Olympiad problems

1998 Belarus Team Selection Test, 1

Two circles $S_1$ and $S_2$ intersect at different points $P,Q$. The arc of $S_1$ lying inside $S_2$ measures $2a$ and the arc of $S_2$ lying inside $S_1$ measures $2b$. Let $T$ be any point on $S_1$. Let $R,S$ be another points of intersection of $S_2$ with $TP$ and $TQ$ respectively. Let $a+2b<\pi$ . Find the locus of the intersection points of $PS$ and $RQ$. S.Shikh

2008 Austria Beginners' Competition, 4

Tags: geometry , angles , concurrency , concurrent

Let $ABC$ be an acute-angled triangle with the property that the bisector of $\angle BAC$, the altitude through $B$ and the perpendicular bisector of $AB$ intersect in one point. Determine the angle $\alpha = \angle BAC$.

1997 Estonia National Olympiad, 2

Tags: Triangle , angles , geometry

Side lengths $a,b,c$ of a triangle satisfy $\frac{a^3+b^3+c^3}{a+b+c}= c^2$. Find the measure of the angle opposite to side $c$.

1949-56 Chisinau City MO, 23

Tags: distance , angles , geometry

Inside the angle $ABC$ of $60^o$, point $O$ is selected, which is located at distances from the sides of the angle $a$ and $b$, respectively. Determine the distance from the top of the angle to this point.

Durer Math Competition CD Finals - geometry, 2017.D4

Tags: geometry , angles

The convex quadrilateral $ABCD$ is has angle $A$ equal to $60^o$ , angle bisector of $A$ the diagonal $AC$ and $\angle ACD= 40^o$ and $\angle ACB = 120^o$. Inside the quadrilateral the point $P$ lies such that $\angle PDA = 40^o$ and $\angle PBA = 10^o$; a) Find the angle $\angle DPB$? b) Prove that $P$ lies on the diagonal $AC$.

1986 Austrian-Polish Competition, 1

Tags: geometry , similar triangles , tangent circles , angles

A non-right triangle $A_1A_2A_3$ is given. Circles $C_1$ and $C_2$ are tangent at $A_3, C_2$ and $C_3$ are tangent at $A_1$, and $C_3$ and $C_1$ are tangent at $A_2$. Points $O_1,O_2,O_3$ are the centers of $C_1, C_2, C_3$, respectively. Supposing that the triangles $A_1A_2A_3$ and $O_1O_2O_3$ are similar, determine their angles.

2015 China Northern MO, 2

Tags: geometry , angles , diameter , circumcircle

It is known that $\odot O$ is the circumcircle of $\vartriangle ABC$ wwith diameter $AB$. The tangents of $\odot O$ at points $B$ and $C$ intersect at $P$ . The line perpendicular to $PA$ at point $A$ intersects the extension of $BC$ at point $D$. Extend $DP$ at length $PE = PB$. If $\angle ADP = 40^o$ , find the measure of $\angle E$.

1982 All Soviet Union Mathematical Olympiad, 334

Tags: geometry , trigonometry , 3D geometry , tetrahedron , angles

Given a point $M$ inside a right tetrahedron. Prove that at least one tetrahedron edge is seen from the $M$ in an angle, that has a cosine not greater than $-1/3$. (e.g. if $A$ and $B$ are the vertices, corresponding to that edge, $cos(\widehat{AMB}) \le -1/3$)

Kyiv City MO Juniors Round2 2010+ geometry, 2019.8.4

Tags: geometry , angles , equal segments

In the triangle $ABC$ it is known that$\angle A = 75^o, \angle C = 45^o$. On the ray $BC$ beyond the point $C$ the point $T$ is taken so that $BC = CT$. Let $M$ be the midpoint of the segment $AT$. Find the measure of the $\angle BMC$. (Anton Trygub)

2011 District Olympiad, 3

Tags: geometry , 3D geometry , prism , areas , angles

Let $ABCA'B'C'$ a right triangular prism with the bases equilateral triangles. A plane $\alpha$ containing point $A$ intersects the rays $BB'$ and $CC'$ at points E and $F$, so that $S_ {ABE} + S_{ACF} = S_{AEF}$. Determine the measure of the angle formed by the plane $(AEF)$ with the plane $(BCC')$.

2017 Sharygin Geometry Olympiad, P18

Tags: geometry , Lemoine point , angles

Let $L$ be the common point of the symmedians of triangle $ABC$, and $BH$ be its altitude. It is known that $\angle ALH = 180^o -2\angle A$. Prove that $\angle CLH = 180^o - 2\angle C$.

