Olympiad problems

2019 Flanders Math Olympiad, 3

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 $\angle CBP$.

2014 Harvard-MIT Mathematics Tournament, 8

Tags: geometry , circumcircle , symmedian , Angle Chasing , radical axis

Let $ABC$ be an acute triangle with circumcenter $O$ such that $AB=4$, $AC=5$, and $BC=6$. Let $D$ be the foot of the altitude from $A$ to $BC$, and $E$ be the intersection of $AO$ with $BC$. Suppose that $X$ is on $BC$ between $D$ and $E$ such that there is a point $Y$ on $AD$ satisfying $XY\parallel AO$ and $YO\perp AX$. Determine the length of $BX$.

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 $\angle BMC$ .

2003 Bosnia and Herzegovina Team Selection Test, 4

Tags: geometry , isosceles , orthogonal , altitudes , Angle Chasing

In triangle $ABC$ $AD$ and $BE$ are altitudes. Let $L$ be a point on $ED$ such that $ED$ is orthogonal to $BL$. If $LB^2=LD\cdot LE$ prove that triangle $ABC$ is isosceles

2019 Sharygin Geometry Olympiad, 6

Tags: geometry , angle bisector , median , circumcircle , Angle Chasing

Let $AK$ and $AT$ be the bisector and the median of an acute-angled triangle $ABC$ with $AC > AB$. The line $AT$ meets the circumcircle of $ABC$ at point $D$. Point $F$ is the reflection of $K$ about $T$. If the angles of $ABC$ are known, find the value of angle $FDA$.

2025 Bulgarian Spring Mathematical Competition, 9.2

Tags: geometry , Angle Chasing , similarity , circumcircle

Let $ABC$ be an acute scalene triangle inscribed in a circle $ \Gamma $. The angle bisector of $ \angle BAC $ intersects $ BC $ at $ L $ and $ \Gamma $ at $ S $. The point $ M $ is the midpoint of $ AL $. Let $ AD $ be the altitude in $ \triangle ABC $, and the circumcircle of $ \triangle DSL $ intersects $ \Gamma $ again at $ P $. Let $ N $ be the midpoint of $ BC $, and let $ K $ be the reflection of $ D $ with respect to $ N $. Prove that the triangles $ \triangle MPS $ and $ \triangle ADK $ are similar.

2006 Singapore Junior Math Olympiad, 4

Tags: geometry , Angle Chasing , angle bisector , incenter

In $\vartriangle ABC$, the bisector of $\angle B$ meets $AC$ at $D$ and the bisector of $\angle C$ meets $AB$ at $E$. These bisectors intersect at $O$ and $OD = OE$. If $AD \ne AE$, prove that $\angle A = 60^o$.

2010 Contests, 3

Tags: geometry , circumcircle , cyclic quadrilateral , radical axis , geometry proposed , Angle Chasing

We are given a cyclic quadrilateral $ABCD$ with a point $E$ on the diagonal $AC$ such that $AD=AE$ and $CB=CE$. Let $M$ be the center of the circumcircle $k$ of the triangle $BDE$. The circle $k$ intersects the line $AC$ in the points $E$ and $F$. Prove that the lines $FM$, $AD$ and $BC$ meet at one point. [i](4th Middle European Mathematical Olympiad, Individual Competition, Problem 3)[/i]

2018 Estonia Team Selection Test, 5

Tags: geometry , IMO Shortlist , Angle Chasing , projective geometry , similar triangles , ISL 2017 , G3

Let $O$ be the circumcenter of an acute triangle $ABC$. Line $OA$ intersects the altitudes of $ABC$ through $B$ and $C$ at $P$ and $Q$, respectively. The altitudes meet at $H$. Prove that the circumcenter of triangle $PQH$ lies on a median of triangle $ABC$.

2019 Belarus Team Selection Test, 8.1

Tags: IMO Shortlist , geometry , geometry solved , symmetry , Angle Chasing , Miquel point

Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.

2015 Romania National Olympiad, 4

Tags: geometry , angles , Angle Chasing

Consider $\vartriangle ABC$ where $\angle ABC= 60 ^o$. Points $M$ and $D$ are on the sides $(AC)$, respectively $(AB)$, such that $\angle BCA = 2 \angle MBC$, and $BD = MC$. Determine $\angle DMB$.

2012 Indonesia TST, 3

Tags: geometry , geometry proposed , Angle Chasing , complex numbers , cyclic quadrilateral

The incircle of a triangle $ABC$ is tangent to the sides $AB,AC$ at $M,N$ respectively. Suppose $P$ is the intersection between $MN$ and the bisector of $\angle ABC$. Prove that $BP$ and $CP$ are perpendicular.

2001 Greece Junior Math Olympiad, 4

Tags: geometry , equal segments , Angle Chasing

Let $ABC$ be a triangle with altitude $AD$ , angle bisectors $AE$ and $BZ$ that intersecting at point $I$. From point $I$ let $IT$ be a perpendicular on $AC$. Also let line $(e)$ be perpendicular on $AC$ at point $A$. Extension of $ET$ intersects line $(e)$ at point $K$. Prove that $AK=AD$.

2014 India PRMO, 16

Tags: geometry , incenter , incircle , angles , Angle Chasing

In a triangle $ABC$, let $I$ denote the incenter. Let the lines $AI,BI$ and $CI$ intersect the incircle at $P,Q$ and $R$, respectively. If $\angle BAC = 40^o$, what is the value of $\angle QPR$ in degrees ?

