Navvye/Olympiad-Problems-CSV huggingface dataset:
2005 Iran MO (3rd Round)
Suppose $A\subseteq \mathbb R^m$ is closed and non-empty. Let $f:A\to A$ is a lipchitz function with constant less than 1. (ie there exist $c<1$ that $|f(x)-f(y)|<c|x-y|,\ \forall x,y \in A)$. Prove that there exists a unique point $x\in A$ such that $f(x)=x$.
2005 Iran MO (3rd Round)
From each vertex of triangle $ABC$ we draw 3 arbitary parrallell lines, and from each vertex we draw a perpendicular to these lines. There are 3 rectangles that one of their diagnals is triangle's side. We draw their other diagnals and call them $\ell_1$, $\ell_2$ and $\ell_3$.



a) Prove that $\ell_1$, $\ell_2$ and $\ell_3$ are concurrent at a point $P$.



b) Find the locus of $P$ as we move the 3 arbitary lines.
2005 Iran MO (3rd Round)
Suppose $O$ is circumcenter of triangle $ABC$. Suppose $\frac{S(OAB)+S(OAC)}2=S(OBC)$. Prove that the distance of $O$ (circumcenter) from the radical axis of the circumcircle and the 9-point circle is $$\frac {a^2}{\sqrt{9R^2-(a^2+b^2+c^2)}}$$
2005 Iran MO (3rd Round)
Prove that in acute-angled traingle ABC if $r$ is inradius and $R$ is radius of circumcircle then: $$a^2+b^2+c^2\geq 4(R+r)^2$$
2005 Iran MO (3rd Round)
Suppose in triangle $ABC$ incircle touches the side $BC$ at $P$ and $\angle APB=\alpha$. Prove that : $$\frac1{p-b}+\frac1{p-c}=\frac2{rtg\alpha}$$
2005 Iran MO (3rd Round)
Suppose $H$ and $O$ are orthocenter and circumcenter of triangle $ABC$. $\omega$ is circumcircle of $ABC$. $AO$ intersects with $\omega$ at $A_1$. $A_1H$ intersects with $\omega$ at $A'$ and $A''$ is the intersection point of $\omega$ and $AH$. We define points $B',\ B'',\ C'$ and $C''$ similiarly. Prove that $A'A'',B'B''$ and $C'C''$ are concurrent in a point on the Euler line of triangle $ABC$.
2005 Iran MO (3rd Round)
Find all $n,p,q\in \mathbb N$ that:$$2^n+n^2=3^p7^q$$
2005 Iran MO (3rd Round)
Let $a\in\mathbb N$ and $m=a^2+a+1$. Find the number of $0\leq x\leq m$ that:$$x^3\equiv1(\mbox{mod}\ m)$$
2005 Iran MO (3rd Round)
$p(x)$ is an irreducible polynomial in $\mathbb Q[x]$ that $\mbox{deg}\ p$ is odd. $q(x),r(x)$ are polynomials with rational coefficients that $p(x)|q(x)^2+q(x).r(x)+r(x)^2$. Prove that $$p(x)^2|q(x)^2+q(x).r(x)+r(x)^2$$
2005 Iran MO (3rd Round)
$k$ is an integer. We define the sequence $\{a_n\}_{n=0}^{\infty}$ like this:

$$a_0=0,\ \ \ a_1=1,\ \ \ a_n=2ka_{n-1}-(k^2+1)a_{n-2}\ \ (n \geq 2)$$

$p$ is a prime number that $p\equiv 3(\mbox{mod}\ 4)$

a) Prove that $a_{n+p^2-1}\equiv a_n(\mbox{mod}\ p)$

b) Prove that $a_{n+p^3-p}\equiv a_n(\mbox{mod}\ p^2)$
2005 Iran MO (3rd Round)
Let $a,b,c\in \mathbb N$ be such that $a,b\neq c$. Prove that there are infinitely many prime numbers $p$ for which there exists $n\in\mathbb N$ that $p|a^n+b^n-c^n$.
2005 Iran MO (3rd Round)
We call the set $A\in \mathbb R^n$ CN if and only if for every continuous $f:A\to A$ there exists some $x\in A$ such that $f(x)=x$.

a) Example: We know that $A = \{ x\in\mathbb R^n | |x|\leq 1 \}$ is CN.

b) The circle is not CN.



Which one of these sets are CN?



