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 .

Click for solution Simple. one can easily show that for the sum of the three square roots to be an integer, each of the terms should be a perfect square. So 2005/(x+y) = (p/q)^2 ,... Now 2005= 5(401) none of them being perfect squares. Hence x+y = 2005 n1^2 y+z=2005n2^2 z+x=2005 n3^2 and finally 1/n1 + 1/n2 +1/n3 is an integer. Note that the triple (n1,n2,n3) satisfying the above is a solution for (x,y,z) iff n1+n2+n3 be an even number. checking the cases we see easily that (n1,n2,n3) = (2,2,4),(2,4,2),(4,2,2) -Ali

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.