Olympiad problems

2022 Durer Math Competition Finals, 5

On a circle $k$, we marked four points $(A, B, C, D)$ and drew pairwise their connecting segments. We denoted angles as seen on the diagram. We know that $\alpha_1 : \alpha_2 = 2 : 5$, $\beta_1 : \beta_2 = 7 : 11$, and $\gamma_1 : \gamma_2 = 10 : 3$. If $\delta_1 : \delta_2 = p : q$, where $p$ and $q$ are coprime positive integers, then what is $p$? [img]https://cdn.artofproblemsolving.com/attachments/c/e/b532dd164a7cf99cea7b3b7d98f81796622da5.png[/img]

2004 May Olympiad, 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.

2021 Yasinsky Geometry Olympiad, 2

Tags: geometry , angles

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)

Durer Math Competition CD 1st Round - geometry, 2023.C7

Tags: geometry , pentagon , angles

Let $ABCDE$ be a regular pentagon. We drew two circles around $A$ and $B$ with radius $AB$. Let $F$ mark the intersection of the two circles that is inside the pentagon. Let $G$ mark the intersection of lines $EF$ and $AD$. What is the degree measure of angle $AGE$?

2019 Novosibirsk Oral Olympiad in Geometry, 3

Tags: geometry , angles , equal segments

Equal line segments are marked in triangle $ABC$. Find its angles. [img]https://cdn.artofproblemsolving.com/attachments/0/2/bcb756bba15ba57013f1b6c4cbe9cc74171543.png[/img]

Estonia Open Junior - geometry, 2019.2.1

Tags: geometry , angles , pentagon , Equilateral

A pentagon can be divided into equilateral triangles. Find all the possibilities that the sizes of the angles of this pentagon can be.

1935 Moscow Mathematical Olympiad, 012

Tags: geometry , 3D geometry , cone , pyramid , angles

The unfolding of the lateral surface of a cone is a sector of angle $120^o$. The angles at the base of a pyramid constitute an arithmetic progression with a difference of $15^o$. The pyramid is inscribed in the cone. Consider a lateral face of the pyramid with the smallest area. Find the angle $\alpha$ between the plane of this face and the base.

2001 Estonia National Olympiad, 3

Tags: angles , Sum , geometry

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]

1953 Moscow Mathematical Olympiad, 252

Tags: angles , concurrent , Line , geometry

Given triangle $\vartriangle A_1A_2A_3$ and a straight line $\ell$ outside it. The angles between the lines $A_1A_2$ and $A_2A_3, A_1A_2$ and $A_2A_3, A_2A_3$ and $A_3A_1$ are equal to $a_3, a_1$ and $a_2$, respectively. The straight lines are drawn through points $A_1, A_2, A_3$ forming with $\ell$ angles of $\pi -a_1, \pi -a_2, \pi -a_3$, respectively. All angles are counted in the same direction from $\ell$ . Prove that these new lines meet at one point.

2001 Swedish Mathematical Competition, 4

Tags: geometry , angles

$ABC$ is a triangle. A circle through $A$ touches the side $BC$ at $D$ and intersects the sides $AB$ and $AC$ again at $E, F$ respectively. $EF$ bisects $\angle AFD$ and $\angle ADC = 80^o$. Find $\angle ABC$.

1950 Moscow Mathematical Olympiad, 176

Tags: angles , sidelengths , geometric inequality , inequalities

Let $a, b, c$ be the lengths of the sides of a triangle and $A, B, C$, the opposite angles. Prove that $$Aa + Bb + Cc \ge \frac{Ab + Ac + Ba + Bc + Ca + Cb}{2}$$

1994 Tournament Of Towns, (437) 3

Tags: geometry , median , angles

The median $AD$ of triangle $ABC$ intersects its inscribed circle (with center $O$) at the points $X$ and $Y$. Find the angle $XOY$ if $AC = AB + AD$. (A Fedotov)

1985 Greece National Olympiad, 3

Tags: geometry , angles

Interior in alake there are two points $A,B$ from which we can see every other point of the lake. Prove that also from any other point of the segment $AB$, we can see all points of the lake.

Novosibirsk Oral Geo Oly IX, 2023.3

Tags: geometry , equal segments , angles , isosceles

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 $\angle CFA$.

