Olympiad problems

1999 Nordic, 2

Consider $7$-gons inscribed in a circle such that all sides of the $7$-gon are of different length. Determine the maximal number of $120^\circ$ angles in this kind of a $7$-gon.

1985 IMO Shortlist, 22

Tags: angle , triangle , circumcircle , geometry

A circle with center $O$ passes through the vertices $A$ and $C$ of the triangle $ABC$ and intersects the segments $AB$ and $BC$ again at distinct points $K$ and $N$ respectively. Let $M$ be the point of intersection of the circumcircles of triangles $ABC$ and $KBN$ (apart from $B$). Prove that $\angle OMB=90^{\circ}$.

2018 Peru Iberoamerican Team Selection Test, P2

Tags: equal segments , geometry , angle , angle chasing

Let $ABC$ be a triangle with $AB = AC$ and let $D$ be the foot of the height drawn from $A$ to $BC$. Let $P$ be a point inside the triangle $ADC$ such that $\angle APB> 90^o$ and $\angle PAD + \angle PBD = \angle PCD$. The $CP$ and $AD$ lines are cut at $Q$ and the $BP$ and $AD$ lines cut into $R$. Let $T$ be a point in segment $AB$ such that $\angle TRB = \angle DQC$ and let S be a point in the extension of the segment $AP$ (on the $P$ side) such that $\angle PSR = 2 \angle PAR$. Prove that $RS = RT$.

1986 Austrian-Polish Competition, 1

Tags: similar triangles , angle , geometry , tangent circle

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.

Champions Tournament Seniors - geometry, 2005.2

Tags: symmedian , angle , circumcircle , equal angles , geometry

Given a triangle $ABC$, the line passing through the vertex $A$ symmetric to the median $AM$ wrt the line containing the bisector of the angle $\angle BAC$ intersects the circle circumscribed around the triangle $ABC$ at points $A$ and $K$. Let $L$ be the midpoint of the segment $AK$. Prove that $\angle BLC=2\angle BAC$.

1988 IMO Shortlist, 17

Tags: convex polygon , angle , pentagon , geometry

In the convex pentagon $ ABCDE,$ the sides $ BC, CD, DE$ are equal. Moreover each diagonal of the pentagon is parallel to a side ($ AC$ is parallel to $ DE$, $ BD$ is parallel to $ AE$ etc.). Prove that $ ABCDE$ is a regular pentagon.

2016 Germany National Olympiad (4th Round), 5

Tags: geometry , angle , trigonometry , cyclic quadrilateral

Let $A,B,C,D$ be points on a circle with radius $r$ in this order such that $|AB|=|BC|=|CD|=s$ and $|AD|=s+r$. Find all possible values of the interior angles of the quadrilateral $ABCD$.

1991 Tournament Of Towns, (305) 2

Tags: isosceles , angle , geometry

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)

1952 Moscow Mathematical Olympiad, 208

Tags: incircle , geometry , angle , acute

The circle is inscribed in $\vartriangle ABC$. Let $L, M, N$ be the tangent points of the circle with sides $AB, AC, BC$, respectively. Prove that $\angle MLN$ is always an acute angle.

2018 Istmo Centroamericano MO, 5

Tags: angle , isosceles , geometry

Let $ABC$ be an isosceles triangle with $CA = CB$. Let $D$ be the foot of the alttiude from $C$, and $\ell$ be the external angle bisector at $C$. Take a point $N$ on $\ell$ so that $AN> AC$ , on the same side as $A$ wrt $CD$. The bisector of the angle $NAC$ cuts $\ell$'at $F$. Show that $\angle NCD + \angle BAF> 180^o.$

2015 Iran Geometry Olympiad, 3

Tags: angle , elementary , geometry

In the figure below, we know that $AB = CD$ and $BC = 2AD$. Prove that $\angle BAD = 30^o$. [img]https://3.bp.blogspot.com/-IXi_8jSwzlU/W1R5IydV5uI/AAAAAAAAIzo/2sREnDEnLH8R9zmAZLCkVCGeMaeITX9YwCK4BGAYYCw/s400/IGO%2B2015.el3.png[/img]

2019 Romania National Olympiad, 3

Tags: geometry , angle , equal segments , equal angles

Let $ABC$ be a triangle in which $\angle ABC = 45^o$ and $\angle BAC > 90^o$. Let $O$ be the midpoint of the side $[BC]$. Consider the point $M \in (AC)$ such that $\angle COM =\angle CAB$. Perpendicular from $M$ on $AC$ intersects line $AB$ at point $P$. a) Find the measure of the angle $\angle BCP$. b) Show that if $\angle BAC = 105^o$, then $PB = 2MO$.

2013 Tournament of Towns, 5

Tags: right angle , angle , geometry , equal segments

In a quadrilateral $ABCD$, angle $B$ is equal to $150^o$, angle $C$ is right, and sides $AB$ and $CD$ are equal. Determine the angle between $BC$ and the line connecting the midpoints of sides $BC$ and $AD$.

