Navvye/Olympiad-Problems-CSV huggingface dataset:
2010 Germany Team Selection Test
Find the largest possible integer $k$, such that the following statement is true:

Let $2009$ arbitrary non-degenerated triangles be given. In every triangle the three sides are coloured, such that one is blue, one is red and one is white. Now, for every colour separately, let us sort the lengths of the sides. We obtain

$$ \left. \begin{array}{rcl}

 & b_1 \leq b_2\leq\ldots\leq b_{2009} & \textrm{the lengths of the blue sides }\\
 & r_1 \leq r_2\leq\ldots\leq r_{2009} & \textrm{the lengths of the red sides }\\
 \textrm{and } & w_1 \leq w_2\leq\ldots\leq w_{2009} & \textrm{the lengths of the white sides }\\
 \end{array}\right.$$

Then there exist $k$ indices $j$ such that we can form a non-degenerated triangle with side lengths $b_j$, $r_j$, $w_j$.



Proposed by Michal Rolinek, Czech Republic
2010 Germany Team Selection Test
Let $P$ be a polygon that is convex and symmetric to some point $O$. Prove that for some parallelogram $R$ satisfying $P\subset R$ we have $$\frac{|R|}{|P|}\leq \sqrt 2$$

where $|R|$ and $|P|$ denote the area of the sets $R$ and $P$, respectively.



Proposed by Witold Szczechla, Poland
2010 Germany Team Selection Test
On a $999\times 999$ board a limp rook can move in the following way: From any square it can move to any of its adjacent squares, i.e. a square having a common side with it, and every move must be a turn, i.e. the directions of any two consecutive moves must be perpendicular. A non-intersecting route of the limp rook consists of a sequence of pairwise different squares that the limp rook can visit in that order by an admissible sequence of moves. Such a non-intersecting route is called cyclic, if the limp rook can, after reaching the last square of the route, move directly to the first square of the route and start over.

How many squares does the longest possible cyclic, non-intersecting route of a limp rook visit?



Proposed by Nikolay Beluhov, Bulgaria
2010 Germany Team Selection Test
Let $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f(a)-f(b)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f(c)$ for some positive integer $c$.



Proposed by Juhan Aru, Estonia
2010 Germany Team Selection Test
Five identical empty buckets of $2$-liter capacity stand at the vertices of a regular pentagon. Cinderella and her wicked Stepmother go through a sequence of rounds: At the beginning of every round, the Stepmother takes one liter of water from the nearby river and distributes it arbitrarily over the five buckets. Then Cinderella chooses a pair of neighbouring buckets, empties them to the river and puts them back. Then the next round begins. The Stepmother goal's is to make one of these buckets overflow. Cinderella's goal is to prevent this. Can the wicked Stepmother enforce a bucket overflow?



Proposed by Gerhard Woeginger, Netherlands
2010 Germany Team Selection Test
Let $ABCD$ be a circumscribed quadrilateral. Let $g$ be a line through $A$ which meets the segment $BC$ in $M$ and the line $CD$ in $N$. Denote by $I_1$, $I_2$ and $I_3$ the incenters of $\triangle ABM$, $\triangle MNC$ and $\triangle NDA$, respectively. Prove that the orthocenter of $\triangle I_1I_2I_3$ lies on $g$.



Proposed by Nikolay Beluhov, Bulgaria
2010 Germany Team Selection Test
A sequence $\left(a_n\right)$ with $a_1 = 1$ satisfies the following recursion: In the decimal expansion of $a_n$ (without trailing zeros) let $k$ be the smallest digest then $a_{n+1} = a_n + 2^k.$ How many digits does $a_{9 \cdot 10^{2010}}$ have in the decimal expansion?
2010 Germany Team Selection Test
Let $ABC$ be a triangle with incenter $I$ and let $X$, $Y$ and $Z$ be the incenters of the triangles $BIC$, $CIA$ and $AIB$, respectively. Let the triangle $XYZ$ be equilateral. Prove that $ABC$ is equilateral too.



