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Part of 2024 Oral Moscow Geometry Olympiad

Problems(2)

Excenters, incenters are concyclic

Source: Oral Moscow Geometry Olympiad 2024, 8-9.2

9/3/2024
The bisector BLBL was drawn in the triangle ABCABC. Let the points I1I_1 and I2I_2 be centers of the circles inscribed in the triangles ABLABL and CBLCBL, and the points J1J_1 and J2J_2 be centers of the excircles of these triangles touching the side BLBL. Prove that the points I1I_1, I2I_2, J1J_1 and J2J_2 lie on the same circle.
geometryincenter
3D problem on Moscow geometry olympiad

Source: Oral Moscow Geometry Olympiad 2024, 10-11.2

9/3/2024
Petya drew a pentagon ABCDEABCDE on the plane. After that, Vasya marked all the points SS in a given half-space relative to the plane of the pentagon so that in the pyramid SABCDSABCD exactly two side faces are perpendicular to the plane of the base ABCDABCD, and the height is 11. How many points could have Vasya?
geometry3d