Maximum-entropy distribution

Among all probability distributions $(p_1, \dots, p_n)$ on $\{1, \dots, n\}$, which one maximizes the entropy $H = -\sum_i p_i \ln p_i$? What changes if you also require the mean $\sum_i i\,p_i$ to equal a given $m$? Describe the answer for a six-sided die constrained to have mean 4.5.