\section*{Exercise 1 - 141021}

Klaus checks his vocabulary knowledge in Russian during the holidays. When he determines the number \(z_{1}\) of words he still knows among the 2555 words he has carefully entered into his vocabulary book over time, and then the number \(z_{2}\) of the remaining words, he notices when writing down these two numbers that \(z_{1}>z_{2}\) and that he used exactly two digits for writing, alternating each time, with the first digit being the same in both numbers.

Determine \(z_{1}\) and \(z_{2}\)!
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When adding the two numbers \(z_{1}\) and \(z_{2}\), there could not have been a carry at any place, otherwise, the units and tens place of their sum 2555 would not be the same.

From the fact that exactly the last three digits of 2555 are the same, we can further conclude that \(z_{1}\) must be a four-digit number and \(z_{2}\) must be a three-digit number. If the decimal representation of \(z_{1}\) is of the form xyxy, where \(x\) and \(y\) are the two alternating digits, then the decimal representation of \(z_{2}\) must be of the form \(x y x\), where \(x=2\) and \(x+y=5\), so \(y=3\) must hold. Indeed, the two numbers \(z_{1}=2323\) and \(z_{2}=232\) satisfy all the conditions of the problem.