Found problems: 628
2018 Hanoi Open Mathematics Competitions, 8
Let $ABCD$ be rhombus, with $\angle ABC = 80^o$: Let $E$ be midpoint of $BC$ and $F$ be perpendicular projection of $A$ onto $DE$. Find the measure of $\angle DFC$ in degree.
2005 All-Russian Olympiad Regional Round, 10.1
The cosines of the angles of one triangle are respectively equal to the sines of the angles of the other triangle. Find the largest of these six angles of triangles.
Novosibirsk Oral Geo Oly IX, 2017.7
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 $1100^o$
2011 Junior Balkan Team Selection Tests - Romania, 4
The measure of the angle $\angle A$ of the acute triangle $ABC$ is $60^o$, and $HI = HB$, where $I$ and $H$ are the incenter and the orthocenter of the triangle $ABC$. Find the measure of the angle $\angle B$.
Kyiv City MO Juniors 2003+ geometry, 2018.7.41
In the quadrilateral $ABCD$ point $E$ - the midpoint of the side $AB$, point $F$ - the midpoint of the side $BC$, point $G$ - the midpoint $AD$ . It turned out that the segment $GE$ is perpendicular to $AB$, and the segment $GF$ is perpendicular to the segment $BC$. Find the value of the angle $GCD$, if it is known that $\angle ADC = 70 {} ^ \circ$.
2021 Junior Balkan Team Selection Tests - Moldova, 5
Let $ABC$ be the triangle with $\angle ABC = 76^o$ and $\angle ACB = 72^o$. Points $P$ and $Q$ lie on the sides $(AB)$ and $(AC)$, respectively, such that $\angle ABQ = 22^o$ and $\angle ACP = 44^o$. Find the measure of angle $\angle APQ$.
2008 Brazil Team Selection Test, 1
Let $AB$ be a chord, not a diameter, of a circle with center $O$. The smallest arc $AB$ is divided into three congruent arcs $AC$, $CD$, $DB$. The chord $AB$ is also divided into three equal segments $AC'$, $C'D'$, $D'B$. Let $P$ be the intersection point of between the lines $CC'$ and $DD'$. Prove that $\angle APB = \frac13 \angle AOB$.
2015 Dutch Mathematical Olympiad, 3 seniors
Points $A, B$, and $C$ are on a line in this order. Points $D$ and $E$ lie on the same side of this line, in such a way that triangles $ABD$ and $BCE$ are equilateral. The segments $AE$ and $CD$ intersect in point $S$. Prove that $\angle ASD = 60^o$.
[asy]
unitsize(1.5 cm);
pair A, B, C, D, E, S;
A = (0,0);
B = (1,0);
C = (2.5,0);
D = dir(60);
E = B + 1.5*dir(60);
S = extension(C,D,A,E);
fill(A--B--D--cycle, gray(0.8));
fill(B--C--E--cycle, gray(0.8));
draw(interp(A,C,-0.1)--interp(A,C,1.1));
draw(A--D--B--E--C);
draw(A--E);
draw(C--D);
draw(anglemark(D,S,A,5));
dot("$A$", A, dir(270));
dot("$B$", B, dir(270));
dot("$C$", C, dir(270));
dot("$D$", D, N);
dot("$E$", E, N);
dot("$S$", S, N);
[/asy]
2010 Belarus Team Selection Test, 2.1
Point $D$ is marked inside a triangle $ABC$ so that $\angle ADC = \angle ABC + 60^o$, $\angle CDB =\angle CAB + 60^o$, $\angle BDA = \angle BCA + 60^o$. Prove that $AB \cdot CD = BC \cdot AD = CA \cdot BD$.
(A. Levin)
2003 Junior Balkan Team Selection Tests - Moldova, 7
The triangle $ABC$ is isosceles with $AB=BC$. The point F on the side $[BC]$ and the point $D$ on the side $AC$ are the feets of the the internals bisectors drawn from $A$ and altitude drawn from $B$ respectively so that $AF=2BD$. Fine the measure of the angle $ABC$.
