Navvye/Olympiad-Problems-CSV huggingface dataset:
2010 Iran MO (2nd Round)
In triangle $ABC$ we havev $\angle A=\frac{\pi}{3}$. Construct $E$ and $F$ on continue of $AB$ and $AC$ respectively such that $BE=CF=BC$. Suppose that $EF$ meets circumcircle of $\triangle ACE$ in $K$. ($K\not \equiv E$). Prove that $K$ is on the bisector of $\angle A$.
2010 Iran MO (2nd Round)
A school has $n$ students and some super classes are provided for them. Each student can participate in any number of classes that he/she wants. Every class has at least two students participating in it. We know that if two different classes have at least two common students, then the number of the students in the first of these two classes is different from the number of the students in the second one. Prove that the number of classes is not greater that $\left(n-1\right)^2$.
2009 Iran MO (2nd Round)
Let $ p(x) $ be a quadratic polynomial for which :

$$  |p(x)| \leq 1 \qquad \forall x \in \{-1,0,1\} $$

Prove that:

$$ \ |p(x)|\leq\frac{5}{4} \qquad \forall x \in [-1,1]$$
2009 Iran MO (2nd Round)
In some of the $ 1\times1 $ squares of a square garden $ 50\times50 $ we've grown apple, pomegranate and peach trees (At most one tree in each square). We call a $ 1\times1 $ square a room and call two rooms neighbor if they have one common side. We know that a pomegranate tree has at least one apple neighbor room and a peach tree has at least one apple neighbor room and one pomegranate neighbor room. We also know that an empty room (a room in which there’s no trees) has at least one apple neighbor room and one pomegranate neighbor room and one peach neighbor room.

Prove that the number of empty rooms is not greater than $ 1000. $
2009 Iran MO (2nd Round)
Let $ ABC $ be a triangle and the point $ D $ is on the segment $ BC $ such that $ AD $ is the interior bisector of $ \angle A $. We stretch $ AD $ such that it meets the circumcircle of $ \Delta ABC $ at $ M $. We draw a line from $ D $ such that it meets the lines $ MB,MC $ at $ P,Q $, respectively ($ M $ is not between $ B,P $ and also is not between $ C,Q $).

Prove that $ \angle PAQ\geq\angle BAC $.
2009 Iran MO (2nd Round)
We have a $ (n+2)\times n $ rectangle and we’ve divided it into $ n(n+2) \ \ 1\times1 $ squares. $ n(n+2) $ soldiers are standing on the intersection points ($ n+2 $ rows and $ n $ columns). The commander shouts and each soldier stands on its own location or gaits one step to north, west, east or south so that he stands on an adjacent intersection point. After the shout, we see that the soldiers are standing on the intersection points of a $ n\times(n+2) $ rectangle ($ n $ rows and $ n+2 $ columns) such that the first and last row are deleted and 2 columns are added to the right and left (To the left $1$ and $1$ to the right).

Prove that $ n $ is even.
2009 Iran MO (2nd Round)
Let $ a_1<a_2<\cdots<a_n $ be positive integers such that for every distinct $1\leq{i,j}\leq{n}$ we have $ a_j-a_i $ divides $ a_i $.

Prove that

$$  ia_j\leq{ja_i} \qquad \text{ for }  1\leq{i}<j\leq{n} $$
2009 Iran MO (2nd Round)
$11$ people are sitting around a circle table, orderly (means that the distance between two adjacent persons is equal to others) and $11$ cards with numbers $1$ to $11$ are given to them. Some may have no card and some may have more than $1$ card. In each round, one [and only one] can give one of his cards with number $ i $ to his adjacent person if after and before the round, the locations of the cards with numbers $ i-1,i,i+1 $ don’t make an acute-angled triangle.

(Card with number $0$ means the card with number $11$ and card with number $12$ means the card with number $1$!)

Suppose that the cards are given to the persons regularly clockwise. (Mean that the number of the cards in the clockwise direction is increasing.)

