Navvye/Olympiad-Problems-CSV huggingface dataset:
2005 France Team Selection Test
In an international meeting of $n \geq 3$ participants, 14 languages are spoken. We know that:



- Any 3 participants speak a common language.



- No language is spoken more that by the half of the participants.



What is the least value of $n$?
2005 France Team Selection Test
Let $X$ be a non empty subset of $\mathbb{N} = \{1,2,\ldots \}$. Suppose that for all $x \in X$, $4x \in X$ and $\lfloor \sqrt{x} \rfloor \in X$. Prove that $X=\mathbb{N}$.
2005 France Team Selection Test
Let $ABC$ be a triangle such that $BC=AC+\frac{1}{2}AB$. Let $P$ be a point of $AB$ such that $AP=3PB$.



Show that $\widehat{PAC} = 2 \widehat{CPA}.$
2005 France Team Selection Test
Let $P$ be a polynom of degree $n \geq 5$ with integer coefficients given by $P(x)=a_{n}x^n+a_{n-1}x^{n-1}+\cdots+a_0 \quad$ with $a_i \in \mathbb{Z}$, $a_n \neq 0$.



Suppose that $P$ has $n$ different integer roots (elements of $\mathbb{Z}$) : $0,\alpha_2,\ldots,\alpha_n$. Find all integers $k \in \mathbb{Z}$ such that $P(P(k))=0$.
2004 France Team Selection Test
If $n$ is a positive integer, let $A = \{n,n+1,...,n+17 \}$.

Does there exist some values of $n$ for which we can divide $A$ into two disjoints subsets $B$ and $C$ such that the product of the elements of $B$ is equal to the product of the elements of $C$?
2004 France Team Selection Test
Each point of the plane with two integer coordinates is the center of a disk with radius $ \frac {1} {1000}$.

Prove that there exists an equilateral triangle whose vertices belong to distinct disks.

Prove that such a triangle has side-length greater than 96.
2004 France Team Selection Test
Let $n$ be a positive integer, and $a_1,...,a_n, b_1,..., b_n$ be $2n$ positive real numbers such that

$a_1 + ... + a_n = b_1 + ... + b_n = 1$.



Find the minimal value of

$ \frac {a_1^2} {a_1 + b_1} + \frac {a_2^2} {a_2 + b_2} + ...+ \frac {a_n^2} {a_n + b_n}$.
2004 France Team Selection Test
Let $P$ be the set of prime numbers. Consider a subset $M$ of $P$ with at least three elements. We assume that, for each non empty and finite subset $A$ of $M$, with $A \neq M$, the prime divisors of the integer $( \prod_{p \in A} ) - 1$ belong to $M$.

Prove that $M = P$.
2003 France Team Selection Test
A lattice point in the coordinate plane with origin $O$ is called invisible if the segment $OA$ contains a lattice point other than $O,A$. Let $L$ be a positive integer. Show that there exists a square with side length $L$ and sides parallel to the coordinate axes, such that all points in the square are invisible.
2003 France Team Selection Test
$M$ is an arbitrary point inside $\triangle ABC$. $AM$ intersects the circumcircle of the triangle again at $A_1$. Find the points $M$ that minimise $\frac{MB\cdot MC}{MA_1}$.
2003 France Team Selection Test
Let $B$ be a point on a circle $S_1$, and let $A$ be a point distinct from $B$ on the tangent at $B$ to $S_1$. Let $C$ be a point not on $S_1$ such that the line segment $AC$ meets $S_1$ at two distinct points. Let $S_2$ be the circle touching $AC$ at $C$ and touching $S_1$ at a point $D$ on the opposite side of $AC$ from $B$.  Prove that the circumcentre of triangle $BCD$ lies on the circumcircle of triangle $ABC$.
2003 France Team Selection Test
$10$ cities are connected by one-way air routes in a way so that each city can be reached from any other by several connected flights. Let $n$ be the smallest number of flights needed for a tourist to visit every city and return to the starting city. Clearly $n$ depends on the flight schedule. Find the largest $n$ and the corresponding flight schedule.
2003 France Team Selection Test
Let $p_1,p_2,\ldots,p_n$ be distinct primes greater than $3$. Show that $2^{p_1p_2\cdots p_n}+1$ has at least $4^n$ divisors.
2002 France Team Selection Test
In an acute-angled triangle $ABC$, $A_1$ and $B_1$ are the feet of the altitudes from $A$ and $B$ respectively, and $M$ is the midpoint of $AB$.

