Let $I$ be the incenter of the triangle $ABC$, et let $A',B',C'$ be the symmetric of $I$ with respect to the lines $BC,CA,AB$ respectively. It is known that $B$ belongs to the circumcircle of $A'B'C'$. Find $\widehat {ABC}$. Pierre.

Let $\omega (n)$ denote the number of prime divisors of the integer $n>1$. Find the least integer $k$ such that the inequality $2^{\omega (n) } \leq k \cdot n^{\frac 1 4}$ holds for all $n > 1.$ Pierre.

Two players write alternatively some integers on the blackboard. The rules are the following : - The first player write $1$. - At each of the other turns, the player has to write $a+1$ or $2a$ where $a$ is any number already wrote in the blackboard and $2a \leq 1000.$ - One cannot write a number which has already been written, and no number is erased. - The player who writes $1000$ is the winner. Determine which player has a winning strategy. Pierre.

Let $x,y,z$ be positive real numbers such that $x^2+y^2+z^2 = 25.$ Find the minimum of $\frac {xy} z + \frac {yz} x + \frac {zx} y .$ Pierre.

Let $I$ be the incenter of the triangle $ABC$. Let $A_1,A_2$ be two distinct points on the line $BC$, let $B_1,B_2$ be two distinct points on the line $CA$, and let $C_1,C_2$ be two distinct points on the line $BA$ such that $AI = A_1I = A_2I$ and $BI = B_1I = B_2I$ and $CI = C_1I = C_2I$. Prove that $A_1A_2+B_1B_2+C_1C_2 = p$ where $p$ denotes the perimeter of $ABC.$ Pierre.

On each unit square of a $9 \times 9$ square, there is a bettle. Simultaneously, at the whistle, each bettle moves from its unit square to another one which has only a common vertex with the original one (thus in diagonal). Some bettles can go to the same unit square. Determine the minimum number of empty unit squares after the moves. Pierre.

Prove that a prime of the form $2^{2^n}+1$ cannot be the difference of two fifth powers of two positive integers. Pierre.

Let $f$ be a function from the set $Q$ of the rational numbers onto itself such that $f(x+y)=f(x)+f(y)+2547$ for all rational numbers $x,y$. Moreover $f(2004) = 2547$. Determine $f(2547).$ Pierre.