Prove that the cards can’t be gathered at one person.
2008 Iran MO (2nd Round)
In how many ways, can we draw $n-3$ diagonals of a $n$-gon with equal sides and equal angles such that:

$i)$ none of them intersect each other in the polygonal.

$ii)$ each of the produced triangles has at least one common side with the polygonal.
2008 Iran MO (2nd Round)
Let $I_a$ be the $A$-excenter of $\Delta ABC$ and the $A$-excircle of $\Delta ABC$ be tangent to the lines $AB,AC$ at $B',C'$, respectively. $ I_aB,I_aC$ meet $B'C'$ at $P,Q$, respectively. $M$ is the meet point of $BQ,CP$. Prove that the length of the perpendicular from $M$ to $BC$ is equal to $r$ where $r$ is the radius of incircle of $\Delta ABC$.
2008 Iran MO (2nd Round)
Let $a,b,c,$ and $d$ be real numbers such that at least one of $c$ and $d$ is non-zero. Let $ f:\mathbb{R}\to\mathbb{R}$ be a function defined as $f(x)=\frac{ax+b}{cx+d}$. Suppose that for all $x\in\mathbb{R}$, we have $f(x) \neq x$. Prove that if there exists some real number $a$ for which $f^{1387}(a)=a$, then for all $x$ in the domain of $f^{1387}$, we have $f^{1387}(x)=x$. Notice that in this problem,

$$f^{1387}(x)=\underbrace{f(f(\cdots(f(x)))\cdots)}_{\text{1387 times}}.$$

Hint. Prove that for every function $g(x)=\frac{sx+t}{ux+v}$, if the equation $g(x)=x$ has more than $2$ roots, then $g(x)=x$ for all $x\in\mathbb{R}-\left\{\frac{-v}{u}\right\}$.
2008 Iran MO (2nd Round)
$\mathbb{N}$ is the set of positive integers and $a\in\mathbb{N}$. We know that for every $n\in\mathbb{N}$, $4(a^n+1)$ is a perfect cube. Prove that $a=1$.
2008 Iran MO (2nd Round)
We want to choose telephone numbers for a city. The numbers have $10$ digits and $0$ isn’t used in the numbers. Our aim is: We don’t choose some numbers such that every $2$ telephone numbers are different in more than one digit OR every $2$ telephone numbers are different in a digit which is more than $1$. What is the maximum number of telephone numbers which can be chosen? In how many ways, can we choose the numbers in this maximum situation?
2008 Iran MO (2nd Round)
In triangle $ABC$, $H$ is the foot of perpendicular from $A$ to $BC$. $O$ is the circumcenter of $\Delta ABC$. $T,T'$ are the feet of perpendiculars from $H$ to $AB,AC$, respectively. We know that $AC=2OT$. Prove that $AB=2OT'$.
2007 Iran MO (2nd Round)
In triangle $ABC$, $\angle A=90^{\circ}$ and $M$ is the midpoint of $BC$. Point $D$ is chosen on segment $AC$ such that $AM=AD$ and $P$ is the second meet point of the circumcircles of triangles $\Delta AMC,\Delta BDC$. Prove that the line $CP$ bisects $\angle ACB$.
2007 Iran MO (2nd Round)
Two vertices of a cube are $A,O$ such that $AO$ is the diagonal of one its faces. A $n-$run is a sequence of $n+1$ vertices of the cube such that each $2$ consecutive vertices in the sequence are $2$ ends of one side of the cube. Is the $1386-$runs from $O$ to itself less than $1386-$runs from $O$ to $A$ or more than it?
2007 Iran MO (2nd Round)
In a city, there are some buildings. We say the building $A$ is dominant to the building $B$ if the line that connects upside of $A$ to upside of $B$ makes an angle more than $45^{\circ}$ with earth. We want to make a building in a given location. Suppose none of the buildings are dominant to each other. Prove that we can make the building with a height such that again, none of the buildings are dominant to each other. (Suppose the city as a horizontal plain and each building as a perpendicular line to the plain.)
2007 Iran MO (2nd Round)
Prove that for every positive integer $n$, there exist $n$ positive integers such that the sum of them is a perfect square and the product of them is a perfect cube.
2007 Iran MO (2nd Round)
Tow circles $C,D$ are exterior tangent to each other at point $P$. Point $A$ is in the circle $C$. We draw $2$ tangents $AM,AN$ from $A$ to the circle $D$ ($M,N$ are the tangency points.). The second meet points of $AM,AN$ with $C$ are $E,F$, respectively. Prove that $\frac{PE}{PF}=\frac{ME}{NF}$.
2007 Iran MO (2nd Round)
Farhad has made a machine. When the machine starts, it prints some special numbers. The property of this machine is that for every positive integer $n$, it prints exactly one of the numbers $n,2n,3n$. We know that the machine prints $2$. Prove that it doesn't print $13824$.
2006 Iran MO (2nd round)
Let $C_1,C_2$ be two circles such that the center of $C_1$ is on the circumference of $C_2$. Let $C_1,C_2$ intersect each other at points $M,N$. Let $A,B$ be two points on the circumference of $C_1$ such that $AB$ is the diameter of it. Let lines $AM,BN$ meet $C_2$ for the second time at $A',B'$, respectively. Prove that $A'B'=r_1$ where $r_1$ is the radius of $C_1$.
2006 Iran MO (2nd round)
Determine all polynomials $P(x,y)$ with real coefficients such that

