Navvye/Olympiad-Problems-CSV huggingface dataset:
1992 IberoAmerican
Let $\{a_{n}\}_{n \geq 0}$ and $\{b_{n}\}_{n \geq 0}$ be two sequences of integer numbers such that:



i. $a_{0}=0$, $b_{0}=8$.



ii. For every $n \geq 0$, $a_{n+2}=2a_{n+1}-a_{n}+2$, $b_{n+2}=2b_{n+1}-b_{n}$.



iii. $a_{n}^{2}+b_{n}^{2}$ is a perfect square for every $n \geq 0$.



Find at least two values of the pair $(a_{1992},\, b_{1992})$.
1992 IberoAmerican
Given a circle $\Gamma$ and the positive numbers $h$ and $m$,  construct with straight edge and compass a trapezoid inscribed in $\Gamma$, such that it has altitude  $h$ and the sum of its parallel sides is $m$.
1992 IberoAmerican
In a triangle $ABC$, points $A_{1}$ and $A_{2}$ are chosen in the prolongations beyond $A$ of segments $AB$ and $AC$, such that $AA_{1}=AA_{2}=BC$. Define analogously points $B_{1}$, $B_{2}$, $C_{1}$, $C_{2}$. If $[ABC]$ denotes the area of triangle $ABC$, show that $[A_{1}A_{2}B_{1}B_{2}C_{1}C_{2}] \geq 13 [ABC]$.
1991 IberoAmerican
Each vertex of a cube is assigned an 1 or a -1, and each face is assigned the product of the numbers assigned to its vertices. Determine the possible values the sum of these 14 numbers can attain.
1991 IberoAmerican
A square is divided in four parts by two perpendicular lines, in such a way that three of these parts have areas equal to 1. Show that the square has area equal to 4.
1991 IberoAmerican
Let $f: \ [0,\ 1] \rightarrow \mathbb{R}$ be an increasing function satisfying the following conditions:



a) $f(0)=0$;



b) $f\left(\frac{x}{3}\right)=\frac{f(x)}{2}$;



c) $f(1-x)=1-f(x)$.



Determine $f\left(\frac{18}{1991}\right)$.
1991 IberoAmerican
Find a positive integer $n$ with five non-zero different digits, which satisfies to be equal to the sum of all the three-digit numbers that can be formed using the digits of $n$.
1991 IberoAmerican
Let $P(x,\, y)=2x^{2}-6xy+5y^{2}$. Let us say an integer number $a$ is a value of $P$ if there exist integer numbers $b$, $c$ such that $P(b,\, c)=a$.



a) Find all values of $P$ lying between 1 and 100.



b) Show that if $r$ and $s$ are values of $P$, then so is $rs$.
1991 IberoAmerican
Let $M$, $N$ and $P$ be three non-collinear points. Construct using straight edge and compass a triangle for which $M$ and $N$ are the midpoints of two of its sides, and $P$ is its orthocenter.
1990 IberoAmerican
Let $f$ be a function defined for the non-negative integers, such that:



a) $f(n)=0$ if $n=2^{j}-1$ for some $j \geq 0$.

b) $f(n+1)=f(n)-1$ otherwise.



i) Show that for every $n \geq 0$ there exists $k \geq 0$ such that $f(n)+n=2^{k}-1$.

ii) Find $f(2^{1990})$.
1990 IberoAmerican
Let $ABC$ be a triangle. $I$ is the incenter, and the incircle is tangent to $BC$, $CA$, $AB$ at $D$, $E$, $F$, respectively. $P$ is the second point of intersection of $AD$ and the incircle. If $M$ is the midpoint of $EF$, show that $P$, $I$, $M$, $D$ are concyclic.
1990 IberoAmerican
Let $b$, $c$ be integer numbers, and define $f(x)=(x+b)^2-c$.



i) If $p$ is a prime number such that $c$ is divisible by $p$ but not by $p^{2}$, show that for every integer $n$, $f(n)$ is not divisible by $p^{2}$.



