OLYMPIC-REVENGE 2018 Problem 3

In a mathematical challenge, positive real numbers $a_{1}\geq a_{2} \geq ... \geq a_{n}$ and an initial sequence of positive real numbers $(b_{1}, b_{2},...,b_{n+1})$ are given to Secco. Let $C$ a non-negative real number. In a sequence $(x_{1},x_{2},...,x_{n+1})$ , consider the following operation: Subtract $1$ of some $x_{j}$ , $j \in \{1,2,...,n+1\}$ , add $C$ to $x_{n+1}$ and replace $(x_{1},x_{2},...,x_{j-1})$ for $(x_{1}+a_{\sigma (1)}, x_{2}+a_{\sigma (2)}, ..., x_{j-1}+a_{\sigma (j-1)})$ , where $\sigma$ is a permutation of $(1,2,...,j-1)$ . Secco's goal is to make all terms of sequence $(b_{k})$ negative after a finite number of operations. Find all values of $C$ , depending of $a_{1}, a_{2},..., a_{n}, b_{1}, b_{2}, ..., b_{n+1}$ , for which Secco can attain his goal.