Question 92
$M = \{1, \dots, 2010\}$ included blue, yellow, and red, and each number was colored in only one color.
$$
S_1 = \{(x, y, z) \in M^3 \mid x, y, z \text{ are of the same color and } x + y + z \equiv 0 \pmod{2010}\}
$$
$$
S_2 = \{(x, y, z) \in M^3 \mid x, y, z \text{ is a different color and } x + y + z \equiv 0 \pmod{2010}\}
$$
confirm $2|S_1| \gt |S_2|$.
$$
S_1 = \{(x, y, z) \in M^3 \mid x, y, z \text{ are of the same color and } x + y + z \equiv 0 \pmod{2010}\}
$$
$$
S_2 = \{(x, y, z) \in M^3 \mid x, y, z \text{ is a different color and } x + y + z \equiv 0 \pmod{2010}\}
$$
confirm $2|S_1| \gt |S_2|$.