Olympiad problems

2010 Oral Moscow Geometry Olympiad, 1

Two equilateral triangles $ABC$ and $CDE$ have a common vertex (see fig). Find the angle between straight lines $AD$ and $BE$. [img]https://1.bp.blogspot.com/-OWpqpAqR7Zw/Xzj_fyqhbFI/AAAAAAAAMao/5y8vCfC7PegQLIUl9PARquaWypr8_luAgCLcBGAsYHQ/s0/2010%2Boral%2Bmoscow%2Bgeometru%2B8.1.gif[/img]

2013 Junior Balkan Team Selection Tests - Moldova, 4

Tags: angle , algebra

A train from stop $A$ to stop $B$ is traveled in $X$ minutes ($0 <X <60$). It is known that when starting from $A$, as well as when arriving at $B$, the angle formed by the hour and the minute had measure equal to $X$ degrees. Find $X $.

I Soros Olympiad 1994-95 (Rus + Ukr), 11.3

Tags: geometry , geometric inequality , angle

It is known that in the triangle $ABC$, $ 2 \angle BAC + 3 \angle ABC= 180^o$. Prove that $4(BC + CA)< 5AB$.

2014 Czech-Polish-Slovak Junior Match, 5

Tags: combinatorics , angle , combinatorial geometry , polygon , sum , square

A square is given. Lines divide it into $n$ polygons. What is he the largest possible sum of the internal angles of all polygons?

2017 Estonia Team Selection Test, 10

Tags: geometry , semicircle , angle

Let $ABC$ be a triangle with $AB = \frac{AC}{2 }+ BC$. Consider the two semicircles outside the triangle with diameters $AB$ and $BC$. Let $X$ be the orthogonal projection of $A$ onto the common tangent line of those semicircles. Find $\angle CAX$.

2017 BMT Spring, 4

Tags: geometry , angle , acute

$2$ darts are thrown randomly at a circular board with center $O$, such that each dart has an equal probability of hitting any point on the board. The points at which they land are marked $A$ and $B$. What is the probability that $\angle AOB$ is acute?

Novosibirsk Oral Geo Oly IX, 2020.5

Tags: angle bisector , angle , geometry

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

2014 German National Olympiad, 6

Tags: geometry , angle , circumscribed , quadrilateral

Let $ABCD$ be a circumscribed quadrilateral and $M$ the centre of the incircle. There are points $P$ and $Q$ on the lines $MA$ and $MC$ such that $\angle CBA= 2\angle QBP.$ Prove that $\angle ADC = 2 \angle PDQ.$

1992 Chile National Olympiad, 5

Tags: side , angle , geometry

In the $\triangle ABC $, points $ M, I, H $ are feet, respectively, of the median, bisector and height, drawn from $ A $. It is known that $ BC = 2 $, $ MI = 2-\sqrt {3} $ and $ AB > AC $. a) Prove that $ I$ lies between $ M $ and $ H $. b) Calculate $ AB ^ 2-AC ^ 2 $. c) Determine $ \dfrac {AB} {AC} $. d) Find the measure of all the sides and angles of the triangle.

Brazil L2 Finals (OBM) - geometry, 2004.2

Tags: geometry , angle , 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 Romania National Olympiad, 3

Tags: geometry , pyramid , angle , 3d geometry

Let $VABC$ be a regular triangular pyramid with base $ABC$, of center $O$. Points $I$ and $H$ are the center of the inscribed circle, respectively the orthocenter $\vartriangle VBC$. Knowing that $AH = 3 OI$, determine the measure of the angle between the lateral edge of the pyramid and the plane of the base.

2009 Postal Coaching, 5

Tags: geometry , angle , equal angles , sum

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 $PA_1 + PA_3 + PA_5 + PA_7 +PA_9 = PA_2 + PA_4 + PA_6 + PA_8 + PA_{10}$.

