Olympiad problems

2017 Yasinsky Geometry Olympiad, 5

The four points of a circle are in the following order: $A, B, C, D$. Extensions of chord $AB$ beyond point $B$ and of chord $CD$ beyond point $C$ intersect at point $E$, with $\angle AED= 60^o$. If $\angle ABD =3 \angle BAC$ , prove that $AD$ is the diameter of the circle.

I Soros Olympiad 1994-95 (Rus + Ukr), 11.2

Tags: angles , perpendicular , geometry , rectangle

Given a rectangle $ABCD$ with $AB> BC$. On the side $CD$, take a point $L$ such that $BL$ and $AC$ are perpendicular. Let $K$ be the intersection point of segments $BL$ and $AC$. It is known that segments $AL$. and $DK$ are perpendicular. Find $\angle ACB.$

Novosibirsk Oral Geo Oly VIII, 2021.2

Tags: geometry , angles

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.

2015 Junior Regional Olympiad - FBH, 1

Tags: angles , geometry

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

2005 Sharygin Geometry Olympiad, 24

Tags: combinatorial geometry , geometry , angles , 3D geometry

A triangle is given, all the angles of which are smaller than $\phi$, where $\phi <2\pi / 3$. Prove that in space there is a point from which all sides of the triangle are visible at an angle $\phi$.

2019 Oral Moscow Geometry Olympiad, 6

Tags: geometry , 3D geometry , angles , solid geometry

The sum of the cosines of the flat angles of the trihedral angle is $-1$. Find the sum of its dihedral angles.

1965 Swedish Mathematical Competition, 1

Tags: Geometric Inequalities , altitudes , geometry , angles , max

The feet of the altitudes in the triangle $ABC$ are $A', B', C'$. Find the angles of $A'B'C'$ in terms of the angles $A, B, C$. Show that the largest angle in $A'B'C'$ is at least as big as the largest angle in $ABC$. When is it equal?

1980 IMO, 3

Tags: geometry , 3D geometry , tetrahedron , sphere , triangle inequality , angles , IMO Shortlist

Prove that the sum of the six angles subtended at an interior point of a tetrahedron by its six edges is greater than 540°.

1999 Portugal MO, 6

Tags: geometry , angles , equal angles

In the triangle $[ABC], D$ is the midpoint of $[AB]$ and $E$ is the trisection point of $[BC]$ closer to $C$. If $\angle ADC= \angle BAE$ , find the measue of $\angle BAC$ .

Kyiv City MO Seniors Round2 2010+ geometry, 2012.10.4

Tags: geometry , angles , geometric inequality

In the triangle $ABC$ with sides $BC> AC> AB$ the angles between altiude and median drawn from one vertex are considered. Find out at which vertex this angle is the largest of the three. (Rozhkova Maria)

2000 German National Olympiad, 3

Tags: Equilateral , angles , geometry

Suppose that an interior point $O$ of a triangle $ABC$ is such that the angles $\angle BAO,\angle CBO, \angle ACO$ are all greater than or equal to $30^o$. Prove that the triangle $ABC$ is equilateral.

2019 Saudi Arabia JBMO TST, 3

Tags: geometry , equal angles , angles

Consider a triangle $ABC$ and let $M$ be the midpoint of the side $BC$. Suppose $\angle MAC = \angle ABC$ and $\angle BAM = 105^o$. Find the measure of $\angle ABC$.

2016 Sharygin Geometry Olympiad, 5

Tags: geometry , angles

The center of a circle $\omega_2$ lies on a circle $\omega_1$. Tangents $XP$ and $XQ$ to $\omega_2$ from an arbitrary point $X$ of $\omega_1$ ($P$ and $Q$ are the touching points) meet $\omega_1$ for the second time at points $R$ and $S$. Prove that the line $PQ$ bisects the segment $RS$.

Kyiv City MO Juniors 2003+ geometry, 2020.9.41

Tags: geometry , angles , equal segments

The points $A, B, C, D$ are selected on the circle as followed so that $AB = BC = CD$. Bisectors of $\angle ABD$ and $\angle ACD$ intersect at point $E$. Find $\angle ABC$, if it is known that $AE \parallel CD$.

2024 Polish MO Finals, 1

Tags: geometry , geometry proposed , rectangle , angles

Let $X$ be an interior point of a rectangle $ABCD$. Let the bisectors of $\angle DAX$ and $\angle CBX$ intersect in $P$. A point $Q$ satisfies $\angle QAP=\angle QBP=90^\circ$. Show that $PX=QX$.

