I don't know who is the first person to start this NOT mathematically sound joke:

Study = no fail
no study = fail
study + no study = fail + no fail
study (1+no) = fail (1+no)
study = fail

The joke is a fail!
First at sight, wow, what a amazing proof. But really, not so...
"no" could be -1
and there is a lot evidence support no is -1
yes correspond to true, which in turn can be expressed by 1
no is the negation of yes. so it's likely be -1

if it's -1, the result is NOT true, because (no+1) = 0
if ab = ac, b=c only if a does not equal to 0
division by zero... that's fail!

What does this mean? study and fail have no relationship? nope, this means study = -fail. so, actually, the real answer for the study = fail proof should be.
|study| = |fail|
so if I absolutely study, I will absolutely fail.
Here is the new proof if you like to show off to your friends.

study = no fail
no study = fail

if no does not equal to -1.

study + no study = fail + no fail
study (1+no) = fail (1+no)
study = fail

if no does equal -1
study = -1*fail
study = -fail

thus
|study| = |fail|

A special case, no = 0. Which makes study = fail and -fail. in that case, study = 0, study=nothing. and thing can be any number, like the definition of "thing" in english. That's Zen.
btw... if no equals everything but 1 or -1, study = fail = 0... I just make that special case so it sounds nice...
read the comment to see how I arrive that.

Some other note. It is ok to divide by zero in some sets. For example, the real projective line. But still, study = fail can't be observed. Since a/0 = unsigned infinity in real projective line.