Navvye/Olympiad-Problems-CSV huggingface dataset:
2016 Iran Team Selection Test
Let $AD,BF,CE$ be altitudes of triangle $ABC$.$Q$ is a point on $EF$ such that $QF=DE$ and $F$ is between $E,Q$.$P$ is a point on $EF$ such that $EP=DF$ and $E$ is between $P,F$.Perpendicular bisector of $DQ$ intersect with $AB$ at $X$ and perpendicular bisector of $DP$ intersect with $AC$ at $Y$.Prove that midpoint of $BC$ lies on $XY$.
2015 Iran Team Selection Test
Find all polynomials $P,Q\in \Bbb{Q}\left [ x \right ]$ such that

$$P(x)^3+Q(x)^3=x^{12}+1.$$
2015 Iran Team Selection Test
$I_b$ is the $B$-excenter of the triangle $ABC$ and $\omega$ is the circumcircle of this triangle. $M$ is the middle of arc $BC$ of $\omega$ which doesn't contain $A$. $MI_b$ meets $\omega$ at $T\not =M$. Prove that

$$ TB\cdot TC=TI_b^2.$$
2015 Iran Team Selection Test
Let $ b_1<b_2<b_3<\dots $ be the sequence of all natural numbers which are sum of squares of two natural numbers.

Prove that there exists infinite natural numbers like $m$ which $b_{m+1}-b_m=2015$ .
2015 Iran Team Selection Test
$n$ is a fixed natural number. Find the least $k$ such that for every set $A$ of $k$ natural numbers, there exists a subset of $A$ with an even number of elements which the sum of it's members is divisible by $n$.
2015 Iran Team Selection Test
Let $A$ be a subset of the edges of an $n\times n $ table. Let $V(A)$ be the set of vertices from the table which are connected to at least on edge from $A$ and $j(A)$ be the number of the connected components of graph $G$ which it's vertices are the set $V(A)$ and it's edges are the set $A$. Prove that for every natural number $l$:

$$\frac{l}{2}\leq min_{|A|\geq l}(|V(A)|-j(A)) \leq  \frac{l}{2}+\sqrt{\frac{l}{2}}+1$$
2015 Iran Team Selection Test
$ABCD$ is a circumscribed and inscribed quadrilateral. $O$ is the circumcenter of the quadrilateral. $E,F$ and $S$ are the intersections of $AB,CD$ , $AD,BC$ and $AC,BD$ respectively. $E'$ and $F'$ are points on $AD$ and $AB$ such that $A\hat{E}E'=E'\hat{E}D$ and $A\hat{F}F'=F'\hat{F}B$. $X$ and $Y$ are points on $OE'$ and $OF'$ such that $\frac{XA}{XD}=\frac{EA}{ED}$ and $\frac{YA}{YB}=\frac{FA}{FB}$. $M$ is the midpoint of arc $BD$ of $(O)$ which contains $A$.

Prove that the circumcircles of triangles $OXY$ and $OAM$ are coaxal with the circle with diameter $OS$.
2015 Iran Team Selection Test
$a,b,c,d$ are positive numbers such that $\sum_{cyc} \frac{1}{ab} =1$. Prove that :

$abcd+16 \geq 8 \sqrt{(a+c)(\frac{1}{a} + \frac{1}{c})}+8\sqrt{(b+d)(\frac{1}{b}+\frac{1}{d})}$
2015 Iran Team Selection Test
In triangle $ABC$(with incenter $I$) let the line parallel to $BC$ from $A$ intersect circumcircle of $\triangle ABC$ at $A_1$ let $AI\cap BC=D$ and $E$ is tangency point of incircle with $BC$ let $ EA_1\cap \odot (\triangle ADE)=T$ prove that $AI=TI$.
2015 Iran Team Selection Test
Let $\triangle ABC$ be an acute triangle. Point $Z$ is on $A$ altitude and points $X$ and $Y$ are on the $B$ and $C$ altitudes out of the triangle respectively, such that:

