Let x_1, x_2, ..., x_n be arbitrary real numbers such that the total sum x_1+...+x_n is strictly positive. Prove that there exists an index k in {1,...,n} such that each of the following n sums x_k, x_k+x_{k+1}, x_k+x_{k+1}+x_{k+2}, ..., x_k+...+x_n, x_k+...+x_n+x_1, x_k+...+x_n+x_1+x_2, ..., x_k+...+x_n+x_1+...+x_{k-1} is strictly positive. (In other words, all the sums x_k+...+x_{k+l}, l=0,1,...,n-1, with indices counted modulo n, must be positive.)