2. Diana wrote a two-digit number, and appended to it a two-digit number that was a permutation of the digits of the first number. It turned out that the difference between the first and second numbers is equal to the sum of the digits of the first number. What four-digit number was written?
# Answer: 5445.

Solution. The difference between such numbers is always divisible by 9, since $10a + b - (10b + a) = 9(a - b)$. This difference equals the sum of two digits, so it is no more than 18. However, it cannot be 18, because then both the first and second numbers would be 99. Therefore, the difference between the two numbers is 9, and the tens digits of the two numbers differ by no more than 1. But the sum of the tens digits of the two numbers is the sum of the digits of the first number, and it equals 9. Clearly, only the digits 4 and 5 fit.

Comment. Correct solution - 20 points. Incomplete proof - 10 points. Solution by trial and error, without proving that there are no other options - 5 points. The answer includes the number 4554 - 1 point is deducted. Some progress in the solution - 5 points. The solution is started, but there is no noticeable progress - 1 point. The correct answer is given without explanation - 0 points.