Olympiad problems

VI Soros Olympiad 1999 - 2000 (Russia), 9.5

Tags: geometry , angles

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 $\angle BMC$.

Kyiv City MO Juniors Round2 2010+ geometry, 2017.8.2

Tags: angles , geometry , right triangle , isosceles

Triangle $ABC$ is right-angled and isosceles with a right angle at the vertex $C$. On rays $CB$ on vertex $B$ is selected point F, on rays $BA$ on vertex $A$ is selected point G so that $AG = BF.$ The ray $GD$ is drawn so that it intersects with ray $AC$ at point $D$ with $\angle FGD = 45^o$. Find $\angle FDG$. (Bogdan Rublev)

2019 Flanders Math Olympiad, 3

Tags: geometry , angles , Angle Chasing

In triangle $\vartriangle ABC$ holds $\angle A= 40^o$ and $\angle B = 20^o$ . The point $P$ lies on the line $AC$ such that $C$ is between $A$ and $P$ and $| CP | = | AB | - | BC |$. Calculate the $\angle CBP$.

2017 Junior Regional Olympiad - FBH, 3

Tags: Triangle , Compare , angles

In acute triangle $ABC$ holds $\angle BAC=80^{\circ}$, and altitudes $h_a$ and $h_b$ intersect in point $H$. if $\angle AHB = 126^{\circ}$, which side is the smallest, and which is the biggest in $ABC$

2014 Oral Moscow Geometry Olympiad, 6

Tags: geometry , angles , Angle Chasing , right triangle , isosceles

Inside an isosceles right triangle $ABC$ with hypotenuse $AB$ a point $M$ is taken such that the angle $\angle MAB$ is $15 ^o$ larger than the angle $\angle MAC$ , and the angle $\angle MCB$ is $15^o$ larger than the angle $\angle MBC$. Find the angle $\angle BMC$ .

2013 Junior Balkan Team Selection Tests - Moldova, 4

Tags: algebra , angles

A train from stop $A$ to stop $B$ is traveled in $X$ minutes ($0 <X <60$). It is known that when starting from $A$, as well as when arriving at $B$, the angle formed by the hour and the minute had measure equal to $X$ degrees. Find $X $.

1994 Tournament Of Towns, (401) 3

Tags: geometry , incenter , geometric inequality , angles

Let $O$ be a point inside a convex polygon $A_1A_2... A_n$ such that $$\angle OA_1A_n \le \angle OA_1A_2, \angle OA_2A_1 \le \angle OA_2A_3, ..., \angle OA_{n-1}A_{n-2} \le \angle OA_{n-1}A_n, \angle OA_nA_{n-1} \le \angle OA_nA_1$$ and all of these angles are acute. Prove that $O$ is the centre of the circle inscribed in the polygon. (V Proizvolov)

1979 IMO Shortlist, 17

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 $PQR.$

2020 Greece Junior Math Olympiad, 2

Tags: geometry , angles , altitudes

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 $ABC$

VI Soros Olympiad 1999 - 2000 (Russia), 10.2

Tags: angles , geometry

In the triangle $ABC$, the point $X$ is the projection of the touchpoint of the inscribed circle to the side $BC$ on the middle line parallel to $BC$. It is known that $\angle BAC \ge 60^o$. Prove that the angle $BXC$ is obtuse.

2008 Thailand Mathematical Olympiad, 1

Tags: geometry , trigonometry , angles

Let $\vartriangle ABC$ be a triangle with $\angle BAC = 90^o$ and $\angle ABC = 60^o$. Point $E$ is chosen on side $BC$ so that $BE : EC = 3 : 2$. Compute $\cos\angle CAE$.

2008 Oral Moscow Geometry Olympiad, 3

Tags: geometry , angles , hexagon

In the regular hexagon $ABCDEF$ on the line $AF$, the point $X$ is taken so that the angle $XCD$ is $45^o$. Find the angle $\angle FXE$. (Kiev Olympiad)

1957 Moscow Mathematical Olympiad, 361

Tags: geometry , angles , min , geometric inequality

The lengths, $a$ and $b$, of two sides of a triangle are known. (a) What length should the third side be, in order for the largest angle of the triangle to be of the least possible value? (b) What length should the third side be in order for the smallest angle of the triangle to be of the greatest possible value?

