Navvye/Olympiad-Problems-CSV huggingface dataset:
2015 German National Olympiad
To prepare a stay abroad, students meet at a workshop including an excursion. To promote interaction between the students, they are to be distributed to two busses such that not too many of the students in the same bus know each other.

Every student knows all those who know her. The number of such pairwise acquaintances is $k$.



Prove that it is possible to distribute the students such that the number of pairwise acquaintances in each bus is at most $\frac{k}{3}$.
2015 German National Olympiad
Let $k$ be a positive integer. Define $n_k$ to be the number with decimal representation $70...01$ where there are exactly $k$ zeroes. Prove the following assertions:



a) None of the numbers $n_k$ is divisible by $13$.

b) Infinitely many of the numbers $n_k$ are divisible by $17$.
2015 German National Olympiad
Let $ABCD$ be a convex quadrilateral such that the circle with diameter $AB$ touches the line $CD$. Prove that that the circle with diameter $CD$ touches the line $AB$ if and only if $BC$ and $AD$ are parallel.
2015 German National Olympiad
Prove that for all $x,y,z>0$, the inequality

$$\frac{x+y+z}{3}+\frac{3}{\frac{1}{x}+\frac{1}{y}+\frac{1}{z}} \ge 5 \sqrt[3]{\frac{xyz}{16}}$$holds. Determine if equality can hold and if so, in which cases it occurs.
2009 German National Olympiad
Find all non-negative real numbers $a$ such that the equation $$ \sqrt[3]{1+x}+\sqrt[3]{1-x}=a $$ has at least one real solution $x$ with $0 \leq x \leq 1$.

For all such $a$, what is $x$?
2009 German National Olympiad
Find all positive interger $ n$ so that $ n^3-5n^2+9n-6$ is perfect square number.
2009 German National Olympiad
Let $ ABCD$ be a (non-degenerate) quadrangle and $ N$ the intersection of $ AC$ and $ BD$. Denote by $ a,b,c,d$ the length of the altitudes from $ N$ to $ AB,BC,CD,DA$, respectively.



Prove that $ \frac{1}{a}+\frac{1}{c} = \frac{1}{b}+\frac{1}{d}$ if $ ABCD$ has an incircle.



Extension: Prove that the converse is true, too.



[If this has already been posted, I humbly apologize. A quick search turned up nothing.]
2009 German National Olympiad
Let $a$ and $b$ be two fixed positive real numbers. Find all real numbers $x$, such that inequality holds $$\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{a+b-x}} < \frac{1}{\sqrt{a}} + \frac{1}{\sqrt{b}}$$
2009 German National Olympiad
Let a triangle $ ABC$. $ E,F$ in segment $ AB$ so that $ E$ lie between $ AF$ and half of circle with diameter $ EF$ is tangent with $ BC,CA$ at $ G,H$. $ HF$ cut $ GE$ at $ S$, $ HE$ cut $ FG$ at $ T$. Prove that $ C$ is midpoint of $ ST$.
2009 German National Olympiad
Let a sequences: $ x_0\in [0;1],x_{n+1}=\frac56-\frac43 \Big|x_n-\frac12\Big|$. Find the "best" $ |a;b|$ so that for all $ x_0$ we have $ x_{2009}\in [a;b]$
2001 German National Olympiad
Determine all real numbers $q$ for which the equation $x^4 -40x^2 +q = 0$ has four real solutions which form an arithmetic progression
2001 German National Olympiad
Determine the maximum possible number of points you can place in a rectangle with lengths $14$ and $28$ such that any two of those points are more than $10$ apart from each other.
2001 German National Olympiad
Wiebke and Stefan play the following game on a rectangular sheet of paper. They start with a rectangle with $60$ rows and $40$ columns and cut it in turns into smaller rectangles. The cuttings must be made along the gridlines, and a player in turn may cut only one smaller rectangle. By that, Stefan makes only vertical cuts, while Wiebke makes only horizontal cuts. A player who cannot make a regular move loses the game.

