Navvye/Olympiad-Problems-CSV huggingface dataset:
2021 Iran MO (2nd Round)
1400 real numbers are given. Prove that one can choose three of them like $x,y,z$ such that :

$$\left|\frac{(x-y)(y-z)(z-x)}{x^4+y^4+z^4+1}\right| < 0.009$$
2021 Iran MO (2nd Round)
Is it possible to arrange 1400 positive integer ( not necessarily distinct )  ,at least one of them being 2021 , around a circle such that any number on this circle equals to the sum of gcd of the two previous numbers and two next numbers? for example , if $a,b,c,d,e$ are five consecutive numbers on this circle , $c=\gcd(a,b)+\gcd(d,e)$
2020 Iran MO (2nd Round)
Let $S$ is a finite set with $n$ elements. We divided $AS$ to $m$ disjoint parts such that if $A$, $B$, $A \cup B$ are in the same part, then $A=B.$ Find the minimum value of $m$.
2020 Iran MO (2nd Round)
let $x,y,z$ be  positive reals , such that $x+y+z=1399$ find the

$$\max( [x]y + [y]z + [z]x ) $$( $[a]$ is the biggest integer not exceeding $a$)
2020 Iran MO (2nd Round)
let $\omega_1$ be a circle with $O_1$ as its center , let $\omega_2$ be a circle passing through $O_1$ with center $O_2$ let $A$ be one of the intersection of $\omega_1$ and $\omega_2$ let $x$ be a line  tangent line to $\omega_1$ passing from $A$ let $\omega_3$ be a circle passing through $O_1,O_2$ with its center on the line $x$ and intersect $\omega_2$ at $P$ (not $O_1$)  prove that the reflection of $P$ through $x$ is on $\omega_1$
2020 Iran MO (2nd Round)
Let $\omega_1$ and $\omega_2$ be two circles that intersect at point $A$ and $B$. Define point $X$ on $\omega_1$ and point $Y$ on $\omega_2$ such that the line $XY$ is tangent to both circles and is closer to $B$. Define points $C$ and $D$ the reflection of $B$ WRT $X$ and $Y$ respectively. Prove that the angle $\angle{CAD}$ is less than $90^{\circ}$
2020 Iran MO (2nd Round)
Call a pair of integers $a$ and $b$ square makers , if $ab+1$ is a perfect square.

Determine for which $n$ is it possible to divide the set $\{1,2, \dots , 2n\}$ into $n$ pairs of square makers.
2020 Iran MO (2nd Round)
Divide a circle into $2n$ equal sections. We call a circle filled if it is filled with the numbers $0,1,2,\dots,n-1$. We call a filled circle  good if it has the following properties:



$i$. Each number $0 \leq a \leq n-1$ is used exactly twice

$ii$. For any $a$ we have that there are exactly $a$ sections between the two sections that have the number $a$ in them.



Here is an example of a good filling for $n=5$ (View attachment)

Prove that there doesn’t exist a good filling for $n=1399$
2019 Iran MO (2nd Round)
We have a rectangle with it sides being a mirror.A light Ray enters from one of the corners of the rectangle and after being reflected several times enters to the opposite corner it started.Prove that at some time the light Ray passed  the center of rectangle(Intersection of diagonals.)
2019 Iran MO (2nd Round)
$ABC$ is an isosceles triangle ($AB=AC$).

Point $X$ is an arbitary point on $BC$.

$Z \in AC$ and $Y \in AB$ such that $\angle BXY = \angle ZXC$. A line parallel to $YZ$ passes trough $B$ and cuts $YZ$ at $T$. Prove that $AT$ bisects $\angle A$.
2019 Iran MO (2nd Round)
$x_1,x_2,...,x_n>1$ are natural numbers and $n \geq 3$

Prove that : $(x_1x_2...x_n)^2 \ne x_1^3 + x_2^3 +...+x_n^3$
2019 Iran MO (2nd Round)
Consider a circle with diameter $AB$ and let $C,D$ be points on its circumcircle such that $C,D$ are not in the same side of $AB$.Consider the parallel line to $AC$ passing from $D$  and let it intersect $AB$ at $E$.Similarly consider the paralell line to $AD$ passing from $C$ and let it intersect $AB$ at $F$.The perpendicular line to $AB$ at $E$ intersects $BC$ at $X$ and the perpendicular line to $AB$ at $F$ intersects $DB$ at $Y$.Prove that the permiter of triangle $AXY$ is twice $CD$.



