Navvye/Olympiad-Problems-CSV huggingface dataset:
2019 Iran Team Selection Test
Consider triangle $ABC$ with orthocenter $H$. Let points $M$ and $N$ be the midpoints of segments $BC$ and $AH$. Point $D$ lies on line $MH$ so that $AD\parallel BC$ and point $K$ lies on line $AH$ so that $DNMK$ is cyclic. Points $E$ and $F$ lie on lines $AC$ and $AB$ such that $\angle EHM=\angle C$ and $\angle FHM=\angle B$. Prove that points $D,E,F$ and $K$ lie on a circle.



Proposed by Alireza Dadgarnia
2019 Iran Team Selection Test
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x,y\in \mathbb{R}$:

$$f\left(f(x)^2-y^2\right)^2+f(2xy)^2=f\left(x^2+y^2\right)^2$$

Proposed by Ali Behrouz - Mojtaba Zare Bidaki
2019 Iran Team Selection Test
$\{a_{n}\}_{n\geq 0}$ and $\{b_{n}\}_{n\geq 0}$ are two sequences of positive integers that $a_{i},b_{i}\in \{0,1,2,\cdots,9\}$. There is an integer number $M$ such that $a_{n},b_{n}\neq 0$ for all $n\geq M$ and for each $n\geq 0$

$$(\overline{a_{n}\cdots a_{1}a_{0}})^{2}+999 \mid(\overline{b_{n}\cdots b_{1}b_{0}})^{2}+999 $$prove that $a_{n}=b_{n}$ for $n\geq 0$.

(Note that $(\overline{x_nx_{n-1}\dots x_0}) = 10^n\times x_n + \dots + 10\times x_1 + x_0$.)



Proposed by Yahya Motevassel
2019 Iran Team Selection Test
$S$ is a subset of Natural numbers which has infinite members.

$$S’=\left\{x^y+y^x: \, x,y\in S, \, x\neq y\right\}$$Prove the set of prime divisors of $S’$ has also infinite members

Proposed by Yahya Motevassel
2019 Iran Team Selection Test
In a triangle $ABC$, $\angle A$ is $60^\circ$. On sides $AB$ and $AC$ we make two equilateral triangles (outside the triangle $ABC$) $ABK$ and $ACL$. $CK$ and $AB$ intersect at $S$ , $AC$ and $BL$ intersect at $R$ , $BL$ and  $CK$ intersect at $T$. Prove the radical centre of circumcircle of triangles $BSK, CLR$ and $BTC$ is on the median of vertex $A$ in triangle $ABC$.

Proposed by Ali Zamani
2019 Iran Team Selection Test
Numbers $m$ and $n$ are given positive integers. There are $mn$ people in a party, standing in the shape of an $m\times n$ grid. Some of these people are police officers and the rest are the guests. Some of the guests may be criminals. The goal is to determine whether there is a criminal between the guests or not.

Two people are considered \textit{adjacent} if they have a common side. Any police officer can see their adjacent people and for every one of them, know that they're criminal or not. On the other hand, any criminal will threaten exactly one of their adjacent people (which is likely an officer!) to murder. A threatened officer will be too scared, that they deny the existence of any criminal between their adjacent people.

Find the least possible number of officers such that they can take position in the party, in a way that the goal is achievable. (Note that the number of criminals is unknown and it is possible to have zero criminals.)



Proposed by Abolfazl Asadi
2019 Iran Team Selection Test
Let $1<t<2$ be a real number. Prove that for all sufficiently large positive integers like $d$, there is a monic polynomial $P(x)$ of degree $d$, such that all of its coefficients are either $+1$ or $-1$ and

$$\left|P(t)-2019\right| <1.$$Proposed by Navid Safaei
2019 Iran Team Selection Test
Let $P$ be a simple polygon completely in $C$, a circle with radius $1$, such that $P$ does not pass through the center of $C$. The perimeter of $P$ is $36$. Prove that there is a radius of $C$ that intersects $P$ at least $6$ times, or there is a circle which is concentric with $C$ and have at least $6$ common points with $P$.



