Olympiad problems

2003 IMO, 1

Tags: algorithm , IMO , IMO 2003 , combinatorics , greedy , independent sets , IMO Shortlist

Let $A$ be a $101$-element subset of the set $S=\{1,2,\ldots,1000000\}$. Prove that there exist numbers $t_1$, $t_2, \ldots, t_{100}$ in $S$ such that the sets \[ A_j=\{x+t_j\mid x\in A\},\qquad j=1,2,\ldots,100 \] are pairwise disjoint.

2003 IMO Shortlist, 1

Tags: algorithm , IMO , IMO 2003 , combinatorics , greedy , independent sets , IMO Shortlist

Let $A$ be a $101$-element subset of the set $S=\{1,2,\ldots,1000000\}$. Prove that there exist numbers $t_1$, $t_2, \ldots, t_{100}$ in $S$ such that the sets \[ A_j=\{x+t_j\mid x\in A\},\qquad j=1,2,\ldots,100 \] are pairwise disjoint.

2004 India IMO Training Camp, 1

Tags: geometry , angle bisector , cyclic quadrilateral , quadrilateral , concurrency , IMO , IMO 2003

Let $ABCD$ be a cyclic quadrilateral. Let $P$, $Q$, $R$ be the feet of the perpendiculars from $D$ to the lines $BC$, $CA$, $AB$, respectively. Show that $PQ=QR$ if and only if the bisectors of $\angle ABC$ and $\angle ADC$ are concurrent with $AC$.

2003 IMO, 5

Tags: inequalities , arithmetic sequence , IMO , IMO 2003 , IMO Shortlist

Let $n$ be a positive integer and let $x_1\le x_2\le\cdots\le x_n$ be real numbers. Prove that \[ \left(\sum_{i,j=1}^{n}|x_i-x_j|\right)^2\le\frac{2(n^2-1)}{3}\sum_{i,j=1}^{n}(x_i-x_j)^2. \] Show that the equality holds if and only if $x_1, \ldots, x_n$ is an arithmetic sequence.

2003 IMO Shortlist, 6

Tags: geometry , IMO , IMO 2003 , hexagon , equal angles , IMO Shortlist , Waldemar Pompe

Each pair of opposite sides of a convex hexagon has the following property: the distance between their midpoints is equal to $\dfrac{\sqrt{3}}{2}$ times the sum of their lengths. Prove that all the angles of the hexagon are equal.

2003 IMO Shortlist, 6

Tags: number theory , polynomial , Divisibility , IMO , IMO 2003 , IMO Shortlist , Hi

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 $q$.

2003 IMO, 3

Tags: geometry , IMO , IMO 2003 , hexagon , equal angles , IMO Shortlist , Waldemar Pompe

Each pair of opposite sides of a convex hexagon has the following property: the distance between their midpoints is equal to $\dfrac{\sqrt{3}}{2}$ times the sum of their lengths. Prove that all the angles of the hexagon are equal.

2003 IMO, 6

Tags: number theory , polynomial , Divisibility , IMO , IMO 2003 , IMO Shortlist , Hi

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 $q$.

2003 IMO, 4

Tags: geometry , angle bisector , cyclic quadrilateral , quadrilateral , concurrency , IMO , IMO 2003

Let $ABCD$ be a cyclic quadrilateral. Let $P$, $Q$, $R$ be the feet of the perpendiculars from $D$ to the lines $BC$, $CA$, $AB$, respectively. Show that $PQ=QR$ if and only if the bisectors of $\angle ABC$ and $\angle ADC$ are concurrent with $AC$.

2004 India IMO Training Camp, 1

Tags: geometry , angle bisector , cyclic quadrilateral , quadrilateral , concurrency , IMO , IMO 2003

Let $ABCD$ be a cyclic quadrilateral. Let $P$, $Q$, $R$ be the feet of the perpendiculars from $D$ to the lines $BC$, $CA$, $AB$, respectively. Show that $PQ=QR$ if and only if the bisectors of $\angle ABC$ and $\angle ADC$ are concurrent with $AC$.

2003 IMO Shortlist, 4

Tags: inequalities , arithmetic sequence , IMO , IMO 2003 , IMO Shortlist

Let $n$ be a positive integer and let $x_1\le x_2\le\cdots\le x_n$ be real numbers. Prove that \[ \left(\sum_{i,j=1}^{n}|x_i-x_j|\right)^2\le\frac{2(n^2-1)}{3}\sum_{i,j=1}^{n}(x_i-x_j)^2. \] Show that the equality holds if and only if $x_1, \ldots, x_n$ is an arithmetic sequence.

2003 IMO Shortlist, 1

Tags: geometry , angle bisector , cyclic quadrilateral , quadrilateral , concurrency , IMO , IMO 2003

Let $ABCD$ be a cyclic quadrilateral. Let $P$, $Q$, $R$ be the feet of the perpendiculars from $D$ to the lines $BC$, $CA$, $AB$, respectively. Show that $PQ=QR$ if and only if the bisectors of $\angle ABC$ and $\angle ADC$ are concurrent with $AC$.