Navvye/Olympiad-Problems-CSV huggingface dataset:
2009 Turkey Team Selection Test
Quadrilateral $ ABCD$ has an inscribed circle which centered at $ O$ with radius $ r$. $ AB$ intersects $ CD$ at $ P$; $ AD$ intersects $ BC$ at $ Q$ and the diagonals $ AC$ and $ BD$ intersects each other at $ K$. If the distance from $ O$ to the line $ PQ$ is $ k$, prove that $ OK\cdot\ k = r^2$.
2009 Turkey Team Selection Test
For which $ p$ prime numbers, there is an integer root of the polynominal $ 1 + p + Q(x^1)\cdot\ Q(x^2)\ldots\ Q(x^{2p - 2})$ such that $ Q(x)$ is a polynominal with integer coefficients?
2009 Turkey Team Selection Test
In a triangle $ ABC$ incircle touches the sides $ AB$, $ AC$ and $ BC$ at $ C_1$, $ B_1$ and $ A_1$ respectively. Prove that $ \sqrt {\frac {AB_1}{AB}} + \sqrt {\frac {BC_1}{BC}} + \sqrt {\frac {CA_1}{CA}}\leq\frac {3}{\sqrt {2}}$ is true.
2009 Turkey Team Selection Test
In a class of $ n\geq 4$ some students are friends. In this class any $ n - 1$ students can be seated in a round table such that every student is sitting next to a friend of him in both sides, but $ n$ students can not be seated in that way. Prove that the minimum value of $ n$ is $ 10$.
2008 Turkey Team Selection Test
In an $ ABC$ triangle such that $ m(\angle B)>m(\angle C)$, the internal and external bisectors of vertice $ A$ intersects $ BC$ respectively at points $ D$ and $ E$. $ P$ is a variable point on $ EA$ such that $ A$ is on $ [EP]$. $ DP$ intersects $ AC$ at $ M$ and $ ME$ intersects $ AD$ at $ Q$. Prove that all $ PQ$ lines have a common point as $ P$ varies.
2008 Turkey Team Selection Test
A graph has $ 30$ vertices, $ 105$ edges and $ 4822$ unordered edge pairs whose endpoints are disjoint. Find the maximal possible difference of degrees of two vertices in this graph.
2008 Turkey Team Selection Test
The equation $ x^3-ax^2+bx-c=0$ has three (not necessarily different) positive real roots. Find the minimal possible value of $ \frac{1+a+b+c}{3+2a+b}-\frac{c}{b}$.
2008 Turkey Team Selection Test
The sequence $ (x_n)$ is defined as; $ x_1=a$, $ x_2=b$ and for all positive integer $ n$, $ x_{n+2}=2008x_{n+1}-x_n$. Prove that there are some positive integers $ a,b$ such that $ 1+2006x_{n+1}x_n$ is a perfect square for all positive integer $ n$.
2008 Turkey Team Selection Test
$ D$ is a point on the edge $ BC$ of triangle $ ABC$ such that $ AD=\frac{BD^2}{AB+AD}=\frac{CD^2}{AC+AD}$. $ E$ is a point such that $ D$ is on $ [AE]$ and $ CD=\frac{DE^2}{CD+CE}$. Prove that $ AE=AB+AC$.
2008 Turkey Team Selection Test
There are $ n$ voters and $ m$ candidates. Every voter makes a certain arrangement list of all candidates (there is one person in every place $ 1,2,...m$) and votes for the first $ k$ people in his/her list. The candidates with most votes are selected and say them winners. A poll profile is all of this $ n$ lists.

