Olympiad problems

1997 All-Russian Olympiad Regional Round, 8.3

On sides $AB$ and $BC$ of an equilateral triangle $ABC$ are taken points $D$ and $K$, and on the side $AC$ , points $E$ and $M$ so that $DA + AE = KC +CM = AB$. Prove that the angle between lines $DM$ and $KE$ is equal to $60^o$.

2021 Iranian Geometry Olympiad, 3

Tags: geometry , perimeter , angles , semicircle

As shown in the following figure, a heart is a shape consist of three semicircles with diameters $AB$, $BC$ and $AC$ such that $B$ is midpoint of the segment $AC$. A heart $\omega$ is given. Call a pair $(P, P')$ bisector if $P$ and $P'$ lie on $\omega$ and bisect its perimeter. Let $(P, P')$ and $(Q,Q')$ be bisector pairs. Tangents at points $P, P', Q$, and $Q'$ to $\omega$ construct a convex quadrilateral $XYZT$. If the quadrilateral $XYZT$ is inscribed in a circle, find the angle between lines $PP'$ and $QQ'$. [img]https://cdn.artofproblemsolving.com/attachments/3/c/8216889594bbb504372d8cddfac73b9f56e74c.png[/img] [i]Proposed by Mahdi Etesamifard - Iran[/i]

Estonia Open Senior - geometry, 2020.2.5

Tags: geometry , angles , ratio

The bisector of the interior angle at the vertex $B$ of the triangle $ABC$ and the perpendicular line on side $BC$ passing through the vertex $C$ intersects at $D$. Let $M$ and $N$ be the midpoints of the segments $BC$ and $BD$, respectively, with $N$ on the side $AC$. Find all possibilities of the angles of the triangles $ABC$, if it is known that $\frac{| AM |}{| BC |}=\frac{|CD|}{|BD|}$. .

1996 Estonia National Olympiad, 2

Tags: geometry , trapezoid , angles , equal segments

Three sides of a trapezoid are equal, and a circle with the longer base as a diameter halves the two non-parallel sides. Find the angles of the trapezoid.

2007 District Olympiad, 2

Tags: geometry , 3D geometry , rectangle , angles , rectnagle , perpendicular bisector , midpoint

Consider a rectangle $ABCD$ with $AB = 2$ and $BC = \sqrt3$. The point $M$ lies on the side $AD$ so that $MD = 2 AM$ and the point $N$ is the midpoint of the segment $AB$. On the plane of the rectangle rises the perpendicular MP and we choose the point $Q$ on the segment $MP$ such that the measure of the angle between the planes $(MPC)$ and $(NPC)$ shall be $45^o$, and the measure of the angle between the planes $(MPC)$ and $(QNC)$ shall be $60^o$. a) Show that the lines $DN$ and $CM$ are perpendicular. b) Show that the point $Q$ is the midpoint of the segment $MP$.

Brazil L2 Finals (OBM) - geometry, 2005.6

Tags: angle bisector , angles , geometry , Angle Chasing

The angle $B$ of a triangle $ABC$ is $120^o$. Let $M$ be a point on the side $AC$ and $K$ a point on the extension of the side $AB$, such that $BM$ is the internal bisector of the angle $\angle ABC$ and $CK$ is the external bisector corresponding to the angle $\angle ACB$ . The segment $MK$ intersects $BC$ at point $P$. Prove that $\angle APM = 30^o$.

2012 India Regional Mathematical Olympiad, 7

Tags: geometry , Angle Chasing , circle , angles

On the extension of chord $AB$ of a circle centroid at $O$ a point $X$ is taken and tangents $XC$ and $XD$ to the circle are drawn from it with $C$ and $D$ lying on the circle, let $E$ be the midpoint of the line segment $CD$. If $\angle OEB = 140^o$ then determine with proof the magnitude of $\angle AOB$.

Kyiv City MO Seniors 2003+ geometry, 2018.10.4

Tags: geometry , angles , altitudes

In the acute-angled triangle $ABC$, the altitudes $BP$ and $CQ$ were drawn, and the point $T$ is the intersection point of the altitudes of $\Delta PAQ$. It turned out that $\angle CTB = 90 {} ^ \circ$. Find the measure of $\angle BAC$. (Mikhail Plotnikov)

Kyiv City MO Juniors Round2 2010+ geometry, 2013.8.3

Tags: geometry , angles , equal angles , parallel

Inside $\angle BAC = 45 {} ^ \circ$ the point $P$ is selected that the conditions $\angle APB = \angle APC = 45 {} ^ \circ $ are fulfilled. Let the points $M$ and $N$ be the projections of the point $P$ on the lines $AB$ and $AC$, respectively. Prove that $BC\parallel MN $. (Serdyuk Nazar)

1983 IMO, 2

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 $\angle O_1AO_2=\angle M_1AM_2$.

2003 District Olympiad, 1

Tags: 3D geometry , geometry , angles , planes , Equilateral

Let $ABC$ be an equilateral triangle. On the plane $(ABC)$ rise the perpendiculars $AA'$ and $BB'$ on the same side of the plane, so that $AA' = AB$ and $BB' =\frac12 AB$. Determine the measure the angle between the planes $(ABC)$ and $(A'B'C')$.

