IMO 1987 SL 9

Origin: HUN

Problem

Does there exist a set M in usual Euclidean space such that for every plane \lambda the intersection M \cap\lambda is finite and nonempty?

Solution

The answer is yes. Consider the curve C = {(x, y, z) | x = t, y = t3, z = t5, t \inR}. Any plane defined by an equation of the form ax+by+cz+d = 0 intersects the curve C at points (t, t3, t5) with t satisfying ct5 + bt3 + at + d = 0. This last equation has at least one but only finitely many solutions.