Navvye/Olympiad-Problems-CSV huggingface dataset:
2018 Turkey Team Selection Test
For integers $a, b$, call the lattice point with coordinates $(a,b)$ basic if $gcd(a,b)=1$. A graph takes the basic points as vertices and the edges are drawn in such way: There is an edge between $(a_1,b_1)$ and $(a_2,b_2)$ if and only if $2a_1=2a_2\in \{b_1-b_2, b_2-b_1\}$ or $2b_1=2b_2\in\{a_1-a_2, a_2-a_1\}$. Some of the edges will be erased, such that the remaining graph is a forest. At least how many edges must be erased to obtain this forest? At least how many trees exist in such a forest?
2018 Turkey Team Selection Test
For integers $m\geq 3$, $n$ and $x_1,x_2, \ldots , x_m$ if $x_{i+1}-x_i \equiv x_i-x_{i-1} (mod n) $ for every $2\leq i \leq m-1$, we say that the $m$-tuple $(x_1,x_2,\ldots , x_m)$ is an arithmetic sequence in $(mod n)$. Let $p\geq 5$ be a prime number and $1<a<p-1$ be an integer. Let ${a_1,a_2,\ldots , a_k}$ be the set of all possible remainders when positive powers of $a$ are divided by $p$. Show that if a permutation of ${a_1,a_2,\ldots , a_k}$ is an arithmetic sequence in $(mod p)$, then $k=p-1$.
2018 Turkey Team Selection Test
For a triangle $T$ and a line $d$, if the feet of perpendicular lines from a point in the plane to the edges of $T$ all lie on $d$, say $d$ focuses $T$. If the set of lines focusing $T_1$ and the set of lines focusing $T_2$ are the same, say $T_1$ and $T_2$ are equivalent. Prove that, for any triangle in the plane, there exists exactly one equilateral triangle which is equivalent to it.
2017 Turkey Team Selection Test
$m, n $ are positive integers and $p$ is a prime number. Find all triples $(m, n, p)$ satisfying $(m^3+n)(n^3+m)=p^3$
2017 Turkey Team Selection Test
There are two-way flights between some of the $2017$ cities in a country, such that given two cities, it is possible to reach  one from the other. No matter how the flights are appointed, one can define $k$ cities as "special city", so that there is a direct flight from each city to at least one "special city". Find the minimum value of $k$.
2017 Turkey Team Selection Test
At the $ABC$ triangle the midpoints of $BC, AC, AB$ are respectively $D, E, F$ and the triangle tangent to the incircle at $G$, $H$ and $I$ in the same order.The midpoint of $AD$ is $J$. $BJ$ and $AG$ intersect at point $K$. The $C-$centered circle passing through $A$ cuts the $[CB$ ray at point $X$. The line passing through $K$ and parallel to the $BC$ and $AX$ meet at $U$. $IU$ and $BC$ intersect at the $P$ point. There is $Y$ point chosen at incircle. $PY$ is tangent to incircle at point $Y$. Prove that $D, E, F, Y$ are cyclic.
2017 Turkey Team Selection Test
Each two of $n$ students, who attended an activity, have different ages. It is given that each student shook hands with at least one student, who did not shake hands with anyone younger than the other. Find all possible values of $n$.
2017 Turkey Team Selection Test
For all positive real numbers $a,b,c$ with $a+b+c=3$, show that

