At the New Year's school party in the city of Lzheretsark, 301 students came. Some of them always tell the truth, while the rest always lie. Each of the 200 students said: "If I leave the hall, then among the remaining students, the majority will be liars." Each of the other students stated: "If I leave the hall, then among the remaining students, there will be twice as many liars as those who tell the truth." How many liars were at the party?
Let's call those who tell the truth knights. We will consider the statement of each of the two hundred students as the first phrase, and the statement of the remaining as the second.

First method. Note that not all students are liars, otherwise there would be no one to say the first phrase. If the second phrase was spoken by a knight, then at the party there are 200 liars and 101 knights. In this case, among those who spoke the first phrase, there are liars. But in this case, a liar could not have said that if he left the hall, then among the remaining students, the majority would be liars. Therefore, there are no knights among those who spoke the second phrase. If the first phrase was spoken by a knight, then there are no more than 150 knights. This means that there is a liar who said the first phrase. But then there are no more than 151 liars. Since there are 301 students in total, there are exactly 150 knights and 151 liars among them.

Second method. Among those who spoke the first phrase, there must be both knights and liars. Indeed, if they were all knights, then this statement would be false, since $199>101$, and if they were all liars, then it would turn out that they told the truth. Therefore, the first phrase was spoken by both liars and knights. If the first phrase was spoken by a liar, then there are no more than 151 liars. If the first phrase was spoken by a knight, then there are no fewer than 151 liars. Therefore, at the party, there are 151 liars.

## Answer

151 liars.