Kyiv City MO Juniors 2003+ geometry, 2016.8.5

Tags: geometry , angle bisector , angles

In the triangle $ABC$ the angle bisectors $AD$ and $BE$ are drawn. Prove that $\angle ACB = 60 {} ^ \circ$ if and only if $AE + BD = AB$. (Hilko Danilo)

2009 Balkan MO Shortlist, G1

Tags: geometry , angles , Angle Chasing , angle bisector , equal segments

In the triangle $ABC, \angle BAC$ is acute, the angle bisector of $\angle BAC$ meets $BC$ at $D, K$ is the foot of the perpendicular from $B$ to $AC$, and $\angle ADB = 45^o$. Point $P$ lies between $K$ and $C$ such that $\angle KDP = 30^o$. Point $Q$ lies on the ray $DP$ such that $DQ = DK$. The perpendicular at $P$ to $AC$ meets $KD$ at $L$. Prove that $PL^2 = DQ \cdot PQ$.

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

Tags: geometry , isosceles , right triangle , angles

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 $?

2009 Postal Coaching, 4

Tags: algebra , angles , min , max , number theory

For positive integers $n \ge 3$ and $r \ge 1$, define $$P(n, r) = (n - 2)\frac{r^2}{2} - (n - 4) \frac{r}{2}$$ We call a triple $(a, b, c)$ of natural numbers, with $a \le b \le c$, an $n$-gonal Pythagorean triple if $P(n, a)+P(n, b) = P(n, c)$. (For $n = 4$, we get the usual Pythagorean triple.) (a) Find an $n$-gonal Pythagorean triple for each $n \ge 3$. (b) Consider all triangles $ABC$ whose sides are $n$-gonal Pythagorean triples for some $n \ge 3$. Find the maximum and the minimum possible values of angle $C$.

2017 Baltic Way, 15

Tags: geometry , geometry proposed , angles , polygon

Let $n \ge 3$ be an integer. What is the largest possible number of interior angles greater than $180^\circ$ in an $n$-gon in the plane, given that the $n$-gon does not intersect itself and all its sides have the same length?

Ukrainian From Tasks to Tasks - geometry, 2012.4

Tags: geometry , trapezoid , angles

Let $ABCD$ be an isosceles trapezoid ($AD\parallel BC$), $\angle BAD = 80^o$, $\angle BDA = 60^o$. Point $P$ lies on $CD$ and $\angle PAD = 50^o$. Find $\angle PBC$

2009 District Olympiad, 4

Tags: angles , geometry , isosceles , Equilateral

Let $ABC$ be an equilateral $ABC$. Points $M, N, P$ are located on the sides $AC, AB, BC$, respectively, such that $\angle CBM= \frac{1}{2} \angle AMN = \frac{1}{3} \angle BNP$ and $\angle CMP = 90 ^o$. a) Show that $\vartriangle NMB$ is isosceles. b) Determine $\angle CBM$.

Russian TST 2015, P2

Tags: geometry , angles

The triangle $ABC$ is given. Let $P_1$ and $P_2$ be points on the side $AB$ such that $P_2$ lies on the segment $BP_1$ and $AP_1 = BP_2$. Similarly, $Q_1$ and $Q_2$ are points on the side $BC$ such that $Q_2$ lies on the segment $BQ_1$ and $BQ_1 = CQ_2$. The segments $P_1Q_2$ and $P_2Q_1$ intersect at the point $R{}$, and the circles $P_1P_2R$ and $Q_1Q_2R$ intersect a second time at the point $S{}$ lying inside the triangle $P_1Q_1R$. Let $M{}$ be the midpoint of the segment $AC$. Prove that the angles $P_1RS$ and $Q_1RM$ are equal.

1980 Austrian-Polish Competition, 3

Tags: geometry , 3D geometry , tetrahedron , sphere , triangle inequality , angles , IMO Shortlist

Prove that the sum of the six angles subtended at an interior point of a tetrahedron by its six edges is greater than 540°.

2020 BMT Fall, Tie 1

Tags: geometry , angles

An [i]exterior [/i] angle is the supplementary angle to an interior angle in a polygon. What is the sum of the exterior angles of a triangle and dodecagon ($12$-gon), in degrees?