$$P(x+y,x-y)=2P(x,y) \qquad \forall x,y\in\mathbb{R}.$$
2006 Iran MO (2nd round)
In the night, stars in the sky are seen in different time intervals. Suppose for every $k$ stars ($k>1$), at least $2$ of them can be seen in one moment. Prove that we can photograph $k-1$ pictures from the sky such that each of the mentioned stars is seen in at least one of the pictures.

(The number of stars is finite. Define the moments that the $n^{th}$ star is seen as $[a_n,b_n]$ that $a_n<b_n$.)
2006 Iran MO (2nd round)
a.) Let $m>1$ be a positive integer. Prove there exist finite number of positive integers $n$ such that $m+n|mn+1$.



b.) For positive integers $m,n>2$, prove that there exists a sequence $a_0,a_1,\cdots,a_k$ from positive integers greater than $2$ that $a_0=m$, $a_k=n$ and $a_i+a_{i+1}|a_ia_{i+1}+1$ for $i=0,1,\cdots,k-1$.
2006 Iran MO (2nd round)
Let $ABCD$ be a convex cyclic quadrilateral. Prove that:

$a)$ the number of points on the circumcircle of $ABCD$, like $M$, such that $\frac{MA}{MB}=\frac{MD}{MC}$ is $4$.

$b)$ The diagonals of  the quadrilateral which is made with these points are perpendicular to each other.
2006 Iran MO (2nd round)
Some books are placed on each other. Someone first, reverses the upper book. Then he reverses the $2$ upper books. Then he reverses the $3$ upper books and continues like this. After he reversed all the books, he starts this operation from the first. Prove that after finite number of movements, the books become exactly like their initial configuration.
2005 Iran MO (2nd round)
Let $n,p>1$ be positive integers and $p$ be prime. We know that $n|p-1$ and $p|n^3-1$. Prove that $4p-3$ is a perfect square.
2005 Iran MO (2nd round)
In triangle $ABC$, $\angle A=60^{\circ}$. The point $D$ changes on the segment $BC$. Let $O_1,O_2$ be the circumcenters of the triangles $\Delta ABD,\Delta ACD$, respectively. Let $M$ be the meet point of $BO_1,CO_2$ and let $N$ be the circumcenter of $\Delta DO_1O_2$. Prove that, by changing $D$ on $BC$, the line $MN$ passes through a constant point.
2005 Iran MO (2nd round)
In one galaxy, there exist more than one million stars. Let $M$ be the set of the distances between any $2$ of them. Prove that, in every moment, $M$ has at least $79$ members. (Suppose each star as a point.)
2005 Iran MO (2nd round)
We have a $2\times n$ rectangle. We call each $1\times1$ square a room and we show the room in the $i^{th}$ row and $j^{th}$ column as $(i,j)$. There are some coins in some rooms of the rectangle. If there exist more than $1$ coin in each room, we can delete $2$ coins from it and add $1$ coin to its right adjacent room OR we can delete $2$ coins from it and add $1$ coin to its up adjacent room. Prove that there exists a finite configuration of allowable operations such that we can put a coin in the room $(1,n)$.
2005 Iran MO (2nd round)
$BC$ is a diameter of a circle and the points $X,Y$ are on the circle such that $XY\perp BC$. The points $P,M$ are on $XY,CY$ (or their stretches), respectively, such that $CY||PB$ and $CX||PM$. Let $K$ be the meet point of the lines $XC,BP$. Prove that $PB\perp MK$.
2005 Iran MO (2nd round)
Find all functions $f:\mathbb{R}^{+}\to \mathbb{R}^{+}$ such that for all positive real numbers $x$ and $y$, the following equation holds:

