Two non-rolling circles $C_1$ and $C_2$ with centers $O_1$ and $O_2$ and radii $2R$ and $R$, respectively, are given on the plane. Find the locus of the centers of gravity of triangles in which one vertex lies on $C_1$ and the other two lie on $C_2$.

 (B. Frenkin)
1. **Define the circles and their properties:**
   - Circle \( C_1 \) has center \( O_1 \) at \((0,0)\) and radius \( 2R \).
   - Circle \( C_2 \) has center \( O_2 \) at \((a,0)\) and radius \( R \).

2. **Define the points on the circles:**
   - Let \( A \) be a point on \( C_1 \).
   - Let \( B \) and \( C \) be points on \( C_2 \).

3. **Define the centroid \( F \) of triangle \( \triangle ABC \):**
   - The centroid \( F \) of a triangle with vertices \( A(x_1, y_1) \), \( B(x_2, y_2) \), and \( C(x_3, y_3) \) is given by:
     \[
     F\left( \frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3} \right)
     \]

4. **Express the coordinates of \( A \), \( B \), and \( C \):**
   - Let \( A \) be \((2R \cos \theta, 2R \sin \theta)\) for some angle \(\theta\).
   - Let \( B \) be \((a + R \cos \phi, R \sin \phi)\) for some angle \(\phi\).
   - Let \( C \) be \((a + R \cos \psi, R \sin \psi)\) for some angle \(\psi\).

5. **Calculate the coordinates of the centroid \( F \):**
   \[
   F_x = \frac{2R \cos \theta + (a + R \cos \phi) + (a + R \cos \psi)}{3}
   \]
   \[
   F_y = \frac{2R \sin \theta + R \sin \phi + R \sin \psi}{3}
   \]

6. **Simplify the expressions for \( F_x \) and \( F_y \):**
   \[
   F_x = \frac{2R \cos \theta + a + R \cos \phi + a + R \cos \psi}{3} = \frac{2R \cos \theta + 2a + R (\cos \phi + \cos \psi)}{3}
   \]
   \[
   F_y = \frac{2R \sin \theta + R \sin \phi + R \sin \psi}{3} = \frac{2R \sin \theta + R (\sin \phi + \sin \psi)}{3}
   \]

7. **Analyze the locus of \( F \):**
   - The term \( \frac{2R \cos \theta}{3} \) varies as \( \theta \) varies, and it ranges from \(-\frac{2R}{3}\) to \(\frac{2R}{3}\).
   - The term \( \frac{2a}{3} \) is constant.
   - The term \( \frac{R (\cos \phi + \cos \psi)}{3} \) varies as \( \phi \) and \( \psi \) vary, and it ranges from \(-\frac{2R}{3}\) to \(\frac{2R}{3}\).

   Therefore, the \( x \)-coordinate of \( F \) ranges from:
   \[
   \frac{2a}{3} - \frac{4R}{3} \quad \text{to} \quad \frac{2a}{3} + \frac{4R}{3}
   \]

   Similarly, the \( y \)-coordinate of \( F \) ranges from:
   \[
   -\frac{4R}{3} \quad \text{to} \quad \frac{4R}{3}
   \]

8. **Conclude the locus of \( F \):**
   - The locus of \( F \) is a circle centered at \( \left( \frac{2a}{3}, 0 \right) \) with radius \( \frac{4R}{3} \).

The final answer is \(\boxed{\text{The locus of the centers of gravity of triangles is a circle centered at } \left( \frac{2a}{3}, 0 \right) \text{ with radius } \frac{4R}{3}.}\)