Olympiad problems

2012 Estonia Team Selection Test, 3

In a cyclic quadrilateral $ABCD$ we have $|AD| > |BC|$ and the vertices $C$ and $D$ lie on the shorter arc $AB$ of the circumcircle. Rays $AD$ and $BC$ intersect at point $K$, diagonals $AC$ and $BD$ intersect at point $P$. Line $KP$ intersects the side $AB$ at point $L$. Prove that $\angle ALK$ is acute.

1997 All-Russian Olympiad Regional Round, 9.1

Tags: geometry , combinatorics , combinatorial geometry , acute

A regular $1997$-gon is divided into triangles by non-intersecting diagonals. Prove that exactly one of them is acute-angled.

1970 IMO, 3

Tags: acute , triangle , point set , counting , combinatorial geometry

Given $100$ coplanar points, no three collinear, prove that at most $70\%$ of the triangles formed by the points have all angles acute.

1955 Moscow Mathematical Olympiad, 304

Tags: geometry , acute , excenter

The centers $O_1, O_2$ and $O_3$ of circles exscribed about $\vartriangle ABC$ are connected. Prove that $O_1O_2O_3$ is an acute-angled one.

Novosibirsk Oral Geo Oly VIII, 2019.7

Tags: acute , square , geometry

The square was cut into acute -angled triangles. Prove that there are at least eight of them.

1990 All Soviet Union Mathematical Olympiad, 518

Tags: equilateral , acute , combinatorial geometry , combinatorics

An equilateral triangle of side $n$ is divided into $n^2$ equilateral triangles of side $1$. A path is drawn along the sides of the triangles which passes through each vertex just once. Prove that the path makes an acute angle at at least $n$ vertices.

1956 Moscow Mathematical Olympiad, 320

Tags: acute , geometry , angle

Prove that there are no four points $A, B, C, D$ on a plane such that all triangles $\vartriangle ABC,\vartriangle BCD, \vartriangle CDA, \vartriangle DAB$ are acute ones.

2022 Indonesia TST, G

Tags: geometry , orthocenter , collinear , perpendicular , extension , acute , triangle

Given an acute triangle $ABC$. with $H$ as its orthocenter, lines $\ell_1$ and $\ell_2$ go through $H$ and are perpendicular to each other. Line $\ell_1$ cuts $BC$ and the extension of $AB$ on $D$ and $Z$ respectively. Whereas line $\ell_2$ cuts $BC$ and the extension of $AC$ on $E$ and $X$ respectively. If the line through $D$ and parallel to $AC$ and the line through $E$ parallel to $AB$ intersects at $Y$, prove that $X,Y,Z$ are collinear.

2019 Novosibirsk Oral Olympiad in Geometry, 7

Tags: geometry , acute , square

The square was cut into acute -angled triangles. Prove that there are at least eight of them.

1967 Czech and Slovak Olympiad III A, 2

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

Let $ABCD$ be a tetrahedron such that \[AB^2+CD^2=AC^2+BD^2=AD^2+BC^2.\] Show that at least one of its faces is an acute triangle.

2007 Sharygin Geometry Olympiad, 5

Tags: geometry , centroid , acute , median

Medians $AA'$ and $BB'$ of triangle $ABC$ meet at point $M$, and $\angle AMB = 120^o$. Prove that angles $AB'M$ and $BA'M$ are neither both acute nor both obtuse.

2016 Auckland Mathematical Olympiad, 5

Tags: combinatorial geometry , acute , regular polygon

A regular $2017$-gon is partitioned into triangles by a set of non-intersecting diagonals. Prove that among those triangles only one is acute-angled.

2003 All-Russian Olympiad Regional Round, 9.8

Tags: geometry , combinatorics , combinatorial geometry , convex , convex polygon , acute

Prove that a convex polygon can be cut by disjoint diagonals into acute triangles in at least one way.

2013 Brazil Team Selection Test, 2

Tags: geometry , acute , cyclic quadrilateral , angle

Let $ABCD$ be a convex cyclic quadrilateral with $AD > BC$, A$B$ not being diameter and $C D$ belonging to the smallest arc $AB$ of the circumcircle. The rays $AD$ and $BC$ are cut at $K$, the diagonals $AC$ and $BD$ are cut at $P$ and the line $KP$ cuts the side $AB$ at point $L$. Prove that angle $\angle ALK$ is acute.