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Part of 2013 Serbia Additional Team Selection Test

Problems(1)

Serbia additional TST 2013

Source:

5/22/2015
Let p>3p > 3 be a given prime number. For a set S⊆ZS \subseteq \mathbb{Z} and a∈Na \in \mathbb{N} , define Sa={x∈{0,1,2,...,p−1}S_a = \{ x \in \{ 0,1, 2,...,p-1 \} | (∃s∈S)x≡pa⋅s}(\exists_s \in S) x \equiv_p a \cdot s \} . (a)(a) How many sets S⊆{1,2,...,p−1}S \subseteq \{ 1, 2,...,p-1 \} are there for which the sequence S1,S2,...,Sp−1S_1 , S_2 , ..., S_{p-1} contains exactly two distinct terms? (b)(b) Determine all numbers k∈Nk \in \mathbb{N} for which there is a set S⊆{1,2,...,p−1} S \subseteq \{ 1, 2,...,p-1 \} such that the sequence S1,S2,...,Sp−1S_1 , S_2 , ..., S_{p-1} contains exactly kk distinct terms.
Proposed by Milan Basic and Milos Milosavljevic
SubsetsTSTalgebra