$\angle AYB=\angle BZC=\angle CXA=90$

Prove that $X$,$Y$ and $Z$ are collinear, if and only if the length of the tangent drawn from $A$ to the nine point circle of $\triangle ABC$ is equal with the sum of the lengths of the tangents drawn from $B$ and $C$ to the nine point circle of $\triangle ABC$.
2015 Iran Team Selection Test
We call a permutation $(a_1, a_2,\cdots , a_n)$ of the set $\{ 1,2,\cdots, n\}$ "good" if for any three natural numbers $i <j <k$, $n\nmid a_i+a_k-2a_j$ find all natural numbers $n\ge 3$ such that there exist a "good" permutation of a set $\{1,2,\cdots, n\}$.
2015 Iran Team Selection Test
If $a,b,c$ are positive real numbers such that $a+b+c=abc$ prove that

$$\frac{abc}{3\sqrt{2}}\left ( \sum_{cyc}\frac{\sqrt{a^3+b^3}}{ab+1} \right )\geq \sum_{cyc}\frac{a}{a^2+1}$$
2015 Iran Team Selection Test
Point $A$ is outside of a given circle $\omega$. Let the tangents from $A$ to $\omega$ meet $\omega$ at $S, T$ points $X, Y$ are midpoints of $AT, AS$ let the tangent from $X$ to $\omega$ meet $\omega$ at $R\neq T$. points $P, Q$ are midpoints of $XT, XR$ let $XY\cap PQ=K, SX\cap TK=L$ prove that quadrilateral $KRLQ$ is cyclic.
2015 Iran Team Selection Test
Assume that $a_1, a_2, a_3$ are three given positive integers consider the following sequence:

$a_{n+1}=\text{lcm}[a_n, a_{n-1}]-\text{lcm}[a_{n-1}, a_{n-2}]$ for $n\ge 3$

Prove that there exist a positive integer $k$ such that $k\le a_3+4$ and $a_k\le 0$.

($[a, b]$ means the least positive integer such that$ a\mid[a,b], b\mid[a, b]$ also because $\text{lcm}[a, b]$ takes only nonzero integers this sequence is defined until we find a zero number in the sequence)
2015 Iran Team Selection Test
$a_1,a_2,\cdots ,a_n,b_1,b_2,\cdots ,b_n$ are $2n$ positive real numbers such that $a_1,a_2,\cdots ,a_n$ aren't all equal. And assume that we can divide $a_1,a_2,\cdots ,a_n$ into two subsets with equal sums.similarly $b_1,b_2,\cdots ,b_n$ have these two conditions. Prove that there exist a simple  $2n$-gon with sides $a_1,a_2,\cdots ,a_n,b_1,b_2,\cdots ,b_n$ and parallel to coordinate axises Such that the lengths of horizontal sides are among $a_1,a_2,\cdots ,a_n$ and the lengths of vertical sides are among $b_1,b_2,\cdots ,b_n$.(simple polygon is a polygon such that it doesn't intersect itself)
2015 Iran Team Selection Test
Prove that for each natural number $d$, There is a monic and unique polynomial of degree $d$ like $P$ such that $P(1)$≠$0$ and for each sequence like $a_{1}$,$a_{2}$, $...$ of real numbers that the recurrence relation below is true for them, there is a natural number $k$ such that $0=a_{k}=a_{k+1}= ...$ :

$P(n)a_{1} + P(n-1)a_{2} + ... + P(1)a_{n}=0$

$n>1$
2015 Iran Team Selection Test
$AH$ is the altitude of triangle $ABC$ and $H^\prime$ is the reflection of $H$ trough the midpoint of $BC$. If the tangent lines to the circumcircle of $ABC$ at $B$ and $C$, intersect each other at $X$ and the perpendicular line to $XH^\prime$ at $H^\prime$, intersects $AB$ and $AC$ at $Y$ and $Z$ respectively, prove that $\angle ZXC=\angle YXB$.
2014 Iran Team Selection Test
suppose that $O$ is the circumcenter of acute triangle $ABC$.

we have circle with center $O$ that is tangent too $BC$ that named $w$

suppose that $X$ and $Y$ are the points of intersection of the tangent from $A$ to $w$ with line $BC$($X$ and $B$ are in the same side of $AO$)

$T$ is  the intersection of the line tangent to circumcirle of $ABC$ in $B$ and the line from $X$ parallel to $AC$.

