Olympiad problems

Indonesia MO Shortlist - geometry, g1

Given a cyclic quadrilateral $ABCD$ so that $AB = AD$ and $AB + BC <CD$. Prove that the angle $ABC$ is more than $120$ degrees.

1997 Tournament Of Towns, (534) 6

Tags: geometry , isosceles triangle , isosceles , angle

Let $P$ be a point inside the triangle $ABC$ such that $AB = BC$, $\angle ABC = 80^o$, $\angle PAC = 40^o$ and $\angle ACP = 30^o$. Find $\angle BPC$. (G Galperin)

Estonia Open Junior - geometry, 2010.1.2

Tags: geometry , equal angles , equal segments , angle

Given a convex quadrangle $ABCD$ with $|AD| = |BD| = |CD|$ and $\angle ADB = \angle DCA$, $\angle CBD = \angle BAC$, find the sizes of the angles of the quadrangle.

1985 Tournament Of Towns, (106) 6

Tags: geometry , altitude , angle bisector , angle

In triangle $ABC, AH$ is an altitude ($H$ is on $BC$) and $BE$ is a bisector ($E$ is on $AC$) . We are given that angle $BEA$ equals $45^o$ .Prove that angle $EHC$ equals $45^o$ . (I. Sharygin , Moscow)

2017 Bundeswettbewerb Mathematik, 2

Tags: gon , angle , acute , combinatorics

What is the maximum number of acute interior angles a non-overlapping planar $2017$-gon can have?

2006 Sharygin Geometry Olympiad, 9

Tags: geometry , concurrency , angle , circles

$L(a)$ is the line connecting the points of the unit circle corresponding to the angles $a$ and $\pi - 2a$. Prove that if $a + b + c = 2\pi$, then the lines $L (a), L (b)$ and $L (c)$ intersect at one point.

1996 Tournament Of Towns, (483) 1

Tags: angle , geometry

In an acute-angled triangle, each angle is an integral number of degrees, and the smallest angle is one-fifth of the largest one. Find these angles. (G Galperin)

Estonia Open Senior - geometry, 2011.1.5

Tags: geometry , angle

Given a triangle $ABC$ where $|BC| = a, |CA| = b$ and $|AB| = c$, prove that the equality $\frac{1}{a + b}+\frac{1}{b + c}=\frac{3}{a + b + c}$ holds if and only if $\angle ABC = 60^o$.

2011 Oral Moscow Geometry Olympiad, 5

Tags: angle chasing , perpendicular , angle , geometry

In a convex quadrilateral $ABCD, AC\perp BD, \angle BCA = 10^o,\angle BDA = 20^o, \angle BAC = 40^o$. Find $\angle BDC$.

2020-21 KVS IOQM India, 27

Tags: geometry , reflection , orthocenter , angle

Let $ABC$ be an acute-angled triangle and $P$ be a point in its interior. Let $P_A,P_B$ and $P_c$ be the images of $P$ under reflection in the sides $BC,CA$, and $AB$, respectively. If $P$ is the orthocentre of the triangle $P_AP_BP_C$ and if the largest angle of the triangle that can be formed by the line segments$ PA, PB$. and $PC$ is $x^o$, determine the value of $x$.

2016 Portugal MO, 4

Tags: tangent , geometry , parallelogram , equal angles , angle

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]

Kyiv City MO 1984-93 - geometry, 1985.10.2

Tags: 3d geometry , cube , angle , geometry

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 ?

2020 Argentina National Olympiad, 3

Tags: right triangle , isosceles , angle , geometry

Let $ABC$ be a right isosceles triangle with right angle at $A$. Let $E$ and $F$ be points on A$B$ and $AC$ respectively such that $\angle ECB = 30^o$ and $\angle FBC = 15^o$. Lines $CE$ and $BF$ intersect at $P$ and line $AP$ intersects side $BC$ at $D$. Calculate the measure of angle $\angle FDC$.