(a) Who has a winning strategy if Stefan makes the first move?

(b) Who has a winning strategy if Wiebke makes the first move?
2001 German National Olympiad
In how many ways can the ”Nikolaus’ House” (see the picture) be drawn? Edges may not be erased nor duplicated, and no additional edges may be drawn.
2001 German National Olympiad
The Fibonacci sequence is given by $x_1 = x_2 = 1$ and $x_{k+2} = x_{k+1} + x_k$ for each $k \in N$.

(a) Prove that there are Fibonacci numbes  that end in a $9$ in the decimal system.

(b) Determine for which $n$ can a Fibonacci number end in $n$ $9$-s in the decimal system.
2001 German National Olympiad
In a pyramid $SABCD$ with the base $ABCD$ the triangles $ABD$ and $BCD$ have equal areas. Points $M,N,P,Q$ are the midpoints of the edges $AB,AD,SC,SD$ respectively. Find the ratio between the volumes of the pyramids $SABCD$ and $MNPQ$
2001 German National Olympiad
Let $ABC$ be a triangle with $\angle A = 90^o$ and $\angle B < \angle C$. The tangent at $A$ to the circumcircle $k$ of $\vartriangle ABC$ intersects line $BC$ at $D$. Let $E$ be the reflection of $A$ in $BC$. Also, let $X$ be the feet of the perpendicular from $A$ to $BE$ and let $Y$ be the midpoint of $AX$. Line $BY$ meets $k$ again at $Z$. Prove that line $BD$ is tangent to the circumcircle of $\vartriangle ADZ$.
2000 German National Olympiad
For each real parameter $a$, find the number of real solutions to the system

$|x|+|y| = 1$,

$x^2 +y^2 = a$.
2000 German National Olympiad
For an integer $n \ge 2$, find all real numbers $x$ for which the polynomial $f(x) = (x-1)^4 +(x-2)^4 +...+(x-n)^4$ takes its minimum value.
2000 German National Olympiad
Suppose that an interior point $O$ of a triangle $ABC$ is such that the angles $\angle BAO,\angle CBO, \angle ACO$ are all greater than or equal to $30^o$. Prove that the triangle $ABC$ is equilateral.
2000 German National Olympiad
Find all nonnegative solutions $(x,y,z)$ to the system

$\sqrt{x+y}+\sqrt{z} = 7$,

$\sqrt{x+z}+\sqrt{y} = 7$,

$\sqrt{y+z}+\sqrt{x} = 5$.
2000 German National Olympiad
(a) Let be given $2n$ distinct points on a circumference, $n$ of which are red and $n$ are blue. Prove that one can join these points pairwise by $n$ segments so that no two segments intersect and the endpoints of each segments have different colors.

(b) Show that the statement from (a) remains valid if the points are in an arbitrary position in the plane so that no three of them are collinear.
2000 German National Olympiad
A sequence ($a_n$) satisfies the following conditions:

(i) For each $m \in N$ it holds that $a_{2^m} = 1/m$.

(ii) For each natural $n \ge  2$ it holds that $a_{2n-1}a_{2n} = a_n$.

(iii) For all integers $m,n$ with $2m > n \ge  1$ it holds that $a_{2n}a_{2n+1} = a_{2^m+n}$.

Determine $a_{2000}$. You may assume that such a sequence exists.
1999 German National Olympiad
Find all $x,y$  which satisfy the equality $x^2 +xy+y^2 = 97$, when $x,y$ are

a) natural numbers,

b) integers

.
1999 German National Olympiad
Determine all real numbers $x$ for which $1+\frac{x}{2} -\frac{x^2}{8} \le \sqrt{1+x} \le  1+\frac{x}{2}$
1999 German National Olympiad
A mathematician investigates methods of finding area of a convex quadrilateral obtains the following formula for the area $A$ of a quadrilateral with consecutive sides $a,b,c,d$:

$A =\frac{a+c}{2}\frac{b+d}{2}$ (1) and $A = \sqrt{(p-a)(p-b)(p-c)(p-d)}$ (2) where $p = (a+b+c+d)/2$.

