Let $ ABC$ be an acute triangle and $ CL$ be the angle bisector of $ \angle ACB$. The point $ P$ lies on the segment $CL$ such that $ \angle APB=\pi-\frac{_1}{^2}\angle ACB$. Let $ k_1$ and $ k_2$ be the circumcircles of the triangles $ APC$ and $ BPC$. $ BP\cap k_1=Q, AP\cap k_2=R$. The tangents to $ k_1$ at $ Q$ and $ k_2$ at $ B$ intersect at $ S$ and the tangents to $ k_1$ at $ A$ and $ k_2$ at $ R$ intersect at $ T$. Prove that $ AS=BT.$

Is it possible to find $2008$ infinite arithmetical progressions such that there exist finitely many positive integers not in any of these progressions, no two progressions intersect and each progression contains a prime number bigger than $2008$?

Let $n\in\mathbb{N}$ and $0\leq a_1\leq a_2\leq\ldots\leq a_n\leq\pi$ and $b_1,b_2,\ldots ,b_n$ are real numbers for which the following inequality is satisfied : \[\left|\sum_{i=1}^{n} b_i\cos(ka_i)\right|<\frac{1}{k}\] for all $ k\in\mathbb{N}$. Prove that $ b_1=b_2=\ldots =b_n=0$.

Find the smallest natural number $ k$ for which there exists natural numbers $ m$ and $ n$ such that $ 1324 + 279m + 5^n$ is $ k$-th power of some natural number.

Let $n$ be a fixed natural number. Find all natural numbers $ m$ for which \[\frac{1}{a^n}+\frac{1}{b^n}\ge a^m+b^m\] is satisfied for every two positive numbers $ a$ and $ b$ with sum equal to $2$.

Let $M$ be the set of the integer numbers from the range $[-n, n]$. The subset $P$ of $M$ is called a base subset if every number from $M$ can be expressed as a sum of some different numbers from $P$. Find the smallest natural number $k$ such that every $k$ numbers that belongs to $M$ form a base subset.