1) $A=\{x\in\mathbb R^3| |x|=1\}$



2) The cross $\{(x,y)\in\mathbb R^2|xy=0,\ |x|+|y|\leq1\}$



3) Graph of the function $f:[0,1]\to \mathbb R$ defined by

$$f(x)=\sin\frac 1x\ \mbox{if}\ x\neq0,\ f(0)=0$$
2005 Iran MO (3rd Round)
$n$ vectors are on the plane. We can move each vector forward and backeard on the line that the vector is on it. If there are 2 vectors that their endpoints concide we can omit them and replace them with their sum (If their sum is nonzero). Suppose with these operations with 2 different method we reach to a vector. Prove that these vectors are on a common line
2005 Iran MO (3rd Round)
$f(n)$ is the least number that there exist a $f(n)-$mino that contains every $n-$mino.

Prove that $10000\leq f(1384)\leq960000$.

Find some bound for $f(n)$
2005 Iran MO (3rd Round)
a) Year 1872 Texas

3 gold miners found a peice of gold. They have a coin that with possibility of $\frac 12$ it will come each side, and they want to give the piece of gold to one of themselves depending on how the coin will come. Design a fair method (It means that each of the 3 miners will win the piece of gold with possibility of $\frac 13$) for the miners.



b) Year 2005, faculty of Mathematics, Sharif university of Technolgy

Suppose $0<\alpha<1$ and we want to find a way for people name $A$ and $B$ that the possibity of winning of $A$ is $\alpha$. Is it possible to find this way?



c) Year 2005 Ahvaz, Takhti Stadium

Two soccer teams have a contest. And we want to choose each player's side with the coin, But we don't know that our coin is fair or not. Find a way to find that coin is fair or not?



d) Year 2005,summer

In the National mathematical Oympiad in Iran. Each student has a coin and must find a way that the possibility of coin being TAIL is $\alpha$ or no. Find a way for the student.
2005 Iran MO (3rd Round)
An airplane wants to go from a point on the equator, and at each moment it will go to the northeast with speed $v$. Suppose the radius of earth is $R$.

a) Will the airplane reach to the north pole? If yes how long it will take to reach the north pole?

b) Will the airplne rotate finitely many times around the north pole? If yes how many times?
2005 Iran MO (3rd Round)
We define a relation between subsets of $\mathbb R ^n$. $A \sim B\Longleftrightarrow$ we can partition $A,B$ in sets $A_1,\dots,A_n$ and $B_1,\dots,B_n$(i.e $\displaystyle A=\bigcup_{i=1} ^n A_i,\ B=\bigcup_{i=1} ^n B_i, 

A_i\cap A_j=\emptyset,\ B_i\cap B_j=\emptyset$) and $A_i\simeq B_i$.

Say the the following sets have the relation $\sim$ or not ?



a) Natural numbers and composite numbers.

b) Rational numbers and rational numbers with finite digits in base 10.

c) $\{x\in\mathbb Q|x<\sqrt 2\}$ and $\{x\in\mathbb Q|x<\sqrt 3\}$

d) $A=\{(x,y)\in\mathbb R^2|x^2+y^2<1\}$ and $A\setminus \{(0,0)\}$
2005 Iran MO (3rd Round)
For each $m\in \mathbb N$ we define $rad\ (m)=\prod p_i$, where $m=\prod p_i^{\alpha_i}$.



abc Conjecture

Suppose $\epsilon >0$ is an arbitrary number, then there exist $K$ depinding on $\epsilon$ that for each 3 numbers $a,b,c\in\mathbb Z$ that $gcd (a,b)=1$ and $a+b=c$ then: $$ max\{|a|,|b|,|c|\}\leq K(rad\ (abc))^{1+\epsilon} $$



Now prove each of the following statements by using the $abc$ conjecture :

a) Fermat's last theorem for $n>N$ where $N$ is some natural number.

b) We call $n=\prod p_i^{\alpha_i}$ strong if and only $\alpha_i\geq 2$.

c) Prove that there are finitely many $n$ such that $n,\ n+1,\ n+2$ are strong.

