geometry problems from Iranian Geometry Olympiad (IGO)

with aops links in the names

by Mahan Tajrobekar

by Mahdi Etesami Fard

by Davood Vakili

by Ali Zooelm

by Mahdi Etesami Fard

by Mahdi Etesami Fard

by Morteza Saghafian

by Davood Vakili

by Mahdi Etesami Fard

by Iman Maghsoudi

by Morteza Saghafian

by Davood Vakili

by Iman Maghsoudi

by Ali Golmakani

by Morteza Saghafian

by Alexey Zaslavsky (Russia)

by Morteza Saghafian

by Morteza Saghafian

by Mahdi Etesami Fard

by Nikolai Beluhov (Bulgaria) and Morteza Saghafian

by Davood Vakili

by Mahdi Etesami Fard

by Davood Vakili

by Hooman Fattahimoghaddam

by Iman Maghsoudi

by Morteza Saghafian

by Boris Frenkin (Russia)

There are three rectangles in the following figure. The lengths of some segments are shown.

Find the length of the segment $XY$ .

with aops links in the names

Iranian Geometry Olympiad problems 2014-2017 ΕΝ in pdf

Iranian Geometry Olympiad all 2014-2019 in pdf with solutions

Iranian Geometry Olympiad all 2014-2019 in pdf with solutions

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 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

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

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

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 $BM +CM > AY$ .

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

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

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

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

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 H

_{1}and H_{2}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 △ABC and CI intersects AB at D. In circumcircle of △ABC, T is midpoint of
arc BAC and BI intersect this circle at M. If MD intersects AT at N, prove
that BM $\perp$ CN.

by Ali Zooelm

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:

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

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

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

by
Morteza Saghafian
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.

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

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.
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

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

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

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

Two circles ω

_{1}and ω_{2}(with centers O_{1}and O_{2}respectively) intersect at A and B. The point X lies on ω_{2}. Let point Y be a point on ω_{1}such that <XBY = 90^{o}. Let X΄ be the second point of intersection of the line O_{1}X and ω_{2}and K be the second point of intersection of X΄Y and ω_{2}. Prove that X is the midpoint of arc AK.by Davood Vakili

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

Let H be the orthocenter of the triangle ABC. Let l

_{1}and l_{2}be two lines passing through H and perpendicular to each other. l_{1}intersects BC and extension of AB at D and Z respectively, and l_{2}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

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 A

_{1},A_{2},A_{3},A_{4}. Find the points B_{1},B_{2},B_{3},B_{4}and C_{1},C_{2},C_{3},C_{4}similarly. Suppose that these 12 points lie on two circles. Prove that the triangle ABC is isosceles.by Morteza Saghafian

Rectangles ABA

_{1}B_{2}, BCB_{1}C_{2}, CAC_{1}A_{2}lie outside triangle ABC. Let C΄ be a point such that C΄A_{1}_{ $\perp$ }A_{1}C_{2}and C΄B_{2}$\perp$ B_{2}C_{1}. Points A΄ and B΄ are defined similarly. Prove that lines AA΄, BB΄, CC΄ concur.by Alexey Zaslavsky (Russia)

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

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

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 $ABCD$.

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

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

In trapezoid ABCD with AB // CD, ω

_{1}and ω_{2 }are two circles with diameters AD and BC, respectively. Let X and Y be two arbitrary points on ω_{1}and ω_{2}, respectively. Show that the length of segment XY is not more than half of the perimeter of ABCD.
by Mahdi Etesami Fard

Let two circles C

_{1}and C_{2}intersect in points A and B. The tangent to C_{1}at A intersects C_{2}in P and the line PB intersects C_{1}for the second time in Q (suppose that Q is outside C_{2}). The tangent to C_{2}from Q intersects C_{1}and C_{2}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

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

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

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

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

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

Let P be the intersection point of sides AD and BC of a convex quadrilateral
ABCD. Suppose that I

_{1}and I_{2}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 AI_{1}B and DHC are tangent together if and only if the circumcircles of triangles AOB and DI_{2}C are tangent together.by Hooman Fattahimoghaddam

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 ω

_{1}is a circle passing through D and tangent to AC at P. Also suppose that ω_{2}is a circle passing through C and tangent to BD at P. Let X be the intersection point of ω_{1}and AD, and Y be the intersection point of ω_{2}and BC. Suppose that the circles ω_{1}and ω_{2}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 XQYby Iman Maghsoudi

Do there exist six points X

_{1},X_{2},Y_{1},Y_{2},Z_{1},Z_{2}in the plane such that all of the triangles X_{i}Y_{j}Z_{k}are similar for 1 ≤ i, j, k ≤ 2 ?by Morteza Saghafian

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

Find the angles of triangle ABC.

by Morteza Saghafian

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

by Alireza Cheraghi

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

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

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

Two circles ω

_{1}, ω_{2 }intersect at A, B. An arbitrary line through B meets ω_{1}, ω_{2 }at C, D respectively. The points E, F are chosen on ω_{1}, ω_{2 }respectively so that CE = CB, BD = DF. Suppose that BF meets ω_{1}at P, and BE meets ω_{2 }at Q. Prove that A, P, Q are collinear.
by Iman Maghsoudi

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)

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

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

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

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

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

Three circles ω

_{1}, ω_{2}, ω_{3}are tangent to line*l*at points A, B, C (B lies between A, C) and ω_{2 }is externally tangent to the other two. Let X, Y be the intersection points of ω_{2 }with the other common external tangent of ω_{1}, ω_{3}. The perpendicular line through B to*l*meets ω_{2 }again at Z. Prove that the circle with diameter AC touches ZX, ZY .
by Iman Maghsoudi - Siamak Ahmadpour

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)

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

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 - Mahdi Etesamifard

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

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

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

Find the length of the segment $XY$ .

by Hirad Aalipanah

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

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

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 - Morteza Saghafian

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

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 $PX = PX'$.

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 Morteza Saghafian

by Fatemeh Sajadi

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

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)

$ABCD$ 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)

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.)

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.

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.$

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.

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

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 $AXY$.

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.

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?

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 $\Gamma$.

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 Hirad Alipanah

by Morteza Saghafian

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

by Iman Maghsoudi

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

by Iman Maghsoudi

by Morteza Saghafian

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

by Iman Maghsoudi

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

$\angle ABC = 90^\circ$. 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

by Boris Frenkin - Russia

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

by Tran Quan - Vietnam

Let points $A, B$ and $C$ lie on the parabola $\Delta$ such that the point $H$, orthocenter of triangle $ABC$, coincideswith 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

source: igo-official.ir

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