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: 721

1987 Tournament Of Towns, (151) 2
Tags: rhombus , locus , angle , geometry
Find the locus of points $M$ inside the rhombus $ABCD$ such that the sum of angles $AMB$ and $CMD$ equals $180^o$ .

2009 Singapore Junior Math Olympiad, 1
Tags: angle , geometry
In $\vartriangle ABC, \angle A= 2 \angle B$. Let $a,b,c$ be the lengths of its sides $BC,CA,AB$, respectively. Prove that $a^2 = b(b + c)$.

2011 Bundeswettbewerb Mathematik, 1
Tags: square , hexagon , tiling , combinatorial geometry , tiles , angle , geometry
Prove that you can't split a square into finitely many hexagons, whose inner angles are all less than $180^o$.

2017 BMT Spring, 17
Tags: geometry , angle
Triangle $ABC$ is drawn such that $\angle A = 80^o$, $\angle B = 60^o$, and $\angle C = 40^o$. Let the circumcenter of $\vartriangle ABC$ be $O$, and let $\omega$ be the circle with diameter $AO$. Circle $\omega$ intersects side $AC$ at point $P$. Let M be the midpoint of side $BC$, and let the intersection of $\omega$ and $PM$ be $K$. Find the measure of $\angle MOK$.

2016 Saudi Arabia IMO TST, 2
Tags: convex , hexagon , equal segments , equal angles , geometry , angle
Let $ABCDEF$ be a convex hexagon with $AB = CD = EF$, $BC =DE = FA$ and $\angle A+\angle B = \angle C +\angle D = \angle E +\angle F$. Prove that $\angle A=\angle C=\angle E$ and $\angle B=\angle D=\angle F$. Tran Quang Hung

2008 Hanoi Open Mathematics Competitions, 7
Tags: geometry , convex , pentagon , angle
The figure $ABCDE$ is a convex pentagon. Find the sum $\angle DAC + \angle EBD +\angle ACE +\angle BDA + \angle CEB$?

2018 EGMO, 5
Tags: triangle , power of a point , angle , geometry
Let $\Gamma $ be the circumcircle of triangle $ABC$. A circle $\Omega$ is tangent to the line segment $AB$ and is tangent to $\Gamma$ at a point lying on the same side of the line $AB$ as $C$. The angle bisector of $\angle BCA$ intersects $\Omega$ at two different points $P$ and $Q$. Prove that $\angle ABP = \angle QBC$.

1980 IMO Shortlist, 15
Tags: geometry , 3d geometry , tetrahedron , sphere , triangle inequality , angle
Prove that the sum of the six angles subtended at an interior point of a tetrahedron by its six edges is greater than 540°.

Kyiv City MO Seniors 2003+ geometry, 2020.10.5.1
Tags: geometry , semicircle , equal angles , angle
Let $\Gamma$ be a semicircle with diameter $AB$. On this diameter is selected a point $C$, and on the semicircle are selected points $D$ and $E$ so that $E$ lies between $B$ and $D$. It turned out that $\angle ACD = \angle ECB$. The intersection point of the tangents to $\Gamma$ at points $D$ and $E$ is denoted by $F$. Prove that $\angle EFD=\angle ACD+ \angle ECB$.

1975 IMO Shortlist, 8
Tags: geometry , trigonometry , triangle , angle
In the plane of a triangle $ABC,$ in its exterior$,$ we draw the triangles $ABR, BCP, CAQ$ so that $\angle PBC = \angle CAQ = 45^{\circ}$, $\angle BCP = \angle QCA = 30^{\circ}$, $\angle ABR = \angle RAB = 15^{\circ}$. Prove that [b]a.)[/b] $\angle QRP = 90\,^{\circ},$ and [b]b.)[/b] $QR = RP.$

1996 Estonia National Olympiad, 3
Tags: diagonal , geometry , cyclic quadrilateral , ratio , angle
The vertices of the quadrilateral $ABCD$ lie on a single circle. The diagonals of this rectangle divide the angles of the rectangle at vertices $A$ and $B$ and divides the angles at vertices $C$ and $D$ in a $1: 2$ ratio. Find angles of the quadrilateral $ABCD$.

2022 Yasinsky Geometry Olympiad, 1
Tags: geometry , median , angle
In the triangle $ABC$, the median $AM$ is extended to the intersection with the circumscribed circle at point $D$. It is known that $AB = 2AM$ and $AD = 4AM$. Find the angles of the triangle $ABC$. (Gryhoriy Filippovskyi)

2012 Cuba MO, 9
Tags: geometry , angle
Let $O$ be a point interior to triangle $ABC$ such that $\angle BAO = 30^o$, $\angle CBO = 20^o$ and $\angle ABO = \angle ACO = 40^o$ . Knowing that triangle $ABC$ is not equilateral, find the measures of its interior angles.