d) Prove that there are finitely many rational numbers $\frac pq$ such that: $$ \Big| \sqrt[3]{2}-\frac pq \Big|<\frac{2^ {1384}}{q^3}  $$
2005 Iran MO (3rd Round)
Suppose we have some proteins that each protein is a sequence of 7 "AMINO-ACIDS" $A,\ B,\ C,\ H,\ F,\ N$. For example $AFHNNNHAFFC$ is a protein. There are some steps that in each step an amino-acid will change to another one. For example with the step $NA\rightarrow N$ the protein $BANANA$ will cahnge to $BANNA$("in Persian means workman"). We have a set of allowed steps that each protein can change with these steps. For example with the

set of steps:

$\\ 1)\ AA\longrightarrow A\\ 2)\ AB\longrightarrow BA\\ 3)\ A\longrightarrow \mbox{null}$

Protein $ABBAABA$ will change like this:

$\\ ABB\underline{AA}BA\\ \underline{AB}BABA\\ B\underline{AB}ABA\\ BB\underline{AA}BA\\ BB\underline{AB}A\\ BBB\underline{AA}\\ BBB\underline{A}\\ BBB$

You see after finite steps this protein will finish it steps.

Set of allowed steps that for them there exist a protein that may have infinitely many steps is dangerous. Which of the following allowed sets are dangerous?

a) $NO\longrightarrow OONN$

b) $\left\{\begin{array}{c}HHCC\longrightarrow HCCH\\ CC\longrightarrow CH\end{array}\right.$

c) Design a set of allowed steps that change $\underbrace{AA\dots A}_{n}\longrightarrow\underbrace{BB\dots B}_{2^{n}}$

d) Design a set of allowed steps that change $\underbrace{A\dots A}_{n}\underbrace{B\dots B}_{m}\longrightarrow\underbrace{CC\dots C}_{mn}$

You see from $c$ and $d$ that we acn calculate the functions $F(n)=2^{n}$ and $G(M,N)=mn$ with these steps. Find some other calculatable functions with these steps. (It has some extra mark.)
2004 Iran MO (3rd Round)
We say $m \circ n$ for natural m,n $\Longleftrightarrow$

nth number of binary representation of m is 1 or mth number of binary representation of n is 1.

and we say $m \bullet n$ if and only if $m,n$ doesn't have the relation $\circ$

We say $A \subset \mathbb{N}$ is golden $\Longleftrightarrow$

$\forall U,V \subset A$ that are finite and arenot empty and $U \cap V = \emptyset$,There exist $z \in A$ that $\forall x \in U,y \in V$ we have $z \circ x ,z \bullet y$

Suppose $\mathbb{P}$ is set of prime numbers.Prove if $\mathbb{P}=P_1 \cup ... \cup P_k$ and $P_i \cap P_j = \emptyset$ then one of $P_1,...,P_k$ is golden.
2004 Iran MO (3rd Round)
$A$ is a compact convex set in plane. Prove that there exists a point $O \in A$, such that for every line $XX'$  passing through $O$, where $X$ and $X'$ are boundary points of $A$, then

$$ \frac12 \leq \frac {OX}{OX'} \leq 2.$$
2004 Iran MO (3rd Round)
Suppose $V= \mathbb{Z}_2^n$ and for a vector $x=(x_1,..x_n)$ in $V$ and permutation $\sigma$.We have $x_{\sigma}=(x_{\sigma(1)},...,x_{\sigma(n)})$

Suppose $ n=4k+2,4k+3$ and $f:V \to V$ is injective and if $x$ and $y$ differ in more than $n/2$ places then $f(x)$ and $f(y)$ differ in more than $n/2$ places.

Prove there exist permutaion $\sigma$ and vector $v$ that $f(x)=x_{\sigma}+v$
2004 Iran MO (3rd Round)
We have finite white and finite black points that for each 4 oints there is a line that white points and black points are at different sides of this line.Prove there is a line that all white points and black points are at different side of this line.
2004 Iran MO (3rd Round)
assume that k,n are two positive  integer $k\leq n$count the number of permutation $\{\ 1,\dots ,n\}\ $ st for any $1\leq i,j\leq k$and any positive integer m we have $f^m(i)\neq j$ ($f^m$ meas iterarte function,)
2004 Iran MO (3rd Round)
assume that we have a n*n table we fill it with 1,...,n such that each number exists exactly n times prove that there exist a row or column such that at least $\sqrt{n}$ diffrent number are contained.
2004 Iran MO (3rd Round)
Suppose $F$ is a polygon with lattice vertices and sides parralell to x-axis and y-axis.Suppose $S(F),P(F)$ are area and perimeter of $F$.

