MathDB

3

Part of 2023 Bundeswettbewerb Mathematik

Problems(2)

2023 Bundeswettbewerb Mathematik

Source: Bundeswettbewerb Mathematik 2023 Round 1 P3

3/8/2023
Given two parallelograms ABCDABCD and AECFAECF with common diagonal ACAC, where EE and FF lie inside parallelogram ABCDABCD. Show: The circumcircles of the triangles AEBAEB, BFCBFC, CEDCED and DFADFA have one point in common.
bundeswettbewerbgeometry
Possible values of gamma when two areas are equal

Source: Bundeswettbewerb Mathematik 2023, Round 2 - Problem 3

9/8/2023
Let ABCABC be a triangle with incenter II. Let MbM_b and MaM_a be the midpoints of ACAC and BCBC, respectively. Let B′B' be the point of intersection of lines MbIM_bI and BCBC, and let A′A' be the point of intersection of lines MaIM_aI and ACAC.
If triangles ABCABC and A′B′CA'B'C have the same area, what are the possible values of ∠ACB\angle ACB?
geometrygeometry proposedanglesarea of a triangle