However, these formulas are not valid for all convex quadrilaterals. Prove that (1) holds if and only if the quadrilateral is a rectangle, while (2) holds if and only if the quadrilateral is cyclic.
1999 German National Olympiad
A convex polygon $P$ is placed inside a unit square $Q$. Prove that the perimeter of $P$ does not exceed $4$.
1999 German National Olympiad
Consider the following inequality for real numbers $x,y,z$: $|x-y|+|y-z|+|z-x| \le a \sqrt{x^2 +y^2 +z^2}$ .

(a) Prove that the inequality is valid for $a = 2\sqrt2$

(b) Assuming that $x,y,z$ are nonnegative, show that the inequality is also valid for $a = 2$.
1999 German National Olympiad
Suppose that an isosceles right-angled triangle is divided into $m$ acute-angled triangles. Find the smallest possible $m$ for which this is possible.
1999 German National Olympiad
Determine all pairs ($m,n$) of natural numbers for which $4^m + 5^n$ is a perfect square.
1998 German National Olympiad
Find all possible numbers of lines in a plane which intersect in exactly $37$ points.
1998 German National Olympiad
Two pupils $A$ and $B$ play the following game. They begin with a pile of $1998$ matches and $A$ plays first. A player who is on turn must take a nonzero square number of matches from the pile. The winner is the one who makes the last move. Decide who has the winning strategy and give one such strategy.
1998 German National Olympiad
For each nonnegative integer $k$ find all nonnegative integers $x,y,z$ such that $x^2 +y^2 +z^2 = 8^k$
1998 German National Olympiad
Let $a$ be a positive real number. Then prove that the polynomial

$$ p(x)=a^3x^3+a^2x^2+ax+a $$

has integer roots if and only if $a=1$ and determine those roots.
1998 German National Olympiad
A sequence ($a_n$) is given by $a_0 = 0, a_1 = 1$ and $a_{k+2} = a_{k+1} +a_k$ for all integers $k \ge 0$.

Prove that the inequality   $\sum_{k=0}^n \frac{a_k}{2^k}< 2$ holds for all positive integers $n$.
1998 German National Olympiad
Find all real pairs $(x,y)$ that solve the system of equations \begin{align} x^5 &= 21x^3+y^3

\\ y^5 &= x^3+21y^3. \end{align}
1998 German National Olympiad
Prove that the following statement holds for all odd integers $n \ge 3$:

If a quadrilateral $ABCD$ can be partitioned by lines into $n$ cyclic quadrilaterals, then $ABCD$ is itself cyclic.
1997 German National Olympiad
Prove that there are no perfect squares $a,b,c$ such that $ab-bc = a$.
1997 German National Olympiad
For a positive integer $k$, let us denote by $u(k)$ the greatest odd divisor of $k$.

Prove that, for each $n \in N$, $\frac{1}{2^n} \sum_{k = 1}^{2^n} \frac{u(k)}{k}> \frac{2}{3}$.
1997 German National Olympiad
In a convex quadrilateral $ABCD$ we are given that $\angle CBD = 10^o$, $\angle CAD = 20^o$, $\angle ABD = 40^o$, $\angle BAC = 50^o$. Determine the angles $\angle BCD$ and $\angle ADC$.
1997 German National Olympiad
Find all real solutions $(x,y,z)$ of the system of equations

$x^3 = 2y-1$,

$y^3 = 2z-1$,

$z^3 = 2x-1$.
1997 German National Olympiad
We are given $n$ discs in a plane, possibly overlapping, whose union has the area $1$. Prove that we can choose some of them which are mutually disjoint and have the total area greater than $1/9$.
1997 German National Olympiad
Let us define $f$ and $g$ by $f(x) = x^5 +5x^4 +5x^3 +5x^2 +1$, $g(x) = x^5 +5x^4 +3x^3 -5x^2 -1$.

