Some of you forgot to answer the second question (i.e. describe the solutions of xn = yn). In my opinion, the simplest way to do it is still by induction on n, distinguishing between the odd and even cases.
Some of you tried to write direct proofs that |A| = |B|, e.g. by explicitly giving a formula for both quantity, using the binomial coefficients. However, the exercise specifically asked for a bijection. Also, binomial coefficents haven't yet been introduced in this course: if you want to use them, you have to define them and prove everything you are using about them...
When you write down a function, it is not enough to say "f(x) is either g(x) or h(x)", you have to make it clear when you are in one case, and when you are in the other, otherwise f is not well-defined (or, depending on your point of view, is not a function).
As always, do not forget to prove all your claims: some of you just wrote down a correct function, but did not explain why this is a bijection.
Most of you had the correct idea, some forgot to explain why the functions involved were or were not surjective.
Again, some answers were a bit short on explanation.
Most of you got this one right !