About the real reason why paper doesn't fall in HALF-FILLED inverted cup I just happen to see a show about air pressure host by Brian Cox. As the link:http://imgur.com/EsRadFi shows, he flipped a half-filled water cup upside down then the paper on cup doesn't drop:
It puzzled me, because the cup is half-filled, the pressure shouldn't balance since inside water pressure+atmosphere pressure should be bigger than outside atmosphere pressure only. 
All the following answers seems not right, now I came the answer with myself: Treat the air inside as gas of a certain density. It's then obvious that the pressure of inside of the paper always equal to the outside unless the pressure exerted by gravity of {water+gas} system beyond the outside atmosphere pressure. The key is that inside is not {water pressure+atmosphere pressure} but {1: water pressure+ 2: small air pressure by inside air's own gravity+ 3: big atmosphere pressure-(1.+2.)}=atmosphere pressure. The third force came from the top inside of the flipped cup. So the paper wouldn't drop until (1.+2.)> atmosphere pressure, even when the paper is rigid body, in constrast to Steeven or others' result.
 A: When the cup is turned upside down, the water wants to fall out. The air-filled cavity is therefore stretched a bit as the gravity pulls down the water. The air expands a bit. This reduces the air pressure inside the cup, since increasing volume reduces pressure. This is hinted in the ideal gas equation:
$$pV=nRT$$
Soon this lower pressure pulls upwards with the same force as the weight of the water pulls downwards. The water is now kept in place and the pressure inside is lower than atmospheric pressure outside.
If you disturb the water by shaking the cup, suddenly different forces work on different parts of the water surface as the water level is not constant. If the water amount at any point is displaced enough for a breach to appear so outside air comes in, then atmospharic pressure is restored inside the cup, the water is not kept up anymore and it falls out.
A: 
How does air pressure on a paper sheet hold water in an half-filled inverted cup?

A simplified model of the experiment you have seen is illustrated in the attached picture. You initially have a piston with a certain quantity  of air at 1 atm inside and a piston of mass m and surface S somewhere in the cylinder. Then you let the piston fall. It will reach a static equilibrium when:
$P_{AirCylinder}*S + mg=1 atm * S$
Solving the equation you get:
$P_{AirCylinder} = 1 atm - mg/S$

