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Show that there are elements a, b such that 11|a^3+ab^2+b^3

Source: Middle European Mathematical Olympiad 2011 - Team Compt. T-7

September 6, 2011

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Problem Statement

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