1992 Tournament Of Towns, (342) 4

Tags: geometry , geometric inequality , angle

(a) In triangle $ABC$, angle $A$ is greater than angle $B$. Prove that the length of side $BC$ is greater than half the length of side $AB$. (b) In the convex quadrilateral $ABCD$, the angle at $A$ is greater than the angle at $C$ and the angle at $D$ is greater than the angle at $B$. Prove that the length of side $BC$ is greater than half of the length of side $AD$. (F Nazarov)

2014 Costa Rica - Final Round, 1

Tags: geometry , collinear , angle

Consider the following figure where $AC$ is tangent to the circle of center $O$, $\angle BCD = 35^o$, $\angle BAD = 40^o$ and the measure of the minor arc $DE$ is $70^o$. Prove that points $B, O, E$ are collinear. [img]https://cdn.artofproblemsolving.com/attachments/4/0/fd5f8d3534d9d0676deebd696d174999c2ad75.png[/img]

1995 Swedish Mathematical Competition, 5

Tags: cyclic quadrilateral , geometry , angle , cyclic

On a circle with center $O$ and radius $r$ are given points $A,B,C,D$ in this order such that $AB, BC$ and $CD$ have the same length $s$ and the length of $AD$ is $s+ r$.Assume that $s < r$. Determine the angles of quadrilateral $ABCD$.

1998 Belarus Team Selection Test, 1

Tags: angle bisector , geometry , perpendicular bisector , angle , midpoint

Let $O$ be a point inside an acute angle with the vertex $A$ and $H, N$ be the feet of the perpendiculars drawn from $O$ onto the sides of the angle. Let point $B$ belong to the bisector of the angle, $K$ be the foot of the perpendicular from $B$ onto either side of the angle. Denote by $P,F$ the midpoints of the segments $AK,HN$ respectively. Known that $ON + OH = BK$, prove that $PF$ is perpendicular to $AB$. Ya. Konstantinovski

2017 Junior Regional Olympiad - FBH, 3

Tags: angle , triangle , compare

In acute triangle $ABC$ holds $\angle BAC=80^{\circ}$, and altitudes $h_a$ and $h_b$ intersect in point $H$. if $\angle AHB = 126^{\circ}$, which side is the smallest, and which is the biggest in $ABC$

2021 Saudi Arabia Training Tests, 13

Tags: geometry , angle , equal angles

Let $ABCD$ be a quadrilateral with $\angle A = \angle B = 90^o$, $AB = AD$. Denote $E$ as the midpoint of $AD$, suppose that $CD = BC + AD$, $AD > BC$. Prove that $\angle ADC = 2\angle ABE$.

2019 District Olympiad, 2

Tags: geometry , equal angles , equal segments , angle

Consider $D$ the midpoint of the base $[BC]$ of the isosceles triangle ABC in which $\angle BAC < 90^o$. On the perpendicular from $B$ on the line $BC$ consider the point $E$ such that $\angle EAB= \angle BAC$, and on the line passing though $C$ parallel to the line $AB$ we consider the point $F$ such that $F$ and $D$ are on different side of the line $AC$ and $\angle FAC = \angle CAD$. Prove that $AE = CF$ and $BF = EF$

2016 Iranian Geometry Olympiad, 5

Tags: angle , quadrilateral , 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

2016 ASMT, 2

Tags: geometry , angle

Points $D$ and $E$ are chosen on the exterior of $\vartriangle ABC$ such that $\angle ADC = \angle BEC = 90^o$. If $\angle ACB = 40^o$, $AD = 7$, $CD = 24$, $CE = 15$, and $BE = 20$, what is the measure of $\angle ABC $ in,degrees?

Kyiv City MO Juniors Round2 2010+ geometry, 2012.8.5

Tags: incenter , angle , equilateral , geometry

In the triangle $ABC$ on the sides $AB$ and $AC$ outward constructed equilateral triangles $ABD$ and $ACE$. The segments $CD$ and $BE$ intersect at point $F$. It turns out that point $A$ is the center of the circle inscribed in triangle $ DEF$. Find the angle $BAC$. (Rozhkova Maria)

2022 Dutch BxMO TST, 2

Tags: geometry , equal angles , angle

Let $ABC$ be an acute triangle, and let $D$ be the foot of the altitude from $A$. The circle with centre $A$ passing through $D$ intersects the circumcircle of triangle $ABC$ in $X$ and $Y$ , in such a way that the order of the points on this circumcircle is: $A, X, B, C, Y$ . Show that $\angle BXD = \angle CYD$.

2002 Kazakhstan National Olympiad, 1

Tags: geometry , angle , incenter , equal segments

Let $ O $ be the center of the inscribed circle of the triangle $ ABC $, tangent to the side of $ BC $. Let $ M $ be the midpoint of $ AC $, and $ P $ be the intersection point of $ MO $ and $ BC $. Prove that $ AB = BP $ if $ \angle BAC = 2 \angle ACB $.

2000 Tournament Of Towns, 3

Tags: geometry , 3d geometry , prism , angle

In each lateral face of a pentagonal prism at least one of the four angles is equal to $f$. Find all possible values of $f$. (A Shapovalov)