MathDB

2

Part of 2018 Iranian Geometry Olympiad

Problems(3)

2018 IGO Elementary Level P2

Source:

9/20/2018
Convex hexagon A1A2A3A4A5A6A_1A_2A_3A_4A_5A_6 lies in the interior of convex hexagon B1B2B3B4B5B6B_1B_2B_3B_4B_5B_6 such that A1A2∥B1B2A_1A_2 \parallel B_1B_2, A2A3∥B2B3A_2A_3 \parallel B_2B_3,..., A6A1∥B6B1A_6A_1 \parallel B_6B_1. Prove that the areas of simple hexagons A1B2A3B4A5B6A_1B_2A_3B_4A_5B_6 and B1A2B3A4B5A6B_1A_2B_3A_4B_5A_6 are equal. (A simple hexagon is a hexagon which does not intersect itself.)
Proposed by Hirad Aalipanah - Mahdi Etesamifard
IGO2018 igoIrangeometry
angle relations in a convex ABCD given, double segment wanted

Source: Iranian Geometry Olympiad 2018 IGO Intermediate p2

9/19/2018
In convex quadrilateral ABCDABCD, the diagonals ACAC and BDBD meet at the point PP. We know that ∠DAC=90o\angle DAC = 90^o and 2∠ADB=∠ACB2 \angle ADB = \angle ACB. If we have ∠DBC+2∠ADC=180o \angle DBC + 2 \angle ADC = 180^o prove that 2AP=BP2AP = BP.
Proposed by Iman Maghsoudi
geometryanglesright angle
equal segments staring with an acure triangle, <A=45^o, O,H

Source: Iranian Geometry Olympiad 2018 IGO Advanced p2

9/19/2018
In acute triangle ABC,∠A=45oABC, \angle A = 45^o. Points O,HO,H are the circumcenter and the orthocenter of ABCABC, respectively. DD is the foot of altitude from BB. Point XX is the midpoint of arc AHAH of the circumcircle of triangle ADHADH that contains DD. Prove that DX=DODX = DO.
Proposed by Fatemeh Sajadi
geometryequal segmentsacute triangleCircumcenterorthocenter