Solve in real numbers the system: $$\begin{cases} a+b+c=0 \\ ab^3+bc^3+ca^3=0 \end{cases}$$

In triangle $ABC$, points $M$, $N$ are the midpoints of sides $AB$, $AC$ respelctively. Let $D$ and $E$ be two points on line segment $BN$ such that $CD \parallel ME$ and $BD <BE$. Prove that $BD=2\cdot EN$.

Find the number of rectangles who have the following properties: a) Have for vertices, points $(x,y)$ of plane $Oxy$ with $x,y$ non negative integers and $ x \le 8$ , $y\le 8$ b) Have sides parallel to axes c) Have area $E$, with $30<E\le 40$

Find all positive integers $a,b$ with $a>1$ such that, $b$ is a divisor of $a-1$ and $2a+1$ is a divisor of $5b-3$.