DE-BUNDESWETTBEWERB 2003

Find all triples $\left(x,\ y,\ z\right)$ of integers satisfying the following system of equations:

$x^3-4x^2-16x+60=y$ ; $y^3-4y^2-16y+60=z$ ; $z^3-4z^2-16z+60=x$ .

Determine all positive integers which cannot be represented as $\frac{a}{b}+\frac{a+1}{b+1}$ with $a,b$ being positive integers.

The sequence $\{a_1,a_2,\ldots\}$ is recursively defined by $a_1 = 1$ , $a_2 = 1$ , $a_3 = 2$ , and $$a_{n+3} = \frac 1{a_n}\cdot (a_{n+1}a_{n+2}+7), \ \forall \ n > 0.$$ Prove that all elements of the sequence are integers.

Consider a cyclic quadrilateral $ABCD$ , and let $S$ be the intersection of $AC$ and $BD$ . Let $E$ and $F$ the orthogonal projections of $S$ on $AB$ and $CD$ respectively. Prove that the perpendicular bisector of segment $EF$ meets the segments $AD$ and $BC$ at their midpoints.

Let $p$ and $q$ be two positive integers that have no common divisor. The set of integers shall be partioned into three subsets $A$ , $B$ , $C$ such that for each integer $z$ in each of the sets $A$ , $B$ , $C$ there is exactly one of the numbers $z$ , $z+p$ and $z+q$ .

a) Prove that such a decomposition is possible if and only if $p+q$ is divisible by $3$ .

b) In the case we omit the restriction that $p$ , $q$ may not have a common divisor, prove that for $p \neq q$ the number $\frac{p+q}{\gcd(p,q)}$ is divisible by 3.