$$a^3b+b^3c+c^3a+9\geq 4(ab+bc+ca).$$
2017 Turkey Team Selection Test
Prove that no pair of different positive integers $(m, n)$ exist, such that $\frac{4m^{2}n^{2}-1}{(m^{2}-n^2)^{2}}$ is an integer.
2017 Turkey Team Selection Test
Let $a$ be a real number. Find the number of functions $f:\mathbb{R}\rightarrow \mathbb{R}$  depending on $a$, such that $f(xy+f(y))=f(x)y+a$ holds for every $x, y\in \mathbb{R}$.
2017 Turkey Team Selection Test
In a triangle $ABC$ the bisectors through vertices $B$ and $C$ meet the sides $\left [ AC \right ]$ and $\left [ AB \right ]$ at $D$ and $E$ respectively. Let $I_{c}$ be the center of the excircle which is tangent to the side $\left [ AB \right ]$ and $F$ the midpoint of $\left [ BI_{c} \right ]$. If $\left | CF \right |^2=\left | CE \right |^2+\left | DF \right |^2$, show that $ABC$ is an equilateral triangle.
2017 Turkey Team Selection Test
Let $S$ be a set of finite number of points in the plane any 3 of which are not linear and any 4 of which are not concyclic. A coloring of all the points in $S$ to red and white is called discrete coloring if there exists a circle which encloses all red points and excludes all white points. Determine the number of discrete colorings for each set $S$.
2016 Turkey Team Selection Test
In an acute triangle $ABC$, a point $P$ is taken on the $A$-altitude. Lines $BP$ and $CP$ intersect the sides $AC$ and $AB$ at points $D$ and $E$, respectively. Tangents drawn from points $D$ and $E$ to the circumcircle of triangle $BPC$ are tangent to it at points $K$ and $L$, respectively, which are in the interior of triangle $ABC$. Line $KD$ intersects the circumcircle of triangle $AKC$ at point $M$ for the second time, and line $LE$ intersects the circumcircle of triangle $ALB$ at point $N$ for the second time. Prove that$$ \frac{KD}{MD}=\frac{LE}{NE} \iff \text{Point P is the orthocenter of triangle ABC}$$
2016 Turkey Team Selection Test
In a class with $23$ students, each pair of students have watched a movie together. Let the set of movies watched by a student be his movie collection. If every student has watched every movie at most once, at least how many different movie collections can these students have?
2016 Turkey Team Selection Test
A sequence of real numbers $a_0, a_1, \dots$ satisfies the condition$$\sum\limits_{n=0}^{m}a_n\cdot(-1)^n\cdot\dbinom{m}{n}=0$$for all large enough positive integers $m$. Prove that there exists a polynomial $P$ such that $a_n=P(n)$ for all $n\ge0$.
2016 Turkey Team Selection Test
Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that for all $m,n \in \mathbb{N}$ holds $f(mn)=f(m)f(n)$ and $m+n \mid f(m)+f(n)$ .
2016 Turkey Team Selection Test
In a triangle $ABC$ with $AB=AC$, let $D$ be the midpoint of $[BC]$. A line passing through $D$ intersects $AB$ at $K$, $AC$ at $L$. A point $E$ on $[BC]$ different from $D$, and a point $P$ on $AE$ is taken such that $\angle KPL=90^\circ-\frac{1}{2}\angle KAL$ and $E$ lies between $A$ and $P$. The circumcircle of triangle $PDE$ intersects $PK$ at point $X$, $PL$ at point $Y$ for the second time. Lines $DX$ and $AB$ intersect at $M$, and lines $DY$ and $AC$ intersect at $N$. Prove that the points $P,M,A,N$ are concyclic.
2016 Turkey Team Selection Test
$A_1, A_2,\dots A_k$ are different subsets of the set $\{1,2,\dots ,2016\}$. If $A_i\cap A_j$ forms an arithmetic sequence for all $1\le i <j\le k$, what is the maximum value of $k$?
2016 Turkey Team Selection Test
All angles of the convex $n$-gon $A_1A_2\dots A_n$ are obtuse, where $n\ge5$. For all $1\le i\le n$, $O_i$ is the circumcenter of triangle $A_{i-1}A_iA_{i+1}$ (where $A_0=A_n$ and $A_{n+1}=A_1$). Prove that the closed path $O_1O_2\dots O_n$ doesn't form a convex $n$-gon.
2016 Turkey Team Selection Test
$p$ is a prime. Let $K_p$ be the set of all polynomials with coefficients from the set $\{0,1,\dots ,p-1\}$ and degree less than $p$. Assume that for all pairs of polynomials $P,Q\in K_p$ such that $P(Q(n))\equiv n\pmod p$ for all integers $n$, the degrees of $P$ and $Q$ are equal. Determine all primes $p$ with this condition.
2015 Turkey Team Selection Test
Let $l, m, n$ be positive integers and $p$ be prime. If $p^{2l-1}m(mn+1)^2 + m^2$ is a perfect square, prove that $m$ is also a perfect square.
2015 Turkey Team Selection Test
There are $2015$ points on a plane and no two distances between them are equal. We call the closest $22$ points to a point its $neighbours$. If $k$ points share the same neighbour, what is the maximum value of $k$?
2015 Turkey Team Selection Test
Let $m, n$ be positive integers. Let $S(n,m)$ be  the number of sequences of length $n$ and consisting of $0$ and $1$ in which there exists a $0$ in any consecutive $m$ digits. Prove that





