\section*{Problem 1 - 041041}

The 30 prize winners of a student competition are to be awarded with newly published specialist books.

Three different types of books, valued at \(30 \mathrm{M}, 24 \mathrm{M}\) and \(18 \mathrm{M}\) respectively, are available. At least one book of each type must be purchased.

What are the possible combinations of books if a total of \(600 \mathrm{M}\) is available for the awards, and this amount should be spent in full?
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We first take out one book each for \(30 \mathrm{M}, 24 \mathrm{M}\), and \(18 \mathrm{M}\). There remain 27 books for a total of \(600 \mathrm{M} - 30 \mathrm{M} - 24 \mathrm{M} - 18 \mathrm{M} = 528 \mathrm{M}\).

27 books at \(18 \mathrm{M}\) each cost \(486 \mathrm{M}\), so there are still \(528 \mathrm{M} - 486 \mathrm{M} = 42 \mathrm{M}\) left, which can be used to buy more expensive books instead of the \(18 \mathrm{M}\) books.

We now distinguish based on how many books at \(30 \mathrm{M}\) are bought:

a) 4 or more are not possible, as this would incur additional costs of at least 48 Marks

b) 3 books cost 36 Marks additionally, leaving 6 Marks, with which exactly one \(18 \mathrm{M}\) book can be replaced by a 24 M book.

c) 2 books at \(30 \mathrm{M}\) and three books at \(24 \mathrm{M}\)

d) 1 book at \(30 \mathrm{M}\) and five books at \(24 \mathrm{M}\)

e) no book at \(30 \mathrm{M}\) and seven books at \(24 \mathrm{M}\)

In total, there are four possibilities

\begin{tabular}{ccc}
\(30 \mathrm{M}\) & \(24 \mathrm{M}\) & \(18 \mathrm{M}\) \\
4 & 2 & 24 \\
3 & 4 & 23 \\
2 & 6 & 22 \\
1 & 8 & 21
\end{tabular}

A check confirms that in all four cases, the conditions of the problem are satisfied.