Determine all prime numbers $p$ such that, for at least one integer $x, 0 \le x < p-1$, both $f(x)$ and $g(x)$ are divisible by $p$. For each such $p$, find all $x$ with this property.
1997 German National Olympiad
An approximate construction of a regular pentagon goes as follows. Inscribe an arbitrary convex pentagon $P_1P_2P_3P_4P_5$ in a circle. Now choose an arror bound $\epsilon > 0$ and apply the following procedure.

(a) Denote $P_0 = P_5$ and $P_6 = P_1$ and construct the midpoint $Q_i$ of the circular arc $P_{i-1}P_{i+1}$ containing $P_i$.

(b) Rename the vertices $Q_1,...,Q_5$ as $P_1,...,P_5$.

(c) Repeat this procedure until the difference between the lengths of the longest and the shortest among the arcs $P_iP_{i+1}$ is less than $\epsilon$.

Prove this procedure must end in a finite time for any choice of $\epsilon$ and the points $P_i$.
1996 German National Olympiad
Find all natural numbers $n$ with the following property:

Given the decimal writing of $n$, adding a few digits one can obtain the decimal writing of $1996n$.
1996 German National Olympiad
Let $a$ and $b$ be positive real numbers smaller than $1$. Prove that the following two statements are equivalent:

(i) $a+b = 1$,

(ii) Whenever $x,y$ are positive real numbers such that $x < 1, y < 1, ax+by < 1$, the following inequlity holds:

$\frac{1}{1-ax-by} \le \frac{a}{1-x} + \frac{b}{1-y}$
1996 German National Olympiad
Let be given an arbitrary tetrahedron $ABCD$ with volume $V$. Consider all lines which pass through the barycenter $S$ of the tetrahedron and intersect the edges $AD,BD,CD$ at points $A',B',C$ respectively. It is known that among the obtained tetrahedra there exists one with the minimal volume. Express this minimal volume in terms of $V$
1996 German National Olympiad
Find all pairs of real numbers $(x,y)$ which satisfy the system

$x-y = 7$

$\sqrt[3]{x^2}+\sqrt[3]{xy}+\sqrt[3]{y^2} = 7$
1996 German National Olympiad
Given two non-intersecting chords $AB$ and $CD$ of a circle $k$ and a length $a <CD$. Determine a point $X$ on $k$ with the following property: If lines $XA$ and $XB$ intersect $CD$ at points $P$ and $Q$ respectively, then $PQ = a$. Show how to construct all such points $X$ and prove that the obtained points indeed have the desired property.
1996 German National Olympiad
Prove the following statement:

If a polynomial $p(x) = x^3 + Ax^2 + Bx +C$ has three real roots at least two of which are distinct, then $A^2 +B^2 +18C > 0$.
1996 German National Olympiad
Each point of a plane is colored in one of three colors: red, black and blue. Prove that there exists a rectangle in this plane whose vertices all have the same color.
1965 German National Olympiad
For a given positive real parameter $p$, solve the equation $\sqrt{p+x}+\sqrt{p-x }= x$.
1965 German National Olympiad
Determine which of the prime numbers $2,3,5,7,11,13,109,151,491$ divide $z =1963^{1965} -1963$.
1965 German National Olympiad
Two parallelograms $ABCD$ and  $A'B'C'D'$ are given in space. Points  $A'',B'',C'',D''$ divide the segments $AA',BB',CC',DD'$ in the same ratio. What can be said about the quadrilateral $A''B''C''D''$?
1965 German National Olympiad
Find the locus of points in the plane, the sum of whose distances from the sides of a regular polygon is five times the inradius of the pentagon.
1965 German National Olympiad
Determine all triples of nonzero decimal digits $(x,y,z)$ for which the equality $\sqrt{ \underbrace{xxx...x}_{2n}- \underbrace{yy...y}_{n}}= \underbrace{zzz...z}_{n}$ holds for at least two different natural numbers $n$.
1965 German National Olympiad
Let $\alpha,\beta, \gamma$ be the angles of a triangle. Prove that $\cos\alpha, + \cos\beta + \cos\gamma \le \frac{3}{2} $ and find the cases of equality.
2020 Greece National Olympiad
Find all non constant polynomials $P(x),Q(x)$ with real coefficients such that: $P((Q(x))^3)=xP(x)(Q(x))^3$
2020 Greece National Olympiad
Given a line segment $AB$ and a point $C$ lies inside it such that $AB=3 \cdot AC$ . Construct a parallelogram $ACDE$ such that $AC=DE=CE>AR$. Let $Z$ be a point on $AC$ such that $\angle AEZ=\angle ACE =\omega$. Prove that the line passing through point $B$ and perpendicular on side $EC$, and the line passing through point $D$  and perpendicular on side $AB$, intersect on point , let it be $K$, lying on line $EZ$.
2020 Greece National Olympiad
On the board there are written in a row, the integers from $1$ until $2030$ (included that) in an increasing order.

