MathDB

6

Part of 2019 Brazil National Olympiad

Problems(2)

10 concurrent lines

Source: Brazil National Olympiad 2019 #6

11/14/2019
Let A1A2A3A4A5A_1A_2A_3A_4A_5 be a convex, cyclic pentagon with ∠Ai+∠Ai+1>180∘\angle A_i + \angle A_{i+1} >180^{\circ} for all i∈{1,2,3,4,5}i \in \{1,2,3,4,5\} (all indices modulo 55 in the problem). Define BiB_i as the intersection of lines Ai−1AiA_{i-1}A_i and Ai+1Ai+2A_{i+1}A_{i+2}, forming a star. The circumcircles of triangles Ai−1Bi−1AiA_{i-1}B_{i-1}A_i and AiBiAi+1A_iB_iA_{i+1} meet again at Ci≠AiC_i \neq A_i, and the circumcircles of triangles Bi−1AiBiB_{i-1}A_iB_i and BiAi+1Bi+1B_iA_{i+1}B_{i+1} meet again at Di≠BiD_i \neq B_i. Prove that the ten lines AiCi,BiDiA_iC_i, B_iD_i, i∈{1,2,3,4,5}i \in \{1,2,3,4,5\}, have a common point.
geometry
Points in Cartesian plane

Source: Brazil National Olympiad 2019 - level 2 - #6

11/23/2019
In the Cartesian plane, all points with both integer coordinates are painted blue. Blue colon they are said to be mutually visible if the line segment connecting them has no other blue dots. Prove that There is a set of 2019 2019 blue dots that are mutually visible two by two.
Brazilian Math Olympiad