FKMO 2009

$a,b,c$ are the length of three sides of a triangle. Let $A= \frac{a^2 +bc}{b+c}+\frac{b^2 +ca}{c+a}+\frac{c^2 +ab}{a+b}$ , $B=\frac{1}{\sqrt{(a+b-c)(b+c-a)}}+\frac{1}{\sqrt{(b+c-a)(c+a-b)}}$ $+\frac{1}{\sqrt{(c+a-b)(a+b-c)}}$ . Prove that $AB \ge 9$ .

$ABC$ is an obtuse triangle. (angle $B$ is obtuse) Its circumcircle is $O$ . A tangent line for $O$ passing $C$ meets with $AB$ at $B_1$ . Let $O_1$ be a circumcenter of triangle $AB_1C$ . $B_2$ is a point on the segment $BB_1$ . Let $C_1$ be a contact point of the tangent line for $O$ passing $B_2$ , which is more closer to $C$ . Let $O_2$ be a circumcenter of triangle $AB_2C_1$ . Prove that if $OO_2$ and $AO_1$ is perpendicular, then five points $O,O_2,O_1,C_1,C$ are concyclic.

2008 white stones and 1 black stone are in a row. An 'action' means the following: select one black stone and change the color of neighboring stone(s). Find all possible initial position of the black stone, to make all stones black by finite actions.

$ABC$ is an acute triangle. (angle $C$ is bigger than angle $B$ ) Let $O$ be a center of the circle which passes $B$ and tangents to $AC$ at $C$ . $O$ meets the segment $AB$ at $D$ . $CO$ meets the circle $(O)$ again at $P$ , a line, which passes $P$ and parallel to $AO$ , meets $AC$ at $E$ , and $EB$ meets the circle $(O)$ again at $L$ . A perpendicular bisector of $BD$ meets $AC$ at $F$ and $LF$ meets $CD$ at $K$ . Prove that two lines $EK$ and $CL$ are parallel.

There is a $m \times (m-1)$ board. (i.e. there are $m+1$ horizontal lines and $m$ vertical lines) A stone is put on an intersection of the lowest horizontal line. Now two players move this stone with the following rules. (i) Each players move the stone to a neighboring intersection along a segment, by turns. (ii) A segment, which is already passed by the stone, cannot be used more. (iii) One who cannot move the stone anymore loses. Prove that there is a winning strategy for the former player.

Find all pairs of two positive integers $(m,n)$ satisfying $3^m - 7^n = 2$ .