Question bank IMO 2003

Let $A$ be a $101$-element subset of $S=\{1,2,\dots,10^6\}$. Prove that there exist numbers $t_1$, $t_2, \dots, t_{100}$ in $S$ such that the sets $$
A_j=\{x+t_j\mid x\in A\},\qquad j=1,2,\dots,100
$$ are pairwise disjoint.

Determine all pairs of positive integers $(a,b)$ such that $$
\frac{a^2}{2ab^2-b^3+1}
$$ is a positive integer.

Each pair of opposite sides of convex hexagon has the property that the distance between their midpoints is $\frac{\sqrt3}{2}$ times the sum of their lengths. Prove that the hexagon is equiangular.

Let $ABCD$ be a cyclic quadrilateral. Let $P$, $Q$ and $R$ be the feet of perpendiculars from $D$ to lines $\overline{BC}$, $\overline{CA}$ and $\overline{AB}$, respectively. Show that $PQ = QR$ if and only if the bisectors of angles $ABC$ and $ADC$ meet on segment $\overline{AC}$.

Let $n\gt2$ be an integer and let $x_1 \le x_2 \le \dots \le x_n$ be real numbers. Prove that $$
\left(\sum_{i=1}^{n}\sum_{j=1}^{n} |x_i - x_j|\right)^2
\le \frac{2(n^2-1)}{3}\sum_{i=1}^{n}\sum_{j=1}^{n} (x_i - x_j)^2
$$ with equality if and only if $x_1$, $x_2$, …, $x_n$ form an arithmetic sequence.

Let $p$ be a prime number. Prove that there exists a prime number $q$ such that for every integer $n$, the number $n^p-p$ is not divisible by $q$.