Question 104
Let $k \gt 1$ be a given positive integer, and let $a^{(1)} = (a_1^{(1)}, \dots, a_k^{(1)}), \dots, a^{(n)} = (a_1^{(n)}, \dots, a_k^{(n)})$ be $n$ distinct points in $k$-dimensional space. Define the *score* of $a^{(i)}$ as
$$
\prod_{j=1}^{k} \#\{i' \mid 1 \le i' \le n \text{ such that } \pi_j(a^{(i')}) = \pi_j(a^{(i)})\},
$$
where $\#S$ denotes the number of elements of the set $S$, and $\pi_j : \mathbb{R}^k \to \mathbb{R}^{k-1}$ is the projection mapping a point of $k$-dimensional space to the point obtained by deleting its $j$-th coordinate. Find the largest real number $t$ such that (for all $n$ and all such points) the $t$-th power mean of the scores of all the $a^{(i)}$ is $\le n$.
Note: The $t$-th power mean of positive real numbers $x_1, \dots, x_n$ is defined as
$$
\left( \frac{x_1^t + \dots + x_n^t}{n} \right)^{\frac{1}{t}} \quad (\text{if } t \neq 0); \qquad \sqrt[n]{x_1 \cdots x_n} \quad (\text{if } t = 0).
$$
$$
\prod_{j=1}^{k} \#\{i' \mid 1 \le i' \le n \text{ such that } \pi_j(a^{(i')}) = \pi_j(a^{(i)})\},
$$
where $\#S$ denotes the number of elements of the set $S$, and $\pi_j : \mathbb{R}^k \to \mathbb{R}^{k-1}$ is the projection mapping a point of $k$-dimensional space to the point obtained by deleting its $j$-th coordinate. Find the largest real number $t$ such that (for all $n$ and all such points) the $t$-th power mean of the scores of all the $a^{(i)}$ is $\le n$.
Note: The $t$-th power mean of positive real numbers $x_1, \dots, x_n$ is defined as
$$
\left( \frac{x_1^t + \dots + x_n^t}{n} \right)^{\frac{1}{t}} \quad (\text{if } t \neq 0); \qquad \sqrt[n]{x_1 \cdots x_n} \quad (\text{if } t = 0).
$$