Question 37
Let $n$ be a fixed positive integer. Let $X$ be a finite set and let $f_1, f_2, \dots, f_n$ and $g_1, g_2, \dots, g_n: X \to [0, 1]$ be functions satisfying
$$
\sum_{x \in X} f_i(x) = \sum_{x \in X} g_j(x) = S \quad \text{and} \quad \sum_{x \in X} f_i(x)g_j(x) = |i - j|
$$
for all $1 \le i, j \le n$. Here $[0, 1] = \{0 \le t \le 1\}$ is the unit interval.
(1) Prove that $S \ge n - 1$.
(2) For $S = n - 1$, find an example of a finite set $X$ and functions $f_1, f_2, \dots, f_n$ and $g_1, g_2, \dots, g_n: X \to [0, 1]$ satisfying the condition of the problem.
(Otgonbayar Uuye)
$$
\sum_{x \in X} f_i(x) = \sum_{x \in X} g_j(x) = S \quad \text{and} \quad \sum_{x \in X} f_i(x)g_j(x) = |i - j|
$$
for all $1 \le i, j \le n$. Here $[0, 1] = \{0 \le t \le 1\}$ is the unit interval.
(1) Prove that $S \ge n - 1$.
(2) For $S = n - 1$, find an example of a finite set $X$ and functions $f_1, f_2, \dots, f_n$ and $g_1, g_2, \dots, g_n: X \to [0, 1]$ satisfying the condition of the problem.
(Otgonbayar Uuye)