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(A,B,C) pairwise disjoint, union {1,2,....,n}, same sum of elemements div3

Source: Rioplatense Olympiad 2014 level 3 P1

September 6, 2018

Setsremaindernumber theorycombinatorics

Problem Statement

Let

n \ge 3

be a positive integer. Determine, in terms of

n

, how many triples of sets

(A,B,C)

satisfy the conditions:

\bullet

A, B

and

C

are pairwise disjoint , that is,

A \cap B = A \cap C= B \cap C= \emptyset

.

\bullet

A \cup B \cup C= \{ 1 , 2 , ... , n \}

.

\bullet

The sum of the elements of

A

, the sum of the elements of

B

and the sum of the elements of

C

leave the same remainder when divided by

3

.
Note: One or more of the sets may be empty.

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