Navvye/Olympiad-Problems-CSV huggingface dataset:
2019 Iran MO (3rd Round)
Given a cyclic quadrilateral $ABCD$. There is a point $P$ on side $BC$ such that $\angle PAB=\angle PDC=90^\circ$. The medians of vertexes $A$ and $D$ in triangles $PAB$ and $PDC$ meet at $K$ and the bisectors of $\angle PAB$ and $\angle PDC$ meet at $L$. Prove that $KL\perp BC$.
2019 Iran MO (3rd Round)
Consider an acute-angled triangle $ABC$ with $AB=AC$ and $\angle A>60^\circ$. Let $O$ be the circumcenter of $ABC$. Point $P$ lies on circumcircle of $BOC$ such that $BP\parallel AC$ and point $K$ lies on segment $AP$ such that $BK=BC$. Prove that $CK$ bisects the arc $BC$ of circumcircle of $BOC$.
2019 Iran MO (3rd Round)
Consider a triangle $ABC$ with circumcenter $O$ and incenter $I$. Incircle touches sides $BC,CA$ and $AB$ at $D, E$ and $F$. $K$ is a point such that $KF$ is tangent to circumcircle of $BFD$ and $KE$ is tangent to circumcircle of $CED$. Prove that $BC,OI$ and $AK$ are concurrent.
2019 Iran MO (3rd Round)
Given a number $k\in \mathbb{N}$. $\{a_{n}\}_{n\geq 0}$ and $\{b_{n}\}_{n\geq 0}$ are two sequences of positive integers that $a_{i},b_{i}\in \{1,2,\cdots,9\}$. For all $n\geq 0$

$$\left.\overline{a_{n}\cdots a_{1}a_{0}}+k \ \middle| \ \overline{b_{n}\cdots b_{1}b_{0}}+k \right. .$$Prove that there is a number $1\leq t \leq 9$ and $N\in \mathbb{N}$ such that $b_n=ta_n$ for all $n\geq N$.

(Note that $(\overline{x_nx_{n-1}\dots x_0}) = 10^n\times x_n + \dots + 10\times x_1 + x_0$)
2019 Iran MO (3rd Round)
Prove that for any positive integers $m>n$, there is infinitely many positive integers $a,b$ such that set of prime divisors of $a^m+b^n$ is equal to set of prime divisors of $a^{2019}+b^{1398}$.
2019 Iran MO (3rd Round)
Let $S$ be an infinite set of positive integers and define:



$T=\{ x+y|x,y \in S , x \neq y \} $



Suppose that there are only finite primes $p$ so that:



1.$p \equiv 1 \pmod 4$



2.There exists a positive integer $s$ so that $p|s,s \in T$.



Prove that there are infinity many primes that divide at least one term of $S$.
2019 Iran MO (3rd Round)
Let $A_1,A_2, \dots A_k$ be points on the unit circle.Prove that:



$\sum\limits_{1\le i<j \le k} d(A_i,A_j)^2 \le k^2 $



Where $d(A_i,A_j)$ denotes the distance between $A_i,A_j$.
2019 Iran MO (3rd Round)
$P(x)$ is a monoic polynomial with integer coefficients so that there exists monoic integer coefficients polynomials $p_1(x),p_2(x),\dots ,p_n(x)$ so that for any natural number $x$ there exist an index $j$ and a natural number $y$ so that $p_j(y)=P(x)$ and also $deg(p_j) \ge deg(P)$ for all $j$.Show that there exist an index $i$ and an integer $k$ so that $P(x)=p_i(x+k)$.
2019 Iran MO (3rd Round)
Let $a,b,c$ be non-zero distinct real numbers so that there exist functions $f,g:\mathbb{R}^{+} \to \mathbb{R}$ so that:



$af(xy)+bf(\frac{x}{y})=cf(x)+g(y)$



For all positive real $x$ and large enough $y$.