a) Prove that $MA_1$ is tangent to the circumcircle of triangle $A_1B_1C$.

b) Prove that the circumcircles of triangles $A_1B_1C,BMA_1$, and $AMB_1$ have a common point.
2002 France Team Selection Test
Consider the set $S$ of integers $k$ which are products of four distinct primes. Such an integer $k=p_1p_2p_3p_4$ has $16$ positive divisors $1=d_1<d_2<\ldots <d_{15}<d_{16}=k$. Find all elements of $S$ less than $2002$ such that $d_9-d_8=22$.
2002 France Team Selection Test
Let $n$ be a positive integer and let $(a_1,a_2,\ldots ,a_{2n})$ be a permutation of $1,2,\ldots ,2n$ such that the numbers $|a_{i+1}-a_i|$ are pairwise distinct for $i=1,\ldots ,2n-1$.

Prove that $\{a_2,a_4,\ldots ,a_{2n}\}=\{1,2,\ldots ,n\}$ if and only if $a_1-a_{2n}=n$.
2002 France Team Selection Test
There are three colleges in a town. Each college has $n$ students. Any student of any college knows $n+1$ students of the other two colleges. Prove that it is possible to choose a student from each of the three colleges so that all three students would know each other.
2002 France Team Selection Test
Let $ ABC$ be a non-equilateral triangle. Denote by $ I$ the incenter and by $ O$ the circumcenter of the triangle $ ABC$. Prove that $ \angle AIO\leq\frac{\pi}{2}$ holds if and only if $ 2\cdot BC\leq AB+AC$.
2002 France Team Selection Test
Let $p\ge 3$ be a prime number. Show that there exist $p$ positive integers $a_1,a_2,\ldots ,a_p$ not exceeding $2p^2$ such that the $\frac{p(p-1)}{2}$ sums $a_i+a_j\ (i<j)$ are all distinct.
2000 France Team Selection Test
Points $P,Q,R,S$ lie on a circle and $\angle PSR$ is right. $H,K$ are the projections of $Q$ on lines $PR,PS$. Prove that $HK$ bisects segment $ QS$.
2000 France Team Selection Test
A function from the positive integers to the positive integers satisfies these properties

1. $f(ab)=f(a)f(b)$ for any two coprime positive integers $a,b$.

2. $f(p+q)=f(p)+f(q)$ for any two primes $p,q$.

Prove that $f(2)=2, f(3)=3, f(1999)=1999$.
2000 France Team Selection Test
$a,b,c,d$ are positive reals with sum $1$. Show that $\frac{a^2}{a+b}+\frac{b^2}{b+c}+\frac{c^2}{c+d}+\frac{d^2}{d+a} \ge \frac{1}{2}$ with equality iff $a=b=c=d=\frac{1}{4}$.
2000 France Team Selection Test
Some squares of a $1999\times 1999$ board are occupied with pawns. Find the smallest number of pawns for which it is possible that for each empty square, the total number of pawns in the row or column of that square is at least $1999$.
2000 France Team Selection Test
$A,B,C,D$ are points on a circle in that order. Prove that $|AB-CD|+|AD-BC| \ge 2|AC-BD|$.
2000 France Team Selection Test
Find all nonnegative integers $x,y,z$ such that $(x+1)^{y+1} + 1= (x+2)^{z+1}$.
2005 Georgia Team Selection Test
1. The transformation $ n \to 2n - 1$ or $ n \to 3n - 1$, where $ n$ is a positive integer, is called the 'change' of $ n$. Numbers $ a$ and $ b$ are called 'similar', if there exists such positive integer, that can be got by finite number of 'changes' from both $ a$ and $ b$. Find all positive integers 'similar' to $ 2005$ and less than $ 2005$.
2005 Georgia Team Selection Test
In triangle $ ABC$ we have  $ \angle{ACB} = 2\angle{ABC}$ and there exists the point $ D$ inside the triangle such that $ AD = AC$ and $ DB = DC$. Prove that $ \angle{BAC} = 3\angle{BAD}$.
2005 Georgia Team Selection Test
Let $ x,y,z$ be positive real numbers,satisfying equality $ x^{2}+y^{2}+z^{2}=25$. Find the minimal possible value of the expression $ \frac{xy}{z} + \frac{yz}{x} + \frac{zx}{y}$.
2005 Georgia Team Selection Test
Find all polynomials with real coefficients, for which the equality

