Question 140
Let $Q_n$ be the set of $n$-tuples $x = (x_1, ..., x_n)$, $x_i \in \{0, 1, 2\}$, $i = 1, 2, ..., n$. A triad $(x, y, z)$, where $x = (x_1, x_2, ..., x_n)$, $y = (y_1, y_2, ..., y_n)$, $z = (z_1, z_2, ..., z_n)$, of distinct elements of $Q_n$ is called good if there exists at least one $i \in \{1, 2, ..., n\}$ for which the sets $\{x_i, y_i, z_i\}$ and $\{0, 1, 2\}$ are equal. A subset $A$ of $Q_n$ is called good if every three elements of $A$ form a good triad. Prove that every good subset of $Q_n$ has at most $2\left(\frac{3}{2}\right)^n$ elements.