5. Given a circle and two points $P$ and $Q$ inside it, inscribe a right triangle such that its legs pass through $P$ and $Q$. For which positions of $P$ and $Q$ does the problem have no solution?

SOLUTION:
Just draw the circle with diameter $P Q$, the eventual intersections with the initial circle centered at $O$ provide us with two points $A$ and $B$ which, when connected with $P$ and $Q$, define the solutions.

There is no solution when both circles do not intersect, that is when

$$
O M+\frac{P Q}{2}<r
$$

where $r$ is the radius of the given circle.