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

2010 Junior Balkan Team Selection Tests - Romania, 2
Tags: midpoints , geometry , angles , equal angles
Let $ABC$ be a triangle and $D, E, F$ the midpoints of the sides $BC, CA, AB$ respectively. Show that $\angle DAC = \angle ABE$ if and only if $\angle AFC = \angle BDA$

1997 Tournament Of Towns, (534) 6
Tags: geometry , angles , Isosceles Triangle , isosceles
Let $P$ be a point inside the triangle $ABC$ such that $AB = BC$, $\angle ABC = 80^o$, $\angle PAC = 40^o$ and $\angle ACP = 30^o$. Find $\angle BPC$. (G Galperin)

1993 Rioplatense Mathematical Olympiad, Level 3, 6
Tags: right triangle , geometry , pentagon , equal segments , angles
Let $ABCDE$ be pentagon such that $AE = ED$ and $BC = CD$. It is known that $\angle BAE + \angle EDC + \angle CB A = 360^o$ and that $P$ is the midpoint of $AB$. Show that the triangle $ECP$ is right.

1957 Moscow Mathematical Olympiad, 362
Tags: geometry , angles , incircle , circumcircle , similar , similar triangles
(a) A circle is inscribed in a triangle. The tangent points are the vertices of a second triangle in which another circle is inscribed. Its tangency points are the vertices of a third triangle. The angles of this triangle are identical to those of the first triangle. Find these angles. (b) A circle is inscribed in a scalene triangle. The tangent points are vertices of another triangle, in which a circle is inscribed whose tangent points are vertices of a third triangle, in which a third circle is inscribed, etc. Prove that the resulting sequence does not contain a pair of similar triangles.

Estonia Open Senior - geometry, 2011.1.5
Tags: geometry , angles
Given a triangle $ABC$ where $|BC| = a, |CA| = b$ and $|AB| = c$, prove that the equality $\frac{1}{a + b}+\frac{1}{b + c}=\frac{3}{a + b + c}$ holds if and only if $\angle ABC = 60^o$.

1998 All-Russian Olympiad Regional Round, 8.7
Tags: geometry , angles
Let $O$ be the center of a circle circumscribed about an acute angle triangle $ABC$, $S_A$, $S_B$, $S_C$ - circles with center O, tangent to sides $BC$, $CA$, $AB$ respectively. Prove that the sum of three angles : between the tangents to $S_A$ drawn from point $A$, to $S_B$ from point $B$ and to $S_C$ - from point $C$, is equal to $180^o$.

2010 Junior Balkan Team Selection Tests - Romania, 4
Tags: angles , geometry , equal segments , equal angles , isosceles
Let $ABC$ be an isosceles triangle with $AB = AC$ and let $n$ be a natural number, $n>1$. On the side $AB$ we consider the point $M$ such that $n \cdot AM = AB$. On the side $BC$ we consider the points $P_1, P_2, ....., P_ {n-1}$ such that $BP_1 = P_1P_2 = .... = P_ {n-1} C = \frac{1}{n} BC$. Show that: $\angle {MP_1A} + \angle {MP_2A} + .... + \angle {MP_ {n-1} A} = \frac{1} {2} \angle {BAC}$.

1953 Czech and Slovak Olympiad III A, 2
Tags: geometry , angles , Triangle
Let $\alpha,\beta,\gamma$ be angles of a triangle. Two of them can be expressed using an auxiliary angle $\varphi$ in a way that $$\alpha=\varphi+\frac\pi4,\quad\beta=\pi-3\varphi.$$ Show that $\alpha>\gamma.$

1955 Moscow Mathematical Olympiad, 288
Tags: geometry , area , bisect segment , angles , right triangle
We are given a right triangle $ABC$ and the median $BD$ drawn from the vertex $B$ of the right angle. Let the circle inscribed in $\vartriangle ABD$ be tangent to side $AD$ at $K$. Find the angles of $\vartriangle ABC$ if $K$ divides $AD$ in halves.

