Navvye/Olympiad-Problems-CSV huggingface dataset:
2007 Estonia Team Selection Test
Let $D$ be the foot of the altitude of triangle $ABC$ drawn from vertex $A$. Let $E$ and $F$ be points symmetric to $D$ w.r.t. lines $AB$ and $AC$, respectively. Let $R_1$ and $R_2$ be the circumradii of triangles $BDE$ and $CDF$, respectively, and let $r_1$ and $r_2$ be the inradii of the same triangles. Prove that $|S_{ABD} - S_{ACD}| > |R_1r_1 - R_2r_2|$
2007 Estonia Team Selection Test
Let $n$ be a natural number, $n > 2$. Prove that if $\frac{b^n-1}{b-1}$ is a prime power for some positive integer $b$ then $n$ is prime.
2007 Estonia Team Selection Test
In square $ABCD,$ points $E$ and $F$ are chosen in the interior of sides $BC$ and $CD$, respectively. The line drawn from $F$ perpendicular to $AE$ passes through the intersection point $G$ of $AE$ and diagonal $BD$. A point $K$ is chosen on $FG$ such that $|AK|= |EF|$. Find $\angle EKF.$
2007 Estonia Team Selection Test
Find all continuous functions $f: R \to R$ such that for all reals $x$ and $y$, $f(x+f(y)) = y+f(x+1)$.
2007 Estonia Team Selection Test
Consider a $10 \times 10$ grid. On every move, we colour $4$ unit squares that lie in the intersection of some two rows and two columns. A move is allowed if at least one of the $4$ squares is previously uncoloured. What is the largest possible number of moves that can be taken to colour the whole grid?
2006 Estonia Team Selection Test
Let $k$ be any fixed positive integer. Let's look at integer pairs $(a, b)$, for which the quadratic equations $x^2 - 2ax + b = 0$ and $y^2 + 2ay + b = 0$ are real solutions (not necessarily different), which can be denoted by $x_1, x_2$ and $y_1, y_2$, respectively, in such an order that the equation $x_1 y_1 - x_2 y_2 = 4k$.

a) Find the largest possible value of the second component $b$ of such a pair of numbers ($a, b)$.

b) Find the sum of the other components of all such pairs of numbers.
2006 Estonia Team Selection Test
The center of the circumcircle of the acute triangle $ABC$ is $O$. The line $AO$ intersects $BC$ at $D$. On the sides $AB$ and $AC$ of the triangle, choose points $E$ and $F$, respectively, so that the points $A, E, D, F$ lie on the same circle. Let $E'$ and $F'$ projections of points $E$ and $F$ on side $BC$ respectively. Prove that length of the segment $E'F'$ does not depend on the position of points $E$ and $F$.
2006 Estonia Team Selection Test
A grid measuring $10 \times 11$ is given. How many "crosses" covering five unit squares can be placed on the grid?

(pictured right) so that no two of them cover the same square?
2006 Estonia Team Selection Test
The side $AC$ of an acute triangle $ABC$ is the diameter of the circle $c_1$ and side $BC$ is the diameter of the circle $c_2$. Let $E$ be the foot of the altitude drawn from the vertex $B$ of the triangle and $F$ the foot of the altitude drawn from the vertex $A$. In addition, let $L$ and $N$ be the points of intersection of the line $BE$ with the circle $c_1$ (the point $L$ lies on the segment $BE$) and the points of intersection of $K$ and $M$ of line $AF$ with circle $c_2$ (point $K$ is in section $AF$). Prove that $K LM N$ is a cyclic quadrilateral.
2006 Estonia Team Selection Test
Let $a_1, a_2, a_3, ...$ be a sequence of positive real numbers. Prove that for any positive integer $n$ the inequality holds $\sum_{i=1}^n b_i^2  \le 4 \sum_{i=1}^n a_i^2$ where $b_i$ is the arithmetic mean of the numbers $a_1, a_2, ..., a_n$
2006 Estonia Team Selection Test
Denote by $d(n)$ the number of divisors of the positive integer $n$. A positive integer $n$ is called highly divisible if $d(n) > d(m)$ for all positive integers $m < n$.