$S$ is  the intersection of the line tangent to circumcirle of $ABC$ in $C$ and the line from $Y$ parallel to $AB$.

prove that $ST$ is tangent $ABC$.
2014 Iran Team Selection Test
find all polynomials with integer coefficients that $P(\mathbb{Z})= ${$p(a):a\in \mathbb{Z}$} has a Geometric progression.
2014 Iran Team Selection Test
we named a $n*n$ table $selfish$ if we number  the row and column with $0,1,2,3,...,n-1$.(from left to right an from up to down)

for every {$ i,j\in{0,1,2,...,n-1}$} the number of cell $(i,j)$ is equal to the number of number $i$ in the row $j$.

for example we have such table for $n=5$

1       0       3       3       4

1       3       2       1       1

0       1       0       1       0

2       1       0       0       0

1       0       0       0       0

prove that for $n>5$ there is no $selfish$ table
2014 Iran Team Selection Test
Find the maximum number of Permutation of set {$1,2,3,...,2014$} such that for every 2 different number $a$ and $b$ in this set  at last in one of the permutation

$b$ comes exactly after $a$
2014 Iran Team Selection Test
$n$ is a natural number. for every positive real numbers $x_{1},x_{2},...,x_{n+1}$ such that $x_{1}x_{2}...x_{n+1}=1$ prove that:

$\sqrt[x_{1}]{n}+...+\sqrt[x_{n+1}]{n} \geq n^{\sqrt[n]{x_{1}}}+...+n^{\sqrt[n]{x_{n+1}}}$
2014 Iran Team Selection Test
$I$ is the incenter of triangle $ABC$. perpendicular from $I$ to $AI$ meet $AB$ and $AC$ at ${B}'$ and ${C}'$ respectively .

Suppose that ${B}''$ and ${C}''$ are points on half-line $BC$ and $CB$ such that $B{B}''=BA$ and $C{C}''=CA$.

Suppose that the second intersection of circumcircles of $A{B}'{B}''$ and $A{C}'{C}''$ is $T$.

Prove that the circumcenter of $AIT$ is on the $BC$.
2014 Iran Team Selection Test
Consider a tree with $n$ vertices, labeled with $1,\ldots,n$ in a way that no label is used twice. We change the labeling in the following way - each time we pick an edge that hasn't been picked before and swap the labels of its endpoints. After performing this action $n-1$ times, we get another tree with its labeling a permutation of the first graph's labeling.

Prove that this permutation contains exactly one cycle.
2014 Iran Team Selection Test
Point $D$ is an arbitary point on side $BC$ of triangle $ABC$. $I$,$I_1$ and$I_2$ are the incenters of triangles $ABC$,$ABD$ and $ACD$ respectively. $M\not=A$ and $N\not=A$ are the intersections of circumcircle of triangle $ABC$ and circumcircles of triangles $IAI_1$ and $IAI_2$ respectively. Prove that regardless of point $D$, line $MN$ goes through a fixed point.
2014 Iran Team Selection Test
prove for all $k> 1$ equation $(x+1)(x+2)...(x+k)=y^{2}$ has finite solutions.
2014 Iran Team Selection Test
$n$ is a natural number. We shall call a permutation $a_1,\dots,a_n$ of $1,\dots,n$ a quadratic(cubic) permutation if $\forall 1\leq i \leq n-1$ we have  $a_ia_{i+1}+1$ is a perfect square(cube).

$(a)$ Prove that for infinitely many natural numbers $n$ there exists a quadratic permutation.

$(b)$ Prove that for no natural number $n$ exists a cubic permutation.
2014 Iran Team Selection Test
if $x,y,z>0$ are postive real numbers such that $x^{2}+y^{2}+z^{2}=x^{2}y^{2}+y^{2}z^{2}+z^{2}x^{2}$

prove that $$((x-y)(y-z)(z-x))^{2}\leq 2((x^{2}-y^{2})^{2}+(y^{2}-z^{2})^{2}+(z^{2}-x^{2})^{2})$$
2014 Iran Team Selection Test
Consider $n$ segments in the plane which no two intersect and between their $2n$ endpoints no three are collinear. Is the following statement true?

