7. 99 people - knights and liars (knights always tell the truth, while liars always lie) - are standing in a row. Each of them said one of two phrases: "To the left of me, there are twice as many knights as liars" or "To the left of me, there are as many knights as liars." In reality, there were more knights than liars, and more than 50 people said the first phrase. How many liars said the first phrase?
Answer: 49.

Solution:

There are more knights than half, so:

1) Either they alternate like this: KRK...LK, but this option does not work: the phrase "there are twice as many knights as liars" cannot be said by any knight, but there are more than 50 such phrases.

2) Or some two knights stand next to each other. Two knights standing next to each other cannot both say the same phrases, and they can say different phrases only in one case: the third says that there are as many knights as liars before him, and the fourth says that there are twice as many knights as liars before him (this can be derived from simple equations). There are no other places where knights stand next to each other, so to have at least 50 knights, the picture can only be like this: KLRKRL...LK. All knights, except the first and the fourth from the left, definitely say that there are as many knights as liars to their left, as they cannot say anything else. There are 48 of them. To have more than 50 opposite phrases, all the remaining ones must say exactly these. Therefore, all liars, who are a total of 49, say the first phrase.