MathDB

2020 Iranian Geometry Olympiad

Part of Iranian Geometry Olympiad

Subcontests

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2020 IGO Intermediate P3

In acute-angled triangle ABCABC (AC>ABAC > AB), point HH is the orthocenter and point MM is the midpoint of the segment BCBC. The median AMAM intersects the circumcircle of triangle ABCABC at XX. The line CHCH intersects the perpendicular bisector of BCBC at EE and the circumcircle of the triangle ABCABC again at FF. Point JJ lies on circle ω\omega, passing through X,E,X, E, and FF, such that BCHJBCHJ is a trapezoid (CB∥HJCB \parallel HJ). Prove that JBJB and EMEM meet on ω\omega.
Proposed by Alireza Dadgarnia

2020 IGO Elementary P3

According to the figure, three equilateral triangles with side lengths a,b,ca,b,c have one common vertex and do not have any other common point. The lengths x,yx, y, and zz are defined as in the figure. Prove that 3(x+y+z)>2(a+b+c)3(x+y+z)>2(a+b+c). Proposed by Mahdi Etesamifard

2020 IGO Advanced P3

Assume three circles mutually outside each other with the property that every line separating two of them have intersection with the interior of the third one. Prove that the sum of pairwise distances between their centers is at most 222\sqrt{2} times the sum of their radii. (A line separates two circles, whenever the circles do not have intersection with the line and are on different sides of it.) [color=#45818E]Note. Weaker results with 222\sqrt{2} replaced by some other cc may be awarded points depending on the value of c>22c>2\sqrt{2} Proposed by Morteza Saghafian
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2020 IGO Elementary P4

Let PP be an arbitrary point in the interior of triangle △ABC\triangle ABC. LinesBP‾\overline{BP} and CP‾\overline{CP} intersect AC‾\overline{AC} and AB‾\overline{AB} at EE and FF, respectively. Let KK and LL be the midpoints of the segments BFBF and CECE, respectively. Let the lines through LL and KK parallel to CF‾\overline{CF} and BE‾\overline{BE} intersect BC‾\overline{BC} at SS and TT, respectively; moreover, denote by MM and NN the reflection of SS and TT over the points LL and KK, respectively. Prove that as PP moves in the interior of triangle △ABC\triangle ABC, line MN‾\overline{MN} passes through a fixed point. Proposed by Ali Zamani

2020 IGO Intermediate P4

Triangle ABCABC is given. An arbitrary circle with center JJ, passing through BB and CC, intersects the sides ACAC and ABAB at EE and FF, respectively. Let XX be a point such that triangle FXBFXB is similar to triangle EJCEJC (with the same order) and the points XX and CC lie on the same side of the line ABAB. Similarly, let YY be a point such that triangle EYCEYC is similar to triangle FJBFJB (with the same order) and the points YY and BB lie on the same side of the line ACAC. Prove that the line XYXY passes through the orthocenter of the triangle ABCABC.
Proposed by Nguyen Van Linh - Vietnam

2020 IGO Advanced P4

Convex circumscribed quadrilateral ABCDABCD with its incenter II is given such that its incircle is tangent to AD‾,DC‾,CB‾,\overline{AD},\overline{DC},\overline{CB}, and BA‾\overline{BA} at K,L,M,K,L,M, and NN. Lines AD‾\overline{AD} and BC‾\overline{BC} meet at EE and lines AB‾\overline{AB} and CD‾\overline{CD} meet at FF. Let KM‾\overline{KM} intersects AB‾\overline{AB} and CD‾\overline{CD} at X,YX,Y, respectively. Let LN‾\overline{LN} intersects AD‾\overline{AD} and BC‾\overline{BC} at Z,TZ,T, respectively. Prove that the circumcircle of triangle △XFY\triangle XFY and the circle with diameter EIEI are tangent if and only if the circumcircle of triangle △TEZ\triangle TEZ and the circle with diameter FIFI are tangent. Proposed by Mahdi Etesamifard
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2020 IGO Elementary P2

A parallelogram ABCDABCD is given (AB≠BCAB \neq BC). Points EE and GG are chosen on the line CD‾\overline{CD} such that AC‾\overline{AC} is the angle bisector of both angles ∠EAD\angle EAD and ∠BAG\angle BAG. The line BC‾\overline{BC} intersects AE‾\overline{AE} and AG‾\overline{AG} at FF and HH, respectively. Prove that the line FG‾\overline{FG} passes through the midpoint of HEHE. Proposed by Mahdi Etesamifard