2018 Dutch BxMO TST, 4

Tags: geometry , angles , circumcircle , Angle Chasing

In a non-isosceles triangle $\vartriangle ABC$ we have $\angle BAC = 60^o$. Let $D$ be the intersection of the angular bisector of $\angle BAC$ with side $BC, O$ the centre of the circumcircle of $\vartriangle ABC$ and $E$ the intersection of $AO$ and $BC$. Prove that $\angle AED + \angle ADO = 90^o$.

2018 Peru Iberoamerican Team Selection Test, P2

Tags: geometry , angles , Angle Chasing , equal segments

Let $ABC$ be a triangle with $AB = AC$ and let $D$ be the foot of the height drawn from $A$ to $BC$. Let $P$ be a point inside the triangle $ADC$ such that $\angle APB> 90^o$ and $\angle PAD + \angle PBD = \angle PCD$. The $CP$ and $AD$ lines are cut at $Q$ and the $BP$ and $AD$ lines cut into $R$. Let $T$ be a point in segment $AB$ such that $\angle TRB = \angle DQC$ and let S be a point in the extension of the segment $AP$ (on the $P$ side) such that $\angle PSR = 2 \angle PAR$. Prove that $RS = RT$.

2013 Saudi Arabia GMO TST, 2

Tags: combinatorics , combinatorial geometry , geometry , angles , equal angles , polygon

Find all values of $n$ for which there exists a convex cyclic non-regular polygon with $n$ vertices such that the measures of all its internal angles are equal.

1975 IMO, 3

Tags: geometry , trigonometry , Triangle , angles , IMO , IMO 1975

In the plane of a triangle $ABC,$ in its exterior$,$ we draw the triangles $ABR, BCP, CAQ$ so that $\angle PBC = \angle CAQ = 45^{\circ}$, $\angle BCP = \angle QCA = 30^{\circ}$, $\angle ABR = \angle RAB = 15^{\circ}$. Prove that [b]a.)[/b] $\angle QRP = 90\,^{\circ},$ and [b]b.)[/b] $QR = RP.$

2017 Flanders Math Olympiad, 2

Tags: geometry , circumcircle , angles , Angle Chasing

In triangle $\vartriangle ABC$, $\angle A = 50^o, \angle B = 60^o$ and $\angle C = 70^o$. The point $P$ is on the side $[AB]$ (with $P \ne A$ and $P \ne B$). The inscribed circle of $\vartriangle ABC$ intersects the inscribed circle of $\vartriangle ACP$ at points $U$ and $V$ and intersects the inscribed circle of $\vartriangle BCP$ at points $X$ and $Y$. The rights $UV$ and $XY$ intersect in $K$. Calculate the $\angle UKX$.

2011 AMC 12/AHSME, 10

Tags: geometry , rectangle , trigonometry , trig identities , Law of Sines , AMC , angles

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 $

1964 All Russian Mathematical Olympiad, 041

Tags: geometry , angles

The two heights in the triangle are not less than the respective sides. Find the angles.

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 $|CA| = |CB| + |CD|$.

2011 Saudi Arabia BMO TST, 3

Tags: geometry , Equilateral , angles

Consider a triangle $ABC$. Let $A_1$ be the symmetric point of $A$ with respect to the line $BC$, $B_1$ the symmetric point of $B$ with respect to the line $CA$, and $C_1$ the symmetric point of $C$ with respect to the line $AB$. Determine the possible set of angles of triangle $ABC$ for which $A_1B_1C_1$ is equilateral.

1996 Singapore Senior Math Olympiad, 2

Tags: max , semicircle , triangle area , Sum , geometry , trigonometry , angles

Let $180^o < \theta_1 < \theta_2 <...< \theta_n = 360^o$. For $i = 1,2,..., n$, $P_i = (\cos \theta_i^o, \sin \theta_i^o)$ is a point on the circle $C$ with centre $(0,0)$ and radius $1$. Let $P$ be any point on the upper half of $C$. Find the coordinates of $P$ such that the sum of areas $[PP_1P_2] + [PP_2P_3] + ...+ [PP_{n-1}P_n]$ attains its maximum.

2017 Estonia Team Selection Test, 10

Tags: semicircle , geometry , angles

Let $ABC$ be a triangle with $AB = \frac{AC}{2 }+ BC$. Consider the two semicircles outside the triangle with diameters $AB$ and $BC$. Let $X$ be the orthogonal projection of $A$ onto the common tangent line of those semicircles. Find $\angle CAX$.