Navvye/Olympiad-Problems-CSV huggingface dataset:
2012 Iran Team Selection Test
Suppose $p$ is an odd prime number. We call the polynomial $f(x)=\sum_{j=0}^n a_jx^j$ with integer coefficients $i$-remainder if $ \sum_{p-1|j,j>0}a_{j}\equiv i\pmod{p}$. Prove that the set $\{f(0),f(1),...,f(p-1)\}$ is a complete residue system modulo $p$ if and only if polynomials $f(x), (f(x))^2,...,(f(x))^{p-2}$ are $0$-remainder and the polynomial $(f(x))^{p-1}$ is $1$-remainder.



Proposed by Yahya Motevassel
2012 Iran Team Selection Test
Let $n$ be a natural number. Suppose $A$ and $B$ are two sets, each containing $n$ points in the plane, such that no three points of a set are collinear. Let $T(A)$ be the number of broken lines, each containing $n-1$ segments, and such that it doesn't intersect itself and its vertices are points of $A$. Define $T(B)$ similarly. If the points of $B$ are vertices of a convex $n$-gon (are in convex position), but the points of $A$ are not, prove that $T(B)<T(A)$.



Proposed by Ali Khezeli
2012 Iran Team Selection Test
Let $O$ be the circumcenter of the acute triangle $ABC$. Suppose points $A',B'$ and $C'$ are on sides $BC,CA$ and $AB$ such that circumcircles of triangles $AB'C',BC'A'$ and $CA'B'$ pass through $O$. Let $\ell_a$ be the radical axis of the circle with center $B'$ and radius $B'C$ and circle with center $C'$ and radius $C'B$. Define $\ell_b$ and $\ell_c$ similarly. Prove that lines $\ell_a,\ell_b$ and $\ell_c$ form a triangle such that it's orthocenter coincides with orthocenter of triangle $ABC$.



Proposed by Mehdi E'tesami Fard
2011 Iran Team Selection Test
In acute triangle $ABC$ angle $B$ is greater than$C$. Let $M$ is midpoint of $BC$. $D$ and $E$ are the feet of the altitude from $C$ and $B$ respectively. $K$ and $L$ are midpoint of $ME$ and $MD$ respectively. If $KL$ intersect the line through $A$ parallel to $BC$ in $T$, prove that $TA=TM$.
2011 Iran Team Selection Test
Find all natural numbers $n$ greater than $2$ such that there exist $n$ natural numbers $a_{1},a_{2},\ldots,a_{n}$ such that they are not all equal, and the sequence $a_{1}a_{2},a_{2}a_{3},\ldots,a_{n}a_{1}$ forms an arithmetic progression with nonzero common difference.
2011 Iran Team Selection Test
There are $n$ points on a circle ($n>1$). Define an "interval" as an arc of a circle such that it's start and finish are from those points. Consider a family of intervals $F$ such that for every element of $F$ like $A$ there is almost one other element of $F$ like $B$ such that $A \subseteq B$ (in this case we call $A$ is sub-interval of $B$). We call an interval maximal if it is not a sub-interval of any other interval. If $m$ is the number of maximal elements of $F$ and $a$ is  number of non-maximal elements of $F,$ prove that $n\geq m+\frac a2.$
2011 Iran Team Selection Test
Define a finite set $A$ to be 'good' if it satisfies the following conditions:



(a) For every three disjoint element of $A,$ like $a,b,c$ we have $\gcd(a,b,c)=1;$



(b) For every two distinct $b,c\in A,$ there exists an $a\in A,$ distinct from $b,c$ such that $bc$ is divisible by $a.$





Find all good sets.
2011 Iran Team Selection Test
Find all surjective functions $f: \mathbb R \to \mathbb R$ such that for every $x,y\in \mathbb R,$ we have

$$f(x+f(x)+2f(y))=f(2x)+f(2y).$$
2011 Iran Team Selection Test
The circle $\omega$ with center $O$ has given. From an arbitrary point $T$ outside of $\omega$ draw tangents $TB$ and $TC$ to it. $K$ and $H$ are on $TB$ and $TC$ respectively.



