Olympiad problems

1969 IMO Longlists, 71

$(YUG 3)$ Let four points $A_i (i = 1, 2, 3, 4)$ in the plane determine four triangles. In each of these triangles we choose the smallest angle. The sum of these angles is denoted by $S.$ What is the exact placement of the points $A_i$ if $S = 180^{\circ}$?

2008 Tournament Of Towns, 1

Tags: angles , combinatorics

A triangle has an angle of measure $\theta$. It is dissected into several triangles. Is it possible that all angles of the resulting triangles are less than $\theta$, if (a) $\theta = 70^o$ ? (b) $\theta = 80^o$ ?

1935 Moscow Mathematical Olympiad, 003

Tags: pyramid , geometry , 3D geometry , angles

The base of a pyramid is an isosceles triangle with the vertex angle $\alpha$. The pyramid’s lateral edges are at angle $\phi$ to the base. Find the dihedral angle $\theta$ at the edge connecting the pyramid’s vertex to that of angle $\alpha$.

2013 Czech And Slovak Olympiad IIIA, 5

Tags: angles , parallelogram , parallel , geometry

Given the parallelogram $ABCD$ such that the feet $K, L$ of the perpendiculars from point $D$ on the sides $AB, BC$ respectively are internal points. Prove that $KL \parallel AC$ when $|\angle BCA| + |\angle ABD| = |\angle BDA| + |\angle ACD|$.

2014 Flanders Math Olympiad, 3

Tags: geometry , equal segments , angles , Angle Chasing

Let $PQRS$ be a quadrilateral with $| P Q | = | QR | = | RS |$, $\angle Q= 110^o$ and $\angle R = 130^o$ . Determine $\angle P$ and $\angle S$ .

2000 Tournament Of Towns, 3

Tags: 3D geometry , prism , geometry , angles

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)

2020 Novosibirsk Oral Olympiad in Geometry, 7

Tags: geometry , angle bisector , angles

You are given a quadrilateral $ABCD$. It is known that $\angle BAC = 30^o$, $\angle D = 150^o$ and, in addition, $AB = BD$. Prove that $AC$ is the bisector of angle $C$.

2014 Junior Balkan Team Selection Tests - Romania, 4

Tags: geometry , angles , ratio

Let $ABCD$ be a quadrilateral with $\angle A + \angle C = 60^o$. If $AB \cdot CD = BC \cdot AD$, prove that $AB \cdot CD = AC \cdot BD$. Leonard Giugiuc

1986 Tournament Of Towns, (131) 7

Tags: combinatorics , combinatorial geometry , arc , angles , points , circle

On the circumference of a circle are $21$ points. Prove that among the arcs which join any two of these points, at least $100$ of them must subtend an angle at the centre of the circle not exceeding $120^o$ . ( A . F . Sidorenko)

Russian TST 2020, P3

Tags: IMO Shortlist , combinatorics , IMO Shortlist 2019 , angles , combinatorial geometry

Let $n>1$ be an integer. Suppose we are given $2n$ points in the plane such that no three of them are collinear. The points are to be labelled $A_1, A_2, \dots , A_{2n}$ in some order. We then consider the $2n$ angles $\angle A_1A_2A_3, \angle A_2A_3A_4, \dots , \angle A_{2n-2}A_{2n-1}A_{2n}, \angle A_{2n-1}A_{2n}A_1, \angle A_{2n}A_1A_2$. We measure each angle in the way that gives the smallest positive value (i.e. between $0^{\circ}$ and $180^{\circ}$). Prove that there exists an ordering of the given points such that the resulting $2n$ angles can be separated into two groups with the sum of one group of angles equal to the sum of the other group.

2021 Saudi Arabia Training Tests, 15

Tags: angles , geometry

Let $ABC$ be convex quadrilateral and $X$ lying inside it such that $XA \cdot XC^2 = XB \cdot XD^2$ and $\angle AXD + \angle BXC = \angle CXD$. Prove that $\angle XAD + \angle XCD = \angle XBC + \angle XDC$.