$$(x+y)f(f(x)y)=x^2f(f(x)+f(y)).$$
2004 Iran MO (2nd round)
$ABC$ is a triangle and $\angle A=90^{\circ}$. Let $D$ be the meet point of the interior bisector of $\angle A$ and $BC$. And let $I_a$ be the $A-$excenter of $\triangle ABC$. Prove that:

$$\frac{AD}{DI_a}\leq\sqrt{2}-1.$$
2004 Iran MO (2nd round)
Let $f:\mathbb{R}^{\geq 0}\to\mathbb{R}$ be a function such that $f(x)-3x$ and $f(x)-x^3$ are ascendant functions. Prove that $f(x)-x^2-x$ is an ascendant function, too.

(We call the function $g(x)$ ascendant, when for every $x\leq{y}$ we have  $g(x)\leq{g(y)}$.)
2004 Iran MO (2nd round)
The road ministry has assigned $80$ informal companies to repair $2400$ roads. These roads connect $100$ cities to each other. Each road is between $2$ cities and there is at most $1$ road between every $2$ cities. We know that each company repairs $30$ roads that it has agencies in each $2$ ends of them. Prove that there exists a city in which $8$ companies have agencies.
2004 Iran MO (2nd round)
$\mathbb{N}$ is the set of positive integers. Determine all functions $f:\mathbb{N}\to\mathbb{N}$ such that for every pair $(m,n)\in\mathbb{N}^2$ we have that:

$$f(m)+f(n) \ | \ m+n .$$
2004 Iran MO (2nd round)
The interior bisector of $\angle A$ from $\triangle ABC$ intersects the side $BC$ and the circumcircle of $\Delta ABC$ at $D,M$, respectively. Let $\omega$ be a circle with center $M$ and radius $MB$. A line passing through $D$, intersects $\omega$ at $X,Y$. Prove that $AD$ bisects $\angle XAY$.
2004 Iran MO (2nd round)
We have a $m\times n$ table and $m\geq{4}$ and we call a $1\times 1$ square a room. When we put an alligator coin in a room, it menaces all the rooms in his column and his adjacent rooms in his row. What's the minimum number of alligator coins required, such that each room is menaced at least by one alligator coin? (Notice that all alligator coins are vertical.)
2003 Iran MO (2nd round)
We call the positive integer $n$ a $3-$stratum number if we can divide the set of its positive divisors into $3$ subsets such that the sum of each subset is equal to the others.

$a)$ Find a $3-$stratum number.

