Navvye/Olympiad-Problems-CSV huggingface dataset:
2008 Iran MO (3rd Round)
A graph is called a self-intersecting graph if it is isomorphic to a graph whose every edge is a segment and every two edges intersect. Notice that no edge contains a vertex except its two endings.

a) Find all $ n$'s for which the cycle of length $ n$ is self-intersecting.

b) Prove that in a self-intersecting graph $ |E(G)|\leq|V(G)|$.

c) Find all self-intersecting graphs.

//cdn.artofproblemsolving.com/images/852ccdfd535666606739dbf8ac5778da24d14d6b.png
2008 Iran MO (3rd Round)
In an old script found in ruins of Perspolis is written:

This script has been finished in a year whose 13th power is 

258145266804692077858261512663

You should know that if you are skilled in Arithmetics you will know the year this script is finished easily.

Find the year the script is finished. Give a reason for your answer.
2007 Iran MO (3rd Round)
Let $ a,b$ be two complex numbers. Prove that roots of $ z^{4}+az^{2}+b$ form a rhombus with origin as center, if and only if $ \frac{a^{2}}{b}$ is a non-positive real number.
2007 Iran MO (3rd Round)
$ a,b,c$ are three different positive real numbers. Prove that:$$ \left|\frac{a+b}{a-b}+\frac{b+c}{b-c}+\frac{c+a}{c-a}\right|>1$$
2007 Iran MO (3rd Round)
Find the largest real $ T$ such that for each non-negative real numbers $ a,b,c,d,e$ such that $ a+b=c+d+e$: $$ \sqrt{a^{2}+b^{2}+c^{2}+d^{2}+e^{2}}\geq T(\sqrt a+\sqrt b+\sqrt c+\sqrt d+\sqrt e)^{2}$$
2007 Iran MO (3rd Round)
a) Let $ n_{1},n_{2},\dots$ be a sequence of natural number such that $ n_{i}\geq2$ and $ \epsilon_{1},\epsilon_{2},\dots$ be a sequence such that $ \epsilon_{i}\in\{1,2\}$. Prove that the sequence: $$ \sqrt[n_{1}]{\epsilon_{1}+\sqrt[n_{2}]{\epsilon_{2}+\dots+\sqrt[n_{k}]{\epsilon_{k}}}}$$is convergent and its limit is in $ (1,2]$. Define $ \sqrt[n_{1}]{\epsilon_{1}+\sqrt[n_{2}]{\epsilon_{2}+\dots}}$ to be this limit.

b) Prove that for each $ x\in(1,2]$ there exist sequences $ n_{1},n_{2},\dots\in\mathbb N$ and $ n_{i}\geq2$ and $ \epsilon_{1},\epsilon_{2},\dots$, such that $ n_{i}\geq2$ and $ \epsilon_{i}\in\{1,2\}$, and $ x=\sqrt[n_{1}]{\epsilon_{1}+\sqrt[n_{2}]{\epsilon_{2}+\dots}}$
2007 Iran MO (3rd Round)
Prove that for two non-zero polynomials $ f(x,y),g(x,y)$ with real coefficients the system:

$$ \left\{\begin{array}{c}f(x,y)=0\\ g(x,y)=0\end{array}\right.$$

has finitely many solutions in $ \mathbb C^{2}$ if and only if $ f(x,y)$ and $ g(x,y)$ are coprime.
2007 Iran MO (3rd Round)
Let $ ABC$, $ l$ and $ P$ be arbitrary triangle, line and point. $ A',B',C'$ are reflections of $ A,B,C$ in point $ P$. $ A''$ is a point on $ B'C'$ such that $ AA''\parallel l$. $ B'',C''$ are defined similarly. Prove that $ A'',B'',C''$ are collinear.
2007 Iran MO (3rd Round)
a) Let $ ABC$ be a triangle, and $ O$ be its circumcenter. $ BO$ and $ CO$ intersect with $ AC,AB$ at $ B',C'$. $ B'C'$ intersects the circumcircle at two points $ P,Q$. Prove that $ AP=AQ$ if and only if $ ABC$ is isosceles.