2020 IGO Intermediate P2

Let ABCABC be an isosceles triangle (AB=ACAB = AC) with its circumcenter OO. Point NN is the midpoint of the segment BCBC and point MM is the reflection of the point NN with respect to the side ACAC. Suppose that TT is a point so that ANBTANBT is a rectangle. Prove that ∠OMT=12∠BAC\angle OMT = \frac{1}{2} \angle BAC.
Proposed by Ali Zamani

2020 IGO Advanced P2

Let △ABC\triangle ABC be an acute-angled triangle with its incenter II. Suppose that NN is the midpoint of the arc \overarc{BAC} of the circumcircle of triangle △ABC\triangle ABC, and PP is a point such that ABPCABPC is a parallelogram.Let QQ be the reflection of AA over NN and RR the projection of AA on QI‾\overline{QI}. Show that the line AI‾\overline{AI} is tangent to the circumcircle of triangle △PQR\triangle PQR Proposed by Patrik Bak - Slovakia
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2020 IGO Elementary P1

By a fold of a polygon-shaped paper, we mean drawing a segment on the paper and folding the paper along that. Suppose that a paper with the following figure is given. We cut the paper along the boundary of the shaded region to get a polygon-shaped paper. Start with this shaded polygon and make a rectangle-shaped paper from it with at most 5 number of folds. Describe your solution by introducing the folding lines and drawing the shape after each fold on your solution sheet. (Note that the folding lines do not have to coincide with the grid lines of the shape.) Proposed by Mahdi Etesamifard

2020 IGO Intermediate P1

A trapezoid ABCDABCD is given where ABAB and CDCD are parallel. Let MM be the midpoint of the segment ABAB. Point NN is located on the segment CDCD such that ∠ADN=12∠MNC\angle ADN = \frac{1}{2} \angle MNC and ∠BCN=12∠MND\angle BCN = \frac{1}{2} \angle MND. Prove that NN is the midpoint of the segment CDCD.
Proposed by Alireza Dadgarnia

2020 IGO Advanced P1

Let M,N,PM,N,P be midpoints of BC,ACBC,AC and ABAB of triangle △ABC\triangle ABC respectively. EE and FF are two points on the segment BC‾\overline{BC} so that ∠NEC=12∠AMB\angle NEC = \frac{1}{2} \angle AMB and ∠PFB=12∠AMC\angle PFB = \frac{1}{2} \angle AMC. Prove that AE=AFAE=AF. Proposed by Alireza Dadgarnia
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2020 IGO Elementary P5

We say two vertices of a simple polygon are visible from each other if either they are adjacent, or the segment joining them is completely inside the polygon (except two endpoints that lie on the boundary). Find all positive integers nn such that there exists a simple polygon with nn vertices in which every vertex is visible from exactly 44 other vertices. (A simple polygon is a polygon without hole that does not intersect itself.) Proposed by Morteza Saghafian

2020 IGO Intermediate P5

Find all numbers n≥4n \geq 4 such that there exists a convex polyhedron with exactly nn faces, whose all faces are right-angled triangles. (Note that the angle between any pair of adjacent faces in a convex polyhedron is less than 180∘180^\circ.)
Proposed by Hesam Rajabzadeh

2020 IGO Advanced P5

Consider an acute-angled triangle △ABC\triangle ABC (AC>ABAC>AB) with its orthocenter HH and circumcircle Γ\Gamma.Points MM,PP are midpoints of BCBC and AHAH respectively.The line AM‾\overline{AM} meets Γ\Gamma again at XX and point NN lies on the line BC‾\overline{BC} so that NX‾\overline{NX} is tangent to Γ\Gamma. Points JJ and KK lie on the circle with diameter MPMP such that ∠AJP=∠HNM\angle AJP=\angle HNM (BB and JJ lie one the same side of AH‾\overline{AH}) and circle ω1\omega_1, passing through K,HK,H, and JJ, and circle ω2\omega_2 passing through K,MK,M, and NN, are externally tangent to each other. Prove that the common external tangents of ω1\omega_1 and ω2\omega_2 meet on the line NH‾\overline{NH}. Proposed by Alireza Dadgarnia