What are some examples of math contest problems that can be solved by using a nonrigorous, 'cheap' shortcut?
For instance, a problem on the 2011 AMC went:
A raft and a motorboat left dock A and started downstream. The raft traveled at the speed of the current. The motorboat maintained a constant speed with respect to the river. The motorboat went to point B then immediately turned back, meeting the raft 9 hours after leaving dock. How long did it take the motorboat to travel from A to B?
An attentive student may notice that the question does not mention the speed of the current, so it must not affect the answer. Then setting the current to be 0, he gets 4.5 hours trivially.
Another example might be the trivial 'derivation' of the probability of a random fill of a Ferrers diagram is a Young tableau. Assume all probabilities are independent, then multiply the individual probability for each hook; this gives the correct formula, but the proof is completely wrong (the probabilities are obviously not independent).
I'm looking for problems in which an otherwise non-rigorous step, or a false intuition leads to the correct answer.