SILK-ROAD 2003

Let $a_1, a_2, ....., a_{2003}$ be sequence of reals number. Call $a_k$ $leading$ element, if at least one of expression $a_k; a_k+a_{k+1}; a_k+a_{k+1}+a_{k+2}; ....; a_k+a{k+1}+a_{k+2}+....+a_{2003}$ is positive. Prove, that if exist at least one $leading$ element, then sum of all $leading$ 's elements is positive.

Official solution here

Let $s=\frac{AB+BC+AC}{2}$ be half-perimeter of triangle $ABC$ . Let $L$ and $N$ be a point's on ray's $AB$ and $CB$ , for which $AL=CN=s$ . Let $K$ is point, symmetric of point $B$ by circumcenter of $ABC$ . Prove, that perpendicular from $K$ to $NL$ passes through incenter of $ABC$ .

Solution for problem here

Let $0<a<b<1$ be reals numbers and $$g(x)=\left\{\begin{array}{cc}x+1-a,&\text{ if } 0<x<bb-a, & \text{ if } x=a x-a, & \text{ if } a<x<b1-a ,&\text{ if } x=b x-a ,&\text{ if } b<x<1 \end{array}\right.$$ Give that there exist $n+1$ reals numbers $0<x_0<x_1<...<x_n<1$ , for which $g^{[n]}(x_i)=x_i \ (0 \leq i \leq n)$ . Prove that there exists a positive integer $N$ , such that $g^{[N]}(x)=x$ for all $0<x<1$ .

( $g^{[n]}(x)= \underbrace{g(g(....(g(x))....))}_{\text{n times}}$ )

Official solution here

Find $\sum_{k \in A} \frac{1}{k-1}$ where $A= \{ m^n : m,n \in \mathbb{Z} m,n \geq 2 \}$ .

Problem was post earlier here , but solution not gives and olympiad doesn't indicate, so I post it again :blush:

Official solution here