National Math Olympiad

If $n\ge 3$ is even, $n=2k$, then such a polygon exists since every regular $2k$-gon satisfies the condition.

If $n=3$ or $n=5$, then such a polygon does not exist. Indeed, no two sides in a triangle are parallel. If every side of a pentagon would be parallel to some other side, we could find three parallel sides and two of these should be adjacent. This is not possible.

For $n=7$ such a polygon exists as shown by the figure.

Let us prove by induction that for odd positive integers $n$, $n\ge 7$, a polygon with the required property exists. Assume that for some integer $k$ a $k$-gon with this property exists. Choose a vertex at which the inner angle is less than $180^{\circ }$. Now, cut away a small parallelogram as shown in the figure. We obtain a $(k+2)$-gon which has the required property. A few examples are shown in the figure below. $n=9$ $n=11$ $n=13$