1999 Tournament Of Towns, 1
Tags: geometry , incenter , similar triangles , similar , angles
The incentre of a triangle is joined by three segments to the three vertices of the triangle, thereby dividing it into three smaller triangles. If one of these three triangles is similar to the original triangle, find its angles. (A Shapovalov)

2018 Dutch BxMO TST, 4
Tags: geometry , angles , circumcircle , Angle Chasing
In a non-isosceles triangle $\vartriangle ABC$ we have $\angle BAC = 60^o$. Let $D$ be the intersection of the angular bisector of $\angle BAC$ with side $BC, O$ the centre of the circumcircle of $\vartriangle ABC$ and $E$ the intersection of $AO$ and $BC$. Prove that $\angle AED + \angle ADO = 90^o$.

2015 Iran Geometry Olympiad, 2
Tags: geometry , Elementary , angles
Let $ABC$ be a triangle with $\angle A = 60^o$. The points $M,N,K$ lie on $BC,AC,AB$ respectively such that $BK = KM = MN = NC$. If $AN = 2AK$, find the values of $\angle B$ and $\angle C$. by Mahdi Etesami Fard

2022 Yasinsky Geometry Olympiad, 3
Tags: geometry , isosceles right triangle , angles
In an isosceles right triangle $ABC$ with a right angle $C$, point $M$ is the midpoint of leg $AC$. At the perpendicular bisector of $AC$, point $D$ was chosen such that $\angle CDM = 30^o$, and $D$ and $B$ lie on different sides of $AC$. Find the angle $\angle ABD$. (Volodymyr Petruk)

2007 Junior Balkan Team Selection Tests - Moldova, 6
Tags: angles , trinomial , 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 $18^o$

2010 Silk Road, 1
Tags: geometry , convex quadrilateral , angles , right triangle
In a convex quadrilateral it is known $ABCD$ that $\angle ADB + \angle ACB = \angle CAB + \angle DBA = 30^{\circ}$ and $AD = BC$. Prove that from the lengths $DB$, $CA$ and $DC$, you can make a right triangle.

VII Soros Olympiad 2000 - 01, 10.8
Tags: geometry , trigonometry , angles , geometric inequality
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 $2 \sin 18^o$ .

2008 Junior Balkan Team Selection Tests - Moldova, 3
Tags: geometry , rhombus , congruent , angles
Rhombuses $ABCD$ and $A_1B_1C_1D_1$ are equal. Side $BC$ intersects sides $B_1C_1$ and $C_1D_1$ at points $M$ and $N$ respectively. Side $AD$ intersects sides $A_1B_1$ and $A_1D_1$ at points $Q$ and $P$ respectively. Let $O$ be the intersection point of lines $MP$ and $QN$. Find $\angle A_1B_1C_1$ , if $\angle QOP = \frac12 \angle B_1C_1D_1$.

1998 Bundeswettbewerb Mathematik, 3
Tags: geometry , angles
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$.

Denmark (Mohr) - geometry, 2018.5
Tags: geometry , angles , angle bisector
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]

Kyiv City MO 1984-93 - geometry, 1988.7.1
Tags: geometry , trapezoid , Equilateral , angles
An isosceles trapezoid is divided by each diagonal into two isosceles triangles. Determine the angles of the trapezoid.

Novosibirsk Oral Geo Oly VII, 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.

1987 Tournament Of Towns, (151) 2
Tags: geometry , rhombus , Locus , angles
Find the locus of points $M$ inside the rhombus $ABCD$ such that the sum of angles $AMB$ and $CMD$ equals $180^o$ .

2022 May Olympiad, 3
Tags: geometry , angles
Let $ABCD$ be a square, $E$ a point on the side $CD$, and $F$ a point inside the square such that that triangle $BFE$ is isosceles and $\angle BFE = 90^o$ . If $DF=DE$, find the measure of angle $\angle FDE$.

1950 Moscow Mathematical Olympiad, 178
Tags: trigonometry , angles
Let $A$ be an arbitrary angle,let $B$ and $C$ be acute angles. Is there an angle $x$ such that $$\sin x =\frac{\sin B \cdot \sin C}{1 - \cos B \cdot \cos C \cdot \cos A} ?$$

2010 Oral Moscow Geometry Olympiad, 5
Tags: square , angles , equal segments
Points $K$ and $M$ are taken on the sides $AB$ and $CD$ of square $ABCD$, respectively, and on the diagonal $AC$ - point $L$ such that $ML = KL$. Let $P$ be the intersection point of the segments $MK$ and $BD$. Find the angle $\angle KPL$.