Navvye/Olympiad-Problems-CSV huggingface dataset:
2008 Germany Team Selection Test
For every integer $ k \geq 2,$ prove that $ 2^{3k}$ divides the number

$$ \binom{2^{k + 1}}{2^{k}} - \binom{2^{k}}{2^{k - 1}}

$$

but $ 2^{3k + 1}$ does not.



Author: Waldemar Pompe, Poland
2008 Germany Team Selection Test
Determine all functions $ f: \mathbb{R} \mapsto \mathbb{R}$ with $ x,y \in \mathbb{R}$ such that



$$ f(x - f(y)) = f(x+y) + f(y)$$
2008 Germany Team Selection Test
Let $ a_1, a_2, \ldots, a_{100}$ be nonnegative real numbers such that $ a^2_1 + a^2_2 + \ldots + a^2_{100} = 1.$ Prove that

$$ a^2_1 \cdot a_2 + a^2_2 \cdot a_3 + \ldots + a^2_{100} \cdot a_1 < \frac {12}{25}.

$$

Author: Marcin Kuzma, Poland
2008 Germany Team Selection Test
Point $ P$ lies on side $ AB$ of a convex quadrilateral $ ABCD$. Let $ \omega$ be the incircle of triangle $ CPD$, and let $ I$ be its incenter. Suppose that $ \omega$ is tangent to the incircles of triangles $ APD$ and $ BPC$ at points $ K$ and $ L$, respectively. Let lines $ AC$ and $ BD$ meet at $ E$, and let lines $ AK$ and $ BL$ meet at $ F$. Prove that points $ E$, $ I$, and $ F$ are collinear.



Author: Waldemar Pompe, Poland
2008 Germany Team Selection Test
Given is a convex polygon $ P$ with $ n$ vertices. Triangle whose vertices lie on vertices of $ P$ is called good  if all its sides are unit length. Prove that there are at most $ \frac {2n}{3}$ good triangles.



Author: Vyacheslav Yasinskiy, Ukraine
2007 Germany Team Selection Test
Let $ n > 1, n \in \mathbb{Z}$ and $ B =\{1,2,\ldots, 2^n\}.$ A subset $ A$ of $ B$ is called weird if it contains exactly one of the distinct elements $ x,y \in B$ such that the sum of $ x$ and $ y$ is a power of two. How many weird subsets does $ B$ have?
2007 Germany Team Selection Test
Find all quadruple $ (m,n,p,q) \in \mathbb{Z}^4$ such that $$ p^m q^n = (p+q)^2 + 1.$$
2007 Germany Team Selection Test
A point $ P$ in the interior of triangle $ ABC$ satisfies



$$ \angle BPC - \angle BAC = \angle CPA - \angle CBA = \angle APB - \angle ACB.$$



Prove that $$ \bar{PA} \cdot \bar{BC} = \bar{PB} \cdot \bar{AC} = \bar{PC} \cdot \bar{AB}.$$
2007 Germany Team Selection Test
The sequence of real numbers $a_0,a_1,a_2,\ldots$ is defined recursively by $$a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.$$Show that $ a_{n} > 0$ for all $ n\geq 1$.



Proposed by Mariusz Skalba, Poland
2007 Germany Team Selection Test
Let $ ABCDE$ be a convex pentagon such that

$$ \angle BAC = \angle CAD = \angle DAE\qquad \text{and}\qquad \angle ABC = \angle ACD = \angle ADE.

$$The diagonals $BD$ and $CE$ meet at $P$.  Prove that the line $AP$ bisects the side $CD$.



Proposed by Zuming Feng, USA
2007 Germany Team Selection Test
For $ x \in (0, 1)$ let $ y \in (0, 1)$ be the number whose $ n$-th digit after the decimal point is the $ 2^{n}$-th digit after the decimal point of $ x$. Show that if $ x$ is rational then so is $ y$.