Prove that there exists a function $h:\mathbb{R}^{+} \to \mathbb{R}$ so that:



$f(xy)+f(\frac{x}{y})=2f(x)+h(y)$



For all positive real $x$ and large enough $y$.
2019 Iran MO (3rd Round)
A bear is in the center of the left down corner of a $100*100$ square .we call a cycle in this grid a bear cycle if it visits each square exactly ones and gets back to the place it started.Removing a row or column with compose the bear cycle into number of pathes.Find the minimum $k$ so that in any bear cycle we can remove a row or column so that the maximum length of the remaining pathes is at most $k$.
2019 Iran MO (3rd Round)
Let $T$ be a triangulation of a $100$-gon.We construct $P(T)$ by copying the same $100$-gon and drawing a diagonal if it was not drawn in $T$ an there is a quadrilateral with this diagonal and two other vertices so that all the sides and diagonals(Except the one we are going to draw) are present in $T$.Let $f(T)$ be the number of intersections of diagonals in $P(T)$.Find the minimum and maximum of $f(T)$.
2019 Iran MO (3rd Round)
Consider a triangle $ABC$ with incenter $I$. Let $D$ be the intersection of $BI,AC$ and $CI$ intersects the circumcircle of $ABC$ at $M$. Point $K$ lies on the line $MD$ and $\angle KIA=90^\circ$. Let $F$ be the reflection of $B$ about $C$. Prove that $BIKF$ is cyclic.
2019 Iran MO (3rd Round)
In acute-angled triangle $ABC$ altitudes $BE,CF$ meet at $H$. A perpendicular line is drawn from $H$ to $EF$ and intersects the arc $BC$ of circumcircle of $ABC$ (that doesn’t contain $A$) at $K$. If $AK,BC$ meet at $P$, prove that $PK=PH$.
2019 Iran MO (3rd Round)
Given an inscribed pentagon $ABCDE$ with circumcircle $\Gamma$. Line $\ell$ passes through vertex $A$ and is tangent to $\Gamma$. Points $X,Y$ lie on $\ell$ so that $A$ lies between $X$ and $Y$. Circumcircle of triangle $XED$ intersects segment $AD$ at $Q$ and circumcircle of triangle $YBC$ intersects segment $AC$ at $P$. Lines $XE,YB$ intersects each other at $S$ and lines $XQ, Y P$ at $Z$. Prove that circumcircle of triangles $XY Z$ and $BES$ are tangent.
2019 Iran MO (3rd Round)
Find all functions $f:\mathbb{N} \to \mathbb{N}$ so that for any distinct positive integers $x,y,z$ the value of $x+y+z$ is a perfect square if and only if $f(x)+f(y)+f(z)$ is a perfect square.
2019 Iran MO (3rd Round)
Call a polynomial $P(x)=a_nx^n+a_{n-1}x^{n-1}+\dots  a_1x+a_0$ with integer coefficients primitive if and only if $\gcd(a_n,a_{n-1},\dots a_1,a_0) =1$.



a)Let $P(x)$ be a primitive polynomial with degree less than $1398$ and $S$ be a set of primes greater than $1398$.Prove that there is a positive integer $n$ so that $P(n)$ is not divisible by any prime in $S$.



b)Prove that there exist a primitive polynomial $P(x)$ with degree less than $1398$ so that for any set $S$ of primes less than $1398$ the polynomial $P(x)$ is always divisible by product of elements of $S$.
2019 Iran MO (3rd Round)
Let $a,m$ be positive integers such that $Ord_m (a)$ is odd and for any integers $x,y$ so that



1.$xy \equiv a \pmod m$



2.$Ord_m(x) \le Ord_m(a)$



3.$Ord_m(y) \le Ord_m(a)$



We have either $Ord_m(x)|Ord_m(a)$ or $Ord_m(y)|Ord_m(a)$.prove that $Ord_m(a)$ contains at most one prime factor.
2018 Iran MO (3rd Round)
Alice and Bob are play a game in a $(2n)*(2n)$ chess boared.Alice starts from the top right point moving 1 unit in every turn.Bob starts from the down left square and does the same.The goal of Alice is to make a closed shape ending in its start position and cannot reach any point that was reached before by any of players .if a players cannot move the other player keeps moving.what is the maximum are of the shape that the first player can build with every strategy of second player.
2018 Iran MO (3rd Round)
There are 8 points in the plane.we write down the area of each triangle having all vertices amoung these points(totally 56 numbers).Let them be $a_1,a_2,\dots a_{56}$.Prove that there is a choice of plus or minus such that:

$$\pm a_1 \pm a_2 \dots \pm a_{56}=0$$
2018 Iran MO (3rd Round)
Find the smallest positive integer $n$ such that we can write numbers $1,2,\dots ,n$ in a 18*18 board such that:

i)each number appears at least once

ii)In each row or column,there are no two numbers having difference 0 or 1
2018 Iran MO (3rd Round)
Let $n$ be a positive integer.Consider all $2^n$ binary strings of length $n$.We say two of these strings are neighbors if they differ in exactly 1 digit.We have colored  $m$ strings.In each moment,we can color any uncolored string which is neighbor with at least 2 colored strings.After some time,all the strings are colored.Find the minimum possible value of $m$.
2018 Iran MO (3rd Round)
Incircle of triangle $ABC$ is tangent to sides $BC,CA,AB$ at $D,E,F$,respectively.Points $P,Q$ are inside angle $BAC$ such that $FP=FB,FP||AC$ and $EQ=EC,EQ||AB$.Prove that $P,Q,D$ are collinear.
2018 Iran MO (3rd Round)
Two intersecting circles $\omega_1$ and $\omega_2$ are given.Lines $AB,CD$ are common tangents of $\omega_1,\omega_2$($A,C \in \omega_1 ,B,D \in \omega_2$)

Let $M$ be the midpoint of $AB$.Tangents through $M$ to $\omega_1$ and $\omega_2$(other than $AB$) intersect $CD$ at $X,Y$.Let $I$ be the incenter of $MXY$.Prove that $IC=ID$.
2018 Iran MO (3rd Round)
$H$ is the orthocenter of acude triangle $ABC$.Let $\omega$ be the circumcircle of $BHC$ with center $O'$.$\Omega$ is the nine-point circle of $ABC$.$X$ is an arbitrary point on arc $BHC$ of $\omega$ and $AX$ intersects $\Omega$ at $Y$.$P$ is a point on $\Omega$ such that $PX=PY$.Prove that $O'PX=90$.
2018 Iran MO (3rd Round)
for acute triangle $\triangle ABC$ with orthocenter $H$, and $E,F$ the feet of altitudes for $B,C$, we have $P$ on $EF$ such as that $HO \perp HP$. $Q$ is on segment $AH$ so $HM \perp PQ$. prove $QA=3QH$
2018 Iran MO (3rd Round)
$n\ge 2 $ is an integer.Prove that the number of natural numbers $m$ so that $0 \le  m \le n^2-1,x^n+y^n \equiv m (mod n^2)$ has no solutions is at least $\binom{n}{2}$
2018 Iran MO (3rd Round)
Prove that for every prime number $p$ there exist infinity many natural numbers $n$ so that they satisfy:



$2^{2^{2^{ \dots ^{2^n}}}} \equiv n^{2^{2^{\dots ^{2}}}} (mod p)$



Where in both sides $2$ appeared $1397$ times
2018 Iran MO (3rd Round)
Find all functions $f:\mathbb{N}\to \mathbb{N}$ so that for every natural numbers $m,n$ :$f(n)+2mn+f(m)$ is a perfect square.
2018 Iran MO (3rd Round)
Prove that for any natural numbers$a,b$ there exist infinity many prime numbers $p$ so that $Ord_p(a)=Ord_p(b)$(Proving that there exist infinity prime numbers $p$ so that $Ord_p(a) \ge Ord_p(b)$ will get a partial mark)
2018 Iran MO (3rd Round)
For positive real numbers$a,b,c$such that $ab+ac+bc=1$ prove that:



$\prod\limits_{cyc} (\sqrt{bc}+\frac{1}{2a+\sqrt{bc}}) \ge 8abc$
2018 Iran MO (3rd Round)
Find all functions $f: \mathbb{R}^{\ge 0} \to \mathbb{R}^{\ge 0}$ such that:



$f(x^3+xf(xy))=f(xy)+x^2f(x+y) \forall x,y \in \mathbb{R}^{\ge 0}$
2018 Iran MO (3rd Round)
A)Let $x,y$ be two complex numbers on the unit circle so that:



$\frac{\pi }{3} \le \arg (x)-\arg (y) \le \frac{5 \pi }{3}$



Prove that for any $z \in \mathbb{C}$ we have:



$|z|+|z-x|+|z-y| \ge |zx-y|$



B)Let $x,y$ be two complex numbers so that:



$\frac{\pi }{3} \le \arg (x)-\arg (y) \le \frac{2 \pi }{3}$



Prove that for any $z \in \mathbb{C}$ we have:



$|z|+|z-y|+|z-x| \ge | \frac{\sqrt{3}}{2} x +(y-\frac{x}{2})i|$
2018 Iran MO (3rd Round)
Let $P(x)$ be a non-zero polynomial with real coefficient so thay $P(0)=0$.Prove that for any positive real number $M$ there exist a positive integer $d$ so that for any monoic polynomial $Q(x)$ with degree at least $d$ the number of integers $k$ so that $|P(Q(k))| \le M$ is at most equal to the degree of $Q$.
2017 Iran MO (3rd round)
Find all polynomials $P(x)$ and $Q(x)$ with real coefficients such that

$$P(Q(x))=P(x)^{2017}$$for all real numbers $x$.
2017 Iran MO (3rd round)
Let $a,b,c$ and $d$ be positive real numbers such that $a^2+b^2+c^2+d^2 \ge 4$. Prove that

$$(a+b)^3+(c+d)^3+2(a^2+b^2+c^2+d^2) \ge 4(ab+bc+cd+da+ac+bd)$$
2017 Iran MO (3rd round)
Find all functions $f:\mathbb{R^+}\rightarrow\mathbb{R^+}$ such that

$$\frac{x+f(y)}{xf(y)}=f(\frac{1}{y}+f(\frac{1}{x}))$$for all positive real numbers $x$ and $y$.
2017 Iran MO (3rd round)
Let $n$ be a positive integer. Consider prime numbers $p_1,\dots ,p_k$. Let $a_1,\dots,a_m$ be all positive integers less than $n$ such that are not divisible by $p_i$ for all $1 \le i \le n$. Prove that if $m\ge 2$ then

$$\frac{1}{a_1}+\dots+\frac{1}{a_m}$$is not an integer.
2017 Iran MO (3rd round)
Consider a sequence $\{a_i\}^\infty_{i\ge1}$ of positive integers. For all positvie integers $n$ prove that there exists infinitely many positive integers $k$ such that there is no pair $(m,t)$ of positive integers where $m>n$ and

$$kn+a_n=tm(m+1)+a_m$$
2017 Iran MO (3rd round)
Let $k$ be a positive integer. Find all functions $f:\mathbb{N}\to \mathbb{N}$ satisfying the following two conditions:



• For infinitely many prime numbers $p$ there exists a positve integer $c$ such that $f(c)=p^k$.



• For all positive integers $m$ and $n$, $f(m)+f(n)$ divides $f(m+n)$.
2017 Iran MO (3rd round)
Let $ABC$ be a triangle. Suppose that $X,Y$ are points in the plane such that $BX,CY$ are tangent to the circumcircle of $ABC$, $AB=BX,AC=CY$ and $X,Y,A$ are in the same side of $BC$. If $I$ be the incenter of $ABC$ prove that $\angle BAC+\angle XIY=180$.
2017 Iran MO (3rd round)
Let $ABCD$ be a trapezoid ($AB<CD,AB\parallel CD$) and $P\equiv AD\cap BC$. Suppose that $Q$ be a point inside $ABCD$ such that $\angle QAB=\angle QDC=90-\angle BQC$. Prove that $\angle PQA=2\angle QCD$.
2017 Iran MO (3rd round)
Let $ABC$ be an acute-angle triangle. Suppose that $M$ be the midpoint of $BC$ and $H$ be the orthocenter of $ABC$. Let $F\equiv BH\cap AC$ and $E\equiv CH\cap AB$. Suppose that $X$ be a point on $EF$ such that $\angle XMH=\angle HAM$ and $A,X$ are in the distinct side of $MH$. Prove that $AH$ bisects $MX$.
2017 Iran MO (3rd round)
There's a tape with $n^2$ cells labeled by $1,2,\ldots,n^2$. Suppose that $x,y$ are two distinct positive integers less than or equal to $n$. We want to color the cells of the tape such that any two cells with label difference of $x$ or $y$ have different colors. Find the minimum number of colors needed to do so.
2017 Iran MO (3rd round)
An angle is considered as a point and two rays coming out of it.

Find the largest value on $n$ such that it is possible to place $n$ $60$ degree angles on the plane in a way that any pair of these angles have exactly $4$ intersection points.
2017 Iran MO (3rd round)
Ali has $6$ types of $2\times2$ squares with cells colored in white or black, and has presented them to Mohammad as forbidden tiles.

