3. Variant 1.

A natural number is written on the board. Five schoolchildren said the following about it:

- Pete: “This number is greater than $10$.”
- Kolya: “This number is not less than $11$.”
- Vasya: “This number is greater than $12$.”
- Dima: “This number is less than $12$.”
- Oleg: “This number is not greater than $12$.”

Find the maximum possible number of correct statements.
Answer: 4

Solution. All five answers could not have been correct, since Vasya's and Dima's answers contradict each other. Four answers could have been correct if the number 11 was written on the board.