For (a) I would use a polarisation identity, . If for all x, then each term the right side of the identity will be 0, and so for all x and y. It easily follows that C=0. Note that this result is true for all n×n complex matrices, they don't need to be hermitian.

(b) is kind of obvious. Just take . (Of course, you have to check that B and C are hermitian.)

For (c), choose some particular values for x. If x has a 1 for the j'th coordinate and zeros everywhere else then x*Ax is just the (j,j)-element of A. So if that is real then A has real elements on the diagonal. Now take x to have two nonzero elements, in the j'th and k'th coordinates. If both these coordinates are 1, then the condition x*Ax is real tells you that the imaginary part of the sum of the (j,k)- and (k,j)-elements is zero. If the j'th coordinate of x is 1 and the k'th coordinate is i, it tells you that the real parts of the (j,k)- and (k,j)-elements of A are the same. Put those results together and they tell you that A is hermitian.