Proposed by  Mirsaleh Bahavarnia, Iran
2010 Germany Team Selection Test
Let $P(x)$ be a non-constant polynomial with integer coefficients. Prove that there is no function $T$ from the set of integers into the set of integers such that the number of integers $x$ with $T^n(x)=x$ is equal to $P(n)$ for every $n\geq 1$, where $T^n$ denotes the $n$-fold application of $T$.



Proposed by Jozsef Pelikan, Hungary
2010 Germany Team Selection Test
Let $ABC$ be a triangle. The incircle of $ABC$ touches the sides $AB$ and $AC$ at the points $Z$ and $Y$, respectively. Let $G$ be the point where the lines $BY$ and $CZ$ meet, and let $R$ and $S$ be points such that the two quadrilaterals $BCYR$ and $BCSZ$ are parallelogram.

Prove that $GR=GS$.



Proposed by Hossein Karke Abadi, Iran
2010 Germany Team Selection Test
For an integer $m\geq 1$, we consider partitions of a $2^m\times 2^m$ chessboard into rectangles consisting of cells of chessboard, in which each of the $2^m$ cells along one diagonal forms a separate rectangle of side length $1$. Determine the smallest possible sum of rectangle perimeters in such a partition.



Proposed by Gerhard Woeginger, Netherlands
2010 Germany Team Selection Test
Find all functions $f$ from the set of real numbers into the set of real numbers which satisfy for all $x$, $y$ the identity $$ f\left(xf(x+y)\right) = f\left(yf(x)\right) +x^2$$



Proposed by Japan
2010 Germany Team Selection Test
In the plane we have points $P,Q,A,B,C$ such triangles $APQ,QBP$ and $PQC$ are similar accordantly (same direction). Then let $A'$ ($B',C'$ respectively) be the intersection of lines $BP$ and $CQ$ ($CP$ and $AQ;$ $AP$ and $BQ,$ respectively.) Show that the points $A,B,C,A',B',C'$ lie on a circle.
2010 Germany Team Selection Test
We are given $m,n \in \mathbb{Z}^+.$ Show the number of solution $4-$tuples $(a,b,c,d)$ of the system



$$

ab + bc + cd - (ca + ad + db) = m\\
2 \left(a^2 + b^2 + c^2 + d^2 \right) - (ab + ac + ad + bc + bd + cd) = n

$$



is divisible by 10.
2010 Germany Team Selection Test
Find all functions $f: \mathbb{R} \to \mathbb{R}$  such that

$$f(x)f(y) = (x+y+1)^2 \cdot f \left( \frac{xy-1}{x+y+1} \right)$$ $\forall x,y \in \mathbb{R}$ with $x+y+1 \neq 0$ and $f(x) > 1$ $\forall x > 0.$
2010 Germany Team Selection Test
Let $a \in \mathbb{R}.$ Show that for $n \geq 2$ every non-real root $z$ of polynomial $X^{n+1}-X^2+aX+1$ satisfies the condition $|z| > \frac{1}{\sqrt[n]{n}}.$
2010 Germany Team Selection Test
Determine all $n \in \mathbb{Z}^+$ such that a regular hexagon (i.e. all sides equal length, all interior angles same size) can be partitioned in finitely many $n-$gons such that they can be composed into $n$ congruent regular hexagons in a non-overlapping way upon certain rotations and translations.
2010 Germany Team Selection Test
Let $f$ be any function that maps the set of real numbers into the set of real numbers. Prove that there exist real numbers $x$ and $y$ such that $$f\left(x-f(y)\right)>yf(x)+x$$



Proposed by Igor Voronovich, Belarus
2010 Germany Team Selection Test
For any integer $n\geq 2$, let $N(n)$ be the maxima number of triples $(a_i, b_i, c_i)$, $i=1, \ldots, N(n)$, consisting of nonnegative integers $a_i$, $b_i$ and $c_i$ such  that the following two conditions are satisfied:



$a_i+b_i+c_i=n$ for all $i=1, \ldots, N(n)$,

If $i\neq j$ then $a_i\neq a_j$, $b_i\neq b_j$ and $c_i\neq c_j$



Determine $N(n)$ for all $n\geq 2$.



