MEMO 2011 ekipno problem 7
Dodao/la:
arhiva28. travnja 2012. Let
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
and
![B](/media/m/c/e/e/ceebc05be717fa6aab8e71b02fe3e4e3.png)
be disjoint nonempty sets with
![A \cup B = \{1, 2,3, \ldots, 10\}](/media/m/9/9/0/990c024d8743eec88778f3e8c745f1c1.png)
. Show that there exist elements
![a \in A](/media/m/9/a/a/9aa30d597018c8ce95d2627d51aa9c61.png)
and
![b \in B](/media/m/e/7/f/e7f33ea34443e6a11b6dd8c6485d8751.png)
such that the number
![a^3 + ab^2 + b^3](/media/m/5/d/1/5d1d902eff286cf1078fe2459eba6ced.png)
is divisible by
![11](/media/m/0/d/2/0d2d0ab9a023da1d30a2ddc91cbc38db.png)
.
%V0
Let $A$ and $B$ be disjoint nonempty sets with $A \cup B = \{1, 2,3, \ldots, 10\}$. Show that there exist elements $a \in A$ and $b \in B$ such that the number $a^3 + ab^2 + b^3$ is divisible by $11$.
Izvor: Srednjoeuropska matematička olimpijada 2011, ekipno natjecanje, problem 7