Remark:This problem is proved to be wrong due to a typo in the exam papers you can find the correct version here.
2019 Iran MO (2nd Round)
Ali and Naqi are playing a game. At first, they have Polynomial $P(x) = 1+x^{1398}$.

Naqi starts. In each turn one can choice natural number $k \in [0,1398]$ in his trun, and add $x^k$ to the polynomial. For example after 2 moves $P$ can be : $P(x) = x^{1398} + x^{300} + x^{100} +1$. If after Ali's turn, there exist $t \in R$ such that $P(t)<0$ then Ali loses the game. Prove that Ali can play forever somehow he never loses the game!
2019 Iran MO (2nd Round)
Consider lattice points of a $6*7$ grid.We start with two points $A,B$.We say two points $X,Y$ connected if one can reflect several times WRT points $A,B$ and reach from $X$ to $Y$.Over all choices of $A,B$ what is the minimum number of connected components?
2018 Iran MO (2nd Round)
Let $P $ be the intersection of $AC $ and $BD $ in isosceles trapezoid  $ABCD $ ($AB\parallel CD$ , $BC=AD $) . The circumcircle of triangle $ABP $ inersects $BC $ for the second time at $X $. Point $Y $ lies on $AX $ such that $DY\parallel BC $. Prove that $\hat {YDA} =2.\hat {YCA} $.
2018 Iran MO (2nd Round)
Let $n$ be odd natural number and $x_1,x_2,\cdots,x_n$ be pairwise distinct numbers. Prove that someone can divide the difference of these number into two sets with equal sum.

( $X=\{\mid x_i-x_j \mid | i<j\}$ )
2018 Iran MO (2nd Round)
Let $a>k$ be natural numbers and $r_1<r_2<\dots r_n,s_1<s_2<\dots <s_n$ be sequences of natural numbers such that:



$(a^{r_1}+k)(a^{r_2}+k)\dots (a^{r_n}+k)=(a^{s_1}+k)(a^{s_2}+k)\dots (a^{s_n}+k)$



Prove that these sequences are equal.
2018 Iran MO (2nd Round)
Find all functions $f:\Bbb {R} \rightarrow \Bbb {R} $ such that:

$$f(x+y)f(x^2-xy+y^2)=x^3+y^3$$for all reals $x, y $.
2018 Iran MO (2nd Round)
Lamps of the hall switch by only five keys. Every key is connected to one or more lamp(s). By switching every key, all connected lamps will be switched too. We know that no two keys have same set of connected lamps with each other. At first all of the lamps are off. Prove that someone can switch just three keys to turn at least two lamps on.
2018 Iran MO (2nd Round)
Two circles $\omega_1,\omega_2$ intersect at $P,Q $. An arbitrary line passing through $P $ intersects $\omega_1 , \omega_2$ at $A,B $ respectively. Another line parallel to  $AB $ intersects $\omega_1$ at $D,F $ and $\omega_2$ at $E,C $ such that $E,F $ lie between  $C,D $.Let $X\equiv AD\cap BE $ and $Y\equiv BC\cap AF $. Let $R $ be the reflection of $P $ about $CD$. Prove that:

a. $R $ lies on $XY $.

