MathDB

3

Part of 2024 Bundeswettbewerb Mathematik

Problems(2)

Circumcircles in a parallelogram

Source: Bundeswettbewerb Mathematik 2024, Round 1 - Problem 3

3/8/2024
Let ABCDABCD be a parallelogram whose diagonals intersect in MM. Suppose that the circumcircle of ABMABM intersects the segment ADAD in a point E≠AE \ne A and that the circumcircle of EMDEMD intersects the segment BEBE in a point F≠EF \ne E. Show that ∠ACB=∠DCF\angle ACB=\angle DCF.
geometrycircumcirclegeometry proposedAngle Chasingparallelogram
Bound sum of ratios of areas in triangle

Source: Bundeswettbewerb Mathematik 2024, Round 2 - Problem 3

9/13/2024
Let ABCABC be a triangle. For a point PP in its interior, we draw the threee lines through PP parallel to the sides of the triangle. This partitions ABCABC in three triangles and three quadrilaterals.
Let VAV_A be the area of the quadrilateral which has AA as one vertex. Let DAD_A be the area of the triangle which has a part of BCBC as one of its sides. Define VB,DBV_B, D_B and VC,DCV_C, D_C similarly.
Determine all possible values of DAVA+DBVB+DCVC\frac{D_A}{V_A}+\frac{D_B}{V_B}+\frac{D_C}{V_C}, as PP varies in the interior of the triangle.
ratiogeometrygeometry proposedarea of a triangleinequalities