Iranian Geometry Olymiad Á.

Ì°íõòóëãà

Àãóóëãà
1 1st Iranian Geometry Olympiad 2014 2
1.1 Junior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Senior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 2st Iranian Geometry Olympiad 2015 3
2.1 Elementary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.3 Advanced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
3 3st Iranian Geometry Olympiad 2016 5
3.1 Elementary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.2 Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.3 Advanced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4 4st Iranian Geometry Olympiad 2017 6
4.1 Elementary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
4.2 medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
4.3 Advenced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
5 5st Iranian Geometry Olympiad 2018 8
5.1 Elementary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
5.2 Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
5.3 Advanced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
6 6st Iranian Geometry Olympiad 2019 10
6.1 Elementary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
6.2 Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
6.3 Advanced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
7 7st Iranian Geometry Olympiad 2020 12
7.1 Elementary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
7.2 Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
7.3 Advanced . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

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1 1st Iranian Geometry Olympiad 2014

1.1 Junior

1. In a right triangle ABC we have ∠A = 90◦ , ∠C = 30◦ . 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 M N ⊥ BC .

by Mahdi Etesami Fard

2. The inscribed circle of 4ABC 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 SSBM D
CN D
= DK
DL

by Mahdi Etesami Fard

3. Each of Mahdi and Morteza has drawn an inscribed 93-gon. Denote the rst one by
4. In a triangle ABC we have ∠C = ∠A + 90◦ . 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
∠EBC = ∠A, ∠EDC = 12 ∠A . Prove that ∠CED = ∠ABC .

5. Two points X, Y lie on the arc BC of the circumcircle of 4ABC (this arc does not contain
A) such that ∠BAX = ∠CAY . Let M denotes the midpoint of the chord AX . Show that
BM + CM > AY .

1.2 Senior

1. In a right triangle ABC we have ∠A = 90◦ , ∠C = 30◦ . 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 M N ⊥ BC .
2. In a quadrilateral ABCD we have ∠B = ∠D = 60◦ . 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 ⇔ ∠M XB = 60◦ .
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 = ∠XY M .
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◦ . Two points E, F respectively on AB, AC at the same side of OP are chosen such
that ∠EXP = ∠ACX, ∠F XO = ∠ABX . If K, L denote the intersection points of EF with the
circumcircle of 4ABC , show thatOP is tangent to the circumcircle of 4KLX .
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 to BC intersect AC, AB at E, F respectively. Let M be the
intersection point of P F and EQ. If H1 and H2 denote the orthocenter of 4BF P and 4CEQ
respectively, show that AM ⊥ H1 H2

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2 2st Iranian Geometry Olympiad 2015

2.1 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◦ and there isn't any hole in it. For example:

2. Let ABC be a triangle with ∠A = 60◦ . The points M, N, K lie on BC, AC, AB respectively such
that BK = KM = M N = N C . If AN = 2AK , nd the values of ∠B and ∠C .
3. In the gure below, we know that AB = CD and BC = 2AD. Prove that ∠BAD = 30◦ .

4. In rectangle ABCD, the points M, N, P, Q lie on AB, BC, CD, DA respectively such that the
area of triangles AQM, BM N, CN P, DP Q are equal. Prove that the quadrilateral M N P Q is
parallelogram.
5. Do there exist 6 circles in the plane such that every circle passes through centers of exactly 3
other circles?

2.2 medium

1. In the gure below, the points P, A, B lie on a circle. The point Q lies inside the circle such that
∠P AQ = 90◦ and P Q = BQ. Prove that the value of ∠AQB − ∠P QA is equal to the arc AB .

2. 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 reection of H with respect to
ED. Prove that the line BF passes through circumcenter of ABC .

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3. 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 ⊥ BC .
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◦ .
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?

2.3 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◦ . Let X be the second point of
0

intersection of the line O1 X and ω2 and K be the second point of intersection of X Y and ω2 .
0

Prove that X is the midpoint of arc AK.