b) Prove the same statement if $ O$ is replaced by $ I$, the incenter.
2007 Iran MO (3rd Round)
Let $ I$ be incenter of triangle $ ABC$, $ M$ be midpoint of side $ BC$, and $ T$ be the intersection point of $ IM$ with incircle, in such a way that $ I$ is between $ M$ and $ T$. Prove that $ \angle BIM-\angle CIM=\frac{3}2(\angle B-\angle C)$, if and only if $ AT\perp BC$.
2007 Iran MO (3rd Round)
Let $ ABC$ be a triangle, and $ D$ be a point where incircle touches side $ BC$. $ M$ is midpoint of $ BC$, and $ K$ is a point on $ BC$ such that $ AK\perp BC$. Let $ D'$ be a point on $ BC$ such that $ \frac{D'M}{D'K}=\frac{DM}{DK}$. Define $ \omega_{a}$ to be circle with diameter $ DD'$. We define $ \omega_{B},\omega_{C}$ similarly. Prove that every two of these circles are tangent.
2007 Iran MO (3rd Round)
Let $ ABC$ be a triangle. Squares $ AB_{c}B_{a}C$, $ CA_{b}A_{c}B$ and $ BC_{a}C_{b}A$ are outside the triangle. Square $ B_{c}B_{c}'B_{a}'B_{a}$ with center $ P$ is outside square $ AB_{c}B_{a}C$. Prove that $ BP,C_{a}B_{a}$ and $ A_{c}B_{c}$ are concurrent.
2007 Iran MO (3rd Round)
Let $ n$ be a natural number, such that $ (n,2(2^{1386}-1))=1$. Let $ \{a_{1},a_{2},\dots,a_{\varphi(n)}\}$ be a reduced residue system for $ n$. Prove that:$$ n|a_{1}^{1386}+a_{2}^{1386}+\dots+a_{\varphi(n)}^{1386}$$
2007 Iran MO (3rd Round)
Let $ m,n$ be two integers such that $ \varphi(m) =\varphi(n) = c$. Prove that there exist natural numbers $ b_{1},b_{2},\dots,b_{c}$ such that $ \{b_{1},b_{2},\dots,b_{c}\}$ is a reduced residue system with both $ m$ and $ n$.
2007 Iran MO (3rd Round)
Let $ n$ be a natural number, and $ n = 2^{2007}k+1$, such that $ k$ is an odd number. Prove that $$ n\not|2^{n-1}+1$$
2007 Iran MO (3rd Round)
Find all integer solutions of $$ x^{4}+y^{2}=z^{4}$$
2007 Iran MO (3rd Round)
A hyper-primitive root is a k-tuple $ (a_{1},a_{2},\dots,a_{k})$ and $ (m_{1},m_{2},\dots,m_{k})$ with the following property:

For each $ a\in\mathbb N$, that $ (a,m) = 1$, has a unique representation in the following form:

$$ a\equiv a_{1}^{\alpha_{1}}a_{2}^{\alpha_{2}}\dots a_{k}^{\alpha_{k}}\pmod{m}\qquad 1\leq\alpha_{i}\leq m_{i}$$

Prove that for each $ m$ we have a hyper-primitive root.
2007 Iran MO (3rd Round)
Something related to this problem:

Prove that for a set $ S\subset\mathbb N$, there exists a sequence $ \{a_{i}\}_{i = 0}^{\infty}$ in $ S$ such that for each $ n$, $ \sum_{i = 0}^{n}a_{i}x^{i}$ is irreducible in $ \mathbb Z[x]$ if and only if $ |S|\geq2$.

By Omid Hatami
2007 Iran MO (3rd Round)
Consider two polygons $ P$ and $ Q$. We want to cut $ P$ into some smaller polygons and put them together in such a way to obtain $ Q$. We can translate the pieces but we can not rotate them or reflect them. We call $ P,Q$ equivalent if and only if we can obtain $ Q$ from $ P$(which is obviously an equivalence relation).

//cdn.artofproblemsolving.com/images/23f6eb64435caf975ce94f6548dc52f4cc527e87.png

a) Let $ P,Q$ be two rectangles with the same area(their sides are not necessarily parallel). Prove that $ P$ and $ Q$ are equivalent.

b) Prove that if two triangles are not translation of each other, they are not equivalent.

c) Find a necessary and sufficient condition for polygons $ P,Q$ to be equivalent.
2007 Iran MO (3rd Round)
We call the mapping $ \Delta:\mathbb Z\backslash\{0\}\longrightarrow\mathbb N$, a degree mapping if and only if for each $ a,b\in\mathbb Z$ such that $ b\neq0$ and $ b\not|a$ there exist integers $ r,s$ such that $ a = br+s$, and $ \Delta(s) <\Delta(b)$.

