Problem 748

Any 3 points determine a triangle. It is not possible for 3 points to lie on the same line, because then there would be two lines forming a $0^{\circ}$ angle, which contradicts the fact that the smallest enclosed angle is $60^{\circ}$. Each interior angle of the triangles is also the angle between some pair of lines, and thus they cannot be less than $60^{\circ}$. This can only happen if any selected triangle is equilateral. However, in space, at most 4 points can be arranged such that the distance between any two of them is the same, meaning that any three of them form an equilateral triangle, so $n$ can be at most 4.

For 3 or 4 points, such an arrangement exists if the points are the vertices of an equilateral triangle or a tetrahedron. In this case, intersecting lines form a $60^{\circ}$ angle, while non-intersecting lines are perpendicular to each other. Therefore, $n$ can be either 3 or 4.