Navvye/Olympiad-Problems-CSV huggingface dataset:
2012 Iran MO (3rd Round)
In a tree with $n$ vertices, for each vertex $x_i$, denote the longest paths passing through it by $l_i^1,l_i^2,...,l_i^{k_i}$. $x_i$ cuts those longest paths into two parts with $(a_i^1,b_i^1),(a_i^2,b_i^2),...,(a_i^{k_i},b_i^{k_i})$ vertices respectively. If $\max_{j=1,...,k_i} \{a_i^j\times b_i^j\}=p_i$, find the maximum and minimum values of $\sum_{i=1}^{n} p_i$.



Proposed by Sina Rezaei
2012 Iran MO (3rd Round)
a) Prove that for all $m,n\in \mathbb N$ there exists a natural number $a$ such that if we color every $3$-element subset of the set $\mathcal A=\{1,2,3,...,a\}$ using $2$ colors red and green, there exists an $m$-element subset of $\mathcal A$ such that all $3$-element subsets of it are red or there exists an $n$-element subset of $\mathcal A$ such that all $3$-element subsets of it are green.



b) Prove that for all $m,n\in \mathbb N$ there exists a natural number $a$ such that if we color every $k$-element subset ($k>3$) of the set $\mathcal A=\{1,2,3,...,a\}$ using $2$ colors red and green, there exists an $m$-element subset of $\mathcal A$ such that all $k$-element subsets of it are red or there exists an $n$-element subset of $\mathcal A$ such that all $k$-element subsets of it are green.
2012 Iran MO (3rd Round)
Suppose $0<m_1<...<m_n$ and $m_i \equiv i (\mod 2)$. Prove that the following polynomial has at most $n$ real roots. ($\forall 1\le i \le n: a_i \in \mathbb R$).

$$a_0+a_1x^{m_1}+a_2x^{m_2}+...+a_nx^{m_n}.$$
2012 Iran MO (3rd Round)
Suppose $N\in \mathbb N$ is not a perfect square, hence we know that the continued fraction of $\sqrt{N}$ is of the form $\sqrt{N}=[a_0,\overline{a_1,a_2,...,a_n}]$. If $a_1\neq 1$ prove that $a_i\le 2a_0$.
2012 Iran MO (3rd Round)
Suppose $p$ is a prime number and $a,b,c \in \mathbb Q^+$ are rational numbers;



a) Prove that $\mathbb Q(\sqrt[p]{a}+\sqrt[p]{b})=\mathbb Q(\sqrt[p]{a},\sqrt[p]{b})$.



b) If $\sqrt[p]{b} \in \mathbb Q(\sqrt[p]{a})$, prove that for a nonnegative integer $k$ we have $\sqrt[p]{\frac{b}{a^k}}\in \mathbb Q$.



c) If $\sqrt[p]{a}+\sqrt[p]{b}+\sqrt[p]{c} \in \mathbb Q$, then prove that numbers $\sqrt[p]{a},\sqrt[p]{b}$ and $\sqrt[p]{c}$ are rational.
2012 Iran MO (3rd Round)
Suppose $f(z)=z^n+a_1z^{n-1}+...+a_n$ for which $a_1,a_2,...,a_n\in \mathbb C$. Prove that the following polynomial has only one positive real root like $\alpha$

$$x^n+\Re(a_1)x^{n-1}-|a_2|x^{n-2}-...-|a_n|$$



and the following polynomial has only one positive real root like $\beta$

$$x^n-\Re(a_1)x^{n-1}-|a_2|x^{n-2}-...-|a_n|.$$

And roots of the polynomial $f(z)$ satisfy $-\beta \le \Re(z) \le \alpha$.
2012 Iran MO (3rd Round)
Let $p$ be an odd prime number and let $a_1,a_2,...,a_n \in \mathbb Q^+$ be rational numbers. Prove that $$\mathbb Q(\sqrt[p]{a_1}+\sqrt[p]{a_2}+...+\sqrt[p]{a_n})=\mathbb Q(\sqrt[p]{a_1},\sqrt[p]{a_2},...,\sqrt[p]{a_n}).$$
2012 Iran MO (3rd Round)
Let $G$ be a simple undirected graph with vertices $v_1,v_2,...,v_n$. We denote the number of acyclic orientations of $G$ with $f(G)$.



a) Prove that $f(G)\le f(G-v_1)+f(G-v_2)+...+f(G-v_n)$.



b) Let $e$ be an edge of the graph $G$. Denote by $G'$ the graph obtained by omiting $e$ and making it's two endpoints as one vertex. Prove that $f(G)=f(G-e)+f(G')$.



c) Prove that for each $\alpha >1$, there exists a graph $G$ and an edge $e$ of it such that



$\frac{f(G)}{f(G-e)}<\alpha$.



