Navvye/Olympiad-Problems-CSV huggingface dataset:
2009 Turkey MO (2nd round)
Let $\Gamma$ be the circumcircle of a triangle $ABC,$ and let $D$ and $E$ be two points different from the vertices on the sides $AB$ and $AC,$ respectively. Let $A'$ be the second point where $\Gamma$ intersects the bisector of the angle $BAC,$ and let $P$ and $Q$ be the second points where $\Gamma$ intersects the lines $A'D$ and $A'E,$ respectively. Let $R$ and $S$ be the second points of intersection of the lines $AA'$ and the circumcircles of the triangles $APD$ and $AQE,$ respectively.

Show that the lines $DS, \: ER$ and the tangent line to $\Gamma$ through $A$ are concurrent.
2009 Turkey MO (2nd round)
Alice, who works for the Graph County Electric Works, is commissioned to wire the newly erected utility poles in $k$ days. Each day she either chooses a pole and runs wires from it to as many poles as she wishes, or chooses at most $17$ pairs of poles and runs wires between each pair. Bob, who works for the Graph County Paint Works, claims that, no matter how many poles there are and how Alice connects them, all the poles can be painted using not more than $2009$ colors in such a way that no pair of poles connected by a wire is the same color. Determine the greatest value of $k$ for which Bob's claim is valid.
2009 Turkey MO (2nd round)
Let $H$ be the orthocenter of an acute triangle $ABC,$ and let $A_1, \: B_1, \: C_1$ be the feet of the altitudes belonging to the vertices $A, \: B, \: C,$ respectively. Let $K$ be a point on the smaller $AB_1$ arc of the circle with diameter $AB$ satisfying the condition $\angle HKB = \angle C_1KB.$ Let $M$ be the point of intersection of the line segment $AA_1$ and the circle with center $C$ and radius $CL$ where $KB \cap CC_1=\{L\}.$ Let $P$ and $Q$ be the points of intersection of the line $CC_1$ and the circle with center $B$ and radius $BM.$ Show that $A, \: K, \: P, \: Q$ are concyclic.
2009 Turkey MO (2nd round)
Show that

$$ \frac{(b+c)(a^4-b^2c^2)}{ab+2bc+ca}+\frac{(c+a)(b^4-c^2a^2)}{bc+2ca+ab}+\frac{(a+b)(c^4-a^2b^2)}{ca+2ab+bc} \geq 0 $$

for all positive real numbers $a, \: b , \: c.$
2009 Turkey MO (2nd round)
If $1<k_1<k_2<...<k_n$ and $a_1,a_2,...,a_n$ are integers such that for every integer $N,$ $k_i \mid N-a_i$ for some $1 \leq i \leq n,$ find the smallest possible value of $n.$
2008 Turkey MO (2nd round)
Given an acute angled triangle $ ABC$ , $ O$ is the circumcenter and $ H$ is the orthocenter.Let $ A_1$,$ B_1$,$ C_1$ be the midpoints of the sides $ BC$,$ AC$ and $ AB$ respectively. Rays $ [HA_1$,$ [HB_1$,$ [HC_1$ cut the circumcircle of $ ABC$ at $ A_0$,$ B_0$ and $ C_0$ respectively.Prove that $ O$,$ H$ and $ H_0$ are collinear if $ H_0$ is the orthocenter of $ A_0B_0C_0$
2008 Turkey MO (2nd round)
$ a - )$ Find all prime $ p$ such that $ \dfrac{7^{p - 1} - 1}{p}$  is a perfect square



$ b - )$ Find all prime $ p$ such that $ \dfrac{11^{p - 1} - 1}{p}$  is a perfect square
2008 Turkey MO (2nd round)
Let a.b.c be positive reals such that their sum is 1. Prove that



$ \frac{a^{2}b^{2}}{c^{3}(a^{2}-ab+b^{2})}+\frac{b^{2}c^{2}}{a^{3}(b^{2}-bc+c^{2})}+\frac{a^{2}c^{2}}{b^{3}(a^{2}-ac+c^{2})}\geq \frac{3}{ab+bc+ac}$
2008 Turkey MO (2nd round)
$ f: \mathbb N \times \mathbb Z \rightarrow \mathbb Z$  satisfy the given conditions