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◦ .
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 Y D k AC
and Y E k AB . Prove that X,Y,Z are collinear.
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.
5. Rectangles ABA1 B2 , BCB1 C2 , CAC1 A2 lie outside triangle ABC. Let C be a point such
0

that C A1 ⊥ A1 C2 and C B2 ⊥ B2 C1 . Points A and B are dened similarly. Prove that lines
0 0 0 0

AA , BB , CC concur.
0 0 0

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3 3st Iranian Geometry Olympiad 2016

3.1 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 nd the shortest path between A and B. Only draw the path and write its length.

2. Let ω be the circumcircle of triangle ABC with AC > AB . Let X be a point on AC and Y be a
point on the circle ω , such that CX = CY = AB . (The points A and Y lie on dierent sides of
the line BC). The line XY intersects ω for the second time in point P. Show that P B = P C .
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.
4. In a right-angled triangle ABC (∠A = 90◦ ), 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 , nd all possible values for angles B and C .
5. Let ABCD be a convex quadrilateral with these properties: ∠ADC = 135◦ and ∠ADB −
∠ABD = 2∠DAB = 4∠CBD. If BC = 2CD, prove that AB = BC + AD.

3.2 Medium

1. In trapezoid ABCD with AB k 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.
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
dierent sides of the line PQ). Show that AD is bisector of the angle ∠CAP .
3. Find all positive integers N such that there exists a triangle which can be dissected into N similar
quadrilaterals.
4. Let ω be the circumcircle of right-angled triangle ABC (∠A = 90◦ ). 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 ∠P KC = 90◦ .
5. Let the circles ω and ω intersect in points A and B. Tangent to circle ω at A intersects ω in
0 0

C and tangent to circle ω at A intersects ω in D. Suppose that the internal bisector of ∠CAD
0

intersects ω and ω at E and F, respectively, and the external bisector of ∠CAD intersects ω
0

and ω in X and Y , respectively. Prove that the perpendicular bisector of XY is tangent to the
0

circumcircle of triangle BEF.

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3.3 Advanced

1. Let the circles ω and ω intersect in A and B. Tangent to circle ω at A intersects ω in C and
0 0

tangent to circle ω at A intersects ω in D. Suppose that the segment CD intersects ω and ω in

0

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,
0

P and Q lie on the same side of the line CD). Prove that the points A, P and Q are collinear.
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◦ (assume that X and C lie on opposite sides
of the line BM). Show that ∠XM B = 2∠M BC .
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 AI1 B and DHC
are tangent together if and only if the circumcircles of triangles AOB and DI2 C are tangent
together.
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 ω2
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 .
5. Do there exist six points X1 , X2 , Y1 , Y2 , Z1 , Z2 in the plane such that all of the triangles Xi Yj Zk
are similar for 1 ≤ i, j, k ≤ 2 ?

4 4st Iranian Geometry Olympiad 2017

4.1 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.

2. Find the angles of triangle ABC.

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Iranian Geometry Olymiad Á.Ì°íõòóëãà

3. In the regular pentagon ABCDE , the perpendicular at C to CD meets AB at F. Prove that

AE + AF = BE .

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?
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 .

4.2 medium

1. Let ABC be an acute-angled triangle with A = 60◦ . Let E, F be the feet of altitudes through B,
C respectively. Prove that CE − BF = 32 (AC − AB).
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 ω1 at P, and BE meets ω2 at Q. Prove that A, P, Q are collinear.

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?
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 .
5. 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
0

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
0

the circumcenter of triangle AP Q.

4.3 Advenced

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 (dierent from AC) intersects AB at Y . If YI and
BC intersect at point Z, prove that AB = BZ.
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.
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.
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 .
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
0

A BCD. Points B , C , D are dened similarly. Prove that A , B , C , D are coplanar and the
0 0 0 0 0 0 0 0

plane A B C D touches S.
0 0 0 0

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5 5st Iranian Geometry Olympiad 2018

5.1 Elementary

1. As shown below, there is a 40 × 30 paper with a lled 10 × 5 rectangle inside of it. We want to
cut out the lled 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 lled 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 nal answer. There is no need to
prove the answer.