Proposed by Morteza Saghafian
2012 Iran MO (3rd Round)
Suppose $S$ is a convex figure in plane with area $10$. Consider a chord of length $3$ in $S$ and let $A$ and $B$ be two points on this chord which divide it into three equal parts. For a variable point $X$ in $S-\{A,B\}$, let $A'$ and $B'$ be the intersection points of rays $AX$ and $BX$ with the boundary of $S$. Let $S'$ be those points $X$ for which $AA'>\frac{1}{3} BB'$. Prove that the area of $S'$ is at least $6$.



Proposed by Ali Khezeli
2012 Iran MO (3rd Round)
Prove that for each $n \in \mathbb N$ there exist natural numbers $a_1<a_2<...<a_n$ such that $\phi(a_1)>\phi(a_2)>...>\phi(a_n)$.



Proposed by Amirhossein Gorzi
2012 Iran MO (3rd Round)
We have $n$ bags each having $100$ coins. All of the bags have $10$ gram coins except one of them which has $9$ gram coins. We have a balance which can show weights of things that have weight of at most $1$ kilogram. At least how many times shall we use the balance in order to find the different bag?



Proposed By Hamidreza Ziarati
2012 Iran MO (3rd Round)
We call the three variable polynomial $P$ cyclic if $P(x,y,z)=P(y,z,x)$. Prove that cyclic three variable polynomials $P_1,P_2,P_3$ and $P_4$ exist such that for each cyclic three variable polynomial $P$, there exists a four variable polynomial $Q$ such that $P(x,y,z)=Q(P_1(x,y,z),P_2(x,y,z),P_3(x,y,z),P_4(x,y,z))$.



Solution by Mostafa Eynollahzade and Erfan Salavati
2012 Iran MO (3rd Round)
a) Prove that $a>0$ exists such that for each natural number $n$, there exists a convex $n$-gon $P$ in plane with lattice points as vertices such that the area of $P$ is less than $an^3$.



b) Prove that there exists $b>0$ such that for each natural number $n$ and each $n$-gon $P$ in plane with lattice points as vertices, the area of $P$ is not less than $bn^2$.



c) Prove that there exist $\alpha,c>0$ such that for each natural number $n$ and each $n$-gon $P$ in plane with lattice points as vertices, the area of $P$ is not less than $cn^{2+\alpha}$.



Proposed by Mostafa Eynollahzade
2012 Iran MO (3rd Round)
The city of Bridge Village has some highways. Highways are closed curves that have intersections with each other or themselves in $4$-way crossroads. Mr.Bridge Lover, mayor of the city, wants to build a bridge on each crossroad in order to decrease the number of accidents. He wants to build the bridges in such a way that in each highway, cars pass above a bridge and under a bridge alternately. By knowing the number of highways determine that this action is possible or not.



Proposed by Erfan Salavati
2012 Iran MO (3rd Round)
a) Does there exist an infinite subset $S$ of the natural numbers, such that $S\neq \mathbb{N}$, and such that for each natural number $n\not \in S$, exactly $n$ members of $S$ are coprime with $n$?



b)  Does there exist an infinite subset $S$ of the natural numbers, such that for each natural number $n\in S$, exactly $n$ members of $S$ are coprime with $n$?



Proposed by Morteza Saghafian
2011 Iran MO (3rd Round)
(a) We say that a hyperplane $H$ that is given with this equation

$$H=\{(x_1,\dots,x_n)\in \mathbb R^n \mid a_1x_1+ \dots +a_nx_n=b\}$$

($a=(a_1,\dots,a_n)\in \mathbb R^n$ and $b\in \mathbb R$ constant) bisects the finite set $A\subseteq \mathbb R^n$ if each of the two halfspaces $H^+=\{(x_1,\dots,x_n)\in \mathbb R^n \mid a_1x_1+ \dots +a_nx_n>b\}$ and $H^-=\{(x_1,\dots,x_n)\in \mathbb R^n \mid a_1x_1+ \dots +a_nx_n<b\}$ have at most $\lfloor \tfrac{|A|}{2}\rfloor$ points of $A$.



