Navvye/Olympiad-Problems-CSV huggingface dataset:
1989 Iran MO (2nd round)
(a) Let $n$ be a positive integer, prove that

$$ \sqrt{n+1} - \sqrt{n} < \frac{1}{2 \sqrt n}$$



(b) Find a positive integer $n$ for which

$$ \bigg\lfloor 1 +\frac{1}{\sqrt 2} +\frac{1}{\sqrt 3} +\frac{1}{\sqrt 4} +  \cdots +\frac{1}{\sqrt n} \bigg\rfloor =12$$
1989 Iran MO (2nd round)
A sphere $S$ with center $O$ and radius $R$ is given. Let $P$ be a fixed point on this sphere. Points $A,B,C$ move on the sphere $S$ such that we have $\angle APB = \angle BPC = \angle CPA = 90^\circ.$ Prove that the plane of triangle $ABC$ passes through a fixed point.
1989 Iran MO (2nd round)
Let $\{a_n\}_{n \geq 1}$ be a sequence in which $a_1=1$ and $a_2=2$ and

$$a_{n+1}=1+a_1a_2a_3 \cdots a_{n-1}+(a_1a_2a_3 \cdots a_{n-1} )^2 \qquad \forall n \geq 2.$$

Prove that

$$\lim_{n \to \infty} \biggl( \frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+\cdots + \frac{1}{a_n} \biggr) =2$$
1989 Iran MO (2nd round)
In a sport competition, $m$ teams have participated. We know that each two teams have competed exactly one time and the result is winning a team and losing the other team (i.e. there is no equal result). Prove that there exists a team $x$ such that for each team $y,$ either $x$ wins $y$ or there exists a team $z$ for which $x$ wins $z$ and $z$ wins $y.$



[i.e. prove that in every tournament there exists a king.]
1989 Iran MO (2nd round)
Let $n$ be a positive integer. Prove that the polynomial

$$P(x)= \frac{x^n}{n!}+\frac{x^{n-1}}{(n-1)!}+...+x+1 $$

Does not have any rational root.
1989 Iran MO (2nd round)
A line $d$ is called faithful to triangle $ABC$ if $d$ be in plane of triangle $ABC$ and the reflections of $d$ over the sides of $ABC$ be concurrent. Prove that for any two triangles with acute angles lying in the same plane, either there exists exactly one faithful line to both of them, or there exist infinitely faithful lines to them.
1988 Iran MO (2nd round)
(a) Prove that for all positive integers $m,n$ we have

$$\sum_{k=1}^n k(k+1)(k+2)\cdots (k+m-1)=\frac{n(n+1)(n+2) \cdots (n+m)}{m+1}$$



(b) Let $P(x)$ be a polynomial with rational coefficients and degree $m.$ If $n$ tends to infinity, then prove that

$$\frac{\sum_{k=1}^n P(k)}{n^{m+1}}$$

Has a limit.
1988 Iran MO (2nd round)
In a cyclic quadrilateral $ABCD$, let $I,J$ be the midpoints of diagonals $AC, BD$ respectively and let $O$ be the center of the circle inscribed in $ABCD.$ Prove that $I, J$ and $O$ are collinear.
1988 Iran MO (2nd round)
Let $f : \mathbb N \to \mathbb N$ be a function satisfying

$$f(f(m)+f(n))=m+n, \quad \forall m,n \in \mathbb N.$$Prove that $f(x)=x$ for all $x \in \mathbb N$.
1988 Iran MO (2nd round)
Let $\{a_n \}_{n=1}^{\infty}$ be a sequence such that $a_1=\frac 12$ and

$$a_n=\biggl( \frac{2n-3}{2n} \biggr) a_{n-1} \qquad \forall n \geq 2.$$

Prove that for every positive integer $n,$ we have $\sum_{k=1}^n a_k <1.$
1988 Iran MO (2nd round)
In tetrahedron $ABCD$ let $h_a, h_b, h_c$ and $h_d$ be the lengths of the altitudes from each vertex to the opposite side of that vertex. Prove that

