China Team Selection Test

Find the smallest positive real constant $a$, such that for any three points $A, B, C$ on the unit circle, there exists an equilateral triangle $PQR$ with side length $a$ such that all of $A, B, C$ lie on the interior or boundary of $\triangle PQR$.

To determine the smallest such $a$, consider the following construction and proof:

1. Proof of Optimality:
- Consider a triangle $ABC$ inscribed in the unit circle with angles $\angle A = 20^\circ$ and $\angle B = \angle C = 80^\circ$.
- The smallest equilateral triangle $PQR$ containing $\triangle ABC$ must have side length $\frac{4}{\sqrt{3}} \sin^2 80^\circ$.

2. Proof of Sufficiency:
- For any triangle $ABC$ inscribed in the unit circle, we can always find an equilateral triangle $PQR$ with side length $\frac{4}{\sqrt{3}} \sin^2 80^\circ$ that contains $\triangle ABC$.
- This is shown by considering different cases based on the angles of $\triangle ABC$ and constructing appropriate equilateral triangles $PQR$ that contain $\triangle ABC$.

Therefore, the smallest positive real constant $a$ such that any three points $A, B, C$ on the unit circle can be enclosed by an equilateral triangle $PQR$ with side length $a$ is:

$$a = \frac{4}{\sqrt{3}} \sin^2 80^\circ.$$

The answer is: $\boxed{\frac{4}{\sqrt{3}} \sin^2 80^\circ}$.