Found problems: 721
Novosibirsk Oral Geo Oly VIII, 2021.4
Angle bisectors $AD$ and $BE$ are drawn in triangle $ABC$. It turned out that $DE$ is the bisector of triangle $ADC$. Find the angle $BAC$.
2010 Flanders Math Olympiad, 3
In a triangle $ABC$, $\angle B= 2\angle A \ne 90^o$ . The inner bisector of $B$ intersects the perpendicular bisector of $[AC]$ at a point $D$. Prove that $AB \parallel CD$.
Kyiv City MO 1984-93 - geometry, 1990.8.2
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?
1990 IMO Longlists, 25
The incenter of the triangle $ ABC$ is $ K.$ The midpoint of $ AB$ is $ C_1$ and that of $ AC$ is $ B_1.$ The lines $ C_1K$ and $ AC$ meet at $ B_2,$ the lines $ B_1K$ and $ AB$ at $ C_2.$ If the areas of the triangles $ AB_2C_2$ and $ ABC$ are equal, what is the measure of angle $ \angle CAB?$
Kyiv City MO Juniors Round2 2010+ geometry, 2010.89.3
In the acute-angled triangle $ABC$ the angle$ \angle B = 30^o$, point $H$ is the intersection point of its altitudes. Denote by $O_1, O_2$ the centers of circles inscribed in triangles $ABH ,CBH$ respectively. Find the degree of the angle between the lines $AO_2$ and $CO_1$.
1993 Rioplatense Mathematical Olympiad, Level 3, 6
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.
2021 Yasinsky Geometry Olympiad, 3
The segments $AC$ and $BD$ are perpendicular, and $AC$ is twice as large as $BD$ and intersects $BD$ in it in the midpoint. Find the value of the angle $BAD$, if we know that $\angle CAD = \angle CDB$.
(Gregory Filippovsky)
2011 Saudi Arabia IMO TST, 2
Let $ABC$ be a non-isosceles triangle with circumcenter $O$, incenter $I$, and orthocenter $H$. Prove that angle $\angle OIH$ is obtuse.
2010 Silk Road, 1
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.
2019 Auckland Mathematical Olympiad, 1
Given a convex quadrilateral $ABCD$ in which $\angle BAC = 20^o$, $\angle CAD = 60^o$, $\angle ADB = 50^o$ , and $\angle BDC = 10^o$. Find $\angle ACB$.
2008 Balkan MO Shortlist, G3
We draw two lines $(\ell_1) , (\ell_2)$ through the orthocenter $H$ of the triangle $ABC$ such that each one is dividing the triangle into two figures of equal area and equal perimeters. Find the angles of the triangle.
2015 Gulf Math Olympiad, 2
a) Let $UVW$ , $U'V'W'$ be two triangles such that $ VW = V'W' , UV = U'V' , \angle WUV = \angle W'U'V'.$
Prove that the angles $\angle VWU , \angle V'W'U'$ are equal or supplementary.
b) $ABC$ is a triangle where $\angle A$ is [b]obtuse[/b]. take a point $P$ inside the triangle , and extend $AP,BP,CP$ to meet the sides $BC,CA,AB$ in $K,L,M$ respectively. Suppose that $PL = PM .$
1) If $AP$ bisects $\angle A$ , then prove that $AB = AC$ .
2) Find the angles of the triangle $ABC$ if you know that $AK,BL,CM$ are angle bisectors of the triangle $ABC$ and that $2AK = BL$.
Brazil L2 Finals (OBM) - geometry, 2007.1
Let $ABC$ be a triangle with circumcenter $O$. Let $P$ be the intersection of straight lines $BO$ and $AC$ and $\omega$ be the circumcircle of triangle $AOP$. Suppose that $BO = AP$ and that the measure of the arc $OP$ in $\omega$, that does not contain $A$, is $40^o$. Determine the measure of the angle $\angle OBC$.
[img]https://3.bp.blogspot.com/-h3UVt-yrJ6A/XqBItXzT70I/AAAAAAAAL2Q/7LVv0gmQWbo1_3rn906fTn6wosY1-nIfwCK4BGAYYCw/s1600/2007%2Bomb%2Bl2.png[/img]
1982 All Soviet Union Mathematical Olympiad, 334
Given a point $M$ inside a right tetrahedron. Prove that at least one tetrahedron edge is seen from the $M$ in an angle, that has a cosine not greater than $-1/3$. (e.g. if $A$ and $B$ are the vertices, corresponding to that edge, $cos(\widehat{AMB}) \le -1/3$)
1948 Moscow Mathematical Olympiad, 155
What is the greatest number of rays in space beginning at one point and forming pairwise obtuse angles?
2013 Czech And Slovak Olympiad IIIA, 5
Given the parallelogram $ABCD$ such that the feet $K, L$ of the perpendiculars from point $D$ on the sides $AB, BC$ respectively are internal points. Prove that $KL \parallel AC$ when $|\angle BCA| + |\angle ABD| = |\angle BDA| + |\angle ACD|$.
1991 Spain Mathematical Olympiad, 4
The incircle of $ABC$ touches the sides $BC,CA,AB$ at $A' ,B' ,C'$ respectively. The line $A' C'$ meets the angle bisector of $\angle A$ at $D$. Find $\angle ADC$.
2011 Romania National Olympiad, 4
Consider $\vartriangle ABC$ where $\angle ABC= 60 ^o$. Points $M$ and $D$ are on the sides $(AC)$, respectively $(AB)$, such that $\angle BCA = 2 \angle MBC$, and $BD = MC$. Determine $\angle DMB$.
2015 Oral Moscow Geometry Olympiad, 2
The square $ABCD$ and the equilateral triangle $MKL$ are located as shown in the figure. Find the angle $\angle PQD$.
[img]https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjQKgjvzy1WhwkMJbcV_C0iveelYmm75FpaGlWgZ-Ap_uQUiegaKYafelo-J_3rMgKMgpMp5soYc1LVYLI8H4riC6R-f8eq2DiWTGGII08xQkwu7t2KVD4pKX4_IN-gC7DVRhdVZSjbaj2S/s1600/oral+moscow+geometry+2015+8.9+p2.png[/img]
2005 Sharygin Geometry Olympiad, 9.5
It is given that for no side of the triangle from the height drawn to it, the bisector and the median it is impossible to make a triangle. Prove that one of the angles of the triangle is greater than $135^o$
2007 District Olympiad, 2
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 $MP$.
2015 India PRMO, 20
$20.$ The circle $\omega$ touches the circle $\Omega$ internally at point $P.$ The centre $O$ of $\Omega$ is outside $\omega.$ Let $XY$ be a diameter of $\Omega$ which is also tangent to $\omega.$ Assume $PY>PX.$ Let $PY$ intersect $\omega$ at $z.$ If $YZ=2PZ,$ what is the magnitude of $\angle{PYX}$ in degrees $?$
2019 BMT Spring, 2
Let $A, B, C$ be unique collinear points$ AB = BC =\frac13$. Let $P$ be a point that lies on the circle centered at $B$ with radius $\frac13$ and the circle centered at $C$ with radius $\frac13$ . Find the measure of angle $\angle PAC$ in degrees.
1978 Czech and Slovak Olympiad III A, 3
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 Junior Balkan Team Selection Tests - Moldova, 2
Inside the parallelogram $ABCD$, point $E$ is chosen, such that $AE = DE$ and $\angle ABE = 90^o$. Point $F$ is the midpoint of the side $BC$ . Find the measure of the angle $\angle DFE$.