Found problems: 44
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.
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.
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)
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.)
Find all possibilities: how many acute angles can there be in a convex polygon?
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.
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.
Cut a square into eight acute-angled triangles.
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.
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.
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.
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 $\triangle BDE$.
Prove that from any five points in the plane it is possible to choose three points that are not vertices of an acute triangle.
Prove that a convex polygon can be cut by disjoint diagonals into acute triangles in at least one way.
$2$ 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?
Given $100$ coplanar points, no three collinear, prove that at most $70\%$ of the triangles formed by the points have all angles acute.
Inside the triangle $ABC$ is point $P$, such that $BP > AP$ and $BP > CP$. Prove that $\angle ABC$ is acute.
A regular $1997$-gon is divided into triangles by non-intersecting diagonals. Prove that exactly one of them is acute-angled.
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.
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.
* 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.
The square was cut into acute -angled triangles. Prove that there are at least eight of them.
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.
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.
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.