Olympiad problems

Estonia Open Junior - geometry, 2020.2.5

The circle $\omega_2$ passing through the center $O$ of the circle $\omega_1$, is tangent to the circle $\omega_2$ at the point $A$. On the circle $\omega_2$, the point $C$ is taken so that the ray $AC$ intersects the circle $\omega_1$ for second time at point $D$, the ray $OC$ intersects the circle $\omega_1$ at point $E$ and the lines $DE$ and $AO$ are parallel. Find the size of the angle $DAE$.

Novosibirsk Oral Geo Oly VIII, 2023.4

Tags: geometry , equal segments , isosceles , angle

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 $\angle CFA$.

1997 Singapore Senior Math Olympiad, 2

Tags: semicircle , geometry , angle chasing , angle

Figure shows a semicircle with diameter $AD$. The chords $AC$ and $BD$ meet at $P$. $Q$ is the foot of the perpendicular from $P$ to $AD$. find $\angle BCQ$ in terms of $\theta$ and $\phi$ . [img]https://cdn.artofproblemsolving.com/attachments/a/2/2781050e842b2dd01b72d246187f4ed434ff69.png[/img]

2021 Novosibirsk Oral Olympiad in Geometry, 5

Tags: geometry , pentagon , angle

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 $\angle ECD = 40^o.$

Novosibirsk Oral Geo Oly VII, 2021.2

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.

2024 Abelkonkurransen Finale, 4a

Tags: circumcircle , geometry , angle

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 $\angle BAC$.

1964 All Russian Mathematical Olympiad, 041

Tags: angle , geometry

The two heights in the triangle are not less than the respective sides. Find the angles.

2021 Czech-Polish-Slovak Junior Match, 5

Tags: hexagon , geometry , angle

A regular heptagon $ABCDEFG$ is given. The lines $AB$ and $CE$ intersect at $ P$. Find the measure of the angle $\angle PDG$.

2021 Iranian Geometry Olympiad, 5

Tags: geometry , angle

Let $A_1, A_2, . . . , A_{2021}$ be $2021$ points on the plane, no three collinear and $$\angle A_1A_2A_3 + \angle A_2A_3A_4 +... + \angle A_{2021}A_1A_2 = 360^o,$$ in which by the angle $\angle A_{i-1}A_iA_{i+1}$ we mean the one which is less than $180^o$ (assume that $A_{2022} =A_1$ and $A_0 = A_{2021}$). Prove that some of these angles will add up to $90^o$. [i]Proposed by Morteza Saghafian - Iran[/i]

Geometry Mathley 2011-12, 11.3

Tags: geometry , equal angles , equal ratio , angle

Let $ABC$ be a triangle such that $AB = AC$ and let $M$ be a point interior to the triangle. If $BM$ meets $AC$ at $D$. show that $\frac{DM}{DA}=\frac{AM}{AB}$ if and only if $\angle AMB = 2\angle ABC$. Michel Bataille

VII Soros Olympiad 2000 - 01, 10.5

Tags: geometry , incenter , locus , angle

An acute-angled triangle $ABC$ is given. Points $A_1, B_1$ and $C_1$ are taken on its sides $BC, CA$ and $AB$, respectively, such that $\angle B_1A_1C_1 + 2 \angle BAC = 180^o$, $\angle A_1C_1B_1 + 2 \angle ACB = 180^o$, $\angle C_1B_1A_1 + 2 \angle CBA = 180^o$. Find the locus of the centers of the circles inscribed in triangles $A_1B_1C_1$ (all kinds of such triangles are considered).

1985 Greece National Olympiad, 3

Tags: geometry , angle

Interior in alake there are two points $A,B$ from which we can see every other point of the lake. Prove that also from any other point of the segment $AB$, we can see all points of the lake.

2018 Hanoi Open Mathematics Competitions, 8

Tags: rhombus , angle chasing , angle , geometry

Let $ABCD$ be rhombus, with $\angle ABC = 80^o$: Let $E$ be midpoint of $BC$ and $F$ be perpendicular projection of $A$ onto $DE$. Find the measure of $\angle DFC$ in degree.