a) Prove that the following mapping is a degree mapping:

$$ \delta(n)=\mbox{Number of digits in the binary representation of }n$$

b) Prove that there exist a degree mapping $ \Delta_{0}$ such that for each degree mapping $ \Delta$ and for each $ n\neq0$, $ \Delta_{0}(n)\leq\Delta(n)$.

c) Prove that $ \delta =\Delta_{0}$

//cdn.artofproblemsolving.com/images/39ddfbca4e48a6470691cdacea9497ee0fb000d7.png
2007 Iran MO (3rd Round)
We call a set $ A$ a good set if it has the following properties:

1. $ A$ consists circles in plane.

2. No two element of $ A$ intersect.

Let $ A,B$ be two good sets. We say $ A,B$ are equivalent if we can reach from $ A$ to $ B$ by moving circles in $ A$, making them bigger or smaller in such a way that during these operations  each circle does not intersect with other circles.

Let $ a_{n}$ be the number of inequivalent good subsets with $ n$ elements. For example $ a_{1}= 1,a_{2}= 2,a_{3}= 4,a_{4}= 9$.

//cdn.artofproblemsolving.com/images/6c5805b464e1eb07d76555aeae63c5a28410de81.png

If there exist $ a,b$ such that $ Aa^{n}\leq a_{n}\leq Bb^{n}$, we say growth ratio of $ a_{n}$ is larger than $ a$ and is smaller than $ b$.

a) Prove that growth ratio of $ a_{n}$ is larger than 2 and is smaller than 4.

b) Find better bounds for upper and lower growth ratio of $ a_{n}$.
2007 Iran MO (3rd Round)
In the following triangular lattice distance of two vertices is length of the shortest path between them. Let $ A_{1},A_{2},\dots,A_{n}$ be constant vertices of the lattice. We want to find a vertex in the lattice whose sum of distances from vertices is minimum. We start from an arbitrary vertex. At each step we check all six neighbors and if sum of distances from vertices of one of the neighbors is less than sum of distances from vertices at the moment we go to that neighbor. If we have more than one choice we choose arbitrarily, as seen in the attached picture.



Obviusly the algorithm finishes

a) Prove that when we can not make any move we have reached to the problem's answer.

b) Does this algorithm reach to answer for each connected graph?
2007 Iran MO (3rd Round)
Look at these fractions. At firs step we have $ \frac{0}{1}$ and $ \frac{1}{0}$, and at each step we write $ \frac{a+b}{c+d}$ between $ \frac{a}{b}$ and $ \frac{c}{d}$, and we do this forever

$$ \begin{array}{ccccccccccccccccccccccccc}\frac{0}{1}&&&&&&&&\frac{1}{0}\\ \frac{0}{1}&&&&\frac{1}{1}&&&&\frac{1}{0}\\ \frac{0}{1}&&\frac{1}{2}&&\frac{1}{1}&&\frac{2}{1}&&\frac{1}{0}\\ \frac{0}{1}&\frac{1}{3}&\frac{1}{2}&\frac{2}{3}&\frac{1}{1}&\frac{3}{2}&\frac{2}{1}&\frac{3}{1}&\frac{1}{0}\\ &&&&\dots\end{array}$$

a) Prove that each of these fractions is irreducible.

b) In the plane we have put infinitely many circles of diameter 1, over each integer on the real line, one circle. The inductively we put circles that each circle is tangent to two adjacent circles and real line, and we do this forever. Prove that points of tangency of these circles are exactly all the numbers in part a(except $ \frac{1}{0}$).

//cdn.artofproblemsolving.com/images/f8ca50c91d67a99bdb77dd5dab3d66101f71815a.png

c) Prove that in these two parts all of positive rational numbers appear.

If you don't understand the numbers, look at here.
2007 Iran MO (3rd Round)
Scientist have succeeded to find new numbers between real numbers with strong microscopes. Now real numbers are extended in a new larger system we have an order on it (which if induces normal order on $ \mathbb R$), and also 4 operations addition, multiplication,... and these operation have all properties the same as $ \mathbb R$.

//cdn.artofproblemsolving.com/images/2593fa57e5f86a443c4e208aaa98236869fa9c70.png

a) Prove that in this larger system there is a number which is smaller than each positive integer and is larger than zero.

b) Prove that none of these numbers are root of a polynomial in $ \mathbb R[x]$.
2007 Iran MO (3rd Round)
A ring is the area between two circles with the same center, and width of a ring is the difference between the radii of two circles.

