1. Find all injective functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for every real number $x$ and positive integer $n$,

$$
\left|\sum_{i=1}^{n} i(f(x+i+1)-f(f(x+i)))\right|<2016
$$

(Macedonia, FYR)
1. Denote $S(x, n)=\sum_{i=1}^{n} i\{f(x+i+1)-f(f(x+i))\}$. Fix $x \in \mathbb{R}$. Subtracting the inequalities $|S(x-n, n)|<2016$ and $|S(x-n, n-1)|<2016$ we obtain

$$
n \cdot|f(x+1)-f(f(x))|=|S(x-n, n)-S(x-n, n-1)|<4032 \quad \text { for each } n
$$

This is possible only if $f(f(x))=f(x+1)$, which by injectivity implies $f(x)=x+1$.