BALKAN 1984

Let $n \geq 2$ be a positive integer and $a_{1},\ldots , a_{n}$ be positive real numbers such that $a_{1}+...+a_{n}= 1$ . Prove that: $$\frac{a_{1}}{1+a_{2}+\cdots +a_{n}}+\cdots +\frac{a_{n}}{1+a_{1}+a_{2}+\cdots +a_{n-1}}\geq \frac{n}{2n-1}$$

Let $ABCD$ be a cyclic quadrilateral and let $H_{A}, H_{B}, H_{C}, H_{D}$ be the orthocenters of the triangles $BCD$ , $CDA$ , $DAB$ and $ABC$ respectively. Show that the quadrilaterals $ABCD$ and $H_{A}H_{B}H_{C}H_{D}$ are congruent.

Show that for any positive integer $m$ , there exists a positive integer $n$ so that in the decimal representations of the numbers $5^{m}$ and $5^{n}$ , the representation of $5^{n}$ ends in the representation of $5^{m}$ .

Let $a,b,c$ be positive real numbers. Find all real solutions $(x,y,z)$ of the system: $$ax+by=(x-y)^{2} by+cz=(y-z)^{2} cz+ax=(z-x)^{2}$$