//cdn.artofproblemsolving.com/images/174a41c25fc95a577a44913940e6beca8f1bc5a7.png

a) Can we put uncountable disjoint rings of width 1(not necessarily same) in the space such that each two of them can not be separated.

//cdn.artofproblemsolving.com/images/e732be471dc7f7f921d9624932a363000d021601.png

b) What's the answer if 1 is replaced with 0?
2007 Iran MO (3rd Round)
In this question you must make all numbers of a clock, each with using 2, exactly 3 times and Mathematical symbols. You are not allowed to use English alphabets and words like $ \sin$ or $ \lim$ or $ a,b$ and no other digits.

//cdn.artofproblemsolving.com/images/adf787c1e725eae7b29edd128ec7405065322fc7.png
2006 Iran MO (3rd Round)
$n$ is a natural number. $d$ is the least natural number that for each $a$ that $gcd(a,n)=1$ we know $a^{d}\equiv1\pmod{n}$. Prove that there exist a natural number that $\mbox{ord}_{n}b=d$
2006 Iran MO (3rd Round)
$n$ is a natural number that $\frac{x^{n}+1}{x+1}$ is irreducible over $\mathbb Z_{2}[x]$. Consider a vector in $\mathbb Z_{2}^{n}$ that it has odd number of $1$'s (as entries) and at least one of its entries are $0$. Prove that these vector and its translations are a basis for $\mathbb Z_{2}^{n}$
2006 Iran MO (3rd Round)
$L$ is a fullrank lattice in $\mathbb R^{2}$ and $K$ is a sub-lattice of $L$, that $\frac{A(K)}{A(L)}=m$. If $m$ is the least number that for each $x\in L$, $mx$ is in $K$. Prove that there exists a basis $\{x_{1},x_{2}\}$ for $L$ that $\{x_{1},mx_{2}\}$ is a basis for $K$.
2006 Iran MO (3rd Round)
$a,b,c,t$ are antural numbers and $k=c^{t}$ and $n=a^{k}-b^{k}$.

a) Prove that if $k$ has at least $q$ different prime divisors, then $n$ has at least $qt$ different prime divisors.

b)Prove that $\varphi(n)$ id divisible by $2^{\frac{t}{2}}$
2006 Iran MO (3rd Round)
For each $n$, define $L(n)$ to be the number of natural numbers $1\leq a\leq n$ such that $n\mid a^{n}-1$. If $p_{1},p_{2},\ldots,p_{k}$ are the prime divisors of $n$, define $T(n)$ as $(p_{1}-1)(p_{2}-1)\cdots(p_{k}-1)$.

a) Prove that for each $n\in\mathbb N$ we have $n\mid L(n)T(n)$.

b) Prove that if $\gcd(n,T(n))=1$ then $\varphi(n) | L(n)T(n)$.
2006 Iran MO (3rd Round)
a) $P(x),R(x)$ are polynomials with rational coefficients and $P(x)$ is not the zero polynomial. Prove that there exist a non-zero polynomial $Q(x)\in\mathbb Q[x]$ that $$P(x)\mid Q(R(x)).$$b) $P,R$ are polynomial with integer coefficients and $P$ is monic. Prove that there exist a monic polynomial $Q(x)\in\mathbb Z[x]$ that $$P(x)\mid Q(R(x)).$$
2006 Iran MO (3rd Round)
For positive numbers $x_{1},x_{2},\dots,x_{s}$, we know that $\prod_{i=1}^{s}x_{k}=1$. Prove that for each $m\geq n$ $$\sum_{k=1}^{s}x_{k}^{m}\geq\sum_{k=1}^{s}x_{k}^{n}$$
2006 Iran MO (3rd Round)
Find all real polynomials that $$p(x+p(x))=p(x)+p(p(x))$$
2006 Iran MO (3rd Round)
Find all real $x,y,z$ that $$\left\{\begin{array}{c}x+y+zx=\frac12\\ \\ y+z+xy=\frac12\\ \\ z+x+yz=\frac12\end{array}\right.$$
2006 Iran MO (3rd Round)
$p(x)$ is a real polynomial that for each $x\geq 0$, $p(x)\geq 0$. Prove that there are real polynomials $A(x),B(x)$ that $p(x)=A(x)^{2}+xB(x)^{2}$
2006 Iran MO (3rd Round)
Find the biggest real number $ k$ such that for each right-angled triangle with sides $ a$, $ b$, $ c$, we have

$$ a^{3}+b^{3}+c^{3}\geq k\left(a+b+c\right)^{3}.$$
2006 Iran MO (3rd Round)
$P,Q,R$ are non-zero polynomials that for each $z\in\mathbb C$, $P(z)Q(\bar z)=R(z)$.