Kyiv City MO 1984-93 - geometry, 1991.7.5

Tags: angle , rectangle , geometry

Inside the rectangle $ABCD$ is taken a point $M$ such that $\angle BMC + \angle AMD = 180^o$. Determine the sum of the angles $BCM$ and $DAM$.

Kyiv City MO 1984-93 - geometry, 1988.7.1

Tags: equilateral , angle , trapezoid , geometry

An isosceles trapezoid is divided by each diagonal into two isosceles triangles. Determine the angles of the trapezoid.

1998 All-Russian Olympiad Regional Round, 8.7

Tags: angle , geometry

Let $O$ be the center of a circle circumscribed about an acute angle triangle $ABC$, $S_A$, $S_B$, $S_C$ - circles with center O, tangent to sides $BC$, $CA$, $AB$ respectively. Prove that the sum of three angles : between the tangents to $S_A$ drawn from point $A$, to $S_B$ from point $B$ and to $S_C$ - from point $C$, is equal to $180^o$.

2006 Sharygin Geometry Olympiad, 9

Tags: angle , concurrency , circles , geometry

$L(a)$ is the line connecting the points of the unit circle corresponding to the angles $a$ and $\pi - 2a$. Prove that if $a + b + c = 2\pi$, then the lines $L (a), L (b)$ and $L (c)$ intersect at one point.

1963 IMO Shortlist, 3

Tags: geometric inequality , geometry , polygon , angle , inequalities

In an $n$-gon $A_{1}A_{2}\ldots A_{n}$, all of whose interior angles are equal, the lengths of consecutive sides satisfy the relation \[a_{1}\geq a_{2}\geq \dots \geq a_{n}. \] Prove that $a_{1}=a_{2}= \ldots= a_{n}$.

1953 Moscow Mathematical Olympiad, 252

Tags: concurrency , angle , line , geometry

Given triangle $\vartriangle A_1A_2A_3$ and a straight line $\ell$ outside it. The angles between the lines $A_1A_2$ and $A_2A_3, A_1A_2$ and $A_2A_3, A_2A_3$ and $A_3A_1$ are equal to $a_3, a_1$ and $a_2$, respectively. The straight lines are drawn through points $A_1, A_2, A_3$ forming with $\ell$ angles of $\pi -a_1, \pi -a_2, \pi -a_3$, respectively. All angles are counted in the same direction from $\ell$ . Prove that these new lines meet at one point.

1996 Singapore MO Open, 2

Tags: geometry , angle , square

In the following figure, $ABCD$ is a square of unit length and $P, Q$ are points on $AD$ and $AB$ respectively. Find $\angle PCQ$ if $|AP| + |AQ| + |PQ| = 2$. [img]https://cdn.artofproblemsolving.com/attachments/2/c/2f40db978c1d3fcbc0161f874b5cbec926058e.png[/img]

Estonia Open Senior - geometry, 2001.2.3

Tags: geometry , cyclic , angle , hexagon

Let us call a convex hexagon $ABCDEF$ [i]boring [/i] if $\angle A+ \angle C + \angle E = \angle B + \angle D + \angle F$. a) Is every cyclic hexagon boring? b) Is every boring hexagon cyclic?

2003 Cuba MO, 2

Tags: angle , circles , circumcircle , geometry

Let $A$ be a point outside the circle $\omega$ . The tangents from $A$ touch the circle at $B$ and $C$. Let $P$ be an arbitrary point on extension of $AC$ towards $C$, $Q$ the projection of $C$ onto $PB$ and $E$ the second intersection point of the circumcircle of $ABP$ with the circle $\omega$ . Prove that $\angle PEQ = 2\angle APB$

2019 Oral Moscow Geometry Olympiad, 6

Tags: geometry , 3d geometry , solid geometry , angle

The sum of the cosines of the flat angles of the trihedral angle is $-1$. Find the sum of its dihedral angles.

Russian TST 2015, P2

Tags: angle , geometry

The triangle $ABC$ is given. Let $P_1$ and $P_2$ be points on the side $AB$ such that $P_2$ lies on the segment $BP_1$ and $AP_1 = BP_2$. Similarly, $Q_1$ and $Q_2$ are points on the side $BC$ such that $Q_2$ lies on the segment $BQ_1$ and $BQ_1 = CQ_2$. The segments $P_1Q_2$ and $P_2Q_1$ intersect at the point $R{}$, and the circles $P_1P_2R$ and $Q_1Q_2R$ intersect a second time at the point $S{}$ lying inside the triangle $P_1Q_1R$. Let $M{}$ be the midpoint of the segment $AC$. Prove that the angles $P_1RS$ and $Q_1RM$ are equal.

Kyiv City MO 1984-93 - geometry, 1985.10.2

Tags: geometry , 3d geometry , cube , angle

Segment $AB$ on the surface of the cube is the shortest polyline on the surface that connects $A$ and $B$. Triangle $ABC$ consisted of such segments $AB, BC,CA$. What may be the sum of angles of such triangle if none of the vertex is on the edge of the cube ?