MEMO 2008

Determine all functions $f: \mathbb{R} \mapsto \mathbb{R}$ such that $$x f(x + xy) = x f(x) + f \left( x^2 \right) f(y) \forall x,y \in \mathbb{R}.$$

On a blackboard there are $n \geq 2, n \in \mathbb{Z}^{+}$ numbers. In each step we select two numbers from the blackboard and replace both of them by their sum. Determine all numbers $n$ for which it is possible to yield $n$ identical number after a finite number of steps.

Let $ABC$ be an acute-angled triangle. Let $E$ be a point such $E$ and $B$ are on distinct sides of the line $AC,$ and $D$ is an interior point of segment $AE.$ We have $\angle ADB = \angle CDE,$ $\angle BAD = \angle ECD,$ and $\angle ACB = \angle EBA.$ Prove that $B, C$ and $E$ lie on the same line.

Prove: If the sum of all positive divisors of $n \in \mathbb{Z}^{+}$ is a power of two, then the number/amount of the divisors is a power of two.