Solve the following equation in $\mathbb{R}^+$ : \[\left\{\begin{matrix} \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2010\\ x+y+z=\frac{3}{670} \end{matrix}\right.\]

Solve in $\mathbb{R}$ the equation : $(x+1)^5 + (x+1)^4(x-1) + (x+1)^3(x-1)^2 +$ $ (x+1)^2(x-1)^3 + (x+1)(x-1)^4 + (x-1)^5 =$ $ 0$.

Let $a$ and $b$ be two real numbers and let$M(a,b)=\max\left \{ 3a^{2}+2b; 3b^{2}+2a\right \}$. Find the values of $a$ and $b$ for which $M(a,b)$ is minimal.

Let $ABC$ be a triangle. $F$ and $L$ are two points on the side $[AC]$ such that $AF=LC< AC/2$. Find the mesure of the angle $\angle FBL$ knowing that $AB^{2}+BC^{2}=AL^{2}+LC^{2}$.

Prove that \[2010< \frac{2^{2}+1}{2^{2}-1}+\frac{3^{2}+1}{3^{2}-1}+...+\frac{2010^{2}+1}{2010^{2}-1}< 2010+\frac{1}{2}.\]

Compute the sum \[S=1+2+3-4-5+6+7+8-9-10+\dots-2010\] where every three consecutive $+$ are followed by two $-$.

Solve in $\mathbb{R}^{3}$ the following system \[\left\{\begin{matrix} \sqrt{x^{2}-y}=z-1\\ \sqrt{y^{2}-z}=x-1\\ \sqrt{z^{2}-x}=y-1 \end{matrix}\right.\]

Let $ABC$ be a triangle with area $1$ and $P$ the middle of the side $[BC]$. $M$ and $N$ are two points of $[AB]-\left \{ A,B \right \} $ and $[AC]-\left \{ A,C \right \}$ respectively such that $AM=2MB$ and$CN=2AN$. The two lines $(AP)$ and $(MN)$ intersect in a point $D$. Find the area of the triangle $ADN$.

Find the maximum value of the real constant $C$ such that $x^{2}+y^{2}+1\geq C(x+y)$, and $ x^{2}+y^{2}+xy+1\geq C(x+y)$ for all reals $x,y$.

Prove that the equation $x^{2}+p|x| = qx - 1 $ has 4 distinct real solutions if and only if $p+|q|+2<0$ ($p$ and $q$ are two real parameters).

Find all functions $f : \mathbb{R} \to \mathbb{R} $ which verify the relation \[(x-2)f(y)+f(y+2f(x))= f(x+yf(x)), \qquad \forall x,y \in \mathbb R.\]

Let $ABCD$ be a convex quadrilateral with angles $\angle ABC$ and $\angle BCD$ not less than $120^{\circ}$. Prove that \[AC + BD> AB+BC+CD\]

Find all positive integers n such that : $-2^{0}+2^{1}-2^{2}+2^{3}-2^{4}+...-(-2)^{n}=4^{0}+4^{1}+4^{2}+...+4^{2010}$

One integer was removed from the set $S=\left \{ 1,2,3,...,n \right \}$ of the integers from $1$ to $n$. The arithmetic mean of the other integers of $S$ is equal to $\frac{163}{4}$. What integer was removed ?

When dividing an integer $m$ by a positive integer $n$, $(0< n\leq 100)$, a student finds $\frac{m}{n}= 0,167a_{1}a_{2}...$. Prove that the student made a mistake while computing.

$ (C)$ and $(C')$ are two circles which intersect in $A$ and $B$. $(D)$ is a line that moves and passes through $A$, intersecting $(C)$ in P and $(C')$ in P'. Prove that the bisector of $[PP']$ passes through a non-moving point.

Given positive reals $a,b,c;$ show that we have \[\left(a+\frac 1b\right)\left(b+\frac 1c\right)\left(c+\frac 1a\right)\geq 8.\]

Let $\alpha , \beta ,\gamma$ be the angles of a triangle $ABC$ of perimeter $ 2p $ and $R$ is the radius of its circumscribed circle. $(a)$ Prove that \[\cot^{2}\alpha +\cot^{2}\beta+\cot^{2}\gamma\geq 3\left(9\cdot \frac{R^{2}}{p^{2}} - 1\right).\] $(b)$ When do we have equality?

Two circles are tangent to each other internally at a point $\ T $. Let the chord $\ AB $ of the larger circle be tangent to the smaller circle at a point $\ P $. Prove that the line $\ TP $ bisects $\ \angle ATB $.

The diagonals of a trapezoid $ ABCD $ whose bases are $ [AB] $ and $ [CD] $ intersect at $P.$ Prove that \[S_{PAB} + S_{PCD} > S_{PBC} + S_{PDA},\] Where $S_{XYZ} $ denotes the area of $\triangle XYZ $.

Let $a$ and $b$ be two positive real numbers such that $a+b=ab$. Prove that $\frac{a}{b^{2}+4}+\frac{b}{a^{2}+4}\geq \frac{1}{2}$.