Two highly divisible integers $m$ and $n$ with $m < n$ are called consecutive if there exists no highly divisible integer $s$ satisfying $m < s < n$.



(a) Show that there are only finitely many pairs of consecutive highly divisible

integers of the form $(a, b)$ with $a\mid b$.



(b) Show that for every prime number $p$ there exist infinitely many positive highly divisible integers $r$ such that $pr$ is also highly divisible.
2005 Estonia Team Selection Test
On a plane, a line $\ell$ and two circles $c_1$ and $c_2$ of different radii are given such that  $\ell$  touches both circles at point $P$. Point $M \ne P$ on $\ell$ is chosen so that the angle $Q_1MQ_2$ is as large as possible where $Q_1$ and $Q_2$ are the tangency points of the tangent lines drawn from $M$ to $c_i$ and $c_2$, respectively, differing from $\ell$ . Find $\angle PMQ_1 + \angle PMQ_2$·
2005 Estonia Team Selection Test
On the planet Automory, there are infinitely many inhabitants. Every Automorian loves exactly one Automorian and honours exactly one Automorian. Additionally, the following can be noticed:

$\bullet$ each Automorian is loved by some Automorian;

$\bullet$ if Automorian $A$ loves Automorian $B$, then also all Automorians honouring $A$ love $B$,

$\bullet$if Automorian $A$ honours Automorian $B$, then also all Automorians loving $A$ honour $B$.

Is it correct to claim that every Automorian honours and loves the same Automorian?
2005 Estonia Team Selection Test
Find all pairs $(x, y)$ of positive integers satisfying the equation $(x + y)^x = x^y$.
2005 Estonia Team Selection Test
Find all pairs $(a, b)$ of real numbers such that the roots of polynomials $6x^2 -24x -4a$ and $x^3 + ax^2 + bx - 8$ are all non-negative real numbers.
2005 Estonia Team Selection Test
On a horizontal line, $2005$ points are marked, each of which is either white or black. For every point, one finds the sum of the number of white points on the right of it and the number of black points on the left of it. Among the $2005$ sums, exactly one number occurs an odd number of times. Find all possible values of this number.
2005 Estonia Team Selection Test
Let $\Gamma$ be a circle and let $d$ be a line such that $\Gamma$ and $d$ have no common points. Further, let $AB$ be a diameter of the circle $\Gamma$; assume that this diameter $AB$ is perpendicular to the line $d$, and the point $B$ is nearer to the line $d$ than the point $A$. Let $C$ be an arbitrary point on the circle $\Gamma$, different from the points $A$ and $B$. Let $D$ be the point of intersection of the lines $AC$ and $d$. One of the two tangents from the point $D$ to the circle $\Gamma$ touches this circle $\Gamma$ at a point $E$; hereby, we assume that the points $B$ and $E$ lie in the same halfplane with respect to the line $AC$. Denote by $F$ the point of intersection of the lines $BE$ and $d$. Let the line $AF$ intersect the circle $\Gamma$ at a point $G$, different from $A$.



Prove that the reflection of the point $G$ in the line $AB$ lies on the line $CF$.
2004 Estonia Team Selection Test
Let $k > 1$ be a fixed natural number. Find all polynomials $P(x)$ satisfying the condition $P(x^k) = (P(x))^k$ for all real numbers $x$.
2004 Estonia Team Selection Test
Let $O$ be the circumcentre of the acute triangle $ABC$ and let lines $AO$ and $BC$ intersect at point $K$. On sides $AB$ and $AC$, points $L$ and $M$ are chosen such that $|KL|= |KB|$ and $|KM| = |KC|$. Prove that segments $LM$ and $BC$ are parallel.
2004 Estonia Team Selection Test
For which natural number $n$ is it possible to draw $n$ line segments between vertices of a regular $2n$-gon so that every vertex is an endpoint for exactly one segment and these segments have pairwise different lengths?
2004 Estonia Team Selection Test
Denote $f(m) =\sum_{k=1}^m (-1)^k cos \frac{k\pi}{2 m + 1}$