Proposed by Dan Schwarz, Romania
2010 Germany Team Selection Test
Let $a$, $b$, $c$ be positive real numbers such that $ab+bc+ca\leq 3abc$. Prove that

$$\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}+3\leq \sqrt{2}\left(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\right)$$



Proposed by Dzianis Pirshtuk, Belarus
2010 Germany Team Selection Test
Find all positive integers $n$ such that there exists a sequence of positive integers $a_1$, $a_2$,$\ldots$, $a_n$ satisfying: $$a_{k+1}=\frac{a_k^2+1}{a_{k-1}+1}-1$$ for every $k$ with $2\leq k\leq n-1$.



Proposed by North Korea
2009 Germany Team Selection Test
In the plane we consider rectangles whose sides are parallel to the coordinate axes and have positive length. Such a rectangle will be called a box. Two boxes intersect if they have a common point in their interior or on their boundary. Find the largest $ n$ for which there exist $ n$ boxes $ B_1$, $ \ldots$, $ B_n$ such that $ B_i$ and $ B_j$ intersect if and only if $ i\not\equiv j\pm 1\pmod n$.



Proposed by Gerhard Woeginger, Netherlands
2009 Germany Team Selection Test
Let $ k$ and $ n$ be integers with $ 0\le k\le n - 2$. Consider a set $ L$ of $ n$ lines in the plane such that no two of them are parallel and no three have a common point. Denote by $ I$ the set of intersections of lines in $ L$. Let $ O$ be a point in the plane not lying on any line of $ L$. A point $ X\in I$ is colored red if the open line segment $ OX$ intersects at most $ k$ lines in $ L$. Prove that $ I$ contains at least $ \dfrac{1}{2}(k + 1)(k + 2)$ red points.



Proposed by Gerhard Woeginger, Netherlands
2009 Germany Team Selection Test
Let $ a$, $ b$, $ c$, $ d$ be positive real numbers such that $ abcd = 1$ and $ a + b + c + d > \dfrac{a}{b} + \dfrac{b}{c} + \dfrac{c}{d} + \dfrac{d}{a}$. Prove that

$$ a + b + c + d < \dfrac{b}{a} + \dfrac{c}{b} + \dfrac{d}{c} + \dfrac{a}{d}$$

Proposed by Pavel Novotný, Slovakia
2009 Germany Team Selection Test
Let $ ABCD$ be a chordal/cyclic quadrilateral. Consider points $ P,Q$ on $ AB$ and $ R,S$ on $ CD$ with

$$ \overline{AP}: \overline{PB} = \overline{CS}: \overline{SD}, \quad \overline{AQ}: \overline{QB} = \overline{CR}: \overline{RD}.$$

How to choose $ P,Q,R,S$ such that $ \overline{PR} \cdot \overline{AB} + \overline{QS} \cdot \overline{CD}$ is minimal?
2009 Germany Team Selection Test
Let $ \left(a_n \right)_{n \in \mathbb{N}}$ defined by $ a_1 = 1,$ and $ a_{n + 1} = a^4_n - a^3_n + 2a^2_n + 1$ for $ n \geq 1.$ Show that there is an infinite number of primes $ p$ such that none of the $ a_n$ is divisible by $ p.$
2009 Germany Team Selection Test
Find all functions $ f: \mathbb{R} \mapsto \mathbb{R}$ such that $ \forall x,y,z \in \mathbb{R}$ we have: If

$$ x^3 + f(y) \cdot x + f(z) = 0,$$

then

$$ f(x)^3 + y \cdot f(x) + z = 0.$$
2009 Germany Team Selection Test
Let $ I$ be the incircle centre of triangle $ ABC$ and $ \omega$ be a circle within the same triangle with centre $ I.$ The perpendicular rays from $ I$ on the sides $ \overline{BC}, \overline{CA}$ and $ \overline{AB}$ meets $ \omega$ in $ A', B'$ and $ C'.$ Show that the three lines $ AA', BB'$ and $ CC'$ have a common point.
2009 Germany Team Selection Test
Tracy has been baking a rectangular cake whose surface is dissected by grid lines in square fields. The number of rows is $ 2^n$ and the number of columns is $ 2^{n + 1}$ where $ n \geq 1, n \in \mathbb{N}.$ Now she covers the fields with strawberries such that each row has at least $ 2n + 2$ of them. Show that there four pairwise distinct strawberries $ A,B,C$ and $ D$ which satisfy those three conditions:



(a) Strawberries $ A$ and $ B$ lie in the same row and $ A$ further left than $ B.$ Similarly $ D$ lies in the same row as $ C$ but further left.



(b) Strawberries $ B$ and $ C$ lie in the same column.



(c) Strawberries $ A$ lies further up and further left than $ D.$
2009 Germany Team Selection Test
Prove that for any four positive real numbers $ a$, $ b$, $ c$, $ d$ the inequality

$$ \frac {(a - b)(a - c)}{a + b + c} + \frac {(b - c)(b - d)}{b + c + d} + \frac {(c - d)(c - a)}{c + d + a} + \frac {(d - a)(d - b)}{d + a + b}\ge 0$$

holds. Determine all cases of equality.



Author: Darij Grinberg (Problem Proposal), Christian Reiher (Solution), Germany
2009 Germany Team Selection Test
Consider cubes of edge length 5 composed of 125 cubes of edge length 1 where each of the 125 cubes is either coloured black or white. A cube of edge length 5 is called "big", a cube od edge length is called "small". A posititve integer $ n$ is called "representable" if there is a big cube with exactly $ n$ small cubes where each row of five small cubes has an even number of black cubes whose centres lie on a line with distances $ 1,2,3,4$ (zero counts as even number).



(a) What is the smallest and biggest representable number?

(b) Construct 45 representable numbers.
2009 Germany Team Selection Test
Let $ S\subseteq\mathbb{R}$ be a set of real numbers. We say that a pair $ (f, g)$ of functions from $ S$ into $ S$ is a Spanish Couple on $ S$, if they satisfy the following conditions:

(i) Both functions are strictly increasing, i.e. $ f(x) < f(y)$ and $ g(x) < g(y)$ for all $ x$, $ y\in S$ with $ x < y$;



(ii) The inequality $ f\left(g\left(g\left(x\right)\right)\right) < g\left(f\left(x\right)\right)$ holds for all $ x\in S$.



Decide whether there exists a Spanish Couple



on the set $ S = \mathbb{N}$ of positive integers;

on the set $ S = \{a - \frac {1}{b}: a, b\in\mathbb{N}\}$





Proposed by Hans Zantema, Netherlands
2009 Germany Team Selection Test
For every $ n\in\mathbb{N}$ let $ d(n)$ denote the number of (positive) divisors of $ n$. Find all functions $ f: \mathbb{N}\to\mathbb{N}$ with the following properties:



$ d\left(f(x)\right) = x$ for all $ x\in\mathbb{N}$.

$ f(xy)$ divides $ (x - 1)y^{xy - 1}f(x)$ for all $ x$, $ y\in\mathbb{N}$.





Proposed by Bruno Le Floch, France
2009 Germany Team Selection Test
Let $ A,B,C,M$ points in the plane and no three of them are on a line. And let $ A',B',C'$ points such that $ MAC'B, MBA'C$ and $ MCB'A$ are parallelograms:



(a) Show that

$$ \overline{MA} + \overline{MB} + \overline{MC} < \overline{AA'} + \overline{BB'} + \overline{CC'}.$$

