This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 628

2023 Romania National Olympiad, 4
Tags: geometry , angles
Let $ABC$ be a triangle with $\angle BAC = 90^{\circ}$ and $\angle ACB = 54^{\circ}.$ We construct bisector $BD (D \in AC)$ of angle $ABC$ and consider point $E \in (BD)$ such that $DE = DC.$ Show that $BE = 2 \cdot AD.$

2002 Junior Balkan Team Selection Tests - Romania, 3
Tags: geometry , angles , Angle Chasing , equal segments
Let $ABC$ be an isosceles triangle such that $AB = AC$ and $\angle A = 20^o$. Let $M$ be the foot of the altitude from $C$ and let $N$ be a point on the side $AC$ such that $CN =\frac12 BC$. Determine the measure of the angle $AMN$.

1996 Tournament Of Towns, (485) 3
Tags: geometry , angles , right triangle
The two tangents to the incircle of a right-angled triangle $ABC$ that are perpendicular to the hypotenuse $AB$ intersect it at the points $P$ and $Q$. Find $\angle PCQ$. (M Evdokimov,)

1978 Czech and Slovak Olympiad III A, 3
Tags: algebra , trigonometry , system of equations , angles
Let $\alpha,\beta,\gamma$ be angles of a triangle. Determine all real triplets $x,y,z$ satisfying the system \begin{align*} x\cos\beta+\frac1z\cos\alpha &=1, \\ y\cos\gamma+\frac1x\cos\beta &=1, \\ z\cos\alpha+\frac1y\cos\gamma &=1. \end{align*}

2021 Novosibirsk Oral Olympiad in Geometry, 4
Tags: geometry , angle bisector , angles
Angle bisectors $AD$ and $BE$ are drawn in triangle $ABC$. It turned out that $DE$ is the bisector of triangle $ADC$. Find the angle $BAC$.

II Soros Olympiad 1995 - 96 (Russia), 9.5
Tags: geometry , angles
Angle $A$ of triangle $ABC$ is $33^o$. A straight line passing through $A$ perpendicular to $AC$ intersects straight line $BC$ at point $D$ so that $CD = 2AB$. What is angle $C$ of triangle $ABC$? (Please list all options.)

2017 Hanoi Open Mathematics Competitions, 11
Tags: geometry , inequalities , geometric inequality , angles , Equilateral
Let $ABC$ be an equilateral triangle, and let $P$ stand for an arbitrary point inside the triangle. Is it true that $| \angle PAB - \angle PAC| \ge | \angle PBC - \angle PCB|$ ?

2006 Sharygin Geometry Olympiad, 9.4
Tags: hexagon , cut , geometry , angles
In a non-convex hexagon, each angle is either $90$ or $270$ degrees. Is it true that for some lengths of the sides it can be cut into two hexagons similar to it and unequal to each other?

1983 IMO Shortlist, 23
Tags: geometry , circles , angles , midpoint , IMO , IMO 1983
Let $A$ be one of the two distinct points of intersection of two unequal coplanar circles $C_1$ and $C_2$ with centers $O_1$ and $O_2$ respectively. One of the common tangents to the circles touches $C_1$ at $P_1$ and $C_2$ at $P_2$, while the other touches $C_1$ at $Q_1$ and $C_2$ at $Q_2$. Let $M_1$ be the midpoint of $P_1Q_1$ and $M_2$ the midpoint of $P_2Q_2$. Prove that $\angle O_1AO_2=\angle M_1AM_2$.

2005 Estonia Team Selection Test, 1
Tags: circles , angles , tangent , geometry
On a plane, a line $\ell$ and two circles $c_1$ and $c_2$ of different radii are given such that $\ell$ touches both circles at point $P$. Point $M \ne P$ on $\ell$ is chosen so that the angle $Q_1MQ_2$ is as large as possible where $Q_1$ and $Q_2$ are the tangency points of the tangent lines drawn from $M$ to $c_i$ and $c_2$, respectively, differing from $\ell$ . Find $\angle PMQ_1 + \angle PMQ_2$·

I Soros Olympiad 1994-95 (Rus + Ukr), 11.3
Tags: geometry , angles , geometric inequality
It is known that in the triangle $ABC$, $ 2 \angle BAC + 3 \angle ABC= 180^o$. Prove that $4(BC + CA)< 5AB$.

