KöMaL 2022 A839

We are given a finite, simple, non-directed graph. Ann writes positive real numbers on each edge of the graph such that for all vertices the following is true: the sum of the numbers written on the edges incident to a given vertex is less than one. Bob wants to write non-negative real numbers on the vertices in the following way: if the number written at vertex $v$ is $v_0$, and Ann's numbers on the edges incident to $v$ are $e_1,e_2, \dots, e_k$, and the numbers on the other endpoints of these edges are $v_1, v_2, \dots, v_k$, then $v_0=\sum\limits_{i=1}^{k}e_iv_i+2022$. Prove that Bob can always number the vertices in this way regardless of the graph and the numbers chosen by Ann.