Suppose that $A_1,\dots,A_n$ are finite subsets of $\mathbb R^n$. Prove that there exists a hyperplane $H$ in $\mathbb R^n$  that bisects all of them at the same time.



(b) Suppose that the points in $B=A_1\cup \dots \cup A_n$ are in general position.  Prove that there exists a hyperplane $H$ such that $H^+\cap A_i$ and $H^-\cap A_i$ contain exactly $\lfloor \tfrac{|A_i|}{2}\rfloor$ points of $A_i$.



(c) With the help of part (b), show that the following theorem is true: Two robbers want to divide an open necklace that has $d$ different kinds of stones, where the number of stones of each kind is even, such that each of the robbers receive the same number of stones of each kind.  Show that the two robbers can accomplish this by cutting the necklace in at most $d$ places.
2011 Iran MO (3rd Round)
Prove that these three statements are equivalent:



(a) For every continuous function $f:S^n \to \mathbb R^n$, there exists an $x\in S^n$ such that $f(x)=f(-x)$.



(b) There is no antipodal mapping $f:S^n \to S^{n-1}$.



(c) For every covering of $S^n$ with closed sets $A_0,\dots,A_n$, there exists an index $i$ such that $A_i\cap -A_i\neq \emptyset$.
2011 Iran MO (3rd Round)
prove that if graph $G$ is a tree, then there is a vertex that is common between all of the longest paths.



proposed by Sina Rezayi
2011 Iran MO (3rd Round)
prove that the number of permutations such that the order of each element is a multiple of $d$ is $\frac{n!}{(\frac{n}{d})!d^{\frac{n}{d}}} \prod_{i=0}^{\frac{n}{d}-1} (id+1)$.



proposed by Mohammad Mansouri
2011 Iran MO (3rd Round)
Suppose that $p(n)$ is the number of partitions of a natural number $n$. Prove that there exists $c>0$ such that $P(n)\ge n^{c \cdot \log n}$.



proposed by Mohammad Mansouri
2011 Iran MO (3rd Round)
We say the point $i$ in the permutation $\sigma$ ongoing if for every $j<i$ we have $\sigma (j)<\sigma (i)$.



a) prove that the number of permutations of the set $\{1,....,n\}$ with exactly $r$ ongoing points is $s(n,r)$.



b) prove that the number of $n$-letter words with letters $\{a_1,....,a_k\},a_1<.....<a_k$. with exactly $r$ ongoing points is $\sum_{m}\dbinom{k}{m} S(n,m) s(m,r)$.
2011 Iran MO (3rd Round)
Suppose that $n$ is a natural number. we call the sequence $(x_1,y_1,z_1,t_1),(x_2,y_2,z_2,t_2),.....,(x_s,y_s,z_s,t_s)$ of $\mathbb Z^4$ good if it satisfies these three conditions:



i) $x_1=y_1=z_1=t_1=0$.



ii) the sequences $x_i,y_i,z_i,t_i$ be strictly increasing.



iii) $x_s+y_s+z_s+t_s=n$. (note that $s$ may vary).



Find the number of good sequences.



proposed by Mohammad Ghiasi
2011 Iran MO (3rd Round)
Every bacterium has a horizontal body with natural length and some nonnegative number of vertical feet, each with nonnegative (!) natural length, that lie below its body. In how many ways can these bacteria fill an $m\times n$ table such that no two of  them overlap?



proposed by Mahyar Sefidgaran
2011 Iran MO (3rd Round)
Suppose that $S\subseteq \mathbb Z$ has the following property: if $a,b\in S$, then $a+b\in S$. Further, we know that $S$ has at least one negative element and one positive element. Is the following statement true?