For which positive integers $m$ is $f(m)$ rational?
2004 Estonia Team Selection Test
Find all natural numbers $n$ for which the number of all positive divisors of the number lcm $(1,2,..., n)$ is equal to $2^k$ for some non-negative integer $k$.
2004 Estonia Team Selection Test
Call a convex polyhedron a footballoid  if it has the following properties.

(1) Any face is either a regular pentagon or a regular hexagon.

(2) All neighbours of a pentagonal face are hexagonal (a neighbour  of a face is a face that has a common edge with it).
2003 Estonia Team Selection Test
Two treasure-hunters found a treasure containing coins of value $a_1< a_2 < ... < a_{2003}$ (the quantity of coins of each value is unlimited). The first treasure-hunter forms all the possible sets of different coins containing odd number of elements, and takes the most valuable coin of each such set. The second treasure-hunter forms all the possible sets of different coins containing even number of elements, and takes the most valuable coin of each such set. Which one of them is going to have more money and how much more?



(H. Nestra)
2003 Estonia Team Selection Test
Let $n$ be a positive integer. Prove that if the number overbrace $\underbrace{\hbox{99...9}}_{\hbox{n}}$ is divisible by $n$, then the number $\underbrace{\hbox{11...1}}_{\hbox{n}}$ is also divisible by $n$.



(H. Nestra)
2003 Estonia Team Selection Test
Let $N$ be the set of all non-negative integers and for each $n \in N$ denote $n'= n +1$. The function $A : N^3 \to N$ is defined as follows:

(i) $A(0, m, n) = m'$ for all   $m,  n \in N$

(ii)  $A(k', 0, n) =\left\{ \begin{array}{ll}

      n & if \, \, k = 0 \\
      0 & if  \, \,k = 1, \\
      1 & if  \, \, k > 1 \end{array} \right.$  for all  $k,  n \in N$

(iii) $A(k', m', n) = A(k, A(k',m,n), n)$ for all  $k,m,  n \in N$.

Compute $A(5, 3, 2)$.



(H. Nestra)
2003 Estonia Team Selection Test
A deck consists of $2^n$ cards. The deck is shuffled using the following operation: if the cards are initially in the order

$a_1,a_2,a_3,a_4,...,a_{2^n-1},a_{2^n}$ then after shuffling the order becomes $a_{2^{n-1}+1},a_1,a_{2^{n-1}+2},a_2,...,a_{2^n},a_{2^{n-1}}$ .

Find the smallest number of such operations after which the original order of the cards is restored.



(R. Palm)
2003 Estonia Team Selection Test
Let $a, b, c$ be positive real numbers satisfying the condition  $\frac{1}{ab}+\frac{1}{ac}+\frac{1}{bc}=1$ . Prove the inequality    $$\frac{a}{\sqrt{1+a^2}}+\frac{b}{\sqrt{1+b^2}}+\frac{c}{\sqrt{1+c^2}} \le \frac{3\sqrt3}{2}$$When does the equality hold?



(L. Parts)
2003 Estonia Team Selection Test
Let $ABC$ be an acute-angled triangle, $O$ its circumcenter and $H$ its orthocenter. The orthogonal projection of the vertex $A$ to the line $BC$ lies on the perpendicular bisector of the segment $AC$. Compute $\frac{CH}{BO}$ .



