Olympiad problems

2000 Tournament Of Towns, 3

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)

Novosibirsk Oral Geo Oly VII, 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 $\angle A'B'C'$.

2010 Contests, 3

Tags: geometry , isosceles , equal segments , angle

Consider triangle $ABC$ with $AB = AC$ and $\angle A = 40 ^o$. The points $S$ and $T$ are on the sides $AB$ and $BC$, respectively, so that $\angle BAT = \angle BCS= 10 ^o$. The lines $AT$ and $CS$ intersect at point $P$. Prove that $BT = 2PT$.

2003 Abels Math Contest (Norwegian MO), 3

Tags: geometry , angle

Let $ABC$ be a triangle with $AC> BC$, and let $S$ be the circumscribed circle of the triangle. $AB$ divides $S$ into two arcs. Let $D$ be the midpoint of the arc containing $C$. (a) Show that $\angle ACB +2 \cdot \angle ACD = 180^o$. (b) Let $E$ be the foot of the altitude from $D$ on $AC$. Show that $BC +CE = AE$.

2016 Ecuador NMO (OMEC), 3

Tags: equal segments , geometry , angle , equilateral

Let $A, B, C, D$ be four different points on a line $\ell$, such that $AB = BC = CD$. In one of the semiplanes determined by the line $\ell$, the points $P$ and $Q$ are chosen in such a way that the triangle $CPQ$ is equilateral with its vertices named clockwise. Let $M$ and $N$ be two points on the plane such that the triangles $MAP$ and $NQD$ are equilateral (the vertices are also named clockwise). Find the measure of the angle $\angle MBN$.

1998 Belarus Team Selection Test, 1

Tags: geometry , locus , circles , angle

Two circles $S_1$ and $S_2$ intersect at different points $P,Q$. The arc of $S_1$ lying inside $S_2$ measures $2a$ and the arc of $S_2$ lying inside $S_1$ measures $2b$. Let $T$ be any point on $S_1$. Let $R,S$ be another points of intersection of $S_2$ with $TP$ and $TQ$ respectively. Let $a+2b<\pi$ . Find the locus of the intersection points of $PS$ and $RQ$. S.Shikh

2011 Chile National Olympiad, 2

Tags: geometry , circles , angle , tangent

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 Swedish Mathematical Competition, 2

Tags: geometry , equal angles , equal segments , angle

Let $D$ be the point on side $AC$ of a triangle $ABC$ such that $BD$ bisects $\angle B$, and $E$ be the point on side $AB$ such that $3\angle ACE = 2\angle BCE$. Suppose that $BD$ and $CE$ intersect at a point $P$ with $ED = DC = CP$. Determine the angles of the triangle.

1995 Tournament Of Towns, (469) 3

Tags: geometry , angle bisector , angle

Let $AK$, $BL$ and $CM$ be the angle bisectors of a triangle $ABC$, with $K$ on $BC$. Let $P$ and $Q$ be the points on the lines $BL$ and $CM$ respectively such that $AP = PK$ and $AQ = QK$. Prove that $\angle PAQ = 90^o -\frac12 \angle B AC.$ (I Sharygin)

1998 Bundeswettbewerb Mathematik, 3

Tags: geometry , angle

A triangle $ABC$ satisfies $BC = AC +\frac12 AB$. Point $P$ on side $AB$ is taken so that $AP = 3PB$. Prove that $ \angle PAC = 2\angle CPA$.

2021 Iranian Geometry Olympiad, 3

Tags: geometry , perimeter , semicircle , angle

As shown in the following figure, a heart is a shape consist of three semicircles with diameters $AB$, $BC$ and $AC$ such that $B$ is midpoint of the segment $AC$. A heart $\omega$ is given. Call a pair $(P, P')$ bisector if $P$ and $P'$ lie on $\omega$ and bisect its perimeter. Let $(P, P')$ and $(Q,Q')$ be bisector pairs. Tangents at points $P, P', Q$, and $Q'$ to $\omega$ construct a convex quadrilateral $XYZT$. If the quadrilateral $XYZT$ is inscribed in a circle, find the angle between lines $PP'$ and $QQ'$. [img]https://cdn.artofproblemsolving.com/attachments/3/c/8216889594bbb504372d8cddfac73b9f56e74c.png[/img] [i]Proposed by Mahdi Etesamifard - Iran[/i]

2003 Estonia National Olympiad, 3

Tags: angle , right triangle , geometry

Let $ABC$ be a triangle with $\angle C = 90^o$ and $D$ a point on the ray $CB$ such that $|AC| \cdot |CD| = |BC|^2$. A parallel line to $AB$ through $D$ intersects the ray $CA$ at $E$. Find $\angle BEC$.

