\section*{Problem 3 - 021033}

Find a three-digit number that has the following properties:

a) The number is divisible by 9 and 11.

b) If the first and last digits are swapped, the result is \(\frac{2}{9}\) of the original number.

How many solutions are there?
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The sought number must be divisible by 11 and by 9. Due to the latter, the cross sum is also divisible by 9. When the digits are rearranged, the cross sum of the number does not change. The newly formed number is thus also divisible by 9.

Since it is obtained from the original number by multiplying with \(\frac{2}{9}\), the initial number must be divisible by 9 twice. The sought number is therefore not only divisible by 11.9 but even by 11.9.9.

The only three-digit number that meets this condition is 891. It is simultaneously the only solution to the problem.