18) On an island of knights and scoundrels, the latter are twice as numerous as the former. Each inhabitant of the island is asked how many knights there are. While each knight provides the correct answer, the first scoundrel gives the exact number increased by 1, the second scoundrel the exact number increased by 2, and so on. Knowing that the sum of all the answers is 1140, how many inhabitants are there on the island?
(A) 30
(B) 45
(C) 60
(D) 90
(E) 99.
18) The answer is (B). Let $c$ be the number of knights and $f$ be the number of rogues. We need to calculate $c+f$. The sum $s$ of the answers is (with $c$ terms in the first parenthesis)

$$
s=(c+\ldots+c)+(c+1)+\ldots+(c+f)=c^{2}+c f+f(f+1) / 2 .
$$

Given $f=2 c$ and $s=1140$, we have

$$
3 c^{2}+c(2 c+1)=1140 \quad \text { i.e., } 5 c^{2}+c-1140=0 \text {. }
$$

The positive solution is $c=15$, so $c+f=3 c=45$.