Olympiad problems

2009 Sharygin Geometry Olympiad, 5

Let $n$ points lie on the circle. Exactly half of triangles formed by these points are acute-angled. Find all possible $n$. (B.Frenkin)

2024 Regional Olympiad of Mexico Southeast, 2

Tags: acute triangle , circumradius , perpendicular , geometry

Let $ABC$ be an acute triangle with circumradius $R$. Let $D$ be the midpoint of $BC$ and $F$ the midpoint of $AB$. The perpendicular to $AC$ through $F$ and the perpendicular to $BC$ through $B$ intersect at $N$. Prove that $ND = R$.

2017 Sharygin Geometry Olympiad, P5

Tags: locus , acute triangle , incenter , geometry

A segment $AB$ is fixed on the plane. Consider all acute-angled triangles with side $AB$. Find the locus of а) the vertices of their greatest angles, b) their incenters.

2017 Sharygin Geometry Olympiad, P17

Tags: geometry , acute triangle , construction

Using a compass and a ruler, construct a point $K$ inside an acute-angled triangle $ABC$ so that $\angle KBA = 2\angle KAB$ and $ \angle KBC = 2\angle KCB$.

1988 Austrian-Polish Competition, 6

Tags: acute , 3d geometry , orthogonal , ray , acute triangle , angle , geometry

Three rays $h_1,h_2,h_3$ emanating from a point $O$ are given, not all in the same plane. Show that if for any three points $A_1,A_2,A_3$ on $h_1,h_2,h_3$ respectively, distinct from $O$, the triangle $A_1A_2A_3$ is acute-angled, then the rays $h_1,h_2,h_3$ are pairwise orthogonal.

2005 USAMO, 3

Tags: circumcircle , acute , acute triangle , conic , geometry

Let $ABC$ be an acute-angled triangle, and let $P$ and $Q$ be two points on its side $BC$. Construct a point $C_{1}$ in such a way that the convex quadrilateral $APBC_{1}$ is cyclic, $QC_{1}\parallel CA$, and $C_{1}$ and $Q$ lie on opposite sides of line $AB$. Construct a point $B_{1}$ in such a way that the convex quadrilateral $APCB_{1}$ is cyclic, $QB_{1}\parallel BA$, and $B_{1}$ and $Q$ lie on opposite sides of line $AC$. Prove that the points $B_{1}$, $C_{1}$, $P$, and $Q$ lie on a circle.

1986 ITAMO, 5

Tags: geometry , 3d geometry , tetrahedron , acute triangle

Given an acute triangle $T$ with sides $a,b,c$, find the tetrahedra with base $T$ whose all faces are acute triangles of the same area.