DANUBE 2009

Prove that all the positive integer numbers , except for the powers of $2$ , can be written as the sum of (at least two) consecutive natural numbers .

Let $n$ be a natural number. Determine the minimal number of equilateral triangles of side $1$ to cover the surface of an equilateral triangle of side $n +\frac{1}{2n}$ .

Let $\sigma, \tau$ be two permutations of the quantity $\{1, 2,. . . , n\}$ . Prove that there is a function $f: \{1, 2,. . . , n\} \to \{-1, 1\}$ such that for any $1 \le i \le j \le n$ , we have $\left|\sum_{k=i}^{j} f(\sigma (k)) \right| \le 2$ and $\left|\sum_{k=i}^{j} f(\tau (k))\right| \le 2$