Olympiad problems

2007 Sharygin Geometry Olympiad, 2

Tags: geometry , altitudes , Circumcenter , concurrent , concurrency

Points $A', B', C'$ are the feet of the altitudes $AA', BB'$ and $CC'$ of an acute triangle $ABC$. A circle with center $B$ and radius $BB'$ meets line $A'C'$ at points $K$ and $L$ (points $K$ and $A$ are on the same side of line $BB'$). Prove that the intersection point of lines $AK$ and $CL$ belongs to line $BO$ ($O$ is the circumcenter of triangle $ABC$).

2018 Silk Road, 1

Tags: geometry , altitudes

In an acute-angled triangle $ABC$ on the sides $AB$, $BC$, $AC$ the points $H$, $L$, $K$ so that $CH \perp AB$, $HL \parallel AC$, $HK \parallel BC$. Let $P$ and $Q$ feet of altitudes of a triangle $HBL$, drawn from the vertices $H$ and $B$ respectively. Prove that the feet of the altitudes of the triangle $AKH$, drawn from the vertices $A$ and $H$ lie on the line $PQ$.

2015 Caucasus Mathematical Olympiad, 5

Tags: geometry , altitudes , midpoints , equal angles , equal segments

Let $AA_1$ and $CC_1$ be the altitudes of the acute-angled triangle $ABC$. Let $K,L$ and $M$ be the midpoints of the sides $AB,BC$ and $CA$ respectively. Prove that if $\angle C_1MA_1 =\angle ABC$, then $C_1 K = A_1L$.

2017 Puerto Rico Team Selection Test, 2

Tags: perpendicular bisector , geometry , altitudes

For an acute triangle $ ABC $ let $ H $ be the point of intersection of the altitudes $ AA_1 $, $ BB_1 $, $ CC_1 $. Let $ M $ and $ N $ be the midpoints of the $ BC $ and $ AH $ segments, respectively. Show that $ MN $ is the perpendicular bisector of segment $ B_1C_1 $.

2016 Saint Petersburg Mathematical Olympiad, 3

Tags: geometry , 3D geometry , tetrahedron , sphere , altitudes , concurrent

In a tetrahedron, the midpoints of all the edges lie on the same sphere. Prove that it's altitudes intersect at one point.

2006 Denmark MO - Mohr Contest, 5

Tags: geometry , similar triangles , equal segments , altitudes

We consider an acute triangle $ABC$. The altitude from $A$ is $AD$, the altitude from $D$ in triangle $ABD$ is $DE,$ and the altitude from $D$ in triangle $ACD$ is $DF$. a) Prove that the triangles $ABC$ and $AF E$ are similar. b) Prove that the segment $EF$ and the corresponding segments constructed from the vertices $B$ and $C$ all have the same length.

2012 Sharygin Geometry Olympiad, 7

Tags: geometry , altitudes

The altitudes $AA_1$ and $CC_1$ of an acute-angled triangle $ABC$ meet at point $H$. Point $Q$ is the reflection of the midpoint of $AC$ in line $AA_1$, point $P$ is the midpoint of segment $A_1C_1$. Prove that $\angle QPH = 90^o$. (D.Shvetsov)