Olympiad problems

2020 Ukrainian Geometry Olympiad - December, 1

The three sides of the quadrilateral are equal, the angles between them are equal, respectively $90^o$ and $150^o$. Find the smallest angle of this quadrilateral in degrees.

2016 ASMT, 8

Tags: geometry , rectangle , angles , asmt

In rectangle $ABCD$, point $E$ is chosen on $AB$ and $F$ is the foot of $E$ onto side $CD$ such that the circumcircle of $\vartriangle ABF$ intersects line segments $AD$ and $BC$ at points $G$ and $H$ respectively. Let $S$ be the intersection of $EF$ and $GH$, and $T$ the intersection of lines $EC$ and $DS$. If $\angle SF T = 15^o$ , compute the measure of $\angle CSD$.

1964 German National Olympiad, 5

Tags: geometry , angles

A triangle $ABC$ with $\beta= 45^o$ is given. There is a point $P$ on side $BC$, where $BP : PC =1 : 2$ (inner division) and $\angle APC = 60^o$. Someone claims that you can do it with elementary geometric theorems alone without using the plane trigonometry, the size of the angle $\gamma$ determine.

2021 Lotfi Zadeh Olympiad, 4

Tags: Polygons , angles , Lotfi Zadeh MO

Find the number of sequences of $0, 1$ with length $n$ satisfying both of the following properties: [list] [*] There exists a simple polygon such that its $i$-th angle is less than $180$ degrees if and only if the $i$-th element of the sequence is $1$. [*] There exists a convex polygon such that its $i$-th angle is less than $90$ degrees if and only if the $i$-th element of the sequence is $1$. [/list]

1997 Czech And Slovak Olympiad IIIA, 1

Tags: geometry , angles

Let $ABC$ be a triangle with sides $a,b,c$ and corresponding angles $\alpha,\beta\gamma$ . Prove that if $\alpha = 3\beta$ then $(a^2 -b^2)(a-b) = bc^2$ . Is the converse true?

2003 Cuba MO, 2

Tags: geometry , circles , circumcircle , angles

Let $A$ be a point outside the circle $\omega$ . The tangents from $A$ touch the circle at $B$ and $C$. Let $P$ be an arbitrary point on extension of $AC$ towards $C$, $Q$ the projection of $C$ onto $PB$ and $E$ the second intersection point of the circumcircle of $ABP$ with the circle $\omega$ . Prove that $\angle PEQ = 2\angle APB$

1997 German National Olympiad, 3

Tags: Angle Chasing , angles , geometry

In a convex quadrilateral $ABCD$ we are given that $\angle CBD = 10^o$, $\angle CAD = 20^o$, $\angle ABD = 40^o$, $\angle BAC = 50^o$. Determine the angles $\angle BCD$ and $\angle ADC$.

1988 IMO Shortlist, 17

Tags: geometry , convex polygon , pentagon , angles , IMO Shortlist

In the convex pentagon $ ABCDE,$ the sides $ BC, CD, DE$ are equal. Moreover each diagonal of the pentagon is parallel to a side ($ AC$ is parallel to $ DE$, $ BD$ is parallel to $ AE$ etc.). Prove that $ ABCDE$ is a regular pentagon.

2019 Yasinsky Geometry Olympiad, p5

Tags: geometry , angles , Angle Chasing

In the triangle $ABC$, $\angle ABC = \angle ACB = 78^o$. On the sides $AB$ and $AC$, respectively, the points $D$ and $E$ are chosen such that $\angle BCD = 24^o$, $\angle CBE = 51^o$. Find the measure of angle $\angle BED$.

2016 Novosibirsk Oral Olympiad in Geometry, 3

Tags: geometry , angles , square

A square is drawn on a sheet of grid paper on the sides of the cells $ABCD$ with side $8$. Point $E$ is the midpoint of side $BC$, $Q$ is such a point on the diagonal $AC$ such that $AQ: QC = 3: 1$. Find the angle between straight lines $AE$ and $DQ$.

1999 Harvard-MIT Mathematics Tournament, 11

Tags: HMMT , geometry , circles , angles

Circles $C_1$, $C_2$, $C_3$ have radius $ 1$ and centers $O, P, Q$ respectively. $C_1$ and $C_2$ intersect at $A$, $C_2$ and $C_3$ intersect at $B$, $C_3$ and $C_1$ intersect at $C$, in such a way that $\angle APB = 60^o$ , $\angle BQC = 36^o$ , and $\angle COA = 72^o$ . Find angle $\angle ABC$ (degrees).

2019 Philippine TST, 6

Tags: geometry , angles , parallelogram , isogonal lines

Let $D$ be an interior point of triangle $ABC$. Lines $BD$ and $CD$ intersect sides $AC$ and $AB$ at points $E$ and $F$, respectively. Points $X$ and $Y$ are on the plane such that $BFEX$ and $CEFY$ are parallelograms. Suppose lines $EY$ and $FX$ intersect at a point $T$ inside triangle $ABC$. Prove that points $B$, $C$, $E$, and $F$ are concyclic if and only if $\angle BAD = \angle CAT$.

