Iranian 2014-22 (IGO) 121p

geometry problems from Iranian Geometry Olympiad (IGO)                   [all authors / proposers]
with aops links in the names s

Iranian Geometry Olympiad problems 2014-2017 ΕΝ in pdf
Iranian Geometry Olympiad all 2014-2021 in pdf with solutions

Iranian Geometry Olympiad 2022 aops post collection

Elementary collected inside aops here

Medium /  Intermediate collected inside aops here

2014 - 2022

IGO 2014 Junior 1 / Senior 1

In a right triangle ABC we have <A = 90o, <C = 30o. Denote by C the circle passing through A which is tangent to BC at the midpoint. Assume that C intersects AC and the circumcircle of ABC at N and M respectively. Prove that MN $\perp$ BC.

by Mahdi Etesami Fard

IGO 2014 Junior 2

The inscribed circle of $\triangle ABC$ touches $BC, AC$ and $AB$ at $D,E$ and $F$ respectively. Denote the perpendicular foots from $F, E$ to $BC$ by $K, L$ respectively. Let the second intersection of these perpendiculars with the incircle be $M, N$ respectively. Show that $\frac{{{S}_{\triangle BMD}}}{{{S}_{\triangle CND}}}=\frac{DK}{DL}$

by Mahdi Etesami Fard

IGO 2014 Junior 3

Each of Mahdi and Morteza has drawn an inscribed $93$-gon. Denote the first one by $A_1A_2…A_{93}$ and the second by $B_1B_2…B_{93}$. It is known that $A_iA_{i+1} // B_iB_{i+1}$ for $1 \le i \le 93$ ($A_{93} = A_1, B_{93} = B_1$). Show that $\frac{A_iA_{i+1} }{ B_iB_{i+1}}$ is a constant number independent of $i$.

by Morteza Saghafian

IGO 2014 Junior 4

In a triangle ABC we have $\angle C = \angle A + 90^o$. The point $D$ on the continuation of $BC$ is given such that $AC = AD$. A point $E$ in the side of $BC$ in which $A$ doesn’t lie is chosen such that $\angle EBC = \angle A, \angle EDC = \frac{1}{2} \angle A$ . Prove that $\angle CED =  \angle ABC$.

by Morteza Saghafian

IGO 2014 Junior 5

Two points $X, Y$ lie on the arc $BC$ of the circumcircle of $\triangle ABC$ (this arc does not contain $A$) such that $\angle BAX = \angle CAY$ . Let $M$ denotes the midpoint of the chord $AX$ . Show that   $BM +CM > AY$ .

by Mahan Tajrobekar

IGO 2014 Senior 1 / Junior 1

In a right triangle ABC we have <A = 90^o, <C = 30^o. Denote by C the circle passing through A which is tangent to BC at the midpoint. Assume that C intersects AC and the circumcircle of ABC at N and M respectively. Prove that MN $\perp$ BC.

by Mahdi Etesami Fard

IGO 2014 Senior 2

In a quadrilateral ABCD we have <B = <D = 60^o. Consider the line which is drawn from M, the midpoint of AD, parallel to CD. Assume this line intersects BC at P. A point X lies on CD such that BX = CX. Prove that AB = BP  <MXB = 60^o.

by Davood Vakili

IGO 2014 Senior 3

An acute-angled triangle ABC is given. The circle with diameter BC intersects AB, AC at E, F respectively. Let M be the midpoint of BC and P the intersection point of AM and EF. X is a point on the arc EF and Y the second intersection point of XP with circle mentioned above. Show that <XAY = <XYM.

by Ali Zooelm

IGO 2014 Senior 4

The tangent line to circumcircle of the acute-angled triangle ABC (AC > AB) at A intersects the continuation of BC at P. We denote by O the circumcenter of ABC. X is a point OP such that  <AXP = 90^o. Two points E, F respectively on AB, AC at the same side of OP are chosen such that <EXP = <ACX, <FXO = <ABX. If K, L denote the intersection points of EF with the circumcircle of △ABC, show that OP is tangent to the circumcircle of △KLX.

by Mahdi Etesami Fard

IGO 2014 Senior 5

Two points P, Q lie on the side BC of triangle ABC and have the same distance to the midpoint. The perpendiculars from P, Q tp BC intesect AC, AB at E, F respectively. Let M be the intersection point of PF and EQ. If H₁ and H₂ denote the orthocenter of △BFP and △CEQ respectively, show that AM $\perp$ H1H2.

by Mahdi Etesami Fard

IGO 2014 Shortlist

Suppose that $I$ is incenter of $\vartriangle ABC$ and $CI$ inresects $AB$ at $D$.In circumcircle of $\vartriangle ABC$, $T$ is midpoint of arc $BAC$ and $BI$ intersect this circle at $M$. If $MD$ intersects $AT$ at $N$, prove that: $BM \parallel CN$.

by Ali Zooelm

IGO 2015 Elementary 1

We have four wooden triangles with sides $3, 4, 5$ centimeters. How many convex polygons can we make by all of these triangles? (Just draw the polygons without any proof)

A convex polygon is a polygon which all of it's angles are less than $180^o$ and there isn't any hole in it. For example:  

by Mahdi Etesami Fard

IGO 2015 Elementary 2

Let ABC be a triangle with $\angle A = 60^o$. The points $M,N,K$ lie on $BC,AC,AB$ respectively such that $BK = KM = MN = NC$. If $AN = 2AK$, find the values of $\angle B$ and $\angle C$.

by Mahdi Etesami Fard

IGO 2015 Elementary 3

In the figure below, we know that $AB = CD$ and $BC = 2AD$. Prove that $<BAD = 30^o$.

           by Morteza Saghafian

IGO  2015 Elementary 4

In rectangle $ABCD$, the points $M,N,P,Q$ lie on $AB,BC,CD,DA$ respectively such that the area of triangles $AQM,BMN,CNP,DPQ$ are equal. Prove that the quadrilateral $MNPQ$ is parallelogram.

by Mahdi Etesami Fard

IGO  2015 Elementary 5

Do there exist $6$ circles in the plane such that every circle passes through centers of exactly $3$ other circles?

by Morteza Saghafian

IGO 2015 Medium1

In the figure below, the points P,A,B lie on a circle. The point Q lies inside the circle such that <PAQ = 90^o and PQ = BQ. Prove that the value of <AQB - <PQA is equal to the arc AB.

 by Davood Vakili

IGO 2015 Medium2

In acute-angled triangle ABC, BH is the altitude of the vertex B. The points D and E are midpoints of AB and AC respectively. Suppose that F be the reflection of H with respect to ED. Prove that the line BF passes through circumcenter of ABC.

