IMO Shortlist 2000 problem A6
Dodao/la:
arhiva2. travnja 2012. A nonempty set
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
of real numbers is called a
![B_3](/media/m/4/b/a/4ba38f08b828c32ed1e0c672a3bd9998.png)
-set if the conditions
![a_1, a_2, a_3, a_4, a_5, a_6 \in A](/media/m/b/6/3/b63479deda564ad33f0fd0b8e12d506e.png)
and
![a_1 + a_2 + a_3 = a_4 + a_5 + a_6](/media/m/c/c/f/ccf7801d52caa5e6a1389bba40162916.png)
imply that the sequences
![(a_1, a_2, a_3)](/media/m/4/4/f/44f3af500aa6b523cd09f5be21c2b3a7.png)
and
![(a_4, a_5, a_6)](/media/m/2/0/8/208c221539b3032c4e5ec59762af74ab.png)
are identical up to a permutation. Let
be infinite sequences of real numbers with
![D(A) = D(B),](/media/m/b/8/f/b8f73563f0019aad667332a7847e7e46.png)
where, for a set
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
of real numbers,
![D(X)](/media/m/2/1/0/2101bc1aa668180e8831fe46e63592f8.png)
denotes the difference set
![\{|x-y| | x, y \in X \}.](/media/m/5/d/c/5dcabcd2f6a28e13f7253ca655d1ebc6.png)
Prove that if
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
is a
![B_3](/media/m/4/b/a/4ba38f08b828c32ed1e0c672a3bd9998.png)
-set, then
%V0
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
$$A = \{a(0) = 0 < a(1) < a(2) < \ldots \}, B = \{b(0) = 0 < b(1) < b(2) < \ldots \}$$
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 $\{|x-y| | x, y \in X \}.$ Prove that if $A$ is a $B_3$-set, then $A = B.$
Izvor: Međunarodna matematička olimpijada, shortlist 2000