MathDB

2022 Iranian Geometry Olympiad

Part of Iranian Geometry Olympiad

Subcontests

(5)
5
3

Quadrilateral geo

Let ABCDABCD be a quadrilateral inscribed in a circle ω\omega with center OO. Let PP be the intersection of two diagonals ACAC and BDBD. Let QQ be a point lying on the segment OPOP. Let EE and FF be the orthogonal projections of QQ on the lines ADAD and BCBC, respectively. The points MM and NN lie on the circumcircle of triangle QEFQEF such that QM∥ACQM \parallel AC and QN∥BDQN \parallel BD. Prove that the two lines MEME and NFNF meet on the perpendicular bisector of segment CDCD.
Proposed by Tran Quang Hung, Vietnam

Equilateral triangles and squares

a) Do there exist four equilateral triangles in the plane such that each two have exactly one vertex in common, and every point in the plane lies on the boundary of at most two of them? b) Do there exist four squares in the plane such that each two have exactly one vertex in common, and every point in the plane lies on the boundary of at most two of them? (Note that in both parts, there is no assumption on the intersection of interior of polygons.)
Proposed by Hesam Rajabzadeh

IGO 2022 advanced/free P5

Let ABCABC be an acute triangle inscribed in a circle ω\omega with center OO. Points EE, FF lie on its side ACAC, ABAB, respectively, such that OO lies on EFEF and BCEFBCEF is cyclic. Let RR, SS be the intersections of EFEF with the shorter arcs ABAB, ACAC of ω\omega, respectively. Suppose KK, LL are the reflection of RR about CC and the reflection of SS about BB, respectively. Suppose that points PP and QQ lie on the lines BSBS and RCRC, respectively, such that PKPK and QLQL are perpendicular to BCBC. Prove that the circle with center PP and radius PKPK is tangent to the circumcircle of RCERCE if and only if the circle with center QQ and radius QLQL is tangent to the circumcircle of BFSBFS.
Proposed by Mehran Talaei
4
3

Internal angle bisector problem

Let ADAD be the internal angle bisector of triangle ABCABC. The incircles of triangles ABCABC and ACDACD touch each other externally. Prove that ∠ABC>120∘\angle ABC > 120^{\circ}. (Recall that the incircle of a triangle is a circle inside the triangle that is tangent to its three sides.)
Proposed by Volodymyr Brayman (Ukraine)

Compatible polygons

We call two simple polygons P,QP, Q <spanclass=′latex−italic′>compatible</span><span class='latex-italic'>compatible</span> if there exists a positive integer kk such that each of P,QP, Q can be partitioned into kk congruent polygons similar to the other one. Prove that for every two even integers m,n≥4m, n \geq 4, there are two compatible polygons with mm and nn sides. (A simple polygon is a polygon that does not intersect itself.)
Proposed by Hesam Rajabzadeh

IGO 2022 advanced/free P4

Let ABCDABCD be a trapezoid with AB∥CDAB\parallel CD. Its diagonals intersect at a point PP. The line passing through PP parallel to ABAB intersects ADAD and BCBC at QQ and RR, respectively. Exterior angle bisectors of angles DBADBA, DCADCA intersect at XX. Let SS be the foot of XX onto BCBC. Prove that if quadrilaterals ABPQABPQ, CDQPCDQP are circumcribed, then PR=PSPR=PS.
Proposed by Dominik Burek, Poland
3
3

Convex pentagon with 3 equal sides

Let ABCDEABCDE be a convex pentagon such that AB=BC=CDAB = BC = CD and ∠BDE=∠EAC=30∘\angle BDE = \angle EAC = 30 ^{\circ}. Find the possible values of ∠BEC\angle BEC.
Proposed by Josef Tkadlec (Czech Republic)

Configuration with three similar triangles

Let OO be the circumcenter of triangle ABCABC. Arbitrary points MM and NN lie on the sides ACAC and BCBC, respectively. Points PP and QQ lie in the same half-plane as point CC with respect to the line MNMN, and satisfy △CMN∼△PAN∼△QMB\triangle CMN \sim \triangle PAN \sim \triangle QMB (in this exact order). Prove that OP=OQOP=OQ.
Proposed by Medeubek Kungozhin, Kazakhstan

IGO 2022 P3

In triangle ABCABC (∠A≠90∘)(\angle A\neq 90^\circ), let OO, HH be the circumcenter and the foot of the altitude from AA respectively. Suppose MM, NN are the midpoints of BCBC, AHAH respectively. Let DD be the intersection of AOAO and BCBC and let H′H' be the reflection of HH about MM. Suppose that the circumcircle of OH′DOH'D intersects the circumcircle of BOCBOC at EE. Prove that NONO and AEAE are concurrent on the circumcircle of BOCBOC.
Proposed by Mehran Talaei
2
3

Isosceles Trapezoid

An isosceles trapezoid ABCDABCD (AB∥CD)(AB \parallel CD) is given. Points EE and FF lie on the sides BCBC and ADAD, and the points MM and NN lie on the segment EFEF such that DF=BEDF = BE and FM=NEFM = NE. Let KK and LL be the foot of perpendicular lines from MM and NN to ABAB and CDCD, respectively. Prove that EKFLEKFL is a parallelogram.
Proposed by Mahdi Etesamifard

Two congruent circles and parallel lines

Two circles ω1\omega_1 and ω2\omega_2 with equal radius intersect at two points EE and XX. Arbitrary points C,DC, D lie on ω1,ω2\omega_1, \omega_2. Parallel lines to XC,XDXC, XD from EE intersect ω2,ω1\omega_2, \omega_1 at A,BA, B, respectively. Suppose that CDCD intersect ω1,ω2\omega_1, \omega_2 again at P,QP, Q, respectively. Prove that ABPQABPQ is cyclic.
Proposed by Ali Zamani

IGO 2022 advanced/free P2

We are given an acute triangle ABCABC with AB≠ACAB\neq AC. Let DD be a point of BCBC such that DADA is tangent to the circumcircle of ABCABC. Let EE and FF be the circumcenters of triangles ABDABD and ACDACD, respectively, and let MM be the midpoints EFEF. Prove that the line tangent to the circumcircle of AMDAMD through DD is also tangent to the circumcircle of ABCABC.
Proposed by Patrik Bak, Slovakia
1
3

Angles of Pentagon

Find the angles of the pentagon ABCDEABCDE in the figure below.

Equal segments and angles

Given is a circle ω\omega and a line ℓ\ell tangent to ω\omega at YY. Point XX lies on ℓ\ell to the left of YY. The tangent to ω\omega, perpendicular to ℓ\ell meets ℓ\ell at AA and touches ω\omega at DD. Let BB a point on ℓ\ell, to the right of YY, such that AX=BYAX=BY. The tangent from BB to ω\omega touches the circle at CC. Prove that ∠XDA=∠YDC\angle XDA= \angle YDC.
Note: This is not the official wording (it was just a diagram without any description).

IGO 2022 advanced/free P1

Four points AA, BB, CC and DD lie on a circle ω\omega such that AB=BC=CDAB=BC=CD. The tangent line to ω\omega at point CC intersects the tangent line to ω\omega at AA and the line ADAD at KK and LL. The circle ω\omega and the circumcircle of triangle KLAKLA intersect again at MM. Prove that MA=MLMA=ML.
Proposed by Mahdi Etesamifard