Navvye/Olympiad-Problems-CSV huggingface dataset:
2021 Turkey MO (2nd round)
Initially, one of the two boxes on the table is empty and the other contains $29$ different colored marbles. By starting with the full box and performing moves in order, in each move, one or more marbles are selected from that box and transferred to the other box. At most, how many moves can be made without selecting the same set of marbles more than once?
2021 Turkey MO (2nd round)
If a polynomial with real coefficients of degree $d$ has at least $d$ coefficients equal to $1$ and has $d$ real roots, what is the maximum possible value of $d$?



(Note: The roots of the polynomial do not have to be different from each other.)
2021 Turkey MO (2nd round)
A circle $\Gamma$ is tangent to the side $BC$ of a triangle $ABC$ at $X$ and tangent to the side $AC$ at $Y$. A point $P$ is taken on the side $AB$. Let $XP$ and $YP$ intersect $\Gamma$ at $K$ and $L$ for the second time, $AK$ and $BL$ intersect $\Gamma$ at $R$ and $S$ for the second time. Prove that $XR$ and $YS$ intersect on $AB$.
2021 Turkey MO (2nd round)
Points $D$ and $E$ are taken on $[BC]$ and $[AC]$ of acute angled triangle $ABC$ such that $BD$ and $CE$ are angle bisectors. Projections of $D$ onto $BC$ and $BA$ are $P$ and $Q$, projections of $E$ onto $CA$ and $CB$ are $R$ and $S$. Let $AP \cap CQ=X$, $AS \cap BR=Y$ and $BX \cap CY=Z$. Show that $AZ \perp BC$.
2021 Turkey MO (2nd round)
There are finitely many primes dividing the numbers  $\{ a \cdot b^n + c\cdot d^n : n=1, 2,  3,... \}$ where $a, b, c, d$ are positive integers. Prove that $b=d$.
2021 Turkey MO (2nd round)
In a school, there are 2021 students, each having exactly $k$ friends. There aren't three students such that all three are friends with each other. What is the maximum possible value of $k$?
2020 Turkey MO (2nd round)
Let $n > 1$ be an integer and $X = \{1, 2, \cdots , n^2 \}$. If there exist $x, y$ such that $x^2\mid y$ in all subsets of $X$ with $k$ elements, find the least possible value of $k$.
2020 Turkey MO (2nd round)
Let $P$ be an interior point of acute triangle $\Delta ABC$, which is different from the orthocenter. Let $D$ and $E$ be the feet of altitudes from $A$ to $BP$ and $CP$, and let $F$ and $G$ be the feet of the altitudes from $P$ to sides $AB$ and $AC$. Denote by $X$ the midpoint of $[AP]$, and let the second intersection of the circumcircles of triangles $\Delta DFX$ and $\Delta EGX$ lie on $BC$. Prove that $AP$ is perpendicular to $BC$ or $\angle PBA = \angle PCA$.
2020 Turkey MO (2nd round)
If $x, y, z$ are positive real numbers find the minimum value of