(J. Willemson)
2002 Estonia Team Selection Test
The princess wishes to have a bracelet with $r$ rubies and $s$ emeralds arranged in such order that there exist two jewels on the bracelet such that starting with these and enumerating the jewels in the same direction she would obtain identical sequences of jewels. Prove that it is possible to fulfill the princess’s wish if and only if $r$ and $s$ have a common divisor.
2002 Estonia Team Selection Test
Consider an isosceles triangle $KL_1L_2$ with  $|KL_1|=|KL_2|$ and let $KA, L_1B_1,L_2B_2$ be its angle bisectors. Prove that $\cos  \angle B_1AB_2 < \frac35$
2002 Estonia Team Selection Test
In a certain country there are $10$ cities connected by a network of one-way nonstop flights so that it is possible to fly (using one or more flights) from any city to any other. Let $n$ be the least number of flights needed to complete a trip starting from one of the cities, visiting all others and returning to the starting point. Find the greatest possible value of $n$.
2002 Estonia Team Selection Test
Let $ABCD$ be a cyclic quadrilateral such that $\angle ACB = 2\angle CAD$ and $\angle ACD = 2\angle BAC$. Prove that $|CA| = |CB| + |CD|$.
2002 Estonia Team Selection Test
Let $0 < a < \frac{\pi}{2}$ and $x_1,x_2,...,x_n$ be real numbers such that $\sin x_1 + \sin x_2 +... + \sin x_n  \ge  n  \cdot  sin a $. Prove that $\sin (x_1 - a) + \sin (x_2 - a) + ... + \sin (x_n - a) \ge 0$ .
2002 Estonia Team Selection Test
Place a pebble at each non-positive integer point on the real line, and let $n$ be a fixed positive integer. At each step we choose some n consecutive integer points, remove one of the pebbles located at these points and rearrange all others arbitrarily within these points (placing at most one pebble at each point).

Determine whether there exists a positive integer $n$ such that for any given $N > 0$ we can place a pebble at a point with coordinate greater than $N$ in a finite number of steps described above.
2001 Estonia Team Selection Test
Consider on the coordinate plane all rectangles whose

(i) vertices have integer coordinates;

(ii) edges are parallel to coordinate axes;

(iii) area is $2^k$, where $k = 0,1,2....$

Is it possible to color all points with integer coordinates in two colors so that no such rectangle has all its vertices of the same color?
2001 Estonia Team Selection Test
Point $X$ is taken inside a regular $n$-gon of side length $a$. Let $h_1,h_2,...,h_n$ be the distances from $X$ to the lines defined by the sides of the $n$-gon. Prove that $\frac{1}{h_1}+\frac{1}{h_2}+...+\frac{1}{h_n}>\frac{2\pi}{a}$
2001 Estonia Team Selection Test
Let $k$ be a fixed real number. Find all functions $f: R \to R$ such that $f(x)+ (f(y))^2 = kf(x + y^2)$ for all real numbers $x$ and $y$.
2001 Estonia Team Selection Test
Consider all products by $2, 4, 6, ..., 2000$ of the elements of the set $A =\left\{\frac12, \frac13, \frac14,...,\frac{1}{2000},\frac{1}{2001}\right\}$ . Find the sum of all these products.
2001 Estonia Team Selection Test
Find the exponent of $37$ in the representation of the number $111...... 11$  with $3\cdot 37^{2000}$ digits equals to $1$, as product of prime powers
2001 Estonia Team Selection Test
Let $C_1$ and $C_2$ be the incircle and the circumcircle of the triangle $ABC$, respectively. Prove that, for any point $A'$ on $C_2$, there exist points $B'$ and $C'$ such that $C_1$ and $C_2$ are the incircle and the circumcircle of triangle $A'B'C'$, respectively.
1997 Estonia Team Selection Test
In a triangle $ABC$ points $A_1,B_1,C_1$ are the midpoints of $BC,CA,AB$ respectively,and $A_2,B_2,C_2$ are the midpoints of the altitudes from $A,B,C$ respectively. Show that the lines $A_1A_2,B_1B_2,C_1,C_2$ are concurrent.
1997 Estonia Team Selection Test
Prove that for all positive real numbers $a_1,a_2,\cdots a_n$ $$\frac{1}{\frac{1}{1+a_1}+\frac{1}{1+a_2}+\cdots +\frac{1}{1+a_n}}-\frac{1}{\frac{1}{a_1}+\frac{1}{a_2}+\cdots +\frac{1}{a_n}}\geq \frac{1}{n}$$When does the inequality hold?
1997 Estonia Team Selection Test
There are $n$ boyfriend-girlfriend pairs at a party. Initially all the girls sit at a round table. For the first dance, each boy invites one of the girls to dance with.After each dance, a boy takes the girl he danced with to her seat, and for the next dance he invites the girl next to her in the counterclockwise direction. For which values of $n$ can the girls be selected in such a way that in every dance at least one boy danced with his girlfriend, assuming that there are no less than $n$ dances?
1997 Estonia Team Selection Test
$(a)$ Is it possible to partition the segment $[0,1]$ into two sets $A$ and $B$ and to define a continuous function $f$ such that for every $x\in  A \ f(x)$ is in $B$, and for every $x\in B \ f(x)$ is in $A$?



