# A geometry and algebra problem — RMO training

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 $n>2$ be an integer and $f:P \rightarrow \Re$ a function defined on the set of points in the plane, with the property that for any regular n-gon $A_{1}A_{2} \ldots A_{n}$

$f(A_{1})+f(A_{2})+ \ldots + f(A_{n})=0$.

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 $AA_{1}A_{2}\ldots A_{n-1}$. Let k be an integer, $0 \leq k \leq n-1$. A rotation with center A of angle $\frac{2\pi k }{n}$ sends the polygon $AA_{1}A_{2}\ldots A_{n-1}$ to $A_{k0}A_{k1} \ldots A_{k,n-1}$, where $A_{k0}=A$ and $A_{ki}$ is the image of $A_{i}$ for all $I=1, 2, \ldots, n-1$

From the condition of the statement, we have

$\sum_{k=0}^{n-1} \sum_{i=0}^{n-1}f(A_{ki})=0$.

Observe that in the sum the number $f(A)$ appears n times, therefore,

$nf(A)+ \sum_{k=0}^{n-1} \sum_{i=1}^{n-1}f(A_{kl})=0$

On the other hand, we have

$\sum_{k=0}^{n-1} \sum_{i=1}^{n-1}f(A_{ki})=\sum_{i=1}^{n-1} \sum_{k=0}^{n-1}f(A_{ki})=0$

since the polygons $A_{0i}A_{1i} \ldots A_{n-1,i}$ are all regular n-gons. From the two equalities above we deduce $f(A)=0$, hence, f is the zero function.

More later,

Nalin Pithwa

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