Question 50
Determine the smallest integer $n$, for which there exists integers $x_1, x_2, \dots, x_n$ and positive integers $a_1, a_2, \dots, a_n$ so that:
$$
x_1 + x_2 + \dots + x_n = 0, \quad a_1x_1 + a_2x_2 + \dots + a_nx_n \gt 0, \quad a_1^2x_1 + a_2^2x_2 + \dots + a_n^2x_n \lt 0.
$$
$$
x_1 + x_2 + \dots + x_n = 0, \quad a_1x_1 + a_2x_2 + \dots + a_nx_n \gt 0, \quad a_1^2x_1 + a_2^2x_2 + \dots + a_n^2x_n \lt 0.
$$