Proposed by J.P. Grossman, Canada
2007 Germany Team Selection Test
A sequence of real numbers $ a_{0},\ a_{1},\ a_{2},\dots$ is defined by the formula

$$ a_{i + 1} = \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0;

$$here $a_0$ is an arbitrary real number, $\lfloor a_i\rfloor$ denotes the greatest integer not exceeding $a_i$, and $\left\langle a_i\right\rangle=a_i-\lfloor a_i\rfloor$.  Prove that $a_i=a_{i+2}$ for $i$ sufficiently large.



Proposed by Harmel Nestra, Estionia
2007 Germany Team Selection Test
Let $ ABCD$ be a trapezoid with parallel sides $ AB > CD$. Points $ K$ and $ L$ lie on the line segments $ AB$ and $ CD$, respectively, so that $AK/KB=DL/LC$. Suppose that there are points $ P$ and $ Q$ on the line segment $ KL$ satisfying $$\angle{APB} = \angle{BCD}\qquad\text{and}\qquad \angle{CQD} = \angle{ABC}.$$Prove that the points $ P$, $ Q$, $ B$ and $ C$ are concyclic.



Proposed by Vyacheslev Yasinskiy, Ukraine
2007 Germany Team Selection Test
Let $ a > b > 1$ be relatively prime positive integers. Define the weight of an integer $ c$, denoted by $ w(c)$ to be the minimal possible value of $ |x| + |y|$ taken over all pairs of integers $ x$ and $ y$ such that $$ax + by = c.$$ An integer $ c$ is called a local champion if $ w(c) \geq w(c \pm a)$ and $ w(c) \geq w(c \pm b)$.



Find all local champions and determine their number.



Proposed by Zoran Sunic, USA
2007 Germany Team Selection Test
We have $ n \geq 2$ lamps $ L_{1}, . . . ,L_{n}$ in a row, each of them being either on or off. Every second we simultaneously modify the state of each lamp as follows: if the lamp $ L_{i}$ and its neighbours (only one neighbour for $ i = 1$ or $ i = n$, two neighbours for other $ i$) are in the same state, then $ L_{i}$ is switched off; – otherwise, $ L_{i}$ is switched on.

Initially all the lamps are off except the leftmost one which is on.



$ (a)$ Prove that there are infinitely many integers $ n$ for which all the lamps will eventually be off.

$ (b)$ Prove that there are infinitely many integers $ n$ for which the lamps will never be all off.
2007 Germany Team Selection Test
Let $ n, k \in \mathbb{N}$ with $ 1 \leq k \leq \frac {n}{2} - 1.$ There are $ n$ points given on a circle. Arbitrarily we select $ nk + 1$ chords among the points on the circle. Prove that of these chords there are at least $ k + 1$ chords which pairwise do not have a point in common.
2007 Germany Team Selection Test
Points $ A_{1}$, $ B_{1}$, $ C_{1}$ are chosen on the sides $ BC$, $ CA$, $ AB$ of a triangle $ ABC$ respectively. The circumcircles of triangles $ AB_{1}C_{1}$, $ BC_{1}A_{1}$, $ CA_{1}B_{1}$ intersect the circumcircle of triangle $ ABC$ again at points $ A_{2}$, $ B_{2}$, $ C_{2}$ respectively  ($ A_{2}\neq A, B_{2}\neq B, C_{2}\neq C$).  Points $ A_{3}$, $ B_{3}$, $ C_{3}$ are symmetric to $ A_{1}$, $ B_{1}$, $ C_{1}$ with respect to the midpoints of the sides $ BC$, $ CA$, $ AB$ respectively. Prove that the triangles $ A_{2}B_{2}C_{2}$ and $ A_{3}B_{3}C_{3}$ are similar.
2007 Germany Team Selection Test
We define a sequence $ \left(a_{1},a_{2},a_{3},\ldots \right)$ by

$$ a_{n} = \frac {1}{n}\left(\left\lfloor\frac {n}{1}\right\rfloor + \left\lfloor\frac {n}{2}\right\rfloor + \cdots + \left\lfloor\frac {n}{n}\right\rfloor\right),

$$ where $\lfloor x\rfloor$ denotes the integer part of $x$.



a) Prove that $a_{n+1}>a_n$ infinitely often.

b) Prove that $a_{n+1}<a_n$ infinitely often.