If $ a$ is a candidate, $ R$ and $ R'$ are two poll profiles. $ R'$ is $ a-good$ for $ R$ if and only if for every voter; the people which in a worse position than $ a$ in $ R$ is also in a worse position than $ a$ in $ R'$. We say positive integer $ k$ is monotone if and only if for every $ R$ poll profile and every winner $ a$ for $ R$ poll profile is also a winner for all $ a-good$ $ R'$ poll profiles. Prove that $ k$ is monotone if and only if $ k>\frac{m(n-1)}{n}$.
2007 Turkey Team Selection Test
Find the number of the connected graphs with 6 vertices. (Vertices are considered to be different)
2007 Turkey Team Selection Test
Two different points $A$ and $B$  and a circle $\omega$ that passes through $A$  and $B$  are given. $P$  is a variable point on $\omega$  (different from $A$  and $B$). $M$ is a point such that $MP$  is the bisector of the angle $\angle{APB}$ ($M$ lies outside of $\omega$) and $MP=AP+BP$. Find the geometrical locus of $M$.
2007 Turkey Team Selection Test
Let $a, b, c$  be positive reals such that their sum is $1$. Prove that $$\frac{1}{ab+2c^{2}+2c}+\frac{1}{bc+2a^{2}+2a}+\frac{1}{ac+2b^{2}+2b}\geq \frac{1}{ab+bc+ac}.$$
2007 Turkey Team Selection Test
Let $ABC$ is an acute angled triangle and let $A_{1},\, B_{1},\, C_{1}$ are points respectively on $BC,\,CA,\,AB$ such that $\triangle ABC$ is similar to $\triangle A_{1}B_{1}C_{1}.$

Prove that orthocenter of $A_{1}B_{1}C_{1}$ coincides with circumcenter of $ABC$.
2007 Turkey Team Selection Test
A number $n$ is satisfying the conditions below



i) $n$ is a positive odd integer;



ii) there are some odd integers such that their squares' sum is equal to $n^{4}$.



Find all such numbers.
2007 Turkey Team Selection Test
We write $1$  or $-1$  on each unit square of a $2007 \times 2007$  board. Find the number of writings such that for every square on the board the absolute value of the sum of numbers on the square is less then or equal to $1$.
2006 Turkey Team Selection Test
Find the maximum value for the area of a heptagon with all vertices on a circle and two diagonals perpendicular.
2006 Turkey Team Selection Test
How many ways are there to divide a $2\times n$ rectangle into rectangles having integral sides, where $n$ is a positive integer?
2006 Turkey Team Selection Test
If $x,y,z$ are positive real numbers and $xy+yz+zx=1$ prove that

$$ \frac{27}{4} (x+y)(y+z)(z+x) \geq ( \sqrt{x+y} +\sqrt{ y+z} + \sqrt{z+x} )^2 \geq 6 \sqrt 3. $$
2006 Turkey Team Selection Test
For all integers $n\geq 1$ we define $x_{n+1}=x_1^2+x_2^2+\cdots +x_n^2$, where $x_1$ is a positive integer. Find the least $x_1$ such that 2006 divides $x_{2006}$.
2006 Turkey Team Selection Test
From a point $Q$ on a circle with diameter $AB$ different from $A$ and $B$, we draw a perpendicular to $AB$, $QH$, where $H$ lies on $AB$. The intersection points of the circle of diameter $AB$ and the circle of center $Q$ and radius $QH$ are $C$ and $D$. Prove that $CD$ bisects $QH$.
2006 Turkey Team Selection Test
Each one of 2006 students makes a list with 12 schools among 2006. If we take any 6 students, there are two schools which at least one of them is included in each of 6 lists. A list which includes at least one school from all lists is a good list.



a) Prove that we can always find a good list with 12 elements, whatever the lists are;