by Davood Vakili

IGO 2015 Medium 3

In triangle ABC, the points M,N,K are the midpoints of BC,CA,AB respectively. Let ω_B and ω_C be two semicircles with diameter AC and AB respectively, outside the triangle. Suppose that MK and MN intersect ω_C and ω_B at X and Y respectively. Let the tangents at X and Y to  ω_C and ω_B respectively, intersect at Z. prove that AZ $\perp$ BC.

by Mahdi Etesami Fard

IGO 2015 Medium 4 / Advanced 2

Let ABC be an equilateral triangle with circumcircle ω and circumcenter O. Let P be the point on the arc BC (the arc which A doesn't lie ). Tangent to ω at P intersects extensions of AB and AC at K and L respectively. Show that <KOL > 90^o.

by Iman Maghsoudi

IGO 2015 Medium 5

a) Do there exist 5 circles in the plane such that every circle passes through centers of exactly 3 circles?

b) Do there exist 6 circles in the plane such that every circle passes through centers of exactly 3 circles?

by Morteza Saghafian

IGO 2015 Advanced 1

Two circles ω₁ and ω₂ (with centers O₁ and O₂ respectively) intersect at A and B. The point X lies on ω₂. Let point Y be a point on ω₁ such that <XBY = 90^o. Let X΄ be the second point of intersection of the line O₁X and ω₂ and K be the second point of intersection of X΄Y and ω₂. Prove that X is the midpoint of arc AK.

by Davood Vakili

IGO 2015 Advanced 2 / Medium 4

Let ABC be an equilateral triangle with circumcircle ω and circumcenter O. Let P be the point on the arc BC (the arc which A doesn't lie ). Tangent to ω at P intersects extensions of AB and AC at K and L respectively. Show that <KOL > 90^o.

by Iman Maghsoudi

IGO 2015 Advanced 3

Let H be the orthocenter of the triangle ABC. Let l₁ and l₂ be two lines passing through H and perpendicular to each other. l₁ intersects BC and extension of AB at D and Z respectively, and l₂ intersects BC and extension of AC at E and X respectively. Let Y be a point such that YD//AC and YE//AB. Prove that X,Y,Z are collinear.

by Ali Golmakani

IGO 2015 Advanced 4

In triangle ABC, we draw the circle with center A and radius AB. This circle intersects AC at two points. Also we draw the circle with center A and radius AC and this circle intersects AB at two points. Denote these four points by A₁,A₂,A₃,A₄. Find the points B₁,B₂,B₃,B₄ and C₁,C₂,C₃,C₄ similarly. Suppose that these 12 points lie on two circles. Prove that the triangle ABC is isosceles.

by Morteza Saghafian

IGO 2015 Advanced 5

Rectangles ABA₁B₂, BCB₁C₂, CAC₁A₂ lie outside triangle ABC. Let C΄ be a point such that C΄A_{1 $\perp$}  A₁C₂ and C΄B₂ $\perp$ B₂C₁. Points A΄ and B΄ are defined similarly. Prove that lines AA΄, BB΄, CC΄ concur.

by Alexey Zaslavsky (Russia)

IGO 2016 Elementary 1

Ali wants to move from point $A$ to point $B$. He cannot walk inside the black areas but he is free to move in any direction inside the white areas (not only the grid lines but the whole plane). Help Ali to find the shortest path between $A$ and $B$. Only draw the path and write its length.

by Morteza Saghafian

IGO 2016 Elementary 2

Let $\omega$ be the circumcircle of triangle $ABC$ with $AC > AB$. Let $X$ be a point on $AC$ and $Y$ be a point on the circle $\omega$, such that $CX = CY = AB$. (The points $A$ and $Y$ lie on different sides of the line $BC$). The line $XY$ intersects $\omega$ for the second time in point $P$. Show that $PB = PC$.

by Iman Maghsoudi

IGO 2016 Elementary 3

Suppose that $ABCD$ is a convex quadrilateral with no parallel sides. Make a parallelogram on each two consecutive sides. Show that among these $4$ new points, there is only one point inside the quadrilateral $ABCD$.

by Morteza Saghafian

IG 2016 Elementary 4

In a right-angled triangle $ABC$ ($\angle A = 90^o$), the perpendicular bisector of $BC$ intersects the line $AC$ in $K$ and the perpendicular bisector of $BK$ intersects the line $AB$ in $L$. If the line $CL$ be the internal bisector of angle $C$, find all possible values for angles $B$ and $C$.

by Mahdi Etesami Fard

IGO 2016 Elementary 5

Let ABCD be a convex quadrilateral with these properties: <ADC = 135 ^o and <ADB - <ABD = 2<DAB = 4<CBD. If BC = √2  CD , prove that AB = BC + AD.

by Mahdi Etesami Fard

IGO 2016 Medium 1

In trapezoid ABCD with AB // CD, ω₁ and ω₂ are two circles with diameters AD and BC, respectively. Let X and Y be two arbitrary points on ω₁ and ω₂, respectively. Show that the length of segment XY is not more than half of the perimeter of ABCD.

by Mahdi Etesami Fard

IGO 2016 Medium 2

Let two circles C₁ and C₂ intersect in points A and B. The tangent to C₁ at A intersects C₂ in P and the line PB intersects C₁ for the second time in Q (suppose that Q is outside C₂). The tangent to C₂ from Q intersects C₁ and C₂ in C and D, respectively (The points A and D lie on different sides of the line PQ). Show that AD is bisector of the angle <CAP.

by Iman Maghsoudi

IGO 2016 Medium 3

Find all positive integers N such that there exists a triangle which can be dissected into N similar quadrilaterals.

by Nikolai Beluhov (Bulgaria) and Morteza Saghafian

IGO 2016 Medium 4

Let ω be the circumcircle of right-angled triangle ABC (<A = 90^o). Tangent to ω at point A intersects the line BC in point P. Suppose that M is the midpoint of (the smaller) arc AB, and PM intersects ω for the second time in Q. Tangent to ω at point Q intersects AC in K. Prove that <PKC = 90^o.

by Davood Vakili

IGO 2016 Medium 5

Let the circles ω and ω΄ intersect in points A and B. Tangent to circle ω at A intersects ω΄ in C and tangent to circle ω΄ at A intersects ω in D. Suppose that the internal bisector of <CAD intersects ω and ω΄ at E and F, respectively, and the external bisector of <CAD intersects ω and ω΄ in X and Y , respectively. Prove that the perpendicular bisector of XY is tangent to the circumcircle of triangle BEF.

by Mahdi Etesami Fard

IGO 2016 Advanced 1

Let the circles ω and ω΄ intersect in A and B. Tangent to circle ω at A intersects ω΄ in C and tangent to circle ω΄ at A intersects ω in D. Suppose that the segment CD intersects ω and ω in E and F, respectively (assume that E is between F and C). The perpendicular to AC from E intersects ω΄ in point P and perpendicular to AD from F intersects ω in point Q (The points A, P and Q lie on the same side of the line CD). Prove that the points A, P and Q are collinear.