$$\frac{1}{h_a} <\frac{1}{h_b}+\frac{1}{h_c}+\frac{1}{h_d}.$$
1988 Iran MO (2nd round)
Let $n$ be a positive integer. $1369^n$ positive rational numbers are given with this property: if we remove one of the numbers, then we can divide remain numbers into $1368$ sets with equal number of elements such that the product of the numbers of the sets be equal. Prove that all of the numbers are equal.
1987 Iran MO (2nd round)
Solve the following system of equations in positive integers

$$\left\{\begin{array}{cc}a^3-b^3-c^3=3abc\\ \\ a^2=2(b+c)\end{array}\right.$$
1987 Iran MO (2nd round)
Let $f$ be a real function defined in the interval $[0, +\infty )$ and suppose that there exist two functions $f', f''$ in the interval $[0, +\infty )$ such that

$$f''(x)=\frac{1}{x^2+f'(x)^2 +1} \qquad \text{and} \qquad f(0)=f'(0)=0.$$

Let $g$ be a function for which

$$g(0)=0 \qquad \text{and} \qquad g(x)=\frac{f(x)}{x}.$$

Prove that $g$ is bounded.
1987 Iran MO (2nd round)
In the following diagram, let $ABCD$ be a square and let $M,N,P$ and $Q$ be the midpoints of its sides. Prove that

$$S_{A'B'C'D'} = \frac 15 S_{ABCD}.$$

[asy]

import graph; size(200); real lsf = 0.5; pen dp = linewidth(0.7) + fontsize(10); defaultpen(dp); pen ds = black; pen qqttzz = rgb(0,0.2,0.6); pen qqzzff = rgb(0,0.6,1);

draw((0,4)--(4,4),qqttzz+linewidth(1.6pt)); draw((4,4)--(4,0),qqttzz+linewidth(1.6pt)); draw((4,0)--(0,0),qqttzz+linewidth(1.6pt)); draw((0,0)--(0,4),qqttzz+linewidth(1.6pt)); draw((0,4)--(2,0),qqzzff+linewidth(1.2pt)); draw((2,4)--(4,0),qqzzff+linewidth(1.2pt)); draw((0,2)--(4,4),qqzzff+linewidth(1.2pt)); draw((0,0)--(4,2),qqzzff+linewidth(1.2pt));

dot((0,4),ds); label("$A$", (0.07,4.12), NE*lsf); dot((0,0),ds); label("$D$", (-0.27,-0.37), NE*lsf); dot((4,0),ds); label("$C$", (4.14,-0.39), NE*lsf); dot((4,4),ds); label("$B$", (4.08,4.12), NE*lsf); dot((2,4),ds); label("$M$", (2.08,4.12), NE*lsf); dot((4,2),ds); label("$N$", (4.2,1.98), NE*lsf); dot((2,0),ds); label("$P$", (1.99,-0.49), NE*lsf); dot((0,2),ds); label("$Q$", (-0.48,1.9), NE*lsf); dot((0.8,2.4),ds); label("$A'$", (0.81,2.61), NE*lsf); dot((2.4,3.2),ds); label("$B'$", (2.46,3.47), NE*lsf); dot((3.2,1.6),ds); label("$C'$", (3.22,1.9), NE*lsf); dot((1.6,0.8),ds); label("$D'$", (1.14,0.79), NE*lsf); clip((-4.44,-11.2)--(-4.44,6.41)--(16.48,6.41)--(16.48,-11.2)--cycle);

[/asy]



