Navvye/Olympiad-Problems-CSV huggingface dataset:
2010 Iran MO (3rd Round)
In a triangle $ABC$, $\angle C=45$. $AD$ is the altitude of the triangle. $X$ is on $AD$ such that $\angle XBC=90-\angle B$ ($X$ is in the triangle). $AD$ and $CX$ cut the circumcircle of $ABC$ in $M$ and $N$ respectively. if tangent to circumcircle of $ABC$ at $M$ cuts $AN$ at $P$, prove that $P$,$B$ and $O$ are collinear.(25 points)



the exam time was 4 hours and 30 minutes.
2010 Iran MO (3rd Round)
suppose that $\mathcal F\subseteq X^{(k)}$ and $|X|=n$. we know that for every three distinct elements of $\mathcal F$ like $A,B,C$, at most one of $A\cap B$,$B\cap C$ and $C\cap A$ is $\phi$. for $k\le \frac{n}{2}$ prove that:

a) $|\mathcal F|\le max(1,4-\frac{n}{k})\times \dbinom{n-1}{k-1}$.(15 points)

b) find all cases of equality in a) for $k\le \frac{n}{3}$.(5 points)
2010 Iran MO (3rd Round)
suppose that $\mathcal F\subseteq \bigcup_{j=k+1}^{n}X^{(j)}$ and $|X|=n$. we know that $\mathcal F$ is a sperner family and it's also $H_k$. prove that:

$\sum_{B\in \mathcal F}\frac{1}{\dbinom{n-1}{|B|-1}}\le 1$

(15 points)
2010 Iran MO (3rd Round)
suppose that $\mathcal F\subseteq p(X)$ and $|X|=n$. we know that for every $A_i,A_j\in \mathcal F$ that $A_i\supseteq A_j$ we have $3\le |A_i|-|A_j|$. prove that:

$|\mathcal F|\le \lfloor\frac{2^n}{3}+\frac{1}{2}\dbinom{n}{\lfloor\frac{n}{2}\rfloor}\rfloor$

(20 points)
2010 Iran MO (3rd Round)
suppose that $\mathcal F\subseteq X^{(K)}$ and $|X|=n$. we know that for every three distinct elements of $\mathcal F$ like $A,B$ and $C$ we have $A\cap B \not\subset C$.



a)(10 points) Prove that :

$$|\mathcal F|\le \dbinom{k}{\lfloor\frac{k}{2}\rfloor}+1$$



b)(15 points) if elements of $\mathcal F$ do not necessarily have $k$ elements, with the above conditions show that:

$$|\mathcal F|\le \dbinom{n}{\lceil\frac{n-2}{3}\rceil}+2$$
2010 Iran MO (3rd Round)
suppose that $\mathcal F\subseteq p(X)$ and $|X|=n$. prove that if $|\mathcal F|>\sum_{i=0}^{k-1}\dbinom{n}{i}$ then there exist $Y\subseteq X$ with $|Y|=k$ such that $p(Y)=\mathcal F\cap Y$ that $\mathcal F\cap Y=\{F\cap Y:F\in \mathcal F\}$(20 points)

you can see this problem also here:

COMBINATORIAL PROBLEMS AND EXERCISES-SECOND EDITION-by LASZLO LOVASZ-AMS CHELSEA PUBLISHING- chapter 13- problem 10(c)!!!
2010 Iran MO (3rd Round)
Suppose that $X$ is a set with $n$ elements and $\mathcal F\subseteq X^{(k)}$ and $X_1,X_2,...,X_s$ is a partition of $X$. We know that for every $A,B\in \mathcal F$ and every $1\le j\le s$, $E=B\cap (\bigcup_{i=1}^{j}X_i)\neq A\cap (\bigcup_{i=1}^{j} X_i)=F$ shows that none of $E,F$ contains the other one. Prove that

$$|\mathcal F|\le \max_{\sum\limits_{i=1}^{S}w_i=k}\prod_{j=1}^{s}\binom{|X_j|}{w_j}$$

(15 points)



Exam time was 5 hours and 20 minutes.
2010 Iran MO (3rd Round)
two variable ploynomial



$P(x,y)$ is a two variable polynomial with real coefficients. degree of a monomial means sum of the powers of $x$ and $y$ in it. we denote by $Q(x,y)$ sum of monomials with the most degree in $P(x,y)$.

