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Print34th Hellenic Mathematical Olympiad
Greece geometry
Problem
1. Let be an acute-angled triangle with , inscribed in the circle . The circle with center and radius intersects the circle at point and the extension of the side at . The line intersects the circle at point and is the symmetric point of with respect to . Prove that the quadrilateral is cyclic.
2. We consider three lines of the plane passing through point 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 either on their interior or on their sides.
2. We consider three lines of the plane passing through point 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 either on their interior or on their sides.
Solution
Since the quadrilateral is inscribed in the circle , we have: . Since triangle is isosceles we have . Therefore , and hence the triangle is isosceles and hence
Figure 2
We put . Then from the circle we get , and hence Moreover from the circle we have: From (2) and (3) we find , which means that is bisector of the isosceles triangle . Hence it is perpendicular bisector of , and From (1) and (4), and from the equality , we conclude that , and hence the quadrilateral is inscribed in a circle with center .
Figure 2
We put . Then from the circle we get , and hence Moreover from the circle we have: From (2) and (3) we find , which means that is bisector of the isosceles triangle . Hence it is perpendicular bisector of , and From (1) and (4), and from the equality , we conclude that , and hence the quadrilateral is inscribed in a circle with center .
Techniques
Cyclic quadrilateralsAngle chasingDistance chasingInclusion-exclusionCounting two ways