[$S_{X}$ denotes area of the $X.$]
1987 Iran MO (2nd round)
Calculate the product:

$$A=\sin 1^\circ \times \sin 2^\circ \times \sin 3^\circ \times \cdots \times \sin 89^\circ$$
1987 Iran MO (2nd round)
Find all continuous functions $f: \mathbb R \to \mathbb R$ such that

$$f(x^2-y^2)=f(x)^2 + f(y)^2, \quad \forall x,y \in \mathbb R.$$
1987 Iran MO (2nd round)
Let $L_1, L_2, L_3, L_4$ be four lines in the space such that no three of them are in the same plane. Let $L_1, L_2$ intersect in $A$, $L_2,L_3$ intersect in $B$ and $L_3, L_4$ intersect in $C.$ Find minimum and maximum number of lines in the space that intersect $L_1, L_2, L_3$ and $L_4.$ Justify your answer.
1986 Iran MO (2nd round)
Let $f$ be a function such that

$$f(x)=\frac{(x^2-2x+1) \sin \frac{1}{x-1}}{\sin \pi x}.$$

Find the limit of $f$ in the point $x_0=1.$
1986 Iran MO (2nd round)
(a) Sketch the diagram of the function $f$ if

$$f(x)=4x(1-|x|) , \quad |x| \leq 1.$$



(b) Does there exist derivative of $f$ in the point $x=0 \ ?$



(c) Let $g$ be a function such that

$$g(x)=\left\{\begin{array}{cc}\frac{f(x)}{x} \quad :  x \neq 0\\ \text{ } \\ 4 \ \ \ \ \quad : x=0\end{array}\right.$$

Is the function $g$ continuous in the point $x=0 \ ?$



(d) Sketch the diagram of $g.$
1986 Iran MO (2nd round)
Find the smallest positive integer for which when we move the last right digit of the number to the left, the remaining number be $\frac 32$ times of the original number.
1986 Iran MO (2nd round)
Find all positive integers $n$ for which the number $1!+2!+3!+\cdots+n!$ is a perfect power of an integer.
1986 Iran MO (2nd round)
We have erasers, four pencils, two note books and three pens and we want to divide them between two persons so that every one receives at least one of the above stationery. In how many ways is this possible? [Note that the are not distinct.]
1986 Iran MO (2nd round)
$O$ is a point in the plane. Let $O'$ be an arbitrary point on the axis $Ox$ of the plane and let $M$ be an arbitrary point. Rotate $M$, $90^\circ$ clockwise around $O$ to get the point $M'$ and rotate $M$, $90^\circ$ anticlockwise around $O'$ to get the point $M''.$ Prove that the midpoint of the segment $MM''$ is a fixed point.
1986 Iran MO (2nd round)
In a trapezoid $ABCD$, the legs $AB$ and $CD$ meet in $M$ and the diagonals $AC$ and $BD$ meet in $N.$ Let $AC=a$ and $BC=b.$ Find the area of triangles $AMD$ and $AND$ in terms of $a$ and $b.$
1986 Iran MO (2nd round)
Prove that

$$\arctan \frac 12 +\arctan \frac 13 = \frac{\pi}{4}.$$
1985 Iran MO (2nd round)
Let $\alpha $ be an angle such that $\cos \alpha = \frac pq$, where $p$ and $q$ are two integers. Prove that the number $q^n \cos n \alpha$ is an integer.
1985 Iran MO (2nd round)
Let $x, y$ and $z$ be three positive real numbers for which

$$x^2+y^2+z^2=xy+yz+zx.$$

Find the value of $\frac{\sqrt x}{\sqrt x + \sqrt y+ \sqrt z}.$
1985 Iran MO (2nd round)
Let $f: \mathbb R \to \mathbb R,g: \mathbb R \to \mathbb R$ and $\varphi: \mathbb R \to \mathbb R$ be three ascendant functions such that

$$f(x) \leq g(x) \leq \varphi(x) \qquad \forall x \in \mathbb R.$$

Prove that

$$f(f(x)) \leq g(g(x)) \leq \varphi(\varphi(x)) \qquad \forall x \in \mathbb R.$$



