MathDB

2

Part of 2019 Iranian Geometry Olympiad

Problems(3)

2019 IGO Elementary P2

Source: 6th Iranian Geometry Olympiad (Elementary) P2

9/20/2019
As shown in the figure, there are two rectangles ABCDABCD and PQRDPQRD with the same area, and with parallel corresponding edges. Let points N,N, MM and TT be the midpoints of segments QR,QR, PCPC and ABAB, respectively. Prove that points N,MN,M and TT lie on the same line.
http://s4.picofile.com/file/8372959484/E02.png
Proposed by Morteza Saghafian
IGOIrangeometry
2019 IGO Intermediate P2

Source: 6th Iranian Geometry Olympiad (Intermediate) P2

9/20/2019
Find all quadrilaterals ABCDABCD such that all four triangles DABDAB, CDACDA, BCDBCD and ABCABC are similar to one-another.
Proposed by Morteza Saghafian
IGOIrangeometry
2019 IGO Advanced P2

Source: 6th Iranian Geometry Olympiad (Advanced) P2

9/20/2019
Is it true that in any convex nn-gon with n>3n > 3, there exists a vertex and a diagonal passing through this vertex such that the angles of this diagonal with both sides adjacent to this vertex are acute?
Proposed by Boris Frenkin - Russia
IGOIrangeometrycombinatorics