Question 6
Let $M=\{1,2,\ldots,2008\}$. Color every element blue, yellow, or red, with exactly one color per element and each color used at least once. Define
$$S_1=\{(x,y,z)\in M^3:x,y,z\text{ have the same color},\ x+y+z\equiv0\pmod{2008}\},$$
$$S_2=\{(x,y,z)\in M^3:x,y,z\text{ have pairwise different colors},\ x+y+z\equiv0\pmod{2008}\}.$$
Prove that $2|S_1|>|S_2|$. Here $M^3=M\times M\times M$.
$$S_1=\{(x,y,z)\in M^3:x,y,z\text{ have the same color},\ x+y+z\equiv0\pmod{2008}\},$$
$$S_2=\{(x,y,z)\in M^3:x,y,z\text{ have pairwise different colors},\ x+y+z\equiv0\pmod{2008}\}.$$
Prove that $2|S_1|>|S_2|$. Here $M^3=M\times M\times M$.