Note. The function is $k(x)$ ascendant if for every $ x,y \in D_k, x \leq {y}$ we have  $g(x)\leq{g(y)}$.
1985 Iran MO (2nd round)
Let $x$ and $y$ be two real numbers. Prove that the equations

$$\lfloor x \rfloor + \lfloor y \rfloor =\lfloor x +y \rfloor , \quad \lfloor -x \rfloor + \lfloor -y \rfloor =\lfloor -x-y \rfloor$$

Holds if and only if at least one of $x$ or $y$ be integer.
1985 Iran MO (2nd round)
Let $f: \mathbb R \to \mathbb R$ and $g: \mathbb R \to \mathbb R$ be two functions satisfying

$$\forall x,y \in \mathbb R: \begin{cases} f(x+y)=f(x)f(y),\\ f(x)= x g(x)+1\end{cases} \quad \text{and} \quad \lim_{x \to 0} g(x)=1.$$Find the derivative of $f$ in an arbitrary point $x.$
1985 Iran MO (2nd round)
Inscribe in the triangle $ABC$ a triangle with minimum perimeter.
1985 Iran MO (2nd round)
In the triangle $ABC$ the length of side $AB$, and height $AH$ are known. also we know that $\angle B = 2 \angle C.$ Plot this triangle.
1985 Iran MO (2nd round)
Find the angle between two common sections of the page $2x+y-z=0$ and the cone $4x^2-y^2+3z^2=0.$
1985 Iran MO (2nd round)
Let $G$ be a group and let $a$ be a constant member of it. Prove that

$$G_a = \{x | \exists n \in \mathbb Z , x=a^n\}$$

Is a subgroup of $G.$
1985 Iran MO (2nd round)
In the Archery with an especial gun, the probability of goal is $90 \%.$ If we continue our work until we goal.



i) What is the probability which exactly $3$ balls consumed.



ii) What is the probability which at least $3$ balls consumed.
1985 Iran MO (2nd round)
In The ring $\mathbf R$, we have $\forall x \in \mathbf R : x^2=x$. Prove that in this ring



i) Every element is equals to its additive inverse.



ii) This ring has commutative property.
1985 Iran MO (2nd round)
Let $a,b$ and $c$ be real numbers with $b,c >0.$ Prove that if $ a<b \ ( a>b),$ then

$$\frac{a+c}{b+c} > \frac ab \qquad ( \frac{a+c}{b+c} < \frac ab) $$

And then prove that $\frac{a+c}{b+c}$ is between $1$ and $\frac ab.$
1984 Iran MO (2nd round)
Let $f$ and $g$ be two functions such that

$$f(x)=\frac{1}{\lfloor | x | \rfloor}, \quad g(x)=\frac{1}{|\lfloor  x \rfloor |}.$$

Find the domains of $f$ and $g$ and then prove that

$$\lim_{x \to -1^+} f(x)= \lim_{x \to 1^- } g(x).$$
1984 Iran MO (2nd round)
Consider the function

$$f(x)= \sin \biggl( \frac{\pi}{2} \lfloor x \rfloor \biggr).$$

Find the period of $f$ and sketch diagram of $f$ in one period. Also prove that $\lim_{x \to 1} f(x)$ does not exist.
1984 Iran MO (2nd round)
Let $f : \mathbb R \to \mathbb R$ be a function such that

$$f(x+y)=f(x) \cdot f(y) \qquad \forall x,y \in \mathbb R$$

Suppose that $f(0) \neq 0$ and $f(0)$ exists and it is finite $(f(0) \neq \infty)$. Prove that $f$ has derivative in each point $x \in \mathbb R.$
1984 Iran MO (2nd round)
Find number of terms when we expand $(a+b+c)^{99}$ (in the general case).
1984 Iran MO (2nd round)
Suppose that

$$S_n=\frac 59 \times \frac{14}{20} \times \frac{27}{35} \times \cdots \times \frac{2n^2-n-1}{2n^2+n-1}$$

