KGS math club/solution 11 14

Choose a point, consider the set of 4-sets of other points. These 4-sets define 4 triangles, exactly two or none of which contain the chosen point (check two cases - four points in a convex shape, four points not in a convex shape). Moreover each triangle is counted an odd number of times, as there is an odd number of points to be the fourth in the tuple (besides the chosen point plus the three forming the triangle), so the total count is an even number divided by an odd number.

solution by Warfreak2

Give the points random z co-ordinates, replace the chosen point with a line in the z-direction, and each 4-tuple with a tetrahedron. Then it's clear that the vertical line intersects either zero or two faces of each tetrahedron, and each face belongs to an odd number of tetrahedra.

solution by Warfreak2

Choose a point, move it out to infinity. Each time it passes a line joining two other points, the number of triangles it sits in changes by an even number (there are an even number of other points besides the one point being moved and the two forming that line). At infinity it sits in 0, so its original count must have been even.

solution reported by Warfreak2