ii) Let $q \neq 2$ be a  prime divisor of $c$. If $q$ divides $f(n)$ for some integer $n$, show that for every integer $r$ there exists an integer $n'$ such that $f(n')$ is divisible by $qr$.
1990 IberoAmerican
Let $\Gamma_{1}$ be a circle. $AB$ is a diameter, $\ell$ is the tangent at $B$, and $M$ is a point on $\Gamma_{1}$ other than $A$. $\Gamma_{2}$ is a circle tangent to $\ell$, and also to $\Gamma_{1}$ at $M$.



a) Determine the point of tangency $P$ of $\ell$ and $\Gamma_{2}$ and find the locus of the center of $\Gamma_{2}$ as $M$ varies.



b) Show that there exists a circle that is always orthogonal to $\Gamma_{2}$, regardless of the position of $M$.
1990 IberoAmerican
$A$ and $B$ are two opposite vertices of an $n \times n$ board. Within each small square of the board, the diagonal parallel to $AB$ is drawn, so that the board is divided in $2n^{2}$ equal triangles. A coin moves from $A$ to $B$ along the grid, and for every segment of the grid that it visits, a seed is put in each triangle that contains the segment as a side. The path followed by the coin is such that no segment is visited more than once, and after the coins arrives at $B$, there are exactly two seeds in each of the $2n^{2}$ triangles of the board. Determine all the values of $n$ for which such scenario is possible.
1990 IberoAmerican
Let $f(x)$ be a cubic polynomial with rational coefficients. If the graph of $f(x)$ is tangent to the $x$ axis, prove that the roots of $f(x)$ are all rational.
1989 IberoAmerican
Determine all triples of real numbers that satisfy the following system of equations:

$$x+y-z=-1\\ x^2-y^2+z^2=1\\ -x^3+y^3+z^3=-1$$
1989 IberoAmerican
Let $x,y,z$ be real numbers such that $0\le x,y,z\le\frac{\pi}{2}$. Prove the inequality

$$\frac{\pi}{2}+2\sin x\cos y+2\sin y\cos z\ge\sin 2x+\sin 2y+\sin 2z.$$
1989 IberoAmerican
Let $a,b$ and $c$ be the side lengths of a triangle. Prove that:

$$\frac{a-b}{a+b}+\frac{b-c}{b+c}+\frac{c-a}{c+a}<\frac{1}{16}$$
1989 IberoAmerican
The incircle of the triangle $ABC$ is tangent to sides $AC$ and $BC$ at $M$ and $N$, respectively. The bisectors of the angles at $A$ and $B$ intersect $MN$ at points $P$ and $Q$, respectively. Let $O$ be the incentre of $\triangle ABC$. Prove that $MP\cdot OA=BC\cdot OQ$.
1989 IberoAmerican
Let the function $f$ be defined on the set $\mathbb{N}$ such that



$\text{(i)}\ \ \quad f(1)=1$

$\text{(ii)}\ \quad f(2n+1)=f(2n)+1$

$\text{(iii)}\quad f(2n)=3f(n)$



Determine the set of values taken $f$.
1989 IberoAmerican
Show that the equation $2x^2-3x=3y^2$ has infinitely many solutions in positive integers.
1988 IberoAmerican
The measure of the angles of a triangle are in arithmetic progression and the lengths of its altitudes are as well. Show that such a triangle is equilateral.
1988 IberoAmerican
Let $a,b,c,d,p$ and $q$ be positive integers satisfying $ad-bc=1$ and $\frac{a}{b}>\frac{p}{q}>\frac{c}{d}$.



