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 5 of 5: Two witnessed pairs must overlap

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In plain words

Half the points cannot each start a fresh pair inside a smaller pool without two pairs colliding.

\{A,B\},\{A,C\}\ \text{witnessed by }P \Rightarrow PA=PB=PC

Detailed analysis

None of the $\frac n2$ pairs witnessed by $P$ can contain $P$ itself (that would force the pair's two points to coincide). So these $\frac n2$ pairs all lie among the remaining $n-1$ points of $V$ . If they were pairwise disjoint they would need $2\cdot\frac n2=n$ distinct points, but only $n-1$ are available; hence two of the pairs share a point, say $\{A,B\}$ and $\{A,C\}$ with $B\ne C$ . Then $PA=PB$ and $PA=PC$ , so $PA=PB=PC$ for the three distinct points $A,B,C$ , contradicting that $V$ is center-free. Therefore no balanced center-free set of even size exists, and the answer to (b) is exactly the odd integers $n\ge3$ .