(b) Assume segments $ AA', BB'$ and $ CC'$ have the same length. Show that $ 2 \left(\overline{MA} + \overline{MB} + \overline{MC} \right) \leq \overline{AA'} + \overline{BB'} + \overline{CC'}.$ When do we have equality?
2009 Germany Team Selection Test
For which $ n \geq 2, n \in \mathbb{N}$ are there positive integers $ A_1, A_2, \ldots, A_n$ which are not the same pairwise and have the property that the product $ \prod^n_{i = 1} (A_i + k)$ is a power for each natural number $ k.$
2009 Germany Team Selection Test
In Skinien there 2009 towns where each of them is connected with exactly 1004 other town by a highway. Prove that starting in an arbitrary town one can make a round trip along the highways such that each town is passed exactly once and finally one returns to its starting point.
2009 Germany Team Selection Test
There is given a convex quadrilateral $ ABCD$. Prove that there exists a point $ P$ inside the quadrilateral such that

$$

\angle PAB + \angle PDC = \angle PBC + \angle PAD = \angle PCD + \angle PBA = \angle PDA + \angle PCB = 90^{\circ}

$$if and only if the diagonals $ AC$ and $ BD$ are perpendicular.



Proposed by Dusan Djukic, Serbia
2009 Germany Team Selection Test
Let $p > 7$ be a prime which leaves residue 1 when divided by 6. Let $m=2^p-1,$ then prove $2^{m-1}-1$ can be divided by $127m$ without residue.
2009 Germany Team Selection Test
Let triangle $ABC$ be perpendicular at $A.$ Let $M$ be the midpoint of segment $\overline{BC}.$ Point $D$ lies on side $\overline{AC}$ and satisfies $|AD|=|AM|.$ Let $P \neq C$ be the intersection of the circumcircle of triangles $AMC$ and $BDC.$ Prove that $CP$ bisects the angle at $C$ of triangle $ABC.$
2009 Germany Team Selection Test
Initially, on a board there a positive integer. If board contains the number $x,$ then we may additionally write the numbers $2x+1$ and $\frac{x}{x+2}.$ At some point 2008 is written on the board. Prove, that this number was there from the beginning.
2009 Germany Team Selection Test
The 16 fields of a $4 \times 4$ checker board can be arranged in 18 lines as follows: the four lines, the four columns, the five diagonals from north west to south east and the five diagonals from north east to south west. These diagonals consists of 2,3 or 4 edge-adjacent fields of same colour; the corner fields of the chess board alone do not form a diagonal. Now, we put a token in 10 of the 16 fields. Each of the 18 lines contains an even number of tokens contains a point. What is the highest possible point number when can be achieved by optimal placing of the 10 tokens. Explain your answer.
2008 Germany Team Selection Test
Show that there is a digit unequal to 2 in the decimal represesentation of $ \sqrt [3]{3}$ between the $ 1000000$-th und $ 3141592$-th position after decimal point.
2008 Germany Team Selection Test
Let $ ABCD$ be an isosceles trapezium with $ AB || CD$ and $ \bar{BC} = \bar{AD}.$ The parallel to $ AD$ through $ B$ meets the perpendicular to $ AD$ through $ D$ in point $ X.$ The line through $ A$ drawn which is parallel to $ BD$ meets the perpendicular to $ BD$ through $ D$ in point $ Y.$ Prove that points $ C,X,D$ and $ Y$ lie on a common circle.
2008 Germany Team Selection Test
Prove there is an integer $ k$ for which $ k^3 - 36 k^2 + 51 k - 97$ is a multiple of $ 3^{2008.}$
2008 Germany Team Selection Test
Find all pairs of natural numbers $ (a, b)$ such that $ 7^a - 3^b$ divides $ a^4 + b^2$.



Author: Stephan Wagner, Austria
2008 Germany Team Selection Test
The diagonals of a trapezoid $ ABCD$ intersect at point $ P$. Point $ Q$ lies between the parallel lines $ BC$ and $ AD$ such that $ \angle AQD = \angle CQB$, and line $ CD$ separates points $ P$ and $ Q$. Prove that $ \angle BQP = \angle DAQ$.



Author: Vyacheslav Yasinskiy, Ukraine
2008 Germany Team Selection Test
A rectangle $ D$ is partitioned in several ($ \ge2$) rectangles with sides parallel to those of $ D$. Given that any line parallel to one of the sides of $ D$, and having common points with the interior of $ D$, also has common interior points with the interior of at least one rectangle of the partition; prove that there is at least one rectangle of the partition having no common points with $ D$'s boundary.



