Peter wrote: Prove that on a coordinate plane it is impossible to draw a closed broken line such that coordinates of each vertex are rational, the length of its every edge is equal to $ 1$, the line has an odd number of vertices. We can rephrase it as follows: coordinates of each vertex are integers, the length of its every edge is a fixed natural number, the line has an odd number of vertices. Well, it is only a idea... I'm trying to use modulo 4 to take some things over. The possible pairs for values mod4 is (0,0),(1,0) and (0,1) ($ \delta_x^2+\delta_y^2=N^2$). We can exclude the case (0,0) WLOG (if N is even, then $ \delta_x,\delta_y$ are even too). Now, we can think in how change the mod4 coordinates of broken-line vertices.

That's a good start. As you remark, by Pythagoras' theorem we have $ a^2 + b^2 = c^2$, implying mod 4 that $ c$ is odd iff $ a+b$ is odd. Assume such a configuration exists, and take one with minimal fixed sidelength $ L$. Then neceessarily $ L$ is odd (otherwise rescaling with factor $ 1/2$ gives a smaller configuration with integer coordinates & lengths). Start at a point, after translation we may assume both coordinates are even. Now every time we go along a side, by the first observation, the sum $ x + y$ switches in parity. So, after an odd number of edges, we can never end back on our starting point. Nice invariant huh?