a) If $P,Q,R\in\mathbb R[x]$, prove that $Q$ is constant polynomial.

b) Is the above statement correct for $P,Q,R\in\mathbb C[x]$?
2006 Iran MO (3rd Round)
Suppose that $A\in\mathcal M_{n}(\mathbb R)$ with $\text{Rank}(A)=k$. Prove that $A$ is sum of $k$ matrices $X_{1},\dots,X_{k}$ with $\text{Rank}(X_{i})=1$.
2006 Iran MO (3rd Round)
$f: \mathbb R^{n}\longrightarrow\mathbb R^{m}$ is a non-zero linear map. Prove that there is a base $\{v_{1},\dots,v_{n}m\}$ for $\mathbb R^{n}$ that  the set $\{f(v_{1}),\dots,f(v_{n})\}$ is linearly independent, after ommitting Repetitive elements.
2006 Iran MO (3rd Round)
Suppose $(u,v)$ is an inner product on $\mathbb R^{n}$ and $f: \mathbb R^{n}\longrightarrow\mathbb R^{n}$ is an isometry, that $f(0)=0$.

1) Prove that for each $u,v$ we have $(u,v)=(f(u),f(v)$

2) Prove that $f$ is linear.
2006 Iran MO (3rd Round)
$f: \mathbb R^{n}\longrightarrow\mathbb R^{n}$ is a bijective map, that Image of every $n-1$-dimensional affine space is a $n-1$-dimensional affine space.

1) Prove that Image of every line is a line.

2) Prove that $f$ is an affine map. (i.e. $f=goh$ that $g$ is a translation and $h$ is a linear map.)
2006 Iran MO (3rd Round)
Prove that in triangle $ABC$, radical center of its excircles lies on line $GI$, which $G$ is Centroid of triangle $ABC$, and $I$ is the incenter.
2006 Iran MO (3rd Round)
$ABC$ is a triangle and $R,Q,P$ are midpoints of $AB,AC,BC$. Line $AP$ intersects $RQ$ in $E$ and circumcircle of $ABC$ in $F$. $T,S$ are on $RP,PQ$ such that $ES\perp PQ,ET\perp RP$. $F'$ is on circumcircle of $ABC$ that $FF'$ is diameter. The point of intersection of $AF'$ and $BC$ is $E'$. $S',T'$ are on $AB,AC$ that $E'S'\perp AB,E'T'\perp AC$. Prove that $TS$ and $T'S'$ are perpendicular.
2006 Iran MO (3rd Round)
In triangle $ABC$, if $L,M,N$ are midpoints of $AB,AC,BC$. And $H$ is orthogonal center of triangle $ABC$, then prove that $$LH^{2}+MH^{2}+NH^{2}\leq\frac14(AB^{2}+AC^{2}+BC^{2})$$
2006 Iran MO (3rd Round)
Circle $\Omega(O,R)$ and its chord $AB$ is given. Suppose $C$ is midpoint of arc $AB$. $X$ is an arbitrary point on the cirlce. Perpendicular from $B$ to $CX$ intersects circle again in $D$. Perpendicular from $C$ to $DX$ intersects circle again in $E$. We draw three lines $\ell_{1},\ell_{2},\ell_{3}$ from $A,B,E$ parralell to $OX,OD,OC$. Prove that these lines are concurrent and find locus of concurrncy point.
2006 Iran MO (3rd Round)
$M$ is midpoint of side $BC$ of triangle $ABC$, and $I$ is incenter of triangle $ABC$, and $T$ is midpoint of arc $BC$, that does not contain $A$. Prove that $$\cos B+\cos C=1\Longleftrightarrow MI=MT$$
2006 Iran MO (3rd Round)
Let $A$ be a family of subsets of $\{1,2,\ldots,n\}$ such that no member of $A$ is contained in another. Sperner’s Theorem states that $|A|\leq{n\choose{\lfloor\frac{n}{2}\rfloor}}$. Find all the families for which  the equality holds.
2006 Iran MO (3rd Round)
Let $B$ be a subset of $\mathbb{Z}_{3}^{n}$ with the property that for every two distinct members $(a_{1},\ldots,a_{n})$ and $(b_{1},\ldots,b_{n})$ of $B$ there exist $1\leq i\leq n$ such that $a_{i}\equiv{b_{i}+1}\pmod{3}$. Prove that $|B| \leq 2^{n}$.
2006 Iran MO (3rd Round)
Let $C$ be a (probably infinite) family of subsets of $\mathbb{N}$ such that for every chain $C_{1}\subset C_{2}\subset \ldots$ of members of $C$, there is a member of $C$ containing all of them. Show that there is a member of $C$ such that no other member of $C$ contains it!
2006 Iran MO (3rd Round)
Let $D$ be a family of $s$-element subsets of $\{1.\ldots,n\}$ such that every $k$ members of $D$ have non-empty intersection. Denote by $D(n,s,k)$ the maximum cardinality of such a family.