Kyiv City MO Juniors 2003+ geometry, 2018.8.3
Tags: geometry , rectangle , isosceles , angles
In the isosceles triangle $ABC$ with the vertex at the point $B$, the altitudes $BH$ and $CL$ are drawn. The point $D$ is such that $BDCH$ is a rectangle. Find the value of the angle $DLH$. (Bogdan Rublev)

1995 Czech And Slovak Olympiad IIIA, 5
Tags: angles , tangent circles , geometry
Let $A,B$ be points on a circle $k$ with center $S$ such that $\angle ASB = 90^o$ . Circles $k_1$ and $k_2$ are tangent to each other at $Z$ and touch $k$ at $A$ and $B$ respectively. Circle $k_3$ inside $\angle ASB$ is internally tangent to $k$ at $C$ and externally tangent to $k_1$ and $k_2$ at $X$ and $Y$, respectively. Prove that $\angle XCY = 45^o$

Kyiv City MO Seniors Round2 2010+ geometry, 2021.10.4
Tags: geometry , Concyclic , angles
Inside the quadrilateral $ABCD$ marked a point $O$ such that $\angle OAD+ \angle OBC = \angle ODA + \angle OCB = 90^o$. Prove that the centers of the circumscribed circles around triangles $OAD$ and $OBC$ as well as the midpoints of the sides $AB$ and $CD$ lie on one circle. (Anton Trygub)

1995 Tournament Of Towns, (443) 3
Tags: geometry , Squares , angles
Suppose $L$ is the circle inscribed in the square $T_1$, and $T_2$ is the square inscribed in $L$, so that vertices of $T_1$ lie on the straight lines containing the sides of $T_2$. Find the angles of the convex octagon whose vertices are at the tangency points of $L$ with the sides of $T_1$ and at the vertices of $T_2$. (S Markelov)

2014 Junior Balkan Team Selection Tests - Moldova, 7
Tags: angles , geometry , equal segments , isosceles , right triangle
Let the isosceles right triangle $ABC$ with $\angle A= 90^o$ . The points $E$ and $F$ are taken on the ray $AC$ so that $\angle ABE = 15^o$ and $CE = CF$. Determine the measure of the angle $CBF$.

1996 Israel National Olympiad, 3
Tags: trigonometry , median , angle bisector , angles , geometry , altitude , concurrent
The angles of an acute triangle $ABC$ at $\alpha , \beta, \gamma$. Let $AD$ be a height, $CF$ a median, and $BE$ the bisector of $\angle B$. Show that $AD,CF$ and $BE$ are concurrent if and only if $\cos \gamma \tan\beta = \sin \alpha$ .

2000 ITAMO, 2
Tags: geometry , angles
Let $ABCD$ be a convex quadrilateral, and write $\alpha=\angle DAB$, $\beta=\angle ADB$, $\gamma=\angle ACB$, $\delta= \angle DBC$ and $\epsilon=\angle DBA$. Assuming that $\alpha<\pi/2$, $\beta+\gamma=\pi /2$, and $\delta+2\epsilon=\pi$, prove that $(DB+BC)^2=AD^2+AC^2$.

1997 Tournament Of Towns, (539) 4
Tags: geometry , 3D geometry , tetrahedron , sphere , angles
All edges of a tetrahedron $ABCD$ are equal. The tetrahedron $ABCD$ is inscribed in a sphere. $CC'$ and $DD'$ are diameters. Find the angle between the planes $ABC$' and $ACD'$. (A Zaslavskiy)

2007 Greece JBMO TST, 1
Tags: circumcircle , geometry , angles
Let $ABC$ be a triangle with $\angle A=105^o$ and $\angle C=\frac{1}{4} \angle B$. a) Find the angles $\angle B$ and $\angle C$ b) Let $O$ be the center of the circumscribed circle of the triangle $ABC$ and let $BD$ be a diameter of that circle. Prove that the distance of point $C$ from the line $BD$ is equal to $\frac{BD}{4}$.

2022 Portugal MO, 4
Tags: angles , median , geometry
Let $[AD]$ be a median of the triangle $[ABC]$. Knowing that $\angle ADB = 45^o$ and $\angle A CB = 30^o$, prove that $\angle BAD = 30^o$.

Novosibirsk Oral Geo Oly VIII, 2020.6
Tags: geometry , angle bisector , angles
Angle bisectors $AA', BB'$and $CC'$ are drawn in triangle $ABC$ with angle $\angle B= 120^o$. Find $\angle A'B'C'$.

Kyiv City MO 1984-93 - geometry, 1990.8.2
Tags: geometry , Equilateral , incircle , geometric inequality , min , angles
A line passes through the center $O$ of an equilateral triangle $ABC$ and intersects the side $BC$. At what angle wrt $BC$ should this line be drawn this line so that its segment inside the triangle has the smallest possible length?

1952 Moscow Mathematical Olympiad, 208
Tags: geometry , acute , angles , incircle
The circle is inscribed in $\vartriangle ABC$. Let $L, M, N$ be the tangent points of the circle with sides $AB, AC, BC$, respectively. Prove that $\angle MLN$ is always an acute angle.

Durer Math Competition CD 1st Round - geometry, 2017.C1
Tags: Durer , geometry , angles , Decagon
The vertices of Durer's favorite regular decagon in clockwise order: $D_1, D_2, D_3, . . . , D_{10}$. What is the angle between the diagonals $D_1D_3$ and $D_2D_5$?