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Suppose all the pairs of a positive integers from a finite collection

A=\{a_{1}, a_{2}, \cdots \}

are added together to form a new collection

A^{*}=\{a_{i}+a_{j}\;\; \vert \; 1 \le i < j \le n \}.

For example,

A=\{ 2, 3, 4, 7 \}

would yield

A^{*}=\{ 5, 6, 7, 9, 10, 11 \}

and

B=\{ 1, 4, 5, 6 \}

would give

B^{*}=\{ 5, 6, 7, 9, 10, 11 \}

. These examples show that it's possible for different collections

A

and

B

to generate the same collections

A^{*}

and

B^{*}

. Show that if

A^{*}=B^{*}

for different sets

A

and

B

, then

|A|=|B|

and

|A|=|B|

must be a power of

2

Problem Statement