## Problem I-2

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 $180^{\circ}$.)

(proposed by Dominik Burek, Poland)

Answer. $n-3$.

In the following we write for short 'reflection of $A$ in $P$ ' instead of 'reflection of the vertex $A$ with respect to the midpoint of the segment connecting the two neigbouring vertices of $A$ in the polygon $P$ ?