There exists an integer $d$ such that for every $x\in \mathbb Z$, $x\in S$ if and only if $d|x$.

proposed by Mahyar Sefidgaran
2011 Iran MO (3rd Round)
Let $n$ and $k$ be two natural numbers such that $k$ is even and for each prime $p$ if $p|n$ then $p-1|k$. let $\{a_1,....,a_{\phi(n)}\}$ be all the numbers coprime to $n$. What's the remainder of the number $a_1^k+.....+a_{\phi(n)}^k$ when it's divided by $n$?

proposed by Yahya Motevassel
2011 Iran MO (3rd Round)
Let $k$ be a natural number such that $k\ge 7$. How many $(x,y)$ such that $0\le x,y<2^k$ satisfy the equation $73^{73^x}\equiv 9^{9^y} \pmod {2^k}$?



Proposed by Mahyar Sefidgaran
2011 Iran MO (3rd Round)
Suppose that $n$ is a natural number and $n$ is not divisible by $3$. Prove that

$(n^{2n}+n^n+n+1)^{2n}+(n^{2n}+n^n+n+1)^n+1$

has at least $2d(n)$ distinct prime factors where $d(n)$ is the number of positive divisors of $n$.

proposed by Mahyar Sefidgaran
2011 Iran MO (3rd Round)
Suppose that $k$ is a natural number. Prove that there exists a prime number in $\mathbb Z_{[i]}$ such that every other prime number in $\mathbb Z_{[i]}$ has a distance at least $k$ with it.
2011 Iran MO (3rd Round)
$a$ is an integer and $p$ is a prime number and we have $p\ge 17$. Suppose that $S=\{1,2,....,p-1\}$ and $T=\{y|1\le y\le p-1,ord_p(y)<p-1\}$. Prove that there are at least $4(p-3)(p-1)^{p-4}$ functions $f:S\longrightarrow S$ satisfying

$\sum_{x\in T} x^{f(x)}\equiv a$ $(mod$ $p)$.

proposed by Mahyar Sefidgaran
2011 Iran MO (3rd Round)
We have $4$ circles in plane such that any two of them are tangent to each other. we connect the tangency point of two circles to the tangency point of two other circles. Prove that these three lines are concurrent.



proposed by Masoud Nourbakhsh
2011 Iran MO (3rd Round)
In triangle $ABC$, $\omega$ is its circumcircle and $O$ is the center of this circle. Points $M$ and $N$ lie on sides $AB$ and $AC$ respectively. $\omega$ and the circumcircle of triangle $AMN$ intersect each other for the second time in $Q$. Let $P$ be the intersection point of $MN$ and $BC$. Prove that $PQ$ is tangent to $\omega$ iff $OM=ON$.



proposed by Mr.Etesami
2011 Iran MO (3rd Round)
In triangle $ABC$, $X$ and $Y$ are the tangency points of incircle (with center $I$) with sides $AB$ and $AC$ respectively. A tangent line to the circumcircle of triangle $ABC$ (with center $O$) at point $A$, intersects the extension of $BC$ at $D$. If $D,X$ and $Y$ are collinear then prove that $D,I$ and $O$ are also collinear.



proposed by Amirhossein Zabeti
2011 Iran MO (3rd Round)
A variant triangle has fixed incircle and circumcircle. Prove that the radical center of its three excircles lies on a fixed circle and the circle's center is the midpoint of the line joining circumcenter and incenter.



proposed by Masoud Nourbakhsh
2011 Iran MO (3rd Round)
Given triangle $ABC$, $D$ is the foot of the external angle bisector of $A$, $I$ its incenter and $I_a$ its $A$-excenter. Perpendicular from $I$ to $DI_a$ intersects the circumcircle of triangle in $A'$. Define $B'$ and $C'$ similarly. Prove that $AA',BB'$ and $CC'$ are concurrent.



proposed by Amirhossein Zabeti
2011 Iran MO (3rd Round)
We define the recursive polynomial $T_n(x)$ as follows:

$T_0(x)=1$

$T_1(x)=x$

$T_{n+1}(x)=2xT_n(x)+T_{n-1}(x)$ $\forall n \in \mathbb N$.



a) find $T_2(x),T_3(x),T_4(x)$ and $T_5(x)$.



b) find all the roots of the polynomial $T_n(x)$ $\forall n \in \mathbb N$.



Proposed by Morteza Saghafian
2011 Iran MO (3rd Round)
For nonnegative real numbers $x,y,z$ and $t$ we know that $|x-y|+|y-z|+|z-t|+|t-x|=4$.

