Olympiad problems

1977 Chisinau City MO, 137

Determine the angles of a triangle in which the median, bisector and altitude, drawn from one vertex, divide this angle into four equal parts.

2019 Yasinsky Geometry Olympiad, p4

Tags: geometry , cyclic quadrilateral , Angle Chasing , angles

Find the angles of the cyclic quadrilateral if you know that each of its diagonals is a bisector of one angle and a trisector of the opposite one (the trisector of the angle is one of the two rays that lie in the interior of the angle and divide it into three equal parts). (Vyacheslav Yasinsky)

2020 Iranian Geometry Olympiad, 2

Tags: rectangle , circumcircle , angles , Triangle , geometry , IGO

Let $ABC$ be an isosceles triangle ($AB = AC$) with its circumcenter $O$. Point $N$ is the midpoint of the segment $BC$ and point $M$ is the reflection of the point $N$ with respect to the side $AC$. Suppose that $T$ is a point so that $ANBT$ is a rectangle. Prove that $\angle OMT = \frac{1}{2} \angle BAC$. [i]Proposed by Ali Zamani[/i]

2021 Girls in Mathematics Tournament, 2

Tags: geometry , angles

Let $\vartriangle ABC$ be a triangle in which $\angle ACB = 40^o$ and $\angle BAC = 60^o$ . Let $D$ be a point inside the segment $BC$ such that $CD =\frac{AB}{2}$ and let $M$ be the midpoint of the segment $AC$. How much is the angle $\angle CMD$ in degrees?

Kyiv City MO Juniors 2003+ geometry, 2020.8.5

Tags: geometry , circumcircle , angles

Given a triangle $ABC, O$ is the center of the circumcircle, $M$ is the midpoint of $BC, W$ is the second intersection of the bisector of the angle $C$ with this circle. A line parallel to $BC$ passing through $W$, intersects$ AB$ at the point $K$ so that $BK = BO$. Find the measure of angle $WMB$. (Anton Trygub)

2015 Danube Mathematical Competition, 4

Tags: geometry , rectangle , perpendicular bisector , angles , square

Let $ABCD$ be a rectangle with $AB\ge BC$ Point $M$ is located on the side $(AD)$, and the perpendicular bisector of $[MC]$ intersects the line $BC$ at the point $N$. Let ${Q} =MN\cup AB$ . Knowing that $\angle MQA= 2\cdot \angle BCQ $, show that the quadrilateral $ABCD$ is a square.

1967 IMO Longlists, 25

Tags: geometry , 3D geometry , sphere , angles , IMO Shortlist , IMO Longlist

Three disks of diameter $d$ are touching a sphere in their centers. Besides, every disk touches the other two disks. How to choose the radius $R$ of the sphere in order that axis of the whole figure has an angle of $60^\circ$ with the line connecting the center of the sphere with the point of the disks which is at the largest distance from the axis ? (The axis of the figure is the line having the property that rotation of the figure of $120^\circ$ around that line brings the figure in the initial position. Disks are all on one side of the plane, passing through the center of the sphere and orthogonal to the axis).

2015 Sharygin Geometry Olympiad, P4

Tags: parallelogram , geometry , angles , Equilateral Triangle

In a parallelogram $ABCD$ the trisectors of angles $A$ and $B$ are drawn. Let $O$ be the common points of the trisectors nearest to $AB$. Let $AO$ meet the second trisector of angle $B$ at point $A_1$, and let $BO$ meet the second trisector of angle $A$ at point $B_1$. Let $M$ be the midpoint of $A_1B_1$. Line $MO$ meets $AB$ at point $N$ Prove that triangle $A_1B_1N$ is equilateral.

Novosibirsk Oral Geo Oly VIII, 2023.4

Tags: geometry , equal segments , angles , isosceles

An isosceles triangle $ABC$ with base $AC$ is given. On the rays $CA$, $AB$ and $BC$, the points $D, E$ and $F$ were marked, respectively, in such a way that $AD = AC$, $BE = BA$ and $CF = CB$. Find the sum of the angles $\angle ADB$, $\angle BEC$ and $\angle CFA$.

2007 Sharygin Geometry Olympiad, 1

Tags: geometry , Equilateral , Equilateral Triangle , isosceles , Isosceles Triangle , angles

A triangle is cut into several (not less than two) triangles. One of them is isosceles (not equilateral), and all others are equilateral. Determine the angles of the original triangle.

1986 All Soviet Union Mathematical Olympiad, 440

Tags: geometry , 3D geometry , sphere , tetrahedron , fixed , angles , Sum

Consider all the tetrahedrons $AXBY$, circumscribed around the sphere. Let $A$ and $B$ points be fixed. Prove that the sum of angles in the non-plane quadrangle $AXBY$ doesn't depend on points $X$ and $Y$ .

Ukraine Correspondence MO - geometry, 2003.8

Tags: geometry , angles

In the triangle $ABC$, $D$ is the midpoint of $AB$, and $E$ is the point on the side $BC$, for which $CE = \frac13 BC$. It is known that $\angle ADC =\angle BAE$. Find $\angle BAC$.

1986 Greece Junior Math Olympiad, 3

Tags: geometry , angles

Inside a triangle ABC, consider points $D, E$ such that $\angle ABD =\angle DBE=\angle EBC$ and $\angle ACD=\angle DC E=\angle ECB$. Calculate angles $\angle BDE$, $\angle B EC$, $\angle D E C$ in terms of the angle of the triangle $ABC$.

