16) In a bag there are 20 balls, and on each is written an integer between 0 and 10 (0 and 10 inclusive). The number written on each ball, if not zero, is the sum of the numbers written on all the other balls. Therefore, the balls on which zero is written are:
(A) no more than five,
(B) ten,
(C) thirteen,
(D) sixteen,
(E) at least eighteen.
16. The answer is $\mathbf{( E )}$.

Suppose there are at least three balls with numbers strictly greater than zero; let $p, q$, and $r$ be the three numbers written on these three balls. The number $p$, being the sum of the numbers written on all the other balls, will be equal to the sum of $q, r$, and all the numbers written on the other 17 balls. Therefore, $p \geq q+r$ and since $r>0$ it follows that $p>q$. By swapping the roles of $p$ and $q$, it can be shown in exactly the same way that $q>p$, and thus we arrive at a contradiction. It is therefore absurd to assume that there are at least three balls with strictly positive numbers.