MathDB

5

Part of 2021 Iranian Geometry Olympiad

Problems(3)

<A_1A_2A_3 +< A_2A_3A_4 +... + < A_{2021}A_1A_2 = 360^o

Source: Iranian Geometry Olympiad 2021 IGO Elementary p5

1/25/2022
Let A1,A2,...,A2021A_1, A_2, . . . , A_{2021} be 20212021 points on the plane, no three collinear and ∠A1A2A3+∠A2A3A4+...+∠A2021A1A2=360o,\angle A_1A_2A_3 + \angle A_2A_3A_4 +... + \angle A_{2021}A_1A_2 = 360^o, in which by the angle ∠Ai−1AiAi+1\angle A_{i-1}A_iA_{i+1} we mean the one which is less than 180o180^o (assume that A2022=A1A_{2022} =A_1 and A0=A2021A_0 = A_{2021}). Prove that some of these angles will add up to 90o90^o.
Proposed by Morteza Saghafian - Iran
geometryangles
Igo 2021 intermediate p5

Source: Intermediate p5

12/30/2021
Consider a convex pentagon ABCDEABCDE and a variable point XX on its side CDCD. Suppose that points K,LK, L lie on the segment AXAX such that AB=BKAB = BK and AE=ELAE = EL and that the circumcircles of triangles CXKCXK and DXLDXL intersect for the second time at YY . As XX varies, prove that all such lines XYXY pass through a fixed point, or they are all parallel. Proposed by Josef Tkadlec - Czech Republic
geometry
AD is Euler line of triangle IKL

Source: IGO 2021 Advanced P5

12/30/2021
Given a triangle ABCABC with incenter II. The incircle of triangle ABCABC is tangent to BCBC at DD. Let PP and QQ be points on the side BC such that ∠PAB=∠BCA\angle PAB = \angle BCA and ∠QAC=∠ABC\angle QAC = \angle ABC, respectively. Let KK and LL be the incenter of triangles ABPABP and ACQACQ, respectively. Prove that ADAD is the Euler line of triangle IKLIKL.
Proposed by Le Viet An, Vietnam
geometry