2002 Estonia Team Selection Test, 4

Tags: angles , equal angles , Cyclic , geometry

Let $ABCD$ be a cyclic quadrilateral such that $\angle ACB = 2\angle CAD$ and $\angle ACD = 2\angle BAC$. Prove that $|CA| = |CB| + |CD|$.

May Olympiad L2 - geometry, 2004.3

Tags: geometry , angles

We have a pool table $8$ meters long and $2$ meters wide with a single ball in the center. We throw the ball in a straight line and, after traveling $29$ meters, it stops at a corner of the table. How many times did the ball hit the edges of the table? Note: When the ball rebounds on the edge of the table, the two angles that form its trajectory with the edge of the table are the same.

Denmark (Mohr) - geometry, 1997.3

Tags: geometry , angles , pentagon , reflection

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 $E$.

2018 India PRMO, 24

Tags: geometry , combinatorics , Integers , angles

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

Novosibirsk Oral Geo Oly VIII, 2016.3

Tags: geometry , angles , square

A square is drawn on a sheet of grid paper on the sides of the cells $ABCD$ with side $8$. Point $E$ is the midpoint of side $BC$, $Q$ is such a point on the diagonal $AC$ such that $AQ: QC = 3: 1$. Find the angle between straight lines $AE$ and $DQ$.

2021 Czech-Polish-Slovak Junior Match, 5

Tags: angles , hexagon , geometry

A regular heptagon $ABCDEFG$ is given. The lines $AB$ and $CE$ intersect at $ P$. Find the measure of the angle $\angle PDG$.

Kyiv City MO Seniors Round2 2010+ geometry, 2019.10.3

Tags: geometry , excenters , excenter , angles

Denote in the triangle $ABC$ by $T_A,T_B,T_C$ the touch points of the exscribed circles of $\vartriangle ABC$, tangent to sides $BC, AC$ and $AB$ respectively. Let $O$ be the center of the circumcircle of $\vartriangle ABC$, and $I$ is the center of it's inscribed circle. It is known that $OI\parallel AC$. Prove that $\angle T_A T_B T_C= 90^o - \frac12 \angle ABC$. (Anton Trygub)

2015 Puerto Rico Team Selection Test, 2

Tags: geometry , angles , equal segments

In the triangle $ABC$, let $P$, $Q$, and $R$ lie on the sides $BC$, $AC$, and $AB$ respectively, such that $AQ = AR$, $BP = BR$ and $CP = CQ$. Let $\angle PQR=75^o$ and $\angle PRQ=35^o$. Calculate the measures of the angles of the triangle $ABC$.

1959 Kurschak Competition, 2

Tags: geometry , angles

The angles subtended by a tower at distances $100$, $200$ and $300$ from its foot sum to $90^o$. What is its height?

2004 District Olympiad, 4

Tags: trigonometry , 3D geometry , geometry , palnes , angles , trapezoid , right angle

In the right trapezoid $ABCD$ with $AB \parallel CD, \angle B = 90^o$ and $AB = 2DC$. At points $A$ and $D$ there is therefore a part of the plane $(ABC)$ perpendicular to the plane of the trapezoid, on which the points $N$ and $P$ are taken, ($AP$ and $PD$ are perpendicular to the plane) such that $DN = a$ and $AP = \frac{a}{2}$ . Knowing that $M$ is the midpoint of the side $BC$ and the triangle $MNP$ is equilateral, determine: a) the cosine of the angle between the planes $MNP$ and $ABC$. b) the distance from $D$ to the plane $MNP$

2015 Balkan MO Shortlist, G3

Tags: geometry , combinatorial geometry , angles

A set of points of the plane is called [i] obtuse-angled[/i] if every three of it's points are not collinear and every triangle with vertices inside the set has one angle $ >91^o$. Is it correct that every finite [i] obtuse-angled[/i] set can be extended to an infinite [i]obtuse-angled[/i] set? (UK)

2020 Novosibirsk Oral Olympiad in Geometry, 6

Tags: geometry , angle bisector , angles

Angle bisectors $AA', BB'$and $CC'$ are drawn in triangle $ABC$ with angle $\angle B= 120^o$. Find $\angle A'B'C'$.

1979 IMO Longlists, 47

Tags: Triangle , angles , geometry , IMO Shortlist , IMO 1979

Inside an equilateral triangle $ABC$ one constructs points $P, Q$ and $R$ such that \[\angle QAB = \angle PBA = 15^\circ,\\ \angle RBC = \angle QCB = 20^\circ,\\ \angle PCA = \angle RAC = 25^\circ.\] Determine the angles of triangle $PQR.$