Find $\lim_{n \to \infty} S_n.$
1984 Iran MO (2nd round)
Let $D$ and $D'$ be two lines with the equations

$$\frac{x-1}{2} = \frac{y-1}{3} = \frac{z-1}{4} \quad \text{and} \quad \frac{x+1}{2} = \frac{y+2}{4} = \frac{z-1}{3}.$$

Find the length of their common perpendicular.
1984 Iran MO (2nd round)
Let $B$ and $C$ be two fixed point on the plane $P.$ Find the locus of the points $M$ on the plane $P$ for which $MB^2 + kMC^2 = a^2.$ ($k$ and $a$ are two given numbers and $k>0.$)
1984 Iran MO (2nd round)
Define the operation $\bigoplus$ on the set of real numbers such that

$$x \bigoplus y = x+y-xy \qquad \forall x,y \in \mathbb R.$$

Prove that this operation is associative.
1983 Iran MO (2nd round)
Let $f, g : \mathbb  R \to \mathbb R$ be two functions such that $g\circ f : \mathbb R \to \mathbb R$ is an injective  function. Prove that $f$ is also injective.
1983 Iran MO (2nd round)
Prove that the number $x = \sqrt{1 + \sqrt 2}$ is irrational.
1983 Iran MO (2nd round)
Find a matrix $A_{(2 \times 2)}$ for which

$$ \begin{bmatrix}2 &1 \\ 3 & 2\end{bmatrix} A  \begin{bmatrix}3 & 2 \\ 4 & 3\end{bmatrix} =  \begin{bmatrix}1 & 2 \\ 2 & 1\end{bmatrix}.$$
1983 Iran MO (2nd round)
The point $M$ moves such that the sum of squares of the lengths from $M$ to faces of a cube, is fixed. Find the locus of $M.$
1983 Iran MO (2nd round)
Find the value of $S_n= \arctan \frac 12 + \arctan \frac 18+ \arctan \frac {1}{18} + \cdots + \arctan \frac {1}{2n^2}.$ Also find $\lim_{n \to \infty} S_n.$
1983 Iran MO (2nd round)
Suppose that

$$f(x)=\{\begin{array}{cc}n,& \qquad n \in \mathbb N , x= \frac 1n\\ \text{} \\x, & \mbox{otherwise}\end{array}$$



i) In which points, the function has a limit?



ii) Prove that there does not exist limit of $f$ in the point $x=0.$
1983 Iran MO (2nd round)
Find the sum $\sum_{i=1}^{\infty} \frac{n}{2^n}.$
2021 Iran MO (3rd Round)
An acute triangle $ABC$ is given. Let $D$ be the foot of altitude dropped for $A$. Tangents from $D$ to circles with diameters $AB$ and $AC$ intersects with the said circles at $K$ and $L$, in respective. Point $S$ in the plane is given so that $\angle ABC + \angle ABS = \angle ACB + \angle ACS = 180^\circ$. Prove that $A, K, L$ and $S$ lie on a circle.
2021 Iran MO (3rd Round)
Given an acute triangle $ABC$ let $M$ be the midpoint of $AB$. Point $K$ is given on the other side of line $AC$ from that of point $B$ such that $\angle KMC = 90 ^ \circ $ and $\angle KAC = 180^\circ - \angle ABC$. The tangent to circumcircle of triangle $ABC$ at $A$ intersects line $CK$ at $E$. Prove that the reflection of line $BC$ with respect to $CM$ passes through the midpoint of line segment $ME$.
2021 Iran MO (3rd Round)
Given triangle $ABC$ variable points $X$ and $Y$ are chosen on segments $AB$ and $AC$, respectively. Point $Z$ on line $BC$ is chosen such that $ZX=ZY$. The circumcircle of $XYZ$ cuts the line $BC$ for the second time at $T$. Point $P$ is given on line $XY$ such that $\angle PTZ = 90^ \circ$. Point $Q$ is on the same side of line $XY$ with $A$ furthermore $\angle QXY = \angle ACP$ and $\angle QYX = \angle ABP$. Prove that the circumcircle of triangle $QXY$ passes through a fixed point (as $X$ and $Y$ vary).
2021 Iran MO (3rd Round)
Is it possible to arrange a permutation of Integers on the integer lattice infinite from both sides such that each row is increasing from left to right and each column increasing from up to bottom?
2021 Iran MO (3rd Round)
Positive real numbers $a, b, c$ and $d$ are given such that $a+b+c+d = 4$ prove that

