Olympiad problems

2018 Estonia Team Selection Test, 5

Tags: geometry , IMO Shortlist , Angle Chasing , projective geometry , similar triangles , ISL 2017 , G3

Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.

2018 Bosnia and Herzegovina Team Selection Test, 6

Tags: geometry , IMO Shortlist , Angle Chasing , projective geometry , similar triangles , ISL 2017 , G3

Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.

2018 Estonia Team Selection Test, 5

Tags: geometry , IMO Shortlist , Angle Chasing , projective geometry , similar triangles , ISL 2017 , G3

Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.

2018 Romania Team Selection Tests, 1

Tags: geometry , IMO Shortlist , Angle Chasing , projective geometry , similar triangles , ISL 2017 , G3

Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.

2018 Taiwan TST Round 3, 4

Tags: geometry , IMO Shortlist , Angle Chasing , projective geometry , similar triangles , ISL 2017 , G3

Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.

Russian TST 2018, P2

Tags: geometry , IMO Shortlist , Angle Chasing , projective geometry , similar triangles , ISL 2017 , G3

Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.

2018 Moldova Team Selection Test, 3

Tags: geometry , IMO Shortlist , Angle Chasing , projective geometry , similar triangles , ISL 2017 , G3

Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.

2017 IMO Shortlist, G3

Tags: geometry , IMO Shortlist , Angle Chasing , projective geometry , similar triangles , ISL 2017 , G3

Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.

2018 Thailand TST, 2

Tags: geometry , IMO Shortlist , Angle Chasing , projective geometry , similar triangles , ISL 2017 , G3

Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.