We have the right of  ''movement'' $K$:

We choose any two numbers $a,b$ that are written in consecutive positions and we replace the pair $(a,b)$ by the number $(a-b)^{2020}$.

We repeat the movement $K$, many times until only one number remains written on the board. Determine whether it would be possible, that number to be:

(i) $2020^{2020}$  (ii)$2021^{2020}$
2020 Greece National Olympiad
Find all values of the positive integer $k$ that has the property:

There are no positive integers $a,b$ such that the expression $A(k,a,b)=\frac{a+b}{a^2+k^2b^2-k^2ab}$ is a composite positive number.
2019 Greece National Olympiad
Define the sequnce  ${(a_n)}_{n\ge1}$ by $a_1=1$ and $a_n=5a_{n-1}+3^{n-1}$ for $n\ge2$.

Find the greatest power of $2$ that divides $a_{2^{2019}}$.
2019 Greece National Olympiad
Let $ABC$ be a triangle with $AB<AC<BC$.Let $O$ be the center of it's circumcircle and $D$ be the center

of minor arc $\overarc{AB}$.Line $AD$ intersects $BC$ at $E$ and the circumcircle of $BDE$ intersects $AB$ at

$Z$ ,($Z\not=B$).The circumcircle of $ADZ$ intersects $AC$ at $H$ ,($H\not=A$),prove that $BE=AH$.
2019 Greece National Olympiad
Find all positive rational $(x,y)$ that satisfy the equation : $$yx^y=y+1$$
2019 Greece National Olympiad
Given a $n\times m$ grid we play the following game . Initially we place $M$ tokens in each of $M$ empty cells and at the end of the game we need to fill the whole grid with tokens.For that purpose  we are allowed to make the following move:If an empty cell shares a common side with at least two other cells that contain a token then we can place a token in this cell.Find the minimum value of $M$ in terms of $m,n$ that enables us to win the game.
2018 Greece National Olympiad
Let $(x_n), n\in\mathbb{N}$ be a sequence such that $x_{n+1}=3x_n^3+x_n, \forall n\in\mathbb{N}$

and $x_1=\frac{a}{b}$ where $a,b$ are positive integers such that $3\not|b$. If $x_m$ is a square of a rational number for some positive integer $m$, prove that $x_1$ is also a square of a rational number.
2018 Greece National Olympiad
Let $ABC$ be an acute-angled triangle with $AB<AC<BC$ and $c(O,R)$ the circumscribed circle. Let $D, E$ be points in the small arcs $AC, AB$ respectively. Let $K$ be the intersection point of $BD,CE$ and $N$ the second common point of the circumscribed circles of the triangles $BKE$ and $CKD$. Prove that $A, K, N$ are collinear if and only if $K$ belongs to the symmedian of $ABC$ passing from $A$.
2018 Greece National Olympiad
Let $n,m$ be positive integers such that $n<m$ and $a_1, a_2, ..., a_m$ be different real numbers.

(a) Find all polynomials $P$ with real coefficients and degree at most $n$ such that:

$|P(a_i)-P(a_j)|=|a_i-a_j|$ for all $i,j=\{1, 2, ..., m\}$ such that $i<j$.