Author: Kei Irie, Japan
2008 Germany Team Selection Test
Let $ n > 1$ be an integer. Find all sequences $ a_1, a_2, \ldots a_{n^2 + n}$ satisfying the following conditions:

$$ \text{ (a) } a_i \in \left\{0,1\right\} \text{ for all } 1 \leq i \leq n^2 + n;

$$



$$ \text{ (b) } a_{i + 1} + a_{i + 2} + \ldots + a_{i + n} < a_{i + n + 1} + a_{i + n + 2} + \ldots + a_{i + 2n} \text{ for all } 0 \leq i \leq n^2 - n.

$$

Author: Dusan Dukic, Serbia
2008 Germany Team Selection Test
(i) Determine the smallest number of edges which a graph of $ n$ nodes may have given that adding an arbitrary new edge would give rise to a 3-clique (3 nodes joined pairwise by edges).

(ii) Determine the smallest number of edges which a graph of $ n$ nodes may have given that adding an arbitrary new edge would give rise to a 4-clique (4 nodes joined pairwise by edges).
2008 Germany Team Selection Test
Denote by $ M$ midpoint of side $ BC$ in an isosceles triangle $ \triangle ABC$ with $ AC = AB$. Take a point $ X$ on a smaller arc $ \overarc{MA}$ of circumcircle of triangle $ \triangle ABM$. Denote by $ T$ point inside of angle $ BMA$ such that $ \angle TMX = 90$ and $ TX = BX$.



Prove that $ \angle MTB - \angle CTM$ does not depend on choice of $ X$.



Author: Farzan Barekat, Canada
2008 Germany Team Selection Test
Consider those functions $ f: \mathbb{N} \mapsto \mathbb{N}$ which satisfy the condition

$$ f(m + n) \geq f(m) + f(f(n)) - 1

$$

for all $ m,n \in \mathbb{N}.$ Find all possible values of $ f(2007).$



Author: Nikolai Nikolov, Bulgaria
2008 Germany Team Selection Test
Find all positive integers $ n$ for which the numbers in the set $ S = \{1,2, \ldots,n \}$ can be colored red and blue, with the following condition being satisfied: The set $ S \times S \times S$ contains exactly $ 2007$ ordered triples $ \left(x, y, z\right)$ such that:



(i) the numbers $ x$, $ y$, $ z$ are of the same color,

and

(ii) the number $ x + y + z$ is divisible by $ n$.



Author: Gerhard Wöginger, Netherlands
2008 Germany Team Selection Test
Let $ X$ be a set of 10,000 integers, none of them is divisible by 47. Prove that there exists a 2007-element subset $ Y$ of $ X$ such that $ a - b + c - d + e$ is not divisible by 47 for any $ a,b,c,d,e \in Y.$



Author: Gerhard Wöginger, Netherlands
2008 Germany Team Selection Test
Let $ A_0 = (a_1,\dots,a_n)$ be a finite sequence of real numbers. For each $ k\geq 0$, from the sequence $ A_k = (x_1,\dots,x_k)$ we construct a new sequence $ A_{k + 1}$ in the following way.

1. We choose a partition $ \{1,\dots,n\} = I\cup J$, where $ I$ and $ J$ are two disjoint sets, such that the expression

$$ \left|\sum_{i\in I}x_i - \sum_{j\in J}x_j\right|

$$

attains the smallest value. (We allow $ I$ or $ J$ to be empty; in this case the corresponding sum is 0.) If there are several such partitions, one is chosen arbitrarily.

2. We set $ A_{k + 1} = (y_1,\dots,y_n)$ where $ y_i = x_i + 1$ if $ i\in I$, and $ y_i = x_i - 1$ if $ i\in J$.

Prove that for some $ k$, the sequence $ A_k$ contains an element $ x$ such that $ |x|\geq\frac n2$.



