MathDB

1

Part of 2017 Iranian Geometry Olympiad

Problems(3)

2017 IGO Elementary P1

Source:

9/15/2017
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
geometryIran
2017 IGO Intermediate P1

Source: 4th Iranian Geometry Olympiad (Intermediate) P1

9/15/2017
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
IGOIrangeometry
2017 IGO Advanced P1

Source: 4th Iranian Geometry Olympiad (Advanced) P1

9/15/2017
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
geometryIranIGO