Found problems: 721
How many different (non similar) triangles are there whose angles have an integer number of degrees?
Suppose that tetrahedron $ABCD$ satisfies $\angle BAC+\angle CAD+\angle DAB = \angle ABC+\angle CBD+\angle DBA = 180^o$. Prove that $CD \ge AB$.
Let $A, B, C$ be unique collinear points$ AB = BC =\frac13$. Let $P$ be a point that lies on the circle centered at $B$ with radius $\frac13$ and the circle centered at $C$ with radius $\frac13$ . Find the measure of angle $\angle PAC$ in degrees.
The angles of a convex quadrilateral form an arithmetic sequence in clockwise order, and its side lengths also form an arithmetic sequence (but not necessarily in clockwise order). If the quadrilateral is not a square, and its shortest side has length $1$, then its perimeter is $a + \sqrt{b}4$, where $ a$ and $b$ are positive integers. What is the value of $a + b$?
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.
(i) $ABC$ is a triangle with a right angle at $A$, and $P$ is a point on the hypotenuse $BC$.
The line $AP$ produced beyond $P$ meets the line through $B$ which is perpendicular to $BC$ at $U$.
Prove that $BU = BA$ if, and only if, $CP = CA$.
(ii) $A$ is a point on the semicircle $CB$, and points $X$ and $Y$ are on the line segment $BC$.
The line $AX$, produced beyond $X$, meets the line through $B$ which is perpendicular to $BC$ at $U$.
Also the line $AY$, produced beyond $Y$, meets the line through $C$ which is perpendicular to $BC$ at $V$.
Given that $BY = BA$ and $CX = CA$, determine the angle $\angle VAU$.
In the isosceles triangle $ABC$, with $AB = AC$, the angle bisector of $\angle B$ intersects side $AC$ at $B'$. Suppose that $ B B' + B'A = BC$. Find the angles of the triangle.
A non isosceles triangle $ABC$ is given, in which $\angle A = 120^o$. Let $AL$ be its angle bisector, $AK$ be it's median, drawn from vertex $A$, point $O$ be the center of the circumcircle of this triangle, $F$ be the point of intersection of the lines $OL$ and $AK$. Prove that $\angle BFC = 60^o$.
There are three squares in the picture. Find the sum of angles $ADC$ and $BDC$.
[img]https://cdn.artofproblemsolving.com/attachments/c/9/885a6c6253fca17e24528f8ba8a5d31a18c845.png[/img]
Given a circle with center $O$. Points $A$ and $B$ lie on the circle such that triangle $OBA$ is equilateral. Let $C$ be a point outside the circle with $\angle ACB = 45^{\circ}$. Line $CA$ intersects the circle at point $D$, and the line $CB$ intersects the circle at point $E$. Find $\angle DBE$.
Let $AH$ be an altitude of an equilateral triangle $ABC$. Let $I$ be the incentre of triangle $ABH$, and let $L, K$ and $J$ be the incentres of triangles $ABI,BCI$ and $CAI$ respectively. Determine $\angle KJL$.
Let $ABCD$ be an isosceles trapezoid ($AD\parallel BC$), $\angle BAD = 80^o$, $\angle BDA = 60^o$. Point $P$ lies on $CD$ and $\angle PAD = 50^o$. Find $\angle PBC$
Let $ABC$ be a right triangle with right angle at $ B$ and $\angle C=30^o$. If $M$ is midpoint of the hypotenuse and $I$ the incenter of the triangle, show that $ \angle IMB=15^o$.
In rectangle $ABCD$, point $E$ is chosen on $AB$ and $F$ is the foot of $E$ onto side $CD$ such that the circumcircle of $\vartriangle ABF$ intersects line segments $AD$ and $BC$ at points $G$ and $H$ respectively. Let $S$ be the intersection of $EF$ and $GH$, and $T$ the intersection of lines $EC$ and $DS$. If $\angle SF T = 15^o$ , compute the measure of $\angle CSD$.
The points $A \ne B$ are given on the plane. The point $C$ moves along the plane in such a way that $\angle ACB = \alpha$ , where $\alpha$ is the fixed angle from the interval ($0^o, 180^o$). The circle inscribed in triangle $ABC$ has center the point $I$ and touches the sides $AB, BC, CA$ at points $D, E, F$ accordingly. Rays $AI$ and $BI$ intersect the line $EF$ at points $M$ and $N$, respectively. Show that:
a) the segment $MN$ has a constant length,
b) all circles circumscribed around triangle $DMN$ have a common point
A circle with center $O$ passes through the vertices $A$ and $C$ of the triangle $ABC$ and intersects the segments $AB$ and $BC$ again at distinct points $K$ and $N$ respectively. Let $M$ be the point of intersection of the circumcircles of triangles $ABC$ and $KBN$ (apart from $B$). Prove that $\angle OMB=90^{\circ}$.
Is there a moment in a day when three hands - hour, minute and second - of a clock running correctly form angles of $120^o$ in pairs?
Point $D$ is marked inside a triangle $ABC$ so that $\angle ADC = \angle ABC + 60^o$, $\angle CDB =\angle CAB + 60^o$, $\angle BDA = \angle BCA + 60^o$. Prove that $AB \cdot CD = BC \cdot AD = CA \cdot BD$.
(A. Levin)
Points $A, B, C, D, E$ and F are given on a circle in such a way that the three chords $AB, CD$ and $EF$ intersect in one point. Express angle $\angle EFA$ in terms of angles $\angle ABC$ and $\angle CDE$ (find all possibilities).
In a triangle $ABC$ we have $|AB| = |AC|$ and $\angle BAC = \alpha$. Let $P \ne B$ be a point on $AB$ and $Q$ a point on the altitude drawn from $A$ such that $|PQ| = |QC|$. Find $ \angle QPC$.
In quadrilateral ABCD it is given that $AB = BC = 1, \angle ABC = 100^o$ , and $\angle CDA = 130^o$ . Find the length of $BD$.
Triangle $ABC$ is right-angled and isosceles with a right angle at the vertex $C$. On rays $CB$ on vertex $B$ is selected point F, on rays $BA$ on vertex $A$ is selected point G so that $AG = BF.$ The ray $GD$ is drawn so that it intersects with ray $AC$ at point $D$ with $\angle FGD = 45^o$. Find $\angle FDG$.
(Bogdan Rublev)
The vertices of Durer's favorite regular decagon in clockwise order: $D_1, D_2, D_3, . . . , D_{10}$. What is the angle between the diagonals $D_1D_3$ and $D_2D_5$?
In an isosceles triangle $\vartriangle ABC, \angle ABC = 20^o$ and $BC = AB$. Points $P$ and $Q$ are chosen on sides $BC$ and $AB$, respectively, so that $\angle PAC = 50^o$ and $\angle QCA = 60^o$ . Prove that $\angle PQC = 30^o$ .
On a circle with center $O$ and radius $r$ are given points $A,B,C,D$ in this order such that $AB, BC$ and $CD$ have the same length $s$ and the length of $AD$ is $s+ r$.Assume that $s < r$. Determine the angles of quadrilateral $ABCD$.