MathDB

3

Part of 2021 Durer Math Competition (First Round)

Problems(2)

can from any point of the AB section one see the whole museum?

Source: 2021 Dürer Math Competition Regional E3 https://artofproblemsolving.com/community/c2773609_2021_

1/7/2022
The floor plan of a contemporary art museum is a (not necessarily convex) polygon and its walls are solid. The security guard guarding the museum has two favourite spots (points AA and BB) because one can see the whole area of the museum standing at either point. Is it true that from any point of the ABAB section one can see the whole museum?
geometrypolygon
concyclic circumcenters wanted, starting with 2 ext. tangent circles

Source: 2021 Dürer Math Competition Regional E+3 https://artofproblemsolving.com/community/c2773609_2021_

1/7/2022
Let k1k_1 and k2k_2 be two circles that are externally tangent at point CC. We have a point AA on k1k_1 and a point BB on k2k_2 such that CC is an interior point of segment ABAB. Let k3k_3 be a circle that passes through points AA and BB and intersects circles k1k_1 and k2k_2 another time at points MM and NN respectively. Let k4k_4 be the circumscribed circle of triangle CMNCMN. Prove that the centres of circles k1,k2,k3k_1, k_2, k_3 and k4k_4 all lie on the same circle.
geometrycircumcircle