Proposed by Seyed Reza Hosseini
2019 Iran Team Selection Test
For any positive integer $n$, define the subset $S_n$ of natural numbers as follow

$$ S_n = \left\{x^2+ny^2 : x,y \in \mathbb{Z} \right\}.$$Find all positive integers $n$ such that there exists an element of $S_n$ which doesn't belong to any of the sets $S_1, S_2,\dots,S_{n-1}$.



Proposed by Yahya Motevassel
2019 Iran Team Selection Test
Find all polynomials $P(x,y)$ with real coefficients such that for all real numbers $x,y$ and $z$:

$$P(x,2yz)+P(y,2zx)+P(z,2xy)=P(x+y+z,xy+yz+zx).$$

Proposed by Sina Saleh
2019 Iran Team Selection Test
Hesam chose $10$ distinct positive integers and he gave all pairwise $\gcd$'s and pairwise ${\text lcm}$'s (a total of $90$ numbers) to Masoud. Can Masoud always find the first $10$ numbers, just by knowing these $90$ numbers?



Proposed by Morteza Saghafian
2019 Iran Team Selection Test
In triangle $ABC$, $M,N$ and $P$ are midpoints of sides $BC,CA$ and $AB$. Point $K$ lies on segment $NP$ so that $AK$ bisects $\angle BKC$. Lines $MN,BK$ intersects at $E$ and lines $MP,CK$ intersects at $F$. Suppose that $H$ be the foot of perpendicular line from $A$ to $BC$ and $L$ the second intersection of circumcircle of triangles $AKH, HEF$. Prove that $MK,EF$ and $HL$ are concurrent.



Proposed by Alireza Dadgarnia
2019 Iran Team Selection Test
Given an acute-angled triangle $ABC$ with orthocenter $H$. Reflection of nine-point circle about $AH$ intersects circumcircle at points $X$ and $Y$. Prove that $AH$ is the external bisector of $\angle XHY$.



Proposed by Mohammad Javad Shabani
2019 Iran Team Selection Test
A sub-graph of a complete graph with $n$ vertices is chosen such that the number of its edges is a multiple of $3$ and degree of each vertex is an even number. Prove that we can assign a weight to each triangle of the graph such that for each edge of the chosen sub-graph, the sum of the weight of the triangles that contain that edge equals one, and for each edge that is not in the sub-graph, this sum equals zero.



Proposed by Morteza Saghafian
2019 Iran Team Selection Test
$x,y$ and $z$ are real numbers such that $x+y+z=xy+yz+zx$. Prove that

$$\frac{x}{\sqrt{x^4+x^2+1}}+\frac{y}{\sqrt{y^4+y^2+1}}+\frac{z}{\sqrt{z^4+z^2+1}}\geq \frac{-1}{\sqrt{3}}.$$

Proposed by Navid Safaei
2018 Iran Team Selection Test
Let $A_1, A_2, ... , A_k$ be the subsets of $\left\{1,2,3,...,n\right\}$ such that for all $1\leq i,j\leq k$:$A_i\cap A_j \neq \varnothing$. Prove that there are $n$ distinct positive integers $x_1,x_2,...,x_n$ such that for each $1\leq j\leq k$:

$$lcm_{i \in A_j}\left\{x_i\right\}>lcm_{i \notin A_j}\left\{x_i\right\}$$Proposed by Morteza Saghafian, Mahyar Sefidgaran
2018 Iran Team Selection Test
Determine the least real number $k$ such that the inequality

$$\left(\frac{2a}{a-b}\right)^2+\left(\frac{2b}{b-c}\right)^2+\left(\frac{2c}{c-a}\right)^2+k \geq 4\left(\frac{2a}{a-b}+\frac{2b}{b-c}+\frac{2c}{c-a}\right)$$holds for all real numbers $a,b,c$.



