Problem

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Tags: Subsets, TST, algebra



Let $p > 3$ be a given prime number. For a set $S \subseteq \mathbb{Z}$ and $a \in \mathbb{N}$ , define $S_a = \{ x \in \{ 0,1, 2,...,p-1 \}$ | $(\exists_s \in S) x \equiv_p a \cdot s \}$ . $(a)$ How many sets $S \subseteq \{ 1, 2,...,p-1 \} $ are there for which the sequence $S_1 , S_2 , ..., S_{p-1}$ contains exactly two distinct terms? $(b)$ Determine all numbers $k \in \mathbb{N}$ for which there is a set $ S \subseteq \{ 1, 2,...,p-1 \} $ such that the sequence $S_1 , S_2 , ..., S_{p-1} $ contains exactly $k$ distinct terms. Proposed by Milan Basic and Milos Milosavljevic