Iranian Geometry 2014-18 61p

geometry problems from Iranian Geometry Olympiad (IGO)
with aops links in the names


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
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   $BM +CM > AY$ .

by Mahan Tajrobekar
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
In a quadrilateral ABCD we have <B = <D = 60o. 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 = 60o.

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 = 90o. 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 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 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:  

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.

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

by Morteza Saghafian
In the figure below, the points P,A,B lie on a circle. The point Q lies inside the circle such that <PAQ = 90o and PQ = BQ. Prove that the value of <AQB - <PQA is equal to the arc AB.
 by Davood Vakili
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 ωB and ωC be two semicircles with diameter AC and AB respectively, outside the triangle. Suppose that MK and MN intersect ωC and ωB at X and Y respectively. Let the tangents at X and Y to  ωC and ωB 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 > 90o.

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 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 = 90o. 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
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 > 90o.

by Iman Maghsoudi
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
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
Rectangles ABA1B2, BCB1C2, CAC1A2 lie outside triangle ABC. Let C΄ be a point such that C΄A1 $\perp$  A1C2 and C΄B2 $\perp$ B2C1. Points A΄ and B΄ are defined similarly. Prove that lines AA΄, BB΄, CC΄ concur.

by Alexey Zaslavsky (Russia)
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 $ABCD$.

by Morteza Saghafian
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, 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
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
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 = 90o). 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 = 90o.

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 = 90o (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 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
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
Do there exist six points X1,X2,Y1,Y2,Z1,Z2 in the plane such that all of the triangles XiYjZk 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 = 60o. 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 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
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 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
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 


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
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 $PX = PX'$.


by Morteza Saghafian
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
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)


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