Question 147
Let $n\ge 3$ be a given integer. Determine the smallest positive integer $k$ for which there exist a set $A$ of $k$ real numbers and $n$ pairwise distinct real numbers $x_1,x_2,\ldots,x_n$ such that
$$
x_1+x_2,\quad x_2+x_3,\quad\ldots,\quad x_{n-1}+x_n,\quad x_n+x_1
$$
all belong to $A$.
$$
x_1+x_2,\quad x_2+x_3,\quad\ldots,\quad x_{n-1}+x_n,\quad x_n+x_1
$$
all belong to $A$.