IMO 2003

Problem 1. Let $A$ be a 101-element subset of the set $S=\{1,2,\dots ,1000000\}$. Prove that there exist numbers $t_1,t_2,\dots ,t_{100}$ in $S$ such that the sets

$A_j=\{x+t_j\mid x\in A\}j=1,2,\dots ,100$

are pairwise disjoint.

Problem 2. Determine all pairs of positive integers $(a,b)$ such that

$\frac{a^2}{2a{b}^2-b^3+1}$

is a positive integer.

Problem 3. A convex hexagon is given in which any two opposite sides have the following property: the distance between their midpoints is $\sqrt{3}/2$ times the sum of their lengths. Prove that all the angles of the hexagon are equal.

(A convex $ABCDEF$ has three pairs of opposite sides: $AB$ and $DE$, $BC$ and $EF$, $CD$ and $FA$.)

Problem 4. Let $ABCD$ be a cyclic quadrilateral. Let $P,Q$ and $R$ be the feet of perpendiculars from $D$ to lines $BC,CA$ and $AB$, respectively. Show that $PQ=QR$ if and only if the bisectors of angles $ABC$ and $ADC$ meet on segment $AC$.

Problem 5. Let $n$ be a positive integer and $x_1,x_2,\dots ,x_n$ be real numbers with $x_1\le x_2\le \cdots \le x_n$.

(a) Prove that

(b) Show that the equality holds if and only if $x_1,x_2,\dots ,x_n$ form an arithmetic sequence.

Problem 6. 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 $q$.