MathDB

1

Part of 2020 Iranian Geometry Olympiad

Problems(3)

2020 IGO Elementary P1

Source: 7th Iranian Geometry Olympiad (Elementary) P1

11/4/2020
By a fold of a polygon-shaped paper, we mean drawing a segment on the paper and folding the paper along that. Suppose that a paper with the following figure is given. We cut the paper along the boundary of the shaded region to get a polygon-shaped paper. Start with this shaded polygon and make a rectangle-shaped paper from it with at most 5 number of folds. Describe your solution by introducing the folding lines and drawing the shape after each fold on your solution sheet. (Note that the folding lines do not have to coincide with the grid lines of the shape.) Proposed by Mahdi Etesamifard
geometryIGO
2020 IGO Intermediate P1

Source: 7th Iranian Geometry Olympiad (Intermediate) P1

11/4/2020
A trapezoid ABCDABCD is given where ABAB and CDCD are parallel. Let MM be the midpoint of the segment ABAB. Point NN is located on the segment CDCD such that ∠ADN=12∠MNC\angle ADN = \frac{1}{2} \angle MNC and ∠BCN=12∠MND\angle BCN = \frac{1}{2} \angle MND. Prove that NN is the midpoint of the segment CDCD.
Proposed by Alireza Dadgarnia
trapezoidmidpointgeometryTriangleIGO
2020 IGO Advanced P1

Source: 7th Iranian Geometry Olympiad (Advanced) P1

11/4/2020
Let M,N,PM,N,P be midpoints of BC,ACBC,AC and ABAB of triangle △ABC\triangle ABC respectively. EE and FF are two points on the segment BC‾\overline{BC} so that ∠NEC=12∠AMB\angle NEC = \frac{1}{2} \angle AMB and ∠PFB=12∠AMC\angle PFB = \frac{1}{2} \angle AMC. Prove that AE=AFAE=AF. Proposed by Alireza Dadgarnia
geometryIGOiranian geometry olympiadmidpoints