(for example if $P(x,y)=3x^4y-2x^2y^3+5xy^2+x-5$ then $Q(x,y)=3x^4y-2x^2y^3$.)

suppose that there are real numbers $x_1$,$y_1$,$x_2$ and $y_2$ such that

$Q(x_1,y_1)>0$ , $Q(x_2,y_2)<0$

prove that the set $\{(x,y)|P(x,y)=0\}$ is not bounded.

(we call a set $S$ of plane bounded if there exist positive number $M$ such that the distance of elements of $S$ from the origin is less than $M$.)



time allowed for this question was 1 hour.
2010 Iran MO (3rd Round)
rolling cube



$a$,$b$ and $c$ are natural numbers. we have a $(2a+1)\times (2b+1)\times (2c+1)$ cube. this cube is on an infinite plane with unit squares. you call roll the cube to every side you want. faces of the cube are divided to unit squares and the square in the middle of each face is coloured (it means that if this square goes on a square of the plane, then that square will be coloured.)

prove that if any two of lengths of sides of the cube are relatively prime, then we can colour every square in plane.



time allowed for this question was 1 hour.
2010 Iran MO (3rd Round)
points in plane



set $A$ containing $n$ points in plane is given. a $copy$ of $A$ is a set of points that is made by using transformation, rotation, homogeneity or their combination on elements of $A$. we want to put $n$ $copies$ of $A$ in plane, such that every two copies have exactly one point in common and every three of them have no common elements.

a) prove that if no $4$ points of $A$ make a parallelogram, you can do this only using transformation. ($A$ doesn't have a parallelogram with angle $0$ and a parallelogram that it's two non-adjacent vertices are one!)

b) prove that you can always do this by using a combination of all these things.



time allowed for this question was 1 hour and 30 minutes
2010 Iran MO (3rd Round)
carpeting



suppose that $S$ is a figure in the plane such that it's border doesn't contain any lattice points. suppose that $x,y$ are two lattice points with the distance $1$ (we call a point lattice point if it's coordinates are integers). suppose that we can cover the plane with copies of $S$ such that $x,y$ always go on lattice points ( you can rotate or reverse copies of $S$). prove that the area of $S$ is equal to lattice points inside it.



time allowed for this question was 1 hour.
2010 Iran MO (3rd Round)
interesting sequence



$n$ is a natural number and $x_1,x_2,...$ is a sequence of numbers $1$ and $-1$ with these properties:



it is periodic and its least period number is $2^n-1$. (it means that for every natural number $j$ we have $x_{j+2^n-1}=x_j$ and $2^n-1$ is the least number with this property.)



There exist distinct integers $0\le t_1<t_2<...<t_k<n$ such that for every natural number $j$ we have

$$x_{j+n}=x_{j+t_1}\times x_{j+t_2}\times ... \times x_{j+t_k}$$



Prove that for every natural number $s$ that $s<2^n-1$ we have

$$\sum_{i=1}^{2^n-1}x_ix_{i+s}=-1$$



Time allowed for this question was 1 hours and 15 minutes.
2010 Iran MO (3rd Round)
polyhedral



we call a $12$-gon in plane good whenever:

first, it should be regular, second, it's inner plane must be filled!!, third, it's center must be the origin of the coordinates, forth, it's vertices must have points $(0,1)$,$(1,0)$,$(-1,0)$ and $(0,-1)$.

find the faces of the massivest polyhedral that it's image on every three plane $xy$,$yz$ and $zx$ is a good $12$-gon.

