TR-MO 2012

Find all polynomials with integer coefficients such that for all positive integers $n$ satisfies $P(n!)=|P(n)|!$

Let $ABC$ be a isosceles triangle with $AB=AC$ an $D$ be the foot of perpendicular of $A$ . $P$ be an interior point of triangle $ADC$ such that $m(APB)>90$ and $m(PBD)+m(PAD)=m(PCB)$ . $CP$ and $AD$ intersects at $Q$ , $BP$ and $AD$ intersects at $R$ . Let $T$ be a point on $[AB]$ and $S$ be a point on $[AP$ and not belongs to $[AP]$ satisfying $m(TRB)=m(DQC)$ and $m(PSR)=2m(PAR)$ . Show that $RS=RT$

Find all non-decreasing functions from real numbers to itself such that for all real numbers $x,y$ $f(f(x^2)+y+f(y))=x^2+2f(y)$ holds.

For all positive real numbers $x, y, z$ , show that $\frac{x(2x-y)}{y(2z+x)}+\frac{y(2y-z)}{z(2x+y)}+\frac{z(2z-x)}{x(2y+z)} \geq 1$ is true.

Let $P$ be the set of all $2012$ tuples $(x_1, x_2, \dots, x_{2012})$ , where $x_i \in \{1,2,\dots 20\}$ for each $1\leq i \leq 2012$ . The set $A \subset P$ is said to be decreasing if for each $(x_1,x_2,\dots ,x_{2012} ) \in A$ any $(y_1,y_2,\dots, y_{2012})$ satisfying $y_i \leq x_i (1\leq i \leq 2012)$ also belongs to $A$ . The set $B \subset P$ is said to be increasing if for each $(x_1,x_2,\dots ,x_{2012} ) \in B$ any $(y_1,y_2,\dots, y_{2012})$ satisfying $y_i \geq x_i (1\leq i \leq 2012)$ also belongs to $B$ . Find the maximum possible value of $f(A,B)= \dfrac {|A\cap B|}{|A|\cdot |B|}$ , where $A$ and $B$ are nonempty decreasing and increasing sets ( $\mid \cdot \mid$ denotes the number of elements of the set).

Let $B$ and $D$ be points on segments $[AE]$ and $[AF]$ respectively. Excircles of triangles $ABF$ and $ADE$ touching sides $BF$ and $DE$ is the same, and its center is $I$ . $BF$ and $DE$ intersects at $C$ . Let $P_1, P_2, P_3, P_4, Q_1, Q_2, Q_3, Q_4$ be the circumcenters of triangles $IAB, IBC, ICD, IDA, IAE, IEC, ICF, IFA$ respectively.a) Show that points $P_1, P_2, P_3, P_4$ concylic and points $Q_1, Q_2, Q_3, Q_4$ concylic.b) Denote centers of theese circles as $O_1$ and $O_2$ . Prove that $O_1, O_2$ and $I$ are collinear.