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

1976 Chisinau City MO, 123
Tags: geometry , acute , combinatorics , combinatorial geometry
Five points are given on the plane. Prove that among all the triangles with vertices at these points there are no more than seven acute-angled ones.

1970 IMO, 3
Tags: counting , acute , triangle , point set , 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.

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)

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

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

1963 Kurschak Competition, 2
Tags: inequalities , trigonometry , acute , algebra
$A$ is an acute angle. Show that $$\left(1 +\frac{1}{sen A}\right)\left(1 +\frac{1}{cos A}\right)> 5$$

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

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.

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

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.

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} $

1954 Moscow Mathematical Olympiad, 270
Tags: acute , geometry
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.

1970 IMO Longlists, 58
Tags: combinatorial geometry , acute , triangle , point set , counting
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?

2010 All-Russian Olympiad Regional Round, 11.5
Tags: inequalities , geometry , acute , trigonometry
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 trĪ±iangle is acute-angled.

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

2003 All-Russian Olympiad Regional Round, 9.8
Tags: combinatorial geometry , geometry , combinatorics , convex , convex polygon , acute
Prove that a convex polygon can be cut by disjoint diagonals into acute triangles in at least one way.

2020 Ukrainian Geometry Olympiad - December, 2
Tags: combinatorics , combinatorial geometry , acute , 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.

2002 Estonia National Olympiad, 2
Tags: geometry , angle , acute , obtuse , orthocenter
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.