Proposed by Mohammad Jafari
2018 Iran Team Selection Test
In triangle $ABC$ let $M$ be the midpoint of $BC$. Let $\omega$ be a circle inside of $ABC$ and is tangent to $AB,AC$ at $E,F$, respectively. The tangents from $M$ to $\omega$ meet $\omega$ at $P,Q$ such that $P$ and $B$ lie on the same side of $AM$. Let $X \equiv PM \cap BF $ and $Y \equiv QM \cap CE $. If $2PM=BC$ prove that $XY$ is tangent to $\omega$.



Proposed by Iman Maghsoudi
2018 Iran Team Selection Test
Let $ABC$ be a triangle ($\angle A\neq 90^\circ$). $BE,CF$ are the altitudes of the triangle. The bisector of $\angle A$ intersects $EF,BC$ at $M,N$. Let $P$ be a point such that $MP\perp EF$ and $NP\perp BC$. Prove that $AP$ passes through the midpoint of $BC$.



Proposed by Iman Maghsoudi, Hooman Fattahi
2018 Iran Team Selection Test
Prove that for each positive integer $m$, one can find $m$ consecutive positive integers like $n$ such that the following phrase doesn't be a perfect power:

$$\left(1^3+2018^3\right)\left(2^3+2018^3\right)\cdots \left(n^3+2018^3\right)$$

Proposed by Navid Safaei
2018 Iran Team Selection Test
A simple graph is called "divisibility", if it's possible to put distinct numbers on its vertices such that there is an edge between two vertices if and only if number of one of its vertices is divisible by another one.



A simple graph is called "permutationary", if it's possible to put numbers $1,2,...,n$ on its vertices and there is a permutation $ \pi $ such that there is an edge between vertices $i,j$ if and only if $i>j$ and $\pi(i)< \pi(j)$ (it's not directed!)



Prove that a simple graph is permutationary if and only if its complement and itself are divisibility.



Proposed by Morteza Saghafian

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2018 Iran Team Selection Test
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ that satisfy the following conditions:

a. $x+f(y+f(x))=y+f(x+f(y)) \quad \forall x,y \in \mathbb{R}$

b. The set $I=\left\{\frac{f(x)-f(y)}{x-y}\mid x,y\in \mathbb{R},x\neq y \right\}$ is an interval.



Proposed by Navid Safaei
2018 Iran Team Selection Test
Mojtaba and Hooman are playing a game. Initially Mojtaba draws $2018$ vectors with zero sum. Then in each turn, starting with Mojtaba, the player takes a vector and puts it on the plane. After the first move, the players must put their vector next to the previous vector (the beginning of the vector must lie on the end of the previous vector).

At last, there will be a closed polygon. If this polygon is not self-intersecting, Mojtaba wins. Otherwise Hooman. Who has the winning strategy?



Proposed by Mahyar Sefidgaran, Jafar Namdar
2018 Iran Team Selection Test
Let $a_1,a_2,a_3,\cdots $ be an infinite sequence of distinct integers. Prove that there are infinitely many primes $p$ that distinct positive integers $i,j,k$ can be found such that $p\mid a_ia_ja_k-1$.



Proposed by Mohsen Jamali
2018 Iran Team Selection Test
Call a positive integer "useful but not optimized " (!), if it can be written as a sum of distinct powers of $3$ and powers of $5$.

Prove that there exist infinitely many positive integers which they are not "useful but not optimized".

(e.g. $37=(3^0+3^1+3^3)+(5^0+5^1)$ is a " useful but not optimized" number)



Proposed by Mohsen Jamali
2018 Iran Team Selection Test
Let $\omega$ be the circumcircle of isosceles triangle $ABC$ ($AB=AC$). Points $P$ and $Q$ lie on $\omega$ and $BC$ respectively such that $AP=AQ$ .$AP$ and $BC$ intersect at $R$. Prove that the tangents from $B$ and $C$ to the incircle of $\triangle AQR$ (different from $BC$) are concurrent on $\omega$.



