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