Statement: There exists a simple $2n$-gon such that it's vertices are the $2n$ endpoints of the segments and each segment is either completely inside the polygon or an edge of the polygon.
2014 Iran Team Selection Test
The incircle of a non-isosceles triangle $ABC$ with the center $I$ touches the sides $BC,AC,AB$ at $A_{1},B_{1},C_{1}$ .

let $AI,BI,CI$ meets $BC,AC,AB$ at  $A_{2},B_{2},C_{2}$.

let $A'$ is a point on $AI$ such that $A_{1}A'\perp B_{2}C_{2}$ .$B',C'$ respectively.

prove that two triangle $A'B'C',A_{1}B_{1}C_{1}$ are equal.
2014 Iran Team Selection Test
is there a function $f:\mathbb{N}\rightarrow \mathbb{N}$ such that

$i) \exists n\in \mathbb{N}:f(n)\neq n$

$ii)$ the number of divisors of $m$ is $f(n)$  if and only if the number of divisors of $f(m)$ is $n$
2014 Iran Team Selection Test
let $m,n\in \mathbb{N}$ and $p(x),q(x),h(x)$ are polynomials with real Coefficients such that $p(x)$ is Descending.

and for all $x\in \mathbb{R}$

$p(q(nx+m)+h(x))=n(q(p(x))+h(x))+m$ .

prove that dont exist function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x\in \mathbb{R}$

$f(q(p(x))+h(x))=f(x)^{2}+1$
2014 Iran Team Selection Test
Find all functions $f:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$ such that

$x,y\in \mathbb{R}^{+},$ $$ f\left(\frac{y}{f(x+1)}\right)+f\left(\frac{x+1}{xf(y)}\right)=f(y) $$
2014 Iran Team Selection Test
Given a set $X=\{x_1,\ldots,x_n\}$ of natural numbers in which for all $1< i \leq n$ we have $1\leq x_i-x_{i-1}\leq 2$, call a real number $a$ good if there exists $1\leq j \leq n$ such that $2|x_j-a|\leq 1$. Also a subset of $X$ is called compact if the average of its elements is a good number.

Prove that at least $2^{n-3}$ subsets of $X$ are compact.



Proposed by Mahyar Sefidgaran
2014 Iran Team Selection Test
The incircle of a non-isosceles triangle $ABC$ with the center $I$ touches the sides $BC$ at $D$.

let $X$  is a point on arc $BC$ from circumcircle of triangle $ABC$ such that if $E,F$ are feet of perpendicular from $X$ on $BI,CI$ and $M$ is midpoint of $EF$ we have $MB=MC$.

prove that $\widehat{BAD}=\widehat{CAX}$
2013 Iran Team Selection Test
In acute-angled triangle $ABC$, let $H$ be the foot of perpendicular from $A$ to $BC$ and also suppose that $J$ and $I$ are excenters oposite to the side $AH$ in triangles $ABH$ and $ACH$. If $P$ is the point that incircle touches $BC$, prove that $I,J,P,H$ are concyclic.
2013 Iran Team Selection Test
Find the maximum number of subsets from $\left \{ 1,...,n \right \}$ such that for any two of them like $A,B$ if $A\subset B$ then $\left | B-A \right |\geq 3$.  (Here $\left | X \right |$ is the number of elements of the set $X$.)
2013 Iran Team Selection Test
For nonnegative integers $m$ and $n$, define the sequence $a(m,n)$ of real numbers  as follows.  Set $a(0,0)=2$ and for every natural number $n$, set $a(0,n)=1$ and $a(n,0)=2$.  Then for $m,n\geq1$, define $$ a(m,n)=a(m-1,n)+a(m,n-1). $$ Prove that for every natural number $k$, all the roots of the polynomial $P_{k}(x)=\sum_{i=0}^{k}a(i,2k+1-2i)x^{i}$ are real.
2013 Iran Team Selection Test
$m$ and $n$ are two nonnegative integers. In the Philosopher's Chess, The chessboard is an infinite grid of identical regular hexagons and a new piece named the Donkey moves on it as follows:



Starting from one of the hexagons, the Donkey moves $m$ cells in one of the $6$ directions, then it turns $60$ degrees clockwise and after that moves $n$ cells in this new direction until it reaches it's final cell.



At most how many cells are in the Philosopher's chessboard such that one cannot go from anyone of them to the other with a finite number of movements of the Donkey?



Proposed by Shayan Dashmiz
2013 Iran Team Selection Test
Do there exist natural numbers $a, b$ and $c$ such that $a^2+b^2+c^2$ is divisible by $2013(ab+bc+ca)$?