2. Convex hexagon A1 A2 A3 A4 A5 A6 lies in the interior of convex hexagon B1 B2 B3 B4 B5 B6 such

that A1 A2 ⊥ B1 B2 , A2 A3 ⊥ B2 B3 , ..., A6 A1 ⊥ B6 B1 . Prove that the areas of simple hexagons
A1 B2 A3 B4 A5 B6 and B1 A2 B3 A4 B5 A6 are equal. (A simple hexagon is a hexagon which does not
intersect itself.)
3. In
√ the given gure, ABCD is a parallelogram. We know that ∠D = 60 , AD = 2 and AB =
◦

3 + 1.Point M is the midpoint of AD. Segment CK is the angle bisector of C. Find the angle
CKB.

4. There are two circles with centers O1 , O2 lie inside of circle ω and are tangent to it. Chord AB
of ω is tangent to these two circles such that they lie on opposite sides of this chord. Prove that
∠O1 AO2 + ∠O1 BO2 > 90◦ .

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?

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(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 gure.) Is it possible to have all segments to be
surrounded?

5.2 Medium

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

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

3. Let ω1 , ω2 be two circles with centers O1 and O2 , respectively. These two circles intersect each
other at points A and B. Line O1B intersects ω2 for the second time at point C, and line O2 A
intersects ω1 for the second time at point D . Let X be the second intersection of AC and ω1 .
Also Y is the second intersection point of BD and ω2 . Prove that CX = DY .
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
P0 = P . In each step, connect Pi to the centroid of one of the faces containing it. This line meets
the perimeter of this face again at point Pi+1 . Continue this process with Pi+1 and the other
face containing Pi+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.)
5. uppose that ABCD is a parallelogram such that ∠DAC = 90◦ . 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 ∠P BA = ∠DBH .

5.3 Advanced

1. Two circles ω1 , ω2 intersect each other at points A, B. Let P Q be a common tangent line of these
two circles with P ∈ ω1 and Q ∈ ω2 . An arbitrary point X lies on ω1 . Line AX intersects ω2 for
the second time at Y . Point Y 6= Y lies on ω2 such that QY = QY . Line Y B intersects ω1 for
0 0 0

the second time at X . Prove that P X = P X

0 0

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2. In acute triangle ABC, ∠A = 45◦ . 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.
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.
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.
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.

6 6st Iranian Geometry Olympiad 2019

6.1 Elementary

1. There is a table in the shape of a 8 × 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
reections? (The ball reects with the same angle after contacting the table edges.)

2. As shown in the gure, there are two rectangles ABCD and P QRD with the same area, and
with parallel corresponding edges. Let points N, M and T be the midpoints of segments QR, P
C and AB, respectively. Prove that points N, M and T lie on the same line.

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?
4. Quadrilateral ABCD is given such that
∠DAC = ∠CAB = 60◦ ,
and
AB = BD − AC.
Lines AB and CD intersect each other at point E. Prove that
∠ADB = 2∠BEC

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5. or a convex polygon (i.e. all angles less than 180◦ ) 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?

6.2 Medium

1. Two circles ω1 and ω2 with centers O1 and O2 respectively intersect each other at points A and
B, and point O1 lies on ω2 . Let P be an arbitrary point lying on ω1 . Lines BP, AP and O1 O2 cut
ω2 for the second time at points X, Y and C, respectively. Prove that quadrilateral XPYC is a
parallelogram.
2. Find all quadrilaterals ABCD such that all four triangles DAB, CDA, BCD and ABC are similar
to one-another
3. Three circles ω1 , ω2 and ω3 pass through one common point, say P. The tangent line to ω1 at P
intersects ω2 and ω3 for the second time at points P1,2 and P1,3 , respectively. Points P2,1 , P2,3 , P3,1
and P3,2 are similarly dened. Prove that the perpendicular bisector of segments P1,2 P1,3 , P2,1 P2,3
and P3,1 P3,2 are concurrent.
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
5. Let ABC be a triangle with ∠A = 60◦ . 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 BF P E and CEQF
are parallelograms. Prove that ∠P AQ > 150◦ . (Consider the angle PAQ that does not contain
side AB of the triangle.)