$(b)$ The same question with $[0,1]$ replaced by $[0,1).$
1997 Estonia Team Selection Test
A quadrilateral $ABCD$ is inscribed in a circle. On each of the sides $AB,BC,CD,DA$ one erects a rectangle towards the interior of the quadrilateral, the other side of the rectangle being equal to $CD,DA,AB,BC,$ respectively. Prove that the centers of these four rectangles are vertices of a rectangle.
1997 Estonia Team Selection Test
It is known that for every integer $n > 1$ there is a prime number among the numbers $n+1,n+2,...,2n-1.$ Determine all positive integers $n$ with the following property: Every integer $m > 1$ less than $n$ and coprime to $n$ is prime.
1996 Estonia Team Selection Test
Suppose that $x,y$ and $\frac{x^2+y^2+6}{xy}$ are positive integers . Prove that $\frac{x^2+y^2+6}{xy}$ is a perfect cube.
1996 Estonia Team Selection Test
Let $a,b,c$ be the sides of a triangle, $\alpha ,\beta ,\gamma$ the corresponding angles and $r$ the inradius. Prove that $$a\cdot sin\alpha+b\cdot sin\beta+c\cdot sin\gamma\geq 9r$$
1996 Estonia Team Selection Test
Find all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy for all $x$:



$(i)$ $f(x)=-f(-x);$



$(ii)$ $f(x+1)=f(x)+1;$



$(iii)$ $f\left( \frac{1}{x}\right)=\frac{1}{x^2}f(x)$ for $x\ne 0$
1996 Estonia Team Selection Test
Prove that the polynomial $P_n(x)=1+x+\frac{x^2}{2!}+\cdots +\frac{x^n}{n!}$ has no real zeros if $n$ is even and has exatly one real zero if $n$ is odd
1996 Estonia Team Selection Test
Let $H$ be the orthocenter of an obtuse triangle $ABC$ and $A_1B_1C_1$ arbitrary points on the sides $BC,AC,AB$ respectively.Prove that the tangents drawn from $H$ to the circles with diametrs $AA_1,BB_1,CC_1$ are equal.
1996 Estonia Team Selection Test
Each face of a cube is divided into $n^2$ equal squares. The vertices of the squares are called nodes, so each face has $(n+1)^2$ nodes.



$(a)$ If $n=2$,does there exist a closed polygonal line whose links are sids of the squares and which passes through each node exactly once?



