Although it is known the solution to the first two questions, somebody may have different nice answers, so I include them:
Given two circles in the plane, there is (at least) a line which is tangent to both of them.
Given three spheres in the space, there is a plane which is tangent to all of them.
In general, given $n$ n-spheres in the n-dimensional space, is there a hyperplane which is tangent to all of them?
What other generalizations does this problem admit?
EDIT: As @Noam kindly remarked below, the existence of the tangent objects is not always true. I think that in #2 the hypothesis must be: one of the spheres not in the cone determined by the other two. In #1 the "cone" determined by one circle is the circle itself. So in #3 we need a suitable definition for the "cone" determined by $n-1$ n-spheres.