b. PR is the bisector of $\hat {XPY}$.
2017 Iran MO (2nd Round)
a) Prove that there doesn't exist sequence $a_1,a_2,a_3,... \in \mathbb{N}$ such that: $\forall i<j: gcd(a_i+j,a_j+i)=1$



b) Let $p$ be an odd prime number. Prove that there exist sequence $a_1,a_2,a_3,... \in \mathbb{N}$ such that: $\forall i<j: p \not | gcd(a_i+j,a_j+i)$
2017 Iran MO (2nd Round)
Let $ABCD$ be an isosceles trapezoid such that $AB \parallel CD$. Suppose that there exists a point $P$ in $ABCD$ such that $\angle APB > \angle ADC$ and  $\angle DPC > \angle ABC$. Prove that $$AB+CD>DA+BC.$$
2017 Iran MO (2nd Round)
Let $n$ be a natural number divisible by $3$. We have a $n \times n$ table and each square is colored either black or white. Suppose that for all $m \times m$ sub-tables from the table ($m > 1$), the number of black squares is not more than white squares. Find the maximum number of black squares.
2017 Iran MO (2nd Round)
Let $x,y$ be two positive real numbers such that $x^4-y^4=x-y$. Prove that

$$\frac{x-y}{x^6-y^6}\leq \frac{4}{3}(x+y).$$
2017 Iran MO (2nd Round)
There are five smart kids sitting around a round table. Their teacher says: "I gave a few apples to some of you, and none of you have the same amount of apple. Also each of you will know the amount of apple that the person to your left and the person to your right has."

The teacher tells the total amount of apples, then asks the kids to guess the difference of the amount of apple that the two kids in front of them have.



$a)$ If the total amount of apples is less than $16$, prove that at least one of the kids will guess the difference correctly.

$b)$ Prove that the teacher can give the total of $16$ apples such that no one can guess the difference correctly.
2017 Iran MO (2nd Round)
Let $ABC$ be a triangle and $X$ be a point on its circumcircle. $Q,P$ lie on a line $BC$ such that $XQ\perp AC , XP\perp AB$. Let $Y$ be the circumcenter of $\triangle XQP$.

Prove that $ABC$ is equilateral triangle if and if only $Y$ moves on a circle when $X$ varies on the circumcircle of $ABC$.
2016 Iran MO (2nd Round)
If $0<a\leq b\leq c$ prove that

$$\frac{(c-a)^2}{6c}\leq \frac{a+b+c}{3}-\frac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}$$
2016 Iran MO (2nd Round)
Let $ABC$ be a triangle such that $\angle C=2\angle B$ and $\omega$ be its circumcircle. a tangent from $A$ to $\omega$ intersect $BC$ at $E$. $\Omega$ is a circle passing throw $B$ that is tangent to $AC$ at $C$. Let $\Omega\cap AB=F$. $K$ is a point on $\Omega$ such that $EK$ is tangent to $\Omega$ ($A,K$ aren't in one side of $BC$). Let $M$ be the midpoint of arc $BC$ of $\omega$ (not containing $A$). Prove that $AFMK$ is a cyclic quadrilateral.



[asy]

import graph; size(15.424606256655986cm); 

real labelscalefactor = 0.5; /* changes label-to-point distance */

pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */ real xmin = -7.905629294221492, xmax = 11.618976962434495, ymin = -5.154837585051625, ymax = 4.0091473316396895;  /* image dimensions */

pen uuuuuu = rgb(0.26666666666666666,0.26666666666666666,0.26666666666666666); 

 /* draw figures */

draw(circle((1.4210145017438194,0.18096629151696939), 2.581514123077079)); 

draw(circle((1.4210145017438194,-1.3302878964546825), 2.8984706754484924)); 

draw(circle((-0.7076932767793396,-0.4161825262831505), 2.9101722408015513), linetype("4 4") + red); 

draw((3.996177869179178,0.)--(-3.839514259733819,0.)); 

draw((3.996177869179178,0.)--(0.07833180472267817,2.385828723227042)); 

draw((0.07833180472267817,2.385828723227042)--(-1.154148865691539,0.)); 

draw((-3.839514259733819,0.)--(-0.6807342461448075,-3.3262298939043657)); 

draw((0.07833180472267817,2.385828723227042)--(-3.839514259733819,0.)); 