Estonia Open Junior - geometry, 2014.2.2

Tags: geometry , angle

In a scalene triangle one angle is exactly two times as big as another one and some angle in this triangle is $36^o$. Find all possibilities, how big the angles of this triangle can be.

2015 Junior Regional Olympiad - FBH, 1

Tags: geometry , angle

Find two angles which add to $180^{\circ}$ which difference is $1^{'}$

2016 Iranian Geometry Olympiad, 5

Tags: quadrilateral , angle , geometry

Let $ABCD$ be a convex quadrilateral with these properties: $\angle ADC = 135^o$ and $\angle ADB - \angle ABD = 2\angle DAB = 4\angle CBD$. If $BC = \sqrt2 CD$ , prove that $AB = BC + AD$. by Mahdi Etesami Fard

II Soros Olympiad 1995 - 96 (Russia), 9.5

Tags: geometry , angle

Angle $A$ of triangle $ABC$ is $33^o$. A straight line passing through $A$ perpendicular to $AC$ intersects straight line $BC$ at point $D$ so that $CD = 2AB$. What is angle $C$ of triangle $ABC$? (Please list all options.)

2022 Yasinsky Geometry Olympiad, 3

Tags: geometry , isosceles right triangle , angle

In an isosceles right triangle $ABC$ with a right angle $C$, point $M$ is the midpoint of leg $AC$. At the perpendicular bisector of $AC$, point $D$ was chosen such that $\angle CDM = 30^o$, and $D$ and $B$ lie on different sides of $AC$. Find the angle $\angle ABD$. (Volodymyr Petruk)

1991 Greece National Olympiad, 2

Tags: geometry , angle

Let $\widehat{xOy}$ be an acute angle , $A$ a point on ray $Oy$ and $B$ a point on ray $Ox$ such that $AB \perp OX$ .Prove that there are two points on $Ox$, each of the equidistant from $A$ and $Ox$.

Durer Math Competition CD 1st Round - geometry, 2018.C+2

Tags: geometry , isosceles , right triangle , angle

In an isosceles right-angled triangle $ABC$, the right angle is at $A$. $D$ lies so on the side $BC$ that $2CD = DB$. Let $E$ be the projection of $B$ onto $AD$. What is the measure fof angle $\angle CED $?

2021 Girls in Mathematics Tournament, 2

Tags: geometry , angle

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?

2020 Novosibirsk Oral Olympiad in Geometry, 6

Tags: angle bisector , angle , geometry

Angle bisectors $AA', BB'$and $CC'$ are drawn in triangle $ABC$ with angle $\angle B= 120^o$. Find $\angle A'B'C'$.

2017 Romania National Olympiad, 2

Tags: perpendicular , geometry , angle

Consider the triangle $ABC$, with $\angle A= 90^o, \angle B = 30^o$, and $D$ is the foot of the altitude from $A$. Let the point $E \in (AD)$ such that $DE = 3AE$ and $F$ the foot of the perpendicular from $D$ to the line $BE$. a) Prove that $AF \perp FC$. b) Determine the measure of the angle $AFB$.

2007 Sharygin Geometry Olympiad, 19

Tags: geometry , minimum , geometric inequality , angle , cyclic quadrilateral , area

Into an angle $A$ of size $a$, a circle is inscribed tangent to its sides at points $B$ and $C$. A line tangent to this circle at a point M meets the segments $AB$ and $AC$ at points $P$ and $Q$ respectively. What is the minimum $a$ such that the inequality $S_{PAQ}<S_{BMC}$ is possible?

2006 Thailand Mathematical Olympiad, 4

Tags: geometry , trigonometry , equal segments , angle

Let $P$ be a point outside a circle centered at $O$. From $P$, tangent lines are drawn to the circle, touching the circle at points $A$ and $B$. Ray $\overrightarrow{BO}$ is drawn intersecting the circle again at $C$ and intersecting ray $\overrightarrow{PA}$ at $Q$. If $3QA = 2AP$, what is the value of $\sin \angle CAQ$?