$(b)$ Prove that, for each $n$, such a polygonal line divides the surface area of the cube into two equal parts
2014 France Team Selection Test
Let $n$ be an positive integer. Find the smallest integer $k$ with the following property; Given any real numbers $a_1 , \cdots , a_d $ such that $a_1 + a_2 + \cdots + a_d = n$ and $0 \le a_i \le 1$ for $i=1,2,\cdots ,d$, it is possible to partition these numbers into $k$ groups (some of which may be empty) such that the sum of the numbers in each group is at most $1$.
2014 France Team Selection Test
Two circles $O_1$ and $O_2$ intersect each other at $M$ and $N$. The common tangent to two circles nearer to $M$ touch $O_1$ and $O_2$ at $A$ and $B$ respectively. Let $C$ and $D$ be the reflection of $A$ and $B$ respectively with respect to $M$. The circumcircle of the triangle $DCM$ intersect circles $O_1$ and $O_2$ respectively at points $E$ and $F$ (both distinct from $M$). Show that the circumcircles of triangles $MEF$ and $NEF$ have same radius length.
2014 France Team Selection Test
Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.
2014 France Team Selection Test
Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions  $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that

$$ m^2 + f(n) \mid mf(m) +n $$

for all positive integers $m$ and $n$.
2014 France Team Selection Test
Let $\omega$ be the circumcircle of a triangle $ABC$. Denote by $M$ and $N$ the midpoints of the sides $AB$ and $AC$, respectively, and denote by $T$ the midpoint of the arc $BC$ of $\omega$ not containing $A$. The circumcircles of the triangles $AMT$ and $ANT$ intersect the perpendicular bisectors of $AC$ and $AB$ at points $X$ and $Y$, respectively; assume that $X$ and $Y$ lie inside the triangle $ABC$. The lines $MN$ and $XY$ intersect at $K$. Prove that $KA=KT$.
2014 France Team Selection Test
Let $n$ be a positive integer and $x_1,x_2,\ldots,x_n$ be positive reals. Show that there are numbers $a_1,a_2,\ldots, a_n \in \{-1,1\}$ such that the following holds:

$$a_1x_1^2+a_2x_2^2+\cdots+a_nx_n^2 \ge (a_1x_1+a_2x_2 +\cdots+a_nx_n)^2$$
2012 France Team Selection Test
Let $n$ and $k$ be two positive integers. Consider a group of $k$ people such that, for each group of $n$ people, there is a $(n+1)$-th person that knows them all (if $A$ knows $B$ then $B$ knows $A$).

1) If $k=2n+1$, prove that there exists a person who knows all others.

2)  If $k=2n+2$, give an example of such a group in which no-one knows all others.
2012 France Team Selection Test
Let $ABC$ be an acute-angled triangle with $AB\not= AC$. Let $\Gamma$ be the circumcircle, $H$ the orthocentre and $O$ the centre of $\Gamma$. $M$ is the midpoint of $BC$. The line $AM$ meets $\Gamma$ again at $N$ and the circle with diameter $AM$ crosses $\Gamma$ again at $P$. Prove that the lines $AP,BC,OH$ are concurrent if and only if $AH=HN$.
2012 France Team Selection Test
Let $p$ be a prime number. Find all positive integers $a,b,c\ge 1$ such that:

$$a^p+b^p=p^c.$$
2012 France Team Selection Test
Let $k>1$ be an integer. A function $f:\mathbb{N^*}\to\mathbb{N^*}$ is called $k$-tastrophic when for every integer $n>0$, we have $f_k(n)=n^k$ where $f_k$ is the $k$-th iteration of $f$:

$$f_k(n)=\underbrace{f\circ f\circ\cdots \circ f}_{k\text{ times}}(n)$$

For which $k$ does there exist a $k$-tastrophic function?
2012 France Team Selection Test
Determine all non-constant polynomials $X^n+a_{n-1}X^{n-1}+\cdots +a_1X+a_0$ with integer coefficients for which the roots are exactly the numbers $a_0,a_1,\ldots ,a_{n-1}$ (with multiplicity).
2012 France Team Selection Test
Let $ABCD$ be a convex quadrilateral whose sides $AD$ and $BC$ are not parallel. Suppose that the circles with diameters $AB$ and $CD$ meet at points $E$ and $F$ inside the quadrilateral. Let $\omega_E$ be the circle through the feet of the perpendiculars from $E$ to the lines $AB,BC$ and $CD$. Let $\omega_F$ be the circle through  the feet of the perpendiculars from $F$ to the lines $CD,DA$ and $AB$. Prove that the midpoint of the segment $EF$ lies on the line through the two intersections of $\omega_E$ and $\omega_F$.



