Fakesolving refers to one of two things, both loosely related to each other:

  • When one has seemingly completed a "proof", but has made an assumption somewhere that makes the "proof" not a proof
  • When one takes advantage of the implied generality in a short-answer problem to deliberately assume a certain case, in order to arrive at a number

Definition 1: Not Actually Proof-ing

When writing a proof problem, often in Olympiad contests (such as the USAMO, for example), sometimes one seems to have completed a proof, but didn't actually, due to either making an assumption somewhere or by loss of generality, or making an assumption somewhere that was blatantly not true and pulled out of the air. Most times solvers are aware of the fact that they did not actually prove the condition set in the problem, and are aware that they will most likely not receive full credit on that problem.

Definition 2: Making Assumptions On Purpose

Sometimes in short answer problems, most often found in middle school math contests (e.g. AMC 8, MATHCOUNTS) the problem implies that a condition will hold for a general case. The solver then chooses a certain case to speed up the solving process, losing generality for the sake of arriving at a number; most often a convenient case is chosen that makes the math "pretty".

For example: assuming a "rectangle" is a square where applicable, to make numbers easier and the problem easier.

Of course, this has you lose generality, though, so it's not suitable for proof problems where you need to prove for the general case. However, for short-answers questions where no proof is required (e.g. AMC, SMO) it can be a quite effective way of earning marks, although questions are deliberately set to avoid the possibility of fake-solving.