$f(x)=ax^2+bx+c;a,b,c$ are reals. $M=\{f(2n)|n \text{ is integer}\},N=\{f(2n+1)|n \text{ is integer}\}$ Prove that $M=N$ or $M \cap N = \O $

$[x,y]-[x,z]=y-z$ and $x \neq y \neq z \neq x$ Prove, that $x|y,x|z$

Streets of Moscow are some circles (rings) with common center $O$ and some straight lines from center $O$ to external ring. Point $A,B$ - two crossroads on external ring. Three friends want to move from $A$ to $B$. Dima goes by external ring, Kostya goes from $A$ to $O$ then to $B$. Sergey says, that there is another way, that is shortest. Prove, that he is wrong.

From $2008 \times 2008$ square we remove one corner cell $1 \times 1$. Is number of ways to divide this figure to corners from $3$ cells odd or even ?

$O$ -circumcenter of $ABCD$. $AC$ and $BD$ intersect in $E$, $AD$ and $BC$ in $F$. $X,Y$ - midpoints of $AD$ and $BC$. $O_1$ -circumcenter of $EXY$. Prove that $OF \parallel O_1E$

Some cities in country are connected by road, and from every city goes $\geq 2008$ roads. Every road is colored in one of two colors. Prove, that exists cycle without self-intersections ,where $\geq 504$ roads and all roads are same color.

$f(x)=x^2+x$ $b_1,...,b_{10000}>0$ and $|b_{n+1}-f(b_n)|\leq \frac{1}{1000}$ for $n=1,...,9999$ Prove, that there is such $a_1>0$ that $a_{n+1}=f(a_n);n=1,...,9999$ and $|a_n-b_n|<\frac{1}{100}$

$x,y$ are naturals. $GCM(x^7,y^4)*GCM(x^8,y^5)=xy$ Prove that $xy$ is cube

$ABCD$ is convex quadrilateral with $AB=CD$. $AC$ and $BD$ intersect in $O$. $X,Y,Z,T$ are midpoints of $BC,AD,AC,BD$. Prove, that circumcenter of $OZT$ lies on $XY$.

$f(x),g(x),h(x)$ are square trinomials with discriminant, that equals $2$. And $f(x)+g(x),f(x)+h(x),g(x)+h(x)$ are square trinomials with discriminant, that equals $1$. Prove,that $f(x)+g(x)+h(x)$ has not roots.

Call a set of some cells in infinite chess field as board. Set of rooks on the board call as awesome if no one rook can beat another, but every empty cell is under rook attack. There are awesome set with $2008$ rooks and with $2010$ rooks. Prove, that there are awesome set with $2009$ rooks.

$(x_n)$ is sequence, such that $x_{n+2}=|x_{n+1}|-x_n$. Prove, that it is periodic.

Points $Y,X$ lies on $AB,BC$ of $\triangle ABC$ and $X,Y,A,C$ are concyclic. $AX$ and $CY$ intersect in $O$. Points $M,N$ are midpoints of $AC$ and $XY$. Prove, that $BO$ is tangent to circumcircle of $\triangle MON$

$b,c$ are naturals. $b|c+1$ Prove that exists such natural $x,y,z$ that $x+y=bz,xy=cz$

There are $40$ members of jury, that want to choose problem for contest. There are list with $30$ problems. They want to find such problem, that can be solved at least half members , but not all. Every member solved $26$ problems, and every two members solved different sets of problems. Prove, that they can find problem for contest.

Points $A_1$ and $C_1$ are on $BC$ and $AB$ of acute-angled triangle $ABC$ . $AA_1$ and $CC_1$ intersect in $K$. Circumcircles of $AA_1B,CC_1B$ intersect in $P$ - incenter of $AKC$. Prove, that $P$ - orthocenter of $ABC$

$ABC$ is acute-angled triangle. $AA_1,BB_1,CC_1$ are altitudes. $X,Y$ - midpoints of $AC_1,A_1C$. $XY=BB_1$. Prove that one side of $ABC$ in $\sqrt{2}$ greater than other side.

Discriminants of square trinomials $f(x),g(x),h(x),f(x)+g(x),f(x)+h(x),g(x)+h(x)$ equals $1$. Prove that $f(x)+h(x)+g(x) \equiv 0$