Proposed by Johan Meyer, South Africa
2007 Germany Team Selection Test
Determine the sum of absolute values for the complex roots of $ 20 x^8 + 7i x^7 -7ix + 20.$
2007 Germany Team Selection Test
Circles $ w_{1}$ and $ w_{2}$ with centres $ O_{1}$ and $ O_{2}$ are externally tangent at point $ D$ and internally tangent to a circle $ w$ at points $ E$ and $ F$ respectively. Line $ t$ is the common tangent of $ w_{1}$ and $ w_{2}$ at $ D$. Let $ AB$ be the diameter of $ w$ perpendicular to $ t$, so that $ A, E, O_{1}$ are on the same side of $ t$. Prove that lines $ AO_{1}$, $ BO_{2}$, $ EF$ and $ t$ are concurrent.
2007 Germany Team Selection Test
A cake has the form of an $ n$ x $ n$ square composed of $ n^{2}$ unit squares. Strawberries lie on some of the unit squares so that each row or column contains exactly one strawberry; call this arrangement $\mathcal{A}$.



Let $\mathcal{B}$ be another such arrangement. Suppose that every grid rectangle with one vertex at the top left corner of the cake contains no fewer strawberries of arrangement $\mathcal{B}$ than of arrangement $\mathcal{A}$. Prove that arrangement $\mathcal{B}$ can be obtained from $ \mathcal{A}$ by performing a number of switches, defined as follows:



A switch consists in selecting a grid rectangle with only two strawberries, situated at its top right corner and bottom left corner, and moving these two strawberries to the other two corners of that rectangle.
2007 Germany Team Selection Test
Find all integer solutions of the equation $$\frac {x^{7} - 1}{x - 1} = y^{5} - 1.$$
2007 Germany Team Selection Test
Let $ k \in \mathbb{N}$. A polynomial is called $ k$-valid if all its coefficients are integers between 0 and $ k$ inclusively. (Here we don't consider 0 to be a natural number.)



a.) For $ n \in \mathbb{N}$ let $ a_n$ be the number of 5-valid polynomials $ p$ which satisfy $ p(3) = n.$ Prove that each natural number occurs in the sequence $ (a_n)_n$ at least once but only finitely often.



b.) For $ n \in \mathbb{N}$ let $ a_n$ be the number of 4-valid polynomials $ p$ which satisfy $ p(3) = n.$ Prove that each natural number occurs infinitely often in the sequence $ (a_n)_n$ .
2007 Germany Team Selection Test
Determine all functions $ f: \mathbb{R}^+ \mapsto \mathbb{R}^+$ which satisfy $$ f \left(\frac {f(x)}{yf(x) + 1}\right) = \frac {x}{xf(y)+1} \quad \forall x,y > 0$$
2007 Germany Team Selection Test
Let $ ABC$ be a triangle and $ P$ an arbitrary point in the plane. Let $ \alpha, \beta, \gamma$ be interior angles of the triangle and its area is denoted by $ F.$ Prove:

$$ \ov{AP}^2 \cdot \sin 2\alpha + \ov{BP}^2 \cdot \sin 2\beta + \ov{CP}^2 \cdot \sin 2\gamma \geq 2F

$$

When does equality occur?
2007 Germany Team Selection Test
For a multiple of $ kb$ of $ b$ let $ a \% kb$ be the greatest number such that $ a \% kb = a \bmod b$ which is smaller than $ kb$ and not greater than $ a$ itself. Let $ n \in \mathbb{Z}^ + .$ Determine all integer pairs $ (a,b)$ with:

$$ a\%b + a\%2b + a\%3b + \ldots + a\%nb = a + b

$$
2007 Germany Team Selection Test
Let $ S$ be a finite set of points in the plane such that no three of them are on a line. For each convex polygon $ P$ whose vertices are in $ S$, let $ a(P)$ be the number of vertices of $ P$, and let $ b(P)$ be the number of points of $ S$ which are outside $ P$. A line segment, a point, and the empty set are considered as convex polygons of $ 2$, $ 1$, and $ 0$ vertices respectively. Prove that for every real number $ x$ $$\sum_{P}{x^{a(P)}(1 - x)^{b(P)}} = 1,$$ where the sum is taken over all convex polygons with vertices in $ S$.



