This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 44

1990 All Soviet Union Mathematical Olympiad, 518
Tags: acute , equilateral , 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.

1955 Moscow Mathematical Olympiad, 304
Tags: excenter , acute , geometry
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.

2016 Auckland Mathematical Olympiad, 5
Tags: combinatorial geometry , regular polygon , acute
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.

2022 Indonesia TST, G
Tags: acute , collinear , extension , triangle , orthocenter , perpendicular , geometry
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.

1955 Moscow Mathematical Olympiad, 301
Tags: acute , trihedral angle , 3d geometry , geometry , angle
Given a trihedral angle with vertex $O$. Find whether there is a planar section $ABC$ such that the angles $\angle OAB$, $\angle OBA$, $\angle OBC$, $\angle OCB$, $\angle OAC$, $\angle OCA$ are acute.

1975 Czech and Slovak Olympiad III A, 1
Tags: triangle area , acute , triangle , right triangle , area , geometry
Let $\mathbf T$ be a triangle with $[\mathbf T]=1.$ Show that there is a right triangle $\mathbf R$ such that $[\mathbf R]\le\sqrt3$ and $\mathbf T\subseteq\mathbf R.$ ($[-]$ denotes area of a triangle.)

2019 Novosibirsk Oral Olympiad in Geometry, 7
Tags: acute , square , geometry
The square was cut into acute -angled triangles. Prove that there are at least eight of them.

1956 Moscow Mathematical Olympiad, 344
Tags: acute , combinatorial geometry , graph , geometry
* Let $A, B, C$ be three nodes of a graph paper. Prove that if $\vartriangle ABC$ is an acute one, then there is at least one more node either inside $\vartriangle ABC$ or on one of its sides.

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.

2020 Ukrainian Geometry Olympiad - April, 2
Tags: geometry , acute , angle
Inside the triangle $ABC$ is point $P$, such that $BP > AP$ and $BP > CP$. Prove that $\angle ABC$ is acute.

2007 Sharygin Geometry Olympiad, 5
Tags: centroid , acute , median , geometry
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.

1946 Moscow Mathematical Olympiad, 106
Tags: acute , angle , geometry , maximum
What is the largest number of acute angles that a convex polygon can have?

2011 Swedish Mathematical Competition, 2
Tags: acute , geometric inequality , geometry
Given a triangle $ABC$, let $P$ be a point inside the triangle such that $| BP | > | AP |, | BP | > | CP |$. Show that $\angle ABC <90^o$

1970 IMO Shortlist, 12
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.

Novosibirsk Oral Geo Oly VIII, 2019.7
Tags: geometry , acute , square
The square was cut into acute -angled triangles. Prove that there are at least eight of them.

1956 Moscow Mathematical Olympiad, 320
Tags: acute , angle , geometry
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.

Novosibirsk Oral Geo Oly VII, 2019.7
Tags: geometry , acute , square
Cut a square into eight acute-angled triangles.

1952 Moscow Mathematical Olympiad, 208
Tags: incircle , geometry , angle , acute
The circle is inscribed in $\vartriangle ABC$. Let $L, M, N$ be the tangent points of the circle with sides $AB, AC, BC$, respectively. Prove that $\angle MLN$ is always an acute angle.

2019 Novosibirsk Oral Olympiad in Geometry, 7
Tags: acute , geometry , square
Cut a square into eight acute-angled triangles.

2013 Brazil Team Selection Test, 2
Tags: geometry , acute , angle , cyclic quadrilateral
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.

2014 Belarus Team Selection Test, 4
Tags: acute , combinatorics , combinatorial geometry
Thirty rays with the origin at the same point are constructed on a plane. Consider all angles between any two of these rays. Let $N$ be the number of acute angles among these angles. Find the smallest possible value of $N$. (E. Barabanov)

2013 Korea Junior Math Olympiad, 5
Tags: midpoint , perpendicular , acute , circumcenter , geometry , circumcircle
In an acute triangle $\triangle ABC, \angle A > \angle B$. Let the midpoint of $AB$ be $D$, and let the foot of the perpendicular from $A$ to $BC$ be $E$, and $B$ from $CA$ be $F$. Let the circumcenter of $\triangle DEF$ be $O$. A point $J$ on segment $BE$ satisfi es $\angle ODC = \angle EAJ$. Prove that $AJ \cap DC$ lies on the circumcircle of $\triangle BDE$.

2011 Indonesia TST, 2
Tags: combinatorics , probability , acute , geometry
Let $n$ be a integer and $n \ge 3$, and $T_1T_2...T_n$ is a regular n-gon. Distinct $3$ points $T_i , T_j , T_k$ are chosen randomly. Determine the probability of triangle $T_iT_jT_k$ being an acute triangle.

1964 Poland - Second Round, 6
Tags: point , combinatorial geometry , geometry , acute , combinatorics
Prove that from any five points in the plane it is possible to choose three points that are not vertices of an acute triangle.

1997 All-Russian Olympiad Regional Round, 9.1
Tags: combinatorial geometry , acute , geometry , combinatorics
A regular $1997$-gon is divided into triangles by non-intersecting diagonals. Prove that exactly one of them is acute-angled.