MathDB

2

Part of 2017 Iranian Geometry Olympiad

Problems(3)

2017 IGO Elementary P2

Source:

9/15/2017
Find the angles of triangle ABCABC. [asy] import graph; size(9.115122858763474cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -7.9216637359954705, xmax = 10.308581981531479, ymin = -6.062398124651168, ymax = 9.377503860601273; /* image dimensions */
/* draw figures */ draw((1.1862495478417192,2.0592342833377844)--(3.0842,-3.6348), linewidth(1.6)); draw((2.412546402365528,-0.6953452852662508)--(1.8579031454761883,-0.8802204313959623), linewidth(1.6)); draw((-3.696094000229639,5.5502174997511595)--(1.1862495478417192,2.0592342833377844), linewidth(1.6)); draw((-1.0848977580797177,4.042515022414228)--(-1.4249466943082025,3.5669367606747184), linewidth(1.6)); draw((1.1862495478417192,2.0592342833377844)--(9.086047353374928,-3.589295214974483), linewidth(1.6)); draw((9.086047353374928,-3.589295214974483)--(3.0842,-3.6348), linewidth(1.6)); draw((6.087339936804166,-3.904360930324946)--(6.082907416570757,-3.319734284649538), linewidth(1.6)); draw((3.0842,-3.6348)--(-2.9176473533749285,-3.6803047850255166), linewidth(1.6)); draw((0.08549258342923806,-3.9498657153504615)--(0.08106006319583221,-3.3652390696750536), linewidth(1.6)); draw((-2.9176473533749285,-3.6803047850255166)--(-6.62699301304923,-3.7084282888220432), linewidth(1.6)); draw((-6.62699301304923,-3.7084282888220432)--(-4.815597805533209,2.0137294983122676), linewidth(1.6)); draw((-5.999986761815922,-0.759127399441624)--(-5.442604056766517,-0.9355713910681529), linewidth(1.6)); draw((-4.815597805533209,2.0137294983122676)--(-3.696094000229639,5.5502174997511595), linewidth(1.6)); draw((-4.815597805533209,2.0137294983122676)--(1.1862495478417192,2.0592342833377844), linewidth(1.6)); draw((-1.8168903889624484,2.3287952136627297)--(-1.8124578687290425,1.744168567987322), linewidth(1.6)); draw((-4.815597805533209,2.0137294983122676)--(-2.9176473533749285,-3.6803047850255166), linewidth(1.6)); draw((-3.5893009510093994,-0.7408500702917692)--(-4.1439442078987385,-0.9257252164214806), linewidth(1.6)); label("AA",(-4.440377746205339,7.118654172569505),SE*labelscalefactor,fontsize(14)); label("BB",(-7.868514331571194,-3.218904987952353),SE*labelscalefactor,fontsize(14)); label("CC",(9.165869786409527,-3.0594567746795223),SE*labelscalefactor,fontsize(14)); /* dots and labels */ dot((3.0842,-3.6348),linewidth(3.pt) + dotstyle); dot((9.086047353374928,-3.589295214974483),linewidth(3.pt) + dotstyle); dot((1.1862495478417192,2.0592342833377844),linewidth(3.pt) + dotstyle); dot((-2.9176473533749285,-3.6803047850255166),linewidth(3.pt) + dotstyle); dot((-4.815597805533209,2.0137294983122676),linewidth(3.pt) + dotstyle); dot((-6.62699301304923,-3.7084282888220432),linewidth(3.pt) + dotstyle); dot((-3.696094000229639,5.5502174997511595),linewidth(3.pt) + dotstyle); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy] Proposed by Morteza Saghafian
Irangeometry
2017 IGO Intermediate P2

Source: 4th Iranian Geometry Olympiad (Intermediate) P2

9/15/2017
Two circles ω1,ω2\omega_1,\omega_2 intersect at A,BA,B. An arbitrary line through BB meets ω1,ω2\omega_1,\omega_2 at C,DC,D respectively. The points E,FE,F are chosen on ω1,ω2\omega_1,\omega_2 respectively so that CE=CB, BD=DFCE=CB,\ BD=DF. Suppose that BFBF meets ω1\omega_1 at PP, and BEBE meets ω2\omega_2 at QQ. Prove that A,P,QA,P,Q are collinear.
Proposed by Iman Maghsoudi
IGOIrangeometry
2017 IGO Advanced P2

Source: 4th Iranian Geometry Olympiad (Advanced) P2

9/15/2017
We have six pairwise non-intersecting circles that the radius of each is at least one (no circle lies in the interior of any other circle). Prove that the radius of any circle intersecting all the six circles, is at least one.
Proposed by Mohammad Ali Abam - Morteza Saghafian
IGOgeometry