Olympiad problems

2005 Estonia Team Selection Test, 1

On a plane, a line $\ell$ and two circles $c_1$ and $c_2$ of different radii are given such that $\ell$ touches both circles at point $P$. Point $M \ne P$ on $\ell$ is chosen so that the angle $Q_1MQ_2$ is as large as possible where $Q_1$ and $Q_2$ are the tangency points of the tangent lines drawn from $M$ to $c_i$ and $c_2$, respectively, differing from $\ell$ . Find $\angle PMQ_1 + \angle PMQ_2$·

2015 Sharygin Geometry Olympiad, P5

Tags: geometry , angles , circles

Let a triangle $ABC$ be given. Two circles passing through $A$ touch $BC$ at points $B$ and $C$ respectively. Let $D$ be the second common point of these circles ($A$ is closer to $BC$ than $D$). It is known that $BC = 2BD$. Prove that $\angle DAB = 2\angle ADB.$

2020 Ukrainian Geometry Olympiad - April, 3

Tags: geometry , parallelogram , orthocenter , angles , angle inequalities

Let $H$ be the orthocenter of the acute-angled triangle $ABC$. Inside the segment $BC$ arbitrary point $D$ is selected. Let $P$ be such that $ADPH$ is a parallelogram. Prove that $\angle BCP< \angle BHP$.

1984 Tournament Of Towns, (O76) T3

Tags: geometry , angles , angle bisector

In $\vartriangle ABC, \angle ABC = \angle ACB = 40^o$ . $BD$ bisects $\angle ABC$ , with $D$ located on $AC$. Prove that $BD + DA = BC$.

1985 Brazil National Olympiad, 2

Tags: geometry , combinatorial geometry , angles

Given $n$ points in the plane, show that we can always find three which give an angle $\le \pi / n$.

Indonesia Regional MO OSP SMA - geometry, 2013.5

Tags: geometry , geometric inequality , angles

Given an acute triangle $ABC$. The longest line of altitude is the one from vertex $A$ perpendicular to $BC$, and it's length is equal to the length of the median of vertex $B$. Prove that $\angle ABC \le 60^o$

1992 IMO Longlists, 37

Tags: geometry , circles , Trigonometric inequality , angles , IMO Shortlist , IMO Longlist

Let the circles $C_1, C_2$, and $C_3$ be orthogonal to the circle $C$ and intersect each other inside $C$ forming acute angles of measures $A, B$, and $C$. Show that $A + B +C < \pi.$

1980 IMO Shortlist, 15

Tags: geometry , 3D geometry , tetrahedron , sphere , triangle inequality , angles , IMO Shortlist

Prove that the sum of the six angles subtended at an interior point of a tetrahedron by its six edges is greater than 540°.

2005 Korea Junior Math Olympiad, 2

Tags: geometry , circumcircle , midpoint , fixed , angles

For triangle $ABC, P$ and $Q$ satisfy $\angle BPA + \angle AQC = 90^o$. It is provided that the vertices of the triangle $BAP$ and $ACQ$ are ordered counterclockwise (or clockwise). Let the intersection of the circumcircles of the two triangles be $N$ ($A \ne N$, however if $A$ is the only intersection $A = N$), and the midpoint of segment $BC$ be $M$. Show that the length of $MN$ does not depend on $P$ and $Q$.

2021 Junior Balkan Team Selection Tests - Moldova, 2

Tags: geometry , angles , parallelogram

Inside the parallelogram $ABCD$, point $E$ is chosen, such that $AE = DE$ and $\angle ABE = 90^o$. Point $F$ is the midpoint of the side $BC$ . Find the measure of the angle $\angle DFE$.

2010 Junior Balkan Team Selection Tests - Romania, 2

Tags: angles , equal angles , geometry , convex quadrilateral

Let $ABCD$ be a convex quadrilateral with $\angle BCD= 120^o, \angle {CBA} = 45^o, \angle {CBD} = 15^o$ and $\angle {CAB} = 90^o$. Show that $AB = AD$.

