Found problems: 628
Kyiv City MO 1984-93 - geometry, 1993.9.2
Let $a, b, c$ be the lengths of the sides of a triangle, and let $S$ be its area. We know that $S = \frac14 (c^2 - a^2 - b^2)$. Prove that $\angle C = 135^o$.
1972 Vietnam National Olympiad, 4
Let $ABCD$ be a regular tetrahedron with side $a$. Take $E,E'$ on the edge $AB, F, F'$ on the edge $AC$ and $G,G'$ on the edge AD so that $AE =a/6,AE' = 5a/6,AF= a/4,AF'= 3a/4,AG = a/3,AG'= 2a/3$. Compute the volume of $EFGE'F'G'$ in term of $a$ and find the angles between the lines $AB,AC,AD$ and the plane $EFG$.
2017 Novosibirsk Oral Olympiad in Geometry, 7
A car is driving along a straight highway at a speed of $60$ km per hour. Not far from the highway there is a parallel to him a $100$-meter fence. Every second, the passenger of the car measures the angle at which the fence is visible. Prove that the sum of all the angles he measured is less than $1100^o$
2010 District Olympiad, 3
Consider triangle $ABC$ with $AB = AC$ and $\angle A = 40 ^o$. The points $S$ and $T$ are on the sides $AB$ and $BC$, respectively, so that $\angle BAT = \angle BCS= 10 ^o$. The lines $AT$ and $CS$ intersect at point $P$. Prove that $BT = 2PT$.
2021 Yasinsky Geometry Olympiad, 1
The quadrilateral $ABCD$ is known to have $BC = CD = AC$, and the angle $\angle ABC= 70^o$. Calculate the degree measure of the angle $\angle ADB$.
(Alexey Panasenko)
1992 Tournament Of Towns, (342) 4
(a) In triangle $ABC$, angle $A$ is greater than angle $B$. Prove that the length of side $BC$ is greater than half the length of side $AB$.
(b) In the convex quadrilateral $ABCD$, the angle at $A$ is greater than the angle at $C$ and the angle at $D$ is greater than the angle at $B$. Prove that the length of side $BC$ is greater than half of the length of side $AD$.
(F Nazarov)
1996 All-Russian Olympiad Regional Round, 8.3
Does such a convex (all angles less than $180^o$) pentagon $ABCDE$, such that all angles $ABD$, $BCE$, $CDA$, $DEB$ and $EAC$ are obtuse?
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 $CO_1$.
2006 Junior Balkan Team Selection Tests - Romania, 1
Let $ABC$ be a triangle and $D$ a point inside the triangle, located on the median of $A$. Prove that if $\angle BDC = 180^o - \angle BAC$, then $AB \cdot CD = AC \cdot BD$.
2011 Purple Comet Problems, 22
Five congruent circles have centers at the vertices of a regular pentagon so that each of the circles is tangent to its two neighbors. A sixth circle (shaded in the diagram below) congruent to the other fi ve is placed tangent to two of the five. If this sixth circle is allowed to roll without slipping around the exterior of the figure formed by the other fi ve circles, then it will turn through an angle of $k$ degrees before it returns to its starting position. Find $k$.
[asy]
import graph; size(6cm);
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
filldraw(circle((2.96,2.58), 1),grey);
draw(circle((-1,3), 1));
draw(circle((1,3), 1));
draw(circle((1.62,1.1), 1));
draw(circle((0,-0.08), 1));
draw(circle((-1.62,1.1), 1));
[/asy]
1956 Moscow Mathematical Olympiad, 320
Prove that there are no four points $A, B, C, D$ on a plane such that all triangles $\vartriangle ABC,\vartriangle BCD, \vartriangle CDA, \vartriangle DAB$ are acute ones.
2022 OMpD, 2
Let $ABCD$ be a rectangle. The point $E$ lies on side $ \overline{AB}$ and the point $F$ is lies side $ \overline{AD}$, such that $\angle FEC=\angle CEB$ and $\angle DFC=\angle CFE$. Determine the measure of the angle $\angle FCE$ and the ratio $AD/AB$.
2009 Sharygin Geometry Olympiad, 6
Given triangle $ABC$ such that $AB- BC = \frac{AC}{\sqrt2}$ . Let $M$ be the midpoint of $AC$, and $N$ be the foot of the angle bisector from $B$. Prove that $\angle BMC + \angle BNC = 90^o$.
