1. Two classes with the same number of students took a test. After checking the tests, the strict director, Alexander Vladimirovich, said that he had given 11 more twos than all other grades. Did Alexander Vladimirovich make a mistake?
Solution. Let there be $x$ students in each class, then there are $-2 x$ students in two classes (an even number). Let Alexander Vladimirovich give $y$ other grades (not twos), then he gave $y+11$ twos. Thus, he gave a total of $2 y+11$ grades (an odd number). Since the number of students must match the number of grades, we conclude that Alexander Vladimirovich made a mistake $(2 x \neq 2 y+1, x, y \in \mathbb{N})$. Answer: made a mistake.