Find the minimum of $x^2+y^2+z^2+t^2$.



proposed by Mohammadmahdi Yazdi, Mohammad Ahmadi
2011 Iran MO (3rd Round)
We define the polynomial $f(x)$ in $\mathbb R[x]$ as follows:

$f(x)=x^n+a_{n-2}x^{n-2}+a_{n-3}x^{n-3}+.....+a_1x+a_0$

Prove that there exists an $i$ in the set $\{1,....,n\}$ such that we have

$|f(i)|\ge \frac{n!}{\dbinom{n}{i}}$.



proposed by Mohammadmahdi Yazdi
2011 Iran MO (3rd Round)
For positive real numbers $a,b$ and $c$ we have $a+b+c=3$. Prove

$\frac{a}{1+(b+c)^2}+\frac{b}{1+(a+c)^2}+\frac{c}{1+(a+b)^2}\le \frac{3(a^2+b^2+c^2)}{a^2+b^2+c^2+12abc}$.



proposed by Mohammad Ahmadi
2011 Iran MO (3rd Round)
$f(x)$ is a monic polynomial of degree $2$ with integer coefficients such that $f(x)$ doesn't have any real roots and also $f(0)$ is a square-free integer (and is not $1$ or $-1$). Prove that for every integer $n$ the polynomial $f(x^n)$ is irreducible over $\mathbb Z[x]$.



proposed by Mohammadmahdi Yazdi
2011 Iran MO (3rd Round)
A regular dodecahedron is a convex polyhedra that its faces are regular pentagons. The regular dodecahedron has twenty vertices and there are three edges connected to each vertex. Suppose that we have marked ten vertices of the regular dodecahedron.



a) prove that we can rotate the dodecahedron in such a way that at most four marked vertices go to a place that there was a marked vertex before.



b) prove that the number four in previous part can't be replaced with three.



proposed by Kasra Alishahi
2011 Iran MO (3rd Round)
a) Prove that for every natural numbers $n$ and $k$, we have monic polynomials of degree $n$, with integer coefficients like $A=\{P_1(x),.....,P_k(x)\}$ such that no two of them have a common factor and for every subset of $A$, the sum of elements of $A$ has all its roots real.



b) Are there infinitely many monic polynomial of degree $n$ with integer coefficients like $P_1(x),P_2(x),....$ such that no two of them have a common factor and the sum of a finite number of them has all it's roots real?



proposed by Mohammad Mansouri
2011 Iran MO (3rd Round)
We have connected four metal pieces to each other such that they have formed a tetragon in space and also the angle between two connected metal pieces can vary.

In the case that the tetragon can't be put in the plane, we've marked a point on each of the pieces such that they are all on a plane. Prove that as the tetragon varies, that four points remain on a plane.



proposed by Erfan Salavati
2011 Iran MO (3rd Round)
The escalator of the station champion butcher has this property that if $m$ persons are on it, then it's speed is $m^{-\alpha}$ where $\alpha$ is a fixed positive real number.

Suppose that $n$ persons want to go up by the escalator and the width of the stairs is such that all the persons can stand on a stair. If the length of the escalator is $l$, what's the least time that is needed for these persons to go up? Why?



proposed by Mohammad Ghiasi
2011 Iran MO (3rd Round)
Suppose that $\alpha$ is a real number and $a_1<a_2<.....$ is a strictly increasing sequence of natural numbers such that for each natural number $n$ we have $a_n\le n^{\alpha}$. We call the prime number $q$ golden if there exists a natural number $m$ such that $q|a_m$. Suppose that $q_1<q_2<q_3<.....$ are all the golden prime numbers of the sequence $\{a_n\}$.



a) Prove that if $\alpha=1.5$, then $q_n\le 1390^n$. Can you find a better bound for $q_n$?



b) Prove that if $\alpha=2.4$, then $q_n\le 1390^{2n}$. Can you find a better bound for $q_n$?



part a proposed by mahyar sefidgaran by an idea of this question that the $n$th prime number is less than $2^{2n-2}$

part b proposed by mostafa einollah zade
2011 Iran MO (3rd Round)
We call two circles in the space fighting if they are intersected or they are clipsed.

