= Solution
Let $q$ be the genuine rain probability and $p$ the reported probability. The <linear probability score> has expected reward
$$
s(p,q)=qp+(1-q)(1-p)=1-q+(2q-1)p.
$$
This is linear in $p$, so its optimal report and expected reward are
$$
\boxed{p^*=1\ (q>1/2),\qquad p^*=0\ (q<1/2),
\qquad s(p^*,q)=\max(q,1-q).}
$$
At $q=1/2$ every report scores $1/2$ on average. Truthful reporting scores $q^2+(1-q)^2$, strictly below the optimum for interior $q\ne1/2$. Thus \b[the rule is not proper]: it rewards maximal confidence in the more likely outcome. The expected total over days is $\sum_t\max(q_t,1-q_t)$.
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