$$ P(2P(x)) = 2P(P(x)) + 2(P(x))^{2}$$

holds for any real number $ x$.
2005 Georgia Team Selection Test
Let $ ABCD$ be a convex quadrilateral. Points $ P,Q$ and $ R$ are the feets of the perpendiculars from point $ D$ to lines $ BC, CA$ and $ AB$, respectively. Prove that $ PQ=QR$ if and only if the bisectors of the angles $ ABC$ and $ ADC$ meet on segment $ AC$.
2005 Georgia Team Selection Test
Let $ A$ be the subset of the set of positive integers, having the following $ 2$ properties:



1) If $ a$ belong to $ A$,than all of the divisors of $ a$ also belong to $ A$;



2) If $ a$ and $ b$, $ 1 < a < b$, belong to $ A$, than $ 1 + ab$ is also in $ A$;



Prove that if $ A$ contains at least $ 3$ positive integers, than $ A$ contains all positive integers.
2005 Georgia Team Selection Test
Determine all positive integers $ n$, for  which $ 2^{n-1}n+1$ is a perfect square.
2005 Georgia Team Selection Test
In a convex quadrilateral $ ABCD$ the points $ P$ and $ Q$ are chosen on the sides $ BC$ and $ CD$ respectively so that $ \angle{BAP}=\angle{DAQ}$. Prove that the line, passing through the orthocenters of triangles $ ABP$ and $ ADQ$, is perpendicular to $ AC$ if and only if the triangles $ ABP$ and $ ADQ$ have the same areas.
2005 Georgia Team Selection Test
Let $ a_{0},a_{1},\ldots,a_{n}$ be integers, one of which is nonzero, and all of the numbers are not less than $ - 1$. Prove that if  $$ a_{0} + 2a_{1} + 2^{2}a_{2} + \cdots + 2^{n}a_{n} = 0,$$ then $ a_{0} + a_{1} + \cdots + a_{n} > 0$.
2005 Georgia Team Selection Test
Let $ a,b,c$ be positive numbers, satisfying $ abc\geq 1$.  Prove that

$$ a^{3} + b^{3} + c^{3} \geq ab + bc + ca.$$
2005 Georgia Team Selection Test
On the sides $ AB, BC, CD$ and $ DA$ of the rhombus $ ABCD$, respectively, are chosen points $ E, F, G$ and $ H$ so, that $ EF$ and $ GH$ touch the incircle of the rhombus. Prove that the lines $ EH$ and $ FG$ are parallel.
2005 Georgia Team Selection Test
$ 30$ students participated in the mathematical Olympiad. Each of them was given $ 8$ problems to solve. Jury estimated their work with the following rule:

1) Each problem was worth $ k$ points, if it wasn't solved by exactly $ k$ students;

2) Each student received the maximum possible points in each problem or got $ 0$ in it;

Lasha got the least number of points. What's the maximal number of points he could have?

Remark: 1) means that if the problem was solved by exactly $ k$ students, than each of them got $ 30 - k$ points in it.
2019 Germany Team Selection Test
Let $\mathbb{Q}^+$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}^+\to \mathbb{Q}^+$ satisfying$$f(x^2f(y)^2)=f(x^2)f(y)$$for all $x,y\in\mathbb{Q}^+$
2019 Germany Team Selection Test
Does there exist a subset $M$ of positive integers such that for all positive rational numbers $r<1$ there exists exactly one finite subset of $M$ like $S$ such that sum of reciprocals of elements in $S$ equals $r$.
2018 Germany Team Selection Test
A rectangle $\mathcal{R}$ with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of $\mathcal{R}$ are either all odd or all even.