$$\frac{ab}{a^2-\frac{4}{3}a+\frac{4}{3}} + \frac{bc}{b^2-\frac{4}{3}b+ \frac{4}{3}} + \frac{cd}{c^2-\frac{4}{3}c+ \frac{4}{3}} + \frac{da}{d^2-\frac{4}{3}d+ \frac{4}{3}}\leq 4.$$
2021 Iran MO (3rd Round)
If $a, b, c$ and $d$ are complex non-zero numbers such that

$$2|a-b|\leq |b|, 2|b-c|\leq |c|, 2|c-d| \leq |d| , 2|d-a|\leq |a|.$$Prove that

$$\frac{7}{2} <\left| \frac{b}{a} + \frac{c}{b} + \frac{d}{c} + \frac{a}{d} \right| .$$
2021 Iran MO (3rd Round)
Polynomial $P$ with non-negative real coefficients and function $f:\mathbb{R}^+\to \mathbb{R}^+$ are given such that for all $x, y\in \mathbb{R}^+$ we have

$$f(x+P(x)f(y)) = (y+1)f(x)$$(a) Prove that $P$ has degree at most 1.

(b) Find all function $f$ and non-constant polynomials $P$ satisfying the equality.
2021 Iran MO (3rd Round)
For a natural number $n$, $f(n)$ is defined as the number of positive integers less than $n$ which are neither coprime to $n$ nor a divisor of it. Prove that for each positive integer $k$ there exist only finitely many $n$ satisfying $f(n) = k$.
2021 Iran MO (3rd Round)
Find all functions $f:\mathbb{N} \to \mathbb{N}$ such that for any two positive integers $a$ and $b$ we have $$ f^a(b) + f^b(a) \mid 2(f(ab) +b^2 -1)$$Where $f^n(m)$ is defined in the standard iterative manner.
2021 Iran MO (3rd Round)
$x_1$ is a natural constant. Prove that there does not exist any natural number $m> 2500$ such that the recursive sequence $\{x_i\} _{i=1} ^ \infty $ defined by $x_{n+1} = x_n^{s(n)} + 1$ becomes eventually periodic modulo $m$. (That is there does not exist natural numbers $N$ and $T$ such that for each $n\geq N$, $m\mid x_n - x_{n+T}$).

($s(n)$ is the sum of digits of $n$.)
2021 Iran MO (3rd Round)
Is it possible to arrange natural numbers 1 to 8 on vertices of a cube such that each number divides sum of the three numbers sharing an edge with it?
2021 Iran MO (3rd Round)
Given an acute triangle $ABC$, let $AD$ be an altitude and $H$ the orthocenter. Let $E$ denote the reflection of $H$ with respect to $A$. Point $X$ is chosen on the circumcircle of triangle $BDE$ such that $AC\| DX$ and point $Y$ is chosen on the circumcircle of triangle $CDE$ such that $DY\| AB$. Prove that the circumcircle of triangle $AXY$ is tangent to that of $ABC$.
2021 Iran MO (3rd Round)
Find all functions $f: \mathbb{Q}[x] \to \mathbb{R}$ such that:

(a) for all $P, Q \in \mathbb{Q}[x]$, $f(P \circ Q) = f(Q \circ P);$

(b) for all $P, Q \in \mathbb{Q}[x]$ with $PQ \neq 0$, $f(P\cdot Q) = f(P) + f(Q).$