by Mahdi Etesami Fard

IGO 2016 Advanced 2

In acute-angled triangle ABC, altitude of A meets BC at D, and M is midpoint of AC. Suppose that X is a point such that <AXB = <DXM = 90^o (assume that X and C lie on opposite sides of the line BM). Show that <XMB = 2<MBC.

by Davood Vakili

IGO 2016 Advanced 3

Let P be the intersection point of sides AD and BC of a convex quadrilateral ABCD. Suppose that I₁ and I₂ are the incenters of triangles PAB and PDC, respectively. Let O be the circumcenter of PAB, and H the orthocenter of PDC. Show that the circumcircles of triangles AI₁B and DHC are tangent together if and only if the circumcircles of triangles AOB and DI₂C are tangent together.

by Hooman Fattahimoghaddam

IGO 2016 Advanced 4

In a convex quadrilateral ABCD, the lines AB and CD meet at point E and the lines AD and BC meet at point F. Let P be the intersection point of diagonals AC and BD. Suppose that ω₁ is a circle passing through D and tangent to AC at P. Also suppose that ω₂ is a circle passing through C and tangent to BD at P. Let X be the intersection point of ω₁ and AD, and Y be the intersection point of ω₂ and BC. Suppose that the circles ω₁ and ω₂ intersect each other in Q for the second time. Prove that the perpendicular from P to the line EF passes through the circumcenter of triangle XQY

by Iman Maghsoudi

IGO 2016 Advanced 5

Do there exist six points X₁,X₂,Y₁,Y₂,Z₁,Z₂ in the plane such that all of the triangles X_iY_jZ_k are similar for 1  ≤  i, j, k ≤ 2 ?

by Morteza Saghafian

IGO 2017 Elementary 1

Each side of square ABCD with side length of 4 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.

by Hirad Aalipanah

IGO 2017 Elementary 2

Find the angles of triangle ABC.               

    by Morteza Saghafian

IGO 2017 Elementary 3

In the regular pentagon ABCDE, the perpendicular at C to CD meets AB at F. Prove that  AE + AF = BE.

by Alireza Cheraghi

IGO 2017 Elementary 4

P1, P2, ... , P100 are 100 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 2017?

by Morteza Saghafian

IGO  2017 Elementary 5 / Medium 4

In the isosceles triangle ABC (AB = AC), let l  be a line parallel to BC through A. Let D be an arbitrary point on l. Let E, F be the feet of perpendiculars through A to BD, CD respectively. Suppose that P, Q are the images of E, F on l. Prove that AP + AQ ≤ AB.

by Morteza Saghafian

IGO  2017 Medium1

Let ABC be an acute-angled triangle with A = 60^o. Let E, F be the feet of altitudes through B, C respectively. Prove that CE - BF = 3 / 2 (AC - AB).

by Fatemeh Sajadi

IGO 2017 Medium 2

Two circles ω₁, ω₂  intersect at A, B. An arbitrary line through B meets ω₁, ω₂  at C, D respectively. The points E, F are chosen on ω₁, ω₂  respectively so that CE = CB, BD = DF. Suppose that BF meets ω₁at P, and BE meets ω₂  at Q. Prove that A, P, Q are collinear.

by Iman Maghsoudi

IGO 2017 Medium 3

On the plane, n points are given (n > 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 n?

by Boris Frenkin (Russia)

IGO 2017 Medium4 / Elementary 5

In the isosceles triangle ABC (AB = AC), let l be a line parallel to BC through A. Let D be an arbitrary point on l. Let E, F be the feet of perpendiculars through A to BD, CD respectively. Suppose that P, Q are the images of E, F on l. Prove that AP + AQ ≤ AB.

by Morteza Saghafian

IGO 2017 Medium5

Let X, Y be two points on the side BC of triangle ABC such that 2XY = BC. (X is between B, Y ) Let AA΄ be the diameter of the circumcircle of triangle AXY . Let P be the point where AX meets the perpendicular from B to BC, and Q be the point where AY meets the perpendicular from C to BC. Prove that the tangent line from A΄ to the circumcircle of AXY passes through the circumcenter of triangle APQ.

by Iman Maghsoudi

IGO 2017 Advanced 1

In triangle ABC, the incircle, with center I, touches the side BC at point D. Line DI meets AC at X. The tangent line from X to the incircle (different from AC) intersects AB at Y . If YI and BC intersect at point Z, prove that AB = BZ.

by Hooman Fattahimoghaddam

IGO 2017 Advanced 2

We have six pairwise non-intersecting circles that the radius of each is at least one. Prove that the radius of any circle intersecting all the six circles, is at least one.

by Mohammad Ali Abam - Morteza Saghafian

IGO 2017 Advanced 3

Let O be the circumcenter of triangle ABC. Line CO intersects the altitude through A at point K. Let P, M be the midpoints of AK, AC respectively. If PO intersects BC at Y , and the circumcircle of triangle BCM meets AB at X, prove that BXOY is cyclic.

by Ali Daeinabi - Hamid Pardazi

IGO 2017 Advanced 4

Three circles ω₁, ω₂, ω₃ are tangent to line l at points A, B, C (B lies between A, C) and ω₂ is externally tangent to the other two. Let X, Y be the intersection points of ω₂ with the other common external tangent of ω₁, ω₃. The perpendicular line through B to l meets ω₂ again at Z. Prove that the circle with diameter AC touches ZX, ZY .

by Iman Maghsoudi - Siamak Ahmadpour

IGO 2017 Advanced 5

Sphere S touches a plane. Let A, B, C, D be four points on this plane such that no three of them are collinear. Consider the point A΄ such that S is tangent to the faces of tetrahedron A΄BCD. Points B΄, C΄, D΄ are defined similarly. Prove that A΄, B΄, C΄, D΄ are coplanar and the plane A΄B΄C΄D΄ touches S.

by Alexey Zaslavsky (Russia)

IGO 2018 Elementary 1

As shown below, there is a $40\times30$ paper with a filled $10\times5$ rectangle inside of it. We want to cut out the filled rectangle from the paper using four straight cuts. Each straight cut is a straight line that divides the paper into two pieces, and we keep the piece containing the filled rectangle. The goal is to minimize the total length of the straight cuts. How to achieve this goal, and what is that minimized length? Show the correct cuts and write the final answer. There is no need to prove the answer.

