Problem 831

Answer. There are four funny numbers: 1111, 1091, 1109, 1089. Note. We will show that there are no other funny numbers. Notice that the number $y$ obtained after changing the digits is not less than $x$, and not equal to $x$, but its first digit can be greater than the first digit of the number $x$ by only 1. This, as is easily seen, is possible only if the number $x$ started with 1 and $y=2x$. In this case, the number $y$ must start with 2, meaning that when multiplying $x$ by 2 "in a column," there was no carry from the hundreds place to the thousands place.

By checking the digits from 0 to 9, we find that if there is no carry in a given place when multiplying $x$ by 2 "in a column," then it could only have been 1 (which was then increased by 1) or 9 (which was then decreased by 1), and if there was a carry, then only 0 (which was then increased by 1) or 8 (which was then decreased by 1). Therefore, the number $x$ could end in 1 or 9, and in the tens place, it could be 1 or 9 in the first case, and 0 or 8 in the second case. The content of the hundreds place in each of the four resulting cases is determined uniquely, which gives the four answers.

Criteria. All points are evaluated together. If only one funny number is found - 0 points. If two funny numbers are found - 2 points. If three funny numbers are found - 4 points. If all four funny numbers are found - 7 points. Justification that there are no other funny numbers is not required. The score is not reduced for the absence of justification that the found numbers are indeed funny. If instead of funny numbers, their doubles are listed, and the funny numbers themselves are not listed, the score is not reduced. If the answer includes an extraneous number in addition to the funny numbers - 1 point is deducted.