$$2\sqrt{(x+y+z) \left( \frac{1}{x}+ \frac{1}{y} + \frac{1}{z} \right)} - \sqrt{ \left( 1+ \frac{x}{y} \right)  \left( 1+ \frac{y}{z} \right)}$$
2020 Turkey MO (2nd round)
Let $p$ be a prime number such that $\frac{28^p-1}{2p^2+2p+1}$ is an integer. Find all possible values of number of divisors of $2p^2+2p+1$.
2020 Turkey MO (2nd round)
Find all polynomials with real coefficients such that one can find an integer valued series $a_0, a_1, \dots$ satisfying  $\lfloor P(x) \rfloor = a_{ \lfloor x^2 \rfloor}$ for all $x$ real numbers.
2020 Turkey MO (2nd round)
$2021$ points are given on a circle. Each point is colored by one of the $1,2, \cdots ,k$ colors. For all points and colors $1\leq r \leq k$, there exist an arc such that at least half of the points on it are colored with $r$. Find the maximum possible value of $k$.
2019 Turkey MO (2nd round)
$a, b, c$ are positive real numbers such that $$(\sqrt {ab}-1)(\sqrt {bc}-1)(\sqrt {ca}-1)=1$$At most, how many of the numbers: $$a-\frac {b}{c}, a-\frac {c}{b}, b-\frac {a}{c}, b-\frac {c}{a}, c-\frac {a}{b}, c-\frac {b}{a}$$can be bigger than $1$?
2019 Turkey MO (2nd round)
Let $d(n)$ denote the number of divisors of a positive integer $n$. If $k$ is a given odd number, prove that there exist an increasing arithmetic progression in positive integers $(a_1,a_2,\ldots a_{2019}) $ such that  $gcd(k,d(a_1)d(a_2)\ldots d(a_{2019})) =1$
2019 Turkey MO (2nd round)
There are 2019 students in a school, and some of these students are members of different student clubs. Each student club has an advisory board consisting of 12 students who are members of that particular club. An {\em advisory meeting} (for a particular club) can be realized only when each participant is a member of that club, and moreover, each of the 12 students forming the advisory board are present among the participants. It is known that each subset of at least 12 students in this school  can realize an advisory meeting for exactly one student club. Determine all possible numbers of different student clubs with exactly 27 members.
2019 Turkey MO (2nd round)
In a triangle $\Delta ABC$, $|AB|=|AC|$. Let $M$ be on the minor arc $AC$ of the circumcircle of $\Delta ABC$ different than $A$ and $C$. Let $BM$ and $AC$ meet at $E$ and the bisector of $\angle BMC$ and $BC$ meet at $F$ such that $\angle AFB=\angle CFE$. Prove that the triangle $\Delta ABC$ is equilateral.
2019 Turkey MO (2nd round)
Let $f:\{1,2,\dots,2019\}\to\{-1,1\}$ be a function, such that for every $k\in\{1,2,\dots,2019\}$, there exists an $\ell\in\{1,2,\dots,2019\}$ such that

$$

\sum_{i\in\mathbb{Z}:(\ell-i)(i-k)\geqslant 0} f(i)\leqslant 0.

$$Determine the maximum possible value of

$$

\sum_{i\in\mathbb{Z}:1\leqslant i\leqslant 2019} f(i).

$$
2019 Turkey MO (2nd round)
Given an integer $n>2$ and an integer $a$, if there exists an integer $d$ such that $n\mid a^d-1$ and $n\nmid a^{d-1}+\cdots+1$, we say $a$ is $n-$separating. Given any n>2, let the defect of $n$ be defined as the number of integers $a$ such that $0<a<n$, $(a,n)=1$, and $a$ is not  $n-$separating. Determine all integers $n>2$ whose defect is equal to the smallest possible value.
2018 Turkey MO (2nd Round)
Find all pairs $(x,y)$ of real numbers that satisfy,

\begin{align*}

    x^2+y^2+x+y &= xy(x+y)-\frac{10}{27}\\
    |xy| & \leq \frac{25}{9}.

\end{align*}
2018 Turkey MO (2nd Round)
Let $P$ be a point in the interior of the triangle $ABC$. The lines $AP$, $BP$, and $CP$ intersect the sides $BC$, $CA$, and $AB$ at $D,E$, and $F$, respectively. A point $Q$ is taken on the ray $[BE$ such that $E\in [BQ]$ and $m(\widehat{EDQ})=m(\widehat{BDF})$. If $BE$ and $AD$ are perpendicular, and $|DQ|=2|BD|$, prove that $m(\widehat{FDE})=60^\circ$.
2018 Turkey MO (2nd Round)
A sequence $a_1,a_2,\dots$ satisfy

$$

\sum_{i =1}^n a_{\lfloor \frac{n}{i}\rfloor }=n^{10},

$$for every $n\in\mathbb{N}$.