 /* dots and labels */

dot((3.996177869179178,0.),blue); 

label("$B$", (4.040279615036859,0.10218054796102663), NE * labelscalefactor,blue); 

dot((-1.154148865691539,0.),blue); 

label("$C$", (-1.3803811057738653,-0.14328333373606214), NE * labelscalefactor,blue); 

dot((1.4210145017438194,1.5681827789938092),linewidth(4.pt)); 

label("$F$", (1.4629088572174203,1.6465574703052102), NE * labelscalefactor); 

dot((0.07833180472267817,2.385828723227042),linewidth(3.pt) + blue); 

label("$A$", (-0.04055741817725232,2.5568193649319144), NE * labelscalefactor,blue); 

dot((-3.839514259733819,0.),linewidth(3.pt)); 

label("$E$", (-4.049800819229713,-0.06146203983703255), NE * labelscalefactor); 

dot((1.4210145017438194,-2.40054783156011),linewidth(4.pt) + uuuuuu); 

label("$M$", (1.4117705485305265,-2.6490604593938434), NE * labelscalefactor,uuuuuu); 

dot((-0.6807342461448075,-3.3262298939043657),linewidth(4.pt)); 

label("$K$", (-0.7871767250058992,-3.5490946922831688), NE * labelscalefactor); 

clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); 

 /* end of picture */[/asy]
2016 Iran MO (2nd Round)
A council has $6$ members and decisions are based on agreeing and disagreeing votes. We call a decision making method an Acceptable way to decide if it satisfies the two following conditions:



Ascending condition: If in some case, the final result is positive, it also stays positive if some one changes their disagreeing vote to agreeing vote.

Symmetry condition: If all members change their votes, the result will also change.



 Weighted Voting for example, is an Acceptable way to decide. In which members are allotted with non-negative weights like $\omega_1,\omega_2,\cdots , \omega_6$ and the final decision is made with comparing the weight sum of the votes for, and the votes against. For instance if $\omega_1=2$ and for all $i\ge2, \omega_i=1$, decision is based on the majority of the votes, and in case when votes are equal, the vote of the first member will be the decider.

Give an example of some Acceptable way to decide method that cannot be represented as a Weighted Voting method.
2016 Iran MO (2nd Round)
Let $l_1,l_2,l_3,...,L_n$ be lines in the plane such that no two of them are parallel and no three of them are concurrent. Let $A$ be the intersection point of lines $l_i,l_j$. We call $A$ an "Interior Point" if there are points $C,D$ on $l_i$ and $E,F$ on $l_j$ such that $A$ is between $C,D$ and $E,F$. Prove that there are at least $\frac{(n-2)(n-3)}{2}$ Interior points.($n>2$)

note: by point here we mean the points which are intersection point of two of $l_1,l_2,...,l_n$.
2016 Iran MO (2nd Round)
$ABCD$ is a quadrilateral such that $\angle ACB=\angle ACD$. $T$ is inside of $ABCD$ such that $\angle ADC-\angle ATB=\angle BAC$ and $\angle ABC-\angle ATD=\angle CAD$. Prove that $\angle BAT=\angle DAC$.
2016 Iran MO (2nd Round)
Find all functions $f: \mathbb N \to \mathbb N$ Such that:

1.for all $x,y\in N$:$x+y|f(x)+f(y)$

2.for all $x\geq 1395$:$x^3\geq 2f(x)$
2015 Iran MO (2nd Round)
Consider a cake in the shape of a circle. It's been divided to some inequal parts by its radii. Arash and Bahram want to eat this cake. At the very first, Arash takes one of the parts. In the next steps, they consecutively pick up a piece adjacent to another piece formerly removed. Suppose that the cake has been divided to 5 parts. Prove that Arash can choose his pieces in such a way at least half of the cake is his.
2015 Iran MO (2nd Round)
There's a special computer and it has a memory. At first, it's memory just contains $x$.

We fill up the memory with the following rules.



1) If $f\neq 0$ is in the memory, then we can also put $\frac{1}{f}$ in it.



2) If $f,g$ are in the memory, then we can also put $ f+g$ and $f-g$ in it.