1986 Brazil National Olympiad, 3

Tags: geometry , combinatorial geometry , angles

The Poincare plane is a half-plane bounded by a line $R$. The lines are taken to be (1) the half-lines perpendicular to $R$, and (2) the semicircles with center on $R$. Show that given any line $L$ and any point $P$ not on $L$, there are infinitely many lines through $P$ which do not intersect $L$. Show that if $ ABC$ is a triangle, then the sum of its angles lies in the interval $(0, \pi)$.

1997 Swedish Mathematical Competition, 2

Tags: geometry , angles , equal angles , equal segments

Let $D$ be the point on side $AC$ of a triangle $ABC$ such that $BD$ bisects $\angle B$, and $E$ be the point on side $AB$ such that $3\angle ACE = 2\angle BCE$. Suppose that $BD$ and $CE$ intersect at a point $P$ with $ED = DC = CP$. Determine the angles of the triangle.

2009 Cuba MO, 8

Tags: geometry , angles , isosceles

Let $ABC$ be an isosceles triangle with base $BC$ and $\angle BAC = 20^o$. Let $D$ a point on side $AB$ such that $AD = BC$. Determine $\angle DCA$.

2005 iTest, 29

Tags: geometry , angles

$WHY$ is a triangle with angle $W \ge 90$ degrees. On the side $HY$, two distinct points $M$ and $E$ are chosen such that angle $HWM$ is equivalent to angle $ MWE$ and $HM * YE = HY * ME$. Find the angle $MWY$.

2009 Denmark MO - Mohr Contest, 1

Tags: geometry , rotation , angles

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]

2011 Saudi Arabia IMO TST, 2

Tags: obtuse , angles , geometry

Let $ABC$ be a non-isosceles triangle with circumcenter $O$, incenter $I$, and orthocenter $H$. Prove that angle $\angle OIH$ is obtuse.

Ukrainian TYM Qualifying - geometry, 2016.15

Tags: geometry , angles , Ukrainian TYM

A non isosceles triangle $ABC$ is given, in which $\angle A = 120^o$. Let $AL$ be its angle bisector, $AK$ be it's median, drawn from vertex $A$, point $O$ be the center of the circumcircle of this triangle, $F$ be the point of intersection of the lines $OL$ and $AK$. Prove that $\angle BFC = 60^o$.

Kyiv City MO Juniors Round2 2010+ geometry, 2012.9.4

Tags: geometry , circumcircle , angles , orthocenter , parallel , equal segments

In an acute-angled triangle $ABC$, the point $O$ is the center of the circumcircle, and the point $H$ is the orthocenter. It is known that the lines $OH$ and $BC$ are parallel, and $BC = 4OH $. Find the value of the smallest angle of triangle $ ABC $. (Black Maxim)

Ukraine Correspondence MO - geometry, 2005.4

Tags: geometry , angles , angle bisector , Ukraine Correspondence

The bisectors of the angles $A$ and $B$ of the triangle $ABC$ intersect the sides $BC$ and $AC$ at points $D$ and $E$. It is known that $AE + BD = AB$. Find the angle $\angle C$.

2016 Iranian Geometry Olympiad, 5

Tags: geometry , angles , quadrilateral

Let $ABCD$ be a convex quadrilateral with these properties: $\angle ADC = 135^o$ and $\angle ADB - \angle ABD = 2\angle DAB = 4\angle CBD$. If $BC = \sqrt2 CD$ , prove that $AB = BC + AD$. by Mahdi Etesami Fard

2007 District Olympiad, 1

Tags: geometry , equal segments , angles , Angle Chasing , Circumcenter , circumcircle

Point $O$ is the intersection of the perpendicular bisectors of the sides of the triangle $\vartriangle ABC$ . Let $D$ be the intersection of the line $AO$ with the segment $[BC]$. Knowing that $OD = BD = \frac 13 BC$, find the measures of the angles of the triangle $\vartriangle ABC$.

1946 Moscow Mathematical Olympiad, 111

Tags: angles , maximum , planes , geometry , 3D geometry

Given two intersecting planes $\alpha$ and $\beta$ and a point $A$ on the line of their intersection. Prove that of all lines belonging to $\alpha$ and passing through $A$ the line which is perpendicular to the intersection line of $\alpha$ and $\beta$ forms the greatest angle with $\beta$.

2021 239 Open Mathematical Olympiad, 1

Tags: geometry , circumcircle , angles , arc midpoint , angle bisector

Points $X$ and $Y$ are the midpoints of arcs $AB$ and $BC$ of the circumscribed circle of triangle $ABC$. Point $T$ lies on side $AC$. It turned out that the bisectors of the angles $ATB$ and $BTC$ pass through points $X$ and $Y$ respectively. What angle $B$ can be in triangle $ABC$?

1995 Swedish Mathematical Competition, 5

Tags: geometry , angles , Cyclic , cyclic quadrilateral

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 $ABCD$.