Take 6 points in 3D space in general position: no 3 in a line, no 3 forming a right angle, no 4 in a plane. Draw all 15 lines between them. This will form 20 distinct triangles. (For each of the six points there are ten pairs of other points, but the apparent 60 triangles are each counted three times.) Each triangle can have at most one obtuse angle, so the maximum number of obtuse angles in the whole figure, over all such point sets, is 20.
Conjecture: the minimum number of obtuse angles over all 6-point sets in 3 dimensions is 2.
This has been established experimentally to a high degree of probability by creating 28 million (so far) random 6-point sets and counting the obtuse angles in each set. I do not know how to prove it, and I see no way to find a counterexample except by trying additional millions of point sets, which are not likely to turn up any.
I think this is a new problem, and I have other similar conjectures.