2015 Iran Geometry Olympiad, 2

Tags: geometry , elementary , angle

Let $ABC$ be a triangle with $\angle A = 60^o$. The points $M,N,K$ lie on $BC,AC,AB$ respectively such that $BK = KM = MN = NC$. If $AN = 2AK$, find the values of $\angle B$ and $\angle C$. by Mahdi Etesami Fard

2011 Saudi Arabia BMO TST, 3

Tags: equilateral , angle , geometry

Consider a triangle $ABC$. Let $A_1$ be the symmetric point of $A$ with respect to the line $BC$, $B_1$ the symmetric point of $B$ with respect to the line $CA$, and $C_1$ the symmetric point of $C$ with respect to the line $AB$. Determine the possible set of angles of triangle $ABC$ for which $A_1B_1C_1$ is equilateral.

2008 Brazil Team Selection Test, 1

Tags: geometry , trisector , angle

Let $AB$ be a chord, not a diameter, of a circle with center $O$. The smallest arc $AB$ is divided into three congruent arcs $AC$, $CD$, $DB$. The chord $AB$ is also divided into three equal segments $AC'$, $C'D'$, $D'B$. Let $P$ be the intersection point of between the lines $CC'$ and $DD'$. Prove that $\angle APB = \frac13 \angle AOB$.

2019 Novosibirsk Oral Olympiad in Geometry, 3

Tags: geometry , equal segments , angle

Equal line segments are marked in triangle $ABC$. Find its angles. [img]https://cdn.artofproblemsolving.com/attachments/0/2/bcb756bba15ba57013f1b6c4cbe9cc74171543.png[/img]

Novosibirsk Oral Geo Oly VIII, 2020.6

Tags: geometry , angle bisector , angle

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

1949-56 Chisinau City MO, 23

Tags: distance , geometry , angle

Inside the angle $ABC$ of $60^o$, point $O$ is selected, which is located at distances from the sides of the angle $a$ and $b$, respectively. Determine the distance from the top of the angle to this point.

Estonia Open Junior - geometry, 2000.2.4

Tags: geometry , concyclic , equal angles , angle

In the plane, there is an acute angle $\angle AOB$ . Inside the angle points $C$ and $D$ are chosen so that $\angle AOC = \angle DOB$. From point $D$ the perpendicular on $OA$ intersects the ray $OC$ at point $G$ and from point C the perpendicular on $OB$ intersects the ray $OD$ at point $H$. Prove that the points $C, D, G$ and $H$ are conlyclic.

2009 Postal Coaching, 5

Tags: equal angles , geometry , sum , angle

Let $P$ be an interior point of a circle and $A_1,A_2...,A_{10}$ be points on the circle such that $\angle A_1PA_2 = \angle A_2PA_3 = ... = \angle A_{10}PA_1 = 36^o$. Prove that $PA_1 + PA_3 + PA_5 + PA_7 +PA_9 = PA_2 + PA_4 + PA_6 + PA_8 + PA_{10}$.

1952 Moscow Mathematical Olympiad, 229

Tags: geometry , isosceles , angle , angle chasing

In an isosceles triangle $\vartriangle ABC, \angle ABC = 20^o$ and $BC = AB$. Points $P$ and $Q$ are chosen on sides $BC$ and $AB$, respectively, so that $\angle PAC = 50^o$ and $\angle QCA = 60^o$ . Prove that $\angle PQC = 30^o$ .

2003 Junior Balkan Team Selection Tests - Romania, 4

Tags: geometry , square , angle chasing , equal angles , angle

Let $E$ be the midpoint of the side $CD$ of a square $ABCD$. Consider the point $M$ inside the square such that $\angle MAB = \angle MBC = \angle BME = x$. Find the angle $x$.

Novosibirsk Oral Geo Oly VIII, 2021.3

Tags: angle , geometry

Find the angle $BCA$ in the quadrilateral of the figure. [img]https://cdn.artofproblemsolving.com/attachments/0/2/974e23be54125cde8610a78254b59685833b5b.png[/img]

2018 Dutch BxMO TST, 4

Tags: geometry , circumcircle , angle chasing , angle

In a non-isosceles triangle $\vartriangle ABC$ we have $\angle BAC = 60^o$. Let $D$ be the intersection of the angular bisector of $\angle BAC$ with side $BC, O$ the centre of the circumcircle of $\vartriangle ABC$ and $E$ the intersection of $AO$ and $BC$. Prove that $\angle AED + \angle ADO = 90^o$.