by Morteza Saghafian

IGO 2018 Elementary 2

Convex hexagon $A_1A_2A_3A_4A_5A_6$ lies in the interior of convex hexagon $B_1B_2B_3B_4B_5B_6$ such that $A_1A_2 \parallel B_1B_2$, $A_2A_3 \parallel B_2B_3$,..., $A_6A_1 \parallel B_6B_1$. Prove that the areas of simple hexagons $A_1B_2A_3B_4A_5B_6$ and $B_1A_2B_3A_4B_5A_6$ are equal. (A simple hexagon is a hexagon which does not intersect itself.)

 by Hirad Aalipanah and Mahdi Etesamifard

IGO 2018 Elementary 3

by Morteza SaghafianIn the given figure, $ABCD$ is a parallelogram. We know that $\angle D = 60^\circ$, $AD = 2$ and $AB = \sqrt3 + 1$. Point $M$ is the midpoint of $AD$. Segment $CK$ is the angle bisector of $C$. Find the angle $CKB$.

by Mahdi Etesamifard

IGO 2018 Elementary 4

There are two circles with centers $O_1,O_2$ lie inside of circle $\omega$ and are tangent to it. Chord $AB$ of $\omega$ is tangent to these two circles such that they lie on opposite sides of this chord. Prove that $\angle O_1AO_2 + \angle O_1BO_2 > 90^\circ$.

by Iman Maghsoudi

IGO 2018 Elementary 5

There are some segments on the plane such that no two of them intersect each other (even at the ending points). We say segment $AB$ breaks segment $CD$ if the extension of $AB$ cuts $CD$ at some point between $C$ and $D$.

a) Is it possible that each segment when extended from both ends, breaks exactly one other segment from each way?

b) A segment is called surrounded if from both sides of it, there is exactly one segment that breaks it.

(e.g. segment $AB$ in the figure.) Is it possible to have all segments to be surrounded?

 by Morteza Saghafian 

IGO 2018 Intemediate 1

There are three rectangles in the following figure. The lengths of some segments are shown.
Find the length of the segment $XY$ .

by Hirad Aalipanah

IGO 2018 Intemediate 2

In convex quadrilateral $ABCD$, the diagonals $AC$ and $BD$ meet at the point $P$. We know that  $\angle DAC = 90^o$ and  $2 \angle ADB = \angle ACB$. If we have $\angle DBC + 2 \angle ADC = 180^o$ prove that $2AP = BP$.

by Iman Maghsoudi

IGO 2018 Intemediate 3

Let $\omega_1,\omega_2$ be two circles with centers $O_1$ and $O_2$, respectively. These two circles intersect each other at points $A$ and $B$. Line $O_1B$ intersects $\omega_2$ for the second time at point $C$, and line $O_2A$ intersects $\omega_1$ for the second time at point $D$ . Let $X$ be the second intersection of $AC$ and $\omega_1$. Also $Y$ is the second intersection point of $BD$ and $\omega_2$. Prove that $CX = DY$ .

by Alireza Dadgarnia

IGO 2018 Intemediate 4

We have a polyhedron all faces of which are triangle. Let $P$ be an arbitrary point on one of the edges of this polyhedron such that $P$ is not the midpoint or endpoint of this edge. Assume that $P_0 = P$. In each step, connect $P_i$ to the centroid of one of the faces containing it. This line meets the perimeter of this face again at point $P_{i+1}$. Continue this process with $P_{i+1}$ and the other face containing $P_{i+1}$. Prove that by continuing this process, we cannot pass through all the faces. (The centroid of a triangle is the point of intersection of its medians.)

by Mahdi Etesamifard and Morteza Saghafian

IGO 2018 Intemediate 5

Suppose that $ABCD$ is a parallelogram such that $\angle DAC = 90^o$. Let $H$ be the foot of perpendicular from $A$ to $DC$, also let $P$ be a point along the line $AC$ such that the line $PD$ is tangent to the circumcircle of the triangle $ABD$. Prove that $\angle PBA = \angle DBH$.

by Iman Maghsoudi

IGO 2018 Advanced 1

Two circles $\omega_1,\omega_2$ intersect each other at points $A,B$. Let $PQ$ be a common tangent line of these two circles with $P \in \omega_1$ and $Q  \in  \omega_2$. An arbitrary point $X$ lies on $\omega_1$. Line $AX$ intersects $\omega_2$ for the second time at $Y$ . Point $Y'\ne Y$ lies on $\omega_2$ such that $QY = QY'$. Line $Y'B$ intersects $\omega_1$ for the second time at $X'$. Prove that $PX = PX'$.

by Morteza Saghafian

IGO 2018 Advanced 2

In acute triangle $ABC, \angle A = 45^o$. Points $O,H$ are the circumcenter and the orthocenter of $ABC$, respectively. $D$ is the foot of altitude from $B$. Point $X$ is the midpoint of arc $AH$ of the circumcircle of triangle $ADH$ that contains $D$. Prove that $DX = DO$.

by Fatemeh Sajadi

IGO 2018 Advanced 3

Find all possible values of integer $n > 3$ such that there is a convex $n$-gon in which, each diagonal is the perpendicular bisector of at least one other diagonal.

by Mahdi Etesamifard

IGO 2018 Advanced 4

Quadrilateral $ABCD$ is circumscribed around a circle. Diagonals $AC,BD$ are not perpendicular to each other. The angle bisectors of angles between these diagonals, intersect the segments $AB,BC,CD$ and $DA$ at points $K,L,M$ and $N$. Given that $KLMN$ is cyclic, prove that so is $ABCD$.

by Nikolai Beluhov (Bulgaria)

IGO 2018 Advanced 5

$ABCD$ is a cyclic quadrilateral. A circle passing through $A,B$ is tangent to segment $CD$ at point $E$. Another circle passing through $C,D$ is tangent to $AB$ at point $F$. Point $G$ is the intersection point of $AE,DF$, and point $H$ is the intersection point of $BE,CF$. Prove that the incenters of triangles $AGF,BHF,CHE,DGE$ lie on a circle.

by Le Viet An (Vietnam)

IGO 2019 Elementary 1

There is a table in the shape of a $8\times 5$ rectangle with four holes on its corners. After shooting a ball from points $A, B$ and $C$ on the shown paths, will the ball fall into any of the holes after 6 reflections? (The ball reflects with the same angle after contacting the table edges.)

by Hirad Alipanah

IGO 2019 Elementary 2

As shown in the figure, there are two rectangles $ABCD$ and $PQRD$ with the same area, and with parallel corresponding edges. Let points $N,$ $M$ and $T$ be the midpoints of segments $QR,$ $PC$ and $AB$, respectively. Prove that points $N,M$ and $T$ lie on the same line.

by Morteza Saghafian

IGO 2019 Elementary 3

There are $n>2$ lines on the plane in general position; Meaning any two of them meet, but no three are concurrent. All their intersection points are marked, and then all the lines are removed, but the marked points are remained. It is not known which marked point belongs to which two lines. Is it possible to know which line belongs where, and restore them all?

