Olympiad problems

2017 Yasinsky Geometry Olympiad, 5

The four points of a circle are in the following order: $A, B, C, D$. Extensions of chord $AB$ beyond point $B$ and of chord $CD$ beyond point $C$ intersect at point $E$, with $\angle AED= 60^o$. If $\angle ABD =3 \angle BAC$ , prove that $AD$ is the diameter of the circle.

2002 Estonia National Olympiad, 1

Tags: equal angles , square , geometry , angle

Points $K$ and $L$ are taken on the sides $BC$ and $CD$ of a square $ABCD$ so that $\angle AKB = \angle AKL$. Find $\angle KAL$.

2020 Greece Junior Math Olympiad, 2

Tags: altitude , geometry , angle

Let $ABC$ be an acute-angled triangle with $AB<AC$. Let $D$ be the midpoint of side $BC$ and $BE,CZ$ be the altitudes of the triangle $ABC$. Line $ZE$ intersects line $BC$ at point $O$. (i) Find all the angles of the triangle $ZDE$ in terms of angle $\angle A$ of the triangle $ABC$ (ii) Find the angle $\angle BOZ$ in terms of angles $\angle B$ and $\angle C$ of the triangle $ABC$

2022 Dutch BxMO TST, 2

Tags: geometry , equal angles , angle

Let $ABC$ be an acute triangle, and let $D$ be the foot of the altitude from $A$. The circle with centre $A$ passing through $D$ intersects the circumcircle of triangle $ABC$ in $X$ and $Y$ , in such a way that the order of the points on this circumcircle is: $A, X, B, C, Y$ . Show that $\angle BXD = \angle CYD$.

1973 Dutch Mathematical Olympiad, 3

Tags: geometry , isosceles , angle

The angles $A$ and $B$ of base of the isosceles triangle $ABC$ are equal to $40^o$. Inside $\vartriangle ABC$, $P$ is such that $\angle PAB = 30^o$ and $\angle PBA = 20^o$. Calculate, without table, $\angle PCA$.

2017 Estonia Team Selection Test, 10

Tags: semicircle , geometry , angle

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 $\angle CAX$.

1946 Moscow Mathematical Olympiad, 111

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

Given two intersecting planes $\alpha$ and $\beta$ and a point $A$ on the line of their intersection. Prove that of all lines belonging to $\alpha$ and passing through $A$ the line which is perpendicular to the intersection line of $\alpha$ and $\beta$ forms the greatest angle with $\beta$.

2004 Regional Olympiad - Republic of Srpska, 3

Tags: geometry , isosceles , angle

Let $ABC$ be an isosceles triangle with $\angle A=\angle B=80^\circ$. A straight line passes through $B$ and through the circumcenter of the triangle and intersects the side $AC$ at $D$. Prove that $AB=CD$.

2010 Oral Moscow Geometry Olympiad, 1

Tags: geometry , equilateral , angle

Two equilateral triangles $ABC$ and $CDE$ have a common vertex (see fig). Find the angle between straight lines $AD$ and $BE$. [img]https://1.bp.blogspot.com/-OWpqpAqR7Zw/Xzj_fyqhbFI/AAAAAAAAMao/5y8vCfC7PegQLIUl9PARquaWypr8_luAgCLcBGAsYHQ/s0/2010%2Boral%2Bmoscow%2Bgeometru%2B8.1.gif[/img]

V Soros Olympiad 1998 - 99 (Russia), 10.6

Tags: geometry , angle bisector , angle

In triangle $ABC$, the bisectors of the internal angles $AA_1$ , $BB_1$ and $CC_1$ are drawn ($A_1, B_1$, $C_1$ - on the sides of the triangle). It is known that $\angle AA_1C = \angle AC_1B_1$. Find $\angle BCA$.

2009 District Olympiad, 3

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

Consider the regular quadrilateral prism $ABCDA'B'C 'D'$, in which $AB = a,AA' = \frac{a \sqrt {2}}{2}$, and $M$ is the midpoint of $B' C'$. Let $F$ be the foot of the perpendicular from $B$ on line $MC$, Let determine the measure of the angle between the planes $(BDF)$ and $(HBS)$.

I Soros Olympiad 1994-95 (Rus + Ukr), 11.3

Tags: geometry , geometric inequality , angle

It is known that in the triangle $ABC$, $ 2 \angle BAC + 3 \angle ABC= 180^o$. Prove that $4(BC + CA)< 5AB$.

