Combinatorics · Set partitions · Self referential conditions · Counting bounds · Extremal constructions
Let \(A\) be the set of all integers from \(-20\) to \(20\), that is
\[ A = \{\, a \in \mathbb{Z} \;:\; -20 \le a \le 20 \,\} . \]
Let \(n\) be a positive integer and let \(A_1, A_2, \ldots, A_n\) be pairwise disjoint sets whose union is \(A\) and which satisfy
\[ \lvert A_i \rvert \in A_i \qquad\text{for every } i = 1, 2, \ldots, n, \]
where \(\lvert A_i \rvert\) denotes the number of elements of \(A_i\). Determine the smallest and the largest value that \(n\) can take.
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Slovenian High School Mathematics Competition for Vega Awards (MaSSA), drzavno (national) round 2025, 1. letnik, category A, problem B2. Organized by DMFA Slovenije (Society of Mathematicians, Physicists and Astronomers of Slovenia). Source