Olympiad problems

2019 Switzerland - Final Round, 7

Let $ABC$ be a triangle with $\angle CAB = 2 \angle ABC$. Assume that a point $D$ is inside the triangle $ABC$ exists such that $AD = BD$ and $CD = AC$. Show that $\angle ACB = 3 \angle DCB$.

2007 Alexandru Myller, 2

Tags: geometry , discrete geometry , angles , Plane Geometry , inequalities

$ n $ lines meet at a point. Each one of the $ 2n $ disjoint angles formed around this point by these lines has either $ 7^{\circ} $ or $ 17^{\circ} . $ [b]a)[/b] Find $ n. $ [b]b)[/b] Prove that among these lines there are at least two perpendicular ones.

V Soros Olympiad 1998 - 99 (Russia), 10.6

Tags: geometry , angles

The straight line containing the centers of the circumscribed and inscribed circles of triangle $ABC$ intersects rays $BA$ and $BC$ and forms an angle with the altitude to side $BC$ equal to half the angle $\angle BAC$. What is angle $\angle ABC$?

2018 EGMO, 5

Tags: geometry , Triangle , angles , EGMO , EGMO 2018 , geometry solved , power of a point

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 $\angle ABP = \angle QBC$.

Ukraine Correspondence MO - geometry, 2005.7

Tags: geometry , trapezoid , angles , right angle , Ukraine Correspondence

Let $O$ be the point of intersection of the diagonals of the trapezoid $ABCD$ with the bases $AB$ and $CD$. It is known that $\angle AOB = \angle DAB = 90^o$. On the sides $AD$ and $BC$ take the points $E$ and $F$ so that $EF\parallel AB$ and $EF = AD$. Find the angle $\angle AOE$.

2017 Auckland Mathematical Olympiad, 5

Tags: geometry , angles , orthocenter

The altitudes of triangle $ABC$ intersect at a point $H$.Find $\angle ACB$ if it is known that $AB = CH$.

1980 All Soviet Union Mathematical Olympiad, 298

Tags: geometry , Equilateral , angles

Given equilateral triangle $ABC$. Some line, parallel to $[AC]$ crosses $[AB]$ and $[BC]$ in $M$ and $P$ points respectively. Let $D$ be the centre of $PMB$ triangle, $E$ be the midpoint of the $[AP]$ segment. Find the angles of triangle $DEC$ .

Kyiv City MO 1984-93 - geometry, 1985.10.2

Tags: geometry , 3D geometry , cube , angles

Segment $AB$ on the surface of the cube is the shortest polyline on the surface that connects $A$ and $B$. Triangle $ABC$ consisted of such segments $AB, BC,CA$. What may be the sum of angles of such triangle if none of the vertex is on the edge of the cube ?

2010 Malaysia National Olympiad, 4

Tags: geometry , angles

In the diagram, $\angle AOB = \angle BOC$ and$\angle COD = \angle DOE = \angle EOF$. Given that $\angle AOD = 82^o$ and $\angle BOE = 68^o$. Find $\angle AOF$. [img]https://cdn.artofproblemsolving.com/attachments/b/2/deba6cd740adbf033ad884fff8e13cd21d9c5a.png[/img]

2007 Chile National Olympiad, 6

Tags: geometry , angles , Angle Chasing , circumcircle

Given an $\triangle ABC$ isoceles with base $BC$ we note with $M$ the midpoint of said base. Let $X$ be any point on the shortest arc $AM$ of the circumcircle of $\triangle ABM$ and let $T$ be a point on the inside $\angle BMA$ such that $\angle TMX = 90^o$ and $TX = BX$. Show that $\angle MTB - \angle CTM$ does not depend on $X$.

2011 Chile National Olympiad, 2

Tags: geometry , angles , circles , Tangents

Let $O$ be the center of the circle circumscribed to triangle $ABC$ and let $ S_ {A} $, $ S_ {B} $, $ S_ {C} $ be the circles centered on $O$ that are tangent to the sides $BC, CA, AB$ respectively. Show that the sum of the angle between the two tangents $ S_ {A} $ from $A$ plus the angle between the two tangents $ S_ {B} $ from $B$ plus the angle between the two tangents $ S_ {C} $ from $C$ is $180$ degrees.

1997 All-Russian Olympiad Regional Round, 11.7

Tags: geometry , 3D geometry , pyramid , combinatorics , combinatorial geometry , angles

Are there convex $n$-gonal ($n \ge 4$) and triangular pyramids such that the four trihedral angles of the $n$-gonal pyramid are equal trihedral angles of a triangular pyramid? [hide=original wording] Существуют ли выпуклая n-угольная (n>= 4) и треугольная пирамиды такие, что четыре трехгранных угла n-угольной пирамиды равны трехгранным углам треугольной пирамиды?[/hide]

2002 Estonia National Olympiad, 2

Tags: acute , obtuse , orthocenter , angles , geometry

Let $ABC$ be a non-right triangle with its altitudes intersecting in point $H$. Prove that $ABH$ is an acute triangle if and only if $\angle ACB$ is obtuse.