Find all natural number $n$ such that we can have $x^n$ in the memory.
2015 Iran MO (2nd Round)
Consider a triangle $ABC$ . The points $D,E$ are on sides $AB,AC$ such that $BDEC$ is a cyclic quadrilateral. Let $P$ be the intersection of $BE$ and $CD$. $H$ is a point on $AC$ such that $\angle PHA = 90^{\circ}$. Let $M,N$ be the midpoints of $AP,BC$. Prove that: $ ACD \sim MNH $.
2015 Iran MO (2nd Round)
In quadrilateral $ABCD$ , $AC$ is bisector of $\hat{A}$ and $\widehat{ADC}=\widehat{ACB}$. $X$ and $Y$ are feet of perpendicular from $A$ to $BC$ and $CD$,respectively.Prove that orthocenter of triangle $AXY$ is on $BD$.
2015 Iran MO (2nd Round)
A circle is divided into $2n$ equal by $2n$ points. Ali draws $n+1$ arcs, of length $1,2,\ldots,n+1$. Prove that we can find two arcs, such that one of them is inside in the other one.
2015 Iran MO (2nd Round)
Let $n \ge 50 $ be a natural number. Prove that $n$ is expressible as sum of two natural numbers $n=x+y$, so that for every prime number $p$ such that $ p\mid x$ or $p\mid y $ we have $ \sqrt{n} \ge p $. For example for $n=94$ we have $x=80, y=14$.
2014 Iran MO (2nd Round)
A basket is called "Stuff Basket" if it includes $10$ kilograms of rice and $30$ number of eggs. A market is to distribute $100$ Stuff Baskets. We know that there is totally $1000$ kilograms of rice and $3000$ number of eggs in the baskets, but some of market's baskets include either more or less amount of rice or eggs. In each step, market workers can select two baskets and move an arbitrary amount of rice or eggs between selected baskets. Starting from an arbitrary situation, what's the minimum number of steps that workers provide $100$ Stuff Baskets?
2014 Iran MO (2nd Round)
Let $ABCD$ be a square. Let $N,P$ be two points on sides $AB, AD$, respectively such that $NP=NC$, and let $Q$ be a point on $AN$ such that $\angle QPN = \angle NCB$. Prove that $$ \angle BCQ = \dfrac{1}{2} \angle AQP .$$
2014 Iran MO (2nd Round)
Let $ x,y,z $ be three non-negative real numbers such that $$x^2+y^2+z^2=2(xy+yz+zx). $$ Prove that $$\dfrac{x+y+z}{3} \ge \sqrt[3]{2xyz}.$$
2014 Iran MO (2nd Round)
Find all positive integers $(m,n)$ such that

$$n^{n^{n}}=m^{m}.$$
2014 Iran MO (2nd Round)
A subset $S$ of positive real numbers is called powerful if for any two distinct elements $a, b$ of $S$, at least one of $a^{b}$ or $b^{a}$ is also an element of $S$.

a) Give an example of a four elements powerful set.

