BG-MO 2017

An convex qudrilateral $ABCD$ is given. $O$ is the intersection point of the diagonals $AC$ and $BD$ . The points $A_1,B_1,C_1, D_1$ lie respectively on $AO, BO, CO, DO$ such that $AA_1=CC_1, BB_1=DD_1$ . The circumcircles of $\triangle AOB$ and $\triangle COD$ meet at second time at $M$ and the the circumcircles of $\triangle AOD$ and $\triangle BOC$ - at $N$ . The circumcircles of $\triangle A_1OB_1$ and $\triangle C_1OD_1$ meet at second time at $P$ and the the circumcircles of $\triangle A_1OD_1$ and $\triangle B_1OC_1$ - at $Q$ . Prove that the quadrilateral $MNPQ$ is cyclic.

Let $m>1$ be a natural number and $N=m^{2017}+1$ . On a blackboard, left to right, are written the following numbers: $$N, N-m, N-2m,\dots, 2m+1,m+1, 1.$$ On each move, we erase the most left number, written on the board, and all its divisors (if any). This procces continues till all numbers are deleted. Which numbers will be deleted on the last move.

Let $M$ be a set of $2017$ positive integers. For any subset $A$ of $M$ we define $f(A) := \{x\in M\mid \text{ the number of the members of }A\,, x \text{ is multiple of, is odd }\}$ . Find the minimal natural number $k$ , satisfying the condition: for any $M$ , we can color all the subsets of $M$ with $k$ colors, such that whenever $A\neq f(A)$ , $A$ and $f(A)$ are colored with different colors.

Find all triples (p,a,m); p is a prime number, $a,m\in \mathbb{N}$ , which satisfy: $a\leq 5p^2$ and $(p-1)!+a=p^m$ .

Let $n$ be a natural number and $f(x)$ be a polynomial with real coefficients having $n$ different positive real roots. Is it possible the polynomial: $$x(x+1)(x+2)(x+4)f(x)+a$$ to be presented as the $k$ -th power of a polynomial with real coefficients, for some natural $k\geq 2$ and real $a$ ?

An acute non-isosceles $\triangle ABC$ is given. $CD, AE, BF$ are its altitudes. The points $E', F'$ are symetrical of $E, F$ with respect accordingly to $A$ and $B$ . The point $C_1$ lies on $\overrightarrow{CD}$ , such that $DC_1=3CD$ . Prove that $\angle E'C_1F'=\angle ACB$