Brazil L2 Finals (OBM) - geometry, 2004.2

Tags: geometry , angles , isosceles

In the figure, $ABC$ and $DAE$ are isosceles triangles ($AB = AC = AD = DE$) and the angles $BAC$ and $ADE$ have measures $36^o$. a) Using geometric properties, calculate the measure of angle $\angle EDC$. b) Knowing that $BC = 2$, calculate the length of segment $DC$. c) Calculate the length of segment $AC$ . [img]https://1.bp.blogspot.com/-mv43_pSjBxE/XqBMTfNlRKI/AAAAAAAAL2c/5ILlM0n7A2IQleu9T4yNmIY_1DtrxzsJgCK4BGAYYCw/s400/2004%2Bobm%2Bl2.png[/img]

2015 BMT Spring, 7

Tags: geometry , incircle , angles

In $ \vartriangle ABC$, $\angle B = 46^o$ and $\angle C = 48^o$ . A circle is inscribed in $ \vartriangle ABC$ and the points of tangency are connected to form $PQR$. What is the measure of the largest angle in $\vartriangle P QR$?

2005 Sharygin Geometry Olympiad, 9.5

Tags: geometry , angle bisector , median , construction , altitude , angles

It is given that for no side of the triangle from the height drawn to it, the bisector and the median it is impossible to make a triangle. Prove that one of the angles of the triangle is greater than $135^o$

2010 Sharygin Geometry Olympiad, 6

Tags: geometry , angles , square , midpoints , Angle Chasing , geometry solved , Sharygin Geometry Olympiad

Let $E, F$ be the midpoints of sides $BC, CD$ of square $ABCD$. Lines $AE$ and $BF$ meet at point $P$. Prove that $\angle PDA = \angle AED$.

2004 District Olympiad, 4

Tags: geometry , angles , equal angles , isosceles , right triangle

Consider the isosceles right triangle $ABC$ ($AB = AC$) and the points $M, P \in [AB]$ so that $AM = BP$. Let $D$ be the midpoint of the side $BC$ and $R, Q$ the intersections of the perpendicular from $A$ on$ CM$ with $CM$ and $BC$ respectively. Prove that a) $\angle AQC = \angle PQB$ b) $\angle DRQ = 45^o$

1963 IMO, 3

Tags: inequalities , geometry , polygon , angles , geometric inequality , IMO , IMO 1963

In an $n$-gon $A_{1}A_{2}\ldots A_{n}$, all of whose interior angles are equal, the lengths of consecutive sides satisfy the relation \[a_{1}\geq a_{2}\geq \dots \geq a_{n}. \] Prove that $a_{1}=a_{2}= \ldots= a_{n}$.

1975 IMO Shortlist, 13

Tags: geometry , point set , euclidean distance , Intersection , angles , IMO Shortlist

Let $A_0,A_1, \ldots , A_n$ be points in a plane such that (i) $A_0A_1 \leq \frac{1}{ 2} A_1A_2 \leq \cdots \leq \frac{1}{2^{n-1} } A_{n-1}A_n$ and (ii) $0 < \measuredangle A_0A_1A_2 < \measuredangle A_1A_2A_3 < \cdots < \measuredangle A_{n-2}A_{n-1}A_n < 180^\circ,$ where all these angles have the same orientation. Prove that the segments $A_kA_{k+1},A_mA_{m+1}$ do not intersect for each $k$ and $n$ such that $0 \leq k \leq m - 2 < n- 2.$

2002 Estonia National Olympiad, 1

Tags: equal angles , angles , square , geometry

Points $K$ and $L$ are taken on the sides $BC$ and $CD$ of a square $ABCD$ so that $\angle AKB = \angle AKL$. Find $\angle KAL$.

2010 Abels Math Contest (Norwegian MO) Final, 1a

Tags: geometry , angles , right angle , equal segments

The point $P$ lies on the edge $AB$ of a quadrilateral $ABCD$. The angles $BAD, ABC$ and $CPD$ are right, and $AB = BC + AD$. Show that $BC = BP$ or $AD = BP$.

2004 Regional Olympiad - Republic of Srpska, 3

Tags: geometry , isosceles , angles

Let $ABC$ be an isosceles triangle with $\angle A=\angle B=80^\circ$. A straight line passes through $B$ and through the circumcenter of the triangle and intersects the side $AC$ at $D$. Prove that $AB=CD$.

2022 IFYM, Sozopol, 7

Tags: geometry , angles

Points $M$, $N$, $P$ and $Q$ are midpoints of the sides $AB$, $BC$, $CD$ and $DA$ of the inscribed quadrilateral $ABCD$ with intersection point $O$ of its diagonals. Let $K$ be the second intersection point of the circumscribed circles of $MOQ$ and $NOP$. Prove that $OK\perp AC$.

1988 Austrian-Polish Competition, 6

Tags: geometry , angles , 3D geometry , orthogonal , ray , acute triangle , acute

Three rays $h_1,h_2,h_3$ emanating from a point $O$ are given, not all in the same plane. Show that if for any three points $A_1,A_2,A_3$ on $h_1,h_2,h_3$ respectively, distinct from $O$, the triangle $A_1A_2A_3$ is acute-angled, then the rays $h_1,h_2,h_3$ are pairwise orthogonal.