$a)$ Prove that Mohammad can color the cells of the infinite table (from each $4$ sides.) in black or white such that there's no forbidden tiles in the table.

$b)$ Prove that Ali can present $7$ forbidden tiles such that Mohammad cannot achieve his goal.
2017 Iran MO (3rd round)
Let $ABC$ be a right-angled triangle $\left(\angle A=90^{\circ}\right)$ and $M$ be the midpoint of $BC$. $\omega_1$ is a circle which passes through $B,M$ and touchs $AC$ at $X$. $\omega_2$ is a circle which passes through $C,M$ and touchs $AB$ at $Y$ ($X,Y$ and $A$ are in the same side of $BC$). Prove that $XY$ passes through the midpoint of  arc $BC$ (does not contain $A$) of the circumcircle of $ABC$.
2017 Iran MO (3rd round)
Assume that $P$ be an arbitrary point inside of triangle $ABC$. $BP$ and $CP$ intersects $AC$ and $AB$ in $E$ and $F$, respectively. $EF$ intersects the circumcircle of $ABC$ in $B'$ and $C'$ (Point $E$ is between of $F$ and $B'$). Suppose that $B'P$ and $C'P$ intersects $BC$ in $C''$ and $B''$ respectively. Prove that $B'B''$ and $C'C''$ intersect each other on the circumcircle of $ABC$.
2017 Iran MO (3rd round)
In triangle $ABC$ points $P$ and $Q$ lies on the external bisector of $\angle A$ such that $B$ and $P$ lies on the same side of $AC$. Perpendicular from $P$ to $AB$ and $Q$ to $AC$ intersect at $X$. Points $P'$ and $Q'$ lies on $PB$ and $QC$ such that $PX=P'X$ and $QX=Q'X$. Point $T$ is the midpoint of arc $BC$ (does not contain $A$) of the circumcircle of $ABC$. Prove that $P',Q'$ and $T$ are collinear if and only if $\angle PBA+\angle QCA=90^{\circ}$.
2017 Iran MO (3rd round)
Let $x$ and $y$ be integers and let $p$ be a prime number. Suppose that there exist realatively prime positive integers $m$ and $n$ such that

$$x^m \equiv y^n \pmod p$$Prove that there exists an unique integer $z$ modulo $p$ such that

$$x \equiv z^n \pmod p \quad \text{and} \quad y \equiv z^m \pmod p$$
2017 Iran MO (3rd round)
For prime number $q$ the polynomial $P(x)$ with integer coefficients is said to be factorable if there exist non-constant polynomials $f_q,g_q$ with integer coefficients such that all of the coefficients of the polynomial $Q(x)=P(x)-f_q(x)g_q(x)$ are dividable by $q$ ; and we write:

$$P(x)\equiv f_q(x)g_q(x)\pmod{q}$$For example the polynomials $2x^3+2,x^2+1,x^3+1$ can be factored modulo $2,3,p$ in the following way:



$$\left\{\begin{array}{lll}

X^2+1\equiv (x+1)(-x+1)\pmod{2}\\
2x^3+2\equiv (2x-1)^3\pmod{3}\\
X^3+1\equiv (x+1)(x^2-x+1)

\end{array}\right.$$

Also the polynomial $x^2-2$ is not factorable modulo $p=8k\pm 3$.



a) Find all prime numbers $p$ such that the polynomial $P(x)$ is factorable modulo $p$:

$$P(x)=x^4-2x^3+3x^2-2x-5$$b) Does there exist irreducible polynomial $P(x)$ in $\mathbb{Z}[x]$ with integer coefficients such that for each prime number $p$ , it is factorable modulo $p$?
2017 Iran MO (3rd round)
Let $n$ be a positive integer. Prove that there exists a poisitve integer $m$ such that

$$7^n \mid 3^m+5^m-1$$
2017 Iran MO (3rd round)
Let $\mathbb{R}^{\ge 0}$ be the set of all nonnegative real numbers. Find all functions $f:\mathbb{R}^{\ge 0}  \to \mathbb{R}^{\ge 0}$ such that

