MathDB

1

Part of 2019 Iranian Geometry Olympiad

Problems(3)

2019 IGO Elementary P1

Source:

9/20/2019
There is a table in the shape of a 8×58\times 5 rectangle with four holes on its corners. After shooting a ball from points A,BA, B and CC on the shown paths, will the ball fall into any of the holes after 6 reflections? (The ball reflects with the same angle after contacting the table edges.)
http://s5.picofile.com/file/8372960750/E01.png
Proposed by Hirad Alipanah
IGOIrangeometry
2019 IGO Intermediate P1

Source: 6th Iranian Geometry Olympiad (Intermediate) P1

9/20/2019
Two circles ω1\omega_1 and ω2\omega_2 with centers O1O_1 and O2O_2 respectively intersect each other at points AA and BB, and point O1O_1 lies on ω2\omega_2. Let PP be an arbitrary point lying on ω1\omega_1. Lines BP,APBP, AP and O1O2O_1O_2 cut ω2\omega_2 for the second time at points XX, YY and CC, respectively. Prove that quadrilateral XPYCXPYC is a parallelogram.
Proposed by Iman Maghsoudi
IGOIrangeometryparallelogram
2019 IGO Advanced P1

Source: 6th Iranian Geometry Olympiad (Advanced) P1

9/20/2019
Circles ω1\omega_1 and ω2\omega_2 intersect each other at points AA and BB. Point CC lies on the tangent line from AA to ω1\omega_1 such that ∠ABC=90∘\angle ABC = 90^\circ. Arbitrary line ℓ\ell passes through CC and cuts ω2\omega_2 at points PP and QQ. Lines APAP and AQAQ cut ω1\omega_1 for the second time at points XX and ZZ respectively. Let YY be the foot of altitude from AA to ℓ\ell. Prove that points X,YX, Y and ZZ are collinear.
Proposed by Iman Maghsoudi
IGOIrangeometryAngle Chasing