*** Most of inductions were very chaotic, badly explained, and in a few occasions, they proved for 1 (or 0), and then tried to prove the case "n+1" but they assumed n=1 so they proved for n=1,2 and concluded for all n.

*** A lot of people wrote "2!" in the formula, the ! is irrelevant, and that leads me to thinking that a lot of people might have merely copied the answer without really thinking about it.

*** The biggest one that LOTS of people did is on the last exercise. A LOT of people said : f is continuous at 0 and so it is differentiable. I believe they learned that if it is differentiable at x then it is continuous at x, but they should under no circumstance think the converse is true (absolute value).

***On the same question, people differentiated the function using usual rules, and tried to prove it has a limit at zero, so there are two big mistakes in this one : they checked the limit of the derivative, which does not tell you whether it exists, AND all of them managed to prove that cos(1/x) has a limit at 0, which should not happen.