4. (7 points) In the room, there are 10 lamps. Petya said: «In this room, there are 5 lamps on». Vasya replied: «You are wrong». And added: «In this room, there are three lamps off». Kolya then said: «An even number of lamps are on». It turned out that only one of the four statements is true. How many lamps are on?
# Answer. 9.

Solution. The first and third statements cannot both be false at the same time, otherwise there would be fewer than five lights on and fewer than three lights off, i.e., fewer than eight lights in total, which contradicts the condition. The first and second statements also cannot both be false at the same time.

Thus, among statements 1 and 3, there is a true one, and among statements 1 and 2, there is a true one. Since there is only one true statement, it is statement 1, and the other statements are false.

Therefore, there are fewer than three lights off (since statement 3 is false). Then there are at least eight lights on, and their number is odd (since statement 4 is false). Thus, there are nine.

## Grading Criteria.

- Any complete and correct solution - 7 points.
- Correct and complete solution, but the answer is given to the wrong question (e.g., "1 light off") - 6 points.
- It is stated but not proven that only statement 1 can be true, but then it is correctly deduced from this fact that there are 9 lights on, -3 points.
- It is explained that only statement 1 can be true, but there is one mistake in constructing the negation of one of the statements 2, 3, 4, leading to an incorrect answer (for example, "on" is confused with "off" or "even" is not changed to "odd"), -3 points.
- It is explained that only statement 1 can be true, but the subsequent reasoning is incorrect or there are at least two mistakes in constructing the negations, -2 points.
- The correct answer is given, and it is stated without justification that only statement 1 is true, but it is not explained why no other statement can be true and why no other answer is possible, -2 points.
- Only the answer is given - 0 points.