Find the smallest k that:

$S(F) \leq k.P(F)^2$
2004 Iran MO (3rd Round)
$\mathbb{P}$ is a n-gon with sides $l_1 ,...,l_n$ and vertices on a circle. Prove that no n-gon with this sides has area more than $\mathbb{P}$
2004 Iran MO (3rd Round)
Let $ABC$ be a triangle, and $O$ the center of its circumcircle.

Let a line through the point $O$ intersect the lines $AB$ and $AC$ at the points $M$ and $N$, respectively. Denote by $S$ and $R$ the midpoints of the segments $BN$ and $CM$, respectively.

Prove that $\measuredangle ROS=\measuredangle BAC$.
2004 Iran MO (3rd Round)
$f:\mathbb{R}^2 \to \mathbb{R}^2$ is injective and surjective. Distance of $X$ and $Y$ is not less than distance of $f(X)$ and $f(Y)$. Prove for $A$ in plane:

$$ S(A) \geq S(f(A))$$

where $S(A)$ is area of $A$
2004 Iran MO (3rd Round)
assume that ABC is acute traingle and AA' is median we extend it until it meets circumcircle at A". let $AP_a$ be a diameter of the circumcircle. the pependicular from A' to $AP_a$ meets the tangent to circumcircle  at A"  in the point $X_a$; we define $X_b,X_c$ similary . prove that $X_a,X_b,X_c$ are one a line.
2004 Iran MO (3rd Round)
$\mathbb{N}_{10}$ is generalization of $\mathbb{N}$ that every hypernumber in $\mathbb{N}_{10}$ is something like: $\overline{...a_2a_1a_0}$ with $a_i \in {0,1..9}$

(Notice that $\overline {...000} \in \mathbb{N}_{10}$)

Also we easily have $+,*$ in $\mathbb{N}_{10}$.

first $k$ number of $a*b$= first $k$ nubmer of (first $k$ number of a * first $k$ number of b)

first $k$ number of $a+b$= first $k$ nubmer of (first $k$ number of a + first $k$ number of b)

Fore example $\overline {...999}+ \overline {...0001}= \overline {...000}$

Prove  that every monic polynomial in $\mathbb{N}_{10}[x]$ with degree $d$ has at most $d^2$ roots.
2004 Iran MO (3rd Round)
Suppose $f$ is a polynomial in $\mathbb{Z}[X]$ and m is integer .Consider the sequence $a_i$ like this $a_1=m$ and $a_{i+1}=f(a_i)$ find all polynomials $f$ and alll integers $m$ that for each $i$:

$$ a_i | a_{i+1}$$
2004 Iran MO (3rd Round)
We define $ f: \mathbb{N} \rightarrow \mathbb{N}$, $ f(n) = \sum_{k = 1}^{n}(k,n)$.



a) Show that if $ \gcd(m,n)=1$ then we have $ f(mn)=f(m)\cdot f(n)$;



b) Show that $ \sum_{d|n}f(d) = nd(n)$.
2004 Iran MO (3rd Round)
This problem is easy but nobody solved it.

point $A$ moves in a line with speed $v$ and $B$ moves also with speed $v'$ that at every time the direction of move of $B$ goes from $A$.We know $v \geq v'$.If we know the point of beginning of path of $A$, then $B$ must be where at first that $B$ can catch $A$.
2004 Iran MO (3rd Round)
Let $ABC$ be a triangle . Let point $X$ be in the triangle and $AX$ intersects $BC$ in $Y$ . Draw the perpendiculars $YP,YQ,YR,YS$ to lines $CA,CX,BX,BA$ respectively. Find the necessary and sufficient condition for $X$ such that $PQRS$ be cyclic .
2004 Iran MO (3rd Round)
Let $ p=4k+1$ be a prime. Prove that $ p$ has at least $ \frac{\phi(p-1)}2$ primitive roots.
2004 Iran MO (3rd Round)
Prove that for any $ n$, there is a subset $ \{a_1,\dots,a_n\}$ of $ \mathbb N$ such that for each subset $ S$ of $ \{1,\dots,n\}$, $ \sum_{i\in S}a_i$ has the same set of prime divisors.
2004 Iran MO (3rd Round)
Find all integer solutions of $ p^3=p^2+q^2+r^2$ where $ p,q,r$ are primes.
2004 Iran MO (3rd Round)
$ p(x)$ is a polynomial in $ \mathbb{Z}[x]$ such that for each $ m,n\in \mathbb{N}$ there is an integer $ a$ such that $ n\mid p(a^m)$. Prove that $0$ or $1$ is a root of $ p(x)$.
2004 Iran MO (3rd Round)
$ a_1, a_2, \ldots, a_n$ are integers, not all equal. Prove that there exist infinitely many prime numbers $ p$ such that for some $ k$

