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Show that if $x\in S, y\in S, x\neq y,$ then $\frac{x+y}{2}\notin S$

Source: Moldova TST 2000

August 7, 2023

Problem Statement

Let SS{} be the set of nonnegative integers, which cointain only digits 00 and 11 in base 44 numeral system. a) Show that if x∈S,y∈S,x≠y,x\in S, y\in S, x\neq y, then x+y2∉S\frac{x+y}{2}\notin S. b) Let TT be a set of nonnegative integers such that S⊂T,T≠SS\subset T, T\neq S. Show that there exist x∈T,y∈T,x≠y,x\in T, y\in T, x\neq y, such that x+y2∈T\frac{x+y}{2} \in T.
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