Alternative formulation:



Let $ M$ be a finite point set in the plane and no three points are collinear. A subset $ A$ of $ M$ will be called round if its elements is the set of vertices of a convex $ A -$gon $ V(A).$ For each round subset let $ r(A)$ be the number of points from $ M$ which are exterior from the convex $ A -$gon $ V(A).$ Subsets with $ 0,1$ and 2 elements are always round, its corresponding polygons are the empty set, a point or a segment, respectively (for which all other points that are not vertices of the polygon are exterior). For each round subset $ A$ of $ M$ construct the polynomial

$$ P_A(x) = x^{|A|}(1 - x)^{r(A)}.

$$

Show that the sum of polynomials for all round subsets is exactly the polynomial $ P(x) = 1.$



Proposed by Federico Ardila, Colombia
2007 Germany Team Selection Test
In triangle $ ABC$ we have $ a \geq b$ and $ a \geq c.$ Prove that the ratio of circumcircle radius to incircle diameter is at least as big as the length of the centroidal axis $ s_a$ to the altitude $ a_a.$ When do we have equality?
2007 Germany Team Selection Test
Prove the inequality:

$$\sum_{i < j}{\frac {a_{i}a_{j}}{a_{i} + a_{j}}}\leq \frac {n}{2(a_{1} + a_{2} +\cdots + a_{n})}\cdot \sum_{i < j}{a_{i}a_{j}}$$

for positive reals $ a_{1},a_{2},\ldots,a_{n}$.



Proposed by Dusan Dukic, Serbia
2007 Germany Team Selection Test
An $ (n, k) -$ tournament is a contest with $ n$ players held in $ k$ rounds such that:



$ (i)$ Each player plays in each round, and every two players meet at most once.

$ (ii)$ If player $ A$ meets player $ B$ in round $ i$, player $ C$ meets player $ D$ in round $ i$, and player $ A$ meets player $ C$ in round $ j$, then player $ B$ meets player $ D$ in round $ j$.



Determine all pairs $ (n, k)$ for which there exists an $ (n, k) -$ tournament.



Proposed by Carlos di Fiore, Argentina
2007 Germany Team Selection Test
For all positive integers $n$, show that there exists a positive integer $m$ such that $n$ divides $2^{m} + m$.



Proposed by Juhan Aru, Estonia
2006 Germany Team Selection Test
Let $A$, $B$, $C$, $D$, $E$, $F$ be six points on a circle such that $AE\parallel BD$ and $BC\parallel DF$. Let $X$ be the reflection of the point $D$ in the line $CE$. Prove that the distance from the point $X$ to the line $EF$ equals to the distance from the point $B$ to the line $AC$.
2006 Germany Team Selection Test
In a room, there are $2005$ boxes, each of them containing one or several sorts of fruits, and of course an integer amount of each fruit.



a) Show that we can find $669$ boxes, which altogether contain at least a third of all apples and at least a third of all bananas.



b) Can we always find $669$ boxes, which altogether contain at least a third of all apples, at least a third of all bananas and at least a third of all pears?
2006 Germany Team Selection Test
Is the following statement true?



For each positive integer $n$, we can find eight nonnegative integers $a$, $b$, $c$, $d$, $e$, $f$, $g$, $h$ such that $n=\frac{2^a-2^b}{2^c-2^d}\cdot\frac{2^e-2^f}{2^g-2^h}$.
2006 Germany Team Selection Test
We denote by $\mathbb{R}^+$ the set of all positive real numbers.



Find all functions $f: \mathbb R^ + \rightarrow\mathbb R^ +$ which have the property:

$$f(x)f(y)=2f(x+yf(x))$$

for all positive real numbers $x$ and $y$.



Proposed by Nikolai Nikolov, Bulgaria
2006 Germany Team Selection Test
Consider a $m\times n$ rectangular board consisting of $mn$ unit squares. Two of its unit squares are called adjacent if they have a common edge, and a path is a sequence of unit squares in which any two consecutive squares are adjacent. Two parths are called non-intersecting if they don't share any common squares.