2016 Czech-Polish-Slovak Junior Match, 1

Tags: geometry , midpoints , angles , Angle Chasing

Let $ABC$ be a right-angled triangle with hypotenuse $AB$. Denote by $D$ the foot of the altitude from $C$. Let $Q, R$, and $P$ be the midpoints of the segments $AD, BD$, and $CD$, respectively. Prove that $\angle AP B + \angle QCR = 180^o$. Czech Republic

1985 IMO Shortlist, 2

Tags: geometry , 3D geometry , polyhedron , angles , Euler s Theorem , IMO Shortlist

A polyhedron has $12$ faces and is such that: [b][i](i)[/i][/b] all faces are isosceles triangles, [b][i](ii)[/i][/b] all edges have length either $x$ or $y$, [b][i](iii)[/i][/b] at each vertex either $3$ or $6$ edges meet, and [b][i](iv)[/i][/b] all dihedral angles are equal. Find the ratio $x/y.$

Ukrainian TYM Qualifying - geometry, VIII.2

Tags: 3D geometry , geometry , tetrahedron , angles , Ukrainian TYM

Investigate the properties of the tetrahedron $ABCD$ for which there is equality $$\frac{AD}{ \sin \alpha}=\frac{BD}{\sin \beta}=\frac{CD}{ \sin \gamma}$$ where $\alpha, \beta, \gamma$ are the values of the dihedral angles at the edges $AD, BD$ and $CD$, respectively.

2015 EGMO, 6

Tags: geometry , circumcircle , EGMO , orthocenter , Centroid , angles , EGMO 2015

Let $H$ be the orthocentre and $G$ be the centroid of acute-angled triangle $ABC$ with $AB\ne AC$. The line $AG$ intersects the circumcircle of $ABC$ at $A$ and $P$. Let $P'$ be the reflection of $P$ in the line $BC$. Prove that $\angle CAB = 60$ if and only if $HG = GP'$

2010 Contests, 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 $AD = BP$.

2012 Cuba MO, 9

Tags: geometry , angles

Let $O$ be a point interior to triangle $ABC$ such that $\angle BAO = 30^o$, $\angle CBO = 20^o$ and $\angle ABO = \angle ACO = 40^o$ . Knowing that triangle $ABC$ is not equilateral, find the measures of its interior angles.

1982 All Soviet Union Mathematical Olympiad, 335

Tags: Compare , trigonometry , algebra , angles

Three numbers $a,b,c$ belong to $[0,\pi /2]$ interval with $$\cos a = a, \sin(\cos b) = b, \cos(\sin c ) = c$$ Sort those numbers in increasing order.

2012 Oral Moscow Geometry Olympiad, 1

Tags: geometry , trapezoid , angles , equal segments

In trapezoid $ABCD$, the sides $AD$ and $BC$ are parallel, and $AB = BC = BD$. The height $BK$ intersects the diagonal $AC$ at $M$. Find $\angle CDM$.

Kyiv City MO Juniors 2003+ geometry, 2018.9.51

Tags: geometry , square , angles

Given a circle $\Gamma$ with center at point $O$ and diameter $AB$. $OBDE$ is square, $F$ is the second intersection point of the line $AD$ and the circle $\Gamma$, $C$ is the midpoint of the segment $AF$. Find the value of the angle $OCB$.

V Soros Olympiad 1998 - 99 (Russia), 11.8

Tags: geometry , angles

Inside triangle $ABC$, point $P$ is taken so that angles $\angle ARB= \angle BPC = \angle CPA= 120^o$. Lines $BP$ and $CP$ intersect lines $AC$ and $AB$ at points $M$ and $K$. It is known that the quadrilateral $AMPK$ has same areq with the triangle $BCP$. What is the angle $\angle BAC$?

Kyiv City MO 1984-93 - geometry, 1990.7.3

Tags: geometry , angles

Given a triangle with sides $a, b, c$ that satisfy $\frac{a}{b+c}=\frac{c}{a+b}$. Determine the angles of this triangle, if you know that one of them is equal to $120^0$.

2007 Estonia Team Selection Test, 4

Tags: Angle Chasing , angles , square , geometry

In square $ABCD,$ points $E$ and $F$ are chosen in the interior of sides $BC$ and $CD$, respectively. The line drawn from $F$ perpendicular to $AE$ passes through the intersection point $G$ of $AE$ and diagonal $BD$. A point $K$ is chosen on $FG$ such that $|AK|= |EF|$. Find $\angle EKF.$

Estonia Open Senior - geometry, 2018.2.5

Tags: geometry , angles , reflection , equal angles

Let $A'$ be the result of reflection of vertex $A$ of triangle ABC through line $BC$ and let $B'$ be the result of reflection of vertex $B$ through line $AC$. Given that $\angle BA' C = \angle BB'C$, can the largest angle of triangle $ABC$ be located: a) At vertex $A$, b) At vertex $B$, c) At vertex $C$?

Novosibirsk Oral Geo Oly VII, 2021.4

Tags: angles

It is known about two triangles that for each of them the sum of the lengths of any two of its sides is equal to the sum of the lengths of any two sides of the other triangle. Are triangles necessarily congruent?