Multi-category Brier

A forecaster gives probabilities $\mathbf p = (p_1,p_2,p_3)$ for "Nifty falls more than 1% / moves less than 1% / rises more than 1% this week". The multi-category Brier score is $\sum_k(p_k - y_k)^2$ with $y$ the one-hot outcome. (a) Prove it is strictly proper. (b) Compute the score of $\mathbf p = (0.2, 0.5, 0.3)$ if the index rose 1.4%. (c) Show that the log score $-\ln p_{k^\*}$ ($k^\*$ the realised category) is strictly proper for any number of categories.