a) $B'$ and $C'$ are the second intersection point of $OB$ and $OC$ with $\omega$ respectively. $K'$ and $H'$ are on angle bisectors of $\angle BCO$ and $\angle CBO$ respectively such that $KK' \bot BC$ and $HH'\bot BC$. Prove that $K,H',B'$ are collinear if and only if $H,K',C'$ are collinear.



b) Consider there exist two circle in $TBC$ such that they are tangent two each other at $J$ and both of them are tangent to $\omega$.and one of them is tangent to $TB$ at $K$ and other one is tangent to $TC$ at $H$. Prove that two quadrilateral $BKJI$ and $CHJI$ are cyclic ($I$ is incenter of triangle  $OBC$).
2011 Iran Team Selection Test
Find the locus of points $P$ in an equilateral triangle $ABC$ for which the square root of the distance of $P$ to one of the sides is equal to the sum of the square root of the distance of $P$ to the two other sides.
2011 Iran Team Selection Test
Let $p$ be a prime and $k$ a positive integer such that $k \le p$. We know that $f(x)$ is a polynomial in $\mathbb Z[x]$ such that for all $x \in \mathbb{Z}$ we have $p^k | f(x)$.



(a) Prove that there exist polynomials $A_0(x),\ldots,A_k(x)$ all in $\mathbb Z[x]$ such that

$$ f(x)=\sum_{i=0}^{k} (x^p-x)^ip^{k-i}A_i(x),$$



(b) Find a counter example for each prime $p$ and each $k > p$.
2011 Iran Team Selection Test
We have $n$ points in the plane such that they are not all collinear. We call a line $\ell$ a 'good' line if we can divide those $n$ points in two sets $A,B$ such that the sum of the distances of all points in $A$ to $\ell$ is equal to the sum of the distances of all points in $B$ to $\ell$. Prove that there exist infinitely many points in the plane such that for each of them we have at least $n+1$ good lines passing through them.
2011 Iran Team Selection Test
Find the least value of $k$ such that for all $a,b,c,d \in \mathbb{R}$ the inequality

$$ \begin{array}  c \sqrt{(a^2+1)(b^2+1)(c^2+1)} +\sqrt{(b^2+1)(c^2+1)(d^2+1)} +\sqrt{(c^2+1)(d^2+1)(a^2+1)} +\sqrt{(d^2+1)(a^2+1)(b^2+1)} \\  \ \\ \ge 2( ab+bc+cd+da+ac+bd)-k  \end{array}$$

holds.
2011 Iran Team Selection Test
Let $ABC$ be a triangle and $A',B',C'$ be the midpoints of $BC,CA,AB$ respectively. Let $P$ and $P'$ be points in plane such that $PA=P'A',PB=P'B',PC=P'C'$. Prove that all $PP'$ pass through a fixed point.
2011 Iran Team Selection Test
Suppose that $f : \mathbb{N} \rightarrow \mathbb{N}$ is a function for which the expression $af(a)+bf(b)+2ab$ for all $a,b \in \mathbb{N}$ is always a perfect square. Prove that $f(a)=a$ for all $a \in \mathbb{N}$.
2010 Iran Team Selection Test
Let $f:\mathbb N\rightarrow\mathbb N$ be a non-decreasing function and let $n$ be an arbitrary natural number. Suppose that there are prime numbers $p_1,p_2,\dots,p_n$ and natural numbers $s_1,s_2,\dots,s_n$ such that for each $1\leq i\leq n$ the set $\{f(p_ir+s_i)|r=1,2,\dots\}$ is an infinite arithmetic progression. Prove that there is a natural number $a$ such that

$$f(a+1), f(a+2), \dots, f(a+n)$$

form an arithmetic progression.
2010 Iran Team Selection Test
Find all non-decreasing functions $f:\mathbb R^+\cup\{0\}\rightarrow\mathbb R^+\cup\{0\}$ such that for each $x,y\in \mathbb R^+\cup\{0\}$

$$f\left(\frac{x+f(x)}2+y\right)=2x-f(x)+f(f(y)).$$
2010 Iran Team Selection Test
Find all two-variable polynomials $p(x,y)$ such that for each $a,b,c\in\mathbb R$:

$$p(ab,c^2+1)+p(bc,a^2+1)+p(ca,b^2+1)=0$$
2010 Iran Team Selection Test
$S,T$ are two trees without vertices of degree 2. To each edge is associated a positive number which is called length of this edge. Distance between two arbitrary vertices $v,w$ in this graph is defined by sum of length of all edges in the path between $v$ and $w$. Let $f$ be a bijective function from leaves of $S$ to leaves of $T$, such that for each two leaves $u,v$ of $S$, distance of $u,v$ in $S$ is equal to distance of $f(u), f(v)$ in $T$. Prove that there is a bijective function $g$ from vertices of $S$ to vertices of $T$ such that for each two vertices $u,v$ of $S$, distance of $u,v$ in $S$ is equal to distance of $g(u)$ and $g(v)$ in $T$.
2010 Iran Team Selection Test
Circles $W_1,W_2$ intersect at $P,K$. $XY$ is common tangent of two circles which is nearer to $P$ and $X$ is on $W_1$ and $Y$ is on $W_2$. $XP$ intersects $W_2$ for the second time in $C$ and $YP$ intersects $W_1$ in $B$. Let $A$ be intersection point of $BX$ and $CY$. Prove that if $Q$ is the second intersection point of circumcircles of $ABC$ and $AXY$

$$\angle QXA=\angle QKP$$
2010 Iran Team Selection Test
Let $M$ be an arbitrary point on side $BC$ of triangle $ABC$. $W$ is a circle which is tangent to $AB$ and $BM$ at $T$ and $K$ and is tangent to circumcircle of $AMC$ at $P$. Prove that if $TK||AM$, circumcircles of $APT$ and $KPC$ are tangent together.
2010 Iran Team Selection Test
Without lifting pen from paper, we draw a polygon in such away that from every two adjacent sides one of them is vertical.

In addition, while drawing the polygon all vertical sides have been drawn from up to down. Prove that this polygon has cut itself.
2010 Iran Team Selection Test
Let $ABC$ an isosceles triangle and $BC>AB=AC$. $D,M$ are respectively midpoints of $BC, AB$. $X$ is a point such that $BX\perp AC$ and $XD||AB$. $BX$ and $AD$ meet at $H$. If $P$ is intersection point of $DX$ and circumcircle of $AHX$ (other than $X$), prove that tangent from $A$ to circumcircle of triangle $AMP$ is parallel to $BC$.
2010 Iran Team Selection Test
Sequence of real numbers $a_0,a_1,\dots,a_{1389}$ are called concave if for each $0<i<1389$, $a_i\geq\frac{a_{i-1}+a_{i+1}}2$. Find the largest $c$ such that for every concave sequence of non-negative real numbers:

$$\sum_{i=0}^{1389}ia_i^2\geq c\sum_{i=0}^{1389}a_i^2$$
2010 Iran Team Selection Test
In every $1\times1$ square of an $m\times n$ table we have drawn one of two diagonals. Prove that there is a path including these diagonals either from left side to the right side, or from the upper side to the lower side.
2010 Iran Team Selection Test
Let $O, H$ be circumcenter and orthogonal center of triangle $ABC$. $M,N$ are midpoints of $BH$ and $CH$. $BB'$ is diagonal of circumcircle. If $HONM$ is a cyclic quadrilateral, prove that $B'N=\frac12AC$.
2010 Iran Team Selection Test
Prove that for each natural number $m$, there is a natural number $N$ such that for each $b$ that $2\leq b\leq1389$ sum of digits of $N$ in base $b$ is larger than $m$.
2009 Iran Team Selection Test
Let $ ABC$ be a triangle and $ A'$ , $ B'$ and $ C'$ lie on $ BC$ , $ CA$ and $ AB$ respectively such that the incenter of $ A'B'C'$ and $ ABC$ are coincide and the inradius of $ A'B'C'$ is half of inradius of $ ABC$ . Prove that $ ABC$ is equilateral .
2009 Iran Team Selection Test
Let $ a$ be a fix natural number . Prove that the set of prime divisors of $ 2^{2^{n}} + a$ for $ n = 1,2,\cdots$ is infinite
2009 Iran Team Selection Test
Suppose that $ a$,$ b$,$ c$ be three positive real numbers such that $ a+b+c=3$ . Prove that :



