= Solution
Let $q$ be the true density and $p$ the announced density. The advantage of truthful reporting for the <logarithmic scoring rule> is
$$
\mathbb E_q[\log q(Y)]-\mathbb E_q[\log p(Y)]
=\int q(y)\log\frac{q(y)}{p(y)}\,dy
=D_{\mathrm{KL}}(q\Vert p).
$$
The <Kullback-Leibler divergence> is nonnegative, with equality exactly when the densities agree almost everywhere. For instance, <Jensen inequality> gives $\mathbb E_q[\log(p/q)]\le\log\mathbb E_q[p/q]\le0$; equality requires a constant likelihood ratio on the true support and no remaining mass outside it. Consequently
$$
\boxed{\mathbb E_q[\log q(Y)]\ge\mathbb E_q[\log p(Y)],
\quad\text{with equality iff }p=q\text{ a.e.}}
$$
This proves strict propriety when the expected log scores are well defined, for example with a finite true expected log density. A forecast assigning zero density to a set of positive true probability has score $-\infty$ there and cannot outperform the truth. The hint's strict $>0$ must be corrected to $\ge0$ to allow its equality case.
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