Let $c$ be a positive integer. Prove that, for every positive integer $n$,

$$

\frac{c^{a_n}-c^{a_{n-1}}}{n}

$$is an integer.
2018 Turkey MO (2nd Round)
In a triangle $ABC$, the bisector of the angle $A$ intersects the excircle that is tangential to side $[BC]$ at two points $D$ and $E$ such that $D\in [AE]$. Prove that,

$$

\frac{|AD|}{|AE|}\leq \frac{|BC|^2}{|DE|^2}.

$$
2018 Turkey MO (2nd Round)
Let $a_1,a_2,a_3,a_4$ be positive integers, with the property that it is impossible to assign them around a circle where all the neighbors are coprime. Let $i,j,k\in\{1,2,3,4\}$ with $i \neq j$, $j\neq k$, and $k\neq i $. Determine the maximum number of triples $(i,j,k)$ for which

$$

({\rm gcd}(a_i,a_j))^2|a_k.

$$
2018 Turkey MO (2nd Round)
Initially, there are 2018 distinct boxes on a table. In the first stage, Yazan and Bozan, starting with Yazan, take turns make $2016$ moves each, such that, in each move, the person whose turn selects a pair of boxes that is not written on the board, and writes the pair on the board.



In the second stage, Bozan enumerates the $4032$ pairs with numbers from $1,2,\dots,4032$, in whichever order he wants, and puts $k$ balls in each boxes written contained in the $k^{th}$ pair. Is there a strategy for Bozan that guarantees that the number of balls in each box are distinct?
2017 Turkey MO (2nd round)
A wedding is going to be held in a city with $25$ types of meals, to which some of the $2017$ citizens will be invited. All of the citizens like some meals and each meal is liked by at least one person. A "$suitable$ $list$" is a set of citizens, such that each meal is liked by at least one person in the set. A "$kamber$ $group$" is a set that contains at least one person from each "$suitable$ $list$". Given a "$kamber$ $group$", which has no subset (other than itself) that is also a "$kamber$ $group$", prove that there exists a meal, which is liked by everyone in the group.
2017 Turkey MO (2nd round)
Let $ABCD$ be a quadrilateral such that line $AB$ intersects $CD$ at $X$. Denote circles with inradius $r_1$ and centers $A, B$ as $w_a$ and $w_b$  with inradius $r_2$ and centers $C, D$ as $w_c$ and $w_d$. $w_a$ intersects $w_d$ at $P, Q$. $w_b$ intersects $w_c$ at $R, S$. Prove that if $XA.XB+r_2^2=XC.XD+r_1^2$, then $P,Q,R,S$ are cyclic.
2017 Turkey MO (2nd round)
Denote the sequence $a_{i,j}$ in positive reals such that $a_{i,j}$.$a_{j,i}=1$. Let $c_i=\sum_{k=1}^{n}a_{k,i}$. Prove that $1\ge$$\sum_{i=1}^{n}\dfrac {1}{c_i}$
2017 Turkey MO (2nd round)
Let $d(n)$ be number of prime divisors of $n$. Prove that one can find $k,m$ positive integers for any positive integer $n$ such that $k-m=n$ and $d(k)-d(m)=1$
2017 Turkey MO (2nd round)
Let $x_0,\dots,x_{2017}$ are positive integers and $x_{2017}\geq\dots\geq x_0=1$ such that $A=\{x_1,\dots,x_{2017}\}$ consists of exactly $25$ different numbers. Prove that $\sum_{i=2}^{2017}(x_i-x_{i-2})x_i\geq 623$, and find the number of sequences that holds the case of equality.
2017 Turkey MO (2nd round)
Finite number of $2017$ units long sticks are fixed on a plate. Each stick has a bead that can slide up and down on it. Beads can only stand on integer heights $( 1, 2, 3,..., 2017 )$. Some of the bead pairs are connected with elastic bands. $The$ $young$ $ant$ can go to every bead, starting from any bead by using the elastic bands. $The$ $old$ $ant$ can use an elastic band if the difference in height of the beads which are connected by the band, is smaller than or equal to $1$. If the heights of the beads which are connected to each other are different, we call it $valid$ $situation$. If there exists at least one $valid$ $situation$, prove that we can create a $valid$ $situation$, by arranging the heights of the beads, in which $the$ $old$ $ant$ can go to every bead, starting from any bead.
2015 Turkey MO (2nd round)
$m$ and $n$ are positive integers. If the number $$ k=\dfrac{(m+n)^2}{4m(m-n)^2+4}$$is an integer, prove that $k$ is a perfect square.
2015 Turkey MO (2nd round)
$x$, $y$ and $z$ are real numbers where the sum of any two among them is not $1$. Show that, $$ \dfrac{(x^2+y)(x+y^2)}{(x+y-1)^2}+\dfrac{(y^2+z)(y+z^2)}{(y+z-1)^2} + \dfrac{(z^2+x)(z+x^2)}{(z+x-1)^2} \ge 2(x+y+z) - \dfrac{3}{4}$$Find all triples $(x,y,z)$ of real numbers satisfying the equality case.
2015 Turkey MO (2nd round)
$n$ points are given on a plane where $n\ge4$. All pairs of points are connected with a segment. Find the maximal number of segments which don't intersect with any other segments in their interior.
2015 Turkey MO (2nd round)
In an exhibition where $2015$ paintings are shown, every participant picks a pair of paintings and writes it on the board. Then, Fake Artist (F.A.) chooses some of the pairs on the board, and marks one of the paintings in all of these pairs as "better". And then, Artist's Assistant (A.A.) comes and in his every move, he can mark $A$ better then $C$ in the pair $(A,C)$ on the board if for a painting $B$, $A$ is marked as better than $B$ and $B$ is marked as better than $C$ on the board. Find the minimum possible value of $k$ such that, for any pairs of paintings on the board, F.A can compare $k$ pairs of paintings making it possible for A.A to compare all of the remaining pairs of paintings.



