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: 628

2020 Yasinsky Geometry Olympiad, 2
Tags: geometry , angles , square , Equilateral
An equilateral triangle $BDE$ is constructed on the diagonal $BD$ of the square $ABCD$, and the point $C$ is located inside the triangle $BDE$. Let $M$ be the midpoint of $BE$. Find the angle between the lines $MC$ and $DE$. (Dmitry Shvetsov)

2020 Ukrainian Geometry Olympiad - December, 5
Tags: geometry , angles , altitudes , circumcircle
In an acute triangle $ABC$ with an angle $\angle ACB =75^o$, altitudes $AA_3,BB_3$ intersect the circumscribed circle at points $A_1,B_1$ respectively. On the lines $BC$ and $CA$ select points $A_2$ and $B_2$ respectively suchthat the line $B_1B_2$ is parallel to the line $BC$ and the line $A_1A_2$ is parallel to the line $AC$ . Let $M$ be the midpoint of the segment $A_2B_2$. Find in degrees the measure of the angle $\angle B_3MA_3$.

2013 Tournament of Towns, 5
Tags: geometry , angles , equal segments , right angle
In a quadrilateral $ABCD$, angle $B$ is equal to $150^o$, angle $C$ is right, and sides $AB$ and $CD$ are equal. Determine the angle between $BC$ and the line connecting the midpoints of sides $BC$ and $AD$.

2019 Romania National Olympiad, 3
Tags: angles , geometry , equal segments , equal angles
Let $ABC$ be a triangle in which $\angle ABC = 45^o$ and $\angle BAC > 90^o$. Let $O$ be the midpoint of the side $[BC]$. Consider the point $M \in (AC)$ such that $\angle COM =\angle CAB$. Perpendicular from $M$ on $AC$ intersects line $AB$ at point $P$. a) Find the measure of the angle $\angle BCP$. b) Show that if $\angle BAC = 105^o$, then $PB = 2MO$.

2021 Swedish Mathematical Competition, 1
Tags: geometry , angles , arithmetic sequence
In a triangle, both the sides and the angles form arithmetic sequences. Determine the angles of the triangle.

2006 All-Russian Olympiad Regional Round, 9.6
Tags: geometry , angles , geometric inequality
In an acute triangle $ABC$, the angle bisector$AD$ and altitude $BE$ are drawn. Prove that angle $CED$ is greater than $45^o$.

Kyiv City MO 1984-93 - geometry, 1984.7.3
Tags: geometry , angles
On the extension of the largest side $AC$ of the triangle $ABC$ set aside the segment $CM$ such that $CM = BC$. Prove that the angle $ABM$ is obtuse or right.

1993 Tournament Of Towns, (390) 2
Tags: geometry , angles , right triangle
Points $M$ and $N$ are taken on the hypotenuse $AB$ of a right triangle $ABC$ so that $BC = BM$ and $AC = AN$. Prove that the angle $MCN$ is equal to $45$ degrees. (Folklore)

2017 Junior Balkan Team Selection Tests - Romania, 4
Tags: Equilateral , geometry , angles , equal angles
Let $ABC$ be a right triangle, with the right angle at $A$. The altitude from $A$ meets $BC$ at $H$ and $M$ is the midpoint of the hypotenuse $[BC]$. On the legs, in the exterior of the triangle, equilateral triangles $BAP$ and $ACQ$ are constructed. If $N$ is the intersection point of the lines $AM$ and $PQ$, prove that the angles $\angle NHP$ and $\angle AHQ$ are equal. Miguel Ochoa Sanchez and Leonard Giugiuc

1997 Denmark MO - Mohr Contest, 3
Tags: geometry , angles , pentagon , reflection
About pentagon $ABCDE$ is known that angle $A$ and angle $C$ are right and that the sides $| AB | = 4$, $| BC | = 5$, $| CD | = 10$, $| DE | = 6$. Furthermore, the point $C'$ that appears by mirroring $C$ in the line $BD$, lies on the line segment $AE$. Find angle $E$.

Durer Math Competition CD Finals - geometry, 2019.D3
Tags: geometry , equal segments , equal angles , angles
a) Does there exist a quadrilateral with (both of) the following properties: three of its edges are of the same length, but the fourth one is different, and three of its angles are equal, but the fourth one is different? b) Does there exist a pentagon with (both of) the following properties: four of its edges are of the same length, but the fifth one is different, and four of its angles are equal, but the fifth one is different?

2011 Tournament of Towns, 5
Tags: Sum , combinatorial geometry , combinatorics , lines , angles
In the plane are $10$ lines in general position, which means that no $2$ are parallel and no $3$ are concurrent. Where $2$ lines intersect, we measure the smaller of the two angles formed between them. What is the maximum value of the sum of the measures of these $45$ angles?