Each unit square of the rectangular board can be colored black or white. We speak of a coloring of the board if all its $mn$ unit squares are colored.



Let $N$ be the number of colorings of the board such that there exists at least one black path from the left edge of the board to its right edge. Let $M$ be the number of colorings of the board for which there exist at least two non-intersecting black paths from the left edge of the board to its right edge.



Prove that $N^{2}\geq M\cdot 2^{mn}$.
2006 Germany Team Selection Test
A house has an even number of lamps distributed among its rooms in such a way that there are at least three lamps in every room. Each lamp shares a switch with exactly one other lamp, not necessarily from the same room. Each change in the switch shared by two lamps changes their states simultaneously. Prove that for every initial state of the lamps there exists a sequence of changes in some of the switches at the end of which each room contains lamps which are on as well as lamps which are off.



Proposed by Australia
2006 Germany Team Selection Test
In an acute triangle $ABC$, let $D$, $E$, $F$ be the feet of the perpendiculars from the points $A$, $B$, $C$ to the lines $BC$, $CA$, $AB$, respectively, and let $P$, $Q$, $R$ be the feet of the perpendiculars from the points $A$, $B$, $C$ to the lines $EF$, $FD$, $DE$, respectively.



Prove that $p\left(ABC\right)p\left(PQR\right) \ge \left(p\left(DEF\right)\right)^{2}$, where $p\left(T\right)$ denotes the perimeter of triangle $T$ .



Proposed by Hojoo Lee, Korea
2006 Germany Team Selection Test
Let $ a$, $ b$, $ c$, $ d$, $ e$, $ f$ be positive integers and let $ S = a+b+c+d+e+f$.

Suppose that the number $ S$ divides $ abc+def$ and  $ ab+bc+ca-de-ef-df$. Prove that $ S$ is composite.
2006 Germany Team Selection Test
Four real numbers $ p$, $ q$, $ r$, $ s$ satisfy $ p+q+r+s = 9$ and $ p^{2}+q^{2}+r^{2}+s^{2}= 21$. Prove that there exists a permutation $ \left(a,b,c,d\right)$ of $ \left(p,q,r,s\right)$ such that $ ab-cd \geq 2$.
2006 Germany Team Selection Test
Suppose we have a $n$-gon. Some $n-3$ diagonals are coloured black and some other $n-3$ diagonals are coloured red (a side is not a diagonal), so that no two diagonals of the same colour can intersect strictly inside the polygon, although they can share a vertex. Find the maximum number of intersection points between diagonals coloured differently strictly inside the polygon, in terms of $n$.



Proposed by Alexander Ivanov, Bulgaria
2006 Germany Team Selection Test
Does there exist a natural number $n$ in whose decimal representation each digit occurs at least $2006$ times and which has the property that you can find two different digits in its decimal representation such that the number obtained from $n$ by interchanging these two digits is different from $n$ and has the same set of prime divisors as $n$ ?
2006 Germany Team Selection Test
Find all functions $ f: \mathbb{R}\to\mathbb{R}$ such that $ f(x+y)+f(x)f(y)=f(xy)+2xy+1$ for all real numbers $ x$ and $ y$.



Proposed by B.J. Venkatachala, India
2006 Germany Team Selection Test
Let $ABCD$ be a parallelogram. A variable line $g$ through the vertex $A$ intersects the rays $BC$ and $DC$ at the points $X$ and $Y$, respectively. Let $K$ and $L$ be the $A$-excenters of the triangles $ABX$ and $ADY$. Show that the angle $\measuredangle KCL$ is independent of the line $g$.



