Problem

Source: Indian Postal Coaching 2008 set 6 p6

Tags: combinatorics, number theory, remainder



Consider the set $A = \{1, 2, 3, ..., 2008\}$. We say that a set is of type $r, r \in \{0, 1, 2\}$, if that set is a nonempty subset of $A$ and the sum of its elements gives the remainder $r$ when divided by $3$. Denote by $X_r, r \in \{0, 1, 2\}$ the class of sets of type $r$. Determine which of the classes $X_r, r \in \{0, 1, 2\}$, is the largest.