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 \[g(x)=\left\{\begin{array}{cc}x+1-a,&\mbox{ if } 0<x<b\\b-a, & \mbox{ if } x=a \\x-a, & \mbox{ if } a<x<b\\1-a ,&\mbox{ if } x=b \\ x-a ,&\mbox{ if } b<x<1 \end{array}\right.\] 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 Official solution here