Proposed by Mahan Malihi
2013 Iran Team Selection Test
Points $A, B, C$ and $D$ lie on line $l$ in this order. Two circular arcs $C_1$ and $C_2$, which both lie on one side of line $l$, pass through points $A$ and $B$ and two circular arcs $C_3$ and $C_4$ pass through points $C$ and $D$ such that $C_1$ is tangent to $C_3$ and $C_2$ is tangent to $C_4$. Prove that the common external tangent of $C_2$ and $C_3$ and the common external tangent of $C_1$ and $C_4$ meet each other on line $l$.



Proposed by Ali Khezeli
2013 Iran Team Selection Test
Nonnegative real numbers $p_{1},\ldots,p_{n}$ and $q_{1},\ldots,q_{n}$ are such that $p_{1}+\cdots+p_{n}=q_{1}+\cdots+q_{n}$

Among all the matrices with nonnegative entries having $p_i$ as sum of the $i$-th row's entries and $q_j$ as sum of the $j$-th column's entries, find the maximum sum of the entries on the main diagonal.
2013 Iran Team Selection Test
Find all Arithmetic progressions $a_{1},a_{2},...$ of natural numbers for which there exists natural number $N>1$ such that for every $k\in \mathbb{N}$:



$a_{1}a_{2}...a_{k}\mid a_{N+1}a_{N+2}...a_{N+k}$
2013 Iran Team Selection Test
find all functions $f,g:\mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that $f$ is increasing and also:



$f(f(x)+2g(x)+3f(y))=g(x)+2f(x)+3g(y)$



$g(f(x)+y+g(y))=2x-g(x)+f(y)+y$
2013 Iran Team Selection Test
On each edge of a graph is written a real number,such that for every even tour of this graph,sum the edges with signs alternatively positive and negative is zero.prove that one can assign to each of the vertices of the graph a real number such that sum of the numbers on two adjacent vertices is the number on the edge between them.(tour is a closed path from the edges of the graph that may have repeated edges or vertices)
2013 Iran Team Selection Test
Let $a,b,c$ be sides of a triangle such that $a\geq b \geq c$. prove that:



$\sqrt{a(a+b-\sqrt{ab})}+\sqrt{b(a+c-\sqrt{ac})}+\sqrt{c(b+c-\sqrt{bc})}\geq a+b+c$
2013 Iran Team Selection Test
Let $ABCD$ be a cyclic quadrilateral that inscribed in the circle $\omega$.Let $I_{1},I_{2}$ and $r_{1},r_{2}$ be incenters and radii of incircles of triangles $ACD$ and $ABC$,respectively.assume that $r_{1}=r_{2}$. let $\omega'$ be a circle that touches $AB,AD$ and touches $\omega$ at $T$. tangents from $A,T$ to $\omega$ meet at the point $K$.prove that $I_{1},I_{2},K$ lie on a line.
2013 Iran Team Selection Test
$P$ is an arbitrary point inside acute triangle $ABC$. Let $A_1,B_1,C_1$ be the reflections of point $P$ with respect to sides $BC,CA,AB$. Prove that the centroid of triangle $A_1B_1C_1$ lies inside triangle $ABC$.
2013 Iran Team Selection Test
we are given $n$ rectangles in the plane. Prove that between $4n$ right angles formed by these rectangles there are at least $[4\sqrt n]$ distinct right angles.
2013 Iran Team Selection Test
a) Does there exist a sequence $a_1<a_2<\dots$ of positive integers, such that there is a positive integer $N$ that $\forall m>N$,

$a_m$ has exactly $d(m)-1$ divisors among $a_i$s?



b) Does there exist a sequence $a_1<a_2<\dots$ of positive integers, such that there is a positive integer $N$ that $\forall m>N$,

$a_m$ has exactly $d(m)+1$ divisors among $a_i$s?
2013 Iran Team Selection Test
The function $f:\mathbb Z \to \mathbb Z$ has the property that for all integers $m$ and $n$

$$f(m)+f(n)+f(f(m^2+n^2))=1.$$

We know that integers $a$ and $b$ exist such that $f(a)-f(b)=3$. Prove that integers $c$ and $d$ can be found such that $f(c)-f(d)=1$.



