RU-SPB 2009

$b,c$ 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 $f(x)+h(x)+g(x) \equiv 0$

$x,y$ 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.

$(x_n)$ 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 $\triangle MON$

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 $|a_n-b_n|<\frac{1}{100}$