Find a good necessary and sufficient condition for four distinct points $A,B,A',B'$ such that each circle passing through $A,B$ and each circle passing through $A',B'$ are fighting circles.



proposed by Ali Khezeli
2011 Iran MO (3rd Round)
Suppose that $f:P(\mathbb N)\longrightarrow \mathbb N$ and $A$ is a subset of $\mathbb N$. We call $f$ $A$-predicting if the set $\{x\in \mathbb N|x\notin A, f(A\cup x)\neq x \}$ is finite. Prove that there exists a function that for every subset $A$ of natural numbers, it's $A$-predicting.



proposed by Sepehr Ghazi-Nezami
2011 Iran MO (3rd Round)
We call the sequence $d_1,....,d_n$ of natural numbers, not necessarily distinct, covering if there exists arithmetic progressions like $c_1+kd_1$,....,$c_n+kd_n$ such that every natural number has come in at least one of them. We call this sequence short if we can not delete any of the $d_1,....,d_n$ such that the resulting sequence be still covering.

a) Suppose that $d_1,....,d_n$ is a short covering sequence and suppose that we've covered all the natural numbers with arithmetic progressions $a_1+kd_1,.....,a_n+kd_n$, and suppose that $p$ is a prime number that $p$ divides $d_1,....,d_k$ but it does not divide $d_{k+1},....,d_n$. Prove that the remainders of $a_1,....,a_k$ modulo $p$ contains all the numbers $0,1,.....,p-1$.

b) Write anything you can about covering sequences and short covering sequences in the case that each of $d_1,....,d_n$ has only one prime divisor.



proposed by Ali Khezeli
2010 Iran MO (3rd Round)
suppose that polynomial $p(x)=x^{2010}\pm x^{2009}\pm...\pm x\pm 1$ does not have a real root. what is the maximum number of coefficients to be $-1$?(14 points)
2010 Iran MO (3rd Round)
$a,b,c$ are positive real numbers. prove the following inequality:



$\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{(a+b+c)^2}\ge \frac{7}{25}(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{a+b+c})^2$



(20 points)
2010 Iran MO (3rd Round)
prove that for each natural number $n$ there exist a polynomial with degree $2n+1$ with coefficients in $\mathbb{Q}[x]$ such that it has exactly $2$ complex zeros and it's irreducible in $\mathbb{Q}[x]$.(20 points)
2010 Iran MO (3rd Round)
For each polynomial $p(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ we define it's derivative as this and we show it by $p'(x)$:



$$p'(x)=na_nx^{n-1}+(n-1)a_{n-1}x^{n-2}+...+2a_2x+a_1$$



a) For each two polynomials $p(x)$ and $q(x)$ prove that:(3 points)

$$(p(x)q(x))'=p'(x)q(x)+p(x)q'(x)$$



b) Suppose that $p(x)$ is a polynomial with degree $n$ and $x_1,x_2,...,x_n$ are it's zeros. prove that:(3 points)

$$\frac{p'(x)}{p(x)}=\sum_{i=1}^{n}\frac{1}{x-x_i}$$





c) $p(x)$ is a monic polynomial with degree $n$ and $z_1,z_2,...,z_n$ are it's zeros such that:

$$|z_1|=1, \quad \forall i\in\{2,..,n\}:|z_i|\le1$$



Prove that $p'(x)$ has at least one zero in the disc with length one with the center $z_1$ in complex plane. (disc with length one with the center $z_1$ in complex plane: $D=\{z \in \mathbb C: |z-z_1|\le1\}$)(20 points)
2010 Iran MO (3rd Round)
$x,y,z$ are positive real numbers such that $xy+yz+zx=1$. prove that:

$3-\sqrt{3}+\frac{x^2}{y}+\frac{y^2}{z}+\frac{z^2}{x}\ge(x+y+z)^2$

(20 points)



