Problem 105

When proposition $p$ is true, $\forall x \in \mathbb{R}$, $x^2 - 4x + 2m \geq 0$ always holds. This means that the discriminant $\Delta = 16 - 8m \leq 0$, which implies $m \geq 2$.

Since "$m \geq 3$" is a sufficient but not necessary condition for "$m \geq 2$", it follows that "$m \geq 3$" is a sufficient but not necessary condition for "proposition $p$ is true".

Therefore, the answer is: $\boxed{\text{A}}$.

From the quadratic inequality, we have $\Delta = 16 - 8m \leq 0$, which gives $m \geq 2$. Using the concept of necessary and sufficient conditions, we find that "$m \geq 3$" is a sufficient but not necessary condition for "$m \geq 2$", and thus we obtain the solution. This problem tests the understanding of quadratic inequalities and necessary and sufficient conditions, and it is considered a simple problem.