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 2 of 5: Reaching every even size

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In plain words

One more shared vertex lets a triangle "hinge" onto the construction without breaking balance.

\triangle OCD,\ \triangle ODE \text{ equilateral} \Rightarrow V\cup\{C,D,E\}\text{ balanced}

Detailed analysis

Take any balanced set $V$ from the previous step (possibly $k=0$ , i.e. $V=\{O\}$ ) together with three further points $C,D,E$ on the same circle such that $\triangle OCD$ and $\triangle ODE$ are both equilateral. As before $OC=OD=OE=r$ , so $O$ serves as the associate for any pair among $\{C,D,E\}$ or between them and the earlier spoke points, and $D$ serves the pairs $(O,C)$ and $(O,E)$ while $C$ (resp. $E$ ) serves $(O,D)$ . Adding these three points to $V$ preserves balance and increases the size by $3$ , so $|V\cup\{C,D,E\}|=2k+4$ ranges over every even number $\ge4$ . Together with the odd sizes, part (a) is proved for all $n\ge3$ .