2000 Chile National Olympiad, 4
Tags: geometry , angles
Let $ AD $ be the bisector of a triangle $ ABC $ $ (D \in BC) $ such that $ AB + AD = CD $ and $ AC + AD = BC $. Determine the measure of the angles of $ \vartriangle ABC $

2006 Cuba MO, 6
Tags: geometry , angles , concentric
Two concentric circles of radii $1$ and $2$ have centere the point $O$. The vertex $A$ of the equilateral triangle $ABC$ lies at the largest circle, while the midpoint of side $BC$ lies on the smaller circle. If$ B$,$O$ and $C$ are not collinear, what measure can the angle $\angle BOC$ have?

2017 Estonia Team Selection Test, 10
Tags: semicircle , geometry , angles
Let $ABC$ be a triangle with $AB = \frac{AC}{2 }+ BC$. Consider the two semicircles outside the triangle with diameters $AB$ and $BC$. Let $X$ be the orthogonal projection of $A$ onto the common tangent line of those semicircles. Find $\angle CAX$.

2017 Rioplatense Mathematical Olympiad, Level 3, 5
Tags: geometry , angles
Let $ABC$ be a triangle and $I$ is your incenter, let $P$ be a point in $AC$ such that $PI$ is perpendicular to $AC$, and let $D$ be the reflection of $B$ wrt circumcenter of $\triangle ABC$. The line $DI$ intersects again the circumcircle of $\triangle ABC$ in the point $Q$. Prove that $QP$ is the angle bisector of the angle $\angle AQC$.

Novosibirsk Oral Geo Oly IX, 2021.5
Tags: geometry , pentagon , angles
The pentagon $ABCDE$ is inscribed in the circle. Line segments $AC$ and $BD$ intersect at point $K$. Line segment $CE$ touches the circumcircle of triangle $ABK$ at point $N$. Find the angle $CNK$ if $\angle ECD = 40^o.$

2010 Flanders Math Olympiad, 3
Tags: geometry , angle bisector , parallel , perpendicular , angles
In a triangle $ABC$, $\angle B= 2\angle A \ne 90^o$ . The inner bisector of $B$ intersects the perpendicular bisector of $[AC]$ at a point $D$. Prove that $AB \parallel CD$.

2019 Yasinsky Geometry Olympiad, p1
Tags: angles , geometry , inradius , Angle Chasing
It is known that in the triangle $ABC$ the distance from the intersection point of the angle bisector to each of the vertices of the triangle does not exceed the diameter of the circle inscribed in this triangle. Find the angles of the triangle $ABC$. (Grigory Filippovsky)

Ukrainian From Tasks to Tasks - geometry, 2013.13
Tags: angles , Circumcenter , geometry
In the quadrilateral $ABCD$ it is known that $ABC + DBC = 180^o$ and $ADC + BDC = 180^o$. Prove that the center of the circle circumscribed around the triangle $BCD$ lies on the diagonal $AC$.

1991 IMO Shortlist, 4
Tags: geometry , angles , geometric inequality , Triangle , Brocard , IMO , imo 1991
Let $ \,ABC\,$ be a triangle and $ \,P\,$ an interior point of $ \,ABC\,$. Show that at least one of the angles $ \,\angle PAB,\;\angle PBC,\;\angle PCA\,$ is less than or equal to $ 30^{\circ }$.

2022 Mexican Girls' Contest, 2
Tags: geometry , isosceles , Mexico , angles
Consider $\triangle ABC$ an isosceles triangle such that $AB = BC$. Let $P$ be a point satisfying $$\angle ABP = 80^\circ, \angle CBP = 20^\circ, \textrm{and} \hspace{0.17cm} AC = BP$$ Find all possible values of $\angle BCP$.

1999 May Olympiad, 2
Tags: geometry , parallelogram , pentagon , angles
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.

2019 Yasinsky Geometry Olympiad, p3
Tags: geometry , cyclic quadrilateral , Angle Chasing , angles
In the quadrilateral $ABCD$, the angles $B$ and $D$ are right . The diagonal $AC$ forms with the side $AB$ the angle of $40^o$, as well with side $AD$ an angle of $30^o$. Find the acute angle between the diagonals $AC$ and $BD$.

2009 IMAR Test, 3
Tags: geometry , Angle Chasing , convex quadrilateral , angles , equal angles
Consider a convex quadrilateral $ABCD$ with $AB=CB$ and $\angle ABC +2 \angle CDA = \pi$ and let $E$ be the midpoint of $AC$. Show that $\angle CDE =\angle BDA$. Paolo Leonetti