a)	Find $D(n,s,4)$.

b)	Find $D(n,s,3)$.
2006 Iran MO (3rd Round)
Let $E$ be a family of subsets of $\{1,2,\ldots,n\}$ with the property that for each $A\subset \{1,2,\ldots,n\}$ there exist $B\in F$ such that $\frac{n-d}2\leq |A \bigtriangleup B| \leq \frac{n+d}2$. (where $A \bigtriangleup B = (A\setminus B) \cup (B\setminus A)$ is the symmetric difference). Denote by $f(n,d)$ the minimum cardinality of such a family.

a)	Prove that if $n$ is even then $f(n,0)\leq n$.

b)	Prove that if $n-d$ is even then $f(n,d)\leq \lceil \frac n{d+1}\rceil$.

c)	Prove that if $n$ is even then $f(n,0) = n$
2006 Iran MO (3rd Round)
The National Foundation of Happiness (NFoH) wants to estimate the happiness of people of country. NFoH selected $n$ random persons, and on every morning asked from each of them whether she is happy or not. On any two distinct days, exactly half of the persons gave the same answer. Show that after $k$ days, there were at most $n-\frac{n}{k}$ persons whose “yes” answers equals their “no” answers.
2006 Iran MO (3rd Round)
A regular polyhedron is a polyhedron that is convex and all of its faces are regular polygons. We call a regular polhedron a "Choombam" iff none of its faces are triangles.

a) prove that each choombam can be inscribed in a sphere.

b) Prove that faces of each choombam are polygons of at most 3 kinds. (i.e. there is a set $\{m,n,q\}$ that each face of a choombam is $n$-gon or $m$-gon or $q$-gon.)

c) Prove that there is only one choombam that its faces are pentagon and hexagon. (Soccer ball)

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d) For $n>3$, a prism that its faces are 2 regular $n$-gons and $n$ squares, is a choombam. Prove that except these choombams there are finitely many choombams.
2006 Iran MO (3rd Round)
A liquid is moving in an infinite pipe. For each molecule if it is at point with coordinate $x$ then after $t$ seconds it will be at a point of $p(t,x)$. Prove that if $p(t,x)$ is a polynomial of $t,x$ then speed of all molecules are equal and constant.
2006 Iran MO (3rd Round)
For $A\subset\mathbb Z$ and $a,b\in\mathbb Z$. We define $aA+b: =\{ax+b|x\in A\}$. If $a\neq0$ then we calll $aA+b$ and $A$ to similar sets. In this question the Cantor set $C$ is the number of non-negative integers that in their base-3 representation there is no $1$ digit. You see $$C=(3C)\dot\cup(3C+2)\ \ \ \ \ \ (1)$$ (i.e. $C$ is partitioned to sets $3C$ and $3C+2$). We give another example $C=(3C)\dot\cup(9C+6)\dot\cup(3C+2)$.



A representation of $C$ is a partition of $C$ to some similiar sets. i.e. $$C=\bigcup_{i=1}^{n}C_{i}\ \ \ \ \ \ (2)$$ and $C_{i}=a_{i}C+b_{i}$ are similar to $C$.

We call a representation of $C$ a primitive representation iff union of some of $C_{i}$ is not a set similar and not equal to $C$.

Consider a primitive representation of Cantor set. Prove that

a) $a_{i}>1$.

b) $a_{i}$ are powers of 3.

c) $a_{i}>b_{i}$

d) (1) is the only primitive representation of $C$.
2006 Iran MO (3rd Round)
The image shown below is a cross with length 2. If length of a cross of length $k$ it is called a $k$-cross. (Each $k$-cross ahs $6k+1$ squares.)