Brazil L2 Finals (OBM) - geometry, 2007.1

Tags: geometry , circumcircle , angle chasing , circumcenter , angle

Let $ABC$ be a triangle with circumcenter $O$. Let $P$ be the intersection of straight lines $BO$ and $AC$ and $\omega$ be the circumcircle of triangle $AOP$. Suppose that $BO = AP$ and that the measure of the arc $OP$ in $\omega$, that does not contain $A$, is $40^o$. Determine the measure of the angle $\angle OBC$. [img]https://3.bp.blogspot.com/-h3UVt-yrJ6A/XqBItXzT70I/AAAAAAAAL2Q/7LVv0gmQWbo1_3rn906fTn6wosY1-nIfwCK4BGAYYCw/s1600/2007%2Bomb%2Bl2.png[/img]

2015 Indonesia MO Shortlist, G1

Tags: rectangle , geometry , angle , geometric inequality

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

2021 Yasinsky Geometry Olympiad, 3

Tags: equal angles , geometry , equal segments , angle

The segments $AC$ and $BD$ are perpendicular, and $AC$ is twice as large as $BD$ and intersects $BD$ in it in the midpoint. Find the value of the angle $BAD$, if we know that $\angle CAD = \angle CDB$. (Gregory Filippovsky)

1946 Moscow Mathematical Olympiad, 106

Tags: acute , maximum , geometry , angle

What is the largest number of acute angles that a convex polygon can have?

Denmark (Mohr) - geometry, 2018.5

Tags: geometry , angle bisector , angle

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]

VII Soros Olympiad 2000 - 01, 10.8

Tags: geometry , trigonometry , geometric inequality , angle

There is a set of triangles, in each of which the smallest angle does not exceed $36^o$ . A new one is formed from these triangles according to the following rule: the smallest side of the new one is equal to the sum of the smallest sides of these triangles, its middle side is equal to the sum of the middle sides, and the largest is the sum of the largest ones. Prove that the sine of the smallest angle of the resulting triangle is less than $2 \sin 18^o$ .

1953 Moscow Mathematical Olympiad, 252

Tags: angle , concurrency , line , geometry

Given triangle $\vartriangle A_1A_2A_3$ and a straight line $\ell$ outside it. The angles between the lines $A_1A_2$ and $A_2A_3, A_1A_2$ and $A_2A_3, A_2A_3$ and $A_3A_1$ are equal to $a_3, a_1$ and $a_2$, respectively. The straight lines are drawn through points $A_1, A_2, A_3$ forming with $\ell$ angles of $\pi -a_1, \pi -a_2, \pi -a_3$, respectively. All angles are counted in the same direction from $\ell$ . Prove that these new lines meet at one point.

2021 Science ON grade VII, 2

Tags: geometry , angle

In triangle $ABC$, we have $\angle ABC=\angle ACB=44^o$. Point $M$ is in its interior such that $\angle MBC=16^o$ and $\angle MCB=30^o$. Prove that $\angle MAC=\angle MBC$. [i] (Andra Elena Mircea)[/i]

2017 Auckland Mathematical Olympiad, 5

Tags: geometry , orthocenter , angle

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

2007 Junior Balkan Team Selection Tests - Moldova, 6

Tags: trinomial , angle , quadratic

The lengths of the sides $a, b$ and $c$ of a right triangle satisfy the relations $a <b <c$, and $\alpha$ is the measure of the smallest angle of the triangle. For which real values $k$ the equation $ax^2 + bx + kc = 0$ has real solutions for any measure of the angle $\alpha$ not exceeding $18^o$

Kyiv City MO Juniors 2003+ geometry, 2015.8.3

Tags: geometry , angle bisector , isosceles , angle chasing , angle

In the isosceles triangle $ABC$, $ (AB = BC)$ the bisector $AD$ was drawn, and in the triangle $ABD$ the bisector $DE$ was drawn. Find the values of the angles of the triangle $ABC$, if it is known that the bisectors of the angles $ABD$ and $AED$ intersect on the line $AD$. (Fedak Ivan)

2015 Romania National Olympiad, 4

Tags: geometry , angle chasing , angle

Consider $\vartriangle ABC$ where $\angle ABC= 60 ^o$. Points $M$ and $D$ are on the sides $(AC)$, respectively $(AB)$, such that $\angle BCA = 2 \angle MBC$, and $BD = MC$. Determine $\angle DMB$.

2004 May Olympiad, 3

Tags: angle , geometry

We have a pool table $8$ meters long and $2$ meters wide with a single ball in the center. We throw the ball in a straight line and, after traveling $29$ meters, it stops at a corner of the table. How many times did the ball hit the edges of the table? Note: When the ball rebounds on the edge of the table, the two angles that form its trajectory with the edge of the table are the same.