Mongolian Mathematical Olympiad

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 $$\sum _{x\in X}{f}_i(x)=\sum _{x\in X}{g}_j(x)=S\text{and}\sum _{x\in X}{f}_i(x)g_j(x)=|i-j|$$ 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)