Analysis of the Situation:
The boy is pedaling around a circle of 20 meters radius at a speed of 20 m/s. The combined mass of the boy and the cycle must make an angle with the vertical in order to prevent them from falling. We need to determine this angle.

Calculating the Centripetal Force:
The centripetal force required to keep the boy and the cycle moving in a circle is provided by the friction between the wheels and the ground. This force is given by the formula:
\[ F_c = \frac{mv^2}{r} \]
Where:
- \( F_c \) is the centripetal force
- \( m \) is the mass of the boy and the cycle
- \( v \) is the speed of the boy and the cycle
- \( r \) is the radius of the circle

Calculating the Force Making an Angle with the Vertical:
The combined mass of the boy and the cycle must make an angle with the vertical in order to balance the forces. This angle can be calculated using trigonometry. The force making an angle with the vertical is given by:
\[ F_{vertical} = F_c \cdot \cos(\theta) \]
Where:
- \( F_{vertical} \) is the force making an angle with the vertical
- \( \theta \) is the angle made with the vertical

Equating the Forces:
In order to prevent the boy and the cycle from falling, the force making an angle with the vertical must be equal to the gravitational force acting on them. Therefore, we can set the force making an angle with the vertical equal to the weight of the boy and the cycle:
\[ F_{vertical} = mg \]

Solving for the Angle:
By equating the two expressions for the force making an angle with the vertical, we can solve for the angle \( \theta \). Once we have the angle, we can determine the combined mass of the boy and the cycle that allows them to stay balanced while pedaling around the circle.