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a) Prove that space can be tiled with $1$-crosses.

b) Prove that space can be tiled with $2$-crosses.

c) Prove that for $k\geq5$ space can not be tiled with $k$-crosses.
2006 Iran MO (3rd Round)
A calculating ruler is a ruler for doing algebric calculations. This ruler has three arms, two of them are sationary and one can move freely right and left. Each of arms is gradient. Gradation of each arm depends on the algebric operation ruler does. For eaxample the ruler below is designed for multiplying two numbers. Gradations are logarithmic.

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For working with ruler, (e.g for calculating $x.y$) we must move the middle arm that the arrow at the beginning of its gradation locate above the $x$ in the lower arm. We find $y$ in the middle arm, and we will read the number on the upper arm. The number written on the ruler is the answer.

1) Design a ruler for calculating $x^{y}$. Grade first arm ($x$) and ($y$) from 1 to 10.

2) Find all rulers that do the multiplication in the interval $[1,10]$.

3) Prove that there is not a ruler for calculating $x^{2}+xy+y^{2}$, that its first and second arm are grade from 0 to 10.
2006 Iran MO (3rd Round)
Assume that $C$ is a convex subset of $\mathbb R^{d}$. Suppose that $C_{1},C_{2},\dots,C_{n}$ are translations of $C$ that $C_{i}\cap C\neq\emptyset$ but $C_{i}\cap C_{j}=\emptyset$. Prove that $$n\leq 3^{d}-1$$ Prove that $3^{d}-1$ is the best bound.

P.S. In the exam problem was given for $n=3$.
2006 Iran MO (3rd Round)
We have finite number of distinct shapes in plane. A "convex Kearting" of these shapes is covering plane with convex sets, that each set consists exactly one of the shapes, and sets intersect at most in border.

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In which case Convex kearting is possible?

1) Finite distinct points

2) Finite distinct segments

3) Finite distinct circles
2006 Iran MO (3rd Round)
We mean a traingle in $\mathbb Q^{n}$, 3 points that are not collinear in $\mathbb Q^{n}$



a) Suppose that $ABC$ is triangle in $\mathbb Q^{n}$. Prove that there is a triangle $A'B'C'$ in $\mathbb Q^{5}$ that $\angle B'A'C'=\angle BAC$.

b) Find a natural $m$ that for each traingle that can be embedded in $\mathbb Q^{n}$ it can be embedded in $\mathbb Q^{m}$.

c) Find a triangle that can be embedded in $\mathbb Q^{n}$ and no triangle similar to it can be embedded in $\mathbb Q^{3}$.

d) Find a natural $m'$ that for each traingle that can be embedded in $\mathbb Q^{n}$ then there is a triangle similar to it, that can be embedded in $\mathbb Q^{m}$.

You must prove the problem for $m=9$ and $m'=6$ to get complete mark. (Better results leads to additional mark.)
2005 Iran MO (3rd Round)
Suppose $a,b,c\in \mathbb R^+$. Prove that :$$\left(\frac ab+\frac bc+\frac ca\right)^2\geq (a+b+c)\left(\frac1a+\frac1b+\frac1c\right)$$
2005 Iran MO (3rd Round)
Suppose $\{x_n\}$ is a decreasing sequence that $\displaystyle\lim_{n \rightarrow\infty}x_n=0$. Prove that $\sum(-1)^nx_n$ is convergent
2005 Iran MO (3rd Round)
Find all $\alpha>0$ and $\beta>0$ that for each $(x_1,\dots,x_n)$ and $(y_1,\dots,y_n)\in\mathbb {R^+}^n$ that:$$(\sum x_i^\alpha)(\sum y_i^\beta)\geq\sum x_iy_i$$
2005 Iran MO (3rd Round)
Suppose $P,Q\in \mathbb R[x]$ that $deg\ P=deg\ Q$ and $PQ'-QP'$ has no real root. Prove that for each $\lambda \in \mathbb R$ number of real roots of $P$ and $\lambda P+(1-\lambda)Q$ are equal.
2005 Iran MO (3rd Round)
Suppose $a,b,c \in \mathbb R^+$and $$\frac1{a^2+1}+\frac1{b^2+1}+\frac1{c^2+1}=2$$



Prove that $ab+ac+bc\leq \frac32$