  We present a comprehensive demonstration of how automated reasoning can
assist mathematical research, both in the discovery of conjectures and in their
verification. Our focus is a discrete geometry problem: What is $\mu_{5}(n)$,
the minimum number of convex pentagons induced by $n$ points in the plane? In
the first stage toward tackling this problem, automated reasoning tools guide
discovery and conjectures: we use SAT-based tools to find abstract
configurations of points that would induce few pentagons. Afterward, we use
Operations Research tools to find and visualize realizations of these
configurations in the plane, if they exist. Mathematical thought and intuition
are still vital parts of the process for turning the obtained visualizations
into general constructions. A surprisingly simple upper bound follows from our
constructions: $\mu_{5}(n) \leq \binom{\lfloor n/2 \rfloor}{5} + \binom{\lceil
n/2 \rceil}{5}$, and we conjecture it is optimal. In the second stage, we turn
our focus to verifying this conjecture. Using MaxSAT, we confirm that
$\mu_5(n)$ matches the conjectured values for $n \leq 16$, thereby improving
both the existing lower and upper bounds for $n \in [12, 16]$. Our MaxSAT
results rely on two mathematical theorems with pen-and-paper proofs,
highlighting once again the rich interplay between automated and traditional
mathematics.