Proposed by Ali Zamani, Hooman Fattahi
2018 Iran Team Selection Test
$a_1,a_2,\ldots,a_n$ is a sequence of positive integers that has at least $\frac {2n}{3}+1$ distinct numbers and each positive integer has occurred at most three times in it. Prove that there exists a permutation  $b_1,b_2,\ldots,b_n$ of $a_i $'s such that all the $n$ sums $b_i+b_{i+1}$ are distinct ($1\le i\le n $ , $b_{n+1}\equiv b_1 $)



Proposed by Mohsen Jamali
2018 Iran Team Selection Test
Two circles $\omega_1(O)$ and $\omega_2$ intersect each other at $A,B$ ,and $O$ lies on $\omega_2$. Let $S$ be a point on $AB$ such that $OS\perp AB$. Line $OS$ intersects $\omega_2$  at $P$ (other than $O$). The bisector of $\hat{ASP}$ intersects  $\omega_1$ at $L$ ($A$ and $L$ are on the same side of the line $OP$). Let $K$ be a point on $\omega_2$ such that $PS=PK$ ($A$ and $K$ are on the same side of the line $OP$). Prove that $SL=KL$.



Proposed by Ali Zamani
2018 Iran Team Selection Test
Find the maximum possible value of $k$ for which there exist distinct reals $x_1,x_2,\ldots ,x_k $ greater than $1$ such that for all $1 \leq i, j \leq k$,

$$x_i^{\lfloor x_j \rfloor }= x_j^{\lfloor x_i\rfloor}.$$

Proposed by Morteza Saghafian
2018 Iran Team Selection Test
$n>1$ and distinct positive integers $a_1,a_2,\ldots,a_{n+1}$ are  given. Does there exist a polynomial $p(x)\in\Bbb{Z}[x]$ of degree  $\le n$ that satisfies the following conditions?

a.  $\forall_{1\le i < j\le n+1}: \gcd(p(a_i),p(a_j))>1 $

b.  $\forall_{1\le i < j < k\le n+1}: \gcd(p(a_i),p(a_j),p(a_k))=1 $



Proposed by Mojtaba Zare
2018 Iran Team Selection Test
We say distinct positive integers  $a_1,a_2,\ldots ,a_n $ are "good" if their sum is equal to the sum of all pairwise $\gcd $'s among them. Prove that there are infinitely many $n$ s such that $n$ good numbers exist.



Proposed by Morteza Saghafian
2018 Iran Team Selection Test
$2n-1$ distinct positive real numbers with sum $S $ are given. Prove that there are at least $\binom {2n-2}{n-1}$ different ways to choose $n $ numbers among them such that their sum is at least $\frac {S}{2}$.



Proposed by Amirhossein Gorzi
2018 Iran Team Selection Test
Consider quadrilateral $ABCD $ inscribed in circle $\omega $. $P\equiv AC\cap BD$. $E$, $F$ lie on sides $AB$, $CD$ respectively such that $\hat {APE}=\hat {DPF} $. Circles $\omega_1$, $\omega_2$ are tangent to $\omega$ at $X $, $Y $ respectively and also both tangent to the circumcircle of $\triangle PEF $ at $P $. Prove that: $$\frac {EX}{EY}=\frac {FX}{FY} $$

Proposed by Ali Zamani
2017 Iran Team Selection Test
Let $a,b,c,d$ be positive real numbers with $a+b+c+d=2$. Prove the following inequality:

$$\frac{(a+c)^{2}}{ad+bc}+\frac{(b+d)^{2}}{ac+bd}+4\geq 4\left ( \frac{a+b+1}{c+d+1}+\frac{c+d+1}{a+b+1} \right).$$

Proposed by Mohammad Jafari
2017 Iran Team Selection Test
In the country of Sugarland, there are $13$ students in the IMO team selection camp.

$6$ team selection tests were taken and the results have came out. Assume that no students have the same score on the same test.To select the IMO team, the national committee of math Olympiad have decided to choose a permutation of these $6$ tests and starting from the first test, the person with the highest score between the remaining students will become a member of the team.The committee is having a session to choose the permutation.

