Question 12
A6 (IRE). A nonempty set $A$ of real numbers is called a $B_3$-set if the conditions $a_1,a_2,a_3,a_4,a_5,a_6\in A$ and $a_1+a_2+a_3=a_4+a_5+a_6$ imply that the sequences $(a_1,a_2,a_3)$ and $(a_4,a_5,a_6)$ are identical up to a permutation. Let $A=\{a_0=0\lt a_1\lt a_2\lt \cdots\}$, $B=\{b_0=0\lt b_1\lt b_2\lt \cdots\}$ be infinite sequences of real numbers with $D(A)=D(B)$, where, for a set $X$ of real numbers, $D(X)$ denotes the difference set $\{|x-y|\mid x,y\in X\}$. Prove that if $A$ is a $B_3$-set, then $A=B$.