($P \circ Q$ indicates $P(Q(x))$.)
2021 Iran MO (3rd Round)
Arash and Babak play the following game, taking turns alternatively, on a $1400\times 1401$ table. Arash starts and in his turns he colors $k$, $L$-corners (any three cell of a square). Babak in his turn colors one $2\times 2$ square. Neither player is allowed to recolor any cell. Find all positive integers $k$ for which Arash has a winning strategy.
2020 Iran MO (3rd Round)
find all functions from the reals to themselves. such that for every real $x,y$.

$$f(y-f(x))=f(x)-2x+f(f(y))$$
2020 Iran MO (3rd Round)
let $a_1,a_2,...,a_n$,$b_1,b_2,...,b_n$,$c_1,c_2,...,c_n$ be real numbers. prove that

$$ \sum_{cyc}{ \sqrt{\sum_{i \in \{1,...,n\} }{ (3a_i-b_i-c_i)^2}}} \ge \sum_{cyc}{\sqrt{\sum_{i \in \{1,2,...,n\}}{a_i^2}}}$$
2020 Iran MO (3rd Round)
find all $k$ distinct integers $a_1,a_2,...,a_k$ such that there exists an injective function $f$ from reals to themselves such that for each positive integer $n$ we have

$$\{f^n(x)-x| x \in \mathbb{R} \}=\{a_1+n,a_2+n,...,a_k+n\}$$.
2020 Iran MO (3rd Round)
We call a polynomial $P(x)$ intresting if there are $1398$ distinct positive integers $n_1,...,n_{1398}$ such that

$$P(x)=\sum_{}{x^{n_i}}+1$$Does there exist infinitly many polynomials $P_1(x),P_2(x),...$ such that for each distinct $i,j$ the polynomial $P_i(x)P_j(x)$ is interesting.
2020 Iran MO (3rd Round)
Let $ABCD$ be a Rhombus and let $w$ be it's incircle. Let $M$ be the midpoint of $AB$ the point $K$ is on $w$ and inside $ABCD$ such that $MK$ is tangent to $w$. Prove that $CDKM$ is cyclic.
2020 Iran MO (3rd Round)
Triangle $ABC$ with it's circumcircle $\Gamma$ is given. Points $D$ and $E$ are chosen on segment $BC$ such that $\angle BAD=\angle CAE$. The circle $\omega$ is tangent to $AD$ at $A$ with it's circumcenter lies on $\Gamma$. Reflection of $A$ through $BC$ is $A'$. If the line $A'E$ meet $\omega$ at $L$ and $K$. Then prove either $BL$ and $CK$ or $BK$ and $CL$ meet on $\Gamma$.
2020 Iran MO (3rd Round)
The circle $\Omega$ with center $I_A$, is the $A$-excircle of triangle $ABC$. Which is tangent to $AB,AC$ at $F,E$ respectivly. Point $D$ is the reflection of $A$ through $I_AB$. Lines $DI_A$ and $EF$ meet at $K$. Prove that ,circumcenter of $DKE$ , midpoint of $BC$ and $I_A$ are collinear.
2020 Iran MO (3rd Round)
Triangle $ABC$ is given. Let $O$ be it's circumcenter. Let $I$ be the center of it's incircle.The external angle bisector of $A$ meet $BC$ at $D$. And $I_A$ is the $A$-excenter . The point $K$ is chosen on the line $AI$ such that $AK=2AI$ and $A$ is closer to $K$ than $I$. If the segment $DF$ is the diameter of the circumcircle of triangle $DKI_A$, then prove $OF=3OI$.
2020 Iran MO (3rd Round)
$1)$. Prove a graph with $2n$ vertices and $n+2$ edges has an independent set of size $n$ (there are $n$ vertices such that no two of them are adjacent ).