(A.Akopjan)
2018 District Olympiad, 4
Let $ABC$ be a triangle with $\angle A = 80^o$ and $\angle C = 30^o$. Consider the point $M$ inside the triangle $ABC$ so that $\angle MAC= 60^o$ and $\angle MCA = 20^o$. If $N$ is the intersection of the lines $BM$ and $AC$ to show that a $MN$ is the bisector of the angle $\angle AMC$.
2017 BMT Spring, 17
Triangle $ABC$ is drawn such that $\angle A = 80^o$, $\angle B = 60^o$, and $\angle C = 40^o$. Let the circumcenter of $\vartriangle ABC$ be $O$, and let $\omega$ be the circle with diameter $AO$. Circle $\omega$ intersects side $AC$ at point $P$. Let M be the midpoint of side $BC$, and let the intersection of $\omega$ and $PM$ be $K$. Find the measure of $\angle MOK$.
2021 Novosibirsk Oral Olympiad in Geometry, 5
The pentagon $ABCDE$ is inscribed in the circle. Line segments $AC$ and $BD$ intersect at point $K$. Line segment $CE$ touches the circumcircle of triangle $ABK$ at point $N$. Find the angle $CNK$ if $\angle ECD = 40^o.$
2006 Thailand Mathematical Olympiad, 4
Let $P$ be a point outside a circle centered at $O$. From $P$, tangent lines are drawn to the circle, touching the circle at points $A$ and $B$. Ray $\overrightarrow{BO}$ is drawn intersecting the circle again at $C$ and intersecting ray $\overrightarrow{PA}$ at $Q$. If $3QA = 2AP$, what is the value of $\sin \angle CAQ$?
1996 Tournament Of Towns, (483) 1
In an acute-angled triangle, each angle is an integral number of degrees, and the smallest angle is one-fifth of the largest one. Find these angles.
(G Galperin)
VI Soros Olympiad 1999 - 2000 (Russia), 11.10
In triangle $ABC$, angle $A$ is equal to $a$ and angle $B$ is equal to $2a$. A circle with center at point $C$ of radius $CA$ intersects the line containing the bisector of the exterior angle at vertex $B$, at points $M$ and $N$. Find the angles of triangle $MAN$.
Kyiv City MO Juniors 2003+ geometry, 2008.9.5
In the triangle $ABC$ on the side $AC$ the points $F$ and $L$ are selected so that $AF = LC <\frac{1}{2} AC$. Find the angle $ \angle FBL $ if $A {{B} ^ {2}} + B {{C} ^ {2}} = A {{L} ^ {2}} + L {{C } ^ {2}}$
(Zhidkov Sergey)
Ukraine Correspondence MO - geometry, 2009.11
In triangle $ABC$, the length of the angle bisector $AD$ is $\sqrt{BD \cdot CD}$. Find the angles of the triangle $ABC$, if $\angle ADB = 45^o$.
2020 Durer Math Competition Finals, 11
The convex quadrilateral $ABCD$ has $|AB| = 8$, $|BC| = 29$, $|CD| = 24$ and $|DA| = 53$. What is the area of the quadrilateral if $\angle ABC + \angle BCD = 270^o$?
2016 Argentina National Olympiad, 4
Find the angles of a convex quadrilateral $ABCD$ such that $\angle ABD = 29^o$, $\angle ADB = 41^o$, $\angle ACB = 82^o$ and $\angle ACD = 58^o$
1994 Canada National Olympiad, 5
Let $ABC$ be an acute triangle. Let $AD$ be the altitude on $BC$, and let $H$ be any interior point on $AD$. Lines $BH,CH$, when extended, intersect $AC,AB$ at $E,F$ respectively. Prove that $\angle EDH=\angle FDH$.
Kyiv City MO 1984-93 - geometry, 1991.10.5
Diagonal sections of a regular 8-gon pyramid, which are drawn through the smallest and largest diagonals of the base, are equal. At what angle is the plane passing through the vertex, the pyramids and the smallest diagonal of the base inclined to the base?
[hide=original wording]Діагональні перерізи правильної 8-кутної піраміди, які Проведені через найменшу і найбільшу діагоналі основи, рівновеликі. Під яким кутом до основи нахилена площина, що проходить через вершину, піраміди і найменшу діагональ основи?[/hide]