(it's obvios that centers of these three $12$-gons are the origin of coordinates for three dimensions.)



time allowed for this question is 1 hour.
2010 Iran MO (3rd Round)
interesting function



$S$ is a set with $n$ elements and $P(S)$ is the set of all subsets of $S$ and

$f : P(S) \rightarrow \mathbb N$

is a function with these properties:

for every subset $A$ of $S$ we have $f(A)=f(S-A)$.

for every two subsets of $S$ like $A$ and $B$ we have

$max(f(A),f(B))\ge f(A\cup B)$

prove that number of natural numbers like $x$ such that there exists $A\subseteq  S$ and $f(A)=x$ is less than $n$.



time allowed for this question was 1 hours and 30 minutes.
2010 Iran MO (3rd Round)
numbers $n^2+1$



Prove that there are infinitely many natural numbers of the form $n^2+1$ such that they don't have any divisor of the form $k^2+1$ except $1$ and themselves.



time allowed for this question was 45 minutes.
2010 Iran MO (3rd Round)
Prove that the group of orientation-preserving symmetries of the cube is isomorphic to $S_4$ (the group of permutations of $\{1,2,3,4\}$).(20 points)
2010 Iran MO (3rd Round)
prove the third sylow theorem: suppose that $G$ is a group and $|G|=p^em$ which $p$ is a prime number and $(p,m)=1$. suppose that $a$ is the number of $p$-sylow subgroups of $G$ ($H<G$ that $|H|=p^e$). prove that $a|m$ and $p|a-1$.(Hint: you can use this: every two $p$-sylow subgroups are conjugate.)(20 points)
2010 Iran MO (3rd Round)
suppose that $G<S_n$ is a subgroup of permutations of $\{1,...,n\}$ with this property that for every $e\neq g\in G$ there exist exactly one $k\in \{1,...,n\}$ such that $g.k=k$. prove that there exist one $k\in \{1,...,n\}$ such that for every $g\in G$ we have $g.k=k$.(20 points)
2010 Iran MO (3rd Round)
a) prove that every discrete subgroup of $(\mathbb R^2,+)$ is in one of these forms:

i-$\{0\}$.

ii-$\{mv|m\in \mathbb Z\}$ for a vector $v$ in $\mathbb R^2$.

iii-$\{mv+nw|m,n\in \mathbb Z\}$ for tho linearly independent vectors $v$ and $w$ in $\mathbb R^2$.(lattice $L$)

b) prove that every finite group of symmetries that fixes the origin and the lattice $L$ is in one of these forms: $\mathcal C_i$ or $\mathcal D_i$ that $i=1,2,3,4,6$ ($\mathcal C_i$ is the cyclic group of order $i$ and $\mathcal D_i$ is the dyhedral group of order $i$).(20 points)
2010 Iran MO (3rd Round)
suppose that $p$ is a prime number. find that smallest $n$ such that there exists a non-abelian group $G$ with $|G|=p^n$.



SL is an acronym for Special Lesson. this year our special lesson was Groups and Symmetries.



the exam time was 5 hours.
2009 Iran MO (3rd Round)
Suppose $n>2$ and let $A_1,\dots,A_n$ be points on the plane such that no three are collinear.

(a) Suppose $M_1,\dots,M_n$ be points on segments $A_1A_2,A_2A_3,\dots ,A_nA_1$ respectively. Prove that if $B_1,\dots,B_n$ are points in triangles $M_2A_2M_1,M_3A_3M_2,\dots ,M_1A_1M_n$ respectively then $$|B_1B_2|+|B_2B_3|+\dots+|B_nB_1| \leq |A_1A_2|+|A_2A_3|+\dots+|A_nA_1|$$

Where $|XY|$ means the length of line segment between $X$ and $Y$.



(b) If $X$, $Y$ and $Z$ are three points on the plane then by $H_{XYZ}$ we mean the half-plane that it's boundary is the exterior angle bisector of angle $\hat{XYZ}$ and doesn't contain $X$ and $Z$ ,having $Y$ crossed out.

Prove that if $C_1,\dots ,C_n$ are points in ${H_{A_nA_1A_2},H_{A_1A_2A_3},\dots,H_{A_{n-1}A_nA_1}}$ then $$|A_1A_2|+|A_2A_3|+\dots +|A_nA_1| \leq |C_1C_2|+|C_2C_3|+\dots+|C_nC_1|$$



Time allowed for this problem was 2 hours.
2009 Iran MO (3rd Round)
Permutation $\pi$ of $\{1,\dots,n\}$ is called stable if the set $\{\pi (k)-k|k=1,\dots,n\}$ is consisted of exactly two different elements.

Prove that the number of stable permutation of $\{1,\dots,n\}$ equals to $\sigma (n)-\tau (n)$ in which $\sigma (n)$ is the sum of positive divisors of $n$ and $\tau (n)$ is the number of positive divisors of $n$.



