ZA-SAMO 1999

How many non-congruent triangles with integer sides and perimeter 1999 can be constructed?

$A,\ B,\ C$ and $D$ are points on a given straight line, in that order. Show how to construct a square $PQRS$ , with all of $P,\ Q,\ R$ and $S$ on the same side of $AD$ , such that $A,\ B,\ C$ and $D$ lie on $PQ,\ SR,\ QR$ and $PS$ produced respectively.

The bisector of $\angle{BAD}$ in the parallellogram $ABCD$ intersects the lines $BC$ and $CD$ at the points $K$ and $L$ respectively. Prove that the centre of the circle passing through the points $C,\ K$ and $L$ lies on the circle passing through the points $B,\ C$ and $D$ .

The sequence $L_1,\ L_2,\ L_3,\ \dots$ is defined by $$L_1 = 1,\ \ L_2 = 3,\ \ L_n = L_{n - 1} + L_{n - 2}\textrm{ for }n > 2.$$ Prove that $L_p - 1$ is divisible by $p$ if $p$ is prime.

Let $S$ be the set of all rational numbers whose denominators are powers of 3. Let $a$ , $b$ and $c$ be given non-zero real numbers. Determine all real-valued functions $f$ that are defined for $x \in S$ , satisfy $$f(x) = af(3x) + bf(3x - 1) + cf(3x - 2) \textrm{ if }0 \leq x \leq 1,$$ and are zero elsewhere.

You are at a point $(a,b)$ and you need to reach another point $(c,d)$ . Both points are below the line $x = y$ and have integer coordinates. You can move in steps of length 1, either upwards of to the right, but you may not move to a point on the line $x = y$ . How many different paths are there?