2021 Sharygin Geometry Olympiad, 10-11.6

Tags: equal angles , geometry , trapezoid , angle

The lateral sidelines $AB$ and $CD$ of trapezoid $ABCD$ meet at point $S$. The bisector of angle $ASC$ meets the bases of the trapezoid at points $K$ and $L$ ($K$ lies inside segment $SL$). Point $X$ is chosen on segment $SK$, and point $Y$ is selected on the extension of $SL$ beyond $L$ such a way that $\angle AXC - \angle AYC = \angle ASC$. Prove that $\angle BXD - \angle BYD = \angle BSD$.

Estonia Open Senior - geometry, 2003.1.2

Tags: geometry , trigonometry , 3d geometry , angle

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 $\cos a$.

Swiss NMO - geometry, 2019.7

Tags: geometry , equal angles , equal segments , angle

Let $ABC$ be a triangle with $\angle CAB = 2 \angle ABC$. Assume that a point $D$ is inside the triangle $ABC$ exists such that $AD = BD$ and $CD = AC$. Show that $\angle ACB = 3 \angle DCB$.

VI Soros Olympiad 1999 - 2000 (Russia), 9.5

Tags: geometry , angle

Angle $A$ in triangle $ABC$ is equal to $a$. A circle passing through $A$ and $B$ and tangent to $BC$ intersects the median to side $BC$ (or its extension) at a point $M$ different from $A$. Find the angle $\angle BMC$.

2019 Yasinsky Geometry Olympiad, p4

Tags: geometry , cyclic quadrilateral , angle chasing , angle

Find the angles of the cyclic quadrilateral if you know that each of its diagonals is a bisector of one angle and a trisector of the opposite one (the trisector of the angle is one of the two rays that lie in the interior of the angle and divide it into three equal parts). (Vyacheslav Yasinsky)

2013 Bangladesh Mathematical Olympiad, 9

Tags: geometry , angle , circles

Six points $A, B, C, D, E, F$ are chosen on a circle anticlockwise. None of $AB, CD, EF$ is a diameter. Extended $AB$ and $DC$ meet at $Z, CD$ and $FE$ at $X, EF$ and $BA$ at $Y. AC$ and $BF$ meets at $P, CE$ and $BD$ at $Q$ and $AE$ and $DF$ at $R.$ If $O$ is the point of intersection of $YQ$ and $ZR,$ find the $\angle XOP.$

2017 Novosibirsk Oral Olympiad in Geometry, 7

Tags: geometry , angle

A car is driving along a straight highway at a speed of $60$ km per hour. Not far from the highway there is a parallel to him a $100$-meter fence. Every second, the passenger of the car measures the angle at which the fence is visible. Prove that the sum of all the angles he measured is less than $1100^o$

1997 Tournament Of Towns, (539) 4

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

All edges of a tetrahedron $ABCD$ are equal. The tetrahedron $ABCD$ is inscribed in a sphere. $CC'$ and $DD'$ are diameters. Find the angle between the planes $ABC$' and $ACD'$. (A Zaslavskiy)

1997 Argentina National Olympiad, 2

Tags: geometry , right triangle , isosceles , midpoint , angle

Let $ABC$ be a triangle and $M$ be the midpoint of $AB$. If it is known that $\angle CAM + \angle MCB = 90^o$, show that triangle $ABC$ is isosceles or right.

2007 Denmark MO - Mohr Contest, 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]

2012 India Regional Mathematical Olympiad, 7

Tags: angle chasing , angle , circles , geometry

On the extension of chord $AB$ of a circle centroid at $O$ a point $X$ is taken and tangents $XC$ and $XD$ to the circle are drawn from it with $C$ and $D$ lying on the circle, let $E$ be the midpoint of the line segment $CD$. If $\angle OEB = 140^o$ then determine with proof the magnitude of $\angle AOB$.

2021 Puerto Rico Team Selection Test, 2

Tags: geometry , incenter , right triangle , angle

Let $ABC$ be a right triangle with right angle at $ B$ and $\angle C=30^o$. If $M$ is midpoint of the hypotenuse and $I$ the incenter of the triangle, show that $ \angle IMB=15^o$.

2015 Portugal MO, 2

Tags: geometry , isosceles , angle

Let $[ABC]$ be a triangle and $D$ a point between $A$ and $B$. If the triangles $[ABC], [ACD]$ and $[BCD]$ are all isosceles, what are the possible values of $\angle ABC$?