Is it possible that all $13$ students have a chance of being a team member?



Proposed by Morteza Saghafian
2017 Iran Team Selection Test
In triangle $ABC$ let $I_a$ be the $A$-excenter. Let $\omega$ be an arbitrary circle that passes through $A,I_a$ and intersects the extensions of sides $AB,AC$ (extended from $B,C$) at $X,Y$ respectively. Let $S,T$ be points on segments $I_aB,I_aC$ respectively such that $\angle AXI_a=\angle BTI_a$ and $\angle AYI_a=\angle CSI_a$.Lines $BT,CS$ intersect at $K$. Lines $KI_a,TS$ intersect at $Z$.

Prove that $X,Y,Z$ are collinear.



Proposed by Hooman Fattahi
2017 Iran Team Selection Test
We arranged all the prime numbers in the ascending order: $p_1=2<p_2<p_3<\cdots$.

Also assume that $n_1<n_2<\cdots$ is a sequence of positive integers that for all $i=1,2,3,\cdots$ the equation $x^{n_i} \equiv 2 \pmod {p_i}$ has a solution for $x$.

Is there always a number $x$ that satisfies all the equations?



Proposed by Mahyar Sefidgaran , Yahya Motevasel
2017 Iran Team Selection Test
In triangle $ABC$, arbitrary points $P,Q$ lie on side $BC$ such that $BP=CQ$ and $P$ lies between $B,Q$.The circumcircle of triangle $APQ$ intersects sides $AB,AC$ at $E,F$ respectively.The point $T$ is the intersection of $EP,FQ$.Two lines passing through the midpoint of $BC$ and parallel to $AB$ and $AC$, intersect $EP$ and $FQ$ at points $X,Y$ respectively.

Prove that the circumcircle of triangle $TXY$ and triangle $APQ$ are tangent to each other.



Proposed by Iman Maghsoudi
2017 Iran Team Selection Test
In the unit squares of a transparent $1 \times 100$ tape, numbers $1,2,\cdots,100$ are written in the ascending order.We fold this tape on it's lines with arbitrary order and arbitrary directions until we reach a $1 \times1$ tape with $100$ layers.A permutation of the numbers $1,2,\cdots,100$ can be seen on the tape, from the top to the bottom.

Prove that the number of possible permutations is between $2^{100}$ and $4^{100}$.

(e.g. We can produce all permutations of numbers $1,2,3$ with a $1\times3$ tape)



Proposed by Morteza Saghafian
2017 Iran Team Selection Test
$ABCD$ is a trapezoid with $AB \parallel CD$. The diagonals intersect at $P$. Let $\omega _1$ be a circle passing through $B$ and tangent to $AC$ at $A$. Let $\omega _2$ be a circle passing through $C$ and tangent to $BD$ at $D$. $\omega _3$ is the circumcircle of triangle $BPC$.

Prove that the common chord of circles $\omega _1,\omega _3$ and the common chord of circles $\omega _2, \omega _3$ intersect each other on $AD$.



Proposed by Kasra Ahmadi
2017 Iran Team Selection Test
Find the largest number $n$ that for which there exists $n$ positive integers such that non of them divides another one, but between every three of them, one divides the sum of the other two.



Proposed by Morteza Saghafian
2017 Iran Team Selection Test
There are $27$ cards, each has some amount of ($1$ or $2$ or $3$) shapes (a circle, a square or a triangle) with some color (white, grey or black) on them. We call a triple of cards a match such that all of them have the same amount of shapes or distinct amount of shapes, have the same shape or distinct shapes and have the same color or distinct colors. For instance, three cards shown in the figure are a match be cause they have distinct amount of shapes, distinct shapes but the same color of shapes.

What is the maximum number of cards that we can choose such that non of the triples make a match?