2021 Science ON Juniors, 3

Tags: geometry , circles , Angle Chasing

Circles $\omega_1$ and $\omega_2$ are externally tangent to each other at $P$. A random line $\ell$ cuts $\omega_1$ at $A$ and $C$ and $\omega_2$ at $B$ and $D$ (points $A,C,B,D$ are in this order on $\ell$). Line $AP$ meets $\omega_2$ again at $E$ and line $BP$ meets $\omega_1$ again at $F$. Prove that the radical axis of circles $(PCD)$ and $(PEF)$ is parallel to $\ell$. \\ \\ [i](Vlad Robu)[/i]

2013 Hanoi Open Mathematics Competitions, 7

Tags: Equilateral Triangle , Equilateral , geometry , Angle Chasing

Let $ABC$ be an equilateral triangle and a point M inside the triangle such that $MA^2 = MB^2 +MC^2$. Draw an equilateral triangle $ACD$ where $D \ne B$. Let the point $N$ inside $\vartriangle ACD$ such that $AMN$ is an equilateral triangle. Determine $\angle BMC$.

2025 Sharygin Geometry Olympiad, 1

Tags: geometry , Sharygin Geometry Olympiad , Angle Chasing

Let $I$ be the incenter of a triangle $ABC$, $D$ be an arbitrary point of segment $AC$, and $A_{1}, A_{2}$ be the common points of the perpendicular from $D$ to the bisector $CI$ with $BC$ and $AI$ respectively. Define similarly the points $C_{1}$, $C_{2}$. Prove that $B$, $A_{1}$, $A_{2}$, $I$, $C_{1},$ $C_{2}$ are concyclic. Proposed by:D.Shvetsov

2017 Polish MO Finals, 1

Tags: geometry , circumcircle , geometry solved , Miquel point , Spiral Similarity , Angle Chasing , cyclic quadrilateral

Points $P$ and $Q$ lie respectively on sides $AB$ and $AC$ of a triangle $ABC$ and $BP=CQ$. Segments $BQ$ and $CP$ cross at $R$. Circumscribed circles of triangles $BPR$ and $CQR$ cross again at point $S$ different from $R$. Prove that point $S$ lies on the bisector of angle $BAC$.

2018 Irish Math Olympiad, 2

Tags: geometry , right triangle , Angle Chasing

The triangle $ABC$ is right-angled at $A$. Its incentre is $I$, and $H$ is the foot of the perpendicular from $I$ on $AB$. The perpendicular from $H$ on $BC$ meets $BC$ at $E$, and it meets the bisector of $\angle ABC$ at $D$. The perpendicular from $A$ on $BC$ meets $BC$ at $F$. Prove that $\angle EFD = 45^o$

2022/2023 Tournament of Towns, P3

Tags: Tournament of Towns , geometry , pentagon , Angle Chasing

A pentagon $ABCDE$ is circumscribed about a circle. The angles at the vertices $A{}$, $C{}$ and $E{}$ of the pentagon are equal to $100^\circ$. Find the measure of the angle $\angle ACE$.

Kyiv City MO Juniors Round2 2010+ geometry, 2013.7.3

Tags: geometry , angles , square , Angle Chasing , equal angles

In the square $ABCD$ on the sides $AD$ and $DC$, the points $M$ and $N$ are selected so that $\angle BMA = \angle NMD = 60 { } ^ \circ $. Find the value of the angle $MBN$.

2002 Junior Balkan Team Selection Tests - Romania, 3

Tags: geometry , angles , Angle Chasing , equal segments

Let $ABC$ be an isosceles triangle such that $AB = AC$ and $\angle A = 20^o$. Let $M$ be the foot of the altitude from $C$ and let $N$ be a point on the side $AC$ such that $CN =\frac12 BC$. Determine the measure of the angle $AMN$.

2017 EGMO, 1

Tags: cylic , geometry , Angle Chasing , quadrilateral , EGMO , EGMO 2017 , barycentric coordinates

Let $ABCD$ be a convex quadrilateral with $\angle DAB=\angle BCD=90^{\circ}$ and $\angle ABC> \angle CDA$. Let $Q$ and $R$ be points on segments $BC$ and $CD$, respectively, such that line $QR$ intersects lines $AB$ and $AD$ at points $P$ and $S$, respectively. It is given that $PQ=RS$.Let the midpoint of $BD$ be $M$ and the midpoint of $QR$ be $N$.Prove that the points $M,N,A$ and $C$ lie on a circle.

2012 Indonesia TST, 3

Tags: geometry , geometry proposed , Angle Chasing , complex numbers , cyclic quadrilateral

The incircle of a triangle $ABC$ is tangent to the sides $AB,AC$ at $M,N$ respectively. Suppose $P$ is the intersection between $MN$ and the bisector of $\angle ABC$. Prove that $BP$ and $CP$ are perpendicular.

2012 Sharygin Geometry Olympiad, 5

Tags: Angle Chasing , isosceles , right triangle , geometry

Let $ABC$ be an isosceles right-angled triangle. Point $D$ is chosen on the prolongation of the hypothenuse $AB$ beyond point $A$ so that $AB = 2AD$. Points $M$ and $N$ on side $AC$ satisfy the relation $AM = NC$. Point $K$ is chosen on the prolongation of $CB$ beyond point $B$ so that $CN = BK$. Determine the angle between lines $NK$ and $DM$. (M.Kungozhin)