The following is regarding question $6$, part $b$, in the following link:
https://thol.sunway.edu.my/examdbase/alv/math/p3/math_p3_j96.pdf
Using the principle of moments and considering the case when $ω=0$ (when the system is at rest).
Let $m_1$ be the mass of the pilot, $m_2$ be the mass of the counterweight $B$ and the distance from $O$ to $B$ is $x$, we have
Principle of Moments:
anticlockwise moments = clockwise moments
$$ xm_2g = 10m_1g$$
As the system is balanced when the system rotates and the forces acting towards $O$ are balanced, the moments when the system is at rest must also be balanced.
Is this enough to answer the question without being given any masses or the distance $x$, where we could show quantitatively that the system is balanced when at rest?
When the system is in motion the forces acting towards $O$ are
$$xm_2\omega^2 = 10m_1\omega^2$$
As $\omega$ is equal for both masses, we can cancel $\omega$ and multiply both sides by $g$, which shows, in general, that the moments about $O$, of each weight, are equal when the system is at rest.