Non-Uniform Birthdays Increase Collision Probability

Suppose $d$ days have birthday probabilities $p_1, p_2, \ldots, p_d$ with $\sum_j p_j = 1$ (not necessarily uniform). For $n$ people whose birthdays are independent draws from this distribution:

(a) Show that for $n = 2$, $P(\text{collision}) = \sum_{j=1}^{d} p_j^2 \ge \frac{1}{d}$, with equality if and only if all $p_j = \frac{1}{d}$.

(b) Deduce that the uniform distribution minimizes the collision probability among all distributions on $d$ days. Give a one-line intuitive explanation for why non-uniformity helps collisions.