b) Prove that every finite powerful set has at most four elements.
2014 Iran MO (2nd Round)
Members of "Professionous Riddlous" society have been divided into some groups, and groups are changed in a special way each weekend: In each group, one of the members is specified as the best member, and the best members of all groups separate from their previous group and form a new group. If a group has only one member, that member joins the new group and the previous group will be removed. Suppose that the society has $n$ members at first, and all the members are in one group. Prove that a week will come, after which number of members of each group will be at most $1+\sqrt{2n}$.
2013 Iran MO (2nd Round)
Find all pairs $(a,b)$ of positive integers for which $\gcd(a,b)=1$, and $\frac{a}{b}=\overline{b.a}$. (For example, if $a=92$ and $b=13$, then $b/a=13.92$ )
2013 Iran MO (2nd Round)
Let $n$ be a natural number and suppose that $ w_1, w_2, \ldots , w_n$ are $n$ weights . We call the set of $\{ w_1, w_2, \ldots , w_n\}$ to be a Perfect Set if we can achieve all of the $1,2, \ldots, W$ weights with sums of $ w_1, w_2, \ldots , w_n$, where $W=\sum_{i=1}^n w_i $. Prove that if we delete the maximum weight of a Perfect Set, the other weights make again a Perfect Set.
2013 Iran MO (2nd Round)
Let $M$ be the midpoint of (the smaller) arc $BC$ in circumcircle of triangle $ABC$. Suppose that the altitude drawn from $A$ intersects the circle at $N$. Draw two lines through circumcenter $O$ of $ABC$ paralell to $MB$ and $MC$, which intersect $AB$ and $AC$ at $K$ and $L$, respectively. Prove that $NK=NL$.
2013 Iran MO (2nd Round)
Let $P$ be a point out of circle $C$. Let $PA$ and $PB$ be the tangents to the circle drawn from $C$. Choose a point $K$ on $AB$ . Suppose that the circumcircle of triangle $PBK$ intersects $C$ again at $T$. Let ${P}'$ be the reflection of $P$ with respect to $A$. Prove that

$$ \angle PBT = \angle {P}'KA $$
2013 Iran MO (2nd Round)
Suppose a $m \times n$ table. We write an integer in each cell of the table. In each move, we chose a column, a row, or a diagonal (diagonal is the set of cells which the difference between their row number and their column number is constant) and add either $+1$ or $-1$ to all of its cells. Prove that if for all arbitrary $3 \times 3$ table we can change all numbers to zero, then we can change all numbers of $m \times n$ table to zero.



(Hint: First of all think about it how we can change number of $ 3 \times 3$ table to zero.)
2013 Iran MO (2nd Round)
Let $\{a_n\}_{n=1}^{\infty}$ be a sequence of positive integers for which

$$ a_{n+2} = \left[\frac{2a_n}{a_{n+1}}\right]+\left[\frac{2a_{n+1}}{a_n}\right]. $$

Prove that there exists a positive integer $m$ such that $a_m=4$ and $a_{m+1} \in\{3,4\}$.



Note. $[x]$ is the greatest integer not exceeding $x$.
2012 Iran MO (2nd Round)
Consider a circle $C_1$ and a point $O$ on it. Circle $C_2$ with center $O$, intersects $C_1$ in two points $P$ and $Q$. $C_3$ is a circle which is externally tangent to $C_2$ at $R$ and internally tangent to $C_1$ at $S$ and suppose that $RS$ passes through $Q$. Suppose $X$ and $Y$ are second intersection points of $PR$ and $OR$ with $C_1$. Prove that $QX$ is parallel with $SY$.
2012 Iran MO (2nd Round)
Suppose $n$ is a natural number. In how many ways can we place numbers $1,2,....,n$ around a circle such that each number is a divisor of the sum of it's two adjacent numbers?
2012 Iran MO (2nd Round)
Prove that if $t$ is a natural number then there exists a natural number $n>1$ such that $(n,t)=1$ and none of the numbers $n+t,n^2+t,n^3+t,....$ are perfect powers.
2012 Iran MO (2nd Round)
a) Do there exist $2$-element subsets $A_1,A_2,A_3,...$ of natural numbers such that each natural number appears in exactly one of these sets and also for each natural number $n$, sum of the elements of $A_n$ equals $1391+n$?



b) Do there exist $2$-element subsets $A_1,A_2,A_3,...$ of natural numbers such that each natural number appears in exactly one of these sets and also for each natural number $n$, sum of the elements of $A_n$ equals $1391+n^2$?



Proposed by Morteza Saghafian
2012 Iran MO (2nd Round)
Consider the second degree polynomial $x^2+ax+b$ with real coefficients. We know that the necessary and sufficient condition for this polynomial to have roots in real numbers is that its discriminant, $a^2-4b$ be greater than or equal to zero. Note that the discriminant is also a polynomial with variables $a$ and $b$. Prove that the same story is not true for polynomials of degree $4$: Prove that there does not exist a $4$ variable polynomial $P(a,b,c,d)$ such that:



The fourth degree polynomial $x^4+ax^3+bx^2+cx+d$ can be written as the product of four $1$st degree polynomials if and only if $P(a,b,c,d)\ge 0$. (All the coefficients are real numbers.)