Proposed by Jeck Lim, Singapore
2018 Germany Team Selection Test
A positive integer $d$ and a permutation of positive integers $a_1,a_2,a_3,\dots$ is given such that for all indices $i\geq 10^{100}$, $|a_{i+1}-a_{i}|\leq 2d$ holds. Prove that there exists infinity many indices $j$ such that $|a_j -j|< d$.
2018 Germany Team Selection Test
Determine all integers $ n\geq 2$ having the following property: for any integers $a_1,a_2,\ldots, a_n$ whose sum is not divisible by $n$, there exists an index $1 \leq i \leq n$ such that none of the numbers $$a_i,a_i+a_{i+1},\ldots,a_i+a_{i+1}+\ldots+a_{i+n-1}$$is divisible by $n$. Here, we let $a_i=a_{i-n}$ when $i >n$.



Proposed by Warut Suksompong, Thailand
2017 Germany Team Selection Test
In a convex quadrilateral $ABCD$, $BD$ is the angle bisector of $\angle{ABC}$. The circumcircle of $ABC$ intersects $CD,AD$ in $P,Q$ respectively and the line through $D$ parallel to $AC$ cuts $AB,AC$ in $R,S$ respectively. Prove that point $P,Q,R,S$ lie on a circle.
2016 Germany Team Selection Test
The two circles $\Gamma_1$ and $\Gamma_2$ with the midpoints $O_1$ resp. $O_2$ intersect in the two distinct points $A$ and $B$. A line through $A$ meets $\Gamma_1$ in $C \neq A$ and $\Gamma_2$ in $D \neq A$. The lines $CO_1$ and $DO_2$ intersect in $X$.

Prove that the four points $O_1,O_2,B$ and $X$ are concyclic.
2016 Germany Team Selection Test
The positive integers $a_1,a_2, \dots, a_n$ are aligned clockwise in a circular line with $n \geq 5$. Let $a_0=a_n$ and $a_{n+1}=a_1$. For each $i \in \{1,2,\dots,n \}$ the quotient $$ q_i=\frac{a_{i-1}+a_{i+1}}{a_i} $$is an integer. Prove $$ 2n \leq q_1+q_2+\dots+q_n < 3n. $$
2016 Germany Team Selection Test
In the beginning there are $100$ integers in a row on the blackboard. Kain and Abel then play the following game: A move consists in Kain choosing a chain of consecutive numbers; the length of the chain can be any of the numbers $1,2,\dots,100$ and in particular it is allowed that Kain only chooses a single number. After Kain has chosen his chain of numbers, Abel has to decide whether he wants to add $1$ to each of the chosen numbers or instead subtract $1$ from of the numbers. After that the next move begins, and so on.



If there are at least $98$ numbers on the blackboard that are divisible by $4$ after a move, then Kain has won.



Prove that Kain can force a win in a finite number of moves.
2015 Germany Team Selection Test
Find the least positive integer $n$, such that there is a polynomial $$ P(x) = a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_1x+a_0 $$ with real coefficients that satisfies both of the following properties:

- For $i=0,1,\dots,2n$ it is $2014 \leq a_i \leq 2015$.

- There is a real number $\xi$ with $P(\xi)=0$.
2015 Germany Team Selection Test
A positive integer $n$ is called naughty if it can be written in the form $n=a^b+b$ with integers $a,b \geq 2$.

Is there a sequence of $102$ consecutive positive integers such that exactly $100$ of those numbers are naughty?
2015 Germany Team Selection Test
Let $ABC$ be an acute triangle with $|AB| \neq |AC|$ and the midpoints of segments $[AB]$ and $[AC]$ be $D$ resp. $E$. The circumcircles of the triangles $BCD$ and $BCE$ intersect the circumcircle of triangle $ADE$ in $P$ resp. $Q$ with $P \neq D$ and $Q \neq E$.

