The cells of an $8\times 8$ chessboard are all coloured in white. A move consists in inverting the colours of a rectangle $1 \times 3$ horizontal or vertical (the white cells become black and conversely). Is it possible to colour all the cells of the chessboard in black in a finite number of moves ?

Let $a, b$ and $c$ be natural numbers. Determine the smallest value that the following expression can take: $$\frac{a}{gcd\,\,(a + b, a - c)} + \frac{b}{gcd\,\,(b + c, b - a)} + \frac{c}{gcd\,\,(c + a, c - b)}.$$. Remark: $gcd \,\, (6, 0) = 6$ and $gcd\,\,(3, -6) = 3$.

Determine all natural integers $n$ for which there is no triplet $(a, b, c)$ of natural numbers such that: $$n = \frac{a \cdot \,\,lcm(b, c) + b \cdot lcm \,\,(c, a) + c \cdot lcm \,\, (a, b)}{lcm \,\,(a, b, c)}$$

Let $D$ be a point inside an acute triangle $ABC$, such that $\angle BAD = \angle DBC$ and $\angle DAC = \angle BCD$. Let $P$ be a point on the circumcircle of the triangle $ADB$. Suppose $P$ are itself outside the triangle $ABC$. A line through $P$ intersects the ray $BA$ in $X$ and ray $CA$ in $Y$, so that $\angle XPB = \angle PDB$. Show that $BY$ and $CX$ intersect on $AD$.

Does there exist any function $f: \mathbb{R}^+ \to \mathbb{R}$ such that for every positive real number $x,y$ the following is true : $$f(xf(x)+yf(y)) = xy$$

Let $k$ be the incircle of the triangle $ABC$ with the center of the incircle $I$. The circle $k$ touches the sides $BC, CA$ and $AB$ in points $D, E$ and $F$. Let $G$ be the intersection of the straight line $AI$ and the circle $k$, which lies between $A$ and $I$. Assume $BE$ and $FG$ are parallel. Show that $BD = EF$.

Let $n$ be a natural integer and let $k$ be the number of ways to write $n$ as the sum of one or more consecutive natural integers. Prove that $k$ is equal to the number of odd positive divisors of $n$. Example: $9$ has three positive odd divisors and $9 = 9$, $9 = 4 + 5$, $9 = 2 + 3 + 4$.

Let $a,b,c,d,e$ be positive real numbers. Find the largest possible value for the expression $$\frac{ab+bc+cd+de}{2a^2+b^2+2c^2+d^2+2e^2}.$$

Let $n$ be a positive integer and let $G$ be the set of points $(x, y)$ in the plane such that $x$ and $y$ are integers with $1 \leq x, y \leq n$. A subset of $G$ is called parallelogram-free if it does not contains four non-collinear points, which are the vertices of a parallelogram. What is the largest number of elements a parallelogram-free subset of $G$ can have?

Let $ p \geq 2$ be a prime number. Eduardo and Fernando play the following game making moves alternately: in each move, the current player chooses an index $i$ in the set $\{0,1,2,\ldots, p-1 \}$ that was not chosen before by either of the two players and then chooses an element $a_i$ from the set $\{0,1,2,3,4,5,6,7,8,9\}$. Eduardo has the first move. The game ends after all the indices have been chosen .Then the following number is computed: $$M=a_0+a_110+a_210^2+\cdots+a_{p-1}10^{p-1}= \sum_{i=0}^{p-1}a_i.10^i$$. The goal of Eduardo is to make $M$ divisible by $p$, and the goal of Fernando is to prevent this. Prove that Eduardo has a winning strategy. Proposed by Amine Natik, Morocco