CWMO 2012

Find the smallest positive integer $m$ satisfying the following condition: for all prime numbers $p$ such that $p>3$ ,have $105|9^{ p^2}-29^p+m.$ (September 28, 2012, Hohhot)

Show that among any $n\geq 3$ vertices of a regular $(2n-1)$ -gon we can find $3$ of them forming an isosceles triangle.

Let $A$ be a set of $n$ elements and $A_1, A_2, ... A_k$ subsets of $A$ such that for any $2$ distinct subsets $A_i, A_j$ either they are disjoint or one contains the other. Find the maximum value of $k$

$P$ is a point in the $\Delta ABC$ , $\omega$ is the circumcircle of $\Delta ABC$ . $BP \cap \omega = \left\{ {B,{B_1}} \right\}$ , $CP \cap \omega = \left\{ {C,{C_1}} \right\}$ , $PE \bot AC$ ， $PF \bot AB$ . The radius of the inscribed circle and circumcircle of $\Delta ABC$ is $r,R$ . Prove $\frac{{EF}}{{{B_1}{C_1}}} \geqslant \frac{r}{R}$ .

Attachments:

$O$ is the circumcenter of acute $\Delta ABC$ , $H$ is the Orthocenter. $AD \bot BC$ , $EF$ is the perpendicular bisector of $AO$ , $D,E$ on the $BC$ . Prove that the circumcircle of $\Delta ADE$ through the midpoint of $OH$ .

Attachments:

Define a sequence $\{a_n\}$ by$$a_0=\frac{1}{2},\ a_{n+1}=a_{n}+\frac{a_{n}^2}{2012}, (n=0,\ 1,\ 2,\ \cdots),$$ find integer $k$ such that $a_{k}<1<a_{k+1}.$ (September 29, 2012, Hohhot)

Let $n$ be a positive integer $\geq 2$ . Consider a $n$ by $n$ grid with all entries $1$ . Define an operation on a square to be changing the signs of all squares adjacent to it but not the sign of its own. Find all $n$ such that it is possible after a finite sequence of operations to reach a $n$ by $n$ grid with all entries $-1$

Find all prime number $p$ , such that there exist an infinite number of positive integer $n$ satisfying the following condition: $p|n^{ n+1}+(n+1)^n.$ (September 29, 2012, Hohhot)