$ a)$ $ f(0,0)=1$ , $ f(0,1)=1$ ,



$ b)$ $ \forall k \notin \left\{0,1\right\}$ $ f(0,k)=0$ and



$ c)$ $ \forall n \geq 1$ and $ k$ , $ f(n,k)=f(n-1,k)+f(n-1,k-2n)$



find the sum $ \displaystyle\sum_{k=0}^{\binom{2009}{2}}f(2008,k)$
2008 Turkey MO (2nd round)
A circle $ \Gamma$ and a line $ \ell$ is given in a plane such that $ \ell$ doesn't cut $ \Gamma$.Determine the intersection set of the circles has $ [AB]$ as diameter for all pairs of $ \left\{A,B\right\}$ (lie on $ \ell$) and satisfy $ P,Q,R,S \in \Gamma$ such that $ PQ \cap RS=\left\{A\right\}$ and $ PS \cap QR=\left\{B\right\}$
2008 Turkey MO (2nd round)
There is a connected network with $ 2008$ computers, in which any of the two cycles don't have any common vertex. A hacker and a administrator are playing a game in this network. On the $ 1st$ move hacker selects one computer and hacks it, on the $ 2nd$ move administrator selects another computer and protects it. Then on every $ 2k+1th$ move hacker hacks one more computer(if he can) which wasn't protected by the administrator and is directly connected (with an edge) to a computer which was hacked by the hacker before and on every $ 2k+2th$ move administrator protects  one more computer(if he can) which wasn't hacked by the hacker and is directly connected (with an edge) to a computer which was protected by the administrator before for every $ k>0$. If both of them can't make move, the game ends. Determine the maximum number of computers which the hacker can guarantee to hack at the end of the game.
2007 Turkey MO (2nd round)
In an acute triangle $ABC$, the circle with diameter $AC$ intersects $AB$ and $AC$ at $K$ and $L$ different from $A$ and $C$ respectively. The circumcircle of $ABC$ intersects the line $CK$ at the point $F$ different from $C$ and the line $AL$ at the point $D$ different from $A$. A point $E$ is choosen on the smaller arc of $AC$ of the circumcircle of $ABC$ . Let $N$ be the intersection of the lines $BE$ and $AC$ . If $AF^{2}+BD^{2}+CE^{2}=AE^{2}+CD^{2}+BF^{2}$  prove that $\angle KNB= \angle BNL$ .
2007 Turkey MO (2nd round)
Some unit squares of $ 2007\times 2007 $ square board are colored. Let $ (i,j) $ be a unit square belonging to the $ith$ line and $jth$ column and $ S_{i,j} $ be the set of all colored unit squares $(x,y)$ satisfying $ x\leq i, y\leq j $. At the first step in each colored unit square $(i,j)$ we write the number of colored unit squares in $ S_{i,j} $ . In each step, in each colored unit square $(i,j)$ we write the sum of all numbers written in $ S_{i,j} $ in the previous step. Prove that after finite number of steps, all numbers in the colored unit squares will be odd.
2007 Turkey MO (2nd round)
If  $a,b,c$ are three positive real numbers such that $a+b+c=3$, prove that

$ {\frac{a^{2}+3b^{2}}{ab^{2}(4-ab)}}+{\frac{b^{2}+3c^{2}}{bc^{2}(4-ab)}}+{\frac{c^{2}+3a^{2}}{ca^{2}(4-ca)}}\geq 4 $
2007 Turkey MO (2nd round)
Let $k>1$ be an integer, $p=6k+1$ be a prime number and $m=2^{p}-1$ .



