MathDB

4

Part of 2019 Iranian Geometry Olympiad

Problems(3)

2019 IGO Elementary P4

Source: 6th Iranian Geometry Olympiad (Elementary) P4

9/20/2019
Quadrilateral ABCDABCD is given such that ∠DAC=∠CAB=60∘,\angle DAC = \angle CAB = 60^\circ, and AB=BD−AC.AB = BD - AC. Lines ABAB and CDCD intersect each other at point EE. Prove that ∠ADB=2∠BEC. \angle ADB = 2\angle BEC.
Proposed by Iman Maghsoudi
IGOIrangeometry
2019 IGO Intermediate P4

Source: 6th Iranian Geometry Olympiad (Intermediate) P4

9/20/2019
Let ABCDABCD be a parallelogram and let KK be a point on line ADAD such that BK=ABBK=AB. Suppose that PP is an arbitrary point on ABAB, and the perpendicular bisector of PCPC intersects the circumcircle of triangle APDAPD at points XX, YY. Prove that the circumcircle of triangle ABKABK passes through the orthocenter of triangle AXYAXY.
Proposed by Iman Maghsoudi
IGOIrangeometry
2019 IGO Advanced P4

Source: 6th Iranian Geometry Olympiad (Advanced) P4

9/20/2019
Given an acute non-isosceles triangle ABCABC with circumcircle Γ\Gamma. MM is the midpoint of segment BCBC and NN is the midpoint of arc BCBC of Γ\Gamma (the one that doesn't contain AA). XX and YY are points on Γ\Gamma such that BX∥CY∥AMBX\parallel CY\parallel AM. Assume there exists point ZZ on segment BCBC such that circumcircle of triangle XYZXYZ is tangent to BCBC. Let ω\omega be the circumcircle of triangle ZMNZMN. Line AMAM meets ω\omega for the second time at PP. Let KK be a point on ω\omega such that KN∥AMKN\parallel AM, ωb\omega_b be a circle that passes through BB, XX and tangents to BCBC and ωc\omega_c be a circle that passes through CC, YY and tangents to BCBC. Prove that circle with center KK and radius KPKP is tangent to 3 circles ωb\omega_b, ωc\omega_c and Γ\Gamma.
Proposed by Tran Quan - Vietnam
IGOIrangeometry