MathDB

2017 Iranian Geometry Olympiad

Part of Iranian Geometry Olympiad

Subcontests

(5)
5
2

2017 IGO Intermediate P5

Let X,YX,Y be two points on the side BCBC of triangle ABCABC such that 2XY=BC2XY=BC (XX is between B,YB,Y). Let AA′AA' be the diameter of the circumcirle of triangle AXYAXY. Let PP be the point where AXAX meets the perpendicular from BB to BCBC, and QQ be the point where AYAY meets the perpendicular from CC to BCBC. Prove that the tangent line from A′A' to the circumcircle of AXYAXY passes through the circumcenter of triangle APQAPQ.
Proposed by Iman Maghsoudi

2017 IGO Advanced P5

Sphere SS touches a plane. Let A,B,C,DA,B,C,D be four points on the plane such that no three of them are collinear. Consider the point A′A' such that SS in tangent to the faces of tetrahedron A′BCDA'BCD. Points B′,C′,D′B',C',D' are defined similarly. Prove that A′,B′,C′,D′A',B',C',D' are coplanar and the plane A′B′C′D′A'B'C'D' touches SS.
Proposed by Alexey Zaslavsky (Russia)
4
3

2017 IGO Elementary P4

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

2017 IGO Intermediate P4

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

2017 IGO Advanced P4

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
3
3

2017 IGO Elementary P3

In the regular pentagon ABCDEABCDE, the perpendicular at CC to CDCD meets ABAB at FF. Prove that AE+AF=BEAE+AF=BE.
Proposed by Alireza Cheraghi

2017 IGO Intermediate P3

On the plane, nn points are given (n>2n>2). No three of them are collinear. Through each two of them the line is drawn, and among the other given points, the one nearest to this line is marked (in each case this point occurred to be unique). What is the maximal possible number of marked points for each given nn?
Proposed by Boris Frenkin (Russia)

2017 IGO Advanced P3

Let OO be the circumcenter of triangle ABCABC. Line COCO intersects the altitude from AA at point KK. Let P,MP,M be the midpoints of AKAK, ACAC respectively. If POPO intersects BCBC at YY, and the circumcircle of triangle BCMBCM meets ABAB at XX, prove that BXOYBXOY is cyclic.
Proposed by Ali Daeinabi - Hamid Pardazi
2
3

2017 IGO Elementary P2

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

2017 IGO Intermediate P2

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

2017 IGO Advanced P2

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
1
3

2017 IGO Elementary P1

Each side of square ABCDABCD with side length of 44 is divided into equal parts by three points. Choose one of the three points from each side, and connect the points consecutively to obtain a quadrilateral. Which numbers can be the area of this quadrilateral? Just write the numbers without proof. [asy] import graph; size(4.662701220158751cm); 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 = -4.337693339683693, xmax = 2.323308403400238, ymin = -0.39518100382374105, ymax = 4.44811779880594; /* image dimensions */ pen yfyfyf = rgb(0.5607843137254902,0.5607843137254902,0.5607843137254902); pen sqsqsq = rgb(0.12549019607843137,0.12549019607843137,0.12549019607843137);
filldraw((-3.,4.)--(1.,4.)--(1.,0.)--(-3.,0.)--cycle, white, linewidth(1.6)); filldraw((0.,4.)--(-3.,2.)--(-2.,0.)--(1.,1.)--cycle, yfyfyf, linewidth(2.) + sqsqsq); /* draw figures */ draw((-3.,4.)--(1.,4.), linewidth(1.6)); draw((1.,4.)--(1.,0.), linewidth(1.6)); draw((1.,0.)--(-3.,0.), linewidth(1.6)); draw((-3.,0.)--(-3.,4.), linewidth(1.6)); draw((0.,4.)--(-3.,2.), linewidth(2.) + sqsqsq); draw((-3.,2.)--(-2.,0.), linewidth(2.) + sqsqsq); draw((-2.,0.)--(1.,1.), linewidth(2.) + sqsqsq); draw((1.,1.)--(0.,4.), linewidth(2.) + sqsqsq); label("AA",(-3.434705427329005,4.459844914550807),SE*labelscalefactor,fontsize(14)); label("BB",(1.056779902954702,4.424663567316209),SE*labelscalefactor,fontsize(14)); label("CC",(1.0450527872098359,0.07390362597090126),SE*labelscalefactor,fontsize(14)); label("DD",(-3.551976584777666,0.12081208895036549),SE*labelscalefactor,fontsize(14)); /* dots and labels */ dot((-2.,4.),linewidth(4.pt) + dotstyle); dot((-1.,4.),linewidth(4.pt) + dotstyle); dot((0.,4.),linewidth(4.pt) + dotstyle); dot((1.,3.),linewidth(4.pt) + dotstyle); dot((1.,2.),linewidth(4.pt) + dotstyle); dot((1.,1.),linewidth(4.pt) + dotstyle); dot((0.,0.),linewidth(4.pt) + dotstyle); dot((-1.,0.),linewidth(4.pt) + dotstyle); dot((-3.,1.),linewidth(4.pt) + dotstyle); dot((-3.,2.),linewidth(4.pt) + dotstyle); dot((-3.,3.),linewidth(4.pt) + dotstyle); dot((-2.,0.),linewidth(4.pt) + dotstyle); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); /* end of picture */ [/asy] Proposed by Hirad Aalipanah

2017 IGO Intermediate P1

Let ABCABC be an acute-angled triangle with A=60∘A=60^{\circ}. Let E,FE,F be the feet of altitudes through B,CB,C respectively. Prove that CE−BF=32(AC−AB)CE-BF=\tfrac{3}{2}(AC-AB)
Proposed by Fatemeh Sajadi

2017 IGO Advanced P1

In triangle ABCABC, the incircle, with center II, touches the sides BCBC at point DD. Line DIDI meets ACAC at XX. The tangent line from XX to the incircle (different from ACAC) intersects ABAB at YY. If YIYI and BCBC intersect at point ZZ, prove that AB=BZAB=BZ.
Proposed by Hooman Fattahimoghaddam