11. On the island of Chenonc'è, there are 2009 inhabitants, divided into three clans: the scoundrels who always lie, the knights who never lie, and the pages who lie one day and tell the truth the next, independently of each other. One day, I ask each of the inhabitants how many scoundrels are on the island. The first says: "there is at least 1 scoundrel"; the second says: "there are at least 2 scoundrels";... the 2009th says: "there are at least 2009 scoundrels". I write down the sequence of 2009 answers in the order they were given. The next day, I ask everyone the same question (not necessarily in the same order), and I get a list of answers identical to the one from the previous day. Knowing that there is only one knight on the island, how many pages are there?
(A) None
(B) 670
(C) 1004
(D) 1338
(E) 1339.
11. The answer is (D). Let $f$ be the number of knaves and $p$ be the number of pages. In each of the two lists, there will be exactly $f$ true statements (the first $f$) and exactly $2009-f$ false statements (the remaining ones). Therefore, between the two lists, there are $4018-2f$ false statements. Now, each knave has provided 2 false statements, while each page has provided only one, since on one of the two days they must have told the truth. Therefore, the number of false statements must equal the number of pages plus twice the number of knaves, that is,

$$
4018-2f=2f+p
$$

and since $p+f=2008$ (there is only one knight), we can derive $f=670$ from which $p=2008-670=1338$.