MathDB

4

Part of 2017 Iranian Geometry Olympiad

Problems(3)

2017 IGO Elementary P4

Source: 4th Iranian Geometry Olympiad (Elementary) P4

9/15/2017
P1,P2,…,P100P_1,P_2,\ldots,P_{100} are 100100 points on the plane, no three of them are collinear. For each three points, call their triangle clockwise if the increasing order of them is in clockwise order. Can the number of clockwise triangles be exactly 20172017?
Proposed by Morteza Saghafian
IGOIrangeometry
2017 IGO Intermediate P4

Source: 4th Iranian Geometry Olympiad, P4 of intermediate level, P5 of elementary level

9/15/2017
In the isosceles triangle ABCABC (AB=ACAB=AC), let ll be a line parallel to BCBC through AA. Let DD be an arbitrary point on ll. Let E,FE,F be the feet of perpendiculars through AA to BD,CDBD,CD respectively. Suppose that P,QP,Q are the images of E,FE,F on ll. Prove that AP+AQ≤ABAP+AQ\le AB
Proposed by Morteza Saghafian
IGOIrangeometry
2017 IGO Advanced P4

Source: 4th Iranian Geometry Olympiad (Advanced) P4

9/15/2017
Three circles ω1,ω2,ω3\omega_1,\omega_2,\omega_3 are tangent to line ll at points A,B,CA,B,C (BB lies between A,CA,C) and ω2\omega_2 is externally tangent to the other two. Let X,YX,Y be the intersection points of ω2\omega_2 with the other common external tangent of ω1,ω3\omega_1,\omega_3. The perpendicular line through BB to ll meets ω2\omega_2 again at ZZ. Prove that the circle with diameter ACAC touches ZX,ZYZX,ZY.
Proposed by Iman Maghsoudi - Siamak Ahmadpour
IGOIrangeometry