$$ x+2 \max\{y,f(x),f(z)\} \ge f(f(x))+2 \max\{z,f(y)\}$$for all nonnegative real numbers $x,y$ and $z$.
2017 Iran MO (3rd round)
Let $P(z)=a_d z^d+\dots+ a_1z+a_0$ be a polynomial with complex coefficients. The $reverse$ of $P$ is defined by

$$P^*(z)=\overline{a_0}z^d+\overline{a_1}z^{d-1}+\dots+\overline{a_d}$$(a) Prove that

$$P^*(z)=z^d  \overline{ P\left( \frac{1}{\overline{z}} \right) } $$(b) Let $m$ be a positive integer and let $q(z)$ be a monic nonconstant polynomial with complex coefficients. Suppose that all roots of $q(z)$ lie inside or on the unit circle. Prove that all roots of the polynomial

$$Q(z)=z^m q(z)+ q^*(z)$$lie on the unit circle.
2017 Iran MO (3rd round)
Let $a,b$ and $c$ be positive real numbers. Prove that

$$\sum_{cyc} \frac {a^3b}{(3a+2b)^3} \ge \sum_{cyc} \frac {a^2bc}{(2a+2b+c)^3} $$
2017 Iran MO (3rd round)
There are $100$ points on the circumference of a circle, arbitrarily labelled by $1,2,\ldots,100$. For each three points, call their triangle clockwise if the increasing order of them is in clockwise order. Prove that it is impossible to have exactly $2017$ clockwise triangles.
2017 Iran MO (3rd round)
Two persons are playing the following game on a $n\times m$ table, with drawn lines:

Person $\#1$ starts the game. Each person in their move, folds the table on one of its lines. The one that could not fold the table on their turn loses the game.

Who has a winning strategy?
2017 Iran MO (3rd round)
$30$ volleyball teams have participated in a league.  Any two teams have played a match with each other exactly once. At the end of the league, a match is called unusual if at the end of the league, the winner of the match have a smaller amount of wins than the loser of the match. A team is called astonishing if all its matches are unusual matches.

Find the maximum number of astonishing teams.
2016 Iran MO (3rd Round)
The sequence $(a_n)$ is defined as:

$$a_1=1007$$$$a_{i+1}\geq a_i+1$$Prove the inequality:



$$\frac{1}{2016}>\sum_{i=1}^{2016}\frac{1}{a_{i+1}^{2}+a_{i+2}^2}$$
2016 Iran MO (3rd Round)
Find all function $f:\mathbb{N}\rightarrow\mathbb{N}$ such that for all $a,b\in\mathbb{N}$ ,

$(f(a)+b) f(a+f(b))=(a+f(b))^2$
2016 Iran MO (3rd Round)
Do there exists many infinitely points like $(x_1,y_1),(x_2,y_2),...$ such that for any sequences like {$b_1,b_2,...$} of real numbers there exists a polynomial $P(x,y)\in R[x,y]$ such that we have for all $i$ :

$P(x_{i},y_{i})=b_{i}$
2016 Iran MO (3rd Round)
Let $ABC$ be an arbitrary triangle,$P$ is the intersection point of the altitude from $C$ and the tangent line from $A$ to the circumcircle. The bisector of angle $A$ intersects $BC$ at $D$ . $PD$ intersects $AB$ at $K$, if $H$ is the orthocenter then prove : $HK\perp AD$
2016 Iran MO (3rd Round)
Let $ABC$ be an arbitrary triangle. Let $E,E$ be two points on $AB,AC$ respectively such that their distance to the midpoint of $BC$ is equal. Let $P$ be the second intersection of the triangles $ABC,AEF$ circumcircles . The tangents from $E,F$ to the circumcircle of $AEF$ intersect each other at $K$. Prove that : $\angle KPA = 90$
2016 Iran MO (3rd Round)
Let $ABC$ be a triangle and let $AD,BE,CF$ be its altitudes . $FA_{1},DB_{1},EC_{1}$ are perpendicular segments to $BC,AC,AB$ respectively.

Prove that : $ABC$~$A_{1}B_{1}C_{1}$
2016 Iran MO (3rd Round)
Let $F$ be a subset of the set of positive integers with at least two elements and $P(x)$ be a polynomial with integer coefficients such that for any two distinct elements of $F$ like $a$ and $b$, the following two conditions hold



$a+b \in F$, and

$\gcd(P(a),P(b))=1$.