Prove that:



$(a)$ $q\ge b+d$



$(b)$ If $q=b+d$, then $p=a+c$.
1988 IberoAmerican
Prove that among all possible triangles whose vertices are $3,5$ and $7$ apart from a given point $P$, the ones with the largest perimeter have $P$ as incentre.
1988 IberoAmerican
$\triangle ABC$ is a triangle with sides $a,b,c$. Each side of $\triangle ABC$ is divided in $n$ equal segments. Let $S$ be the sum of the squares of the distances from each vertex to each of the points of division on its opposite side. Show that $\frac{S}{a^2+b^2+c^2}$ is a rational number.
1988 IberoAmerican
Consider all the numbers of the form $x+yt+zt^2$, with $x,y,z$ rational numbers and $t=\sqrt[3]{2}$. Prove that if $x+yt+zt^2\not= 0$, then there exist rational numbers $u,v,w$ such that

$$(x+yt+z^2)(u+vt+wt^2)=1$$
1988 IberoAmerican
Consider all sets of $n$ distinct positive integers, no three of which form an arithmetic progression. Prove that among all such sets there is one which has the largest sum of the reciprocals of its elements.
1987 IberoAmerican
Find the function $f(x)$ such that

$$f(x)^2f\left(\frac{1-x}{x+1}\right) =64x $$

for $x\not=0,x\not=1,x\not=-1$.
1987 IberoAmerican
In a triangle $ABC$, $M$ and $N$ are the respective midpoints of the sides $AC$ and $AB$, and $P$ is the point of intersection of $BM$ and $CN$. Prove that, if it is possible to inscribe a circle in the quadrilateral $AMPN$, then the triangle $ABC$ is isosceles.
1987 IberoAmerican
Prove that if $m,n,r$ are positive integers, and:

$$1+m+n\sqrt{3}=(2+\sqrt{3})^{2r-1}  $$

then $m$ is a perfect square.
1987 IberoAmerican
The sequence $(p_n)$ is defined as follows: $p_1=2$ and for all $n$ greater than or equal to $2$, $p_n$ is the largest prime divisor of the expression $p_1p_2p_3\ldots p_{n-1}+1$.

Prove that every $p_n$ is different from $5$.
1987 IberoAmerican
Let $r,s,t$ be the roots of the equation $x(x-2)(3x-7)=2$. Show that $r,s,t$ are real and positive and determine $\arctan r+\arctan s +\arctan t$.
1987 IberoAmerican
Let $ABCD$ be a convex quadrilateral and let $P$ and $Q$ be the points on the sides $AD$ and $BC$ respectively such that $\frac{AP}{PD}=\frac{BQ}{QC}=\frac{AB}{CD}$.

Prove that the line $PQ$ forms equal angles with the lines $AB$ and $CD$.
1985 IberoAmerican
Find all the triples of integers $ (a, b,c)$ such that:

$$ \begin{array}{ccc}a+b+c &=& 24\\ a^{2}+b^{2}+c^{2}&=& 210\\ abc &=& 440\end{array}$$
1985 IberoAmerican
Let $ P$ be a point in the interior of the equilateral triangle $ \triangle{}ABC$ such that $ PA = 5$, $ PB = 7$, $ PC = 8$. Find the length of the side of the triangle $ ABC$.
1985 IberoAmerican
Find all the roots $ r_{1}$, $ r_{2}$, $ r_{3}$ y $ r_{4}$ of the equation $ 4x^{4}-ax^{3}+bx^{2}-cx+5 = 0$, knowing that they are real, positive and that:

$$ \frac{r_{1}}{2}+\frac{r_{2}}{4}+\frac{r_{3}}{5}+\frac{r_{4}}{8}= 1.$$
1985 IberoAmerican
To each positive integer $ n$ it is assigned a non-negative integer $f(n)$ such that the following conditions are satisfied:



(1) $ f(rs) = f(r)+f(s)$

(2) $ f(n) = 0$, if the first digit (from right to left) of $ n$ is 3.

(3) $ f(10) = 0$.



