discredit wrote: Given a regular $ 2n$-gon, show that each of its sides and diagonals can be assigned in such a way that the sum of the obtained vectors equals zero. One of my favourite problems Solution: If it is possible to assign vectors to each side and diagonal of some regular $ k$-gon,so that their sum equals to zero vector,then we will call this $ k$-gon - "good". We will show that if we have some "good" regular $ n$-gon,then regular $ 2n$-gon is "good",too. Let $ A_1A_2\dots A_{2n}$ be some regular $ 2n$-gon.Divide all its vertexes into two sets $ A_1A_2\dots A_{2n-1}$ and $ A_2A_4\dots A_{2n}$.Obiosuly vertexes from one set form regular $ n$-gon,denote them as $ R_1$ and $ R_2$,and as we know these $ n$-gons are "good".So we can assign vectors to each side and diagonals of $ R_1$ and $ R_2$,so that their sum is equal to $ 0$. Now take each vertex of $ R_1$ and lead vector from it to each vertex from $ R_2$.It is clear that their sum is equal to $ 0$. Now suppose that $ 2n=2^{k}r$,where $ r\in\mathbb{N}$ is odd and $ k\geq 1$. It is enough to show that regular $ r$-gon is "good".Let $ A_1A_2\dots A_r$ be our regular $ r$-gon. For each vertex we will lead a vector from it to next $ \frac{r-1}{2}$ vertexes,in a clockwise direction.As it showed on this $ 7$-gon: Image not found Obiously their sum is equal to $ 0$,because if we turn regular $ r$-gon with respect to its center in angle $ \frac{\pi}{r}$,our vectors will turn to itself.

Erken wrote: ...Now take each vertex of $ R_1$ and lead vector from it to each vertex from $ R_2$.It is clear that their sum is equal to $ 0$... I can't see why this holds. Actually, I didn't look too careful but it seemed untrue for $ 2n = 10$. The rest of the problem is just recognizing that every graph which has all its vertices' degrees even, has a Eulerian path. Direct every vector to the same direction on this path and sum of them are $ 0$.

So you are claiming that, if we divide all vertices of a regular $ 10$-gon into two sets, $ 1,3,5,7,9$ and $ 2,4,6,8,10$ and lead a vector from each vertex of the first set to each vertex of the second set, the total sum of vectors will be non-zero, right? But that has to be untrue, since if you rotate this vector system with respect to the centre of our 10-gon to a $ \frac {2\pi}{5}$ angle, the sum should turn into itself, but that means that the sum is $ 0$.