$$ p\mid a_1^k + \dots + a_n^k.$$
2004 Iran MO (3rd Round)
Suppose that $ \mathcal F$ is a family of subsets of $ X$. $ A,B$ are two subsets of $ X$ s.t. each element of $ \mathcal{F}$ has non-empty intersection with $ A, B$. We know that no subset of $ X$ with $ n - 1$ elements has this property. Prove that there is a representation $ A,B$ in the form $ A = \{a_1,\dots,a_n\}$ and $ B =  \{b_1,\dots,b_n\}$, such that for each $ 1\leq i\leq n$, there is an element of $ \mathcal F$ containing both $ a_i, b_i$.
2004 Iran MO (3rd Round)
$ \mathcal F$ is a family of 3-subsets of set $ X$. Every two distinct elements of $ X$ are exactly in $ k$ elements of $ \mathcal F$. It is known that there is a partition of $ \mathcal F$ to sets $ X_1,X_2$ such that each element of $ \mathcal F$ has non-empty intersection with both $ X_1,X_2$. Prove that $ |X|\leq4$.
2004 Iran MO (3rd Round)
In triangle $ ABC$, points $ M,N$ lie on line $ AC$ such that $ MA=AB$ and $ NB=NC$. Also $ K,L$ lie on line $ BC$ such that $ KA=KB$ and $ LA=LC$. It is know that $ KL=\frac12{BC}$ and $ MN=AC$. Find angles of triangle $ ABC$.
2004 Iran MO (3rd Round)
Finitely many convex subsets of $\mathbb R^3$ are given, such that every three have non-empty intersection. Prove that there exists a line in $\mathbb R^3$ that intersects all of these subsets.
2004 Iran MO (3rd Round)
Finitely many points are given on the surface of a sphere, such that every four of them lie on the surface of open hemisphere. Prove that all points lie on the surface of an open hemisphere.
2004 Iran MO (3rd Round)
$ \Delta_1,\ldots,\Delta_n$ are $ n$ concurrent segments (their lines concur) in the real plane. Prove that if for every three of them there is a line intersecting these three segments, then there is a line that intersects all of the segments.
2004 Iran MO (3rd Round)
Find all prime numbers $p$ such that $ p = m^2 + n^2$ and $p\mid m^3+n^3-4$.
2004 Iran MO (3rd Round)
Incircle of triangle $ ABC$ touches $ AB,AC$ at $ P,Q$. $ BI, CI$ intersect with $ PQ$ at $ K,L$. Prove that circumcircle of $ ILK$ is tangent to incircle of $ ABC$ if and only if $ AB+AC=3BC$.
2004 Iran MO (3rd Round)
Find all polynomials $ p\in\mathbb Z[x]$ such that $ (m,n)=1\Rightarrow (p(m),p(n))=1$
2003 Iran MO (3rd Round)
suppose this equation: x <sup>2</sup> +y <sup>2</sup> +z <sup>2</sup> =w <sup>2</sup>   . show that the solution of this equation ( if w,z have same parity) are in this form:

x=2d(XZ-YW), y=2d(XW+YZ),z=d(X <sup>2</sup> +Y <sup>2</sup> -Z <sup>2</sup> -W <sup>2</sup> ),w=d(X <sup>2</sup> +Y <sup>2</sup> +Z <sup>2</sup> +W <sup>2</sup> )
2003 Iran MO (3rd Round)
assume ABCD a convex quadrilatral. P and Q are on BC and DC respectively such that angle BAP= angle DAQ .prove that [ADQ]=[ABP]

([ABC] means its area ) iff the line which crosses through the orthocenters of these traingles , is perpendicular to AC.
2003 Iran MO (3rd Round)
assume that A is a finite subset of prime numbers, and a is an positive integer.

prove that there are only finitely many positive integers m s.t: prime divisors of a^m-1 are contained in A.
2003 Iran MO (3rd Round)
XOY is angle in the plane.A,B are variable point on OX,OY such that 1/OA+1/OB=1/K (k is constant).draw two circles with diameter OA and OB.prove that common external  tangent to these circles is tangent to the constant circle( ditermine the  radius and the locus of its center).
2003 Iran MO (3rd Round)
Let $p$ be an odd prime number. Let $S$ be the sum of all primitive roots modulo $p$. Show that if $p-1$ isn't squarefree (i. e., if there exist integers $k$ and $m$ with $k>1$ and $p-1=k^2m$), then $S \equiv 0 \mod p$.



