important as exercise! That cannot be stressed enough. Too much exercise, too little rest is the key problem with your plan, and if you expect to up the volume and intensity until your PRMC you're likely to burn out, get injured or both.