Question 45
Let $n \geq 3$ be an integer. We say that a vertex $A_{i}$ ($1 \leq i \leq n$) of a convex polygon $A_{1} A_{2} \ldots A_{n}$ is Bohemian if its reflection with respect to the midpoint of the segment $A_{i-1} A_{i+1}$ (with $A_{0}=A_{n}$ and $A_{n+1}=A_{1}$) lies inside or on the boundary of the polygon $A_{1} A_{2} \ldots A_{n}$. Determine the smallest possible number of Bohemian vertices a convex $n$-gon can have (depending on $n$).
(A convex polygon $A_{1} A_{2} \ldots A_{n}$ has $n$ vertices with all inner angles smaller than $180^{\circ}$.)
(A convex polygon $A_{1} A_{2} \ldots A_{n}$ has $n$ vertices with all inner angles smaller than $180^{\circ}$.)