1999 Ukraine Team Selection Test, 7

Tags: combinatorics , combinatorial geometry , angles

Let $P_1P_2...P_n$ be an oriented closed polygonal line with no three segments passing through a single point. Each point $P_i$ is assinged the angle $180^o - \angle P_{i-1}P_iP_{i+1} \ge 0$ if $P_{i+1}$ lies on the left from the ray $P_{i-1}P_i$, and the angle $-(180^o -\angle P_{i-1}P_iP_{i+1}) < 0$ if $P_{i+1}$ lies on the right. Prove that if the sum of all the assigned angles is a multiple of $720^o$, then the number of self-intersections of the polygonal line is odd

Croatia MO (HMO) - geometry, 2013.7

Tags: geometry , angles , angle bisector , Angle Bisectors

In triangle $ABC$, the angle at vertex $B$ is $120^o$. Let $A_1, B_1, C_1$ be points on the sides $BC, CA, AB$ respectively such that $AA_1, BB_1, CC_1$ are bisectors of the angles of triangle $ABC$. Determine the angle $\angle A_1B_1C_1$.

2022 Junior Balkan Team Selection Tests - Moldova, 8

Tags: geometry , incenter , angles

Let $ABC$ be the triangle and $I$ the center of the circle inscribed in this triangle. The point $M$, located on the tangent taken to the point $B$ to the circumscribed circle of the triangle $ABC$, satisfies the relation $AB = MB$. Point $N$, located on the tangent taken to point $C$ to the same circle, satisfies the relation $AC = NC$. Points $M, A$ and $N$ lie on the same side of the line $BC$. Prove that $$\angle BAC + \angle MIN = 180^o.$$

2022 New Zealand MO, 1

Tags: geometry , rectangle , Equilateral , angles

$ABCD$ is a rectangle with side lengths $AB = CD = 1$ and $BC = DA = 2$. Let $ M$ be the midpoint of $AD$. Point $P$ lies on the opposite side of line $MB$ to $A$, such that triangle $MBP$ is equilateral. Find the value of $\angle PCB$.

2016 Portugal MO, 4

Tags: geometry , parallelogram , Tangents , equal angles , angles

Let $[ABCD]$ be a parallelogram with $AB <BC$ and let $E, F$ be points on the circle that passes through $A, B$ and $C$ such that $DE$ and $DF$ are tangents to this circle. Knowing that $\angle ADE = \angle CDF$ , determine $\angle ABC$. [img]https://cdn.artofproblemsolving.com/attachments/5/e/4140b92730e9d382df49ac05ca4e8ba48332dc.png[/img]

Champions Tournament Seniors - geometry, 2005.2

Tags: geometry , circumcircle , symmedian , angles , equal angles , Champions Tournament

Given a triangle $ABC$, the line passing through the vertex $A$ symmetric to the median $AM$ wrt the line containing the bisector of the angle $\angle BAC$ intersects the circle circumscribed around the triangle $ABC$ at points $A$ and $K$. Let $L$ be the midpoint of the segment $AK$. Prove that $\angle BLC=2\angle BAC$.

2021 Malaysia IMONST 1, 11

Tags: geometry , angles , circles

Given two points $ A$ and $ B$ and two circles, $\Gamma_1$ with center $A$ and passing through $ B$, and $\Gamma_2$ with center $ B$ and passing through $ A$. Line $AB$ meets $\Gamma_2$ at point $C$. Point $D$ lies on $\Gamma_2$ such that $\angle CDB = 57^o$. Line $BD$ meets $\Gamma_1$ at point $E$. What is $\angle CAE$, in degrees?

2021 Durer Math Competition Finals, 16

Tags: geometry , angles , arithmetic sequence

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 $a + b$?

2018 Denmark MO - Mohr Contest, 5

Tags: geometry , angles , angle bisector

In triangle $ABC$ the angular bisector from $A$ intersects the side $BC$ at the point $D$, and the angular bisector from $B$ intersects the side $AC$ at the point $E$. Furthermore $|AE| + |BD| = |AB|$. Prove that $\angle C = 60^o$ [img]https://1.bp.blogspot.com/-8ARqn8mLn24/XzP3P5319TI/AAAAAAAAMUQ/t71-imNuS18CSxTTLzYXpd806BlG5hXxACLcBGAsYHQ/s0/2018%2BMohr%2Bp5.png[/img]

Novosibirsk Oral Geo Oly VIII, 2017.7

Tags: geometry , angles

A car is driving along a straight highway at a speed of $60$ km per hour. Not far from the highway there is a parallel to him a $100$-meter fence. Every second, the passenger of the car measures the angle at which the fence is visible. Prove that the sum of all the angles he measured is less than $1100^o$

2017 Junior Balkan Team Selection Tests - Romania, 2

Tags: geometry , circles , circumcircle , angles

Let $A$ be a point outside the circle $\omega$ . The tangents from $A$ touch the circle at $B$ and $C$. Let $P$ be an arbitrary point on extension of $AC$ towards $C$, $Q$ the projection of $C$ onto $PB$ and $E$ the second intersection point of the circumcircle of $ABP$ with the circle $\omega$ . Prove that $\angle PEQ = 2\angle APB$

2019 District Olympiad, 3

Tags: 3D geometry , geometry , angles , parallelepiped

Consider the rectangular parallelepiped $ABCDA'B'C'D' $ as such the measure of the dihedral angle formed by the planes $(A'BD)$ and $(C'BD)$ is $90^o$ and the measure of the dihedral angle formed by the planes $(AB'C)$ and $(D'B'C)$ is $60^o$. Determine and measure the dihedral angle formed by the planes $(BC'D)$ and $(A'C'D)$.