On a generalization of the plank problem

Let two concentric circles $k$ and $K$ be given on the plane with radii $r$ and $R$, $r<R$. Assume that the disc bounded by $k$ is covered by angular domains whose vertices are within $K$. Then the sum of the angles of these angular domains is greater than or equal to the view angle of $k$ from an arbitrary point of $K$.

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