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Arithmetic progression in sets

Source: CWMI 2015 Q6

August 20, 2015
combinatoricsarithmetic sequence

Problem Statement

For a sequence a1,a2,...,ama_1,a_2,...,a_m of real numbers, define the following sets A={ai∣1≤i≤m} and B={ai+2aj∣1≤i,j≤m,i≠j}A=\{a_i | 1\leq i\leq m\}\ \text{and} \ B=\{a_i+2a_j | 1\leq i,j\leq m, i\neq j\} Let nn be a given integer, and n>2n>2. For any strictly increasing arithmetic sequence of positive integers, determine, with proof, the minimum number of elements of set A△BA\triangle B, where A△BA\triangle B =(A∪B)∖(A∩B).= \left(A\cup B\right) \setminus \left(A\cap B\right).
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