Solve in $(\mathbb{R}_{+}^{*})^{4}$ the following system : $\left\{\begin{matrix} x+y+z+t=4\\ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{t}=5-\frac{1}{xyzt} \end{matrix}\right.$

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $\forall x\in \mathbb{R} \ \ f(x) = max(2xy-f(y))$ where $y\in \mathbb{R}$.

Let $ABC$ be a triangle. The inside bisector of the angle $\angle BAC$ cuts $[BC]$ in $L$ and the circle $(C)$ circumsbribed to the triangle $ABC$ in $D$. The perpendicular to $(AC)$ going through $D$ cuts $[AC]$ in $M$ and the circle $(C)$ in $K$. Find the value of $\frac{AM}{MC}$ knowing that $\frac{BL}{LC}=\frac{1}{2}$.

Compute the sum \[S=1+2+3-4-5+6+7+8-9-10+\dots-2010\] where every three consecutive $+$ are followed by two $-$.

Let $a,b,c$ be three postive real numbers such that $a+b+c=1$. Prove that $9abc\leq ab+ac+bc < 1/4 +3abc$.

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x,y, \in \mathbb{R}$, \[xf(x+xy)=xf(x)+f(x^{2})\cdot f(y).\]

Let $ABC$ be a triangle with area $1$ and $P$ the middle of the side $[BC]$. $M$ and $N$ are two points of $[AB]-\left \{ A,B \right \} $ and $[AC]-\left \{ A,C \right \}$ respectively such that $AM=2MB$ and$CN=2AN$. The two lines $(AP)$ and $(MN)$ intersect in a point $D$. Find the area of the triangle $ADN$.

Find the maximum value of the real constant $C$ such that $x^{2}+y^{2}+1\geq C(x+y)$, and $ x^{2}+y^{2}+xy+1\geq C(x+y)$ for all reals $x,y$.

Prove that the equation $x^{2}+p|x| = qx - 1 $ has 4 distinct real solutions if and only if $p+|q|+2<0$ ($p$ and $q$ are two real parameters).

Find all functions $f : \mathbb{R} \to \mathbb{R} $ which verify the relation \[(x-2)f(y)+f(y+2f(x))= f(x+yf(x)), \qquad \forall x,y \in \mathbb R.\]

Let $ABC$ be a triangle and $I$ the center of its incircle. $P$ is a point inside $ABC$ such that $\angle PBA +\angle PCA = \angle PBC + \angle PCB$. Prove that $AP\geq AI$ with equality iff $P=I$.

Find all positive integers n such that : $-2^{0}+2^{1}-2^{2}+2^{3}-2^{4}+...-(-2)^{n}=4^{0}+4^{1}+4^{2}+...+4^{2010}$

One integer was removed from the set $S=\left \{ 1,2,3,...,n \right \}$ of the integers from $1$ to $n$. The arithmetic mean of the other integers of $S$ is equal to $\frac{163}{4}$. What integer was removed ?

Problem 3 (MAR CP 1992) : From the digits $1,2,...,9$, we write all the numbers formed by these nine digits (the nine digits are all distinct), and we order them in increasing order as follows : $123456789$, $123456798$, ..., $987654321$. What is the $100000th$ number ?

Two circles $C_{1}$ and $C_{2}$ intersect in $A$ and $B$. A line passing through $B$ intersects $C_{1}$ in $C$ and $C_{2}$ in $D$. Another line passing through $B$ intersects $C_{1}$ in $E$ and $C_{2}$ in $F$, $(CF)$ intersects $C_{1}$ and $C_{2}$ in $P$ and $Q$ respectively. Make sure that in your diagram, $B, E, C, A, P \in C_{1}$ and $B, D, F, A, Q \in C_{2}$ in this order. Let $M$ and $N$ be the middles of the arcs $BP$ and $BQ$ respectively. Prove that if $CD=EF$, then the points $C,F,M,N$ are cocylic in this order.

Let $x$, $y$, and $z$ be three real positive numbers such that $x^{2}+y^{2}+z^{2}+2xyz=1$. Prove that $2(x+y+z)\leq 3$.

Let $\alpha , \beta ,\gamma$ be the angles of a triangle $ABC$ of perimeter $ 2p $ and $R$ is the radius of its circumscribed circle. $(a)$ Prove that \[\cot^{2}\alpha +\cot^{2}\beta+\cot^{2}\gamma\geq 3\left(9\cdot \frac{R^{2}}{p^{2}} - 1\right).\] $(b)$ When do we have equality?

Prove that there exist two functions $f,g \colon \mathbb{R} \to \mathbb{R}$, such that $f\circ g$ is strictly decreasing and $g\circ f$ is strictly increasing. (Poland) Andrzej Komisarski and Marcin Kuczma

Let $a, b, c, d, m, n$ be positive integers such that $a^{2}+b^{2}+c^{2}+d^{2}=1989$, $n^{2}=max\left \{ a,b,c,d \right \}$ and $a+b+c+d=m^{2}$. Find the values of $m$ and $n$.