b) Prove that students can make lists such that no shorter list is good.
2005 Turkey Team Selection Test
Find all functions $ f :\mathbb{R}_{0}^{+}\mapsto\mathbb{R}_{0}^{+} $  satisfying the conditions $4f(x)\geq 3x$ and $f(4f(x)-3x)=x$  for all  $x\geq 0$ .
2005 Turkey Team Selection Test
Let $N$ be midpoint of the side $AB$ of a triangle $ABC$ with $\angle A$ greater than $\angle B$. Let $D$ be a point on the ray $AC$ such that $CD=BC$ and $P$ be a point on the ray $DN$ which lies on the same side of $BC$ as $A$ and satisfies the condition $\angle PBC =\angle A$. The lines $PC$ and $AB$ intersect at $E$, and the lines $BC$ and $DP$ intersect at $T$. Determine the value of  $\frac{BC}{TC} - \frac{EA}{EB}$.
2005 Turkey Team Selection Test
Initially the numbers 1 through 2005 are marked. A finite set of marked consecutive integers is called a block if it is not contained in any larger set of marked consecutive integers. In each step we select a set of marked integers which does not contain the first or last element of any block, unmark the selected integers, and mark the same number of consecutive integers starting with the integer two greater than the largest marked integer. What is the minimum number of steps necessary to obtain 2005 single integer blocks?
2005 Turkey Team Selection Test
Show that for any integer $n\geq2$ and all integers $a_{1},a_{2},...,a_{n}$ the product $\prod_{i<j}{(a_{j}-a_{i})}$ is divisible by $\prod_{i<j}{(j-i)}$  .
2005 Turkey Team Selection Test
Let $ABC$ be a triangle such that $\angle A=90$ and $\angle B < \angle C$. The tangent at $A$  to its circumcircle $\Gamma$ meets the line $BC$ at $D$. Let $E$ be the reflection of $A$ across $BC$, $X$ the foot of the perpendicular from $A$ to $BE$, and $Y$ be the midpoint of $AX$. Let the line $BY$ meet $\Gamma$ again at $Z$. Prove that the line $BD$ is tangent to circumcircle of triangle $ADZ$ .
2005 Turkey Team Selection Test
We are given 5040 balls in k different colors, where the number of balls of each color is the same. The balls are put into 2520 bags so that each bag contains two balls of different colors. Find the smallest k such that, however the balls are distributed into the bags, we can arrange the bags around a circle so that no two balls of the same color are in two neighboring bags.
2004 Turkey Team Selection Test
An $11\times 11$ chess board is covered with one $\boxed{ }$ shaped and forty $\boxed{ }\boxed{ }\boxed{ }$ shaped tiles. Determine the squares where $\boxed{}$ shaped tile can be placed.
2004 Turkey Team Selection Test
Show that

$$ 

\min \{ |PA|, |PB|, |PC| \} + |PA| + |PB| + |PC| < |AB|+|BC|+|CA|

$$

if $P$ is a point inside $\triangle ABC$.
2004 Turkey Team Selection Test
Let $n$ be a positive integer. Determine integers, $n+1 \leq r \leq 3n+2$ such that for all integers $a_1,a_2,\dots,a_m,b_1,b_2,\dots,b_m$ satisfying the equations

$$

a_1b_1^k+a_2b_2^k+\dots + a_mb_m^k=0 $$

for every $1 \leq k \leq n$, the condition

$$

r %Error. "divides" is a bad command.

a_1b_1^r+a_2b_2^r+\dots + a_mb_m^r=0 $$

also holds.
2004 Turkey Team Selection Test
Find all possible values of $x-\lfloor x\rfloor$ if $\sin \alpha = 3/5$ and $x=5^{2003}\sin {(2004\alpha)}$.
2004 Turkey Team Selection Test
Let $\triangle ABC$ be an acute triangle, $O$ be its circumcenter, and $D$ be a point different that $A$ and $C$ on the smaller $AC$ arc of its circumcircle. Let $P$ be a point on $[AB]$ satisfying $\widehat{ADP} = \widehat {OBC}$ and $Q$ be a point on $[BC]$ satisfying $\widehat{CDQ}=\widehat {OBA}$. Show that $\widehat {DPQ} = \widehat {DOC}$.
2004 Turkey Team Selection Test
Each student in a classroom has $0,1,2,3,4,5$ or $6$ pieces of candy. At each step the teacher chooses some of the students, and gives one piece of candy to each of them and also to any other student in the classroom who is friends with at least one of these students. A student who receives the seventh piece eats all $7$ pieces. Assume that for every pair of students in the classroom, there is at least one student who is friend swith exactly one of them. Show that no matter how many pieces each student has at the beginning, the teacher can make them to have any desired numbers of pieces after finitely many steps.
2003 Turkey Team Selection Test
Let $M = \{(a,b,c,d)|a,b,c,d \in \{1,2,3,4\} \text{ and } abcd > 1\}$. For each $n\in \{1,2,\dots, 254\}$, the sequence $(a_1, b_1, c_1, d_1)$, $(a_2, b_2, c_2, d_2)$, $\dots$, $(a_{255}, b_{255},c_{255},d_{255})$ contains each element of $M$ exactly once and the equality $$|a_{n+1} - a_n|+|b_{n+1} - b_n|+|c_{n+1} - c_n|+|d_{n+1} - d_n| = 1$$ holds.  If $c_1 = d_1 = 1$, find all possible values of the pair $(a_1,b_1)$.
2003 Turkey Team Selection Test
Let $K$ be the intersection of the diagonals of a convex quadrilateral $ABCD$. Let $L\in [AD]$, $M \in [AC]$, $N \in [BC]$ such that $KL\parallel AB$, $LM\parallel DC$, $MN\parallel AB$. Show that $$\dfrac{Area(KLMN)}{Area(ABCD)} < \dfrac {8}{27}.$$
2003 Turkey Team Selection Test
Is there an arithmetic sequence with



