TR-MO 2007

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 bek -directionally connected to the city $B$ , if : we can orient at most $k$ roads such that afterarbitrary 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 bek - directionally connected. What is the minimal value of $k$ ?