CR-MO 2009

Let $x$ and $y$ positive real numbers such that $(1+x)(1+y)=2$ . Show that $xy+\frac{1}{xy}\geq\ 6$

Prove that for that for every positive integer $n$ , the smallest integer that is greater than $(\sqrt {3} + 1)^{2n}$ is divisible by $2^{n + 1}$ .

Let triangle $ABC$ acutangle, with $m \angle ACB\leq\ m \angle ABC$ . $M$ the midpoint of side $BC$ and $P$ a point over the side $MC$ . Let $C_{1}$ the circunference with center $C$ . Let $C_{2}$ the circunference with center $B$ . $P$ is a point of $C_{1}$ and $C_{2}$ . Let $X$ a point on the opposite semiplane than $B$ respecting with the straight line $AP$ ; Let $Y$ the intersection of side $XB$ with $C_{2}$ and $Z$ the intersection of side $XC$ with $C_{1}$ . Let $m\angle PAX = \alpha$ and $m\angle ABC = \beta$ . Find the geometric place of $X$ if it satisfies the following conditions: $(a) \frac {XY}{XZ} = \frac {XC + CP}{XB + BP}$ $(b) \cos(\alpha) = AB\cdot \frac {\sin(\beta )}{AP}$

Show that the number $3^{{4}^{5}} + 4^{{5}^{6}}$ can be expresed as the product of two integers greater than $10^{2009}$

Suppose the polynomial $x^{n} + a_{n - 1}x^{n - 1} + ... + a_{1} + a_{0}$ can be factorized as $(x + r_{1})(x + r_{2})...(x + r_{n})$ , with $r_{1}, r_{2}, ..., r_{n}$ real numbers. Show that $(n - 1)a_{n - 1}^{2}\geq\ 2na_{n - 2}$

Let $\Delta ABC$ with incircle $\Gamma$ , let $D, E$ and $F$ the tangency points of $\Gamma$ with sides $BC, AC$ and $AB$ , respectively and let $P$ the intersection point of $AD$ with $\Gamma$ . $a)$ Prove that $BC, EF$ and the straight line tangent to $\Gamma$ for $P$ concur at a point $A'$ . $b)$ Define $B'$ and $C'$ in an anologous way than $A'$ . Prove that $A'-B'-C'$