the exam time was 6 hours.
2010 Iran MO (3rd Round)
suppose that $a=3^{100}$ and $b=5454$. how many $z$s in $[1,3^{99})$ exist such that for every $c$ that $gcd(c,3)=1$, two equations $x^z\equiv c$ and $x^b\equiv c$ (mod $a$) have the same number of answers?($\frac{100}{6}$ points)
2010 Iran MO (3rd Round)
$R$ is a ring such that $xy=yx$ for every $x,y\in R$ and if $ab=0$ then $a=0$ or $b=0$. if for every Ideal $I\subset R$ there exist $x_1,x_2,..,x_n$ in $R$ ($n$ is not constant) such that $I=(x_1,x_2,...,x_n)$, prove that every element in $R$ that is not $0$ and it's not a unit, is the product of finite irreducible elements.($\frac{100}{6}$ points)
2010 Iran MO (3rd Round)
If $p$ is a prime number, what is the product of elements like $g$ such that $1\le g\le p^2$ and $g$ is a primitive root modulo $p$ but it's not a primitive root modulo $p^2$, modulo $p^2$?($\frac{100}{6}$ points)
2010 Iran MO (3rd Round)
sppose that $\sigma_k:\mathbb N \longrightarrow \mathbb R$ is a function such that $\sigma_k(n)=\sum_{d|n}d^k$. $\rho_k:\mathbb N \longrightarrow \mathbb R$ is a function such that $\rho_k\ast \sigma_k=\delta$. find a formula for $\rho_k$.($\frac{100}{6}$ points)
2010 Iran MO (3rd Round)
prove that if $p$ is a prime number such that $p=12k+\{2,3,5,7,8,11\}$($k \in \mathbb N \cup \{0\}$), there exist a field with $p^2$ elements.($\frac{100}{6}$ points)
2010 Iran MO (3rd Round)
$g$ and $n$ are natural numbers such that $gcd(g^2-g,n)=1$ and $A=\{g^i|i \in \mathbb N\}$ and $B=\{x\equiv (n)|x\in A\}$(by $x\equiv (n)$ we mean a number from the set $\{0,1,...,n-1\}$ which is congruent with $x$ modulo $n$). if for $0\le i\le g-1$

$a_i=|[\frac{ni}{g},\frac{n(i+1)}{g})\cap B|$

prove that $g-1|\sum_{i=0}^{g-1}ia_i$.( the symbol $|$ $|$ means the number of elements of the set)($\frac{100}{6}$ points)



the exam time was 4 hours
2010 Iran MO (3rd Round)
1. In a triangle $ABC$, $O$ is the circumcenter and $I$ is the incenter. $X$ is the reflection of $I$ to $O$. $A_1$ is foot of the perpendicular from $X$ to $BC$. $B_1$ and $C_1$ are defined similarly. prove that $AA_1$,$BB_1$ and $CC_1$ are concurrent.(12 points)
2010 Iran MO (3rd Round)
in a quadrilateral $ABCD$, $E$ and $F$ are on $BC$ and $AD$ respectively such that the area of triangles $AED$ and $BCF$ is $\frac{4}{7}$ of the area of $ABCD$. $R$ is the intersection point of digonals of $ABCD$. $\frac{AR}{RC}=\frac{3}{5}$ and $\frac{BR}{RD}=\frac{5}{6}$.

a) in what ratio does $EF$ cut the digonals?(13 points)

b) find $\frac{AF}{FD}$.(5 points)
2010 Iran MO (3rd Round)
in a quadrilateral $ABCD$ digonals are perpendicular to each other. let $S$ be the intersection of digonals. $K$,$L$,$M$ and $N$ are reflections of $S$ to $AB$,$BC$,$CD$ and $DA$. $BN$ cuts the circumcircle of $SKN$ in $E$ and $BM$ cuts the circumcircle of $SLM$ in $F$. prove that $EFLK$ is concyclic.(20 points)
2010 Iran MO (3rd Round)
in a triangle $ABC$, $I$ is the incenter. $BI$ and $CI$ cut the circumcircle of $ABC$ at $E$ and $F$ respectively. $M$ is the midpoint of $EF$. $C$ is a circle with diameter $EF$. $IM$ cuts $C$ at two points $L$ and $K$ and the arc $BC$ of circumcircle of $ABC$ (not containing $A$) at $D$. prove that $\frac{DL}{IL}=\frac{DK}{IK}$.(25 points)
2010 Iran MO (3rd Round)
In a triangle $ABC$, $I$ is the incenter. $D$ is the reflection of $A$ to $I$. the incircle is tangent to $BC$ at point $E$. $DE$ cuts $IG$ at $P$ ($G$ is centroid). $M$ is the midpoint of $BC$. prove that

a) $AP||DM$.(15 points)

b) $AP=2DM$. (10 points)