= Solution
{c}
Equality forces $\mathbb EX<\infty$ because its proposed value is at most $a$. The random variable $Y=X-a\mathbf1_{\{X\geq a\}}$ is nonnegative and has expectation zero, so $Y=0$ almost surely. On $\{X<a\}$ it equals the strictly positive variable $X$, making that event null. On $\{X\geq a\}$ it equals $X-a$, forcing $X=a$ there. Hence
$$
\boxed{\mathbb P(X=a)=1.}
$$
This is the <equality in Markov's inequality> under strict positivity. The reasoning even works without the discreteness assumption.
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