Found problems: 64
2020 Iranian Geometry Olympiad, 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 $\angle OMT = \frac{1}{2} \angle BAC$.
[i]Proposed by Ali Zamani[/i]
2019 Iranian Geometry Olympiad, 5
For a convex polygon (i.e. all angles less than $180^\circ$) call a diagonal [i]bisector[/i] if its bisects both area and perimeter of the polygon. What is the maximum number of bisector diagonals for a convex pentagon?
[i]Proposed by Morteza Saghafian[/i]
2017 Iranian Geometry Olympiad, 2
We have six pairwise non-intersecting circles that the radius of each is at least one (no circle lies in the interior of any other circle). Prove that the radius of any circle intersecting all the six circles, is at least one.
[i]Proposed by Mohammad Ali Abam - Morteza Saghafian[/i]
2019 Iranian Geometry Olympiad, 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?
[i]Proposed by Boris Frenkin - Russia[/i]
2020 Iranian Geometry Olympiad, 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.
[i]Proposed by Ali Zamani[/i]
2017 Iranian Geometry Olympiad, 3
In the regular pentagon $ABCDE$, the perpendicular at $C$ to $CD$ meets $AB$ at $F$. Prove that $AE+AF=BE$.
[i]Proposed by Alireza Cheraghi[/i]
2019 Iranian Geometry Olympiad, 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.
[img]http://s4.picofile.com/file/8372959484/E02.png[/img]
[i]Proposed by Morteza Saghafian[/i]
2019 Iranian Geometry Olympiad, 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.
[i]Proposed by Dominik Burek - Poland[/i]
2018 Iranian Geometry Olympiad, 3
In 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 $CKB$.
[i]Proposed by Mahdi Etesamifard[/i]
2021 Iranian Geometry Olympiad, 4
$2021$ 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.)
2020 Iranian Geometry Olympiad, 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 $AE=AF$.
[i]Proposed by Alireza Dadgarnia[/i]
2019 Iranian Geometry Olympiad, 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.
[i]Proposed by Iman Maghsoudi[/i]
2020 Iranian Geometry Olympiad, 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.
[i]Proposed by Mahdi Etesamifard[/i]
2019 Iranian Geometry Olympiad, 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 \[
\angle ADB = 2\angle BEC.
\]
[i]Proposed by Iman Maghsoudi[/i]
2019 Iranian Geometry Olympiad, 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.
[i]Proposed by Mahdi Etesamifard[/i]
2020 Iranian Geometry Olympiad, 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 $ABC$.
[i]Proposed by Nguyen Van Linh - Vietnam[/i]
2021 Indonesia TST, G
let $ w_1 $ and $ w_2 $ two circles such that $ w_1 \cap w_2 = \{ A , B \} $
let $ X $ a point on $ w_2 $ and $ Y $ on $ w_1 $ such that $ BY \bot BX $
suppose that $ O $ is the center of $ w_1 $ and $ X' = w_2 \cap OX $
now if $ K = w_2 \cap X'Y $ prove $ X $ is the midpoint of arc $ AK $
2020 Iranian Geometry Olympiad, 2
Let $\triangle ABC$ be an acute-angled triangle with its incenter $I$. Suppose that $N$ is the midpoint of the arc $\overarc{BAC}$ 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 $\triangle PQR$
[i]Proposed by Patrik Bak - Slovakia[/i]
2019 Iranian Geometry Olympiad, 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 $\Gamma$.
[i]Proposed by Tran Quan - Vietnam[/i]
2017 Iranian Geometry Olympiad, 1
In triangle $ABC$, the incircle, with center $I$, touches the sides $BC$ 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 $AB=BZ$.
[i]Proposed by Hooman Fattahimoghaddam[/i]
2019 Iranian Geometry Olympiad, 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.)
[i]Proposed by Alireza Dadgarnia[/i]
2019 Iranian Geometry Olympiad, 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?
[i]Proposed by Boris Frenkin - Russia[/i]
2015 Iran Geometry Olympiad, 1
let $ w_1 $ and $ w_2 $ two circles such that $ w_1 \cap w_2 = \{ A , B \} $
let $ X $ a point on $ w_2 $ and $ Y $ on $ w_1 $ such that $ BY \bot BX $
suppose that $ O $ is the center of $ w_1 $ and $ X' = w_2 \cap OX $
now if $ K = w_2 \cap X'Y $ prove $ X $ is the midpoint of arc $ AK $
2018 Iranian Geometry Olympiad, 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 $CX = DY$ .
Proposed by Alireza Dadgarnia
2017 Iranian Geometry Olympiad, 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 $n$?
[i]Proposed by Boris Frenkin (Russia)[/i]