Olympiad problems

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, 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.

2004 IMO, 5

Tags: geometry , IMO 2004 , Waldemar Pompe , Isogonal conjugate , IMO , Angle Chasing , easier P5

In a convex quadrilateral $ABCD$, the diagonal $BD$ bisects neither the angle $ABC$ nor the angle $CDA$. The point $P$ lies inside $ABCD$ and satisfies \[\angle PBC=\angle DBA\quad\text{and}\quad \angle PDC=\angle BDA.\] Prove that $ABCD$ is a cyclic quadrilateral if and only if $AP=CP$.

2005 IMO Shortlist, 4

Tags: geometry , IMO Shortlist , IMO 2005 , IMO , Waldemar Pompe , Spiral Similarity , Harry Potter

Let $ABCD$ be a fixed convex quadrilateral with $BC=DA$ and $BC$ not parallel with $DA$. Let two variable points $E$ and $F$ lie of the sides $BC$ and $DA$, respectively and satisfy $BE=DF$. The lines $AC$ and $BD$ meet at $P$, the lines $BD$ and $EF$ meet at $Q$, the lines $EF$ and $AC$ meet at $R$. Prove that the circumcircles of the triangles $PQR$, as $E$ and $F$ vary, have a common point other than $P$.

2005 IMO, 5

Tags: geometry , IMO Shortlist , IMO 2005 , IMO , Waldemar Pompe , Spiral Similarity , Harry Potter

Let $ABCD$ be a fixed convex quadrilateral with $BC=DA$ and $BC$ not parallel with $DA$. Let two variable points $E$ and $F$ lie of the sides $BC$ and $DA$, respectively and satisfy $BE=DF$. The lines $AC$ and $BD$ meet at $P$, the lines $BD$ and $EF$ meet at $Q$, the lines $EF$ and $AC$ meet at $R$. Prove that the circumcircles of the triangles $PQR$, as $E$ and $F$ vary, have a common point other than $P$.

2004 IMO Shortlist, 4

Tags: geometry , IMO 2004 , Waldemar Pompe , Isogonal conjugate , IMO , Angle Chasing , easier P5

In a convex quadrilateral $ABCD$, the diagonal $BD$ bisects neither the angle $ABC$ nor the angle $CDA$. The point $P$ lies inside $ABCD$ and satisfies \[\angle PBC=\angle DBA\quad\text{and}\quad \angle PDC=\angle BDA.\] Prove that $ABCD$ is a cyclic quadrilateral if and only if $AP=CP$.