by Boris Frenkin (Russia)

IGO 2019 Elementary 4

Quadrilateral $ABCD$ is given such that $\angle DAC = \angle CAB = 60^\circ,$ and $AB = BD - AC.$ Lines $AB$ and $CD$ intersect each other at point $E$. Prove that $    \angle ADB = 2\angle BEC.$
  

by Iman Maghsoudi

IGO 2019 Elementary 5

For a convex polygon (i.e. all angles less than $180^\circ$) call a diagonal bisector if its bisects both area and perimeter of the polygon. What is the maximum number of bisector diagonals for a convex pentagon?

by Morteza Saghafian

IGO 2019 Intermediate 1

Two circles $\omega_1$ and $\omega_2$ with centers $O_1$ and $O_2$ respectively intersect each other at points $A$ and $B$, and point $O_1$ lies on $\omega_2$. Let $P$ be an arbitrary point lying on $\omega_1$. Lines $BP, AP$ and $O_1O_2$ cut $\omega_2$ for the second time at points $X$, $Y$ and $C$, respectively. Prove that quadrilateral $XPYC$ is a parallelogram.

by Iman Maghsoudi

IGO 2019 Intermediate 2

Find all quadrilaterals $ABCD$ such that all four triangles $DAB$, $CDA$, $BCD$ and $ABC$ are similar to one-another.

by Morteza Saghafian

IGO 2019 Intermediate 3

Three circles $\omega_1$, $\omega_2$ and $\omega_3$ pass through one common point, say $P$. The tangent line to $\omega_1$ at $P$ intersects $\omega_2$ and $\omega_3$ for the second time at points $P_{1,2}$ and $P_{1,3}$, respectively. Points $P_{2,1}$, $P_{2,3}$, $P_{3,1}$ and $P_{3,2}$ are similarly defined. Prove that the perpendicular bisector of segments $P_{1,2}P_{1,3}$, $P_{2,1}P_{2,3}$ and $P_{3,1}P_{3,2}$ are concurrent.

by Mahdi Etesamifard

IGO 2019 Intermediate 4

Let $ABCD$ be a parallelogram and let $K$ be a point on line $AD$ such that $BK=AB$. Suppose that $P$ is an arbitrary point on $AB$, and the perpendicular bisector of $PC$ intersects the circumcircle of triangle $APD$ at points $X$, $Y$. Prove that the circumcircle of triangle $ABK$ passes through the orthocenter of triangle $AXY$.

by Iman Maghsoudi

IGO 2019 Intermediate 5

Let $ABC$ be a triangle with $\angle A = 60^\circ$. Points $E$ and $F$ are the foot of angle bisectors of vertices $B$ and $C$ respectively. Points $P$ and $Q$ are considered such that quadrilaterals $BFPE$ and $CEQF$ are parallelograms. Prove that $\angle PAQ > 150^\circ$. (Consider the angle $PAQ$ that does not contain side $AB$ of the triangle.)

by Alireza Dadgarnia

IGO 2019 Advanced 1

Circles $\omega_1$ and $\omega_2$ intersect each other at points $A$ and $B$. Point $C$ lies on the tangent line from $A$ to $\omega_1$ such that $\angle ABC = 90^\circ$. Arbitrary line $\ell$ passes through $C$ and cuts $\omega_2$ at points $P$ and $Q$. Lines $AP$ and $AQ$ cut $\omega_1$ for the second time at points $X$ and $Z$ respectively. Let $Y$ be the foot of altitude from $A$ to $\ell$. Prove that points $X, Y$ and $Z$ are collinear.

by Iman Maghsoudi

IGO 2019 Advanced 2

Is it true that in any convex $n$-gon with $n > 3$, there exists a vertex and a diagonal passing through this vertex such that the angles of this diagonal with both sides adjacent to this vertex are acute?

by Boris Frenkin (Russia)

IGO 2019 Advanced 3

Circles $\omega_1$ and $\omega_2$ have centres $O_1$ and $O_2$, respectively. These two circles intersect at points $X$ and $Y$. $AB$ is common tangent line of these two circles such that $A$ lies on $\omega_1$ and $B$ lies on $\omega_2$. Let tangents to $\omega_1$ and $\omega_2$ at $X$ intersect $O_1O_2$ at points $K$ and $L$, respectively. Suppose that line $BL$ intersects $\omega_2$ for the second time at $M$ and line $AK$ intersects $\omega_1$ for the second time at $N$. Prove that lines $AM, BN$ and $O_1O_2$ concur.

by Dominik Burek (Poland)

IGO 2019 Advanced 4

Given an acute non-isosceles triangle $ABC$ with circumcircle $\Gamma$. $M$ is the midpoint of segment $BC$ and $N$ is the midpoint of arc $BC$ of $\Gamma$ (the one that doesn't contain $A$). $X$ and $Y$ are points on $\Gamma$ such that $BX\parallel CY\parallel AM$. Assume there exists point $Z$ on segment $BC$ such that circumcircle of triangle $XYZ$ is tangent to $BC$. Let $\omega$ be the circumcircle of triangle $ZMN$. Line $AM$ meets $\omega$ for the second time at $P$. Let $K$ be a point on $\omega$ such that $KN\parallel AM$, $\omega_b$ be a circle that passes through $B$, $X$ and tangents to $BC$ and $\omega_c$ be a circle that passes through $C$, $Y$ and tangents to $BC$. Prove that circle with center $K$ and radius $KP$ is tangent to 3 circles $\omega_b$, $\omega_c$ and $\Gamma$.

by Tran Quan (Vietnam)

IGO 2019 Advanced 5

Let points $A, B$ and $C$ lie on the parabola $\Delta$ such that the point $H$, orthocenter of triangle $ABC$, coincides with the focus of parabola $\Delta$. Prove that by changing the position of points $A, B$ and $C$ on $\Delta$ so that the orthocenter remain at $H$, inradius of triangle $ABC$ remains unchanged.

by Mahdi Etesamifard

IGO  2020 Elementary 1

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.)

by Mahdi Etesamifard

IGO  2020 Elementary 2

A parallelogram $ABCD$ is given ($AB \neq BC$). Points $E$ and $G$ are chosen on the line $\overline{CD}$ such that $\overline{AC}$ is the angle bisector of both angles $\angle EAD$ and $\angle BAG$. The line $\overline{BC}$ intersects $\overline{AE}$ and $\overline{AG}$ at $F$ and $H$, respectively. Prove that the line $\overline{FG}$ passes through the midpoint of $HE$.

