Consider a finite set $\mathcal{S}$ of points in the plane. The problem involves two specific definitions: a set is balanced if, for any two different points $A$ and $B$ in $\mathcal{S}$, there is a point $C$ in $\mathcal{S}$ such that $AC = BC$. Additionally, the set is centre-free if for any three different points $A$, $B$, and $C$ in $\mathcal{S}$, there is no point $P$ in $\mathcal{S}$ such that $PA = PB = PC$.

### Part (a)

To show that for all integers $n \geq 3$, there exists a balanced set consisting of $n$ points, consider placing the points equally spaced on a circle. This configuration is symmetrical, and for any pair of points $A$ and $B$, the perpendicular bisector of the segment $AB$ will contain a point $C$ on the circle such that $AC = BC$. This holds for any pair of points on the circle.

1. Place $n$ points on a circle such that each point is evenly spaced.
2. For any two points $A$ and $B$, there exists a point $C$ on the circle because of the symmetrical nature of the circle (specifically, the point diametrically opposite to the midpoint of arc $AB$).

Thus, this configuration ensures that the set is balanced for any number of points $n \geq 3$.

### Part (b)

To determine all integers $n \geq 3$ for which there exists a balanced centre-free set consisting of $n$ points, analyze the geometric properties of balanced configurations.

1. Consider the property of being centre-free: if any three points $A$, $B$, and $C$ are considered, no fourth point $P$ can exist such that $PA = PB = PC$.
2. In a balanced configuration using an even number $n$ of points, symmetry can lead to points $P$ equidistant from any three others, which violates the centre-free condition.

By the above consideration, a balanced centre-free set cannot exist if $n$ is even, because having evenly spaced points on a circle for an even $n$ results in symmetry that allows for a central point equidistant from multiple others.

Thus, a balanced centre-free set consisting of $n$ points is only possible for:

$$\boxed{\text{All odd integers } n \geq 3}$$