a. $2003$



b. infinitely many



terms such that each term is a power of a natural number with a degree greater than $1$?
2003 Turkey Team Selection Test
Find the least



a. positive real number



b. positive integer



$t$ such that the equation $(x^2+y^2)^2 + 2tx(x^2 + y^2) = t^2y^2$ has a solution where $x,y$ are positive integers.
2003 Turkey Team Selection Test
Let $A$ be a point on a circle with center $O$ and $B$ be the midpoint of $[OA]$. Let $C$ and $D$ be points on the circle such that they lie on the same side of the line $OA$ and $\widehat{CBO} = \widehat{DBA}$. Show that the reflection of the midpoint of $[CD]$ over $B$ lies on the circle.
2003 Turkey Team Selection Test
For all positive integers $n$, let $p(n)$ be the number of non-decreasing sequences of positive integers such that for each sequence, the sum of all terms of the sequence is equal to $n$. Prove that

$$\dfrac{1+p(1)+p(2) + \dots + p(n-1)}{p(n)} \leq \sqrt {2n}.$$
2002 Turkey Team Selection Test
If $ab(a+b)$ divides $a^2 + ab+ b^2$ for different integers $a$ and $b$, prove that $$|a-b|>\sqrt[3]{ab}.$$
2002 Turkey Team Selection Test
In a triangle $ABC$, the angle bisector of $\widehat{ABC}$ meets $[AC]$ at $D$, and the angle bisector of $\widehat{BCA}$ meets $[AB]$ at $E$. Let $X$ be the intersection of the lines $BD$ and $CE$ where $|BX|=\sqrt 3|XD|$ ve $|XE|=(\sqrt 3 - 1)|XC|$. Find the angles of triangle $ABC$.
2002 Turkey Team Selection Test
A positive integer $n$ and real numbers $a_1,\dots, a_n$ are given. Show that there exists integers $m$ and $k$ such that $$|\sum\limits_{i=1}^m a_i -\sum\limits_{i=m+1}^n a_i | \leq |a_k|.$$
2002 Turkey Team Selection Test
If a function $f$ defined on all real numbers has at least two centers of symmetry, show that this function can be written as sum of a linear function and a periodic function.



[For every real number $x$, if there is a real number $a$ such that $f(a-x) + f(a+x) =2f(a)$, the point $(a,f(a))$ is called a center of symmetry of the function $f$.]
2002 Turkey Team Selection Test
Two circles are internally tangent at a point $A$. Let $C$ be a point on the smaller circle other than $A$. The tangent line to the smaller circle at $C$ meets the bigger circle at $D$ and $E$; and the line $AC$ meets the bigger circle at $A$ and $P$. Show that the line $PE$ is tangent to the circle through $A$, $C$, and $E$.
2002 Turkey Team Selection Test
Consider $2n+1$ points in space, no four of which are coplanar where $n>1$. Each line segment connecting any two of these points is either colored red, white or blue. A subset $M$ of these points is called a connected monochromatic subset, if for each $a,b \in M$, there are points $a=x_0,x_1, \dots, x_l = b$ that belong to $M$ such that the line segments $x_0x_1, x_1x_2, \dots, x_{l-1}x_1$ are all have the same color. No matter how the points are colored, if there always exists a connected monochromatic $k-$subset, find the largest value of $k$. ($l > 1$)
2001 Turkey Team Selection Test
Each one of $2001$ children chooses a positive integer and writes down his number and names of some of other $2000$ children to his notebook. Let $A_c$ be the sum of the numbers chosen by the children who appeared in the notebook of the child $c$. Let $B_c$ be the sum of the numbers chosen by the children who wrote the name of the child $c$ into their notebooks. The number $N_c = A_c - B_c$ is assigned to the child $c$. Determine whether all of the numbers assigned to the children could be positive.
2001 Turkey Team Selection Test
A circle touches to diameter $AB$ of a unit circle with center $O$ at $T$ where $OT>1$. These circles intersect at two different points $C$ and $D$. The circle through $O$, $D$, and $C$ meet the line $AB$ at $P$ different from $O$. Show that