$b)$ Prove that there are infinitely many $3-$stratum numbers.
2003 Iran MO (2nd round)
In a village, there are $n$ houses with $n>2$ and all of them are not collinear. We want to generate a water resource in the village. For doing this, point $A$ is better than point $B$ if the sum of the distances from point $A$ to the houses is less than the sum of the distances from point $B$ to the houses. We call a point ideal if there doesn’t exist any better point than it. Prove that there exist at most $1$ ideal point to generate the resource.
2003 Iran MO (2nd round)
$n$ volleyball teams have competed to each other (each $2$ teams have competed exactly $1$ time.). For every $2$ distinct teams like $A,B$, there exist exactly $t$ teams which have lost their match with $A,B$. Prove that $n=4t+3$. (Notabene that in volleyball, there doesn’t exist tie!)
2003 Iran MO (2nd round)
Let $x,y,z\in\mathbb{R}$ and $xyz=-1$. Prove that:

$$ x^4+y^4+z^4+3(x+y+z)\geq\frac{x^2}{y}+\frac{x^2}{z}+\frac{y^2}{x}+\frac{y^2}{z}+\frac{z^2}{x}+\frac{z^2}{y}. $$
2003 Iran MO (2nd round)
$\angle{A}$ is the least angle in $\Delta{ABC}$. Point $D$ is on the arc $BC$ from the circumcircle of $\Delta{ABC}$. The perpendicular bisectors of the segments $AB,AC$ intersect the line $AD$ at $M,N$, respectively. Point $T$ is the meet point of $BM,CN$. Suppose that $R$ is the radius of the circumcircle of $\Delta{ABC}$. Prove that:

$$ BT+CT\leq{2R}. $$
2003 Iran MO (2nd round)
We have a chessboard and we call a $1\times1$ square a room. A robot is standing on one arbitrary vertex of the rooms. The robot starts to move and in every one movement, he moves one side of a room. This robot has $2$ memories $A,B$. At first, the values of $A,B$ are $0$. In each movement, if he goes up, $1$ unit is added to $A$, and if he goes down, $1$ unit is waned from $A$, and if he goes right, the value of $A$ is added to $B$, and if he goes left, the value of $A$ is waned from $B$. Suppose that the robot has traversed a traverse (!) which hasn’t intersected itself and finally, he has come back to its initial vertex. If $v(B)$ is the value of $B$ in the last of the traverse, prove that in this traverse, the interior surface of the shape that the robot has moved on its circumference is equal to $|v(B)|$.
2002 Iran MO (2nd round)
Let $n \in \mathbb N$ and $A_n$ set of all permutations $(a_1, \ldots, a_n)$ of the set $\{1, 2, \ldots , n\}$ for which

$$k|2(a_1 + \cdots+ a_k), \text{ for all } 1 \leq k \leq n.$$

Find the number of elements of the set $A_n$.



Proposed by Vidan Govedarica, Serbia
2002 Iran MO (2nd round)
A rectangle is partitioned into finitely many small rectangles. We call a point a cross point if it belongs to four different small rectangles. We call a segment on the obtained diagram maximal if there is no other segment containing it. Show that the number of maximal segments plus the number of cross points is $3$ more than the number of small rectangles.
2002 Iran MO (2nd round)
In a convex quadrilateral $ABCD$ with $\angle ABC = \angle ADC = 135^\circ$, points $M$ and $N$ are taken on the rays $AB$ and $AD$ respectively such that $\angle MCD = \angle NCB = 90^\circ$. The circumcircles of triangles $AMN$ and $ABD$ intersect at $A$ and $K$. Prove that $AK \perp KC.$
2002 Iran MO (2nd round)
Let $A$ and $B$ be two fixed points in the plane. Consider all possible convex quadrilaterals $ABCD$ with $AB = BC, AD = DC$, and $\angle ADC = 90^\circ$. Prove that there is a fixed point $P$ such that, for every such quadrilateral $ABCD$ on the same side of $AB$, the line $DC$ passes through $P.$
2002 Iran MO (2nd round)
Let $\delta$ be a symbol such that $\delta \neq 0$ and $\delta^2 = 0$. Define $\mathbb R[\delta] = \{a + b \delta | a, b \in \mathbb R\}$, where $a+ b \delta = c+ d \delta$ if and only if $a = c$ and $b = d$, and define

$$(a + b \delta) + (c + d \delta) = (a + c) + (b + d) \delta,$$$$(a + b \delta) \cdot (c + d \delta) = ac + (ad + bc) \delta.$$