by Mahdi Etesamifard

IGO  2020 Elementary 3

According to the figure, three equilateral triangles with side lengths $a,b,c$ have one

common vertex and do not have any other common point. The lengths $x, y$, and $z$ are defined as

in the figure. Prove that $3(x+y+z)>2(a+b+c)$.

by Mahdi Etesamifard

IGO  2020 Elementary 4

Let $P$ be an arbitrary point in the interior of triangle $\triangle ABC$. Lines$\overline{BP}$ and $\overline{CP}$ intersect $\overline{AC}$ and $\overline{AB}$ at $E$ and $F$, respectively. Let $K$ and $L$ be the midpoints of the segments $BF$ and $CE$, respectively. Let the lines through $L$ and $K$ parallel to $\overline{CF}$ and $\overline{BE}$ intersect $\overline{BC}$ at $S$ and $T$, respectively; moreover, denote by $M$ and $N$ the reflection of $S$ and $T$ over the points $L$ and $K$, respectively. Prove that as $P$ moves in the interior of triangle $\triangle ABC$, line $\overline{MN}$ passes through a fixed point.

by Ali Zamani

IGO  2020 Elementary 5

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 $n$ such that there exists a simple polygon with $n$ vertices in which every vertex is visible from exactly $4$ other vertices.

(A simple polygon is a polygon without hole that does not intersect itself.)

by Morteza Saghafian

IGO  2020 Intermediate 1

A trapezoid $ABCD$ is given where $AB$ and $CD$ are parallel. Let $M$ be the midpoint of the segment $AB$. Point $N$ is located on the segment $CD$ such that $\angle ADN = \frac{1}{2} \angle MNC$ and $\angle BCN = \frac{1}{2} \angle MND$. Prove that $N$ is the midpoint of the segment $CD$.

by Alireza Dadgarnia

IGO  2020 Intermediate 2

Let $ABC$ be an isosceles triangle ($AB = AC$) with its circumcenter $O$. Point $N$ is the midpoint of the segment $BC$ and point $M$ is the reflection of the point $N$ with respect to the side $AC$. Suppose that $T$ is a point so that $ANBT$ is a rectangle. Prove that $\angle OMT = \frac{1}{2} \angle BAC$.

by Ali Zamani

IGO  2020 Intermediate 3

In acute-angled triangle $ABC$ ($AC > AB$), point $H$ is the orthocenter and point $M$ is the midpoint of the segment $BC$. The median $AM$ intersects the circumcircle of triangle $ABC$ at $X$. The line $CH$ intersects the perpendicular bisector of $BC$ at $E$ and the circumcircle of the triangle $ABC$ again at $F$. Point $J$ lies on circle $\omega$, passing through $X, E,$ and $F$, such that $BCHJ$ is a trapezoid ($CB \parallel HJ$). Prove that $JB$ and $EM$ meet on $\omega$.

by Alireza Dadgarnia

IGO  2020 Intermediate 4

Triangle $ABC$ is given. An arbitrary circle with center $J$, passing through $B$ and $C$, intersects the sides $AC$ and $AB$ at $E$ and $F$, respectively. Let $X$ be a point such that triangle $FXB$ is similar to triangle $EJC$ (with the same order) and the points $X$ and $C$ lie on the same side of the line $AB$. Similarly, let $Y$ be a point such that triangle $EYC$ is similar to triangle $FJB$ (with the same order) and the points $Y$ and $B$ lie on the same side of the line $AC$. Prove that the line $XY$ passes through the orthocenter of the triangle $ABC$.

by Nguyen Van Linh (Vietnam)

IGO  2020 Intermediate 5

Find all numbers $n \geq 4$ such that there exists a convex polyhedron with exactly $n$ 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 $180^\circ$.)

by Hesam Rajabzadeh

IGO  2020 Advanced 1

Let $M,N,P$ be midpoints of $BC,AC$ and $AB$ of triangle $\triangle ABC$ respectively. $E$ and $F$ are two points on the segment $\overline{BC}$ so that $\angle NEC = \frac{1}{2} \angle AMB$ and $\angle PFB = \frac{1}{2} \angle AMC$. Prove that $AE=AF$.

by Alireza Dadgarnia

IGO  2020 Advanced 2

Let $\triangle ABC$ be an acute-angled triangle with its incenter $I$. Suppose that $N$ is the midpoint of the arc $BAC$ of the circumcircle of triangle $\triangle ABC$, and $P$ is a point such that $ABPC$ is a parallelogram.Let $Q$ be the reflection of $A$ over $N$ and $R$ the projection of $A$ on $\overline{QI}$. Show that the line $\overline{AI}$ is tangent to the circumcircle of triangle $\triangle PQR$

by Patrik Bak (Slovakia)

IGO  2020 Advanced 3

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 $2\sqrt{2}$ 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.)

Note. Weaker results with $2\sqrt{2}$ replaced by some other $c$ may be awarded points depending on the value of $c>2\sqrt{2}$

by Morteza Saghafian

IGO  2020 Advanced 4

Convex circumscribed quadrilateral $ABCD$ with its incenter $I$ is given such that its incircle is tangent to $\overline{AD},\overline{DC},\overline{CB},$ and $\overline{BA}$ at $K,L,M,$ and $N$. Lines $\overline{AD}$ and $\overline{BC}$ meet at $E$ and lines $\overline{AB}$ and $\overline{CD}$ meet at $F$. Let $\overline{KM}$ intersects $\overline{AB}$ and $\overline{CD}$ at $X,Y$, respectively. Let $\overline{LN}$ intersects $\overline{AD}$ and $\overline{BC}$ at $Z,T$, respectively. Prove that the circumcircle of triangle $\triangle XFY$ and the circle with diameter $EI$ are tangent if and only if the circumcircle of triangle $\triangle TEZ$ and the circle with diameter $FI$ are tangent.

by Mahdi Etesamifard

IGO  2020 Advanced 5

Consider an acute-angled triangle $\triangle ABC$ ($AC>AB$) with its orthocenter $H$ and circumcircle $\Gamma$.Points $M$,$P$ are midpoints of $BC$ and $AH$ respectively.The line $\overline{AM}$ meets $\Gamma$ again at $X$ and point $N$ lies on the line $\overline{BC}$ so that $\overline{NX}$ is tangent to $\Gamma$.