$$|PA|\cdot |PB| = \dfrac {|PT|^2}{|OT|^2}.$$
2001 Turkey Team Selection Test
For all integers $x,y,z$, let $$S(x,y,z) = (xy - xz, yz-yx, zx - zy).$$ Prove that for all integers $a$, $b$ and $c$ with $abc>1$, and for every integer $n\geq n_0$, there exists integers $n_0$ and $k$ with $0<k\leq abc$ such that $$S^{n+k}(a,b,c) \equiv S^n(a,b,c) \pmod {abc}.$$ ($S^1 = S$ and for every integer $m\geq 1$, $S^{m+1} = S \circ S^m.$

$(u_1, u_2, u_3) \equiv (v_1, v_2, v_3) \pmod M \Longleftrightarrow u_i \equiv v_i \pmod M (i=1,2,3).$)
2001 Turkey Team Selection Test
Find all ordered pairs of integers $(x,y)$ such that $5^x = 1 + 4y + y^4$.
2001 Turkey Team Selection Test
Let $H$ be the intersection of the altitudes of an acute triangle $ABC$ and $D$ be the midpoint of $[AC]$. Show that $DH$ passes through one of the intersection point of the circumcircle of $ABC$ and the circle with diameter $[BH]$.
2001 Turkey Team Selection Test
Show that there is no continuous function $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for every real number $x$

$$f(x-f(x)) = \dfrac x2.$$
2000 Turkey Team Selection Test
$(a)$ Prove that for every positive integer $n$, the number of ordered pairs $(x, y)$ of integers satisfying $x^2-xy+y^2 = n$ is divisible by $3.$

$(b)$ Find all ordered pairs of integers satisfying $x^2-xy+y^2=727.$
2000 Turkey Team Selection Test
Let $P(x)=x+1$ and $Q(x)=x^2+1.$ Consider all sequences $\langle(x_k,y_k)\rangle_{k\in\mathbb{N}}$ such that $(x_1,y_1)=(1,3)$ and $(x_{k+1},y_{k+1})$ is either $(P(x_k), Q(y_k))$ or $(Q(x_k),P(y_k))$ for each $k.  $ We say that a positive integer $n$ is nice if $x_n=y_n$ holds in at least one of these sequences. Find all nice numbers.
2000 Turkey Team Selection Test
Show that any triangular prism of inﬁnite length can be cut by a plane such that the resulting intersection is an equilateral triangle.
2000 Turkey Team Selection Test
Points $M,\ N,\ K,\ L$ are taken on the sides $AB,\ BC,\ CD,\ DA$ of a rhombus

$ABCD,$ respectively, in such a way that $MN\parallel LK$ and the distance between $MN$ and $KL$ is equal to the height of $ABCD.$ Show that the circumcircles of the triangles $ALM$ and $NCK$ intersect each other, while those of $LDK$ and $MBN$ do not.
2000 Turkey Team Selection Test
Suppose $f:\mathbb{R} \to \mathbb{R}$ is a function such that

$$|f(x+y)-f(x)-f(y)|\le 1\ \ \ \text{for all} \ \  x, y \in\mathbb R.$$

Prove that there is a function $g:\mathbb{R}\to\mathbb{R}$ such that $|f(x)-g(x)|\le 1$ and $g(x+y)=g(x)+g(y)$ for all $x,y \in\mathbb R.$
1999 Turkey Team Selection Test
Let $m \leq n$ be positive integers and $p$ be a prime. Let $p-$expansions of $m$ and $n$ be

$$m = a_0 + a_1p + \dots + a_rp^r$$$$n = b_0 + b_1p + \dots + b_sp^s$$

respectively, where $a_r, b_s \neq 0$, for all $i \in \{0,1,\dots,r\}$ and for all $j \in \{0,1,\dots,s\}$, we have $0 \leq a_i, b_j \leq p-1$ .

