Question 157
Denote by $M$ the set of the first $2008$ positive integers. The numbers in $M$ are colored blue, yellow and red such that each number is of one color and each color is used at least once. Consider the following sets:
$$
S_1 = \{(x, y, z) \in M^3 \mid x, y, z \text{ are of the same color and } (x + y + z) \equiv 0 \pmod{2008}\};
$$
$$
S_2 = \{(x, y, z) \in M^3 \mid x, y, z \text{ are of the three colors and } (x + y + z) \equiv 0 \pmod{2008}\}.
$$
Prove that $2|S_1| \gt |S_2|$.
$$
S_1 = \{(x, y, z) \in M^3 \mid x, y, z \text{ are of the same color and } (x + y + z) \equiv 0 \pmod{2008}\};
$$
$$
S_2 = \{(x, y, z) \in M^3 \mid x, y, z \text{ are of the three colors and } (x + y + z) \equiv 0 \pmod{2008}\}.
$$
Prove that $2|S_1| \gt |S_2|$.