IMO Shortlist 2012 problem A2
Dodao/la:
arhiva3. studenoga 2013. Let
![\mathbb{Z}](/media/m/7/e/7/7e7a66cf43dd596531b0d3b302075071.png)
and
![\mathbb{Q}](/media/m/3/9/e/39ef065be8e709ef772a872dcb00dc83.png)
be the sets of integers and rationals respectively.
a) Does there exist a partition of
![\mathbb{Z}](/media/m/7/e/7/7e7a66cf43dd596531b0d3b302075071.png)
into three non-empty subsets
![A,B,C](/media/m/6/0/1/6012c28979f41c54e9b40b9fc855aa34.png)
such that the sets
![A+B, B+C, C+A](/media/m/f/5/9/f596b3f8a54cd9428f4834ca22ecf417.png)
are disjoint?
b) Does there exist a partition of
![\mathbb{Q}](/media/m/3/9/e/39ef065be8e709ef772a872dcb00dc83.png)
into three non-empty subsets
![A,B,C](/media/m/6/0/1/6012c28979f41c54e9b40b9fc855aa34.png)
such that the sets
![A+B, B+C, C+A](/media/m/f/5/9/f596b3f8a54cd9428f4834ca22ecf417.png)
are disjoint?
Here
![X+Y](/media/m/d/8/8/d8853b24683d0936bb8454c7b9b432a8.png)
denotes the set
![\{ x+y : x \in X, y \in Y \}](/media/m/f/d/b/fdb20180aa858a472e075a873acc46da.png)
, for
![X,Y \subseteq \mathbb{Z}](/media/m/e/f/0/ef0c91b153d0a5dd05ae31fa93e6b6e8.png)
and for
![X,Y \subseteq \mathbb{Q}](/media/m/e/b/3/eb3891bb8d55795c03466f564b843057.png)
.
%V0
Let $\mathbb{Z}$ and $\mathbb{Q}$ be the sets of integers and rationals respectively.
a) Does there exist a partition of $\mathbb{Z}$ into three non-empty subsets $A,B,C$ such that the sets $A+B, B+C, C+A$ are disjoint?
b) Does there exist a partition of $\mathbb{Q}$ into three non-empty subsets $A,B,C$ such that the sets $A+B, B+C, C+A$ are disjoint?
Here $X+Y$ denotes the set $\{ x+y : x \in X, y \in Y \}$, for $X,Y \subseteq \mathbb{Z}$ and for $X,Y \subseteq \mathbb{Q}$.
Izvor: Međunarodna matematička olimpijada, shortlist 2012