2. Each of the islanders is either a knight, who always tells the truth, or a liar, who always lies (both types are present on the island). Every resident knows whether each other is a knight or a liar. Some of the islanders stated that there is an even number of knights on the island, while all the others claimed that there is an odd number of liars on the island. Could there be exactly 2021 residents on the island?
Answer: No

Solution. Since 2021 is an odd number, on the island there is either an odd number of knights and an even number of liars, or an odd number of liars and an even number of knights. The first case is impossible, since both statements are false and, therefore, no knight could have made either the first or the second statement. In the second case, both statements are true, so no liar could have made either the first or the second statement.