Proposed by Vyacheslev Yasinskiy, Ukraine
2006 Germany Team Selection Test
For any positive integer $n$, let $w\left(n\right)$ denote the number of different prime divisors of the number $n$. (For instance, $w\left(12\right)=2$.) Show that there exist infinitely many positive integers $n$ such that $w\left(n\right)<w\left(n+1\right)<w\left(n+2\right)$.
2006 Germany Team Selection Test
Let $A_{1}$, $B_{1}$, $C_{1}$ be the feet of the altitudes of an acute-angled triangle $ABC$ issuing from the vertices $A$, $B$, $C$, respectively. Let $K$ and $M$ be points on the segments $A_{1}C_{1}$ and $B_{1}C_{1}$, respectively, such that $\measuredangle KAM = \measuredangle A_{1}AC$. Prove that the line $AK$ is the angle bisector of the angle $C_{1}KM$.
2006 Germany Team Selection Test
Does there exist a set $ M$ of points in space such that every plane intersects $ M$ at a finite but nonzero number of points?
2006 Germany Team Selection Test
Let $n\geq 3$ be a fixed integer. Each side and each diagonal of a regular $n$-gon is labelled with a number from the set $\left\{1;\;2;\;...;\;r\right\}$ in a way such that the following two conditions are fulfilled:



1. Each number from the set $\left\{1;\;2;\;...;\;r\right\}$ occurs at least once as a label.



2. In each triangle formed by three vertices of the $n$-gon, two of the sides are labelled with the same number, and this number is greater than the label of the third side.



(a) Find the maximal $r$ for which such a labelling is possible.



(b) Harder version (IMO Shortlist 2005): For this maximal value of $r$, how many such labellings are there?



Easier version (5th German TST 2006) - contains answer to the harder versionEasier version (5th German TST 2006): Show that, for this maximal value of $r$, there are exactly $\frac{n!\left(n-1\right)!}{2^{n-1}}$ possible labellings.



Proposed by Federico Ardila, Colombia
2006 Germany Team Selection Test
Find all positive integers $ n$ such that there exists a unique integer $ a$ such that $ 0\leq a < n!$ with the following property:

$$ n!\mid a^n + 1

$$



Proposed by Carlos Caicedo, Colombia
2006 Germany Team Selection Test
The diagonals $AC$ and $BD$ of a cyclic quadrilateral $ABCD$ meet at a point $X$. The circumcircles of triangles $ABX$ and $CDX$ meet at a point $Y$ (apart from $X$). Let $O$ be the center of the circumcircle of the quadrilateral $ABCD$. Assume that the points $O$, $X$, $Y$ are all distinct.  Show that $OY$ is perpendicular to $XY$.
2006 Germany Team Selection Test
Let $ ABC$ be an equilateral triangle, and $ P,Q,R$ three points in its interior satisfying

$$ \measuredangle PCA = \measuredangle CAR = 15^{\circ},\ \measuredangle RBC = \measuredangle BCQ = 20^{\circ},\  \measuredangle QAB = \measuredangle ABP = 25^{\circ}.$$ Compute the angles of triangle $ PQR$.
2006 Germany Team Selection Test
There are $ n$ markers, each with one side white and the other side black. In the beginning, these $ n$ markers are aligned in a row so that their white sides are all up. In each step, if possible, we choose a marker whose white side is up (but not one of the outermost markers), remove it, and reverse the closest marker to the left of it and also reverse the closest marker to the right of it. Prove that, by a finite sequence of such steps, one can achieve a state with only two markers remaining if and only if $ n - 1$ is not divisible by $ 3$.



Proposed by Dusan Dukic, Serbia
2006 Germany Team Selection Test
Let $n$ be a positive integer, and let $b_{1}$, $b_{2}$, ..., $b_{n}$ be $n$ positive reals. Set $a_{1}=\frac{b_{1}}{b_{1}+b_{2}+...+b_{n}}$ and $a_{k}=\frac{b_{1}+b_{2}+...+b_{k}}{b_{1}+b_{2}+...+b_{k-1}}$ for every $k>1$. Prove the inequality



$a_{1}+a_{2}+...+a_{n}\leq\frac{1}{a_{1}}+\frac{1}{a_{2}}+...+\frac{1}{a_{n}}$.
2006 Germany Team Selection Test
Find all real solutions $x$ of the equation



$\cos\cos\cos\cos x=\sin\sin\sin\sin x$.



