BG-MO 2005

Determine all triples $\left( x,y,z\right)$ of positive integers for which the number $$\sqrt{\frac{2005}{x+y}}+\sqrt{\frac{2005}{y+z}}+\sqrt{\frac{2005}{z+x}}$$ is an integer .

Consider two circles $k_{1},k_{2}$ touching externally at point $T$ . a line touches $k_{2}$ at point $X$ and intersects $k_{1}$ at points $A$ and $B$ . Let $S$ be the second intersection point of $k_{1}$ with the line $XT$ . On the arc $\widehat{TS}$ not containing $A$ and $B$ is chosen a point $C$ . Let $\ CY$ be the tangent line to $k_{2}$ with $Y\in k_{2}$ , such that the segment $CY$ does not intersect the segment $ST$ . If $I=XY\cap SC$ . Prove that :

(a) the points $C,T,Y,I$ are concyclic.

(b) $I$ is the excenter of triangle $ABC$ with respect to the side $BC$ .

Let $M=(0,1)\cap \mathbb Q$ . Determine, with proof, whether there exists a subset $A\subset M$ with the property that every number in $M$ can be uniquely written as the sum of finitely many distinct elements of $A$ .

Let $ABC$ be a triangle with $AC\neq BC$ , and let $A^{\prime }B^{\prime }C$ be a triangle obtained from $ABC$ after some rotation centered at $C$ . Let $M,E,F$ be the midpoints of the segments $BA^{\prime },AC$ and $CB^{\prime }$ respectively. If $EM=FM$ , find $\widehat{EMF}$ .

For positive integers $t,a,b,$ a $(t,a,b)$ -game is a two player game defined by the following rules. Initially, the number $t$ is written on a blackboard. At his first move, the 1st player replaces $t$ with either $t-a$ or $t-b$ . Then, the 2nd player subtracts either $a$ or $b$ from this number, and writes the result on the blackboard, erasing the old number. After this, the first player once again erases either $a$ or $b$ from the number written on the blackboard, and so on. The player who first reaches a negative number loses the game. Prove that there exist infinitely many values of $t$ for which the first player has a winning strategy for all pairs $(a,b)$ with $a+b=2005$ .

Let $a,b$ and $c$ be positive integers such that $ab$ divides $c(c^{2}-c+1)$ and $a+b$ is divisible by $c^{2}+1$ . Prove that the sets $\{a,b\}$ and $\{c,c^{2}-c+1\}$ coincide.