Time allowed for this problem was 75 minutes.
2009 Iran MO (3rd Round)
An arbitary triangle is partitioned to some triangles homothetic with itself. The ratio of homothety of the triangles can be positive or negative.

Prove that sum of all homothety ratios equals to $1$.



Time allowed for this problem was 45 minutes.
2009 Iran MO (3rd Round)
Does there exists two functions $f,g :\mathbb{R}\rightarrow \mathbb{R}$ such that:

$\forall x\not =y : |f(x)-f(y)|+|g(x)-g(y)|>1$



Time allowed for this problem was 75 minutes.
2009 Iran MO (3rd Round)
A ball is placed on a plane and a point on the ball is marked.

Our goal is to roll the ball on a polygon in the plane in a way that it comes back to where it started and the marked point comes to the top of it. Note that We are not allowed to rotate without moving, but only rolling.

Prove that it is possible.



Time allowed for this problem was 90 minutes.
2009 Iran MO (3rd Round)
Let $z$ be a complex non-zero number such that $Re(z),Im(z)\in \mathbb{Z}$.

Prove that $z$ is uniquely representable as $a_0+a_1(1+i)+a_2(1+i)^2+\dots+a_n(1+i)^n$ where $n\geq 0$ and $a_j \in \{0,1\}$ and $a_n=1$.



Time allowed for this problem was 1 hour.
2009 Iran MO (3rd Round)
A sphere is inscribed in polyhedral $P$. The faces of $P$ are coloured with black and white in a way that no two black faces share an edge.

Prove that the sum of surface of black faces is less than or equal to the sum of the surface of the white faces.



Time allowed for this problem was 1 hour.
2009 Iran MO (3rd Round)
Sone of vertices of the infinite grid $\mathbb{Z}^{2}$ are missing. Let's take the remainder as a graph. Connect two edges of the graph if they are the same in one component and their other components have a difference equal to one. Call every connected component of this graph a branch.

Suppose that for every natural $n$ the number of missing vertices in the $(2n+1)\times(2n+1)$ square centered by the origin is less than $\frac{n}{2}$.

Prove that among the branches of the graph, exactly one has an infinite number of vertices.



Time allowed for this problem was 90 minutes.
2009 Iran MO (3rd Round)
1-Let $ \triangle ABC$ be a triangle and $ (O)$ its circumcircle. $ D$ is the midpoint of arc $ BC$ which doesn't contain $ A$. We draw a circle $ W$ that is tangent internally to $ (O)$ at $ D$ and tangent to $ BC$.We draw the tangent $ AT$ from $ A$ to circle $ W$.$ P$ is taken on $ AB$ such that $ AP = AT$.$ P$ and $ T$ are at the same side wrt $ A$.PROVE $ \angle APD = 90^\circ$.
2009 Iran MO (3rd Round)
2-There is given a trapezoid $ ABCD$.We have the following properties:$ AD||BC,DA = DB = DC,\angle BCD = 72^\circ$. A point $ K$ is taken on $ BD$ such that $ AD = AK,K \neq D$.Let $ M$ be the midpoint of $ CD$.$ AM$ intersects $ BD$ at $ N$.PROVE $ BK = ND$.
2009 Iran MO (3rd Round)
3-There is given a trapezoid $ ABCD$  in the plane with $ BC||AD$.We know that the angle bisectors of the angles of the trapezoid are concurrent at $ O$.Let $ T$ be the intersection of the diagonals $ AC,BD$.Let $ Q$ be on $ CD$ such that $ \angle OQD = 90^\circ$.Prove that if the circumcircle of the triangle $ OTQ$ intersects $ CD$ again at $ P$ then $ TP||AD$.
2009 Iran MO (3rd Round)
4-Point $ P$ is taken on the segment $ BC$ of the scalene triangle $ ABC$ such that $ AP \neq AB,AP \neq AC$.$ l_1,l_2$ are the incenters of triangles $ ABP,ACP$ respectively. circles $ W_1,W_2$ are drawn centered at $ l_1,l_2$ and with radius equal to $ l_1P,l_2P$,respectively. $ W_1,W_2$ intersects at $ P$ and $ Q$. $ W_1$ intersects $ AB$ and $ BC$ at $ Y_1( \mbox{the intersection closer to B})$ and $ X_1$,respectively. $ W_2$ intersects $ AC$ and $ BC$ at $ Y_2(\mbox{the intersection closer to C})$ and $ X_2$,respectively.PROVE THE CONCURRENCY OF $ PQ,X_1Y_1,X_2Y_2$.
2009 Iran MO (3rd Round)
5-Two circles $ S_1$ and $ S_2$ with equal radius and intersecting at two points are given in the plane.A line $ l$ intersects $ S_1$ at $ B,D$ and $ S_2$ at $ A,C$(the order of the points on the line are as follows:$ A,B,C,D$).Two circles $ W_1$ and $ W_2$ are drawn such that both of them are tangent externally at $ S_1$ and internally at $ S_2$ and also tangent to $ l$ at both sides.Suppose $ W_1$ and $ W_2$ are tangent.Then PROVE $ AB = CD$.
2008 Iran MO (3rd Round)
Suppose that $ f(x)\in\mathbb Z[x]$ be an irreducible polynomial. It is known that $ f$ has a root of norm larger than $ \frac32$. Prove that if $ \alpha$ is a root of $ f$ then $ f(\alpha^3+1)\neq0$.
2008 Iran MO (3rd Round)
Find the smallest real $ K$ such that for each $ x,y,z\in\mathbb R^{ + }$:

