International Mathematical Olympiad · 2003

Problems

Problem 1Let $A$ be a 101-element subset of $S=\{1,2,\ldots,10^6\}$ . Prove that there exist numbers $t_1,t_2,\ldots,t_{100}$ in $S$ such that the sets $A_j=\{x+t_j\mid x\in A\}$ , $j=1,2,\ldots,100$ , are pairwise disjoint.Solutions: 1
Problem 2Determine all pairs of positive integers $(a,b)$ such that $\frac{a^2}{2ab^2-b^3+1}$ is a positive integer.Solutions: 1
Problem 3Each pair of opposite sides of convex hexagon has the property that the distance $p$ between their midpoints is $\frac{\sqrt3}{2}$ times the sum of their lengths. Prove that the hexagon is equiangular.Solutions: 1
Problem 4Let $ABCD$ be a cyclic quadrilateral. Let $P,Q,R$ be the feet of perpendiculars from $D$ to lines $BC,CA,AB$ , respectively. Show that $PQ=QR$ if and only if the bisectors of angles $ABC$ and $ADC$ meet on segment $AC$ .Solutions: 1
Problem 5Let $n$ be a positive integer and let $x_1,x_2,\ldots,x_n$ 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 $x_1,x_2,\ldots,x_n$ form an arithmetic sequence.Solutions: 1
Problem 6Let $p$ 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 $q$ .Solutions: 1