Proposed by Amin Bahjati
2017 Iran Team Selection Test
A $n+1$-tuple $\left(h_1,h_2, \cdots, h_{n+1}\right)$ where $h_i\left(x_1,x_2, \cdots , x_n\right)$ are $n$ variable polynomials with real coefficients is called good if the following condition holds:

For any $n$ functions $f_1,f_2, \cdots ,f_n : \mathbb R \to \mathbb R$ if for all $1 \le i \le n+1$, $P_i(x)=h_i \left(f_1(x),f_2(x), \cdots, f_n(x) \right)$ is a polynomial with variable $x$, then $f_1(x),f_2(x), \cdots, f_n(x)$ are polynomials.



$a)$ Prove that for all positive integers $n$, there exists a good $n+1$-tuple $\left(h_1,h_2, \cdots, h_{n+1}\right)$ such that the degree of all $h_i$ is more than $1$.



$b)$ Prove that there doesn't exist any integer $n>1$ that for which there is a good $n+1$-tuple $\left(h_1,h_2, \cdots, h_{n+1}\right)$ such that all $h_i$ are symmetric polynomials.



Proposed by Alireza Shavali
2017 Iran Team Selection Test
$k,n$ are two arbitrary positive integers. Prove that there exists at least $(k-1)(n-k+1)$ positive integers that can be produced by $n$ number of $k$'s and using only $+,-,\times, \div$ operations and adding parentheses between them, but cannot be produced using $n-1$ number of $k$'s.



Proposed by Aryan Tajmir
2017 Iran Team Selection Test
Let $k>1$ be an integer. The sequence $a_1,a_2, \cdots$ is defined as:

$a_1=1, a_2=k$ and for all $n>1$ we have: $a_{n+1}-(k+1)a_n+a_{n-1}=0$

Find all positive integers $n$ such that $a_n$ is a power of $k$.



Proposed by Amirhossein Pooya
2017 Iran Team Selection Test
Let $n>1$ be an integer. Prove that there exists an integer $n-1 \ge m \ge \left \lfloor \frac{n}{2} \right \rfloor$ such that the following equation has integer solutions with $a_m>0:$

$$\frac{a_{m}}{m+1}+\frac{a_{m+1}}{m+2}+ \cdots + \frac{a_{n-1}}{n}=\frac{1}{\textrm{lcm}\left ( 1,2, \cdots , n \right )}$$

Proposed by Navid Safaei
2017 Iran Team Selection Test
Let $P$ be a point in the interior of quadrilateral $ABCD$ such that:

$$\angle BPC=2\angle BAC \ \ ,\ \ \angle PCA = \angle PAD \ \ ,\ \ \angle PDA=\angle PAC$$Prove that:

$$\angle PBD= \left | \angle BCA - \angle PCA \right |$$

Proposed by Ali Zamani
2017 Iran Team Selection Test
Find all functions $f: \mathbb {R}^+ \times \mathbb {R}^+ \to \mathbb {R}^+$ that satisfy the following conditions for all positive real numbers $x,y,z:$

$$f\left ( f(x,y),z \right )=x^2y^2f(x,z)$$$$f\left ( x,1+f(x,y) \right ) \ge x^2 + xyf(x,x)$$

Proposed by Mojtaba Zare, Ali Daei Nabi
2017 Iran Team Selection Test
There are $6$ points on the plane such that no three of them are collinear. It's known that between every $4$ points of them, there exists a point that it's power with respect to the circle passing through the other three points is a constant value $k$.(Power of a point in the interior of a circle has a negative value.)

Prove that $k=0$ and all $6$ points lie on a circle.



Proposed by Morteza Saghafian
2017 Iran Team Selection Test
Let $\left \{ c_i \right \}_{i=0}^{\infty}$ be a sequence of non-negative real numbers with $c_{2017}>0$. A sequence of polynomials is defined as

$$P_{-1}(x)=0 \ ,  \ P_0(x)=1 \ ,  \ P_{n+1}(x)=xP_n(x)+c_nP_{n-1}(x).$$Prove that there doesn't exist any integer $n>2017$ and some real number $c$ such that

$$P_{2n}(x)=P_n(x^2+c).$$

Proposed by Navid Safaei
2017 Iran Team Selection Test
In triangle $ABC$ let $O$ and $H$ be the circumcenter and the orthocenter. The point $P$ is the reflection of $A$ with respect to $OH$. Assume that $P$ is not on the same side of $BC$ as $A$. Points $E,F$ lie on $AB,AC$ respectively such that $BE=PC \ ,  CF=PB$. Let $K$ be the intersection point of $AP,OH$. Prove that $\angle EKF = 90 ^{\circ}$



