Olympiad problems

2011 Romania National Olympiad, 4

Consider $\vartriangle ABC$ where $\angle ABC= 60 ^o$. Points $M$ and $D$ are on the sides $(AC)$, respectively $(AB)$, such that $\angle BCA = 2 \angle MBC$, and $BD = MC$. Determine $\angle DMB$.

Novosibirsk Oral Geo Oly VIII, 2022.4

Tags: geometry , angles

In triangle $ABC$, angle $C$ is three times the angle $A$, and side $AB$ is twice the side $BC$. What can be the angle $ABC$?

V Soros Olympiad 1998 - 99 (Russia), 9.4

Tags: geometry , angles

Let $ABC$ be a triangle without obtuse angles, $M$ the midpoint of $BC$, $K$ the midpoint of $BM$. What is the largest value of the angle $\angle KAM$?

2010 Sharygin Geometry Olympiad, 1

Tags: geometry , angle bisector , Angle Chasing , angles , altitude

For each vertex of triangle $ABC$, the angle between the altitude and the bisectrix from this vertex was found. It occurred that these angle in vertices $A$ and $B$ were equal. Furthermore the angle in vertex $C$ is greater than two remaining angles. Find angle $C$ of the triangle.

Estonia Open Junior - geometry, 2010.2.3

Tags: geometry , angles , fixed , Equilateral

On the side $BC$ of the equilateral triangle $ABC$, choose any point $D$, and on the line $AD$, take the point $E$ such that $| B A | = | BE |$. Prove that the size of the angle $AEC$ is of does not depend on the choice of point $D$, and find its size.

2020 Ukrainian Geometry Olympiad - April, 5

Tags: geometry , angles , geometric inequality

Inside the convex quadrilateral $ABCD$ there is a point $M$ such that $\angle AMB = \angle ADM + \angle BCM$ and $\angle AMD = \angle ABM + \angle DCM$. Prove that $AM \cdot CM + BM \cdot DM \ge \sqrt{AB \cdot BC\cdot CD \cdot DA}$

2024 Abelkonkurransen Finale, 4a

Tags: geometry , circumcircle , geometry proposed , angles

The triangle $ABC$ with $AB < AC$ has an altitude $AD$. The points $E$ and $A$ lie on opposite sides of $BC$, with $E$ on the circumcircle of $ABC$. Furthermore, $AD = DE$ and $\angle ADO=\angle CDE$, where $O$ is the circumcentre of $ABC$. Determine $\angle BAC$.

2020 Ukrainian Geometry Olympiad - April, 2

Tags: angles , acute , geometry

Inside the triangle $ABC$ is point $P$, such that $BP > AP$ and $BP > CP$. Prove that $\angle ABC$ is acute.

2008 Balkan MO Shortlist, G3

Tags: geometry , perimeter , angles , area of a triangle , bisector , orthocenter , IMO Shortlist

We draw two lines $(\ell_1) , (\ell_2)$ through the orthocenter $H$ of the triangle $ABC$ such that each one is dividing the triangle into two figures of equal area and equal perimeters. Find the angles of the triangle.

Kyiv City MO Juniors Round2 2010+ geometry, 2014.7.4

Tags: ratio , angles , median , geometry

The median $BM$ is drawn in the triangle $ABC$. It is known that $\angle ABM = 40 {} ^ \circ$ and $\angle CBM = 70 {} ^ \circ $ Find the ratio $AB: BM$.

2021 Malaysia IMONST 2, 1

Tags: geometry , circle , angles

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 $\angle DBE$.

1999 IMO Shortlist, 4

Tags: geometry , trigonometry , Triangle , angles , similarity , IMO Shortlist

For a triangle $T = ABC$ we take the point $X$ on the side $(AB)$ such that $AX/AB=4/5$, the point $Y$ on the segment $(CX)$ such that $CY = 2YX$ and, if possible, the point $Z$ on the ray ($CA$ such that $\widehat{CXZ} = 180 - \widehat{ABC}$. We denote by $\Sigma$ the set of all triangles $T$ for which $\widehat{XYZ} = 45$. Prove that all triangles from $\Sigma$ are similar and find the measure of their smallest angle.

2007 German National Olympiad, 4

Tags: arithmetic sequence , geometry , Triangle , angles

Find all triangles such that its angles form an arithmetic sequence and the corresponding sides form a geometric sequence.