$$S(2015n,n).S(2015m,m)\ge S(2015n,m).S(2015m,n)$$
2015 Turkey Team Selection Test
Let $ABC$ be a triangle such that $|AB|=|AC|$ and let $D,E$ be points on the minor arcs $\overarc{AB}$ and $\overarc{AC}$ respectively. The lines $AD$ and $BC$ intersect at $F$ and the line $AE$ intersects the circumcircle of $\triangle FDE$ a second time at $G$. Prove that the line $AC$ is tangent to the circumcircle of $\triangle ECG$.
2015 Turkey Team Selection Test
We are going to colour the cells of a $2015 \times 2015$ board such that there are none of the following:

$1)$ Three cells with the same colour where two of them are in the same column, and the third is in the same row and to the right of the upper cell,

$2)$ Three cells with the same colour where two of them are in the same column, and the third is in the same row and to the left of the lower cell.

What is the minimum number of colours $k$ required to make such a colouring possible?
2015 Turkey Team Selection Test
Prove that there are infinitely many positive integers $n$ such that $(n!)^{n+2015}$ divides $(n^{2})!$.
2015 Turkey Team Selection Test
Find all the functions $f:R\to R$ such that $$f(x^2) + 4y^2f(y) = (f(x-y) + y^2)(f(x+y) + f(y))$$ for every real $x,y$.
2015 Turkey Team Selection Test
Let $ABC$ be a triangle with incenter $I$ and circumcenter $O$ such that $|AC|>|BC|>|AB|$ and the incircle touches the sides $BC, CA, AB$ at $D, E, F$ respectively. Let the reflection of $A$ with respect to $F$ and $E$ be $F_1$ and $E_1$ respectively. The circle tangent to $BC$ at $D$ and passing through $F_1$ intersects $AB$ a second time at $F_2$ and the circle tangent to $BC$ at $D$ and passing through $E_1$ intersects $AC$ a second time at $E_2$. The midpoints of the segments $|OE|$ and $|IF|$ are $P$ and $Q$ respectively. Prove that $$|AB| + |AC| = 2|BC| \iff PQ\perp E_2F_2 $$.
2015 Turkey Team Selection Test
In a country with $2015$ cities there is exactly one two-way flight between each city. The three flights made between three cities belong to at most two different airline companies. No matter how the flights are shared between some number of companies, if there is always a city in which $k$ flights belong to the same airline, what is the maximum value of $k$?
2014 Turkey Team Selection Test
Find the number of $(a_1,a_2, ... ,a_{2014})$ permutations of the $(1,2, . . . ,2014)$ such that, for all $1\leq i<j\leq2014$, $i+a_i \leq j+a_j$.
2014 Turkey Team Selection Test
Find all $f$ functions from real numbers to itself such that for all real numbers $x,y$ the equation

$$f(f(y)+x^2+1)+2x=y+(f(x+1))^2$$

holds.
2014 Turkey Team Selection Test
Let $r,R$ and $r_a$ be the radii of the incircle, circumcircle and A-excircle of the triangle $ABC$ with $AC>AB$, respectively. $I,O$ and $J_A$ are the centers of these circles, respectively. Let incircle touches the $BC$ at $D$, for a point $E \in (BD)$ the condition $A(IEJ_A)=2A(IEO)$ holds.