(b) If $n,m\ge 2$ does there exist a polynomial $Q$ with real coefficients and degree $n$ such that:

$|Q(a_i)-Q(a_j)|<|a_i-a_j|$ for all $i,j=\{1, 2, ..., m\}$ such that $i<j$

Edit: See #3
2018 Greece National Olympiad
In the plane, there are $n$ points ($n\ge 4$) where no 3 of them are collinear. Let $A(n)$ be the number of parallelograms whose vertices are those points with area $1$. Prove the following inequality:

$A(n)\leq \frac{n^2-3n}{4}$ for all $n\ge 4$
2017 Greece National Olympiad
An acute triangle $ABC$ with $AB<AC<BC$ is inscribed in a circle $c(O,R)$. The circle $c_1(A,AC)$ intersects the circle $c$ at point $D$ and intersects $CB$ at $E$. If the line $AE$ intersects $c$ at $F$ and $G$ lies in $BC$ such that $EB=BG$, prove that $F,E,D,G$ are concyclic.
2017 Greece National Olympiad
Let $A$ be a point in the plane and $3$ lines which pass through this point divide the plane in $6$ regions.

In each region there are $5$ points. We know that no three of the $30$ points existing in these regions are collinear. Prove that there exist at least $1000$ triangles whose vertices are points of those regions such that $A$ lies either in the interior or on the side of the triangle.
2017 Greece National Olympiad
Find all integer triples $(a,b,c)$ with $a>0>b>c$ whose sum equal $0$ such that the number

$$N=2017-a^3b-b^3c-c^3a$$is a perfect square of an integer.
2017 Greece National Olympiad
Let $u$ be the positive root of the equation $x^2+x-4=0$. The polynomial

$$P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_0$$where $n$ is positive integer has non-negative integer coefficients and $P(u)=2017$.

1) Prove that $a_0+a_1+...+a_n\equiv 1\mod 2$.

2) Find the minimum possible value of $a_0+a_1+...+a_n$.
2016 Greece National Olympiad
Find all triplets of nonnegative integers $(x,y,z)$ and $x\leq y$ such that

$x^2+y^2=3 \cdot 2016^z+77$
2016 Greece National Olympiad
Find all monic polynomials $P,Q$ which are non-constant, have real coefficients and they satisfy $2P(x)=Q(\frac{(x+1)^2}{2})-Q(\frac{(x-1)^2}{2})$ and $P(1)=1$ for all real $x$.
2016 Greece National Olympiad
$ABC$ is an acute isosceles triangle $(AB=AC)$ and $CD$ one altitude. Circle $C_2(C,CD)$ meets $AC$ at $K$, $AC$ produced at $Z$ and circle $C_1(B, BD)$ at $E$. $DZ$ meets circle $(C_1)$ at $M$. Show that:

a) $\widehat{ZDE}=45^0$

b) Points $E, M, K$ lie on a line.

c) $BM//EC$
2016 Greece National Olympiad
A square $ABCD$ is divided into $n^2$ equal small (fundamental) squares by drawing lines parallel to its sides.The vertices of the fundamental squares are called vertices of the grid.A rhombus is called nice when:

$\bullet$ It is not a square

$\bullet$ Its vertices are points of the grid

$\bullet$ Its diagonals are parallel to the sides of the square $ABCD$



Find (as a function of $n$) the number of the nice rhombuses ($n$ is a positive integer greater than $2$).
2015 Greece National Olympiad
Find all triplets $(x,y,p)$ of positive integers such that $p$ be a prime number and $\frac{xy^3}{x+y}=p$
2015 Greece National Olympiad
Let $P(x)=ax^3+(b-a)x^2-(c+b)x+c$ and $Q(x)=x^4+(b-1)x^3+(a-b)x^2-(c+a)x+c$ be polynomials of $x$ with $a,b,c$ non-zero real numbers and $b>0$.If $P(x)$ has three distinct real roots $x_0,x_1,x_2$ which are also roots of $Q(x)$ then:

A)Prove that $abc>28$,

B)If $a,b,c$ are non-zero integers with $b>0$,find all their possible values.
2015 Greece National Olympiad
Given is a triangle $ABC$ with $\angle{B}=105^{\circ}$.Let $D$ be a point on $BC$ such that $\angle{BDA}=45^{\circ}$.