2013 Tournament of Towns, 4

Tags: geometry , isosceles , equal segments , angles

Let $ABC$ be an isosceles triangle. Suppose that points $K$ and $L$ are chosen on lateral sides $AB$ and $AC$ respectively so that $AK = CL$ and $\angle ALK + \angle LKB = 60^o$. Prove that $KL = BC$.

1991 IMO, 2

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 }$.

2018 Flanders Math Olympiad, 1

Tags: geometry , angles , Geometric Inequalities

In the triangle $\vartriangle ABC$ we have $| AB |^3 = | AC |^3 + | BC |^3$. Prove that $\angle C> 60^o$ .

1980 Tournament Of Towns, (006) 3

Tags: vector , combinatorial geometry , combinatorics , angles , geometry

We are given $30$ non-zero vectors in $3$ dimensional space. Prove that among these there are two such that the angle between them is less than $45^o$.

2016 Irish Math Olympiad, 2

Tags: geometry , angles , Angle Chasing , angle bisector

In triangle $ABC$ we have $|AB| \ne |AC|$. The bisectors of $\angle ABC$ and $\angle ACB$ meet $AC$ and $AB$ at $E$ and $F$, respectively, and intersect at I. If $|EI| = |FI|$ find the measure of $\angle BAC$.

2019 Polish Junior MO First Round, 2

Tags: geometry , angles

A convex quadrilateral $ABCD$ is given in which $\angle DAB = \angle ABC = 45^o$ and $DA = 3$, $AB = 7\sqrt2$, $BC = 4$. Calculate the length of side $CD$. [img]https://cdn.artofproblemsolving.com/attachments/1/2/046e31a628b3df4d23d3162cb570e1b9cb71e2.png[/img]

2003 Abels Math Contest (Norwegian MO), 3

Tags: angles , geometry

Let $ABC$ be a triangle with $AC> BC$, and let $S$ be the circumscribed circle of the triangle. $AB$ divides $S$ into two arcs. Let $D$ be the midpoint of the arc containing $C$. (a) Show that $\angle ACB +2 \cdot \angle ACD = 180^o$. (b) Let $E$ be the foot of the altitude from $D$ on $AC$. Show that $BC +CE = AE$.

1970 Kurschak Competition, 1

Tags: combinatorics , combinatorial geometry , angles

What is the largest possible number of acute angles in an $n$-gon which is not selfintersecting (no two non-adjacent edges interesect)?

2002 Junior Balkan Team Selection Tests - Moldova, 10

Tags: geometry , equal angles , circles , angles

The circles $C_1$ and $C_2$ intersect at the distinct points $M$ and $N$. Points $A$ and $B$ belong respectively to the circles $C_1$ and $C_2$ so that the chords $[MA]$ and $[MB]$ are tangent at point $M$ to the circles $C_2$ and $C_1$, respectively. To prove it that the angles $\angle MNA$ and $\angle MNB$ are equal.

2003 District Olympiad, 4

Tags: geometry , 3D geometry , tetrahedron , angles , equal angles

a) Let $MNP$ be a triangle such that $\angle MNP> 60^o$. Show that the side $MP$ cannot be the smallest side of the triangle $MNP$. b) In a plane the equilateral triangle $ABC$ is considered. The point $V$ that does not belong to the plane $(ABC)$ is chosen so that $\angle VAB = \angle VBC = \angle VCA$. Show that if $VA = AB$, the tetrahedron $VABC$ is regular. Valentin Vornicu

2020 Estonia Team Selection Test, 2

Tags: combinatorics , geometry , combinatorial geometry , angles , min

Let $n$ be an integer, $n \ge 3$. Select $n$ points on the plane, none of which are three on the same line. Consider all triangles with vertices at selected points, denote the smallest of all the interior angles of these triangles by the variable $\alpha$. Find the largest possible value of $\alpha$ and identify all the selected $n$ point placements for which the max occurs.

2005 Sharygin Geometry Olympiad, 7

Tags: Equilateral , Equilateral Triangle , concentric circles , geometry , angles

Two circles with radii $1$ and $2$ have a common center at the point $O$. The vertex $A$ of the regular triangle $ABC$ lies on the larger circle, and the middpoint of the base $CD$ lies on the smaller one. What can the angle $BOC$ be equal to?