MathDB

1

Part of 2022 Iranian Geometry Olympiad

Problems(3)

Angles of Pentagon

Source: IGO 2022 Elementary P1

12/14/2022
Find the angles of the pentagon ABCDEABCDE in the figure below.
geometrypentagoniranian geometry olympiad
Equal segments and angles

Source: IGO 2022 Intermediate P1

12/13/2022
Given is a circle ω\omega and a line ℓ\ell tangent to ω\omega at YY. Point XX lies on ℓ\ell to the left of YY. The tangent to ω\omega, perpendicular to ℓ\ell meets ℓ\ell at AA and touches ω\omega at DD. Let BB a point on ℓ\ell, to the right of YY, such that AX=BYAX=BY. The tangent from BB to ω\omega touches the circle at CC. Prove that ∠XDA=∠YDC\angle XDA= \angle YDC.
Note: This is not the official wording (it was just a diagram without any description).
geometry
IGO 2022 advanced/free P1

Source: Iranian Geometry Olympiad 2022 P1 Advanced, Free

12/13/2022
Four points AA, BB, CC and DD lie on a circle ω\omega such that AB=BC=CDAB=BC=CD. The tangent line to ω\omega at point CC intersects the tangent line to ω\omega at AA and the line ADAD at KK and LL. The circle ω\omega and the circumcircle of triangle KLAKLA intersect again at MM. Prove that MA=MLMA=ML.
Proposed by Mahdi Etesamifard
geometry