Points $J$ and $K$ lie on the circle with diameter $MP$ such that $\angle AJP=\angle HNM$ ($B$ and $J$ lie one the same side of $\overline{AH}$) and circle $\omega_1$, passing through $K,H$, and $J$, and circle $\omega_2$ passing through $K,M$, and $N$, are externally tangent to each other. Prove that the common external tangents of $\omega_1$ and $\omega_2$ meet on the line $\overline{NH}$.

by Alireza Dadgarnia

2021 IGO Elementary p1

With putting the four shapes drawn in the following figure together make a shape with at least two

reflection symmetries.

by Mahdi Etesamifard

2021 IGO Elementary p2

Points $K, L, M, N$ lie on the sides $AB, BC, CD, DA$ of a square $ABCD$, respectively, such that

the area of $KLMN$ is equal to one half of the area of $ABCD$. Prove that some diagonal of $KLMN$

is parallel to some side of $ABCD$.

by Josef Tkadlec (Czech Republic)

2021 IGO Elementary p3

As shown in the following figure, a heart is a shape consist of three semicircles with diameters $AB$,

$BC$ and $AC$ such that $B$ is midpoint of the segment $AC$. A heart $\omega$ is given. Call a pair

$(P, P')$ bisector if $P$ and $P'$ lie on $\omega$ and bisect its perimeter. Let $(P, P')$ and $(Q,Q')$ be

bisector pairs. Tangents at points $P, P', Q$, and $Q'$ to $\omega$ construct a convex quadrilateral $XYZT$.

If the quadrilateral $XYZT$ is inscribed in a circle, find the angle between lines $PP'$ and $QQ'$.

by Mahdi Etesamifard

2021 IGO Elementary p4

In isosceles trapezoid $ABCD$ ($AB \parallel CD$) points $E$ and $F$ lie on the segment $CD$

in such a way that $D, E, F$ and $C$ are in that order and $DE = CF$. Let $X$ and $Y$ be the

reflection of $E$ and $C$ with respect to $AD$ and $AF$. Prove that circumcircles of triangles

$ADF$ and $BXY$ are concentric.

by Iman Maghsoudi

2021 IGO Elementary p5

Let $A_1, A_2, . . . , A_{2021}$ be $2021$ points on the plane, no three collinear and

$$\angle A_1A_2A_3 + \angle A_2A_3A_4 +... + \angle A_{2021}A_1A_2 = 360^o,$$ in which

by the angle $\angle A_{i-1}A_iA_{i+1}$ we mean the one which is less than $180^o$

(assume that $A_{2022} =A_1$ and $A_0 = A_{2021}$). Prove that some of these angles will add

up to $90^o$.

by Morteza Saghafian

2021 IGO Intermediate p1

Let $ABC$ be a triangle with $AB = AC$. Let $H$ be the orthocenter of $ABC$. Point $E$ is the midpoint of $AC$ and point $D$ lies on the side $BC$ such that $3CD = BC$. Prove that $BE \perp HD$.

by Tran Quang Hung (Vietnam)

2021 IGO Intermediate p2

Let $ABCD$ be a parallelogram. Points $E, F$ lie on the sides $AB, CD$ respectively, such that $\angle EDC = \angle FBC$ and $\angle ECD = \angle FAD$. Prove that $AB \geq 2BC$.

by Pouria Mahmoudkhan Shirazi

2021 IGO Intermediate p3

Given a convex quadrilateral $ABCD$ with $AB = BC$and $\angle ABD = \angle BCD = 90$.Let point $E$ be the intersection of diagonals $AC$ and $BD$. Point $F$ lies on the side $AD$ such that $\frac{AF}{F D}=\frac{CE}{EA}$.. Circle $\omega$ with diameter $DF$ and the circumcircle of triangle $ABF$ intersect for the second time at point $K$. Point $L$ is the second intersection of $EF$ and $\omega$. Prove that the line $KL$ passes through the midpoint of $CE$.

by Mahdi Etesamifard and Amir Parsa Hosseini

2021 IGO Intermediate p4

Let $ABC$ be a scalene acute-angled triangle with its incenter $I$ and circumcircle $\Gamma$. Line $AI$ intersects $\Gamma$ for the second time at $M$. Let $N$ be the midpoint of $BC$ and $T$ be the point on $\Gamma$ such that $IN \perp MT$. Finally, let $P$ and $Q$ be the intersection points of $TB$ and $TC$, respectively, with the line perpendicular to $AI$ at $I$. Show that $PB = CQ$.

by Patrik Bak (Slovakia)

2021 IGO Intermediate p5

Consider a convex pentagon $ABCDE$ and a variable point $X$ on its side $CD$. Suppose that points $K, L$ lie on the segment $AX$ such that $AB = BK$ and $AE = EL$ and that the circumcircles of triangles $CXK$ and $DXL$ intersect for the second time at $Y$ . As $X$ varies, prove that all such lines $XY$ pass through a fixed point, or they are all parallel. 

by Josef Tkadlec (Czech Republic)

2021 IGO Advanced p1

Acute-angled triangle $ABC$ with circumcircle $\omega$ is given. Let $D$ be the midpoint of $AC$, $E$ be the foot of altitude from $A$ to $BC$, and $F$ be the intersection point of $AB$ and $DE$. Point $H$ lies on the arc $BC$ of $\omega$ (the one that does not contain $A$) such that $\angle BHE=\angle ABC$. Prove that $\angle BHF=90^\circ$.

by Harris Leung (Hong Kong)

2021 IGO Advanced p2

Two circles $\Gamma_1$ and $\Gamma_2$ meet at two distinct points $A$ and $B$. A line passing through $A$ meets $\Gamma_1$ and $\Gamma_2$ again at $C$ and $D$ respectively, such that $A$ lies between $C$ and $D$. The tangent at $A$ to $\Gamma_2$ meets $\Gamma_1$ again at $E$. Let $F$ be a point on $\Gamma_2$ such that $F$ and $A$ lie on different sides of $BD$, and $2\angle AFC=\angle ABC$. Prove that the tangent at $F$ to $\Gamma_2$, and lines $BD$ and $CE$ are concurrent.

by Tak Wing Ching (Hong Kong)

2021 IGO Advanced p3

Consider a triangle $ABC$ with altitudes $AD, BE$, and $CF$, and orthocenter $H$. Let the perpendicular line from $H$ to $EF$ intersects $EF, AB$ and $AC$ at $P, T$ and $L$, respectively. Point $K$ lies on the side $BC$ such that $BD=KC$. Let $\omega$ be a circle that passes through $H$ and $P$, that is tangent to $AH$. Prove that circumcircle of triangle $ATL$ and $\omega$ are tangent, and $KH$ passes through the tangency point.

by Mahdi Etesamifard

2021 IGO Advanced p4

$2021$ points on the plane in the convex position, no three collinear and no four concyclic, are given. Prove that there exist two of them such that every circle passing through these two points contains at least $673$ of the other points in its interior. (A finite set of points on the plane are in convex position if the points are the vertices of a convex polygon.)

