On the Island of Misfortune, only knights, who always tell the truth, and liars, who always lie, live. In

the Island's Council - there are 101 deputies. To reduce the budget, it was decided to reduce the Council by one deputy. But each of the deputies stated that if they were removed from the Council, the majority among the remaining deputies would be liars. How many knights and how many liars are in the Council?
Write two inequalities corresponding to the statements of the deputy liar and the deputy knight.

## Solution

Let $P$ be the number of knights in the Duma, and $L$ be the number of liars ($P+L=101$). According to the statement of the deputy knight, $P-1<50$. Therefore, $P<51$ (from this, in particular, it follows that the Duma cannot consist of knights alone). According to the statement of the deputy liar, $L-1 \leq 50$. Therefore, $L \leq 51$ (from this, in particular, it follows that the Duma cannot consist of liars alone).

## Answer

51 liars and 50 knights.