Can you punch a hole in an A4 sheet of paper with your hand? The heading says it all, but let me explain it a little further.
Suppose you hang a sheet of paper (A4 size) from one end leaving the other end free. Now practically it seems impossible to punch a hole through it, but is this theoretically possible? Assume you can not use a bigger size of paper.
So is it theoretically possible to punch a hole in a suspended sheet of paper?
Now, how will the size of the paper affect the result?
I am talking about punching with the hand and not anything else.
 A: In martial arts it is a test of your fast punches.
Hit it fast
The fighter has to twist his hand (mainly karate) in the instant before hitting the target. Momentum as well as angular momentum enforce the punch. The counteracting force will be formed the aerodynamic drag of the paper. Inititially it is Maximum prior to the impact. Be quick before it's deformation will reduce its effective area. Otherwise the paper will move by the impulse of your punch and wrap around your fist.
Hit it effective
The tip of the pen will strike through the paper. Concentrate the force of the punch on a smaller area by sticking up a knuckle of your fist. Fingertips even provide a smaller area. This force per area is a pressure and has to be compared by the material properties (bursting strength, measured in $kPa$) of paper.
Depending of the cellulose fiber length, additives and your physical condition you can punch a piece of paper. Best use a newspaper and train martial arts. 
A: I can easily punch through most unsupported (free in the air) paper and can do the same with cardboard boxes (all the way through both sides effectively spearing the box on my arm) and fingertip striking through all the above with all five fingers. The most difficult target is real playing cards. I cannot do more than dent them. Force and speed compared to the relative strength of the material and its weight and wind resistance all come into play including conditioning and technique. The twisting of the fist in karate is not a stylistic addition to the art. It is purposed to displace flesh upon the strike (much more damage) as well as make the blow more difficult to block/grab and is also used in boxing to blow open guards. 
A: You just have to punch fast enough, such that the inertia of the paper would allow for high enough tensions to build-up. This would tear the paper, instead of acceleration it up to the speed of your fist. You can look at it this way, when your fist makes contact with the paper, then the part making contact will nearly instantly accelerate up to the speed of your fist, since it is being pushed by the fist. The rest of the paper will initially not have moved, but the paper just outside this contact area will experience tension because the neighboring paper (within the contact area) will have moved slightly and will also start to accelerate. This tension will basically grow outwards. I think you can say that an upper limit would be near the speed of sound in paper, since that is how fast this tension could propagate. Making a better guess would probably involve simulations.
If the these speeds are humanly possible is another question. Maybe the 