Proposed by Carlos Yuzo Shine, Brazil
2007 France Team Selection Test
For a positive integer $a$, $a'$ is the integer obtained by the following method: the decimal writing of $a'$ is the inverse of the decimal writing of $a$ (the decimal writing of $a'$ can begin by zeros, but not the one of $a$); for instance if $a=2370$, $a'=0732$, that is $732$.



Let $a_{1}$ be a positive integer, and $(a_{n})_{n \geq 1}$ the sequence defined by $a_{1}$ and the following formula for $n \geq 1$:

$$a_{n+1}=a_{n}+a'_{n}. $$

Can $a_{7}$ be prime?
2007 France Team Selection Test
Let $a,b,c,d$ be positive reals such taht $a+b+c+d=1$.



Prove that: $$6(a^{3}+b^{3}+c^{3}+d^{3})\geq a^{2}+b^{2}+c^{2}+d^{2}+\frac{1}{8}.$$
2007 France Team Selection Test
Let $A,B,C,D$ be four distinct points on a circle such that the lines $(AC)$ and $(BD)$ intersect at $E$, the lines $(AD)$ and $(BC)$ intersect at $F$ and such that $(AB)$ and $(CD)$ are not parallel.



Prove that $C,D,E,F$ are on the same circle if, and only if, $(EF)\bot(AB)$.
2007 France Team Selection Test
Do there exist $5$ points in the space, such that for all $n\in\{1,2,\ldots,10\}$ there exist two of them at distance between them $n$?
2007 France Team Selection Test
Find all functions $f: \mathbb{Z}\rightarrow\mathbb{Z}$ such that for all $x,y \in \mathbb{Z}$:

$$f(x-y+f(y))=f(x)+f(y).$$
2006 France Team Selection Test
Let $ABCD$ be a square and let $\Gamma$ be the circumcircle of $ABCD$. $M$ is a point of $\Gamma$ belonging to the arc $CD$ which doesn't contain $A$. $P$ and $R$ are respectively the intersection points of $(AM)$ with $[BD]$ and $[CD]$, $Q$ and $S$ are respectively the intersection points of $(BM)$ with $[AC]$ and $[DC]$.

Prove that $(PS)$ and $(QR)$ are perpendicular.
2006 France Team Selection Test
Let $a,b,c$ be three positive real numbers such that $abc=1$. Show that:



$$ \displaystyle \frac{a}{(a+1)(b+1)}+\frac{b}{(b+1)(c+1)}+ \frac{c}{(c+1)(a+1)} \geq \frac{3}{4}.  $$



When is there equality?
2006 France Team Selection Test
Let $a$, $b$ be positive integers such that $b^n+n$ is a multiple of $a^n+n$ for all positive integers $n$. Prove that $a=b$.



Proposed by Mohsen Jamali, Iran
2006 France Team Selection Test
Let $M=\{1,2,\ldots,3 \cdot n\}$. Partition $M$ into three sets $A,B,C$ which $card$ $A$ $=$ $card$ $B$ $=$ $card$ $C$ $=$ $n .$

Prove that there exists $a$ in $A,b$ in $B, c$ in $C$ such that or $a=b+c,$ or $b=c+a,$ or $c=a+b$



Edited by orl.
2005 France Team Selection Test
Let $x$, $y$ be two positive integers such that $\displaystyle 3x^2+x=4y^2+y$.

Prove that $x-y$ is a perfect square.
2005 France Team Selection Test
Two right angled triangles are given, such that the incircle of the first one is equal to the circumcircle of the second one. Let $S$ (respectively $S'$) be the area of the first triangle (respectively of the second triangle).



Prove that $\frac{S}{S'}\geq 3+2\sqrt{2}$.