Prove $|AP|=|AQ|$.



(Notation: $|\cdot|$ denotes the length of a segment and $[\cdot]$ denotes the line segment.)
2015 Germany Team Selection Test
Let $ABC$ be an acute triangle with the circumcircle $k$ and incenter $I$. The perpendicular through $I$ in $CI$ intersects segment $[BC]$ in $U$ and $k$ in $V$. In particular $V$ and $A$ are on different sides of $BC$. The parallel line through $U$ to $AI$ intersects $AV$ in $X$.

Prove: If $XI$ and $AI$ are perpendicular to each other, then $XI$ intersects segment $[AC]$ in its midpoint $M$.



(Notation: $[\cdot]$ denotes the line segment.)
2014 Germany Team Selection Test
In Sikinia we only pay with coins that have a value of either $11$ or $12$ Kulotnik. In a burglary in one of Sikinia's banks, $11$ bandits cracked the safe and could get away with $5940$ Kulotnik. They tried to split up the money equally - so that everyone gets the same amount - but it just doesn't worked. After a while their leader claimed that it actually isn't possible.

Prove that they didn't get any coin with the value $12$ Kulotnik.
2014 Germany Team Selection Test
Let $ABCD$ be a convex cyclic quadrilateral with $AD=BD$. The diagonals $AC$ and $BD$ intersect in $E$. Let the incenter of triangle $\triangle BCE$ be $I$. The circumcircle of triangle $\triangle BIE$ intersects side $AE$ in $N$.

Prove

$$ AN \cdot NC = CD \cdot BN. $$
2014 Germany Team Selection Test
Let $a_1 \leq a_2 \leq \cdots$ be a non-decreasing sequence of positive integers. A positive integer $n$ is called good if there is an index $i$ such that $n=\dfrac{i}{a_i}$.

Prove that if $2013$ is good, then so is $20$.
2014 Germany Team Selection Test
In a triangle $ABC$, let $D$ and $E$ be the feet of the angle bisectors of angles $A$ and $B$, respectively. A rhombus is inscribed into the quadrilateral $AEDB$ (all vertices of the rhombus lie on different sides of $AEDB$). Let $\varphi$ be the non-obtuse angle of the rhombus. Prove that $\varphi \le \max \{  \angle BAC, \angle ABC  \}$.
2013 Germany Team Selection Test
$n$ is an odd positive integer and $x,y$ are two rational numbers satisfying $$x^n+2y=y^n+2x.$$Prove that $x=y$.
2013 Germany Team Selection Test
Given a $m\times n$ grid rectangle with $m,n \ge 4$ and a closed path $P$ that is not self intersecting from inner points of the grid, let $A$ be the number of points on $P$ such that $P$ does not turn in them and let $B$ be the number of squares that $P$ goes through two non-adjacent sides of them furthermore let $C$ be the number of squares with no side in $P$. Prove that $$A=B-C+m+n-1.$$
2013 Germany Team Selection Test
Let $ABC$ be an acute-angled triangle with circumcircle $\omega$. Prove that there exists a point $J$ such that for any point $X$ inside $ABC$ if $AX,BX,CX$ intersect $\omega$ in $A_1,B_1,C_1$ and $A_2,B_2,C_2$ be reflections of $A_1,B_1,C_1$ in midpoints of $BC,AC,AB$ respectively then $A_2,B_2,C_2,J$ lie on a circle.
2013 Germany Team Selection Test
Two concentric circles $\omega, \Omega$ with radii $8,13$ are given. $AB$ is a diameter of $\Omega$ and the tangent from $B$ to $\omega$ touches $\omega$ at $D$. What is the length of $AD$.
2013 Germany Team Selection Test
Call admissible a set $A$ of integers that has the following property:

If $x,y \in A$ (possibly $x=y$) then $x^2+kxy+y^2 \in A$ for every integer $k$.