2021 Iranian Geometry Olympiad, 5

Tags: geometry , angles

Let $A_1, A_2, . . . , A_{2021}$ be $2021$ points on the plane, no three collinear and $$\angle A_1A_2A_3 + \angle A_2A_3A_4 +... + \angle A_{2021}A_1A_2 = 360^o,$$ in which by the angle $\angle A_{i-1}A_iA_{i+1}$ we mean the one which is less than $180^o$ (assume that $A_{2022} =A_1$ and $A_0 = A_{2021}$). Prove that some of these angles will add up to $90^o$. [i]Proposed by Morteza Saghafian - Iran[/i]

Novosibirsk Oral Geo Oly IX, 2016.2

Tags: geometry , angles

Bisector of one angle of triangle $ABC$ is equal to the bisector of its external angle at the same vertex (see figure). Find the difference between the other two angles of the triangle. [img]https://cdn.artofproblemsolving.com/attachments/c/3/d2efeb65544c45a15acccab8db05c8314eb5f2.png[/img]

1996 Swedish Mathematical Competition, 4

Tags: pentagon , Increasing , angles , Cyclic , geometry

The angles at $A,B,C,D,E$ of a pentagon $ABCDE$ inscribed in a circle form an increasing sequence. Show that the angle at $C$ is greater than $\pi/2$, and that this lower bound cannot be improved.

2023 Novosibirsk Oral Olympiad in Geometry, 7

Tags: angles , geometry

Triangle $ABC$ is given with angles $\angle ABC = 60^o$ and $\angle BCA = 100^o$. On the sides AB and AC, the points $D$ and $E$ are chosen, respectively, in such a way that $\angle EDC = 2\angle BCD = 2\angle CAB$. Find the angle $\angle BED$.

V Soros Olympiad 1998 - 99 (Russia), 10.6

Tags: geometry , angle bisector , angles

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 $\angle BCA$.

2023 Bundeswettbewerb Mathematik, 3

Tags: geometry , geometry proposed , angles , area of a triangle

Let $ABC$ be a triangle with incenter $I$. Let $M_b$ and $M_a$ be the midpoints of $AC$ and $BC$, respectively. Let $B'$ be the point of intersection of lines $M_bI$ and $BC$, and let $A'$ be the point of intersection of lines $M_aI$ and $AC$. If triangles $ABC$ and $A'B'C$ have the same area, what are the possible values of $\angle ACB$?

1997 Tournament Of Towns, (529) 2

Tags: geometry , angles

One side of a triangle is equal to one third of the sum of the other two. Prove that the angle opposite the first side is the smallest angle of the triangle. (AK Tolpygo)

2015 Sharygin Geometry Olympiad, P2

Tags: geometry , Circumcenter , orthocenter , incenter , angles

Let $O$ and $H$ be the circumcenter and the orthocenter of a triangle $ABC$. The line passing through the midpoint of $OH$ and parallel to $BC$ meets $AB$ and $AC$ at points $D$ and $E$. It is known that $O$ is the incenter of triangle $ADE$. Find the angles of $ABC$.

2010 Contests, 3

Tags: angles , geometry , isosceles , equal segments

Consider triangle $ABC$ with $AB = AC$ and $\angle A = 40 ^o$. The points $S$ and $T$ are on the sides $AB$ and $BC$, respectively, so that $\angle BAT = \angle BCS= 10 ^o$. The lines $AT$ and $CS$ intersect at point $P$. Prove that $BT = 2PT$.

2019 Argentina National Olympiad, 3

Tags: geometry , angles , equal segments

In triangle $ABC$ it is known that $\angle ACB = 2\angle ABC$. Furthermore $P$ is an interior point of the triangle $ABC$ such that $AP = AC$ and $PB = PC$. Prove that $\angle BAC = 3 \angle BAP$.

1955 Moscow Mathematical Olympiad, 301

Tags: trihedral angle , angles , acute , 3D geometry , geometry

Given a trihedral angle with vertex $O$. Find whether there is a planar section $ABC$ such that the angles $\angle OAB$, $\angle OBA$, $\angle OBC$, $\angle OCB$, $\angle OAC$, $\angle OCA$ are acute.

II Soros Olympiad 1995 - 96 (Russia), 9.3

Tags: geometry , angles

Two straight lines are drawn on a plane, intersecting at an angle of $40^o$. A circle with center at point $O$ touches these lines. Let's consider a line, different from the given ones, tangent to the same circle and intersecting the given lines at points $B$ and $C$. What can the angle $\angle BOC$ be equal to?

Ukraine Correspondence MO - geometry, 2017.8

Tags: geometry , trapezoid , min , angles , Ukraine Correspondence

On the midline of the isosceles trapezoid $ABCD$ ($BC \parallel AD$) find the point $K$, for which the sum of the angles $\angle DAK + \angle BCK$ will be the smallest.

1996 Czech And Slovak Olympiad IIIA, 4

Tags: geometry , Product , angles

Points $A$ and $B$ on the rays $CX$ and $CY$ respectively of an acute angle $XCY$ are given so that $CX < CA = CB < CY$. Construct a line meeting the ray $CX$ and the segments $AB,BC$ at $K,L,M$, respectively, such that $KA \cdot YB = XA \cdot MB = LA\cdot LB \ne 0$.