$ \frac{1}{2+a^{2}+b^{2}}+\frac{1}{2+b^{2}+c^{2}}+\frac{1}{2+c^{2}+a^{2}} \leq \frac{3}{4}$
2009 Iran Team Selection Test
Find all polynomials $f$ with integer coefficient such that, for every prime $p$ and natural numbers $u$ and $v$ with the condition:

$$ p \mid uv - 1 $$

we always have $p \mid f(u)f(v) - 1$.
2009 Iran Team Selection Test
$ ABC$ is a triangle and $ AA'$ , $ BB'$ and $ CC'$ are three altitudes of this triangle . Let $ P$ be the feet of perpendicular from $ C'$ to $ A'B'$ , and $ Q$ is a point on $ A'B'$ such that $ QA = QB$ . Prove that :

$ \angle PBQ = \angle PAQ = \angle PC'C$
2009 Iran Team Selection Test
We have a closed path on a vertices of a $ n$×$ n$ square which pass from each vertice exactly once . prove that we have two adjacent vertices such that if we cut the path from these points then length of each pieces is not less than quarter of total path .
2009 Iran Team Selection Test
Suppose three direction on the plane . We draw $ 11$ lines in each direction . Find maximum number of the points on the plane which are on three lines .
2009 Iran Team Selection Test
Find all polynomials $ P(x,y)$ such that for all reals $ x$ and $y$,

$$P(x^{2},y^{2}) =P\left(\frac {(x + y)^{2}}{2},\frac {(x - y)^{2}}{2}\right).$$
2009 Iran Team Selection Test
In triangle $ABC$, $D$, $E$ and $F$ are the points of tangency of incircle with the center of $I$ to $BC$, $CA$ and $AB$ respectively. Let $M$ be the foot of the perpendicular from $D$ to $EF$. $P$ is on $DM$ such that $DP = MP$. If $H$ is the orthocenter of $BIC$, prove that $PH$ bisects $ EF$.
2009 Iran Team Selection Test
Let $ ABC$ be a triangle and $ AB\ne AC$ . $ D$ is a point on $ BC$ such that $ BA = BD$ and $ B$ is between $ C$ and $ D$ . Let $ I_{c}$ be center of the circle which touches $ AB$ and the extensions of $ AC$ and $ BC$ . $ CI_{c}$ intersect the circumcircle of $ ABC$ again at $ T$ .

If $ \angle TDI_{c} = \frac {\angle B + \angle C}{4}$ then find $ \angle A$
2009 Iran Team Selection Test
Let $n$ be a positive integer. Prove that $$ 3^{\dfrac{5^{2^n}-1}{2^{n+2}}} \equiv (-5)^{\dfrac{3^{2^n}-1}{2^{n+2}}} \pmod{2^{n+4}}. $$
2009 Iran Team Selection Test
$ T$ is a subset of $ {1,2,...,n}$ which has this property : for all distinct $ i,j \in T$ , $ 2j$ is not divisible by $ i$ . Prove that :

$ |T| \leq \frac {4}{9}n + \log_2 n + 2$
2008 Iran Team Selection Test
Find all functions $ f: \mathbb R\longrightarrow \mathbb R$ such that for each $ x,y\in\mathbb R$:

$$ f(xf(y)) + y + f(x) = f(x + f(y)) + yf(x)$$
2008 Iran Team Selection Test
Suppose that $ I$ is incenter of triangle $ ABC$ and $ l'$ is a line tangent to the incircle. Let $ l$ be another line such that intersects $ AB,AC,BC$ respectively at $ C',B',A'$. We draw a tangent from $ A'$ to the incircle other than $ BC$, and this line intersects with $ l'$ at $ A_1$. $ B_1,C_1$ are similarly defined. Prove that $ AA_1,BB_1,CC_1$ are concurrent.
2008 Iran Team Selection Test
Suppose that $ T$ is a tree with $ k$ edges. Prove that the $ k$-dimensional cube can be partitioned to graphs isomorphic to $ T$.
2008 Iran Team Selection Test
Let $ P_1,P_2,P_3,P_4$ be points on the unit sphere. Prove that $ \sum_{i\neq j}\frac1{|P_i-P_j|}$ takes its minimum value if and only if these four points are vertices of a regular pyramid.
2008 Iran Team Selection Test
Let $a,b,c > 0$ and $ab+bc+ca = 1$. Prove that:

$$ \sqrt {a^3 + a} + \sqrt {b^3 + b} + \sqrt {c^3 + c}\geq2\sqrt {a + b + c}. $$
2008 Iran Team Selection Test
Prove that in a tournament with 799 teams, there exist 14 teams,  that can be partitioned into groups in a way that all of the teams in the first group have won all of the teams in the second group.
2008 Iran Team Selection Test
Let $ S$ be a set with $ n$ elements, and $ F$ be a family of subsets of $ S$ with $ 2^{n-1}$ elements, such that for each $ A,B,C\in F$, $ A\cap B\cap C$ is not empty. Prove that the intersection of all of the elements of $ F$ is not empty.
2008 Iran Team Selection Test
Find all polynomials $ p$ of one variable with integer coefficients such that if $ a$ and $ b$ are natural numbers such that $ a + b$ is a perfect square, then $ p\left(a\right) + p\left(b\right)$ is also a perfect square.
2008 Iran Team Selection Test
$ I_a$ is the excenter of the triangle $ ABC$ with respect to $ A$, and $ AI_a$ intersects the circumcircle of $ ABC$ at $ T$. Let $ X$ be a point on $ TI_a$ such that $ XI_a^2=XA.XT$. Draw a perpendicular line from $ X$ to $ BC$ so that it intersects $ BC$ in $ A'$. Define $ B'$ and $ C'$ in the same way. Prove that $ AA'$, $ BB'$ and $ CC'$ are concurrent.
2008 Iran Team Selection Test
In the triangle $ ABC$, $ \angle B$ is greater than $ \angle C$. $ T$ is the midpoint of the arc $ BAC$ from the circumcircle of $ ABC$ and $ I$ is the incenter of $ ABC$. $ E$ is a point such that $ \angle AEI=90^\circ$ and $ AE\parallel BC$. $ TE$ intersects the circumcircle of $ ABC$ for the second time in $ P$. If $ \angle B=\angle IPB$, find the angle $ \angle A$.
2008 Iran Team Selection Test
$ k$ is a given natural number. Find all functions $ f: \mathbb{N}\rightarrow\mathbb{N}$ such that for each $ m,n\in\mathbb{N}$ the following holds:	$$ f(m)+f(n)\mid (m+n)^k$$
2008 Iran Team Selection Test
In the acute-angled triangle $ ABC$, $ D$ is the intersection of the altitude passing through $ A$ with $ BC$ and $ I_a$ is the excenter of the triangle with respect to $ A$. $ K$ is a point on the extension of $ AB$ from $ B$, for which $ \angle AKI_a=90^\circ+\frac 34\angle C$. $ I_aK$ intersects the extension of $ AD$ at $ L$. Prove that $ DI_a$ bisects the angle $ \angle AI_aB$ iff $ AL=2R$. ($ R$ is the circumradius of $ ABC$)
2007 Iran Team Selection Test
In triangle $ABC$, $M$ is midpoint of $AC$, and $D$ is a point on $BC$ such that $DB=DM$. We know that $2BC^{2}-AC^{2}=AB.AC$. Prove that $$BD.DC=\frac{AC^{2}.AB}{2(AB+AC)}$$
2007 Iran Team Selection Test
Let $A$ be the largest subset of $\{1,\dots,n\}$ such that for each $x\in A$, $x$ divides at most one other element in $A$. Prove that $$\frac{2n}3\leq |A|\leq \left\lceil \frac{3n}4\right\rceil. $$
2007 Iran Team Selection Test
Find all solutions of the following functional equation: $$f(x^{2}+y+f(y))=2y+f(x)^{2}. $$
2007 Iran Team Selection Test
In an isosceles right-angled triangle shaped billiards table , a ball starts moving from one of the vertices adjacent to hypotenuse. When it reaches to one side then it will reflect its path. Prove that if we reach to a vertex then it is not the vertex at initial position

By Sam Nariman
2007 Iran Team Selection Test
Find all monic polynomials $f(x)$  in $\mathbb Z[x]$ such that $f(\mathbb Z)$ is closed under multiplication.