Proposed by Sahand Seifnashri
2012 Iran MO (2nd Round)
The incircle of triangle $ABC$, is tangent to sides $BC,CA$ and $AB$ in $D,E$ and $F$ respectively. The reflection of $F$ with respect to $B$ and the reflection of $E$ with respect to $C$ are $T$ and $S$ respectively. Prove that the incenter of triangle $AST$ is inside or on the incircle of triangle $ABC$.



Proposed by Mehdi E'tesami Fard
2011 Iran MO (2nd Round)
We have a line and $1390$ points around it such that the distance of each point to the line is less than $1$ centimeters and the distance between any two points is more than $2$ centimeters. prove that there are two points such that their distance is at least $10$ meters ($1000$ centimeters).
2011 Iran MO (2nd Round)
In triangle $ABC$, we have $\angle ABC=60$. The line through $B$ perpendicular to side $AB$ intersects angle bisector of $\angle BAC$ in $D$ and the line through $C$ perpendicular $BC$ intersects angle bisector of $\angle ABC$ in $E$. prove that $\angle BED\le 30$.
2011 Iran MO (2nd Round)
Find all increasing sequences $a_1,a_2,a_3,...$ of natural numbers such that for each $i,j\in \mathbb N$, number of the divisors of $i+j$ and $a_i+a_j$ is equal. (an increasing sequence is a sequence that if $i\le j$, then $a_i\le a_j$.)
2011 Iran MO (2nd Round)
find the smallest natural number $n$ such that there exists $n$ real numbers in the interval $(-1,1)$ such that their sum equals zero and the sum of their squares equals $20$.
2011 Iran MO (2nd Round)
rainbow is the name of a bird. this bird has $n$ colors and it's colors in two consecutive days are not equal. there doesn't exist $4$ days in this bird's life like $i,j,k,l$ such that $i<j<k<l$ and the bird has the same color in days $i$ and $k$ and the same color in days $j$ and $l$ different from the colors it has in days $i$ and $k$. what is the maximum number of days rainbow can live in terms of $n$?
2011 Iran MO (2nd Round)
The line $l$ intersects the extension of $AB$ in $D$ ($D$ is nearer to $B$ than $A$) and the extension of $AC$ in $E$ ($E$ is nearer to $C$ than $A$) of triangle $ABC$. Suppose that reflection of line $l$ to perpendicular bisector of side $BC$ intersects the mentioned extensions in $D'$ and $E'$ respectively. Prove that if $BD+CE=DE$, then $BD'+CE'=D'E'$.
2010 Iran MO (2nd Round)
Let $a,b$ be two positive integers and $a>b$.We know that $\gcd(a-b,ab+1)=1$ and $\gcd(a+b,ab-1)=1$. Prove that $(a-b)^2+(ab+1)^2$ is not a perfect square.
2010 Iran MO (2nd Round)
There are $n$ points in the page such that no three of them are collinear.Prove that number of triangles that vertices of them are chosen from these $n$ points and area of them is 1,is not greater than $\frac23(n^2-n)$.
2010 Iran MO (2nd Round)
Circles $W_1,W_2$ meet at $D$and $P$. $A$ and $B$ are on $W_1,W_2$ respectively, such that $AB$ is tangent to $W_1$ and $W_2$. Suppose $D$ is closer than $P$ to the line $AB$. $AD$ meet circle $W_2$ for second time at $C$. Let $M$ be the midpoint of $BC$. Prove that $\angle{DPM}=\angle{BDC}$.
2010 Iran MO (2nd Round)
Let $P(x)=ax^3+bx^2+cx+d$ be a polynomial with real coefficients such that $$\min\{d,b+d\}> \max\{|{c}|,|{a+c}|\}$$

Prove that $P(x)$ do not have a real root in $[-1,1]$.