If not, then what is $S$ congruent to $\mod p$ ?
2003 Iran MO (3rd Round)
let the incircle of a triangle ABC touch BC,AC,AB at A1,B1,C1 respectively. M and N are the midpoints of AB1 and AC1 respectively. MN meets A1C1 at T . draw two tangents TP and TQ through T to incircle. PQ meets MN at L and B1C1 meets PQ at K . assume I is the center of the incircle .



prove IK is parallel to AL
2003 Iran MO (3rd Round)
$f_{1},f_{2},\dots,f_{n}$ are polynomials with integer coefficients. Prove there exist a reducible $g(x)$ with integer coefficients that $f_{1}+g,f_{2}+g,\dots,f_{n}+g$ are irreducible.
2003 Iran MO (3rd Round)
A positive integer $n$ is said to be a perfect power if $n=a^b$ for some integers $a,b$ with $b>1$.

$(\text{a})$ Find $2004$ perfect powers in arithmetic progression.

$(\text{b})$ Prove that perfect powers cannot form an infinite arithmetic progression.
2003 Iran MO (3rd Round)
Does there exist an infinite set $ S$ such that for every $ a, b \in S$ we have  $ a^2 + b^2 - ab \mid (ab)^2$.
2003 Iran MO (3rd Round)
let p be a prime and a and n be natural numbers such that (p^a -1 )/ (p-1) = 2 ^n

find the number of natural divisors of na. :)
2003 Iran MO (3rd Round)
assume that X is a set of n number.and $0\leq k\leq n$.the maximum number of permutation which acting on $X$ st every two of them have at least k component in common,is $a_{n,k}$.and the maximum nuber of permutation st every two of them have at most k component in common,is $b_{n,k}$.

a)proeve that :$a_{n,k}\cdot b_{n,k-1}\leq n!$

b)assume that p is prime number,determine the exact value of $a_{p,2}$.
2003 Iran MO (3rd Round)
There is a lamp in space.(Consider lamp a point)

Do there exist finite number of equal sphers in space that the light of the lamp can not go to the infinite?(If a ray crash in a sphere it stops)
2003 Iran MO (3rd Round)
here is the most difficult and the most beautiful problem occurs in 21th iranian (2003) olympiad

assume that P is n-gon ,lying on the plane ,we name its edge 1,2,..,n.

if S=s1,s2,s3,.... be a finite or infinite sequence such that  for each i, si is in {1,2,...,n},

we move P on the plane according to the S in this form: at first we reflect P through the s1

( s1 means the edge which iys number is s1)then through s2 and so on like the figure below.

a)show that there exist the infinite sequence S sucth that if we move P according to S we cover all the plane

b)prove that the sequence in a) isn't periodic.

c)assume that P is regular pentagon ,which the radius of its circumcircle is 1,and D is circle ,with radius 1.00001 ,arbitrarily in the plane .does exist a sequence S such that we move P according to S then P reside in D completely?
2003 Iran MO (3rd Round)
n \geq 6 is an integer. evaluate the minimum of f(n) s.t: any graph with n vertices and f(n) edge contains two cycle which are distinct( also they have no comon vertice)?
2003 Iran MO (3rd Round)
Assume $m\times n$ matrix which is filled with just 0, 1 and any two row differ in at least $n/2$ members, show that $m \leq 2n$.