If $a_i \leq b_i$ for all  $i \in \{0,1,\dots,r\}$, we write $ m \prec_p n$. Prove that

$$p \nmid {{n}\choose{m}} \Leftrightarrow m \prec_p n$$.
1999 Turkey Team Selection Test
Let $L$ and $N$ be the mid-points of the diagonals $[AC]$ and $[BD]$ of the cyclic quadrilateral $ABCD$, respectively. If $BD$ is the bisector of the angle $ANC$, then prove that $AC$ is the bisector of the angle $BLD$.
1999 Turkey Team Selection Test
Determine all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that the set

$$\left \{   \frac{f(x)}{x}: x \neq 0 \textnormal{ and } x \in \mathbb{R}\right \}$$

is finite, and for all $x \in \mathbb{R}$

$$f(x-1-f(x)) = f(x) - x - 1$$
1999 Turkey Team Selection Test
Let the area and the perimeter of a cyclic quadrilateral $C$ be $A_C$ and $P_C$, respectively. If the area and the perimeter of the quadrilateral which is tangent to the circumcircle of $C$ at the vertices of $C$ are $A_T$ and $P_T$ , respectively, prove that $\frac{A_C}{A_T} \geq \left (\frac{P_C}{P_T}\right )^2$.
1999 Turkey Team Selection Test
Each of $A$, $B$, $C$, $D$, $E$, and $F$ knows a piece of gossip. They communicate by telephone via a central switchboard, which can connect only two of them at a time. During a conversation, each side tells the other everything he or she knows at that point. Determine the minimum number of calls for everyone to know all six pieces of gossip.
1999 Turkey Team Selection Test
Prove that the plane is not a union of the inner regions of finitely many parabolas. (The outer region of a parabola is the union of the lines not intersecting the parabola. The inner region of a parabola is the set of points of the plane that do not belong to the outer region of the parabola)
1998 Turkey Team Selection Test
Squares $BAXX^{'}$ and $CAYY^{'}$ are drawn in the exterior of a triangle $ABC$ with $AB = AC$. Let $D$ be the midpoint of $BC$, and $E$ and $F$ be the feet of the perpendiculars from an arbitrary point $K$ on the segment $BC$ to $BY$ and $CX$, respectively.

$(a)$ Prove that $DE = DF$ .