Determine all pairs $m,n$ of nonzero integers such that the only admissible set containing both $m$ and $n$ is the set of all integers.



Proposed by Warut Suksompong, Thailand
2013 Germany Team Selection Test
Let $n \geq 1$ be an integer. What is the maximum number of disjoint pairs of elements of the set $\{ 1,2,\ldots , n \}$ such that the sums of the different pairs are different integers not exceeding $n$?
2012 Germany Team Selection Test
Find the least integer $k$ such that for any $2011 \times 2011$ table filled with integers Kain chooses, Abel be able to change at most $k$ cells to achieve a new table in which $4022$ sums of rows and columns are pairwise different.
2012 Germany Team Selection Test
Let $\Gamma$ be the circumcircle of isosceles triangle $ABC$ with vertex $C$. An arbitrary point $M$ is chosen on the  segment $BC$ and point $N$ lies on the ray $AM$ with $M$ between $A,N$ such that $AN=AC$. The circumcircle of $CMN$ cuts $\Gamma$ in $P$ other than $C$ and $AB,CP$ intersect at $Q$. Prove that $\angle BMQ = \angle QMN.$
2012 Germany Team Selection Test
Let $a,b,c$ be positive real numbers with $a^2+b^2+c^2 \geq 3$. Prove that:

$$\frac{(a+1)(b+2)}{(b+1)(b+5)}+\frac{(b+1)(c+2)}{(c+1)(c+5)}+\frac{(c+1)(a+2)}{(a+1)(a+5)} \geq \frac{3}{2}.$$
2012 Germany Team Selection Test
Consider a polynomial $P(x) =  \prod^9_{j=1}(x+d_j),$ where $d_1, d_2, \ldots d_9$ are nine distinct integers. Prove that there exists an integer $N,$ such that for all integers $x \geq N$ the number $P(x)$ is divisible by a prime number greater than 20.



Proposed by Luxembourg
2012 Germany Team Selection Test
Let $ABC$ be an acute triangle. Let $\omega$ be a circle whose centre $L$ lies on the side $BC$. Suppose that $\omega$ is tangent to $AB$ at $B'$ and $AC$ at $C'$. Suppose also that the circumcentre $O$ of triangle $ABC$ lies on the shorter arc $B'C'$ of $\omega$. Prove that the circumcircle of $ABC$ and $\omega$ meet at two points.



Proposed by Härmel Nestra, Estonia
2012 Germany Team Selection Test
Determine all pairs $(f,g)$ of functions from the set of real numbers to itself that satisfy $$g(f(x+y)) = f(x) + (2x + y)g(y)$$ for all real numbers $x$ and $y$.



Proposed by Japan
2011 Germany Team Selection Test
Two circles $\omega , \Omega$ intersect in distinct points $A,B$ a line through $B$ intersects $\omega , \Omega$ in $C,D$ respectively such that $B$ lies between $C,D$ another line through $B$ intersects $\omega , \Omega$ in $E,F$ respectively such that $E$ lies between $B,F$ and $FE=CD$. Furthermore $CF$ intersects $\omega , \Omega$ in $P,Q$ respectively and $M,N$ are midpoints of the arcs $PB,QB$.  Prove that $CNMF$ is a cyclic quadrilateral.
2011 Germany Team Selection Test
Let $n$ be a positive integer prove that $$6\nmid \lfloor (\sqrt[3]{28}-3)^{-n} \rfloor.$$
2011 Germany Team Selection Test
We call a function $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ good if for all $x,y \in \mathbb{Q}^+$ we have: $$f(x)+f(y)\geq 4f(x+y).$$a) Prove that for all good functions  $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ and $x,y,z \in \mathbb{Q}^+$ $$f(x)+f(y)+f(z) \geq 8f(x+y+z)$$b) Does there exists a good functions  $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ and $x,y,z \in \mathbb{Q}^+$ such that $$f(x)+f(y)+f(z) < 9f(x+y+z) ?$$
2011 Germany Team Selection Test
A sequence $x_1, x_2, \ldots$ is defined by $x_1 = 1$ and $x_{2k}=-x_k, x_{2k-1} = (-1)^{k+1}x_k$ for all $k \geq 1.$ Prove that $\forall n \geq 1$ $x_1 + x_2 + \ldots + x_n \geq 0.$