P.S: A.A can decide $A_1>A_n$ if there is a sequence $ A_1 > A_2 > A_3 > \dots > A_{n-1} > A_n$ where $X>Y$ means painting $X$ is better than painting $Y$.
2015 Turkey MO (2nd round)
In a cyclic quadrilateral $ABCD$ whose largest interior angle is $D$, lines $BC$ and $AD$ intersect at point $E$, while lines $AB$ and $CD$ intersect at point $F$. A point $P$ is taken in the interior of quadrilateral $ABCD$ for which $\angle EPD=\angle FPD=\angle BAD$. $O$ is the circumcenter of quadrilateral $ABCD$. Line $FO$ intersects the lines $AD$, $EP$, $BC$ at $X$, $Q$, $Y$, respectively. If $\angle DQX = \angle CQY$, show that $\angle AEB=90^\circ$.
2015 Turkey MO (2nd round)
Find all positive integers $n$ such that for any positive integer $a$ relatively prime to $n$, $2n^2 \mid a^n - 1$.
2014 Turkey MO (2nd round)
In a bag there are $1007$ black and $1007$ white balls, which are randomly numbered $1$ to $2014$. In every step we draw one ball and put it on the table; also if we want to, we may choose two different colored balls from the table and put them in a different bag. If we do that we earn points equal to the absolute value of their differences. How many points can we guarantee to earn after $2014$ steps?
2014 Turkey MO (2nd round)
Find all all positive integers x,y,and z satisfying the equation $x^3=3^y7^z+8$
2014 Turkey MO (2nd round)
Let $D, E, F$ be points on the sides $BC, CA, AB$ of a triangle $ABC$, respectively such that the lines $AD, BE, CF$ are concurrent at the point $P$. Let a line $\ell$  through $A$ intersect the rays $[DE$ and $[DF$ at the points $Q$ and $R$, respectively. Let $M$ and $N$ be points on the rays $[DB$ and $[DC$, respectively such that the equation

$$ \frac{QN^2}{DN}+\frac{RM^2}{DM}=\frac{(DQ+DR)^2-2\cdot RQ^2+2\cdot DM\cdot DN}{MN} $$

holds. Show that the lines $AD$ and $BC$ are perpendicular to each other.
2014 Turkey MO (2nd round)
Let $P$ and $Q$ be the midpoints of non-parallel chords $k_1$ and $k_2$ of a circle $\omega$, respectively. Let the tangent lines of $\omega$ passing through the endpoints of $k_1$ intersect at $A$ and the tangent lines  passing through the endpoints of $k_2$ intersect at $B$. Let the symmetric point of the orthocenter of triangle $ABP$ with respect to the line $AB$ be $R$ and let the feet of the perpendiculars from $R$ to the lines $AP, BP, AQ, BQ$ be $R_1, R_2, R_3, R_4$, respectively. Prove that