( for example the diffrence of this two row is only in one index

110

100)



Edited by Myth
2003 Iran MO (3rd Round)
Segment $ AB$ is fixed in plane. Find the largest $ n$, such that there are $ n$ points $ P_1,P_2,\dots,P_n$ in plane that triangles $ ABP_i$ are similar for $ 1\leq i\leq n$. Prove that all of $ P_i$'s lie on a circle.
2003 Iran MO (3rd Round)
A simple calculator is given to you. (It contains 8 digits and only does the operations +,-,*,/,$ \sqrt{\mbox{}}$) How can you find $ 3^{\sqrt{2}}$ with accuracy of 6 digits.
2003 Iran MO (3rd Round)
In tetrahedron $ ABCD$, radius four circumcircles of four faces are equal. Prove that $ AB=CD$, $ AC=BD$ and $ AD=BC$.
2003 Iran MO (3rd Round)
An integer $ n$ is called a good number if and only if $ |n|$ is not square of another intger. Find all integers $ m$ such that they can be written in infinitely many ways as sum of three different good numbers and product of these three numbers is square of an odd number.
2003 Iran MO (3rd Round)
Suppose that $ M$ is an arbitrary point on side $ BC$ of triangle $ ABC$. $ B_1,C_1$ are points on $ AB,AC$ such that $ MB = MB_1$ and $ MC = MC_1$.  Suppose that $ H,I$ are orthocenter of triangle $ ABC$ and incenter of triangle $ MB_1C_1$. Prove that $ A,B_1,H,I,C_1$ lie on a circle.
2003 Iran MO (3rd Round)
Let $ ABC$ be a triangle. $ W_a$ is a circle with center on $ BC$ passing through $ A$ and perpendicular to circumcircle of $ ABC$. $ W_b,W_c$ are defined similarly. Prove that center of $ W_a,W_b,W_c$ are collinear.
2003 Iran MO (3rd Round)
Let $ a_1=a_2=1$ and $$ a_{n+2}=\frac{n(n+1)a_{n+1}+n^2a_n+5}{n+2}-2$$for each $ n\in\mathbb N$. Find all $ n$ such that $ a_n\in\mathbb N$.
2003 Iran MO (3rd Round)
Find all homogeneous linear recursive sequences such that there is a $ T$  such that $ a_n=a_{n+T}$ for each $ n$.
2003 Iran MO (3rd Round)
$ A,B$ are fixed points. Variable line $ l$ passes through the fixed point $ C$. There are two circles passing through $ A,B$ and tangent to $ l$ at $ M,N$. Prove that circumcircle of $ AMN$ passes through a fixed point.
2003 Iran MO (3rd Round)
Let $ A,B,C,Q$ be fixed points on plane. $ M,N,P$ are intersection points of $ AQ,BQ,CQ$ with $ BC,CA,AB$. $ D',E',F'$ are tangency points of incircle of $ ABC$ with $ BC,CA,AB$. Tangents drawn from $ M,N,P$ (not triangle sides) to incircle of $ ABC$ make triangle $ DEF$. Prove that $ DD',EE',FF'$ intersect at $ Q$.
2003 Iran MO (3rd Round)
Circles $ C_1,C_2$ intersect at $ P$. A line $ \Delta$ is drawn arbitrarily from $ P$ and intersects with $ C_1,C_2$ at $ B,C$. What is locus of $ A$ such that the median of $ AM$ of triangle $ ABC$ has fixed length $ k$.
2003 Iran MO (3rd Round)
$ S\subset\mathbb N$ is called a square set, iff for each $ x,y\in S$, $ xy+1$ is square of an integer.

a) Is $ S$ finite?

b) Find maximum number of elements of $ S$.
2003 Iran MO (3rd Round)
There are $ n$ points in $ \mathbb R^3$ such that every three form an acute angled triangle. Find maximum of $ n$.
2003 Iran MO (3rd Round)
Let $ c\in\mathbb C$ and $ A_c = \{p\in \mathbb C[z]|p(z^2 + c) = p(z)^2 + c\}$.



a) Prove that for each $ c\in C$, $ A_c$ is infinite.



b) Prove that if $ p\in A_1$, and $ p(z_0) = 0$, then $ |z_0| < 1.7$.



c) Prove that each element of $ A_c$ is odd or even.



Let $ f_c = z^2 + c\in \mathbb C[z]$. We see easily that $ B_c: = \{z,f_c(z),f_c(f_c(z)),\dots\}$ is a subset of $ A_c$. Prove that in the following cases $ A_c = B_c$.



d) $ |c| > 2$.



e) $ c\in \mathbb Q\backslash\mathbb Z$.



f) $ c$ is a non-algebraic number



g) $ c$ is a real number and $ c\not\in [ - 2,\frac14]$.