Prove that

$$ED=AC-AB \iff R=2r+r_a.$$
2014 Turkey Team Selection Test
Find all pairs  $(m,n)$ of positive odd integers, such that $n \mid 3m+1$ and $m \mid n^2+3$.
2014 Turkey Team Selection Test
A circle $\omega$ cuts the sides $BC,CA,AB$ of the triangle $ABC$ at $A_1$ and $A_2$; $B_1$ and $B_2$; $C_1$ and $C_2$, respectively. Let $P$ be the center of $\omega$. $A'$ is the circumcenter of the triangle $A_1A_2P$, $B'$ is the circumcenter of the triangle $B_1B_2P$, $C'$ is the circumcenter of the triangle $C_1C_2P$. Prove that $AA', BB'$ and $CC'$ concur.
2014 Turkey Team Selection Test
Prove that for all all non-negative real numbers $a,b,c$ with $a^2+b^2+c^2=1$

$$\sqrt{a+b}+\sqrt{a+c}+\sqrt{b+c} \geq 5abc+2.$$
2014 Turkey Team Selection Test
Let $P$ be a point inside the acute triangle $ABC$ with $m(\widehat{PAC})=m(\widehat{PCB})$. $D$ is the midpoint of the segment $PC$. $AP$ and $BC$ intersect at $E$, and $BP$ and $DE$ intersect at $Q$. Prove that $\sin\widehat{BCQ}=\sin\widehat{BAP}$.
2014 Turkey Team Selection Test
$a_1=-5$, $a_2=-6$ and for all $n \geq 2$ the ${(a_n)^\infty}_{n=1}$ sequence defined as,

$$a_{n+1}=a_n+(a_1+1)(2a_2+1)(3a_3+1)\cdots((n-1)a_{n-1}+1)((n^2+n)a_n+2n+1)).$$

If a prime $p$ divides $na_n+1$ for a natural number n, prove that there is a integer $m$ such that $m^2\equiv5(modp)$
2014 Turkey Team Selection Test
At the bottom-left corner of a $2014\times 2014$ chessboard, there are some green worms and at the top-left corner of the same chessboard, there are some brown worms. Green worms can  move only to right and up, and brown worms can move only to right and down. After a while, the worms make some moves and all of the unit squares of the chessboard become occupied at least once throughout this process. Find the minimum total number of the worms.
2013 Turkey Team Selection Test
Let $\phi(n)$ be the number of positive integers less than $n$ that are relatively prime to $n$, where $n$ is a positive integer. Find all pairs of positive integers $(m,n)$ such that $$2^n + (n-\phi(n)-1)! = n^m+1.$$
2013 Turkey Team Selection Test
We put pebbles on some unit squares of a $2013 \times 2013$ chessboard such that every unit square contains at most one pebble. Determine the minimum number of pebbles on the chessboard, if each $19\times 19$ square formed by unit squares contains at least $21$ pebbles.
2013 Turkey Team Selection Test
Let $O$ be the circumcenter and $I$ be the incenter of an acute triangle $ABC$ with $m(\widehat{B}) \neq m(\widehat{C})$. Let $D$, $E$, $F$ be the midpoints of the sides $[BC]$, $[CA]$, $[AB]$, respectively. Let $T$ be the foot of perpendicular from $I$ to $[AB]$. Let $P$ be the circumcenter of the triangle $DEF$ and $Q$ be the midpoint of $[OI]$. If $A$, $P$, $Q$ are collinear, prove that $$\dfrac{|AO|}{|OD|}-\dfrac{|BC|}{|AT|}=4.$$
2013 Turkey Team Selection Test
Let the incircle of the triangle $ABC$ touch $[BC]$ at $D$ and $I$ be the incenter of the triangle. Let $T$ be midpoint of $[ID]$. Let the perpendicular from $I$ to $AD$ meet $AB$ and $AC$ at $K$ and $L$, respectively. Let the perpendicular from $T$ to $AD$ meet $AB$ and $AC$ at $M$ and $N$, respectively. Show that $|KM|\cdot |LN|=|BM|\cdot|CN|$.
2013 Turkey Team Selection Test
For all real numbers $x,y,z$ such that $-2\leq x,y,z \leq 2$ and $x^2+y^2+z^2+xyz = 4$, determine the least real number $K$ satisfying $$\dfrac{z(xz+yz+y)}{xy+y^2+z^2+1} \leq K.$$
2013 Turkey Team Selection Test
Let $E$ be intersection of the diagonals of convex quadrilateral $ABCD$. It is given that  $m(\widehat{EDC}) = m(\widehat{DEC})=m(\widehat{BAD})$. If $F$ is a point on $[BC]$ such that $m(\widehat{BAF}) + m(\widehat{EBF})=m(\widehat{BFE})$, show that $A$, $B$, $F$, $D$ are concyclic.
2013 Turkey Team Selection Test
Determine all functions $f:\mathbf{R} \rightarrow \mathbf{R}^+$ such that for all real numbers $x,y$ the following conditions hold:



$\begin{array}{rl}

i. & f(x^2) = f(x)^2 -2xf(x) \\
ii. & f(-x) = f(x-1)\\
iii. & 1<x<y \Longrightarrow f(x) < f(y).

\end{array}$
2013 Turkey Team Selection Test
Some cities of a country consisting of $n$ cities are connected by round trip flights so that there are at least $k$ flights from any city and any city is reachable from any city. Prove that for any such flight organization these flights can be distributed among $n-k$ air companies so that one can reach any city from any city by using of at most one flight of each air company.
2012 Turkey Team Selection Test
Let $A=\{1,2,\ldots,2012\}, \: B=\{1,2,\ldots,19\}$ and $S$ be the set of all subsets of $A.$ Find the number of functions $f : S\to B$ satisfying $f(A_1\cap A_2)=\min\{f(A_1),f(A_2)\}$ for all $A_1, A_2 \in S.$
2012 Turkey Team Selection Test
In an acute triangle $ABC,$ let $D$ be a point on the side $BC.$ Let $M_1, M_2, M_3, M_4, M_5$ be the midpoints of the line segments $AD, AB, AC, BD, CD,$ respectively and $O_1, O_2, O_3, O_4$ be the circumcenters of triangles $ABD, ACD, M_1M_2M_4, M_1M_3M_5,$ respectively. If $S$ and $T$ are midpoints of the line segments $AO_1$ and $AO_2,$ respectively, prove that $SO_3O_4T$ is an isosceles trapezoid.
2012 Turkey Team Selection Test
For all positive real numbers $a, b, c$ satisfying $ab+bc+ca \leq 1,$ prove that

$$ a+b+c+\sqrt{3} \geq 8abc \left(\frac{1}{a^2+1}+\frac{1}{b^2+1}+\frac{1}{c^2+1}\right) $$
2012 Turkey Team Selection Test
In a triangle $ABC,$ incircle touches the sides $BC, CA, AB$ at $D, E, F,$ respectively. A circle $\omega$ passing through $A$ and tangent to line $BC$ at $D$ intersects the line segments $BF$ and $CE$ at $K$ and $L,$ respectively. The line passing through $E$ and parallel to $DL$ intersects the line passing through $F$ and parallel to $DK$ at $P.$ If $R_1, R_2, R_3, R_4$ denotes the circumradius of the triangles $AFD, AED, FPD, EPD,$ respectively, prove that $R_1R_4=R_2R_3.$
2012 Turkey Team Selection Test
A positive integer $n$ is called good if for all positive integers $a$ which can be written as $a=n^2 \sum_{i=1}^n {x_i}^2$ where $x_1, x_2, \ldots ,x_n$ are integers, it is possible to express $a$ as $a=\sum_{i=1}^n {y_i}^2$ where $y_1, y_2, \ldots, y_n$ are integers with none of them is divisible by $n.$ Find all good numbers.
2012 Turkey Team Selection Test
Two players $A$ and $B$ play a game on a $1\times m$ board, using $2012$ pieces numbered from $1$ to $2012.$ At each turn, $A$ chooses a piece and $B$ places it to an empty place. After $k$ turns, if all pieces are placed on the board increasingly, then $B$ wins, otherwise $A$ wins. For which values of $(m,k)$ pairs can $B$ guarantee to win?
2012 Turkey Team Selection Test
Let $S_r(n)=1^r+2^r+\cdots+n^r$ where $n$ is a positive integer and $r$ is a rational number. If $S_a(n)=(S_b(n))^c$ for all positive integers $n$ where $a, b$ are positive rationals and $c$ is positive integer then we call $(a,b,c)$ as nice triple. Find all nice triples.
2012 Turkey Team Selection Test
In a plane, the six different points $A, B, C, A', B', C'$ are given such that triangles $ABC$ and $A'B'C'$ are congruent, i.e. $AB=A'B', BC=B'C', CA=C'A'.$ Let $G$ be the centroid of $ABC$ and $A_1$ be an intersection point of the circle with diameter $AA'$ and the circle with center $A'$ and passing through $G.$ Define $B_1$ and $C_1$ similarly. Prove that