$$ \frac{AR_1}{PR_1} \cdot \frac{PR_2}{BR_2} = \frac{AR_3}{QR_3} \cdot \frac{QR_4}{BR_4} $$
2014 Turkey MO (2nd round)
Find all natural numbers $n$ for which there exist non-zero and distinct real numbers $a_1, a_2, \ldots, a_n$ satisfying

$$ \left\{a_i+\dfrac{(-1)^i}{a_i} \, \Big | \, 1 \leq i \leq n\right\} = \{a_i \mid 1 \leq i \leq n\}. $$
2014 Turkey MO (2nd round)
$5$ airway companies operate in a country consisting of $36$ cities. Between any pair of cities exactly one company operates two way flights. If some air company operates between cities $A, B$ and $B, C$ we say that the triple $A, B, C$ is properly-connected. Determine the largest possible value of $k$ such that no matter how these flights are arranged there are at least $k$ properly-connected triples.
2013 Turkey MO (2nd round)
The circle $\omega_1$ with diameter $[AB]$ and the circle $\omega_2$ with center $A$ intersects at points $C$ and $D$. Let $E$ be a point on the circle $\omega_2$, which is outside $\omega_1$ and at the same side as $C$ with respect to the line $AB$. Let the second point of intersection of the line $BE$ with $\omega_2$ be $F$. For a point $K$ on the circle $\omega_1$ which is on the same side as $A$ with respect to the diameter of $\omega_1$ passing through $C$ we have $2\cdot CK \cdot AC =  CE \cdot AB$. Let the second point of intersection of the line $KF$ with $\omega_1$ be $L$. Show that the symmetric of the point $D$ with respect to the line $BE$ is on the circumcircle of the triangle $LFC$.
2013 Turkey MO (2nd round)
Let $m$ be a positive integer.

a.  Show that there exist infinitely many positive integers $k$ such that $1+km^3$ is a perfect cube and $1+kn^3$ is not a perfect cube for all positive integers $n<m$.

b. Let $m=p^r$ where $p \equiv 2 \pmod 3$ is a prime number and $r$ is a positive integer. Find all numbers $k$ satisfying the condition in part a.
2013 Turkey MO (2nd round)
Let $G$ be a simple, undirected, connected graph with $100$ vertices and $2013$ edges. It is given that there exist two vertices $A$ and $B$ such that it is not possible to reach $A$ from $B$ using one or two edges. We color all edges using $n$ colors, such that for all pairs of vertices, there exists a way connecting them with a single color. Find the maximum value of $n$.
2013 Turkey MO (2nd round)
Find all positive integers $m$ and $n$ satisfying $2^n+n=m!$.
2013 Turkey MO (2nd round)
Find the maximum value of $M$ for which for all positive real numbers $a, b, c$ we have

$$ a^3+b^3+c^3-3abc \geq M(ab^2+bc^2+ca^2-3abc) $$
2013 Turkey MO (2nd round)
Let $n$ be a positive integer and $P_1, P_2, \ldots, P_n$ be different points on the plane such that distances between them are all integers. Furthermore, we know that the distances $P_iP_1, P_iP_2, \ldots, P_iP_n$ forms the same sequence  for all $i=1,2, \ldots, n$ when these numbers are arranged in a non-decreasing order. Find all possible values of $n$.
2012 Turkey MO (2nd round)
Find all polynomials with integer coefficients such that for all positive integers $n$ satisfies $P(n!)=|P(n)|!$
2012 Turkey MO (2nd round)
Let $ABC$ be a isosceles triangle with $AB=AC$ an $D$ be the foot of perpendicular of $A$. $P$ be an interior point of triangle $ADC$ such that $m(APB)>90$ and $m(PBD)+m(PAD)=m(PCB)$.

