OSN 2013 Problem 8

Let $A$ be a set of positive integers. $A$ is called "balanced" if [and only if] the number of 3-element subsets of $A$ whose elements add up to a multiple of $3$ is equal to the number of 3-element subsets of $A$ whose elements add up to not a multiple of $3$ .

a. Find a 9-element balanced set. b. Prove that no set of $2013$ elements can be balanced.