Question 27
1. Let $ABC$ be an acute-angled triangle with $AB \lt AC \lt BC$, inscribed in the circle $c(O,R)$. The circle $c_1$ with center $A$ and radius $AC$ intersects the circle $c(O,R)$ at point $D$ and the extension of the side $CB$ at $E$. The line $AE$ intersects the circle $c(O,R)$ at point $F$ and $G$ is the symmetric point of $E$ with respect to $B$. Prove that the quadrilateral $FEDG$ is cyclic.
2. We consider three lines of the plane passing through point $A$ and dividing the plane in 6 sectors. At the interior of each sector there exist 5 points. We suppose that no three of the 30 points existing in the sectors are collinear. Prove that there exist at least 1000 triangles with vertices from the points of the 6 sectors which contain point $A$ either on their interior or on their sides.
2. We consider three lines of the plane passing through point $A$ and dividing the plane in 6 sectors. At the interior of each sector there exist 5 points. We suppose that no three of the 30 points existing in the sectors are collinear. Prove that there exist at least 1000 triangles with vertices from the points of the 6 sectors which contain point $A$ either on their interior or on their sides.