$2)$.Find the number of graphs with $2n$ vertices and $n+3$ edges , such that among any $n$ vertices there is an edge connecting two of them
2020 Iran MO (3rd Round)
For each $n$ find the number of ways one can put the numbers $\{1,2,3,...,n\}$ numbers on the circle, such that if for any $4$ numbers $a,b,c,d$ where $n|a+b-c-d$. The segments joining $a,b$ and $c,d$ do not meet inside the circle. (Two ways are said to be identical , if one can be obtained from rotaiting the other)
2020 Iran MO (3rd Round)
Consider a latin square of size $n$. We are allowed to choose a $1 \times 1$ square in the table, and add $1$ to any number on the same row and column as the chosen square (the original square will be counted aswell) , or we can add $-1$ to all of them instead. Can we with doing finitly many operation , reach any latin square of size $n?$
2020 Iran MO (3rd Round)
What is the maximum number of subsets of size $5$, taken from the set $A=\{1,2,3,...,20\}$ such that any $2$ of them share exactly $1$ element.
2020 Iran MO (3rd Round)
Find all positive integers $n$ such that the following holds.

$$\tau(n)|2^{\sigma(n)}-1$$
2020 Iran MO (3rd Round)
Find all polynomials $P$ with integer coefficients such that all the roots of $P^n(x)$ are integers. (here $P^n(x)$ means $P(P(...(P(x))...))$ where $P$ is repeated $n$ times)
2020 Iran MO (3rd Round)
Find all functions $f$ from positive integers to themselves, such that the followings hold.

$1)$.for each positive integer $n$ we have $f(n)<f(n+1)<f(n)+2020$.

$2)$.for each positive integer $n$ we have $S(f(n))=f(S(n))$ where $S(n)$ is the sum of digits of $n$ in base $10$ representation.
2020 Iran MO (3rd Round)
Prove that for every two positive integers $a,b$ greater than $1$. there exists infinitly many $n$ such that the equation $\phi(a^n-1)=b^m-b^t$ can't hold for any positive integers $m,t$.
2019 Iran MO (3rd Round)
$a,b$ and $c$ are positive real numbers so that $\sum_{\text{cyc}} (a+b)^2=2\sum_{\text{cyc}} a +6abc$. Prove that

$$\sum_{\text{cyc}} (a-b)^2\leq\left|2\sum_{\text{cyc}} a -6abc\right|.$$
2019 Iran MO (3rd Round)
Find all function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for any three real number   $a,b,c$  , if $ a + f(b) + f(f(c)) = 0$ :

$$ f(a)^3 + bf(b)^2 + c^2f(c) = 3abc $$.
2019 Iran MO (3rd Round)
We are given a natural number $d$. Find all open intervals of maximum length $I \subseteq R$ such that for all real numbers $a_0,a_1,...,a_{2d-1}$ inside interval $I$, we have that the polynomial $P(x)=x^{2d}+a_{2d-1}x^{2d-1}+...+a_1x+a_0$ has no real roots.
2019 Iran MO (3rd Round)
Hossna is playing with a $m*n$ grid of points.In each turn she draws segments between points with the following conditions.



**1.** No two segments intersect.



**2.**  Each segment is drawn between two consecutive rows.



**3.** There is at most one segment between any two points.



Find the maximum number of regions Hossna can create.
2019 Iran MO (3rd Round)
Let $n,k$ be positive integers so that $n \ge k$.Find the maximum number of binary sequances of length $n$ so that fixing any arbitary $k$ bits they do not produce all binary sequances of length $k$.For exmple if $k=1$ we can only have one sequance otherwise they will differ in at least one bit which means that bit produces all binary sequances of length $1$.
2019 Iran MO (3rd Round)
Cells of a $n*n$ square are filled with positive integers in the way that in the intersection of the $i-$th column and $j-$th row, the number $i+j$ is written. In every step, we can choose two non-intersecting equal rectangles with one dimension equal to $n$ and swap all the numbers inside these two rectangles with one another. ( without reflection or rotation ) Find the minimum number of moves one should do to reach the position where the intersection of the $i-$th column and $j-$row is written $2n+2-i-j$.