Proposed by Amirhossein Gorzi
2013 Iran Team Selection Test
In triangle $ABC$, $AD$ and $AH$ are the angle bisector and the altitude of vertex $A$, respectively. The perpendicular bisector of $AD$, intersects the semicircles with diameters $AB$ and $AC$ which are drawn outside triangle $ABC$ in $X$ and $Y$, respectively. Prove that the quadrilateral $XYDH$ is concyclic.



Proposed by Mahan Malihi
2013 Iran Team Selection Test
A special kind of parallelogram tile is made up by attaching the legs of two right isosceles triangles of side length $1$. We want to put a number of these tiles on the floor of an $n\times n$ room such that the distance from each vertex of each tile to the sides of the room is an integer and also no two tiles overlap. Prove that at least an area $n$ of the room will not be covered by the tiles.



Proposed by Ali Khezeli
2012 Iran Team Selection Test
Find all positive integers $n \geq 2$ such that for all integers $i,j$ that $ 0 \leq i,j\leq n$  , $i+j$ and $ {n\choose i}+ {n \choose j}$ have same parity.



Proposed by Mr.Etesami
2012 Iran Team Selection Test
Consider $\omega$ is circumcircle of an acute triangle $ABC$. $D$ is midpoint of arc $BAC$ and $I$ is incenter of triangle $ABC$. Let $DI$ intersect $BC$ in $E$ and $\omega$ for second time in $F$.  Let $P$ be a point on line $AF$ such that $PE$ is parallel to $AI$. Prove that $PE$ is bisector of angle $BPC$.



Proposed by Mr.Etesami
2012 Iran Team Selection Test
Let $n$ be a positive integer. Let $S$ be a subset of points on the plane with these conditions:



$i)$  There does not exist $n$ lines in the plane such that every element of $S$ be on at least one of them.



$ii)$ for all $X \in S$ there exists $n$ lines in the plane such that every element of $S - {X} $ be on at least one of them.



Find maximum of $\mid S\mid$.



Proposed by Erfan Salavati
2012 Iran Team Selection Test
Consider $m+1$ horizontal and $n+1$ vertical lines ($m,n\ge 4$) in the plane forming an $m\times n$ table. Cosider a closed path on the segments of this table such that it does not intersect itself and also it passes through all $(m-1)(n-1)$ interior vertices (each vertex is an intersection point of two lines) and it doesn't pass through any of outer vertices. Suppose $A$ is the number of vertices such that the path passes through them straight forward,  $B$ number of the table squares that only their two opposite sides are used in the path, and $C$ number of the table squares that none of their sides is used in the path. Prove that

$$A=B-C+m+n-1.$$



Proposed by Ali Khezeli
2012 Iran Team Selection Test
The function $f:\mathbb R^{\ge 0} \longrightarrow \mathbb R^{\ge 0}$ satisfies the following properties for all $a,b\in \mathbb R^{\ge 0}$:



a) $f(a)=0 \Leftrightarrow a=0$



b) $f(ab)=f(a)f(b)$



c) $f(a+b)\le 2 \max \{f(a),f(b)\}$.



Prove that for all $a,b\in \mathbb R^{\ge 0}$ we have $f(a+b)\le f(a)+f(b)$.



Proposed by Masoud Shafaei
2012 Iran Team Selection Test
The pentagon $ABCDE$ is inscirbed in a circle $w$. Suppose that $w_a,w_b,w_c,w_d,w_e$ are reflections of $w$ with respect to sides $AB,BC,CD,DE,EA$ respectively. Let $A'$ be the second intersection point of $w_a,w_e$ and define $B',C',D',E'$ similarly. Prove that

$$2\le \frac{S_{A'B'C'D'E'}}{S_{ABCDE}}\le 3,$$

where $S_X$ denotes the surface of figure $X$.



Proposed by Morteza Saghafian, Ali khezeli
2012 Iran Team Selection Test
Is it possible to put $\binom{n}{2}$ consecutive natural numbers on the edges of a complete graph with $n$ vertices in a way that for every path (or cycle) of length $3$ where the numbers $a,b$ and $c$ are written on its edges (edge $b$ is between edges $c$ and $a$), $b$ is divisible by the greatest common divisor of the numbers $a$ and $c$?