Prove that $\frac{2^{m-1}-1}{127m}$ is an integer.
2007 Turkey MO (2nd round)
Let $ABC$ be a triangle with $\angle B=90$. The incircle of $ABC$ touches the side $BC$ at $D$. The incenters of triangles $ABD$ and $ADC$ are $X$ and $Z$ , respectively. The lines $XZ$ and $AD$ are intersecting at the point $K$. $XZ$ and circumcircle of $ABC$ are intersecting at $U$ and $V$. Let $M$ be the midpoint of line segment $[UV]$ . $AD$ intersects the circumcircle of $ABC$  at $Y$ other than $A$. Prove that $|CY|=2|MK|$ .
2007 Turkey MO (2nd round)
In a country between each pair of cities there is at most one direct road. There is a connection (using one or more roads) between any two cities even after the elimination of any given city and all roads incident to this city. We say that the city $A$ can be k -directionally connected to the city $B$, if : we can orient at most $k$ roads such that after arbitrary orientation of remaining roads for any fixed road $l$ (directly connecting two cities) there is a path passing through roads in the direction of their orientation starting at $A$, passing through $l$ and ending at $B$ and visiting each city at most once. Suppose that in a country with $n$ cities, any two cities can be k - directionally connected. What is the minimal value of $k$?
2006 Turkey MO (2nd round)
Points $P$ and $Q$ on side $AB$ of a convex quadrilateral $ABCD$ are given such that $AP = BQ.$ The circumcircles of triangles $APD$ and $BQD$ meet again at $K$ and those of $APC$ and $BQC$ meet again at $L$. Show that the points $D,C,K,L$ lie on a circle.
2006 Turkey MO (2nd round)
There are $2006$ students and $14$ teachers in a school. Each student knows at least one teacher (knowing is a symmetric relation). Suppose that, for each pair of a student and a teacher who know each other, the ratio of the number of the students whom the teacher knows to that of the teachers whom the student knows is at least $t.$ Find the maximum possible value of $t.$
2006 Turkey MO (2nd round)
Find all positive integers $n$ for which all coefficients of polynomial $P(x)$ are divisible by $7,$ where

$$P(x) = (x^2 + x + 1)^n - (x^2 + 1)^n - (x + 1)^n - (x^2 + x)^n + x^{2n} + x^n + 1.$$
2006 Turkey MO (2nd round)
$x_{1},...,x_{n}$ are positive reals such that their sum and their squares' sum are equal to $t$. Prove that $\sum_{i\neq{j}}\frac{x_{i}}{x_{j}}\ge\frac{(n-1)^{2}\cdot{t}}{t-1}$
2006 Turkey MO (2nd round)
$ABC$ be a triangle. Its incircle touches the sides $CB, AC, AB$ respectively at $N_{A},N_{B},N_{C}$. The orthic triangle of $ABC$ is $H_{A}H_{B}H_{C}$ with $H_{A}, H_{B}, H_{C}$ are respectively on $BC, AC, AB$. The incenter of $AH_{C}H_{B}$ is $I_{A}$; $I_{B}$ and $I_{C}$ were defined similarly.

Prove that the hexagon $I_{A}N_{B}I_{C}N_{A}I_{B}N_{C}$ has all sides equal.
2006 Turkey MO (2nd round)
Find all the triangles such that its side lenghts, area and its angles' measures (in degrees) are rational.
2005 Turkey MO (2nd round)
For all positive real numbers $a,b,c,d$ prove the inequality



$$\sqrt{a^4+c^4}+\sqrt{a^4+d^4}+\sqrt{b^4+c^4}+\sqrt{b^4+d^4} \ge 2\sqrt{2}(ad+bc)$$
2005 Turkey MO (2nd round)
In a triangle $ABC$ with $AB<AC<BC$, the perpendicular bisectors of $AC$ and $BC$ intersect $BC$ and $AC$ at $K$ and $L$, respectively. Let $O$, $O_1$, and $O_2$ be the circumcentres of triangles $ABC$, $CKL$, and $OAB$, respectively. Prove that $OCO_1O_2$ is a parallelogram.
2005 Turkey MO (2nd round)
Some of the $n + 1$ cities in a country (including the capital city) are connected by one-way or two-way airlines. No two cities are connected by both a one-way airline and a two-way airline, but there may be more than one two-way airline between two cities. If $d_A$ denotes the number of airlines from a city $A$, then $d_A \le n$ for any city $A$ other than the capital city and $d_A + d_B \le n$ for any two cities $A$ and $B$ other than the capital city which are not connected by a two-way airline. Every airline has a return, possibly consisting of several connected flights. Find the largest possible number of two-way airlines and all configurations of airlines for which this largest number is attained.
2005 Turkey MO (2nd round)
Find all triples of nonnegative integers  $(m,n,k)$ satisfying $5^m+7^n=k^3$.
2005 Turkey MO (2nd round)
If $a,b,c$ are the sides of a triangle and $r$ the inradius of the triangle, prove that