2020 German National Olympiad, 1

Tags: geometry , geometry proposed , angles , circle , Maximal

Let $k$ be a circle with center $M$ and let $B$ be another point in the interior of $k$. Determine those points $V$ on $k$ for which $\measuredangle BVM$ becomes maximal.

1946 Moscow Mathematical Olympiad, 106

Tags: acute , angles , maximum , geometry

What is the largest number of acute angles that a convex polygon can have?

1996 May Olympiad, 4

Tags: geometry , angles , square

Let $ABCD$ be a square and let point $F$ be any point on side $BC$. Let the line perpendicular to $DF$, that passes through $B$, intersect line $DC$ at $Q$. What is value of $\angle FQC$?

2015 AMC 10, 19

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

The isosceles right triangle $ABC$ has right angle at $C$ and area $12.5$. The rays trisecting $\angle{ACB}$ intersect $AB$ at $D$ and $E$. What is the area of $\triangle{CDE}$? $\textbf{(A) }\frac{5\sqrt{2}}{3}\qquad\textbf{(B) }\frac{50\sqrt{3}-75}{4}\qquad\textbf{(C) }\frac{15\sqrt{3}}{8}\qquad\textbf{(D) }\frac{50-25\sqrt{3}}{2}\qquad\textbf{(E) }\frac{25}{6}$

1994 Portugal MO, 5

Tags: geometry , circumcircle , angles

Consider a circle $C$ of center $O$ and its inner point $Q$, different from $O$. Where we must place a point $P$ on the circle $C$ so that the angle $\angle OPQ$ is the largest possible?

1974 Chisinau City MO, 82

Tags: geometry , angles , equal angles

Is there a moment in a day when three hands - hour, minute and second - of a clock running correctly form angles of $120^o$ in pairs?

2020 Argentina National Olympiad, 3

Tags: geometry , angles , right triangle , isosceles

Let $ABC$ be a right isosceles triangle with right angle at $A$. Let $E$ and $F$ be points on A$B$ and $AC$ respectively such that $\angle ECB = 30^o$ and $\angle FBC = 15^o$. Lines $CE$ and $BF$ intersect at $P$ and line $AP$ intersects side $BC$ at $D$. Calculate the measure of angle $\angle FDC$.

1985 Tournament Of Towns, (106) 6

Tags: geometry , angles , altitude , angle bisector

In triangle $ABC, AH$ is an altitude ($H$ is on $BC$) and $BE$ is a bisector ($E$ is on $AC$) . We are given that angle $BEA$ equals $45^o$ .Prove that angle $EHC$ equals $45^o$ . (I. Sharygin , Moscow)

2018 Romania National Olympiad, 4

Tags: geometry , 3D geometry , cube , angles , parallel , equal segments

In the rectangular parallelepiped $ABCDA'B'C'D'$ we denote by $M$ the center of the face $ABB'A'$. We denote by $M_1$ and $M_2$ the projections of $M$ on the lines $B'C$ and $AD'$ respectively. Prove that: a) $MM_1 = MM_2$ b) if $(MM_1M_2) \cap (ABC) = d$, then $d \parallel AD$; c) $\angle (MM_1M_2), (A B C)= 45^ o \Leftrightarrow \frac{BC}{AB}=\frac{BB'}{BC}+\frac{BC}{BB'}$.

2001 Bosnia and Herzegovina Team Selection Test, 1

Tags: ratio , geometry , circle , angles , arc

On circle there are points $A$, $B$ and $C$ such that they divide circle in ratio $3:5:7$. Find angles of triangle $ABC$

2013 Brazil Team Selection Test, 2

Tags: geometry , acute , angles , cyclic quadrilateral

Let $ABCD$ be a convex cyclic quadrilateral with $AD > BC$, A$B$ not being diameter and $C D$ belonging to the smallest arc $AB$ of the circumcircle. The rays $AD$ and $BC$ are cut at $K$, the diagonals $AC$ and $BD$ are cut at $P$ and the line $KP$ cuts the side $AB$ at point $L$. Prove that angle $\angle ALK$ is acute.

1988 Tournament Of Towns, (185) 2

Tags: geometry , angles , altitude

In a triangle two altitudes are not smaller than the sides on to which they are dropped. Find the angles of the triangle.