A) If $D$ is the midpoint of $BC$ then prove that $\angle{C}=30^{\circ}$,

B) If $\angle{C}=30^{\circ}$ then prove that $D$ is the midpoint of $BC$
2015 Greece National Olympiad
Square $ABCD$ with side-length $n$ is divided into $n^2$ small (fundamental) squares by drawing lines parallel to its sides (the case $n=5$ is presented on the diagram).The squares' vertices that lie inside (or on the boundary) of the triangle $ABD$ are connected with each other with arcs.Starting from $A$,we move only upwards or to the right.Each movement takes place on the segments that are defined by the fundamental squares and the arcs of the circles.How many possible roots are there in order to reach $C$;
2014 Greece National Olympiad
Find all the polynomials with real coefficients which satisfy $ (x^2-6x+8)P(x)=(x^2+2x)P(x-2)$ for all $x\in \mathbb{R}$.
2014 Greece National Olympiad
Find all the integers $n$ for which $\frac{8n-25}{n+5}$ is cube of a rational number.
2014 Greece National Olympiad
For even positive integer $n$ we put all numbers $1,2,...,n^2$ into the squares of an $n\times n$ chessboard (each number appears once and only once).

Let $S_1$ be the sum of the numbers put in the black squares and $S_2$ be the sum of the numbers put in the white squares. Find all $n$ such that we can achieve $\frac{S_1}{S_2}=\frac{39}{64}.$
2014 Greece National Olympiad
We are given a circle $c(O,R)$ and two points $A,B$ so that $R<AB<2R$.The circle $c_1 (A,r)$ ($0<r<R$) crosses the circle $c$ at C,D ($C$ belongs to the short arc $AB$).From $B$ we consider the tangent lines $BE,BF$ to the circle $c_1$ ,in such way that $E$ lays out of the circle $c$.If $M\equiv EC\cap DF$ show that the quadrilateral $BCFM$ is cyclic.
2013 Greece National Olympiad
Let the sequence of real numbers $(a_n),n=1,2,3...$ with $a_1=2$ and $a_n=\left(\frac{n+1}{n-1} \right)\left(a_1+a_2+...+a_{n-1} \right),n\geq 2$.

Find the term $a_{2013}$.
2013 Greece National Olympiad
Solve in integers the following equation:

$$y=2x^2+5xy+3y^2$$
2013 Greece National Olympiad
We define the sets $A_1,A_2,...,A_{160}$ such that $\left|A_{i} \right|=i$ for all $i=1,2,...,160$. With the elements of these sets we create new sets $M_1,M_2,...M_n$ by the following procedure: in the first step we choose some of the sets $A_1,A_2,...,A_{160}$ and we remove from each of them the same number of elements. These elements that we removed are the elements of $M_1$. In the second step we repeat the same procedure in the sets that came of the implementation of the first step and so we define $M_2$. We continue similarly until there are no more elements in $A_1,A_2,...,A_{160}$, thus defining the sets $M_1,M_2,...,M_n$. Find the minimum value of $n$.
2013 Greece National Olympiad
Let a triangle $ABC$ inscribed in circle $c(O,R)$ and $D$ an arbitrary point on $BC$(different from the midpoint).The circumscribed circle of $BOD$,which is $(c_1)$, meets $c(O,R)$ at $K$ and $AB$ at $Z$.The circumscribed circle of $COD$ $(c_2)$,meets $c(O,R)$ at $M$ and $AC$ at $E$.Finally, the circumscribed circle of $AEZ$ $(c_3)$,meets $c(O,R)$ at $N$.Prove that $\triangle{ABC}=\triangle{KMN}.$