$$ AA_1^2+BB_1^2+CC_1^2 \leq AB^2+BC^2+CA^2 $$
2012 Turkey Team Selection Test
Let $\mathbb{Z^+}$ and $\mathbb{P}$ denote the set of positive integers and the set of prime numbers, respectively. A set $A$ is called $S-\text{proper}$ where $A, S \subset \mathbb{Z^+}$ if there exists a positive integer $N$ such that for all $a \in A$ and for all $0 \leq b <a$ there exist $s_1, s_2, \ldots, s_n \in S$ satisfying $ b \equiv s_1+s_2+\cdots+s_n \pmod a$ and $1 \leq n \leq N.$

Find a subset $S$ of $\mathbb{Z^+}$ for which $\mathbb{P}$ is $S-\text{proper}$ but $\mathbb{Z^+}$ is not.
2011 Turkey Team Selection Test
Let $\mathbb{Q^+}$ denote the set of positive rational numbers. Determine all functions $f: \mathbb{Q^+} \to \mathbb{Q^+}$ that satisfy the conditions



$$ f \left( \frac{x}{x+1}\right) = \frac{f(x)}{x+1}  \qquad \text{and} \qquad f \left(\frac{1}{x}\right)=\frac{f(x)}{x^3}$$



for all $x \in \mathbb{Q^+}.$
2011 Turkey Team Selection Test
Let $I$ be the incenter and $AD$ be a diameter of the circumcircle of a triangle $ABC.$ If the point $E$ on the ray $BA$ and the point $F$ on the ray $CA$ satisfy the condition

$$BE=CF=\frac{AB+BC+CA}{2}$$

show that the lines $EF$ and $DI$ are perpendicular.
2011 Turkey Team Selection Test
Let $A$ and $B$ be sets with $2011^2$ and $2010$ elements, respectively. Show that there is a function $f:A \times A \to B$ satisfying the condition $f(x,y)=f(y,x)$ for all $(x,y) \in A \times A$ such that for every function $g:A \to B$ there exists $(a_1,a_2) \in A \times A$ with $g(a_1)=f(a_1,a_2)=g(a_2)$ and $a_1 \neq a_2.$
2011 Turkey Team Selection Test
Let $D$ be a point different from the vertices on the side $BC$ of a triangle $ABC.$ Let $I, \: I_1$ and $I_2$ be the incenters of the triangles $ABC, \: ABD$ and $ADC,$ respectively. Let $E$ be the second intersection point of the circumcircles of the triangles $AI_1I$ and $ADI_2,$ and $F$ be the second intersection point of the circumcircles of the triangles $AII_2$ and $AI_1D.$ Prove that if $AI_1=AI_2,$ then

$$ \frac{EI}{FI} \cdot \frac{ED}{FD}=\frac{{EI_1}^2}{{FI_1}^2}.$$
2011 Turkey Team Selection Test
Let $a,b,c$ be positive real numbers satisfying $a^2+b^2+c^2 \geq 3.$ Prove that



$$ \frac{(a+1)(b+2)}{(b+1)(b+5)} + \frac{(b+1)(c+2)}{(c+1)(c+5)}+\frac{(c+1)(a+2)}{(a+1)(a+5)} \geq \frac{3}{2} $$
2011 Turkey Team Selection Test
Let $t(n)$ be the sum of the digits in the binary representation of a positive integer $n,$ and let $k \geq 2$ be an integer.



