Olympiad problems

2018 India PRMO, 24

If $N$ is the number of triangles of different shapes (i.e., not similar) whose angles are all integers (in degrees), what is $\frac{N}{100}$?

2019 Oral Moscow Geometry Olympiad, 1

Tags: geometry , square , incircle , angle

Circle inscribed in square $ABCD$ , is tangent to sides $AB$ and $CD$ at points $M$ and $K$ respectively. Line $BK$ intersects this circle at the point $L, X$ is the midpoint of $KL$. Find the angle $\angle MXK $.

1950 Moscow Mathematical Olympiad, 178

Tags: trigonometry , angle

Let $A$ be an arbitrary angle,let $B$ and $C$ be acute angles. Is there an angle $x$ such that $$\sin x =\frac{\sin B \cdot \sin C}{1 - \cos B \cdot \cos C \cdot \cos A} ?$$

Ukraine Correspondence MO - geometry, 2005.4

Tags: geometry , angle bisector , angle

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 $\angle C$.

2015 EGMO, 6

Tags: geometry , circumcircle , orthocenter , centroid , angle

Let $H$ be the orthocentre and $G$ be the centroid of acute-angled triangle $ABC$ with $AB\ne AC$. The line $AG$ intersects the circumcircle of $ABC$ at $A$ and $P$. Let $P'$ be the reflection of $P$ in the line $BC$. Prove that $\angle CAB = 60$ if and only if $HG = GP'$

2005 Sharygin Geometry Olympiad, 11.5

Tags: fixed point , fixed , angle , geometry

The angle and the point $K$ inside it are given on the plane. Prove that there is a point $M$ with the following property: if an arbitrary line passing through intersects the sides of the angle at points $A$ and $B$, then $MK$ is the bisector of the angle $AMB$.

2002 District Olympiad, 3

Tags: 3d geometry , pyramid , polyhedron , angle , geometry

Consider the regular pyramid $VABCD$ with the vertex in $V$ which measures the angle formed by two opposite lateral edges is $45^o$. The points $M,N,P$ are respectively, the projections of the point $A$ on the line $VC$, the symmetric of the point $M$ with respect to the plane $(VBD)$ and the symmetric of the point $N$ with respect to $O$. ($O$ is the center of the base of the pyramid.) a) Show that the polyhedron $MDNBP$ is a regular pyramid. b) Determine the measure of the angle between the line $ND$ and the plane $(ABC) $

1980 IMO Longlists, 15

Tags: geometry , 3d geometry , tetrahedron , sphere , triangle inequality , angle

Prove that the sum of the six angles subtended at an interior point of a tetrahedron by its six edges is greater than 540°.

1985 IMO Longlists, 94

Tags: geometry , circumcircle , angle , triangle

A circle with center $O$ passes through the vertices $A$ and $C$ of the triangle $ABC$ and intersects the segments $AB$ and $BC$ again at distinct points $K$ and $N$ respectively. Let $M$ be the point of intersection of the circumcircles of triangles $ABC$ and $KBN$ (apart from $B$). Prove that $\angle OMB=90^{\circ}$.

Novosibirsk Oral Geo Oly VII, 2021.5

Tags: geometry , equal segments , angle

In an acute-angled triangle $ABC$ on the side $AC$, point $P$ is chosen in such a way that $2AP = BC$. Points $X$ and $Y$ are symmetric to $P$ with respect to vertices $A$ and $C$, respectively. It turned out that $BX = BY$. Find $\angle BCA$.

2021 India National Olympiad, 5

Tags: geometry , equilateral triangle , equal lengths , trigonometry , angle

In a convex quadrilateral $ABCD$, $\angle ABD=30^\circ$, $\angle BCA=75^\circ$, $\angle ACD=25^\circ$ and $CD=CB$. Extend $CB$ to meet the circumcircle of triangle $DAC$ at $E$. Prove that $CE=BD$. [i]Proposed by BJ Venkatachala[/i]

2020 Durer Math Competition Finals, 11

Tags: geometry , angle

The convex quadrilateral $ABCD$ has $|AB| = 8$, $|BC| = 29$, $|CD| = 24$ and $|DA| = 53$. What is the area of the quadrilateral if $\angle ABC + \angle BCD = 270^o$?

1997 Denmark MO - Mohr Contest, 3

Tags: geometry , pentagon , reflection , angle

About pentagon $ABCDE$ is known that angle $A$ and angle $C$ are right and that the sides $| AB | = 4$, $| BC | = 5$, $| CD | = 10$, $| DE | = 6$. Furthermore, the point $C'$ that appears by mirroring $C$ in the line $BD$, lies on the line segment $AE$. Find angle $E$.