$CP$ and $AD$ intersects at $Q$, $BP$ and $AD$ intersects at $R$. Let $T$ be a point on $[AB]$ and $S$ be a point on $[AP$ and not belongs to $[AP]$ satisfying $m(TRB)=m(DQC)$ and $m(PSR)=2m(PAR)$. Show that $RS=RT$
2012 Turkey MO (2nd round)
Find all non-decreasing functions from real numbers to itself such that for all real numbers $x,y$ $f(f(x^2)+y+f(y))=x^2+2f(y)$ holds.
2012 Turkey MO (2nd round)
For all positive real numbers $x, y, z$, show that $ \frac{x(2x-y)}{y(2z+x)}+\frac{y(2y-z)}{z(2x+y)}+\frac{z(2z-x)}{x(2y+z)} \geq 1$ is true.
2012 Turkey MO (2nd round)
Let $P$ be the set of all $2012$ tuples $(x_1, x_2, \dots, x_{2012})$, where $x_i \in \{1,2,\dots 20\}$ for each $1\leq i \leq 2012$. The set $A \subset P$ is said to be decreasing if for each $(x_1,x_2,\dots ,x_{2012} ) \in A$ any $(y_1,y_2,\dots, y_{2012})$ satisfying $y_i \leq x_i (1\leq i \leq 2012)$ also belongs to $A$. The set $B \subset P$ is said to be increasing if for each $(x_1,x_2,\dots ,x_{2012} ) \in B$ any $(y_1,y_2,\dots, y_{2012})$ satisfying $y_i \geq x_i (1\leq i \leq 2012)$ also belongs to $B$. Find the maximum possible value of $f(A,B)= \dfrac {|A\cap B|}{|A|\cdot |B|}$, where $A$ and $B$ are nonempty decreasing and increasing sets ($\mid \cdot \mid$ denotes the number of elements of the set).
2012 Turkey MO (2nd round)
Let $B$ and $D$ be points on segments $[AE]$ and $[AF]$ respectively. Excircles of triangles $ABF$ and $ADE$ touching sides $BF$ and $DE$ is the same, and its center is $I$. $BF$ and $DE$ intersects at $C$. Let $P_1, P_2, P_3, P_4, Q_1, Q_2, Q_3, Q_4$ be the circumcenters of triangles $IAB, IBC, ICD, IDA, IAE, IEC, ICF, IFA$ respectively.



a)  Show that points $P_1, P_2, P_3, P_4$ concylic and points $Q_1, Q_2, Q_3, Q_4$ concylic.

b)  Denote centers of theese circles as $O_1$ and $O_2$. Prove that $O_1, O_2$ and $I$ are collinear.
2011 Turkey MO (2nd round)
$n\geq2$ and $E=\left \{ 1,2,...,n \right \}. A_1,A_2,...,A_k$ are subsets of $E$, such that for all $1\leq{i}<{j}\leq{k}$ Exactly one of $A_i\cap{A_j},A_i'\cap{A_j},A_i\cap{A_j'},A_i'\cap{A_j'}$ is empty set. What is the maximum possible $k$?
2011 Turkey MO (2nd round)
Let $ABC$ be a triangle $D\in[BC]$ (different than $A$ and $B$).$E$ is the midpoint of $[CD]$. $F\in[AC]$ such that $\widehat{FEC}=90$ and $|AF|.|BC|=|AC|.|EC|.$ Circumcircle of $ADC$ intersect $[AB]$ at $G$ different than $A$.Prove that tangent to circumcircle of $AGF$ at $F$ is touch circumcircle of $BGE$ too.
2011 Turkey MO (2nd round)
$x,y,z$ positive real numbers such that $xyz=1$

Prove that:

$\frac{1}{x+y^{20}+z^{11}}+\frac{1}{y+z^{20}+x^{11}}+\frac{1}{z+x^{20}+y^{11}}\leq1$
2011 Turkey MO (2nd round)
$a_{1}=5$ and $a_{n+1}=a_{n}^{3}-2a_{n}^{2}+2$ for all $n\geq1$. $p$ is a prime such that $p=3(mod 4)$ and $p|a_{2011}+1$. Show that $p=3$.
2011 Turkey MO (2nd round)
Let $M$ and $N$ be two regular polygonic area.Define $K(M,N)$ as the midpoints of segments $[AB]$ such that $A$ belong to $M$ and $B$ belong to $N$. Find all situations of $M$ and $N$ such that $K(M,N)$ is a regualr polygonic area too.
2011 Turkey MO (2nd round)
Let $A$ and $B$ two countries which inlude exactly $2011$ cities.There is exactly one flight from a city of $A$ to a city of $B$ and there is no domestic flights (flights are bi-directional).For every city $X$ (doesn't matter from $A$ or from $B$), there exist at most $19$ different airline such that airline have a flight from $X$ to the another city.For an integer $k$, (it doesn't matter how flights arranged ) we can say that there exists at least $k$ cities such that it is possible to trip from one of these $k$ cities to another with same airline.So find the maximum value of $k$.
2010 Turkey MO (2nd round)
In a country, there are some two-way roads between the cities. There are $2010$ roads connected to the capital city. For all cities different from the capital city, there are less than $2010$ roads connected to that city. For two cities, if there are the same number of roads connected to these cities, then this number is even. $k$ roads connected to the capital city will be deleted. It is wanted that whatever the road network is, if we can reach from one city to another at the beginning, then we can reach after the deleting process also. Find the maximum value of $k.$
2010 Turkey MO (2nd round)
Let $P$ be an interior point of the triangle $ABC$ which is not on the median belonging to $BC$ and satisfying $\angle CAP = \angle BCP.  \: BP \cap CA = \{B'\} \: , \: CP \cap AB = \{C'\}$ and $Q$ is the second point of intersection of  $AP$ and the circumcircle of $ABC. \: B'Q$ intersects $CC'$ at $R$ and $B'Q$ intersects the line through $P$ parallel to $AC$ at $S.$ Let $T$ be the point of intersection of lines $B'C'$ and $QB$ and $T$ be on the other side of $AB$ with respect to $C.$ Prove that

$$\angle BAT = \angle BB'Q \: \Longleftrightarrow \: |SQ|=|RB'| $$
2010 Turkey MO (2nd round)
Prove that for all $n \in \mathbb{Z^+}$ and for all positive real numbers satisfying $a_1a_2...a_n=1$

$$ \displaystyle\sum_{i=1}^{n} \frac{a_i}{\sqrt{{a_i}^4+3}} \leq \frac{1}{2}\displaystyle\sum_{i=1}^{n} \frac{1}{a_i} $$
2010 Turkey MO (2nd round)
Let $A$ and $B$ be two points on the circle with diameter $[CD]$ and on the different sides of the line $CD.$ A circle $\Gamma$ passing through $C$ and $D$ intersects $[AC]$ different from the endpoints at $E$ and intersects $BC$ at $F.$ The line tangent to $\Gamma$ at $E$ intersects $BC$ at $P$ and $Q$ is a point on the circumcircle of the triangle $CEP$ different from $E$ and satisfying $|QP|=|EP|. \: AB \cap EF =\{R\}$ and $S$ is the midpoint of $[EQ].$ Prove that $DR$ is parallel to $PS.$
2010 Turkey MO (2nd round)
For integers $a$ and $b$ with $0 \leq a,b < {2010}^{18}$ let $S$ be the set of all polynomials in the form of $P(x)=ax^2+bx.$ For a polynomial $P$ in $S,$ if for all integers n with $0 \leq n <{2010}^{18}$ there exists a polynomial $Q$ in $S$ satisfying $Q(P(n)) \equiv n \pmod {2010^{18}},$ then we call $P$ as a good polynomial.

Find the number of good polynomials.
2010 Turkey MO (2nd round)
Let $K$ be the set of all sides and diagonals of a convex $2010-gon$ in the plane. For a subset $A$ of $K,$ if every pair of line segments belonging to $A$ intersect, then we call $A$ as an intersecting set. Find the maximum possible number of elements of union of two intersecting sets.
2009 Turkey MO (2nd round)
Find all prime numbers $p$ for which $p^3-4p+9$ is a perfect square.