NMC 1996

Show that there exists an integer divisible by 1996 such that the sum of the its decimal digits is 1996 .

Determine all real numbers $x$, such that

$$x^{n}+x^{-n}$$

is an integer for all integers $n$.

The circle whose diameter is the altitude dropped from the vertex $A$ of the triangle $A B C$ intersects the sides $A B$ and $A C$ at $D$ and $E$, respectively $(A \neq D, A \neq E)$. Show that the circumcentre of $A B C$ lies on the altitude dropped from the vertex $A$ of the triangle $A D E$, or on its extension.

Figure 8.

The real-valued function $f$ is defined for positive integers, and the positive integer a satisfies

$$\begin{gathered} f(a)=f(1995), f(a+1)=f(1996), f(a+2)=f(1997) \\ f(n+a)=\frac{f(n)-1}{f(n)+1} \text { for all positive integers } n \end{gathered}$$

(i) Show that $f(n+4 a)=f(n)$ for all positive integers $n$. (ii) Determine the smallest possible $a$.