IL-GILLIS 1996

Let $a$ be a prime number and $n > 2$ an integer. Find all integer solutions of the equation $x^n +ay^n = a^2z^n$ .

Find all polynomials $P(x)$ satisfying $P(x+1)-2P(x)+P(x-1)= x$ for all $x$

The angles of an acute triangle $ABC$ at $\alpha , \beta, \gamma$ . Let $AD$ be a height, $CF$ a median, and $BE$ the bisector of $\angle B$ . Show that $AD,CF$ and $BE$ are concurrent if and only if $\cos \gamma \tan\beta = \sin \alpha$ .

Eight guests arrive to a hotel with four rooms. Each guest dislikes at most three other guests and doesn't want to share a room with any of them (this feeling is mutual). Show that the guests can reside in the four rooms, with two persons in each room

Suppose that the circumradius $R$ and the inradius $r$ of a triangle $ABC$ satisfy $R = 2r$ . Prove that the triangle is equilateral.

Let $x,y,z$ be real numbers with $|x|,|y|,|z| > 2$ . What is the smallest possible value of $|xyz+2(x+y+z)|$ ?

Find all positive integers $a,b,c$ such that $$\begin{cases} a^2 = 4(b+c) a^3 -2b^3 -4c^3 =\frac12 abc \end {cases}$$

Consider the function $f : N \to N$ given by (i) $f(1) = 1$ , (ii) $f(2n) = f(n)$ for any $n \in N$ , (iii) $f(2n+1) = f(2n)+1$ for any $n \in N$ . (a) Find the maximum value of $f(n)$ for $1 \le n \le 1995$ ; (b) Find all values of $f$ on this interval.