MathNet 2014 Question 29

Let $x_1, x_2, \dots, x_5$ be real numbers. Find the least positive integer $n$ with the following property: if there exist $n$ distinct sums of the form $x_p + x_q + x_r$ (with $1 \le p \lt q \lt r \le 5$) which are equal to $0$, then $x_1 = x_2 = \dots = x_5 = 0$.
Bulgaria, 2003