Novosibirsk Oral Geo Oly VII, 2021.5

Tags: geometry , angles , equal segments

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 $\angle BCA$.

1948 Moscow Mathematical Olympiad, 152

Tags: reflection , geometry , angles , geometric transformation , 3D geometry , combinatorial geometry

a) Two legs of an angle $\alpha$ on a plane are mirrors. Prove that after several reflections in the mirrors any ray leaves in the direction opposite the one from which it came if and only if $\alpha = \frac{90^o}{n}$ for an integer $n$. Find the number of reflections. b) Given three planar mirrors in space forming an octant (trihedral angle with right planar angles), prove that any ray of light coming into this mirrored octant leaves it, after several reflections in the mirrors, in the direction opposite to the one from which it came. Find the number of reflections.

2007 Estonia Team Selection Test, 4

Tags: Angle Chasing , angles , square , geometry

In square $ABCD,$ points $E$ and $F$ are chosen in the interior of sides $BC$ and $CD$, respectively. The line drawn from $F$ perpendicular to $AE$ passes through the intersection point $G$ of $AE$ and diagonal $BD$. A point $K$ is chosen on $FG$ such that $|AK|= |EF|$. Find $\angle EKF.$

Geometry Mathley 2011-12, 11.3

Tags: geometry , equal angles , angles , equal ratio

Let $ABC$ be a triangle such that $AB = AC$ and let $M$ be a point interior to the triangle. If $BM$ meets $AC$ at $D$. show that $\frac{DM}{DA}=\frac{AM}{AB}$ if and only if $\angle AMB = 2\angle ABC$. Michel Bataille

Estonia Open Junior - geometry, 2010.1.2

Tags: geometry , angles , equal angles , equal segments

Given a convex quadrangle $ABCD$ with $|AD| = |BD| = |CD|$ and $\angle ADB = \angle DCA$, $\angle CBD = \angle BAC$, find the sizes of the angles of the quadrangle.

Brazil L2 Finals (OBM) - geometry, 2008.5

Tags: geometry , orthocenter , Circumcenter , angles

Let $ABC$ be an acutangle triangle and $O, H$ its circumcenter, orthocenter, respectively. If $\frac{AB}{\sqrt2}=BH=OB$, calculate the angles of the triangle $ABC$ .

2011 Tournament of Towns, 3

Tags: geometry , angles , convex , quadrilateral

In a convex quadrilateral $ABCD, AB = 10, BC = 14, CD = 11$ and $DA = 5$. Determine the angle between its diagonals.

Estonia Open Senior - geometry, 1994.1.4

Tags: angles , ratio

Prove that if $\frac{AC}{BC}=\frac{AB + BC}{AC}$ in a triangle $ABC$ , then $\angle B = 2 \angle A$ .

2005 Abels Math Contest (Norwegian MO), 3a

Tags: angles , isosceles , geometry

In the isosceles triangle $\vartriangle ABC$ is $AB = AC$. Let $D$ be the midpoint of the segment $BC$. The points $P$ and $Q$ are respectively on the lines $AD$ and $AB$ (with $Q \ne B$) so that $PQ = PC$. Show that $\angle PQC =\frac12 \angle A $

2021 Novosibirsk Oral Olympiad in Geometry, 4

Tags: angles

It is known about two triangles that for each of them the sum of the lengths of any two of its sides is equal to the sum of the lengths of any two sides of the other triangle. Are triangles necessarily congruent?

1985 Tournament Of Towns, (090) T1

Tags: geometry , angles

In quadrilateral ABCD it is given that $AB = BC = 1, \angle ABC = 100^o$ , and $\angle CDA = 130^o$ . Find the length of $BD$.

1991 Tournament Of Towns, (305) 2

Tags: geometry , isosceles , angles

In $\vartriangle ABC$, $AB = AC$ and $\angle BAC = 20^o$. A point $D$ lies on the side $AB$ and $AD = BC$. Find $\angle BCD$. (LF. Sharygin, Moscow)

Novosibirsk Oral Geo Oly IX, 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 $DQ$.

2020 Romanian Master of Mathematics Shortlist, G1

Tags: geometry , angles , RMM , RMM 2020 , RMM Shortlist

The incircle of a scalene triangle $ABC$ touches the sides $BC, CA$, and $AB$ at points $D, E$, and $F$, respectively. Triangles $APE$ and $AQF$ are constructed outside the triangle so that \[AP =PE, AQ=QF, \angle APE=\angle ACB,\text{ and }\angle AQF =\angle ABC.\]Let $M$ be the midpoint of $BC$. Find $\angle QMP$ in terms of the angles of the triangle $ABC$. [i]Iran, Shayan Talaei[/i]