Problem:
A cube with side length is suspended above a plane. The vertex closest to the plane is labeled . The three vertices adjacent to vertex are at heights and above the plane. The distance from vertex to the plane can be expressed as , where , and are positive integers. Find .
Solution:
Let the vertices of heights and above the plane be labeled , , and , respectively. Note that these vertices are the vertices of an equilateral triangle with side length . Set up a coordinate system where the given plane is the graph of is at is at , and is at . Suppose vertex is at . Because is a distance from vertex , it follows that . Because the vectors from to to , and to are mutually perpendicular, their dot products are all zero, or 0 and . Thus and . These equations yield and . Substituting and into and simplifying yields the quadratic equation . This equation has solutions . Because the plane is at and vertex lies below the plane, it follows that vertex is at a height . The requested sum is therefore .
Note that the vertex is at a height of about above the plane.
The problems on this page are the property of the MAA's American Mathematics Competitions