I sound like a broken record these days but bad capacitors are rampant these days, and it looks possibly like something in the power supply is gone and the input board isn't getting the power it needs that or a fuse on the power or input board is gone, or both.
Capacitors tend to fail gradually, thus the 30 minute warm up time (which also indicates that it's highly unlikely to be a loose cable of any kind and chances are that no fuses are gone)
There's a good chance you have failing electrolytic capacitors in the power section
Any caps in that look bulged at the top, or bulged/leaking at the bottom need to be replaced no matter what section they are in.
If you repeatedly turn it on, eventually it'll probably stay on, but every time you turn it off, the TV will get harder and harder to start up until one day it just won't.
Sometimes you have to do the opposite to start it up and unplug it for 5 mins, then try.
If you aren't tech savvy, don't worry, people that have never soldered before have managed this repair.
If you are handy with a soldering iron and can identify the power supply, a handful of capacitors will likely fix you right up.
Match the capacitance on the capacitors. Go over voltage if you can, and still have them fit.
IE - it's not a bad idea to replace a 10V cap with a 16V or 25V or even a 50V, but don't replace a 680uF cap with a 500uF or a 1000uF (unless you are positive it's only doing ripple filtering, and even then, you should go OVER, not under the uF rating).
Most of the caps that go are 10V 1000uF or 3300uF.
I found some great videos of the procedure (for many Samsungs with the same issue) on youtube.
http://www.youtube.com/watch?v=Mm51C_RDIZEhttp://www.youtube.com/watch?v=BNHHrgX_6cs&feature=relatedhttp://www.youtube.com/watch?v=8l0AUj8QUkg&NR=1&feature=fvwphttp://www.youtube.com/watch?v=Y7b_nTaZYcU&feature=relatedAs you can see, this issue spans plasma TVs, as well as large and small LCD TVs and monitors.
The parts are cheap, and skill required is minimal.