Question 108
Fix an integer $n \ge 2$. Determine the least possible value the sum
$$
\left\lfloor \frac{x_2 + x_3 + \cdots + x_n}{x_1} \right\rfloor + \left\lfloor \frac{x_1 + x_3 + \cdots + x_n}{x_2} \right\rfloor + \cdots + \left\lfloor \frac{x_1 + x_2 + \cdots + x_{n-1}}{x_n} \right\rfloor
$$
may achieve, as $x_1, x_2, \dots, x_n$ run through all positive real numbers.
$$
\left\lfloor \frac{x_2 + x_3 + \cdots + x_n}{x_1} \right\rfloor + \left\lfloor \frac{x_1 + x_3 + \cdots + x_n}{x_2} \right\rfloor + \cdots + \left\lfloor \frac{x_1 + x_2 + \cdots + x_{n-1}}{x_n} \right\rfloor
$$
may achieve, as $x_1, x_2, \dots, x_n$ run through all positive real numbers.