Prove that $P(x)$ is a constant polynomial.
2016 Iran MO (3rd Round)
Let $P$ be a polynomial with integer coefficients. We say $P$ is good  if there exist infinitely many prime numbers $q$ such that the set $$X=\left\{P(n) \mod q : \quad n\in \mathbb N\right\}$$has at least $\frac{q+1}{2}$ members.

Prove that the polynomial $x^3+x$ is good.
2016 Iran MO (3rd Round)
Let $m$ be a positive integer. The positive integer $a$ is called a golden residue modulo $m$ if $\gcd(a,m)=1$ and $x^x \equiv a \pmod m$ has a solution for $x$. Given a positive integer $n$, suppose that $a$ is a golden residue modulo $n^n$. Show that $a$ is also a golden residue modulo $n^{n^n}$.



Proposed by Mahyar Sefidgaran
2016 Iran MO (3rd Round)
Find the number of all $\text{permutations}$ of $\left \{ 1,2,\cdots ,n \right \}$ like $p$ such that there exists a unique $i \in \left \{ 1,2,\cdots ,n \right \}$ that :



$$p(p(i)) \geq i$$
2016 Iran MO (3rd Round)
Is it possible to divide a $7\times7$ table into a few $\text{connected}$ parts of cells with the same perimeter?

( A group of cells is called $\text{connected}$ if any cell in the group, can reach other cells by passing through the sides of cells.)
2016 Iran MO (3rd Round)
There are $24$ robots on the plane. Each robot has a $70^{\circ}$ field of view. What is the maximum number of observing relations?

(Observing is a one-sided relation)
2016 Iran MO (3rd Round)
Let $P(x) \in \mathbb {Z}[X]$ be a polynomial of degree $2016$ with no rational roots. Prove that there exists a polynomial $T(x) \in \mathbb {Z}[X]$ of degree $1395$ such that for all distinct (not necessarily real) roots of $P(x)$ like $(\alpha ,\beta):$



$$T(\alpha)-T(\beta) \not \in \mathbb {Q}$$

Note: $\mathbb {Q}$ is the set of rational numbers.
2016 Iran MO (3rd Round)
Let $a,b,c \in \mathbb {R}^{+}$ and $abc=1$ prove that:



$\frac {a+b}{(a+b+1)^2}+\frac {b+c}{(b+c+1)^2}+\frac {c+a}{(c+a+1)^2} \geq \frac {2}{a+b+c}$
2016 Iran MO (3rd Round)
Find all functions $f:\mathbb {R}^{+} \rightarrow \mathbb {R}^{+} $ such that for all positive real numbers $x,y:$



$$f(y)f(x+f(y))=f(x)f(xy)$$
2016 Iran MO (3rd Round)
In triangle $ABC$ , $w$ is a circle which passes through $B,C$ and intersects $AB,AC$ at $E,F$ respectively. $BF,CE$ intersect the circumcircle of $ABC$ at $B',C'$ respectively. Let $A'$ be a point on $BC$ such that $\angle C'A'B=\angle B'A'C$ .

Prove that if we change $w$, then all the circumcircles of triangles $A'B'C'$ passes through a common point.
2016 Iran MO (3rd Round)
Given $\triangle ABC$ inscribed in $(O)$ an let $I$ and $I_a$ be it's incenter and $A$-excenter ,respectively.

Tangent lines to $(O)$ at $C,B$ intersect the angle bisector of $A$ at $M,N$ ,respectively.

Second tangent lines through $M,N$ intersect $(O)$ at $X,Y$.



Prove that $XYII_a$ is cyclic.
2016 Iran MO (3rd Round)
Given triangle $\triangle ABC$ and let $D,E,F$ be the foot of angle bisectors of $A,B,C$ ,respectively.

$M,N$ lie on $EF$ such that $AM=AN$. Let $H$ be the foot of $A$-altitude on $BC$.

Points $K,L$ lie on $EF$ such that triangles $\triangle AKL, \triangle HMN$ are correspondingly similiar (with the given order of vertices) such that $AK \not\parallel HM$ and $AK \not\parallel HN$.



Show that: $DK=DL$
2016 Iran MO (3rd Round)
Let $p,q$ be prime numbers ($q$ is odd). Prove that there exists an integer $x$ such that:

$$q |(x+1)^p-x^p$$If and only if $$q \equiv 1 \pmod p$$