Find $f(1985)$. Justify your answer.
1985 IberoAmerican
Given an acute triangle $ABC$, let $D$, $E$ and $F$ be points in the lines $BC$, $AC$ and $AB$ respectively. If the lines $AD$, $BE$ and $CF$ pass through $O$ the centre of the circumcircle of the triangle $ABC$, whose radius is $R$, show that:

$$\frac{1}{AD}+\frac{1}{BE}+\frac{1}{CF}=\frac{2}{R}$$
2021 Iranian Geometry Olympiad
With putting the four shapes drawn in the following figure together make a shape with at least two reflection symmetries.





Proposed by Mahdi Etesamifard - Iran
2021 Iranian Geometry Olympiad
Points $K, L, M, N$ lie on the sides $AB, BC, CD, DA$ of a square $ABCD$, respectively, such that the area of $KLMN$ is equal to one half of the area of $ABCD$. Prove that some diagonal of $KLMN$ is parallel to some side of $ABCD$.



Proposed by Josef Tkadlec - Czech Republic
2021 Iranian Geometry Olympiad
As shown in the following figure, a heart is a shape consist of three semicircles with diameters $AB$, $BC$ and $AC$ such that $B$ is midpoint of the segment $AC$. A heart $\omega$ is given. Call a pair $(P, P')$ bisector if $P$ and $P'$ lie on $\omega$ and bisect its perimeter. Let $(P, P')$ and $(Q,Q')$ be bisector pairs. Tangents at points $P, P', Q$, and $Q'$ to $\omega$ construct a convex quadrilateral $XYZT$. If the quadrilateral $XYZT$ is inscribed in a circle, find the angle between lines $PP'$ and $QQ'$.





Proposed by Mahdi Etesamifard - Iran
2021 Iranian Geometry Olympiad
In isosceles trapezoid $ABCD$ ($AB \parallel CD$) points $E$ and $F$ lie on the segment $CD$ in such a way that $D, E, F$ and $C$ are in that order and $DE = CF$. Let $X$ and $Y$ be the reflection of $E$ and $C$ with respect to $AD$ and $AF$. Prove that circumcircles of triangles $ADF$ and $BXY$ are concentric.



Proposed by Iman Maghsoudi - Iran
2021 Iranian Geometry Olympiad
Let $A_1, A_2, . . . , A_{2021}$ be $2021$ points on the plane, no three collinear and $$\angle A_1A_2A_3 + \angle A_2A_3A_4 +... + \angle A_{2021}A_1A_2 = 360^o,$$in which by the angle $\angle A_{i-1}A_iA_{i+1}$ we mean the one which is less than $180^o$ (assume that $A_{2022} =A_1$ and $A_0 = A_{2021}$). Prove that some of these angles will add up to $90^o$.



Proposed by Morteza Saghafian - Iran
2021 Iranian Geometry Olympiad
Let $ABC$ be a triangle with $AB = AC$. Let $H$ be the orthocenter of $ABC$. Point

$E$ is the midpoint of $AC$ and point $D$ lies on the side $BC$ such that $3CD = BC$. Prove that

$BE \perp HD$.



Proposed by Tran Quang Hung - Vietnam
2021 Iranian Geometry Olympiad
Let $ABCD$ be a parallelogram. Points $E, F$ lie on the sides $AB, CD$ respectively,

such that $\angle EDC = \angle FBC$ and $\angle ECD = \angle FAD$. Prove that $AB \geq 2BC$.



Proposed by Pouria Mahmoudkhan Shirazi - Iran
2021 Iranian Geometry Olympiad
Given a convex quadrilateral $ABCD$ with $AB = BC $and $\angle ABD = \angle BCD = 90$.Let point $E$ be the intersection of diagonals $AC$ and $BD$. Point $F$ lies on the side $AD$ such that

$\frac{AF}{F D}=\frac{CE}{EA}$.. Circle $\omega$ with diameter $DF$ and the circumcircle of triangle $ABF$ intersect for the second time at point $K$. Point $L$ is the second intersection of $EF$ and $\omega$. Prove that the line $KL$ passes through the midpoint of $CE$.

