Problem 1

We say that a finite set

S

of points in the plane is balanced if, for any two different points

A

and

B

S

, there is a point

C

S

such that

AC=BC

. We say that

S

is center-free if for any three different points

A

B

C

S

, there is no point

P

S

such that

PA=PB=PC

. (a) Show that for all integers

n\ge3

, there exists a balanced set consisting of

n

points. (b) Determine all integers

n\ge3

for which there exists a balanced center-free set consisting of

n

points.

Step 3 of 5: The regular

n

-gon is balanced, and center-free exactly when it has no center

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In plain words

On a regular polygon, the perpendicular bisector of any side or diagonal always passes through another vertex when $n$ is odd.

2k\equiv i+j \pmod n \Rightarrow A_iA_k=A_jA_k

Detailed analysis

For odd $n$ , label the vertices of a regular $n$ -gon $A_1,\dots,A_n$ . For any $i\ne j$ , since $\gcd(2,n)=1$ there is a unique $k\in\{1,\dots,n\}$ with $2k\equiv i+j\pmod n$ , and then $k-i\equiv j-k\pmod n$ places $A_k$ on the perpendicular bisector of $A_iA_j$ , i.e. $A_iA_k=A_jA_k$ ; so the regular $n$ -gon is balanced. It is also center-free: if some $P$ satisfied $PA=PB=PC$ for three distinct vertices $A,B,C$ , then $P$ would have to be the circumcenter of the polygon, which is not one of the vertices $A_1,\dots,A_n$ .