By Mohsen Jamali
2007 Iran Team Selection Test
Let $\omega$ be incircle of $ABC$. $P$ and $Q$ are on $AB$ and $AC$, such that $PQ$ is parallel to $BC$ and is tangent to $\omega$. $AB,AC$ touch $\omega$ at $F,E$. Prove that if $M$ is midpoint of $PQ$, and $T$ is intersection point of $EF$ and $BC$, then $TM$ is tangent to $\omega$.

By Ali Khezeli
2007 Iran Team Selection Test
Does there exist a a sequence $a_{0},a_{1},a_{2},\dots$ in $\mathbb N$, such that for each $i\neq j, (a_{i},a_{j})=1$, and for each $n$, the polynomial $\sum_{i=0}^{n}a_{i}x^{i}$ is irreducible in $\mathbb Z[x]$?

By Omid Hatami
2007 Iran Team Selection Test
Suppose $n$ lines in plane are such that no two are parallel and no three are concurrent. For each two lines their angle is a real number in $[0,\frac{\pi}2]$. Find the largest value of the sum of the $\binom n2$ angles between line.

By Aliakbar Daemi
2007 Iran Team Selection Test
$O$ is a point inside triangle $ABC$ such that $OA=OB+OC$. Suppose $B',C'$ be midpoints of arcs $\overarc{AOC}$ and $AOB$. Prove that circumcircles $COC'$ and $BOB'$ are tangent to each other.
2007 Iran Team Selection Test
Find all polynomials of degree 3, such that for each $x,y\geq 0$: $$p(x+y)\geq p(x)+p(y)$$
2007 Iran Team Selection Test
Triangle $ABC$ is isosceles ($AB=AC$). From $A$, we draw a line $\ell$ parallel to $BC$. $P,Q$ are on perpendicular bisectors of $AB,AC$ such that $PQ\perp BC$. $M,N$ are points on $\ell$ such that angles $\angle APM$ and $\angle AQN$ are $\frac\pi2$. Prove that

$$\frac{1}{AM}+\frac1{AN}\leq\frac2{AB}$$
2007 Iran Team Selection Test
Let $P$ be a point in a square whose side are mirror. A ray of light comes from $P$ and with slope $\alpha$. We know that this ray of light never arrives to a vertex. We make an infinite sequence of $0,1$. After each contact of light ray with a horizontal side, we put $0$, and after each contact with a vertical side, we put $1$. For each $n\geq 1$, let $B_{n}$ be set of all blocks of length $n$, in this sequence.

a) Prove that $B_{n}$ does not depend on location of $P$.

b) Prove that if $\frac{\alpha}{\pi}$ is irrational, then $|B_{n}|=n+1$.
2006 Iran Team Selection Test
Suppose that $p$ is a prime number.

Find all natural numbers $n$ such that $p|\varphi(n)$ and for all $a$ such that $(a,n)=1$ we have

$$ n|a^{\frac{\varphi(n)}{p}}-1 $$
2006 Iran Team Selection Test
Suppose $n$ coins are available that their mass is unknown. We have a pair of balances and every time we can choose an even number of coins and put half of them on one side of the balance and put another half on the other side, therefore a comparison will be done. Our aim is determining that the mass of all coins is equal or not. Show that at least $n-1$ comparisons are required.
2006 Iran Team Selection Test
Suppose $ABC$ is a triangle with $M$ the midpoint of $BC$.

Suppose that $AM$ intersects the incircle at $K,L$.

We draw parallel line from $K$ and $L$ to $BC$ and name their second intersection point with incircle $X$ and $Y$. Suppose that $AX$ and $AY$ intersect $BC$ at $P$ and $Q$.

Prove that $BP=CQ$.
2006 Iran Team Selection Test
Let $x_1,x_2,\ldots,x_n$ be real numbers. Prove that

$$ \sum_{i,j=1}^n |x_i+x_j|\geq n\sum_{i=1}^n |x_i| $$
2006 Iran Team Selection Test
Let $ABC$ be a triangle such that it's circumcircle radius is equal to the radius of outer inscribed circle with respect to $A$.

Suppose that the outer inscribed circle with respect to $A$ touches $BC,AC,AB$ at $M,N,L$.