Let $P(x)$ be a polynomial with real coefficients. Show that $P(x)$ has a multiple real root if and only if $P(x)$ has a non-real root in $\mathbb R[\delta].$
2002 Iran MO (2nd round)
Let $G$ be a simple graph with $100$ edges on $20$ vertices. Suppose that we can choose a pair of disjoint edges in $4050$ ways. Prove that $G$ is regular.
2001 Iran MO (2nd round)
Let $n$ be a positive integer and $p$ be a prime number such that $np+1$ is a perfect square. Prove that $n+1$ can be written as the sum of $p$ perfect squares.
2001 Iran MO (2nd round)
Let $ABC$ be an acute triangle. We draw $3$ triangles $B'AC,C'AB,A'BC$ on the sides of $\Delta ABC$ at the out sides such that:

$$ \angle{B'AC}=\angle{C'BA}=\angle{A'BC}=30^{\circ} \ \ \ , \ \ \ \angle{B'CA}=\angle{C'AB}=\angle{A'CB}=60^{\circ} $$

If $M$ is the midpoint of side $BC$, prove that $B'M$ is perpendicular to $A'C'$.
2001 Iran MO (2nd round)
Find all positive integers $n$ such that we can put $n$ equal squares on the plane that their sides are horizontal and vertical and the shape after putting the squares has at least $3$ axises.
2001 Iran MO (2nd round)
Find all polynomials $P$ with real coefficients such that $\forall{x\in\mathbb{R}}$ we have:

$$ P(2P(x))=2P(P(x))+2(P(x))^2. $$
2001 Iran MO (2nd round)
In triangle $ABC$, $AB>AC$. The bisectors of $\angle{B},\angle{C}$ intersect the sides $AC,AB$ at $P,Q$, respectively. Let $I$ be the incenter of $\Delta ABC$. Suppose that $IP=IQ$. How much isthe value of $\angle A$?
2001 Iran MO (2nd round)
Suppose a table with one row and infinite columns. We call each $1\times1$ square a room. Let the table be finite from left. We number the rooms from left to $\infty$. We have put in some rooms some coins (A room can have more than one coin.). We can do $2$ below operations:

$a)$ If in $2$ adjacent rooms, there are some coins, we can move one coin from the left room $2$ rooms to right and delete one room from the right room.

$b)$ If a room whose number is $3$ or more has more than $1$ coin, we can move one of its coins $1$ room to right and move one other coin $2$ rooms to left.



$i)$ Prove that for any initial configuration of the coins, after a finite number of movements, we cannot do anything more.

$ii)$ Suppose that there is exactly one coin in each room from $1$ to $n$. Prove that by doing the allowed operations, we cannot put any coins in the room $n+2$ or the righter rooms.
2000 Iran MO (2nd round)
$21$ distinct numbers are chosen from the set $\{1,2,3,\ldots,2046\}.$ Prove that we can choose three distinct numbers $a,b,c$ among those $21$ numbers such that

$$bc<2a^2<4bc$$
2000 Iran MO (2nd round)
The points $D,E$ and $F$ are chosen on the sides $BC,AC$ and $AB$ of triangle $ABC$, respectively. Prove that triangles $ABC$ and $DEF$ have the same centroid if and only if

$$\frac{BD}{DC} = \frac{CE}{EA}=\frac{AF}{FB}$$
2000 Iran MO (2nd round)
Let $M=\{1,2,3,\ldots, 10000\}.$ Prove that there are $16$ subsets of $M$ such that for every $a \in M,$ there exist $8$ of those subsets that intersection of the sets is exactly $\{a\}.$
2000 Iran MO (2nd round)
Find all positive integers $n$ such that we can divide the set $\{1,2,3,\ldots,n\}$ into three sets with the same sum of members.
2000 Iran MO (2nd round)
In a tetrahedron we know that sum of angles of all vertices is $180^\circ.$ (e.g. for vertex $A$, we have $\angle BAC + \angle CAD + \angle DAB=180^\circ.$)

