KZ-MO 2008

Let $F_n$ be a set of all possible connected figures, that consist of $n$ unit cells. For each element $f_n$ of this set, let $S(f_n)$ be the area of that minimal rectangle that covers $f_n$ and each side of the rectangle is parallel to the corresponding side of the cell. Find $max(S(f_n))$ ,where $f_n\in F_n$ ?

Remark: Two cells are called connected if they have a common edge.

Suppose that $B_1$ is the midpoint of the arc $AC$ , containing $B$ , in the circumcircle of $\triangle ABC$ , and let $I_b$ be the $B$ -excircle's center. Assume that the external angle bisector of $\angle ABC$ intersects $AC$ at $B_2$ . Prove that $B_2I$ is perpendicular to $B_1I_B$ , where $I$ is the incenter of $\triangle ABC$ .

Let $f(x,y,z)$ be the polynomial with integer coefficients. Suppose that for all reals $x,y,z$ the following equation holds: $$f(x,y,z) = - f(x,z,y) = - f(y,x,z) = - f(z,y,x)$$ Prove that if $a,b,c\in\mathbb{Z}$ then $f(a,b,c)$ takes an even value

Find all integer solutions $(a_1,a_2,\dots,a_{2008})$ of the following equation:

$(2008-a_1)^2+(a_1-a_2)^2+\dots+(a_{2007}-a_{2008})^2+a_{2008}^2=2008$

Let $\triangle ABC$ be a triangle and let $K$ be some point on the side $AB$ , so that the tangent line from $K$ to the incircle of $\triangle ABC$ intersects the ray $AC$ at $L$ . Assume that $\omega$ is tangent to sides $AB$ and $AC$ , and to the circumcircle of $\triangle AKL$ . Prove that $\omega$ is tangent to the circumcircle of $\triangle ABC$ as well.

Find the maximum number of planes in the space, such there are $6$ points, that satisfy to the following conditions: **1.**Each plane contains at least $4$ of them **2.**No four points are collinear.