$$\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\le \frac{1}{4r^2} $$
2005 Turkey MO (2nd round)
Suppose that a sequence $(a_n)_{n=1}^{\infty}$ of integers has the following property: For all $n$ large enough (i.e. $n \ge N$ for some $N$ ), $a_n$ equals the number of indices $i$, $1 \le i < n$, such that $a_i + i \ge n$. Find the maximum possible number of integers which occur infinitely many times in the sequence.
2004 Turkey MO (2nd round)
In a triangle $\triangle ABC$ with$\angle B>\angle C$, the altitude, the angle bisector, and the median from $A$ intersect $BC$ at $H, L$ and $D$, respectively. Show that $\angle HAL=\angle DAL$ if and only if $\angle BAC=90^{\circ}$.
2004 Turkey MO (2nd round)
Two-way flights are operated between $80$ cities in such a way that each city is connected to at least $7$ other cities by a direct flight and any two cities are connected by a finite sequence of flights. Find the smallest $k$ such that for any such arrangement of flights it is possible to travel from any city to any other city by a sequence of at most $k$ flights.
2004 Turkey MO (2nd round)
(a) Determine if exist an integer $n$ such that $n^2 -k$ has exactly $10$ positive divisors for each $k = 1, 2, 3.$



(b) Show that the number of positive divisors of $n^2 -4$ is not $10$ for any integer $n.$
2004 Turkey MO (2nd round)
Find all functions $f:\mathbb{Z}\to \mathbb{Z}$ satisfying the condition $f(n)-f(n+f(m))=m$ for all $m,n\in \mathbb{Z}$
2004 Turkey MO (2nd round)
The excircle of a triangle $ABC$ corresponding to $A$ touches the lines $BC,CA,AB$ at $A_1,B_1,C_1$, respectively. The excircle corresponding to $B$ touches $BC,CA,AB$ at $A_2,B_2,C_2$, and the excircle corresponding to $C$ touches $BC,CA,AB$ at $A_3,B_3,C_3$, respectively. Find the maximum possible value of the ratio of the sum of the perimeters of $\triangle A_1B_1C_1$, $\triangle A_2B_2C_2$ and $\triangle A_3B_3C_3$ to the circumradius of $\triangle ABC$.
2004 Turkey MO (2nd round)
Define $K(n,0)=\varnothing $ and, for all nonnegative integers m and n, $K(n,m+1)=\left\{ \left. k \right|\text{ }1\le k\le n\text{ and }K(k,m)\cap K(n-k,m)=\varnothing  \right\}$.  Find the number of elements of $K(2004,2004)$.
2003 Turkey MO (2nd round)
$ n\geq 2$ cars are participating in a rally. The cars leave the start line at different times and arrive at the finish line at different times. During the entire rally each car takes over any other car at most once , the number of cars taken over by each car is different and each car is taken over by the same number of cars. Find all possible values of $ n$
2003 Turkey MO (2nd round)
Let $ABCD$ be a convex quadrilateral and $K,L,M,N$ be points on $[AB],[BC],[CD],[DA]$, respectively. Show that,

$$

\sqrt[3]{s_{1}}+\sqrt[3]{s_{2}}+\sqrt[3]{s_{3}}+\sqrt[3]{s_{4}}\leq 2\sqrt[3]{s}

$$

where $s_1=\text{Area}(AKN)$, $s_2=\text{Area}(BKL)$, $s_3=\text{Area}(CLM)$, $s_4=\text{Area}(DMN)$ and $s=\text{Area}(ABCD)$.
2003 Turkey MO (2nd round)
Let $ f: \mathbb R \rightarrow \mathbb R$ be a function such that



