a) Is the number $ 1111\cdots11$ (with $ 2010$ ones) a prime number? b) Prove that every prime factor of $ 1111\cdots11$ (with $ 2011$ ones) is of the form $ 4022j+1$ where $ j$ is a natural number.

Let $ a\geq 2$ be a real number; with the roots $ x_{1}$ and $ x_{2}$ of the equation $ x^2-ax+1=0$ we build the sequence with $ S_{n}=x_{1}^n + x_{2}^n$. a)Prove that the sequence $ \frac{S_{n}}{S_{n+1}}$, where $ n$ takes value from $ 1$ up to infinity, is strictly non increasing. b)Find all value of $ a$ for the which this inequality hold for all natural values of $ n$ $ \frac{S_{1}}{S_{2}}+\cdots +\frac{S_{n}}{S_{n+1}}>n-1$

Let $ K$ be the circumscribed circle of the trapezoid $ ABCD$ . In this trapezoid the diagonals $ AC$ and $ BD$ are perpendicular. The parallel sides $ AB=a$ and $ CD=c$ are diameters of the circles $ K_{a}$ and $ K_{b}$ respectively. Find the perimeter and the area of the part inside the circle $ K$, that is outside circles $ K_{a}$ and $ K_{b}$.

Let's consider the inequality $ a^3+b^3+c^3<k(a+b+c)(ab+bc+ca)$ where $ a,b,c$ are the sides of a triangle and $ k$ a real number. a) Prove the inequality for $ k=1$. b) Find the smallest value of $ k$ such that the inequality holds for all triangles.