$$ x\sqrt y + y\sqrt z + z\sqrt x\leq K\sqrt {(x + y)(y + z)(z + x)}

$$
2008 Iran MO (3rd Round)
Let $ (b_0,b_1,b_2,b_3)$ be a permutation of the set $ \{54,72,36,108\}$. Prove that $ x^5+b_3x^3+b_2x^2+b_1x+b_0$ is irreducible in $ \mathbb Z[x]$.
2008 Iran MO (3rd Round)
Let $ x,y,z\in\mathbb R^{+}$ and $ x+y+z=3$. Prove that:

$$ \frac{x^3}{y^3+8}+\frac{y^3}{z^3+8}+\frac{z^3}{x^3+8}\geq\frac19+\frac2{27}(xy+xz+yz)$$
2008 Iran MO (3rd Round)
Prove that the following polynomial is irreducible in $ \mathbb Z[x,y]$:

$$ x^{200}y^5+x^{51}y^{100}+x^{106}-4x^{100}y^5+x^{100}-2y^{100}-2x^6+4y^5-2$$
2008 Iran MO (3rd Round)
Let $ k>1$ be an integer. Prove that there exists infinitely many natural numbers such as $ n$ such that: $$ n|1^n+2^n+\dots+k^n$$
2008 Iran MO (3rd Round)
Prove that there exists infinitely many primes $ p$ such that: $$ 13|p^3+1$$
2008 Iran MO (3rd Round)
Let $ P$ be a regular polygon. A regular sub-polygon of $ P$ is a subset of vertices of $ P$ with at least two vertices such that divides the circumcircle to equal arcs. Prove that there is a subset of vertices of $ P$ such that its intersection with each regular sub-polygon has even number of vertices.
2008 Iran MO (3rd Round)
Let $ u$ be an odd number. Prove that $ \frac{3^{3u}-1}{3^u-1}$ can be written as sum of two squares.
2008 Iran MO (3rd Round)
Find all polynomials $ f\in\mathbb Z[x]$ such that for each $ a,b,x\in\mathbb N$

$$ a+b+c|f(a)+f(b)+f(c)$$
2008 Iran MO (3rd Round)
Prove that the number of pairs $ \left(\alpha,S\right)$ of a permutation $ \alpha$ of $ \{1,2,\dots,n\}$ and a subset $ S$ of $ \{1,2,\dots,n\}$ such that

$$ \forall x\in S: \alpha(x)\not\in S$$

is equal to $ n!F_{n + 1}$ in which $ F_n$ is the Fibonacci sequence such that $ F_1 = F_2 = 1$
2008 Iran MO (3rd Round)
Prove that the number permutations $ \alpha$ of $ \{1,2,\dots,n\}$ s.t. there does not exist $ i<j<n$ s.t. $ \alpha(i)<\alpha(j+1)<\alpha(j)$ is equal to the number of partitions of that set.
2008 Iran MO (3rd Round)
Prove that for each $ n$:

$$ \sum_{k=1}^n\binom{n+k-1}{2k-1}=F_{2n}$$
2008 Iran MO (3rd Round)
Let $ S$ be a sequence that:

$$ \left\{

\begin{array}{cc}

S_0=0\hfill\\
S_1=1\hfill\\
S_n=S_{n-1}+S_{n-2}+F_n& (n>1)