Prove that $O$ (Center of circumcircle) is the orthocenter of $MNL$.
2006 Iran Team Selection Test
Let $G$ be a tournoment such that it's edges are colored either red or blue.

Prove that there exists a vertex of $G$ like $v$ with the property that, for every other vertex $u$ there is a mono-color directed path from $v$ to $u$.
2006 Iran Team Selection Test
We have $n$ points in the plane, no three on a line.

We call $k$ of them good if they form a convex polygon and there is no other point in the convex polygon.

Suppose that for a fixed $k$ the number of $k$ good points is $c_k$.

Show that the following sum is independent of the structure of points and only depends on $n$ :

$$ \sum_{i=3}^n (-1)^i c_i $$
2006 Iran Team Selection Test
Let $n$ be a fixed natural number.

a) Find all solutions to the following equation :

$$ \sum_{k=1}^n [\frac x{2^k}]=x-1  $$

b) Find the number of solutions to the following equation ($m$ is a fixed natural) :

$$ \sum_{k=1}^n [\frac x{2^k}]=x-m  $$
2006 Iran Team Selection Test
Let $l,m$ be two parallel lines in the plane.

Let $P$ be a fixed point between them.

Let $E,F$ be variable points on $l,m$ such that the angle $EPF$ is fixed to a number like $\alpha$ where $0<\alpha<\frac{\pi}2$.

(By angle $EPF$ we mean the directed angle)

Show that there is another point (not $P$) such that it sees the segment $EF$ with a fixed angle too.
2006 Iran Team Selection Test
Let $n$ be a fixed natural number.

Find all $n$ tuples of natural pairwise distinct and coprime numbers like $a_1,a_2,\ldots,a_n$ such that for $1\leq i\leq n$ we have

$$ a_1+a_2+\ldots+a_n|a_1^i+a_2^i+\ldots+a_n^i  $$
2006 Iran Team Selection Test
Let $ABC$ be an acute angle triangle.

Suppose that $D,E,F$ are the feet of perpendicluar lines from $A,B,C$ to $BC,CA,AB$.

Let $P,Q,R$ be the feet of perpendicular lines from $A,B,C$ to $EF,FD,DE$.

Prove that

$$ 2(PQ+QR+RP)\geq DE+EF+FD $$
2006 Iran Team Selection Test
Suppose we have a simple polygon (that is it does not intersect itself, but not necessarily convex).

Show that this polygon has a diameter which is completely inside the polygon and the two arcs it creates on the polygon perimeter (the two arcs have 2 vertices in common) both have at least one third of the vertices of the polygon.
2005 Iran Team Selection Test
Suppose that $ a_1$, $ a_2$, ..., $ a_n$ are positive real numbers such that $ a_1 \leq a_2 \leq \dots \leq a_n$. Let

$$ {{a_1 + a_2 + \dots + a_n} \over n} = m; \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ {{a_1^2 + a_2^2 + \dots + a_n^2} \over n} = 1.

$$

Suppose that, for some $ i$, we know $ a_i \leq m$. Prove that:

$$ n - i \geq n \left(m - a_i\right)^2

$$
2005 Iran Team Selection Test
Assume $ABC$ is an isosceles triangle that $AB=AC$ Suppose $P$ is a point on extension of side $BC$. $X$ and $Y$ are points on $AB$ and $AC$ that:

$$PX || AC \ , \ PY ||AB $$

Also $T$ is midpoint of arc $BC$. Prove that $PT \perp XY$
2005 Iran Team Selection Test
Suppose there are 18 lighthouses on the Persian Gulf. Each of the lighthouses lightens an angle with size 20 degrees. Prove that we can choose the directions of the lighthouses such that whole of the blue Persian (always Persian) Gulf is lightened.
2005 Iran Team Selection Test
Find all $f : N \longmapsto N$ that there exist $k \in N$ and a prime $p$ that:

$\forall n \geq k \ f(n+p)=f(n)$ and also if $m \mid n$ then $f(m+1) \mid f(n)+1$
2005 Iran Team Selection Test
Suppose there are $n$ distinct points on plane. There is circle with radius $r$ and center $O$ on the plane. At least one of the points are in the circle. We do the following instructions. At each step we move $O$ to the baricenter of the point in the circle. Prove that location of $O$ is constant after some steps.