$(b)$ Find the locus of the midpoint of $EF$ .
1998 Turkey Team Selection Test
Let the sequence $(a_{n})$ be deﬁned by $a_{1} = t$ and $a_{n+1} = 4a_{n}(1 - a_{n})$ for $n \geq 1$. How many possible values of t are there, if $a_{1998} = 0$?
1998 Turkey Team Selection Test
Let $A = {1, 2, 3, 4, 5}$. Find the number of functions $f$ from the nonempty subsets of $A$ to $A$, such that $f(B) \in B$ for any $B \subset A$, and $f(B \cup C)$ is either $f(B)$ or $f(C)$ for any $B$, $C \subset A$
1998 Turkey Team Selection Test
Suppose $n$ houses are to be assigned to $n$ people. Each person ranks the houses in the order of preference, with no ties. After the assignment is made, it is observed that every other assignment would assign to at least one person a less preferred house. Prove that there is at least one person who received the house he/she preferred most under this assignment.
1998 Turkey Team Selection Test
In a triangle $ABC$, the circle through $C$ touching $AB$ at $A$ and the circle through $B$ touching $AC$ at $A$ have diﬀerent radii and meet again at $D$. Let $E$ be the point on the ray $AB$ such that $AB = BE$. The circle through $A$, $D$, $E$ intersect the ray $CA$ again at $F$ . Prove that $AF = AC$.
1998 Turkey Team Selection Test
Let $f(x_{1}, x_{2}, . . . , x_{n})$ be a polynomial with integer coeﬃcients of degree less than $n$. Prove that if $N$ is the number of $n$-tuples $(x_{1}, . . . , x_{n})$ with $0 \leq x_{i} < 13$ and $f(x_{1}, . . . , x_{n}) = 0 (mod 13)$, then $N$ is divisible by 13.
1997 Turkey Team Selection Test
In a triangle $ABC$ with a right angle at $A$, $H$ is the foot of the altitude from $A$. Prove that the sum of the inradii of the triangles $ABC$, $ABH$, and $AHC$ is equal to $AH$.
1997 Turkey Team Selection Test
The sequences $(a_{n})$, $(b_{n})$ are deﬁned by $a_{1} = \alpha$, $b_{1} = \beta$, $a_{n+1} = \alpha a_{n} - \beta b_{n}$, $b_{n+1} = \beta a_{n} + \alpha b_{n}$ for all $n > 0.$ How many pairs $(\alpha, \beta)$ of real numbers are there such that $a_{1997} = b_{1}$ and $b_{1997} = a_{1}$?
1997 Turkey Team Selection Test
In a football league, whenever a player is transferred from a team $X$ with $x$ players to a team $Y$ with $y$ players, the federation is paid $y-x$ billions liras by $Y$ if $y \geq x$, while the federation pays $x-y$ billions liras to $X$ if $x > y$. A player is allowed to change as many teams as he wishes during a season. Suppose that a season started with $18$ teams of $20$ players each. At the end of the season, $12$ of the teams turn out to have again $20$ players, while the remaining $6$ teams end up with $16,16, 21, 22, 22, 23$ players, respectively. What is the maximal amount the federation may have won during the season?
1997 Turkey Team Selection Test
A convex $ABCDE$ is inscribed in a unit circle, $AE$ being its diameter. If $AB = a$, $BC = b$, $CD = c$, $DE = d$ and $ab = cd =\frac{1}{4}$, compute $AC + CE$ in terms of $a, b, c, d.$
1997 Turkey Team Selection Test
Show that for each prime $p \geq 7$, there exist a positive integer $n$ and integers $x_{i}$, $y_{i}$ $(i = 1, . . . , n)$, not divisible by $p$, such that $x_{i}^{2}+ y_{i}^{2}\equiv x_{i+1}^{2}\pmod{p}$ where $x_{n+1} = x_{1}$
1997 Turkey Team Selection Test
If $x_{1}, x_{2},\ldots ,x_{n}$ are positive real numbers with $x_{1}^2+x_2^{2}+\ldots +x_{n}^{2}=1$, ﬁnd the minimum value of $\sum_{i=1}^{n}\frac{x_{i}^{5}}{x_{1}+x_{2}+\ldots +x_{n}-x_{i}}$.
1996 Turkey Team Selection Test
Let $	\prod_{n=1}^{1996}{(1+nx^{3^n})}= 1+ a_{1}x^{k_{1}}+ a_{2}x^{k_{2}}+...+ a_{m}x^{k_{m}}$



where $a_{1}, a_{1}, . . . , a_{m}$ are nonzero and $k_{1} < k_{2} <...< k_{m}$. Find $a_{1996}$.
1996 Turkey Team Selection Test
In a parallelogram $ABCD$ with $\angle A < 90$, the circle with diameter $AC$ intersects the lines $CB$ and $CD$ again at $E$ and $F$ , and the tangent to this circle at $A$ meets the line $BD$ at $P$ . Prove that the points $P$, $E$, $F$ are collinear.
1996 Turkey Team Selection Test
If $0=x_{1}<x_{2}<...<x_{2n+1}=1$ are real numbers with $x_{i+1}-x_{i} \leq h$ for $1 \leq i \leq 2n$, show that

$\frac{1-h}{2}<\sum_{i=1}^{n}{x_{2i}(x_{2i+1}-x_{2i-1})}\leq \frac{1+h}{2}$
1996 Turkey Team Selection Test
The diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ with $S_{ABC} = S_{ADC}$ intersect at $E$. The lines through $E$ parallel to $AD$, $DC$, $CB$, $BA$

meet $AB$, $BC$, $CD$, $DA$ at $K$, $L$, $M$, $N$, respectively. Compute the ratio $\frac{S_{KLMN}}{S_{ABC}}$