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: 721

2021 Novosibirsk Oral Olympiad in Geometry, 5
Tags: geometry , equal segments , angle
In an acute-angled triangle $ABC$ on the side $AC$, point $P$ is chosen in such a way that $2AP = BC$. Points $X$ and $Y$ are symmetric to $P$ with respect to vertices $A$ and $C$, respectively. It turned out that $BX = BY$. Find $\angle BCA$.

2002 Junior Balkan Team Selection Tests - Romania, 3
Tags: angle chasing , equal segments , angle , geometry
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 $AMN$.

1964 German National Olympiad, 5
Tags: geometry , angle
A triangle $ABC$ with $\beta= 45^o$ is given. There is a point $P$ on side $BC$, where $BP : PC =1 : 2$ (inner division) and $\angle APC = 60^o$. Someone claims that you can do it with elementary geometric theorems alone without using the plane trigonometry, the size of the angle $\gamma$ determine.

2017 Swedish Mathematical Competition, 4
Tags: geometry , angle bisector , angle chasing , angle
Let $D$ be the foot of the altitude towards $BC$ in the triangle $ABC$. Let $E$ be the intersection of $AB$ with the bisector of angle $\angle C$. Assume that the angle $\angle AEC = 45^o$ . Determine the angle $\angle EDB$.

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)

2019 Oral Moscow Geometry Olympiad, 4
Tags: excircle , geometry , angle , right triangle
Given a right triangle $ABC$ ($\angle C=90^o$). The $C$-excircle touches the hypotenuse $AB$ at point $C_1, A_1$ is the touchpoint of $B$-excircle with line $BC, B_1$ is the touchpoint of $A$-excircle with line $AC$. Find the angle $\angle A_1C_1B_1$.

Novosibirsk Oral Geo Oly IX, 2020.5
Tags: geometry , angle bisector , angle
Angle bisectors $AA', BB'$and $CC'$ are drawn in triangle $ABC$ with angle $\angle B= 120^o$. Find $\angle A'B'C'$.

Novosibirsk Oral Geo Oly VIII, 2017.2
Tags: geometry , angle
You are given a convex quadrilateral $ABCD$. It is known that $\angle CAD = \angle DBA = 40^o$, $\angle CAB = 60^o$, $\angle CBD = 20^o$. Find the angle $\angle CDB $.

2020 Novosibirsk Oral Olympiad in Geometry, 7
Tags: angle bisector , angle , geometry
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 $C$.

Ukraine Correspondence MO - geometry, 2016.11
Tags: geometry , trapezoid , square , angle
Inside the square $ABCD$ mark the point $P$, for which $\angle BAP = 30^o$ and $\angle BCP = 15^o$. The point $Q$ was chosen so that $APCQ$ is an isosceles trapezoid ($PC\parallel AQ$). Find the angles of the triangle $CAM$, where $M$ is the midpoint of $PQ$.

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 $\angle BEC$.

May Olympiad L2 - geometry, 1996.4
Tags: geometry , square , angle
Let $ABCD$ be a square and let point $F$ be any point on side $BC$. Let the line perpendicular to $DF$, that passes through $B$, intersect line $DC$ at $Q$. What is value of $\angle FQC$?

2009 Denmark MO - Mohr Contest, 1
Tags: geometry , rotation , angle
In the figure, triangle $ADE$ is produced from triangle $ABC$ by a rotation by $90^o$ about the point $A$. If angle $D$ is $60^o$ and angle $E$ is $40^o$, how large is then angle $u$? [img]https://1.bp.blogspot.com/-6Fq2WUcP-IA/Xzb9G7-H8jI/AAAAAAAAMWY/hfMEAQIsfTYVTdpd1Hfx15QPxHmfDLEkgCLcBGAsYHQ/s0/2009%2BMohr%2Bp1.png[/img]

1995 Swedish Mathematical Competition, 5
Tags: cyclic , cyclic quadrilateral , angle , geometry
On a circle with center $O$ and radius $r$ are given points $A,B,C,D$ in this order such that $AB, BC$ and $CD$ have the same length $s$ and the length of $AD$ is $s+ r$.Assume that $s < r$. Determine the angles of quadrilateral $ABCD$.

2017 Junior Balkan Team Selection Tests - Romania, 2
Tags: circles , circumcircle , angle , geometry
Let $A$ be a point outside the circle $\omega$ . The tangents from $A$ touch the circle at $B$ and $C$. Let $P$ be an arbitrary point on extension of $AC$ towards $C$, $Q$ the projection of $C$ onto $PB$ and $E$ the second intersection point of the circumcircle of $ABP$ with the circle $\omega$ . Prove that $\angle PEQ = 2\angle APB$

2008 Thailand Mathematical Olympiad, 1
Tags: geometry , trigonometry , angle
Let $\vartriangle ABC$ be a triangle with $\angle BAC = 90^o$ and $\angle ABC = 60^o$. Point $E$ is chosen on side $BC$ so that $BE : EC = 3 : 2$. Compute $\cos\angle CAE$.

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

2002 Estonia Team Selection Test, 4
Tags: equal angles , cyclic , geometry , angle
Let $ABCD$ be a cyclic quadrilateral such that $\angle ACB = 2\angle CAD$ and $\angle ACD = 2\angle BAC$. Prove that $|CA| = |CB| + |CD|$.

Ukraine Correspondence MO - geometry, 2017.8
Tags: geometry , trapezoid , min , angle
On the midline of the isosceles trapezoid $ABCD$ ($BC \parallel AD$) find the point $K$, for which the sum of the angles $\angle DAK + \angle BCK$ will be the smallest.

2017 Estonia Team Selection Test, 10
Tags: semicircle , angle , geometry
Let $ABC$ be a triangle with $AB = \frac{AC}{2 }+ BC$. Consider the two semicircles outside the triangle with diameters $AB$ and $BC$. Let $X$ be the orthogonal projection of $A$ onto the common tangent line of those semicircles. Find $\angle CAX$.

2016 Bulgaria JBMO TST, 1
Tags: geometry , geometric inequality , cyclic quadrilateral , angle
The quadrilateral $ABCD$, in which $\angle BAC < \angle DCB$ , is inscribed in a circle $c$, with center $O$. If $\angle BOD = \angle ADC = \alpha$. Find out which values of $\alpha$ the inequality $AB <AD + CD$ occurs.

1994 Tournament Of Towns, (437) 3
Tags: geometry , median , angle
The median $AD$ of triangle $ABC$ intersects its inscribed circle (with center $O$) at the points $X$ and $Y$. Find the angle $XOY$ if $AC = AB + AD$. (A Fedotov)

1981 Bulgaria National Olympiad, Problem 2
Tags: geometry , angle , triangle
Let $ABC$ be a triangle such that the altitude $CH$ and the sides $CA,CB$ are respectively equal to a side and two distinct diagonals of a regular heptagon. Prove that $\angle ACB<120^\circ$.

2023 Novosibirsk Oral Olympiad in Geometry, 7
Tags: geometry , angle
Triangle $ABC$ is given with angles $\angle ABC = 60^o$ and $\angle BCA = 100^o$. On the sides AB and AC, the points $D$ and $E$ are chosen, respectively, in such a way that $\angle EDC = 2\angle BCD = 2\angle CAB$. Find the angle $\angle BED$.

2001 Estonia National Olympiad, 1
Tags: convex , polygon , geometry , angle
The angles of a convex $n$-gon are $a,2a, ... ,na$. Find all possible values of $n$ and the corresponding values of $a$.