In an acute triangle $ABC$, the circle with diameter $AC$ intersects $AB$ and $AC$ at $K$ and $L$ different from $A$ and $C$ respectively. The circumcircle of $ABC$ intersects the line $CK$ at the point $F$ different from $C$ and the line $AL$ at the point $D$ different from $A$. A point $E$ is choosen on the smaller arc of $AC$ of the circumcircle of $ABC$ . Let $N$ be the intersection of the lines $BE$ and $AC$ . If $AF^{2}+BD^{2}+CE^{2}=AE^{2}+CD^{2}+BF^{2}$ prove that $\angle KNB= \angle BNL$ .

Some unit squares of $ 2007\times 2007 $ square board are colored. Let $ (i,j) $ be a unit square belonging to the $ith$ line and $jth$ column and $ S_{i,j} $ be the set of all colored unit squares $(x,y)$ satisfying $ x\leq i, y\leq j $. At the first step in each colored unit square $(i,j)$ we write the number of colored unit squares in $ S_{i,j} $ . In each step, in each colored unit square $(i,j)$ we write the sum of all numbers written in $ S_{i,j} $ in the previous step. Prove that after finite number of steps, all numbers in the colored unit squares will be odd.

If $a,b,c$ are three positive real numbers such that $a+b+c=3$, prove that $ {\frac{a^{2}+3b^{2}}{ab^{2}(4-ab)}}+{\frac{b^{2}+3c^{2}}{bc^{2}(4-ab)}}+{\frac{c^{2}+3a^{2}}{ca^{2}(4-ca)}}\geq 4 $

Let $k>1$ be an integer, $p=6k+1$ be a prime number and $m=2^{p}-1$ . Prove that $\frac{2^{m-1}-1}{127m}$ is an integer.

Let $ABC$ be a triangle with $\angle B=90$. The incircle of $ABC$ touches the side $BC$ at $D$. The incenters of triangles $ABD$ and $ADC$ are $X$ and $Z$ , respectively. The lines $XZ$ and $AD$ are intersecting at the point $K$. $XZ$ and circumcircle of $ABC$ are intersecting at $U$ and $V$. Let $M$ be the midpoint of line segment $[UV]$ . $AD$ intersects the circumcircle of $ABC$ at $Y$ other than $A$. Prove that $|CY|=2|MK|$ .

In a country between each pair of cities there is at most one direct road. There is a connection (using one or more roads) between any two cities even after the elimination of any given city and all roads incident to this city. We say that the city $A$ can be k -directionally connected to the city $B$, if : we can orient at most $k$ roads such that after arbitrary orientation of remaining roads for any fixed road $l$ (directly connecting two cities) there is a path passing through roads in the direction of their orientation starting at $A$, passing through $l$ and ending at $B$ and visiting each city at most once. Suppose that in a country with $n$ cities, any two cities can be k - directionally connected. What is the minimal value of $k$?