Olympiad problems

2002 Estonia Team Selection Test, 4

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

2013 Junior Balkan Team Selection Tests - Moldova, 7

Tags: equal segments , angle , square , geometry

The points $M$ and $N$ are located respectively on the diagonal $(AC)$ and the side $(BC)$ of the square $ABCD$ such that $MN = MD$. Determine the measure of the angle $MDN$.

1997 All-Russian Olympiad Regional Round, 8.3

Tags: equilateral , angle , geometry

On sides $AB$ and $BC$ of an equilateral triangle $ABC$ are taken points $D$ and $K$, and on the side $AC$ , points $E$ and $M$ so that $DA + AE = KC +CM = AB$. Prove that the angle between lines $DM$ and $KE$ is equal to $60^o$.

2013 Dutch Mathematical Olympiad, 3

Tags: angle , diagonal , geometry , parallel

The sides $BC$ and $AD$ of a quadrilateral $ABCD$ are parallel and the diagonals intersect in $O$. For this quadrilateral $|CD| =|AO|$ and $|BC| = |OD|$ hold. Furthermore $CA$ is the angular bisector of angle $BCD$. Determine the size of angle $ABC$. [asy] unitsize(1 cm); pair A, B, C, D, O; D = (0,0); B = 3*dir(180 + 72); C = 3*dir(180 + 72 + 36); A = extension(D, D + (1,0), C, C + dir(180 - 36)); O = extension(A, C, B, D); draw(A--B--C--D--cycle); draw(B--D); draw(A--C); dot("$A$", A, N); dot("$B$", B, SW); dot("$C$", C, SE); dot("$D$", D, N); dot("$O$", O, E); [/asy] Attention: the figure is not drawn to scale.

2011 Saudi Arabia IMO TST, 3

Tags: geometry , angle , isosceles

In acute triangle $ABC$, $\angle A = 20^o$. Prove that the triangle is isosceles if and only if $$\sqrt[3]{a^3 + b^3 + c^3 -3abc} = \min\{b, c\}$$, where $a,b, c$ are the side lengths of triangle $ABC$.

Novosibirsk Oral Geo Oly VIII, 2022.4

Tags: angle , geometry

In triangle $ABC$, angle $C$ is three times the angle $A$, and side $AB$ is twice the side $BC$. What can be the angle $ABC$?

Estonia Open Senior - geometry, 2002.1.4

Tags: angle , geometry , equal segments

In a triangle $ABC$ we have $\angle B = 2 \cdot \angle C$ and the angle bisector drawn from $A$ intersects $BC$ in a point $D$ such that $|AB| = |CD|$. Find $\angle A$.

2006 Sharygin Geometry Olympiad, 25

Tags: angle , geometry , 3d geometry , tetrahedron , ratio

In the tetrahedron $ABCD$ , the dihedral angles at the $BC, CD$, and $DA$ edges are equal to $\alpha$, and for the remaining edges equal to $\beta$. Find the ratio $AB / CD$.

1996 Tournament Of Towns, (485) 3

Tags: right triangle , geometry , angle

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,)

2020 Ukrainian Geometry Olympiad - April, 2

Tags: geometry , acute , angle

Inside the triangle $ABC$ is point $P$, such that $BP > AP$ and $BP > CP$. Prove that $\angle ABC$ is acute.

Brazil L2 Finals (OBM) - geometry, 2012.4

Tags: pentagon , angle chasing , region , equilateral , angle

The figure below shows a regular $ABCDE$ pentagon inscribed in an equilateral triangle $MNP$ . Determine the measure of the angle $CMD$. [img]http://4.bp.blogspot.com/-LLT7hB7QwiA/Xp9fXOsihLI/AAAAAAAAL14/5lPsjXeKfYwIr5DyRAKRy0TbrX_zx1xHQCK4BGAYYCw/s200/2012%2Bobm%2Bl2.png[/img]

2017 Novosibirsk Oral Olympiad in Geometry, 2

Tags: geometry , angle

You are given a convex quadrilateral $ABCD$. It is known that $\angle CAD = \angle DBA = 40^o$, $\angle CAB = 60^o$, $\angle CBD = 20^o$. Find the angle $\angle CDB $.

Estonia Open Senior - geometry, 2003.1.2

Tags: 3d geometry , trigonometry , angle , geometry

Four rays spread out from point $O$ in a $3$-dimensional space in a way that the angle between every two rays is $a$. Find $\cos a$.

2005 Abels Math Contest (Norwegian MO), 3a

Tags: geometry , angle , isosceles

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 $

2016 Saudi Arabia IMO TST, 2

Tags: angle , convex , equal segments , geometry , hexagon , equal angles

Let $ABCDEF$ be a convex hexagon with $AB = CD = EF$, $BC =DE = FA$ and $\angle A+\angle B = \angle C +\angle D = \angle E +\angle F$. Prove that $\angle A=\angle C=\angle E$ and $\angle B=\angle D=\angle F$. Tran Quang Hung

2017 Abels Math Contest (Norwegian MO) Final, 4

Tags: geometry , angle , circumcenter , orthocenter

Let $a > 0$ and $0 < \alpha <\pi$ be given. Let $ABC$ be a triangle with $BC = a$ and $\angle BAC = \alpha$ , and call the cicumcentre $O$, and the orthocentre $H$. The point $P$ lies on the ray from $A$ through $O$. Let $S$ be the mirror image of $P$ through $AC$, and $T$ the mirror image of $P$ through $AB$. Assume that $SATH$ is cyclic. Show that the length $AP$ depends only on $a$ and $\alpha$.

2009 District Olympiad, 4

Tags: geometry , equilateral , angle , isosceles

Let $ABC$ be an equilateral $ABC$. Points $M, N, P$ are located on the sides $AC, AB, BC$, respectively, such that $\angle CBM= \frac{1}{2} \angle AMN = \frac{1}{3} \angle BNP$ and $\angle CMP = 90 ^o$. a) Show that $\vartriangle NMB$ is isosceles. b) Determine $\angle CBM$.

2001 Argentina National Olympiad, 2

Tags: geometry , angle , angle bisector , equal angles

Let $\vartriangle ABC$ be a triangle such that angle $\angle ABC$ is less than angle $\angle ACB$. The bisector of angle $\angle BAC$ cuts side $BC$ at $D$. Let $E$ be on side $AB$ such that $\angle EDB = 90^o$ and $F$ on side $AC$ such that $\angle BED = \angle DEF$. Prove that $\angle BAD = \angle FDC$.

Kyiv City MO Juniors Round2 2010+ geometry, 2013.8.3

Tags: angle , parallel , geometry , equal angles

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)