6.3 Advanced

1. Circles ω1 and ω2 intersect each other at points A and B. Point C lies on the tangent line from
A to ω1 such that ∠ABC = 90◦ . Arbitrary line l passes through C and cuts ω2 at points P and
Q. Lines AP and AQ cut ω1 for the second time at points X and Z respectively. Let Y be the
foot of altitude from A to l. Prove that points X, Y and Z are collinear.
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?
3. Circles ω1 and ω2 have centres O1 and O2 , respectively. These two circles intersect at points X
and Y . AB is common tangent line of these two circles such that A lies on ω1 and B lies on ω2 .
Let tangents to ω1 and ω2 at X intersect O1 O2 at points K and L, respectively. Suppose that
line BL intersects ω2 for the second time at M and line AK intersects ω1 for the second time at
N. Prove that lines AM, BN and O1 O2 concur
4.
5. Let points A, B and C lie on the parabola 4 such that the point H, orthocenter of triangle ABC,
coincides with the focus of parabola 4. Prove that by changing the position of points A, B and
C on 4 so that the orthocenter remain at H, inradius of triangle ABC remains unchanged.

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Iranian Geometry Olymiad Á.Ì°íõòóëãà

7 7st Iranian Geometry Olympiad 2020

7.1 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 gure 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.)

2. A parallelogram ABCD is given (AB 6= BC ). Points E and G are chosen on the line CD such
that AC is the angle bisector of both angles ∠EAD and ∠BAG. The line BC intersects AE and
AG at F and H, respectively. Prove that the line F G passes through the midpoint of HE.
3. According to the gure, 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 dened as in the
gure. Prove that 3(x + y + z) > 2(a + b + c).

4. Let P be an arbitrary point in the interior of triangle ABC. Lines BP and CP intersect AC
and 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 CF and BE intersect BC at S and T,
respectively; moreover, denote by M and N the reection of S and T over the points L and K,
respectively. Prove that as P moves in the interior of triangle ABC, line MN passes through a
xed point.
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.)

7.2 Medium

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 ∠ADN = 12 ∠M N C and ∠BCN =
2 ∠M N D . Prove that N is the midpoint of the segment CD.
1

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Iranian Geometry Olymiad Á.Ì°íõòóëãà

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 reection of the point N with respect to the side AC.
Suppose that T is a point so that ANBT is a rectangle. Prove that ∠OM T = 12 ∠BAC .
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 ω , passing through X, E, and F, such that BCHJ is a trapezoid
(CB k HJ ). Prove that JB and EM meet on ω .
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 EY C is similar to triangle F JB (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.
5. Find all numbers n ≥ 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◦ .)

7.3 Advanced

1. Let M, N, and P be the midpoints of sides BC, AC, and AB of triangle ABC, respectively. E and
F are two points on the segment BC so that N EC = 21 ∠AM B and ∠P F B = 12 ∠AM C . Prove
that AE = AF.
2. Let 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 ABC, and P is a point such that ABP C is a parallelogram.
Let Q be the reection of A over N, and R the projection of A on QI. Show that the line AI is
tangent to the circumcircle of triangle PQR.
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 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 dierent sides of
it.) Note. Weaker results
√ with 2 2 replaced by some other c may be awarded points depending
on the value of c > 2 2.
4. Convex circumscribed quadrilateral ABCD with incenter I is given such that its incircle is tangent
to AD, DC, CB, and BA at K, L, M, and N. Lines AD and BC meet at E and lines AB and
CD meet at F. Let KM intersects AB and CD at X and Y , respectively. Let LN intersects AD
and BC at Z and T, respectively. Prove that the circumcircle of triangle XFY and the circle
with diameter EI are tangent if and only if the circumcircle of triangle TEZ and the circle with
diameter F I are tangent.
5. Consider an acute-angled triangle ABC (AC > AB ) with its orthocenter H and circumcircle Ã.
Points M and P are the midpoints of the segments BC and AH, respectively. The line AM meets
à again at X and point N lies on the line BC so that NX is tangent to Ã. Points J and K lie
on the circle with diameter MP such that ∠AJP = ∠HN M ( B and J lie on the same side of
AH) and circle ω1 , passing through K, H, and J, and circle ω2 , passing through K, M, and N, are
externally tangent to each other. Prove that the common external tangents of ω1 and ω2 meet
on the line NH.

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