Is it possible that for all $1 \leq i \leq n-1$ $\sum_{m=1, m\neq i}^n c_m a_m \leq k a_i (1-c_i)$ Let $a_1, a_2, \dots, a_n$ be positive numbers and $c_1, c_2, \dots, c_n$ be positive numbers such that $\sum_{i=1}^n c_i= 1$. Let $0 < k \leq 1$.
Does there exist any choice of $a_i, c_i$ satisfying the above conditions so that for all $1 \leq i \leq n-1$ the following statement is true:
$$\sum_{m=1, m\neq i}^n c_m a_m \leq k a_i (1-c_i)$$ 

Clue:
What will happen if $a_i \to 0$?
if $a_j$ are  positive, take$a_i$ small enough, but $a_m = 1$ otherwise
