Several Olympiad problems deal with functions defined on certain sets of points. These problems are interesting in that they combine both geometrical and algebraic ideas.

Problem.

Let be an integer and a function defined on the set of points in the plane, with the property that for any regular n-gon

.

Prove that f is the zero function.

Proof:

Core Concept: 

In Euclidean geometry, the only motions permissible are rigid motions — translations, rotations, and reflections.

Solution:

Let A be an arbitrary point. Consider a regular n-gon . Let k be an integer, . A rotation with center A of angle sends the polygon to , where and is the image of for all

From the condition of the statement, we have

.

Observe that in the sum the number appears n times, therefore,

On the other hand, we have

since the polygons are all regular n-gons. From the two equalities above we deduce , hence, f is the zero function.

More later,

Nalin Pithwa