(a) Find the value of the real number $k$, for which the polynomial $P(x)=x^3-kx+2$ has the number $2$ as a root. In addition, for the value of $k$ you will find, write this polynomial as the product of two polynomials with integer coefficients. (b) If the positive real numbers $a,b$ satisfy the equation $$2a+b+\frac{4}{ab}=10,$$find the maximum possible value of $a$.

Let $ABC$ be an isosceles triangle, and point $D$ in its interior such that $$D \hat{B} C=30^\circ, D \hat{B}A=50^\circ, D \hat{C}B=55^\circ$$ (a) Prove that $\hat B=\hat C=80^\circ$. (b) Find the measure of the angle $D \hat{A} C$.

On the board we write a series of $n$ numbers, where $n \geq 40$, and each one of them is equal to either $1$ or $-1$, such that the following conditions both hold: (i) The sum of every $40$ consecutive numbers is equal to $0$. (ii) The sum of every $42$ consecutive numbers is not equal to $0$. We denote by $S_n$ the sum of the $n$ numbers of the board. Find the maximum possible value of $S_n$ for all possible values of $n$.

Find all couples of non-zero integers $(x,y)$ such that, $x^2+y^2$ is a common divisor of $x^5+y$ and $y^5+x$.