Introduction:
In this problem, we will show that if a particle describes a circular orbit under the influence of an attractive central force directed towards a point on the circle, then the force varies as the inverse fifth power of the distance.

Derivation:
To begin, let's consider a particle moving in a circular orbit of radius 'r' under the influence of an attractive central force directed towards a point on the circle. Let's assume this central force is given by F.

Centripetal Force:
The centripetal force required to keep the particle in a circular orbit is given by the equation F_c = m(v^2 / r), where m is the mass of the particle and v is its velocity. This force acts towards the center of the circle.

Central Force:
The central force F acts in the opposite direction of the centripetal force and is responsible for keeping the particle in a circular orbit. Since the central force is directed towards a point on the circle, it will have two components - one along the radius of the circle (F_r) and one perpendicular to the radius (F_θ).

Equilibrium Condition:
For the particle to be in a circular orbit, the net force acting on it must be zero. This gives us the equation F_c + F_r = 0.
Since F_c = m(v^2 / r), we can rewrite the equation as m(v^2 / r) + F_r = 0.

Force Variation:
Now, let's analyze the force variation. Since the particle is in equilibrium, the velocity v is constant. Therefore, the only way for the force to vary is through the radial component F_r.

Force Variation with Distance:
Let's assume that the force F_r varies with the distance r as a power law, i.e., F_r = k / r^n, where k is a constant and n is the power law exponent. Substituting this into the equation m(v^2 / r) + F_r = 0, we get m(v^2 / r) + k / r^n = 0.

Equating Exponents:
To determine the value of n, we equate the exponents on both sides of the equation. The left-hand side has an exponent of -1 (from r in the denominator), and the right-hand side has an exponent of -n (from the power law). Therefore, we have -1 = -n.

Force Variation as Inverse Fifth Power:
Solving the equation, we find that n = 1, which means the force F_r varies as the inverse of the distance, i.e., F_r = k / r^5.
Therefore, the force F acting on the particle varies as the inverse fifth power of the distance, in accordance with the given conditions.

Conclusion:
In conclusion, we have shown that if a particle describes a circular orbit under the influence of an attractive central force directed towards a point on the circle, then the force varies as the inverse fifth power of the distance. This result is derived by analyzing the equilibrium condition and the force components acting on the particle.