\end{array}

\right.$$

such that $ F_n$ is Fibonacci sequence such that $ F_1=F_2=1$. Find $ S_n$ in terms of Fibonacci numbers.
2008 Iran MO (3rd Round)
$ n$ people decide to play a game. There are $ n-1$ ropes and each of its two ends are in hand of one of the players, in such a way that ropes and players form a tree. (Each person can hold more than rope end.)



At each step a player gives one of the rope ends he is holding to another player. The goal is to make a path of length $ n-1$ at the end.



But the game regulations change before game starts. Everybody has to give one of his rope ends only two one of his neighbors. Let $ a$ and $ b$ be minimum steps for reaching to goal in these two games. Prove that $ a=b$ if and only if by removing all players with one rope end (leaves of the tree) the remaining people are on a path. (the remaining graph is a path.)

//cdn.artofproblemsolving.com/images/fef850b1badbfda0191bb614ddd0fa617fd09cc8.png
2008 Iran MO (3rd Round)
Prove that for $ n > 0$ and $ a\neq0$ the polynomial $ p(z) = az^{2n + 1} + bz^{2n} + \bar bz + \bar a$ has a root on unit circle
2008 Iran MO (3rd Round)
Let $ g,f: \mathbb C\longrightarrow\mathbb C$ be two continuous functions such that for each $ z\neq 0$, $ g(z)=f(\frac1z)$. Prove that there is a $ z\in\mathbb C$ such that $ f(\frac1z)=f(-\bar z)$
2008 Iran MO (3rd Round)
For each $ c\in\mathbb C$, let $ f_c(z,0)=z$, and $ f_c(z,n)=f_c(z,n-1)^2+c$ for $ n\geq1$.

a) Prove that if $ |c|\leq\frac14$ then there is a neighborhood $ U$ of origin such that for each $ z\in U$ the sequence $ f_c(z,n),n\in\mathbb N$ is bounded.

b) Prove that if $ c>\frac14$ is a real number there is a neighborhood $ U$ of origin such that for each $ z\in U$  the sequence $ f_c(z,n),n\in\mathbb N$ is unbounded.
2008 Iran MO (3rd Round)
Let $ ABC$ be a triangle with $ BC > AC > AB$. Let $ A',B',C'$ be feet of perpendiculars from $ A,B,C$ to $ BC,AC,AB$, such that $ AA' = BB' = CC' = x$. Prove that:

a) If $ ABC\sim A'B'C'$ then $ x = 2r$

b) Prove that if $ A',B'$ and $ C'$ are collinear, then $ x = R + d$ or $ x = R - d$.



(In this problem $ R$ is the radius of circumcircle, $ r$ is radius of incircle and $ d = OI$)
2008 Iran MO (3rd Round)
Let $ l_a,l_b,l_c$ be three parallel lines passing through $ A,B,C$ respectively. Let $ l_a'$ be reflection of $ l_a$ into $ BC$. $ l_b'$ and $ l_c'$ are defined similarly. Prove that $ l_a',l_b',l_c'$ are concurrent if and only if $ l_a$ is parallel to Euler line of triangle $ ABC$.
2008 Iran MO (3rd Round)
Let $ ABCD$ be a quadrilateral, and $ E$ be intersection points of  $ AB,CD$ and $ AD,BC$ respectively. External bisectors of $ DAB$ and $ DCB$ intersect at $ P$, external bisectors of $ ABC$ and $ ADC$ intersect at $ Q$ and external bisectors of $ AED$ and $ AFB$ intersect at $ R$. Prove that $ P,Q,R$ are collinear.
2008 Iran MO (3rd Round)
Let $ ABC$ be an isosceles triangle with $ AB=AC$, and $ D$ be midpoint of $ BC$, and $ E$ be foot of altitude from $ C$. Let $ H$ be orthocenter of $ ABC$ and $ N$ be midpoint of $ CE$. $ AN$ intersects with circumcircle of triangle $ ABC$ at $ K$. The tangent from $ C$ to circumcircle of $ ABC$ intersects with $ AD$ at $ F$.

