BALKAN-SL 2008 C1

All $n+3$ offices of University of Somewhere are numbered with numbers $0,1,2, \ldots ,$ $n+1,$ $n+2$ for some $n \in \mathbb{N}$ . One day, Professor $D$ came up with a polynomial with real coefficients and power $n$ . Then, on the door of every office he wrote the value of that polynomial evaluated in the number assigned to that office. On the $i$ th office, for $i$ $\in$ $\{0,1, \ldots, n+1 \}$ he wrote $2^i$ and on the $(n+2)$ th office he wrote $2^{n+2}$ $-n-3$ . [list=a] [] Prove that Professor D made a calculation error [] Assuming that Professor D made a calculation error, what is the smallest number of errors he made? Prove that in this case the errors are uniquely determined, find them and correct them. [/list]