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

2013 Korea Junior Math Olympiad, 5
Tags: geometry , circumcircle , acute , circumcenter , perpendicular , midpoint
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 $\triangle BDE$.

2020 Ukrainian Geometry Olympiad - December, 2
Tags: acute , combinatorics , combinatorial geometry , geometry
On a circle noted $n$ points. It turned out that among the triangles with vertices in these points exactly half of the acute. Find all values $n$ in which this is possible.

1952 Moscow Mathematical Olympiad, 208
Tags: geometry , acute , incircle , angle
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.

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

2010 All-Russian Olympiad Regional Round, 11.5
Tags: geometry , acute , trigonometry , inequalities
The angles of the triangle $\alpha, \beta, \gamma$ satisfy the inequalities $$\sin \alpha > \cos \beta, \sin \beta > \cos \gamma, \sin \gamma > \cos \alpha. $$Prove that the trαiangle is acute-angled.

2012 Estonia Team Selection Test, 3
Tags: diagonal , cyclic quadrilateral , acute , geometry , circumcircle
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.

Estonia Open Junior - geometry, 2017.1.5
Tags: convex , polygon , acute , geometry
Find all possibilities: how many acute angles can there be in a convex polygon?

1985 Tournament Of Towns, (105) 5
Tags: geometry , convex , convex polygon , acute , convex polyhedron , 3d geometry
(a) The point $O$ lies inside the convex polygon $A_1A_2A_3...A_n$ . Consider all the angles $A_iOA_j$ where $i, j$ are distinct natural numbers from $1$ to $n$ . Prove that at least $n- 1$ of these angles are not acute . (b) Same problem for a convex polyhedron with $n$ vertices. (V. Boltyanskiy, Moscow)

2017 Bundeswettbewerb Mathematik, 2
Tags: gon , angle , acute , combinatorics
What is the maximum number of acute interior angles a non-overlapping planar $2017$-gon can have?

1954 Moscow Mathematical Olympiad, 270
Tags: geometry , acute
Consider $\vartriangle ABC$ and a point $S$ inside it. Let $A_1, B_1, C_1$ be the intersection points of $AS, BS, CS$ with $BC, AC, AB$, respectively. Prove that at least in one of the resulting quadrilaterals $AB_1SC_1, C_1SA_1B, A_1SB_1C$ both angles at either $C_1$ and $B_1$, or $C_1$ and $A_1$, or $A_1$ and $B_1$ are not acute.

1988 Austrian-Polish Competition, 6
Tags: geometry , 3d geometry , orthogonal , ray , acute triangle , acute , angle
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.

2002 Estonia National Olympiad, 2
Tags: acute , obtuse , orthocenter , angle , geometry
Let $ABC$ be a non-right triangle with its altitudes intersecting in point $H$. Prove that $ABH$ is an acute triangle if and only if $\angle ACB$ is obtuse.

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

1970 IMO Longlists, 58
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.

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

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

1975 Czech and Slovak Olympiad III A, 1
Tags: geometry , acute , triangle , right triangle , area , triangle area
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.)

2011 Indonesia TST, 2
Tags: combinatorics , geometry , probability , acute
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.

1956 Moscow Mathematical Olympiad, 344
Tags: graph , acute , geometry , combinatorial 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.

1946 Moscow Mathematical Olympiad, 115
Tags: trigonometry , inequalities , acute
Prove that if $\alpha$ and $\beta$ are acute angles and $\alpha$ < $\beta$ , then $\frac{tan \alpha}{\alpha} < \frac{tan \beta}{\beta} $

2011 Indonesia TST, 2
Tags: geometry , probability , acute , combinatorics
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 , geometry , acute , combinatorics , combinatorial geometry
Prove that from any five points in the plane it is possible to choose three points that are not vertices of an acute triangle.

2005 USAMO, 3
Tags: geometry , circumcircle , acute , acute triangle , conic
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.

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.

2017 BMT Spring, 4
Tags: geometry , angle , acute
$2$ darts are thrown randomly at a circular board with center $O$, such that each dart has an equal probability of hitting any point on the board. The points at which they land are marked $A$ and $B$. What is the probability that $\angle AOB$ is acute?