Suppose that radical axis of circumcircles of $ CHA$ and $ CKF$ is $ BC$. Find $ \angle BAC$.
2008 Iran MO (3rd Round)
Let $ D,E,F$ be tangency point of incircle of triangle $ ABC$ with sides $ BC,AC,AB$. $ DE$ and $ DF$ intersect the line from $ A$ parallel to $ BC$ at $ K$ and $ L$. Prove that the Euler line of triangle $ DKL$ passes through Feuerbach point of triangle $ ABC$.
2008 Iran MO (3rd Round)
Police want to arrest on the famous criminals of the country whose name is Kaiser. Kaiser is in one of the streets of a square shaped city with $ n$ vertical streets and $ n$ horizontal streets. In the following cases how many police officers are needed to arrest Kaiser?

//cdn.artofproblemsolving.com/images/11140c24e660a067a6665bc14dc7e6f485aeadd6.gif //cdn.artofproblemsolving.com/images/c0f5250d7d372315b41b50e2b31ad1fb5e43702a.gif

a) Each police officer has the same speed as Kaiser and every police officer knows the location of Kaiser anytime.

b) Kaiser has an infinite speed (finite but with no bound) and police officers can only know where he is only when one of them see Kaiser.

Everybody in this problem (including police officers and Kaiser) move continuously and can stop or change his path.
2008 Iran MO (3rd Round)
Consider six arbitrary points in space. Every two points are joined by a segment. Prove that there are two triangles that can not be separated.

//cdn.artofproblemsolving.com/images/09f7a08b3e04245f863cec52e79138c8b04f31b8.png
2008 Iran MO (3rd Round)
a) Prove that there are two polynomials in $ \mathbb Z[x]$ with at least one coefficient larger than 1387 such that coefficients of their product is in the set $ \{-1,0,1\}$.

b) Does there exist a multiple of $ x^2-3x+1$ such that all of its coefficient are in the set $ \{-1,0,1\}$
2008 Iran MO (3rd Round)
=A subset $ S$ of $ \mathbb R^2$ is called an algebraic set if and only if there is a polynomial $ p(x,y)\in\mathbb R[x,y]$ such that

$$ S = \{(x,y)\in\mathbb R^2|p(x,y) = 0\}

$$

Are the following subsets of plane an algebraic sets?

1. A square

//cdn.artofproblemsolving.com/images/bf0d3ddf7d1ca28f9a15e39f2b88f95811c1722c.png

2. A closed half-circle

//cdn.artofproblemsolving.com/images/8efeb14a12e98b8e7945d3ae483a51974b8ae857.png
2008 Iran MO (3rd Round)
a) Suppose that $ RBR'B'$ is a convex quadrilateral such that vertices $ R$ and $ R'$ have red color and vertices $ B$ and $ B'$ have blue color. We put $ k$ arbitrary points of colors blue and red in the quadrilateral such that no four of these $ k+4$ point (except probably $ RBR'B'$) lie one a circle. Prove that exactly one of the following cases occur?

1. There is a path from $ R$ to $ R'$ such that distance of every point on this path from one of red points is less than its distance from all blue points.

2. There is a path from $ B$ to $ B'$ such that distance of every point on this path from one of blue points is less than its distance from all red points.

We call these two paths the blue path and the red path respectively.



Let $ n$ be a natural number. Two people play the following game. At each step one player puts a point in quadrilateral satisfying the above conditions. First player only puts red point and second player only puts blue points. Game finishes when every player has put $ n$ points on the plane. First player's goal is to make a red path from $ R$ to $ R'$ and the second player's goal is to make a blue path from $ B$ to $ B'$.

b) Prove that if $ RBR'B'$ is rectangle then for each $ n$ the second player wins.

c) Try to specify the winner for other quadrilaterals.
2008 Iran MO (3rd Round)
There are five research labs on Mars. Is it always possible to divide Mars to five connected congruent regions such that each region contains exactly on research lab.

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