(Angles are measured in radians.)
2006 Germany Team Selection Test
The lengths of the altitudes of a triangle are positive integers, and the length of the radius of the incircle is a prime number.

Find the lengths of the sides of the triangle.
2006 Germany Team Selection Test
Suppose that $ a_1$, $ a_2$, $ \ldots$, $ a_n$ are integers such that $ n\mid a_1 + a_2 + \ldots + a_n$.

Prove that there exist two permutations $ \left(b_1,b_2,\ldots,b_n\right)$ and $ \left(c_1,c_2,\ldots,c_n\right)$ of $ \left(1,2,\ldots,n\right)$ such that for each integer $ i$ with $ 1\leq i\leq n$, we have

$$ n\mid a_i - b_i - c_i

$$



Proposed by Ricky Liu & Zuming Feng, USA
2005 Germany Team Selection Test
Given the positive numbers $a$ and $b$ and the natural number $n$, find the greatest among the $n + 1$ monomials in the binomial expansion of $\left(a+b\right)^n$.
2005 Germany Team Selection Test
Let $M$ be a set of points in the Cartesian plane, and let $\left(S\right)$ be a set of segments (whose endpoints not necessarily have to belong to $M$) such that one can walk from any point of $M$ to any other point of $M$ by travelling along segments which are in $\left(S\right)$. Find the smallest total length of the segments of $\left(S\right)$ in the cases



a.) $M = \left\{\left(-1,0\right),\left(0,0\right),\left(1,0\right),\left(0,-1\right),\left(0,1\right)\right\}$.

b.) $M = \left\{\left(-1,-1\right),\left(-1,0\right),\left(-1,1\right),\left(0,-1\right),\left(0,0\right),\left(0,1\right),\left(1,-1\right),\left(1,0\right),\left(1,1\right)\right\}$.



In other words, find the Steiner trees of the set $M$ in the above two cases.
2005 Germany Team Selection Test
Let $a$, $b$, $c$, $d$ and $n$ be positive integers such that $7\cdot 4^n = a^2+b^2+c^2+d^2$. Prove that the numbers $a$, $b$, $c$, $d$ are all $\geq 2^{n-1}$.
2005 Germany Team Selection Test
Let $a_0$, $a_1$, $a_2$, ... be an infinite sequence of real numbers satisfying the equation $a_n=\left|a_{n+1}-a_{n+2}\right|$ for all $n\geq 0$, where $a_0$ and $a_1$ are two different positive reals.



Can this sequence $a_0$, $a_1$, $a_2$, ... be bounded?



Proposed by Mihai Bălună, Romania
2005 Germany Team Selection Test
Prove that there doesn't exist any positive integer $n$ such that $2n^2+1,3n^2+1$ and $6n^2+1$ are perfect squares.
2005 Germany Team Selection Test
If $a$, $b$, $c$ are positive reals such that $a+b+c=1$, prove that



$$\frac{1+a}{1-a}+\frac{1+b}{1-b}+\frac{1+c}{1-c}\leq 2\left(\frac{b}{a}+\frac{c}{b}+\frac{a}{c}\right).$$
2005 Germany Team Selection Test
Let ABC be a triangle and let $r, r_a, r_b, r_c$ denote the inradius and ex-radii opposite to the vertices $A, B, C$, respectively. Suppose that $a>r_a, b>r_b, c>r_c$. Prove that

(a) $\triangle ABC$ is acute.

(b) $a+b+c > r+r_a+r_b+r_c$.
2005 Germany Team Selection Test
Let $\tau(n)$ denote the number of positive divisors of the positive integer $n$. Prove that there exist infinitely many positive integers $a$ such that the equation $ \tau(an)=n $ does not have a positive integer solution $n$.
2005 Germany Team Selection Test
If $a$, $b$ ,$c$ are three positive real numbers such that $ab+bc+ca = 1$, prove that $$ \sqrt[3]{ \frac{1}{a} + 6b} + \sqrt[3]{\frac{1}{b} + 6c} + \sqrt[3]{\frac{1}{c} + 6a } \leq \frac{1}{abc}.  $$