Kyiv City MO Juniors 2003+ geometry, 2019.8.3

Tags: equal angles , equal segments , angle , geometry

In the triangle $ABC$ it is known that $2AC=AB$ and $\angle A = 2\angle B$. In this triangle draw the angle bisector $AL$, and mark point $M$, the midpoint of the side $AB$. It turned out that $CL = ML$. Prove that $\angle B= 30^o$. (Hilko Danilo)

1965 Swedish Mathematical Competition, 1

Tags: altitude , geometric inequalities , geometry , max , angle

The feet of the altitudes in the triangle $ABC$ are $A', B', C'$. Find the angles of $A'B'C'$ in terms of the angles $A, B, C$. Show that the largest angle in $A'B'C'$ is at least as big as the largest angle in $ABC$. When is it equal?

1998 All-Russian Olympiad Regional Round, 8.7

Tags: geometry , angle

Let $O$ be the center of a circle circumscribed about an acute angle triangle $ABC$, $S_A$, $S_B$, $S_C$ - circles with center O, tangent to sides $BC$, $CA$, $AB$ respectively. Prove that the sum of three angles : between the tangents to $S_A$ drawn from point $A$, to $S_B$ from point $B$ and to $S_C$ - from point $C$, is equal to $180^o$.

1992 Chile National Olympiad, 5

Tags: side , geometry , angle

In the $\triangle ABC $, points $ M, I, H $ are feet, respectively, of the median, bisector and height, drawn from $ A $. It is known that $ BC = 2 $, $ MI = 2-\sqrt {3} $ and $ AB > AC $. a) Prove that $ I$ lies between $ M $ and $ H $. b) Calculate $ AB ^ 2-AC ^ 2 $. c) Determine $ \dfrac {AB} {AC} $. d) Find the measure of all the sides and angles of the triangle.

2011 AMC 10, 18

Tags: geometry , rectangle , trigonometry , trig identities , law of sines , angle

Rectangle $ABCD$ has $AB=6$ and $BC=3$. Point $M$ is chosen on side $AB$ so that $\angle AMD = \angle CMD$. What is the degree measure of $\angle AMD$? $ \textbf{(A)}\ 15 \qquad \textbf{(B)}\ 30 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 60 \qquad \textbf{(E)}\ 75 $

2020 Yasinsky Geometry Olympiad, 2

Tags: geometry , angle

It is known that the angles of the triangle $ABC$ are $1: 3: 5$. Find the angle between the bisector of the largest angle of the triangle and the line containing the altitude drawn to the smallest side of the triangle.

Kyiv City MO Juniors Round2 2010+ geometry, 2014.7.4

Tags: ratio , median , geometry , angle

The median $BM$ is drawn in the triangle $ABC$. It is known that $\angle ABM = 40 {} ^ \circ$ and $\angle CBM = 70 {} ^ \circ $ Find the ratio $AB: BM$.

1935 Moscow Mathematical Olympiad, 003

Tags: pyramid , geometry , 3d geometry , angle

The base of a pyramid is an isosceles triangle with the vertex angle $\alpha$. The pyramid’s lateral edges are at angle $\phi$ to the base. Find the dihedral angle $\theta$ at the edge connecting the pyramid’s vertex to that of angle $\alpha$.

1992 IMO Longlists, 37

Tags: geometry , circles , trigonometric inequality , angle

Let the circles $C_1, C_2$, and $C_3$ be orthogonal to the circle $C$ and intersect each other inside $C$ forming acute angles of measures $A, B$, and $C$. Show that $A + B +C < \pi.$

2018 Adygea Teachers' Geometry Olympiad, 4

Tags: geometry , 3d geometry , cube , angle , plane

Given a cube $ABCDA_1B_1C_1D_1$ with edge $5$. On the edge $BB_1$ of the cube , point $K$ such thath $BK=4$. a) Construct a cube section with the plane $a$ passing through the points $K$ and $C_1$ parallel to the diagonal $BD_1$. b) Find the angle between the plane $a$ and the plane $BB_1C_1$.

Denmark (Mohr) - geometry, 2007.4

Tags: geometry , circles , angle , tangent circle

The figure shows a $60^o$ angle in which are placed $2007$ numbered circles (only the first three are shown in the figure). The circles are numbered according to size. The circles are tangent to the sides of the angle and to each other as shown. Circle number one has radius $1$. Determine the radius of circle number $2007$. [img]https://1.bp.blogspot.com/-1bsLIXZpol4/Xzb-Nk6ospI/AAAAAAAAMWk/jrx1zVYKbNELTWlDQ3zL9qc_22b2IJF6QCLcBGAsYHQ/s0/2007%2BMohr%2Bp4.png[/img]

Kyiv City MO 1984-93 - geometry, 1990.7.3

Tags: geometry , angle

Given a triangle with sides $a, b, c$ that satisfy $\frac{a}{b+c}=\frac{c}{a+b}$. Determine the angles of this triangle, if you know that one of them is equal to $120^0$.