$ f(tx_1+(1-t)x_2)\leq tf(x_1)+(1-t)f(x_2)$



for all $ x_1 , x_2 \in \mathbb R$ and $ t\in (0,1)$. Show that



$ \sum_{k=1}^{2003}f(a_{k+1})a_k \geq \sum_{k=1}^{2003}f(a_k)a_{k+1}$



for all real numbers $ a_1,a_2,...,a_{2004}$ such that $ a_1\geq a_2\geq ... \geq a_{2003}$ and $ a_{2004}=a_1$
2003 Turkey MO (2nd round)
Suppose that $2^{2n+1}+ 2^{n}+1=x^{k}$, where $k\geq2$ and $n$ are positive integers. Find all possible values of $n$.
2003 Turkey MO (2nd round)
A circle which is tangent to the sides $ [AB]$ and $ [BC]$ of $ \triangle ABC$ is also tangent to its circumcircle at the point $ T$. If $ I$ is the incenter of $ \triangle ABC$ , show that $ \widehat{ATI}=\widehat{CTI}$
2003 Turkey MO (2nd round)
An assignment of either a $ 0$ or a $ 1$ to each unit square of an $ m$x$ n$ chessboard is called $ fair$ if the total numbers of $ 0$s and $ 1$s are equal. A real number $ a$ is called $ beautiful$ if there are positive integers $ m,n$ and a fair assignment for the $ m$x$ n$ chessboard such that for each of the $ m$ rows and $ n$ columns , the percentage of $ 1$s on that row or column is not less than $ a$ or greater than $ 100-a$. Find the largest beautiful number.
2002 Turkey MO (2nd round)
Let $(a_1, a_2,\ldots , a_n)$ be a permutation of $1, 2, \ldots , n,$ where $n  \geq 2.$ For each $k = 1, \ldots , n$, we know that $a_k$ apples are placed at the point $k$ on the real axis. Children named $A,B,C$ are assigned respective points $x_A, x_B, x_C \in  \{1, \ldots , n\}.$ For each $k,$ the children whose points are closest to $ k$ divide $a_k$ apples equally among themselves. We call $(x_A, x_B, x_C)$ a stable configuration if no child’s total share can be increased by assigning a new point to this child and not changing the points of the other two. Determine the values of $n$ for which a stable configuration exists for some distribution $(a_1, \ldots, a_n)$ of the apples.
2002 Turkey MO (2nd round)
Two circles are externally tangent to each other at a point $A$ and internally tangent to a third circle $\Gamma$ at points $B$ and $C.$ Let $D$ be the midpoint of the secant of $\Gamma$ which is tangent to the smaller circles at $A.$ Show that $A$ is the incenter of the triangle $BCD$ if the centers of the circles are not collinear.
2002 Turkey MO (2nd round)
Graph Airlines $ (GA)$ operates flights between some of the cities of the Republic of Graphia. There are at least three $ GA$ flights from each city, and it is possible to travel from any city in Graphia to any city in Graphia using $ GA$ flights. $ GA$ decides to discontinue some of its flights. Show that this can be done in such a way that it is still possible to travel between any two cities using $ GA$ flights, yet at least $ 2/9$ of the cities have only one flight.
2002 Turkey MO (2nd round)
Find all prime numbers $p$ for which the number of ordered pairs of integers $(x, y)$ with $0\leq x, y < p$ satisfying the condition

$$y^2 \equiv  x^3 -  x \pmod p$$

is exactly $p.$
2002 Turkey MO (2nd round)
Let $ABC$ be a triangle, and points $D,E$ are on $BA,CA$ respectively such that $DB=BC=CE$. Let $O,I$ be the circumcenter, incenter of $\triangle ABC$. Prove that the circumradius of $\triangle ADE$ is equal to $OI$.
2002 Turkey MO (2nd round)
Let $n$ be a positive integer and let $T$ denote the collection of points $(x_1, x_2, \ldots, x_n) \in \mathbb R^n$ for which there exists a permutation $\sigma$ of $1, 2, \ldots , n$ such that $x_{\sigma(i)} - x_{\sigma(i+1) } \geq 1$ for each $i=1, 2, \ldots , n.$ Prove that there is a real number $d$ satisfying the following condition:

For every $(a_1, a_2, \ldots, a_n) \in \mathbb R^n$ there exist points $(b_1, \ldots, b_n)$ and $(c_1,\ldots, c_n)$ in $T$ such that, for each $i = 1, . . . , n,$

$$a_i=\frac 12 (b_i+c_i) , \quad |a_i - b_i|  \leq d, \quad \text{and} \quad  |a_i - c_i| \leq d.$$
2001 Turkey MO (2nd round)
Let $ABCD$ be a convex quadrilateral. The perpendicular bisectors of the sides $[AD]$ and $[BC]$ intersect at a point $P$ inside the quadrilateral and the perpendicular bisectors of the sides $[AB]$ and $[CD]$ also intersect at a point $Q$ inside the quadrilateral. Show that, if $\angle APD = \angle BPC$ then $\angle AQB = \angle CQD$
2001 Turkey MO (2nd round)
$(x_{n})_{-\infty<n<\infty}$ is a sequence of real numbers which satisfies $x_{n+1}=\frac{x_{n}^2+10}{7}$ for every $n \in \mathbb{Z}$. If there exist a real upperbound for this sequence, find all the values $x_{0}$ can take.
2001 Turkey MO (2nd round)
One wants to distribute $n$ same sized cakes between $k$ people equally by cutting every cake at most once. If the number of positive divisors of $n$ is denoted as $d(n)$, show that the number of different values of $k$ which makes such distribution possible is $n+d(n)$
2001 Turkey MO (2nd round)
Find all ordered triples of positive integers $(x,y,z)$ such that

$$3^{x}+11^{y}=z^{2}$$
2001 Turkey MO (2nd round)
Two nonperpendicular lines throught the point $A$ and a point $F$ on one of these lines different from $A$ are given. Let $P_{G}$ be the intersection point of tangent lines at $G$ and $F$ to the circle through the point $A$, $F$ and $G$ where $G$ is a point on the given line different from the line $FA$. What is the locus of $P_{G}$ as $G$ varies.
2001 Turkey MO (2nd round)
We wish to color the cells of a $n \times n$ chessboard with $k$ different colors such that for every $i\in \{1,2,...,n\}$, the $2n-1$ cells on $i$. row and $i$. column have all different colors.



a) Prove that for $n=2001$ and $k=4001$, such coloring is not possible.



b) Show that for $n=2^{m}-1$ and $k=2^{m+1}-1$, such coloring is possible.
2000 Turkey MO (2nd round)
A circle with center $O$ and a point $A$ in this circle are given. Let $P_{B}$ is the intersection point of $[AB]$ and the internal bisector of $\angle AOB$ where $B$ is a point on the circle such that $B$ doesn't lie on the line $OA$, Find the locus of $P_{B}$ as $B$ varies.
2000 Turkey MO (2nd round)
Let define $P_{n}(x)=x^{n-1}+x^{n-2}+x^{n-3}+ \dots +x+1$ for every positive integer $n$. Prove that for every positive integer $a$ one can find a positive integer $n$ and polynomials $R(x)$ and $Q(x)$ with integer coefficients such that $$P_{n}(x)= [1+ax+x^{2}R(x)] Q(x).$$
2000 Turkey MO (2nd round)
Let $f(x,y)$ and $g(x,y)$ be defined for every $x,y\in \{1,2,\dots, 2000\}$ and take different values for at most $n$ ordered pairs of $(x,y)$. For every function pairs $f(x,y), g(x,y)$, when $x\notin X$ and $y\notin Y$, it is always possible to find $1000-$element sets $X,Y \subset \{1,2,\dots, 2000\}$ such that $f(x,y) = g(x,y)$. Determine the largest integer that $n$ can take.
2000 Turkey MO (2nd round)
Let $p$ be a prime number. $T(x)$ is a polynomial with integer coefficients and degree from the set $\{0,1,...,p-1\}$ and such that $T(n) \equiv T(m)  (mod p)$ for some integers m and n implies that $ m \equiv n (mod p)$. Determine the maximum possible value of degree of $T(x)$
2000 Turkey MO (2nd round)
A positive real number $a$ and two rays wich intersect at point $A$ are given. Show that all the circles which pass through $A$ and intersect these rays at points $B$ and $C$ where $|AB|+|AC|=a$ have a common point other than $A$.
2000 Turkey MO (2nd round)
Find all continuous functions $f:[0,1]\to [0,1]$ for which there exists a positive integer $n$ such that $f^{n}(x)=x$ for $x \in [0,1]$ where $f^{0} (x)=x$ and $f^{k+1}=f(f^{k}(x))$ for every positive integer $k$.
1999 Turkey MO (2nd round)
Find the number of ordered quadruples $(x,y,z,w)$ of integers with $0\le x,y,z,w\le 36$ such that ${{x}^{2}}+{{y}^{2}}\equiv {{z}^{3}}+{{w}^{3}}\text{ (mod 37)}$.
1999 Turkey MO (2nd round)
Problem-2:

Given a circle with center $O$, the two tangent lines from a point $S$ outside the circle touch the circle at points $P$ and $Q$. Line $SO$ intersects the circle at $A$ and $B$, with $B$ closer to $S$. Let $X$ be an interior point of minor arc  $PB$, and let line $OS$ intersect lines $QX$ and $PX$ at $C$ and $D$, respectively. Prove that

$\frac{1}{\left| AC \right|}+\frac{1}{\left| AD \right|}=\frac{2}{\left| AB \right|}$.
1999 Turkey MO (2nd round)
For any two positive integers $n$ and $p$, prove that there are exactly ${{(p+1)}^{n+1}}-{{p}^{n+1}}$ functions

$f:\left\{ 1,2,...,n \right\}\to \left\{ -p,-p+1,-p+2,....,p-1,p \right\}$

such that $\left| f(i)-f(j) \right|\le p$ for all $i,j\in \left\{ 1,2,...,n \right\}$.
1999 Turkey MO (2nd round)
Find all sequences ${{a}_{1}},{{a}_{2}},...,{{a}_{2000}}$ of real numbers such that $\sum\limits_{n=1}^{2000}{{{a}_{n}}=1999}$ and such that $\frac{1}{2}<{{a}_{n}}<1$ and ${{a}_{n+1}}={{a}_{n}}(2-{{a}_{n}})$ for all $n\ge 1$.
1999 Turkey MO (2nd round)
In an acute triangle $\vartriangle ABC$ with circumradius $R$, altitudes $\overline{AD},\overline{BE},\overline{CF}$ have lengths ${{h}_{1}},{{h}_{2}},{{h}_{3}}$, respectively. If ${{t}_{1}},{{t}_{2}},{{t}_{3}}$ are lengths of the tangents from $A,B,C$, respectively, to the circumcircle of triangle $\vartriangle DEF$, prove that

$\sum\limits_{i=1}^{3}{{{\left( \frac{t{}_{i}}{\sqrt{h{}_{i}}} \right)}^{2}}\le }\frac{3}{2}R$.
1999 Turkey MO (2nd round)
We wish to ﬁnd the sum of $40$ given numbers utilizing $40$ processors. Initially, we have the number $0$ on the screen of each processor. Each processor adds the number on its screen with a number entered directly (only the given numbers could be entered directly to the processors) or transferred from another processor in a unit time. Whenever a number is transferred from a processor to another, the former processor resets. Find the least time needed to ﬁnd the desired sum.
1998 Turkey MO (2nd round)
Let $D$  be the point on the base $BC$ of an isosceles $\vartriangle ABC$ triangle such that $\frac{\left| BD \right|}{\left| DC \right|}=\text{ }2$, and let $P$  be the point on the segment $\left[ AD \right]$ such that $\angle BAC=\angle BPD$. Prove that $\angle DPC=\frac{1}{2}\angle BAC$.
1998 Turkey MO (2nd round)
If $0\le a\le b\le c$ real numbers, prove that $(a+3b)(b+4c)(c+2a)\ge 60abc$.
1998 Turkey MO (2nd round)
The points of a circle are colored by three colors. Prove that there exist infinitely many isosceles triangles inscribed in the circle whose vertices are of the same color.
1998 Turkey MO (2nd round)
Find all positive integers $x$ and $n$ such that ${{x}^{3}}+3367={{2}^{n}}$.
1998 Turkey MO (2nd round)
Variable points $M$ and $N$ are considered on the arms $\left[ OX \right.$ and $\left[ OY \right.$ , respectively, of an angle $XOY$ so that $\left| OM \right|+\left| ON \right|$ is constant. Determine the locus of the midpoint of $\left[ MN \right]$.