IV Soros Olympiad 1997 - 98 (Russia), 9.3
What is angle $B$ of triangle$ ABC$, if it is known that the altitudes drawn from $A$ and $C$ intersect inside the triangle and one of them is divided by of intersection point into equal parts, and the other one in the ratio of $2: 1$, counting from the vertex?
2015 Portugal MO, 2
Let $[ABC]$ be a triangle and $D$ a point between $A$ and $B$. If the triangles $[ABC], [ACD]$ and $[BCD]$ are all isosceles, what are the possible values of $\angle ABC$?
Ukraine Correspondence MO - geometry, 2016.11
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 $PQ$.
2015 Bangladesh Mathematical Olympiad, 6
Trapezoid $ABCD$ has sides $AB=92,BC=50,CD=19,AD=70$ $AB$ is parallel to $CD$ A circle with center $P$ on $AB$ is drawn tangent to $BC$ and $AD$.Given that $AP=\dfrac mn$ (Where $m,n$ are relatively prime).What is $m+n$?
2012 Chile National Olympiad, 4
Consider an isosceles triangle $ABC$, where $AB = AC$. $D$ is a point on the $AC$ side and $P$ a point on the segment $BD$ so that the angle $\angle APC = 90^o$ and $ \angle ABP = \angle BCP $. Determine the ratio $AD: DC$.
Ukrainian TYM Qualifying - geometry, 2014.1
In the triangle $ABC$, one of the angles of which is equal to $48^o$, side lengths satisfy $(a-c)(a+c)^2+bc(a+c)=ab^2$. Express in degrees the measures of the other two angles of this triangle.
Novosibirsk Oral Geo Oly VIII, 2017.2
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 $.
2000 Estonia National Olympiad, 4
Prove that for any triangle the equation holds $a \cdot \cos (\beta + \gamma ) + b \cdot \cos (\gamma +\alpha) + c\cdot \cos (\alpha -\beta) = 0$,
where $a, b, c$ are the sides of the triangle and $\alpha, \beta,\gamma$ according to their angles sizes of opposite angles.
2017 Romania National Olympiad, 2
Consider the triangle $ABC$, with $\angle A= 90^o, \angle B = 30^o$, and $D$ is the foot of the altitude from $A$. Let the point $E \in (AD)$ such that $DE = 3AE$ and $F$ the foot of the perpendicular from $D$ to the line $BE$.
a) Prove that $AF \perp FC$.
b) Determine the measure of the angle $AFB$.
2023 Novosibirsk Oral Olympiad in Geometry, 6
An isosceles triangle $ABC$ with base $AC$ is given. On the rays $CA$, $AB$ and $BC$, the points $D, E$ and $F$ were marked, respectively, in such a way that $AD = AC$, $BE = BA$ and $CF = CB$. Find the sum of the angles $\angle ADB$, $\angle BEC$ and $\angle CFA$.
1995 Singapore Team Selection Test, 2
$ABC$ is a triangle with $\angle A > 90^o$ . On the side $BC$, two distinct points $P$ and $Q$ are chosen such that $\angle BAP = \angle PAQ$ and $BP \cdot CQ = BC \cdot PQ$. Calculate the size of $\angle PAC$.
Novosibirsk Oral Geo Oly VIII, 2016.2
Bisector of one angle of triangle $ABC$ is equal to the bisector of its external angle at the same vertex (see figure). Find the difference between the other two angles of the triangle.
[img]https://cdn.artofproblemsolving.com/attachments/c/3/d2efeb65544c45a15acccab8db05c8314eb5f2.png[/img]
1990 Greece Junior Math Olympiad, 3
Let $A_1A_2A_3...A_{72}$ be a regurar $72$-gon with center $O$. Calculate an extenral angle of that polygon and the angles $\angle A_{45} OA_{46}$, $\angle A_{44} A_{45}A_{46}$. How many diagonals does this polygon have?
May Olympiad L2 - geometry, 2022.3
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 $\angle FDE$.
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.