Author: Omid Hatami, Iran
2008 Germany Team Selection Test
Let $ ABC$ be a fixed triangle, and let $ A_1$, $ B_1$, $ C_1$ be the midpoints of sides $ BC$, $ CA$, $ AB$, respectively. Let $ P$ be a variable point on the circumcircle. Let lines $ PA_1$, $ PB_1$, $ PC_1$ meet the circumcircle again at $ A'$, $ B'$, $ C'$, respectively. Assume that the points $ A$, $ B$, $ C$, $ A'$, $ B'$, $ C'$ are distinct, and lines $ AA'$, $ BB'$, $ CC'$ form a triangle. Prove that the area of this triangle does not depend on $ P$.



Author: Christopher Bradley, United Kingdom
2008 Germany Team Selection Test
Find all surjective functions $ f: \mathbb{N} \to \mathbb{N}$ such that for every $ m,n \in \mathbb{N}$ and every prime $ p,$ the number $ f(m + n)$ is divisible by $ p$ if and only if $ f(m) + f(n)$ is divisible by $ p$.



Author: Mohsen Jamaali and Nima Ahmadi Pour Anari, Iran
2008 Germany Team Selection Test
Determine $ Q \in \mathbb{R}$ which is so big that a sequence with non-negative reals elements $ a_1 ,a_2, \ldots$ which satisfies the following two conditions:



(i) $ \forall m,n \geq 1$ we have $ a_{m + n} \leq 2 \left(a_m + a_n \right)$

(ii) $ \forall k \geq 0$ we have $ a_{2^k} \leq \frac {1}{(k + 1)^{2008}}$



such that for each sequence element we have the inequality $ a_n \leq Q.$
2008 Germany Team Selection Test
Tracey baked a square cake whose surface is dissected in a $ 10 \times 10$ grid. In some of the fields she wants to put a strawberry such that for each four fields that compose a rectangle whose edges run in parallel to the edges of the cake boundary there is at least one strawberry. What is the minimum number of required strawberries?
2008 Germany Team Selection Test
Let $ ABCD$ be an isosceles trapezium. Determine the geometric location of all points $ P$ such that $$ |PA| \cdot |PC| = |PB| \cdot |PD|.$$
2008 Germany Team Selection Test
A sequence $ (S_n), n \geq 1$ of sets of natural numbers with $ S_1 = \{1\}, S_2 = \{2\}$ and

$${ S_{n + 1} = \{k \in }\mathbb{N}|k - 1 \in S_n \text{ XOR } k \in S_{n - 1}\}.

$$

Determine $ S_{1024}.$
2008 Germany Team Selection Test
For three points $ X,Y,Z$ let $ R_{XYZ}$ be the circumcircle radius of the triangle $ XYZ.$ If $ ABC$ is a triangle with incircle centre $ I$ then we have:



$$ \frac{1}{R_{ABI}} + \frac{1}{R_{BCI}} + \frac{1}{R_{CAI}} \leq \frac{1}{\bar{AI}} + \frac{1}{\bar{BI}} + \frac{1}{\bar{CI}}.$$
2008 Germany Team Selection Test
Find all real polynomials $ f$ with $ x,y \in \mathbb{R}$ such that

$$ 2 y f(x + y) + (x - y)(f(x) + f(y)) \geq 0.

$$
2008 Germany Team Selection Test
Let $ ABC$ be an acute triangle, and $ M_a$, $ M_b$, $ M_c$ be the midpoints of the sides $ a$, $ b$, $ c$. The perpendicular bisectors of $ a$, $ b$, $ c$ (passing through $ M_a$, $ M_b$, $ M_c$) intersect the boundary of the triangle again in points $ T_a$, $ T_b$, $ T_c$. Show that if the set of points $ \left\{A,B,C\right\}$ can be mapped to the set $ \left\{T_a, T_b, T_c\right\}$ via a similitude transformation, then two feet of the altitudes of triangle $ ABC$ divide the respective triangle sides in the same ratio. (Here, "ratio" means the length of the shorter (or equal) part divided by the length of the longer (or equal) part.) Does the converse statement hold?