Proposed by Mahdi Etesamifard and Amir Parsa Hosseini - Iran
2021 Iranian Geometry Olympiad
Let $ABC$ be a scalene acute-angled triangle with its incenter $I$ and circumcircle $\Gamma$.

Line $AI$ intersects $\Gamma$ for the second time at $M$. Let $N$ be the midpoint of $BC$ and $T$ be the point

on $\Gamma$ such that $IN \perp MT$. Finally, let $P $ and $Q$ be the intersection points of $TB $ and $TC$,

respectively, with the line perpendicular to $AI$ at $I$. Show that $PB = CQ$.

Proposed by Patrik Bak - Slovakia
2021 Iranian Geometry Olympiad
Consider a convex pentagon $ABCDE$ and a variable point $X$ on its side $CD$.

Suppose that points $K, L$ lie on the segment $AX$ such that $AB = BK$ and $AE = EL$ and that

the circumcircles of triangles $CXK$ and $DXL$ intersect for the second time at $Y$ . As $X$ varies,

prove that all such lines $XY$ pass through a fixed point, or they are all parallel.

Proposed by Josef Tkadlec - Czech Republic
2021 Iranian Geometry Olympiad
Acute-angled triangle $ABC$ with circumcircle $\omega$ is given. Let $D$ be the midpoint of $AC$, $E$ be the foot of altitude from $A$ to $BC$, and $F$ be the intersection point of $AB$ and $DE$. Point $H$ lies on the arc $BC$ of $\omega$ (the one that does not contain $A$) such that $\angle BHE=\angle ABC$. Prove that $\angle BHF=90^\circ$.
2021 Iranian Geometry Olympiad
Two circles $\Gamma_1$ and $\Gamma_2$ meet at two distinct points $A$ and $B$. A line passing through $A$ meets $\Gamma_1$ and $\Gamma_2$ again at $C$ and $D$ respectively, such that $A$ lies between $C$ and $D$. The tangent at $A$ to $\Gamma_2$ meets $\Gamma_1$ again at $E$. Let $F$ be a point on $\Gamma_2$ such that $F$ and $A$ lie on different sides of $BD$, and $2\angle AFC=\angle ABC$. Prove that the tangent at $F$ to $\Gamma_2$, and lines $BD$ and $CE$ are concurrent.
2021 Iranian Geometry Olympiad
Consider a triangle $ABC$ with altitudes $AD, BE$, and $CF$, and orthocenter $H$. Let the perpendicular line from $H$ to $EF$ intersects $EF, AB$ and $AC$ at $P, T$ and $L$, respectively. Point $K$ lies on the side $BC$ such that $BD=KC$. Let $\omega$ be a circle that passes through $H$ and $P$, that is tangent to $AH$. Prove that circumcircle of triangle $ATL$ and $\omega$ are tangent, and $KH$ passes through the tangency point.
2021 Iranian Geometry Olympiad
$2021$ 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.)
2021 Iranian Geometry Olympiad
Given a triangle $ABC$ with incenter $I$. The incircle of triangle $ABC$ is tangent to $BC$

at $D$. Let $P$ and $Q$ be points on the side BC such that $\angle PAB = \angle BCA$ and $\angle QAC = \angle ABC$, respectively. Let $K$ and $L$ be the incenter of triangles $ABP$ and $ACQ$, respectively. Prove that $AD$ is the Euler line of triangle $IKL$.
2020 Iranian Geometry Olympiad
By a fold of a polygon-shaped paper, we mean drawing a segment on the paper and folding the paper along that. Suppose that a paper with the following figure is given. We cut the paper along the boundary of the shaded region to get a polygon-shaped paper.

Start with this shaded polygon and make a rectangle-shaped paper from it with at most 5 number

of folds. Describe your solution by introducing the folding lines and drawing the shape after each fold on your solution sheet.

(Note that the folding lines do not have to coincide with the grid lines of the shape.)

