IMO 2003

Official IMO 2003 problems  ·  0/6 solved.

Problem 1

$S$ is the set ${1, 2, 3, \dots ,1000000}$. Show that for any subset $A$ of $S$ with $101$ elements we can find $100$ distinct elements $x_i$ of $S$, such that the sets ${a + x_i \mid a \in A}$ are all pairwise disjoint.

Problem 2

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

Problem 3

Each pair of opposite sides of convex hexagon has the property that the distance between their midpoints is $\frac{\sqrt{3}}{2}$ times the sum of their lengths. Prove that the hexagon is equiangular.

Problem 4

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 $\overline{AC}$.

Problem 5

Let $n$ be a positive integer and let $x_1 \le x_2 \le \cdots \le x_n$ be real numbers. Prove that

$$ \left( \sum_{i=1}^{n}\sum_{j=i}^{n} |x_i-x_j|\right)^2 \le \frac{2(n^2-1)}{3}\sum_{i=1}^{n}\sum_{j=i}^{n}(x_i-x_j)^2 $$

with equality if and only if $x_1, x_2, ..., x_n$ 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 $q$.