2014 Romania National Olympiad, 2
Tags: geometry , 3d geometry , cube , angle , distance
Let $ABCDA'B'C'D'$ be a cube with side $AB = a$. Consider points $E \in (AB)$ and $F \in (BC)$ such that $AE + CF = EF$. a) Determine the measure the angle formed by the planes $(D'DE)$ and $(D'DF)$. b) Calculate the distance from $D'$ to the line $EF$.

2021 Novosibirsk Oral Olympiad in Geometry, 3
Tags: geometry , angle
Find the angle $BCA$ in the quadrilateral of the figure. [img]https://cdn.artofproblemsolving.com/attachments/0/2/974e23be54125cde8610a78254b59685833b5b.png[/img]

2007 Postal Coaching, 1
Tags: geometry , isosceles , equal angles , angle
Let $ABC$ be an isosceles triangle with $AC = BC$, and let $M$ be the midpoint of $AB$. Let $P$ be a point inside the triangle such that $\angle PAB =\angle PBC$. Prove that $\angle APM + \angle BPC = 180^o$.

2021 Novosibirsk Oral Olympiad in Geometry, 2
Tags: geometry , angle
The extensions of two opposite sides of the convex quadrilateral intersect and form an angle of $20^o$ , the extensions of the other two sides also intersect and form an angle of $20^o$. It is known that exactly one angle of the quadrilateral is $80^o$. Find all of its other angles.

Ukrainian From Tasks to Tasks - geometry, 2012.4
Tags: geometry , trapezoid , angle
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 $\angle PBC$

May Olympiad L1 - geometry, 1999.2
Tags: geometry , parallelogram , pentagon , angle
In a parallelogram $ABCD$ , $BD$ is the largest diagonal. By matching $B$ with $D$ by a bend, a regular pentagon is formed. Calculate the measures of the angles formed by the diagonal $BD$ with each of the sides of the parallelogram.

Estonia Open Junior - geometry, 2019.2.1
Tags: geometry , pentagon , equilateral , angle
A pentagon can be divided into equilateral triangles. Find all the possibilities that the sizes of the angles of this pentagon can be.

1996 Swedish Mathematical Competition, 4
Tags: pentagon , increasing , cyclic , geometry , angle
The angles at $A,B,C,D,E$ of a pentagon $ABCDE$ inscribed in a circle form an increasing sequence. Show that the angle at $C$ is greater than $\pi/2$, and that this lower bound cannot be improved.

2003 Junior Balkan Team Selection Tests - Moldova, 7
Tags: isosceles , equal segments , angle , geometry
The triangle $ABC$ is isosceles with $AB=BC$. The point F on the side $[BC]$ and the point $D$ on the side $AC$ are the feets of the the internals bisectors drawn from $A$ and altitude drawn from $B$ respectively so that $AF=2BD$. Fine the measure of the angle $ABC$.

2017 Flanders Math Olympiad, 2
Tags: geometry , circumcircle , angle chasing , angle
In triangle $\vartriangle ABC$, $\angle A = 50^o, \angle B = 60^o$ and $\angle C = 70^o$. The point $P$ is on the side $[AB]$ (with $P \ne A$ and $P \ne B$). The inscribed circle of $\vartriangle ABC$ intersects the inscribed circle of $\vartriangle ACP$ at points $U$ and $V$ and intersects the inscribed circle of $\vartriangle BCP$ at points $X$ and $Y$. The rights $UV$ and $XY$ intersect in $K$. Calculate the $\angle UKX$.

2011 Saudi Arabia Pre-TST, 3.3
Tags: isosceles , angle , geometry
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.

Russian TST 2018, P2
Tags: geometry , angle
Inside the acute-angled triangle $ABC$, the points $P{}$ and $Q{}$ are chosen so that $\angle ACP = \angle BCQ$ and $\angle CBP =\angle ABQ$. The point $Z{}$ is the projection of $P{}$ onto the line $BC$. The point $Q'$ is symmetric to $Q{}$ with respect to $Z{}$. The points $K{}$ and $L{}$ are chosen on the rays $AB$ and $AC$ respectively, so that $Q'K \parallel QC$ and $Q'L \parallel QB$. Prove that $\angle KPL=\angle BPC$.