by Morteza Saghafian

2021 IGO Advanced p5

Given a triangle $ABC$ with incenter $I$. The incircle of triangle $ABC$ is tangent to $BC$ at $D$. Let $P$ and $Q$ be points on the side BC such that $\angle PAB = \angle BCA$ and $\angle QAC = \angle ABC$, respectively. Let $K$ and $L$ be the incenter of triangles $ABP$ and $ACQ$, respectively. Prove that $AD$ is the Euler line of triangle $IKL$.

by Le Viet An (Vietnam)

2022 IGO Elementary p1

Find the angles of the pentagon $ABCDE$ in the figure below.

by Morteza Saghafian

2022 IGO Elementary p2

An isosceles trapezoid $ABCD$ $(AB \parallel CD)$ is given. Points $E$ and $F$ lie on the sides $BC$ and $AD$, and the points $M$ and $N$ lie on the segment $EF$ such that $DF = BE$ and $FM = NE$. Let $K$ and $L$ be the foot of perpendicular lines from $M$ and $N$ to $AB$ and $CD$, respectively. Prove that $EKFL$ is a parallelogram.

by Mahdi Etesamifard

2022 IGO Elementary p3

Let $ABCDE$ be a convex pentagon such that $AB = BC = CD$ and $\angle BDE = \angle EAC = 30 ^{\circ}$. Find the possible values of $\angle BEC$.

by Josef Tkadlec (Czech Republic)

2022 IGO Elementary p4

Let $AD$ be the internal angle bisector of triangle $ABC$. The incircles of triangles $ABC$ and $ACD$ touch each other externally. Prove that $\angle ABC > 120^{\circ}$. (Recall that the incircle of a triangle is a circle inside the triangle that is tangent to its three sides.)

by Volodymyr Brayman (Ukraine)

2022 IGO Elementary p5

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.)

by Hesam Rajabzadeh

2022 IGO Intermediate p1

In the figure below we have $AX = BY$ . Prove that $\angle XDA = \angle CDY$ .

by Iman Maghsoudi

2022 IGO Intermediate p2

Two circles $\omega_1$ and $\omega_2$ with equal radius intersect at two points $E$ and $X$. Arbitrary points $C, D$ lie on $\omega_1, \omega_2$. Parallel lines to $XC, XD$ from $E$ intersect $\omega_2, \omega_1$ at $A, B$, respectively. Suppose that $CD$ intersect $\omega_1, \omega_2$ again at $P, Q$, respectively. Prove that $ABPQ$ is cyclic.

by Ali Zamani

2022 IGO Intermediate p3

Let $O$ be the circumcenter of triangle $ABC$. Arbitrary points $M$ and $N$ lie on the sides $AC$ and $BC$, respectively. Points $P$ and $Q$ lie in the same half-plane as point $C$ with respect to the line $MN$, and satisfy $\triangle CMN \sim \triangle PAN \sim \triangle QMB$ (in this exact order). Prove that $OP=OQ$.

by Medeubek Kungozhin (Kazakhstan)

2022 IGO Intermediate p4

We call two simple polygons $P, Q$ $\textit{compatible}$ if there exists a positive integer $k$ such that each of $P, Q$ can be partitioned into $k$ congruent polygons similar to the other one. Prove that for every two even integers $m, n \geq 4$, there are two compatible polygons with $m$ and $n$ sides. (A simple polygon is a polygon that does not intersect itself.)

by Hesam Rajabzadeh

2022 IGO Intermediate p5

Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$ with center $O$. Let $P$ be the intersection of two diagonals $AC$ and $BD$. Let $Q$ be a point lying on the segment $OP$. Let $E$ and $F$ be the orthogonal projections of $Q$ on the lines $AD$ and $BC$, respectively. The points $M$ and $N$ lie on the circumcircle of triangle $QEF$ such that $QM \parallel AC$ and $QN \parallel BD$. Prove that the two lines $ME$ and $NF$ meet on the perpendicular bisector of segment $CD$.

by Tran Quang Hung (Vietnam)

2022 IGO Advanced p1

Four points $A$, $B$, $C$ and $D$ lie on a circle $\omega$ such that $AB=BC=CD$. The tangent line to $\omega$ at point $C$ intersects the tangent line to $\omega$ at $A$ and the line $AD$ at $K$ and $L$. The circle $\omega$ and the circumcircle of triangle $KLA$ intersect again at $M$. Prove that $MA=ML$.

by Mahdi Etesamifard

2022 IGO Advanced p2

We are given an acute triangle $ABC$ with $AB\neq AC$. Let $D$ be a point of $BC$ such that $DA$ is tangent to the circumcircle of $ABC$. Let $E$ and $F$ be the circumcenters of triangles $ABD$ and $ACD$, respectively, and let $M$ be the midpoints $EF$. Prove that the line tangent to the circumcircle of $AMD$ through $D$ is also tangent to the circumcircle of $ABC$.

by Patrik Bak (Slovakia)

2022 IGO Advanced p3

In triangle $ABC$ $(\angle A\neq 90^\circ)$, let $O$, $H$ be the circumcenter and the foot of the altitude from $A$ respectively. Suppose $M$, $N$ are the midpoints of $BC$, $AH$ respectively. Let $D$ be the intersection of $AO$ and $BC$ and let $H'$ be the reflection of $H$ about $M$. Suppose that the circumcircle of $OH'D$ intersects the circumcircle of $BOC$ at $E$. Prove that $NO$ and $AE$ are concurrent on the circumcircle of $BOC$.

by Mehran Talaei

2022 IGO Advanced p4

Let $ABCD$ be a trapezoid with $AB\parallel CD$. Its diagonals intersect at a point $P$. The line passing through $P$ parallel to $AB$ intersects $AD$ and $BC$ at $Q$ and $R$, respectively. Exterior angle bisectors of angles $DBA$, $DCA$ intersect at $X$. Let $S$ be the foot of $X$ onto $BC$. Prove that if quadrilaterals $ABPQ$, $CDQP$ are circumcribed, then $PR=PS$.

by Dominik Burek (Poland)

2022 IGO Advanced p5

Let $ABC$ be an acute triangle inscribed in a circle $\omega$ with center $O$. Points $E$, $F$ lie on its side $AC$, $AB$, respectively, such that $O$ lies on $EF$ and $BCEF$ is cyclic. Let $R$, $S$ be the intersections of $EF$ with the shorter arcs $AB$, $AC$ of $\omega$, respectively. Suppose $K$, $L$ are the reflection of $R$ about $C$ and the reflection of $S$ about $B$, respectively. Suppose that points $P$ and $Q$ lie on the lines $BS$ and $RC$, respectively, such that $PK$ and $QL$ are perpendicular to $BC$. Prove that the circle with center $P$ and radius $PK$ is tangent to the circumcircle of $RCE$ if and only if the circle with center $Q$ and radius $QL$ is tangent to the circumcircle of $BFS$.

by Mehran Talaei

source: igo-official.ir