Proposed by Gerhard Wöginger, Austria
2011 Germany Team Selection Test
Let $ABCDE$ be a convex pentagon such that $BC \parallel AE,$ $AB = BC +  AE,$ and $\angle ABC = \angle CDE.$ Let $M$ be the midpoint of $CE,$ and let $O$ be the circumcenter of triangle $BCD.$ Given that $\angle DMO = 90^{\circ},$ prove that $2 \angle BDA = \angle CDE.$



Proposed by Nazar Serdyuk, Ukraine
2011 Germany Team Selection Test
Vertices and Edges of a regular $n$-gon are numbered $1,2,\dots,n$ clockwise such that edge $i$ lies between vertices $i,i+1 \mod n$. Now non-negative integers $(e_1,e_2,\dots,e_n)$ are assigned to corresponding edges and non-negative integers $(k_1,k_2,\dots,k_n)$ are assigned to corresponding vertices such that:

$i$) $(e_1,e_2,\dots,e_n)$ is a permutation of $(k_1,k_2,\dots,k_n)$.

$ii$) $k_i=|e_{i+1}-e_i|$ indexes$\mod n$.



a) Prove that for all $n\geq 3$ such non-zero $n$-tuples exist.

b) Determine for each $m$ the smallest positive integer $n$ such that there is an $n$-tuples stisfying the above conditions and also $\{e_1,e_2,\dots,e_n\}$ contains all $0,1,2,\dots m$.
2010 Germany Team Selection Test
The quadrilateral $ABCD$ is a rhombus with acute angle at $A.$ Points $M$ and $N$ are on segments $\overline{AC}$ and $\overline{BC}$ such that $|DM| = |MN|.$ Let $P$ be the intersection of $AC$ and $DN$ and let $R$ be the intersection of $AB$ and $DM.$ Prove that $|RP| = |PD|.$
2010 Germany Team Selection Test
Prove or disprove that $\forall a,b,c,d \in \mathbb{R}^+$ we have the following inequality:



$$3 \leq \frac{4a+b}{a+4b} + \frac{4b+c}{b+4c} + \frac{4c+a}{c+4a} < \frac{33}{4}$$
2010 Germany Team Selection Test
Determine all $(m,n) \in \mathbb{Z}^+ \times \mathbb{Z}^+$ which satisfy $3^m-7^n=2.$
2010 Germany Team Selection Test
Consider 2009 cards which are lying in sequence on a table. Initially, all cards have their top face white and bottom face black. The cards are enumerated from 1 to 2009. Two players, Amir and Ercole, make alternating moves, with Amir starting. Each move consists of a player choosing a card with the number $k$ such that $k < 1969$ whose top face is white, and then this player turns all cards at positions $k,k+1,\ldots,k+40.$ The last player who can make a legal move wins.



(a) Does the game necessarily end?

(b) Does there exist a winning strategy for the starting player?



Also compare shortlist 2009, combinatorics problem C1.
2010 Germany Team Selection Test
Given a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ meet at $E$ and the lines $AD$ and $BC$ meet at $F$. The midpoints of $AB$ and $CD$ are $G$ and $H$, respectively. Show that $EF$ is tangent at $E$ to the circle through the points $E$, $G$ and $H$.



Proposed by David Monk, United Kingdom
2010 Germany Team Selection Test
A positive integer $N$ is called balanced, if $N=1$ or if $N$ can be written as a product of an even number of not necessarily distinct primes. Given positive integers $a$ and $b$, consider the polynomial $P$ defined by $P(x)=(x+a)(x+b)$.

(a) Prove that there exist distinct positive integers $a$ and $b$ such that all the number $P(1)$, $P(2)$,$\ldots$, $P(50)$ are balanced.

(b) Prove that if $P(n)$ is balanced for all positive integers $n$, then $a=b$.



Proposed by Jorge Tipe, Peru