Proposed by Morteza Saghafian
2012 Iran Team Selection Test
Let $g(x)$ be a polynomial of degree at least $2$ with all of its coefficients positive. Find all functions $f:\mathbb R^+ \longrightarrow \mathbb R^+$ such that

$$f(f(x)+g(x)+2y)=f(x)+g(x)+2f(y) \quad \forall x,y\in \mathbb R^+.$$



Proposed by Mohammad Jafari
2012 Iran Team Selection Test
Suppose $ABCD$ is a parallelogram. Consider circles $w_1$ and $w_2$ such that $w_1$ is tangent to segments $AB$ and $AD$ and $w_2$ is tangent to segments $BC$ and $CD$. Suppose that there exists a circle which is tangent to lines $AD$ and $DC$ and externally tangent to $w_1$ and $w_2$. Prove that there exists a circle which is tangent to lines $AB$ and $BC$ and also externally tangent to circles $w_1$ and $w_2$.



Proposed by Ali Khezeli
2012 Iran Team Selection Test
For positive reals $a,b$ and $c$ with $ab+bc+ca=1$, show that

$$\sqrt{3}({\sqrt{a}+\sqrt{b}+\sqrt{c})\le \frac{a\sqrt{a}}{bc}+\frac{b\sqrt{b}}{ca}+\frac{c\sqrt{c}}{ab}.}$$



Proposed by Morteza Saghafian
2012 Iran Team Selection Test
Points $A$ and $B$ are on a circle $\omega$ with center $O$ such that $\tfrac{\pi}{3}< \angle AOB <\tfrac{2\pi}{3}$. Let $C$ be the circumcenter of the triangle $AOB$. Let $l$ be a line passing through $C$ such that the angle between $l$ and the segment $OC$ is $\tfrac{\pi}{3}$. $l$ cuts tangents in $A$ and $B$ to $\omega$ in $M$ and $N$ respectively. Suppose circumcircles of triangles $CAM$ and $CBN$, cut $\omega$ again in $Q$ and $R$ respectively and theirselves in $P$ (other than $C$). Prove that $OP\perp QR$.



Proposed by Mehdi E'tesami Fard, Ali Khezeli
2012 Iran Team Selection Test
We call a subset $B$ of natural numbers loyal if there exists natural numbers $i\le j$ such that $B=\{i,i+1,\ldots,j\}$. Let $Q$ be the set of all loyal sets. For every subset $A=\{a_1<a_2<\ldots<a_k\}$ of $\{1,2,\ldots,n\}$ we set

$$f(A)=\max_{1\le i \le k-1}{a_{i+1}-a_i}\qquad\text{and}\qquad g(A)=\max_{B\subseteq A, B\in Q} |B|.$$Furthermore, we define $$F(n)=\sum_{A\subseteq \{1,2,\ldots,n\}} f(A)\qquad\text{and}\qquad G(n)=\sum_{A\subseteq \{1,2,\ldots,n\}} g(A).$$Prove that there exists $m\in \mathbb N$ such that for each natural number $n>m$ we have $F(n)>G(n)$.  (By $|A|$ we mean the number of elements of $A$, and if $|A|\le 1$, we define $f(A)$ to be zero).



Proposed by Javad Abedi
2012 Iran Team Selection Test
Consider a regular $2^k$-gon with center $O$ and label its sides clockwise by $l_1,l_2,...,l_{2^k}$. Reflect $O$ with respect to $l_1$, then reflect the resulting point with respect to $l_2$ and do this process until the last side. Prove that the distance between the final point and $O$ is less than the perimeter of the $2^k$-gon.



Proposed by Hesam Rajabzade
2012 Iran Team Selection Test
Do there exist $2000$ real numbers (not necessarily distinct) such that all of them are not zero and if we put any group containing $1000$ of them as the roots of a monic polynomial of degree $1000$, the coefficients of the resulting polynomial (except the coefficient of $x^{1000}$) be a permutation of the $1000$ remaining numbers?



Proposed by Morteza Saghafian
2012 Iran Team Selection Test
Find all integer numbers $x$ and $y$ such that:

$$(y^3+xy-1)(x^2+x-y)=(x^3-xy+1)(y^2+x-y).$$



Proposed by Mahyar Sefidgaran