2003 IMO Problems

Problems of the 2003 IMO.

Day I

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.

Solution

Problem 2

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

Solution

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.

Solution

Day II

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 $\overline{AC}$ .

Solution

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.

Solution

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 $q$ .

Solution

2003 IMO (Problems) • Resources
Preceded by 2002 IMO	1 • 2 • 3 • 4 • 5 • 6	Followed by 2004 IMO
All IMO Problems and Solutions

Page

Toolbox

Search

2003 IMO Problems

Contents

Day I

Problem 1

Problem 2

Problem 3

Day II

Problem 4

Problem 5

Problem 6

See Also