Proposed by Mahdi Etesamifard
2020 Iranian Geometry Olympiad
A parallelogram $ABCD$ is given ($AB \neq BC$). Points $E$ and $G$ are chosen on the line $\overline{CD}$ such that $\overline{AC}$ is the angle bisector of both angles $\angle EAD$ and $\angle BAG$. The line $\overline{BC}$ intersects $\overline{AE}$ and $\overline{AG}$ at $F$ and $H$, respectively. Prove that the line $\overline{FG}$ passes through the midpoint of $HE$.

Proposed by Mahdi Etesamifard
2020 Iranian Geometry Olympiad
According to the figure, three equilateral triangles with side lengths $a,b,c$ have one

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

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

Proposed by Mahdi Etesamifard
2020 Iranian Geometry Olympiad
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.

Proposed by Ali Zamani
2020 Iranian Geometry Olympiad
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.)

Proposed by Morteza Saghafian
2020 Iranian Geometry Olympiad
A trapezoid $ABCD$ is given where $AB$ and $CD$ are parallel. Let $M$ be the midpoint of the segment $AB$. Point $N$ is located on the segment $CD$ such that $\angle ADN = \frac{1}{2} \angle MNC$ and $\angle BCN = \frac{1}{2} \angle MND$. Prove that $N$ is the midpoint of the segment $CD$.



Proposed by Alireza Dadgarnia
2020 Iranian Geometry Olympiad
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 $\angle OMT = \frac{1}{2} \angle BAC$.



Proposed by Ali Zamani
2020 Iranian Geometry Olympiad
In acute-angled triangle $ABC$ ($AC > AB$), point $H$ is the orthocenter and point $M$ is the midpoint of the segment $BC$. The median $AM$ intersects the circumcircle of triangle $ABC$ at $X$. The line $CH$ intersects the perpendicular bisector of $BC$ at $E$ and the circumcircle of the triangle $ABC$ again at $F$. Point $J$ lies on circle $\omega$, passing through $X, E,$ and $F$, such that $BCHJ$ is a trapezoid ($CB \parallel HJ$). Prove that $JB$ and $EM$ meet on $\omega$.





Proposed by Alireza Dadgarnia
2020 Iranian Geometry Olympiad
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 $ABC$.



Proposed by Nguyen Van Linh - Vietnam
2020 Iranian Geometry Olympiad
Find all numbers $n \geq 4$ such that there exists a convex polyhedron with exactly $n$ faces, whose all faces are right-angled triangles.

(Note that the angle between any pair of adjacent faces in a convex polyhedron is less than $180^\circ$.)



Proposed by Hesam Rajabzadeh
2020 Iranian Geometry Olympiad
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 $AE=AF$.

Proposed by Alireza Dadgarnia
2020 Iranian Geometry Olympiad
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 $\triangle PQR$

Proposed by Patrik Bak - Slovakia
2020 Iranian Geometry Olympiad
Assume three circles mutually outside each other with the property that every line separating two of them have intersection with the interior of the third one. Prove that the sum of pairwise distances between their centers is at most $2\sqrt{2}$ times the sum of their radii.

(A line separates two circles, whenever the circles do not have intersection with the line and are on different sides of it.)

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

Proposed by Morteza Saghafian
2020 Iranian Geometry Olympiad
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.

Proposed by Mahdi Etesamifard
2020 Iranian Geometry Olympiad
Consider an acute-angled triangle $\triangle ABC$ ($AC>AB$) with its orthocenter $H$ and circumcircle $\Gamma$.Points $M$,$P$ are midpoints of $BC$ and $AH$ respectively.The line $\overline{AM}$ meets $\Gamma$ again at $X$ and point $N$ lies on the line $\overline{BC}$ so that $\overline{NX}$ is tangent to $\Gamma$.

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

Proposed by Alireza Dadgarnia
2019 Iranian Geometry Olympiad
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.)







Proposed by Hirad Alipanah
2019 Iranian Geometry Olympiad
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.







Proposed by Morteza Saghafian
2019 Iranian Geometry Olympiad
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?



Proposed by Boris Frenkin - Russia