CR-MO 2008

We want to colour all the squares of an $nxn$ board of red or black. The colorations should be such that any subsquare of $2x2$ of the board have exactly two squares of each color. If $n\geq 2$ how many such colorations are possible?

Let $ABC$ be a triangle and let $P$ be a point on the angle bisector $AD$ , with $D$ on $BC$ . Let $E$ , $F$ and $G$ be the intersections of $AP$ , $BP$ and $CP$ with the circumcircle of the triangle, respectively. Let $H$ be the intersection of $EF$ and $AC$ , and let $I$ be the intersection of $EG$ and $AB$ . Determine the geometric place of the intersection of $BH$ and $CI$ when $P$ varies.

Find all polinomials $P(x)$ with real coefficients, such that

$P(\sqrt {3}(a - b)) + P(\sqrt {3}(b - c)) + P(\sqrt {3}(c - a)) = P(2a - b - c) + P( - a + 2b - c) + P( - a - b + 2c)$ for any $a$ , $b$ and $c$ real numbers

Let $x$ , $y$ and $z$ be non negative reals, such that there are not two simultaneously equal to $0$ . Show that

$\frac {x + y}{y + z} + \frac {y + z}{x + y} + \frac {y + z}{z + x} + \frac {z + x}{y + z} + \frac {z + x}{x + y} + \frac {x + y}{z + x}\geq\ 5 + \frac {x^{2} + y^{2} + z^{2}}{xy + yz + zx}$ and determine the equality cases.

Let $p$ be a prime number such that $p-1$ is a perfect square. Prove that the equation $a^{2}+(p-1)b^{2}=pc^{2}$ has infinite many integer solutions $a$ , $b$ and $c$ with $(a,b,c)=1$

Let $O$ be the circumcircle of a $\Delta ABC$ and let $I$ be its incenter, for a point $P$ of the plane let $f(P)$ be the point obtained by reflecting $P'$ by the midpoint of $OI$ , with $P'$ the homothety of $P$ with center $O$ and ratio $\frac{R}{r}$ with $r$ the inradii and $R$ the circumradii,(understand it by $\frac{OP}{OP'}=\frac{R}{r}$ ). Let $A_1$ , $B_1$ and $C_1$ the midpoints of $BC$ , $AC$ and $AB$ , respectively. Show that the rays $A_1f(A)$ , $B_1f(B)$ and $C_1f(C)$ concur on the incircle.