 Proposed by Iman Maghsoudi
2016 Iran Team Selection Test
Let $ABCD$ be a convex quadrilateral, and let $P$, $Q$, $R$, and $S$ be points on the sides $AB$, $BC$, $CD$, and $DA$, respectively. Let the line segment $PR$ and $QS$ meet at $O$. Suppose that each of the quadrilaterals $APOS$, $BQOP$, $CROQ$, and $DSOR$ has an incircle. Prove that the lines $AC$, $PQ$, and $RS$ are either concurrent or parallel to each other.
2016 Iran Team Selection Test
Let $n$ be a fixed positive integer. Find the maximum possible value of $$ \sum_{1 \le r < s \le 2n} (s-r-n)x_rx_s, $$where $-1 \le x_i \le 1$ for all $i = 1, \cdots , 2n$.
2016 Iran Team Selection Test
In a company of people some pairs are enemies. A group of people is called unsociable if the number of members in the group is odd and at least $3$, and it is possible to arrange all its members around a round table so that every two neighbors are enemies. Given that there are at most $2015$ unsociable groups, prove that it is possible to partition the company into $11$ parts so that no two enemies are in the same part.



Proposed by Russia
2016 Iran Team Selection Test
Let $a,b,c,d$ be positive real numbers such that $\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}+\frac{1}{d+1}=2$. Prove that $$\sum_{cyc} \sqrt{\frac{a^2+1}{2}} \geq (3.\sum_{cyc} \sqrt{a}) -8$$
2016 Iran Team Selection Test
Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k \le n$. The rules of the game are:



(i) A player cannot choose a number that has been chosen by either player on any previous turn.

(ii) A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn.

(iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game.



The player $A$ takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies.



Proposed by Finland
2016 Iran Team Selection Test
Let $ABC$ be a triangle with $CA \neq CB$. Let $D$, $F$, and $G$ be the midpoints of the sides $AB$, $AC$, and $BC$ respectively. A circle $\Gamma$ passing through $C$ and tangent to $AB$ at $D$ meets the segments $AF$ and $BG$ at $H$ and $I$, respectively. The points $H'$ and $I'$ are symmetric to $H$ and $I$ about $F$ and $G$, respectively. The line $H'I'$ meets $CD$ and $FG$ at $Q$ and $M$, respectively. The line $CM$ meets $\Gamma$ again at $P$. Prove that $CQ = QP$.



Proposed by El Salvador
2016 Iran Team Selection Test
Let $\mathbb{Z}_{>0}$ denote the set of positive integers. For any positive integer $k$, a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is called $k$-good if $\gcd(f(m) + n, f(n) + m) \le k$ for all $m \neq n$. Find all $k$ such that there exists a $k$-good function.



Proposed by James Rickards, Canada
2016 Iran Team Selection Test
Let $ABC$ be an arbitrary triangle and $O$ is the circumcenter of $\triangle {ABC}$.Points $X,Y$ lie on $AB,AC$,respectively such that the reflection of $BC$ WRT $XY$ is tangent to circumcircle of $\triangle {AXY}$.Prove that the circumcircle of triangle $AXY$ is tangent to circumcircle of triangle $BOC$.
2016 Iran Team Selection Test
Let $p \neq 13$ be a prime number of the form $8k+5$ such that $39$ is a quadratic non-residue modulo $p$. Prove that the equation $$x_1^4+x_2^4+x_3^4+x_4^4 \equiv 0 \pmod p$$has a solution in integers such that $p\nmid x_1x_2x_3x_4$.