a. Show that  there exists a sequence $(a_i)_{i=1}^{\infty}$ of integers such that $a_m \geq 3$ is an odd integer and $t(a_1a_2 \cdots a_m)=k$ for all $m \geq 1.$



b. Show that there is an integer $N$ such that $t(3 \cdot 5 \cdots (2m+1))>k$ for all integers $m \geq N.$
2011 Turkey Team Selection Test
Let $K$ be a point in the interior of an acute triangle $ABC$ and $ARBPCQ$ be a convex hexagon whose vertices lie on the circumcircle $\Gamma$ of the triangle $ABC.$ Let $A_1$ be the second point where the circle passing through $K$ and tangent to $\Gamma$ at $A$ intersects the line $AP.$ The points $B_1$ and $C_1$ are defined similarly. Prove that

$$ \min\left\{\frac{PA_1}{AA_1}, \: \frac{QB_1}{BB_1}, \: \frac{RC_1}{CC_1}\right\} \leq 1.$$
2011 Turkey Team Selection Test
Graphistan has $2011$ cities and Graph Air (GA) is running one-way flights between all pairs of these cities. Determine the maximum possible value of the integer $k$ such that no matter how these flights are arranged it is possible to travel between any two cities in Graphistan riding only GA flights as long as the absolute values of the difference between the number of flights originating and terminating at any city is not more than $k.$
2011 Turkey Team Selection Test
Let $p$ be a prime, $n$ be a positive integer, and let $\mathbb{Z}_{p^n}$ denote the set of congruence classes modulo $p^n.$ Determine the number of functions $f: \mathbb{Z}_{p^n} \to \mathbb{Z}_{p^n}$ satisfying the condition

$$ f(a)+f(b) \equiv f(a+b+pab) \pmod{p^n} $$

for all $a,b \in \mathbb{Z}_{p^n}.$
2010 Turkey Team Selection Test
$D, \: E , \: F$ are points on the sides $AB, \: BC, \: CA,$ respectively, of a triangle $ABC$ such that $AD=AF, \: BD=BE,$ and $DE=DF.$ Let $I$ be the incenter of the triangle $ABC,$ and let $K$ be the point of intersection of the line $BI$ and the tangent line through $A$ to the circumcircle of the triangle $ABI.$ Show that $AK=EK$ if $AK=AD.$
2010 Turkey Team Selection Test
Show that

$$ \sum_{cyc} \sqrt[4]{\frac{(a^2+b^2)(a^2-ab+b^2)}{2}} \leq \frac{2}{3}(a^2+b^2+c^2)\left(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right) $$

for all positive real numbers $a, \: b, \: c.$
2010 Turkey Team Selection Test
A teacher wants to divide the $2010$ questions she asked in the exams during the school year into three folders of $670$ questions and give each folder to a student who solved all $670$ questions in that folder. Determine the minimum number of students in the class that makes this possible for all possible situations in which there are at most two students who did not solve any given question.
2010 Turkey Team Selection Test
Let $0 \leq k < n$ be integers and $A=\{a \: : \: a \equiv k \pmod n \}.$ Find the smallest value of $n$ for which the expression

$$ \frac{a^m+3^m}{a^2-3a+1} $$

does not take any integer values for $(a,m) \in A  \times \mathbb{Z^+}.$
2010 Turkey Team Selection Test
For an interior point $D$ of a triangle $ABC,$ let $\Gamma_D$ denote the circle passing through the points $A, \: E, \: D, \: F$ if these points are concyclic where $BD \cap AC=\{E\}$ and $CD \cap AB=\{F\}.$ Show that all circles $\Gamma_D$ pass through a second common point different from $A$ as $D$ varies.
2010 Turkey Team Selection Test
Let $\Lambda$ be the set of points in the plane whose coordinates are integers and let $F$ be the collection of all functions from $\Lambda$ to $\{1,-1\}.$ We call a function $f$ in $F$ perfect if every function $g$ in $F$ that differs from $f$ at finitely many points satisfies the condition

$$ \sum_{0<d(P,Q)<2010} \frac{f(P)f(Q)-g(P)g(Q)}{d(P,Q)} \geq 0 $$

where $d(P,Q)$ denotes the distance between $P$ and $Q.$ Show that there exist infinitely many perfect functions that are not translates of each other.
2009 Turkey Team Selection Test
Find all $ f: Q^ + \to\ Z$ functions that satisfy $ f \left(\frac {1}{x} \right) = f(x)$ and $ (x + 1)f(x - 1) = xf(x)$ for all rational numbers that are bigger than 1.