Found problems: 721
1996 Tournament Of Towns, (485) 3
The two tangents to the incircle of a right-angled triangle $ABC$ that are perpendicular to the hypotenuse $AB$ intersect it at the points $P$ and $Q$. Find $\angle PCQ$.
(M Evdokimov,)
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 $\angle ACD = 58^o$
2016 Novosibirsk Oral Olympiad in Geometry, 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]
Kyiv City MO 1984-93 - geometry, 1991.7.5
Inside the rectangle $ABCD$ is taken a point $M$ such that $\angle BMC + \angle AMD = 180^o$. Determine the sum of the angles $BCM$ and $DAM$.
VII Soros Olympiad 2000 - 01, 10.8
There is a set of triangles, in each of which the smallest angle does not exceed $36^o$ . A new one is formed from these triangles according to the following rule: the smallest side of the new one is equal to the sum of the smallest sides of these triangles, its middle side is equal to the sum of the middle sides, and the largest is the sum of the largest ones. Prove that the sine of the smallest angle of the resulting triangle is less than $2 \sin 18^o$ .
2013 IMAC Arhimede, 5
Let $\Gamma$ be the circumcircle of a triangle $ABC$ and let $E$ and $F$ be the intersections of the bisectors of $\angle ABC$ and $\angle ACB$ with $\Gamma$. If $EF$ is tangent to the incircle $\gamma$ of $\triangle ABC$, then find the value of $\angle BAC$.
2017 BMT Spring, 4
$2$ darts are thrown randomly at a circular board with center $O$, such that each dart has an equal probability of hitting any point on the board. The points at which they land are marked $A$ and $B$. What is the probability that $\angle AOB$ is acute?
2001 Estonia National Olympiad, 3
There are three squares in the picture. Find the sum of angles $ADC$ and $BDC$.
[img]https://cdn.artofproblemsolving.com/attachments/c/9/885a6c6253fca17e24528f8ba8a5d31a18c845.png[/img]
1999 May Olympiad, 2
In a parallelogram $ABCD$ , $BD$ is the largest diagonal. By matching $B$ with $D$ by a bend, a regular pentagon is formed. Calculate the measures of the angles formed by the diagonal $BD$ with each of the sides of the parallelogram.
1999 Nordic, 2
Consider $7$-gons inscribed in a circle such that all sides of the $7$-gon are of different length. Determine the maximal number of $120^\circ$ angles in this kind of a $7$-gon.
2020 Ukrainian Geometry Olympiad - April, 4
On the sides $AB$ and $AD$ of the square $ABCD$, the points $N$ and $P$ are selected respectively such that $NC=NP$. The point $Q$ is chosen on the segment $AN$ so that $\angle QPN = \angle NCB$. Prove that $2\angle BCQ = \angle AQP$.
2017 Dutch Mathematical Olympiad, 2
A parallelogram $ABCD$ with $|AD| =|BD|$ has been given. A point $E$ lies on line segment $|BD|$ in such a way that $|AE| = |DE|$. The (extended) line $AE$ intersects line segment $BC$ in $F$. Line $DF$ is the angle bisector of angle $CDE$. Determine the size of angle $ABD$.
[asy]
unitsize (3 cm);
pair A, B, C, D, E, F;
D = (0,0);
A = dir(250);
B = dir(290);
C = B + D - A;
E = extension((A + D)/2, (A + D)/2 + rotate(90)*(A - D), B, D);
F = extension(A, E, B, C);
draw(A--B--C--D--cycle);
draw(A--F--D--B);
dot("$A$", A, SW);
dot("$B$", B, SE);
dot("$C$", C, NE);
dot("$D$", D, NW);
dot("$E$", E, S);
dot("$F$", F, SE);
[/asy]
2017 Hanoi Open Mathematics Competitions, 11
Let $ABC$ be an equilateral triangle, and let $P$ stand for an arbitrary point inside the triangle.
Is it true that $| \angle PAB - \angle PAC| \ge | \angle PBC - \angle PCB|$ ?
Estonia Open Junior - geometry, 2002.2.3
In a triangle $ABC$ we have $|AB| = |AC|$ and $\angle BAC = \alpha$. Let $P \ne B$ be a point on $AB$ and $Q$ a point on the altitude drawn from $A$ such that $|PQ| = |QC|$. Find $ \angle QPC$.
2001 All-Russian Olympiad Regional Round, 8.3
All sides of a convex pentagon are equal, and all angles are different. Prove that the maximum and minimum angles are adjacent to the same side of the pentagon.
Estonia Open Senior - geometry, 1994.1.4
Prove that if $\frac{AC}{BC}=\frac{AB + BC}{AC}$ in a triangle $ABC$ , then $\angle B = 2 \angle A$ .
2021 Yasinsky Geometry Olympiad, 2
In the quadrilateral $ABCD$ it is known that $\angle A = 90^o$, $\angle C = 45^o$ . Diagonals $AC$ and $BD$ intersect at point $F$, and $BC = CF$, and the diagonal $AC$ is the bisector of angle $A$. Determine the other two angles of the quadrilateral $ABCD$.
(Maria Rozhkova)
1989 IMO Longlists, 67
Prove that the intersection of a plane and a regular tetrahedron can be an obtuse-angled triangle and that the obtuse angle in any such triangle is always smaller than $ 120^{\circ}.$
2022 IFYM, Sozopol, 7
Points $M$, $N$, $P$ and $Q$ are midpoints of the sides $AB$, $BC$, $CD$ and $DA$ of the inscribed quadrilateral $ABCD$ with intersection point $O$ of its diagonals. Let $K$ be the second intersection point of the circumscribed circles of $MOQ$ and $NOP$. Prove that $OK\perp AC$.
1986 Tournament Of Towns, (131) 7
On the circumference of a circle are $21$ points. Prove that among the arcs which join any two of these points, at least $100$ of them must subtend an angle at the centre of the circle not exceeding $120^o$ .
( A . F . Sidorenko)
Kyiv City MO Juniors Round2 2010+ geometry, 2013.8.3
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)
Novosibirsk Oral Geo Oly IX, 2023.7
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 $\angle BED$.
2020 Ukrainian Geometry Olympiad - December, 3
About the pentagon $ABCDE$ we know that $AB = BC = CD = DE$, $\angle C = \angle D =108^o$, $\angle B = 96^o$. Find the value in degrees of $\angle E$.
2008 Abels Math Contest (Norwegian MO) Final, 4a
Three distinct points $A, B$, and $C$ lie on a circle with centre at $O$. The triangles $AOB, BOC$ , and $COA$ have equal area. What are the possible measures of the angles of the triangle $ABC$ ?
2017 Auckland Mathematical Olympiad, 5
The altitudes of triangle $ABC$ intersect at a point $H$.Find $\angle ACB$ if it is known that $AB = CH$.