Prove that faces of this tetrahedron are four congruent triangles.
2000 Iran MO (2nd round)
Super number is a sequence of numbers $0,1,2,\ldots,9$ such that it has infinitely many digits at left. For example $\ldots 3030304$ is a super number. Note that all of positive integers are super numbers, which have zeros before they're original digits (for example we can represent the number $4$ as $\ldots, 00004$). Like positive integers, we can add up and multiply super numbers. For example:

$$  \begin{array}{cc}&  \ \ \ \ldots 3030304 \\ &+ \ldots4571378\\ &\overline{\qquad \qquad \qquad }\\ & \ \ \ \ldots 7601682 \end{array} $$

And



$$ \begin{array}{cl}&  \ \ \ \ldots 3030304 \\ &\times \ldots4571378\\ &\overline{\qquad \qquad \qquad }\\ & \ \ \ \ldots 4242432 \\  & \ \ \ \ldots 212128 \\ & \ \ \ \ldots 90912 \\ & \ \ \ \ldots 0304 \\ & \ \ \ \ldots 128 \\ & \ \ \ \ldots 20 \\ & \ \ \ \ldots 6 \\ &\overline{\qquad \qquad \qquad } \\ & \ \ \   \ldots 5038912 \end{array}$$

a) Suppose that $A$ is a super number. Prove that there exists a super number $B$ such that $A+B=\stackrel{\leftarrow}{0}$ (Note: $\stackrel{\leftarrow}{0}$ means a super number that all of its digits are zero).



b) Find all super numbers $A$ for which there exists a super number $B$ such that $A \times B=\stackrel{\leftarrow}{0}1$ (Note: $\stackrel{\leftarrow}{0}1$ means the super number $\ldots 00001$).



c) Is this true that if $A \times B= \stackrel{\leftarrow}{0}$, then $A=\stackrel{\leftarrow}{0}$ or $B=\stackrel{\leftarrow}{0}$? Justify your answer.
1999 Iran MO (2nd round)
Does there exist a positive integer that is a power of $2$ and we get another power of $2$ by swapping its digits? Justify your answer.
1999 Iran MO (2nd round)
$ABC$ is a triangle with $\angle{B}>45^{\circ}$ , $\angle{C}>45^{\circ}$. We draw the isosceles triangles $CAM,BAN$ on the sides $AC,AB$ and outside the triangle, respectively, such that $\angle{CAM}=\angle{BAN}=90^{\circ}$. And we draw isosceles triangle $BPC$ on the side $BC$ and inside the triangle such that $\angle{BPC}=90^{\circ}$. Prove that $\Delta{MPN}$ is an isosceles triangle, too, and $\angle{MPN}=90^{\circ}$.
1999 Iran MO (2nd round)
We have a $100\times100$ garden and we’ve plant $10000$ trees in the $1\times1$ squares (exactly one in each.). Find the maximum number of trees that we can cut such that on the segment between each two cut trees, there exists at least one uncut tree.
1999 Iran MO (2nd round)
Find all positive integers $m$ such that there exist positive integers $a_1,a_2,\ldots,a_{1378}$ such that:

$$ m=\sum_{k=1}^{1378}{\frac{k}{a_k}}. $$
1999 Iran MO (2nd round)
Let $ABC$ be a triangle and points $P,Q,R$ be on the sides $AB,BC,AC$, respectively. Now, let $A',B',C'$ be on the segments $PR,QP,RQ$ in a way that $AB||A'B'$ , $BC||B'C'$ and $AC||A'C'$. Prove that:

$$ \frac{AB}{A'B'}=\frac{S_{PQR}}{S_{A'B'C'}}. $$

Where $S_{XYZ}$ is the surface of the triangle $XYZ$.