Olympiad problems

2018 Estonia Team Selection Test, 5

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 $ABC$.

2007 Estonia Team Selection Test, 4

Tags: angle chasing , geometry , angle , square

In square $ABCD,$ points $E$ and $F$ are chosen in the interior of sides $BC$ and $CD$, respectively. The line drawn from $F$ perpendicular to $AE$ passes through the intersection point $G$ of $AE$ and diagonal $BD$. A point $K$ is chosen on $FG$ such that $|AK|= |EF|$. Find $\angle EKF.$

2018 Peru MO (ONEM), 3

Tags: orthocenter , right angle , angle chasing , isosceles , geometry

Let $ABC$ be an acute triangle such that $BA = BC$. On the sides $BA$ and $BC$ points $D$ and $E$ are chosen respectively, such that $DE$ and $AC$ are parallel. Let $H$ be the orthocenter of the triangle $DBE$ and $M$ be the midpoint of $AE$. If $\angle HMC = 90^o$, determine the measure of angle $\angle ABC$.

2011 Romania National Olympiad, 4

Tags: angle chasing , geometry , angle

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 $\angle DMB$.

Kyiv City MO Juniors 2003+ geometry, 2015.8.3

Tags: isosceles , angle chasing , angle , geometry , angle bisector

In the isosceles triangle $ABC$, $ (AB = BC)$ the bisector $AD$ was drawn, and in the triangle $ABD$ the bisector $DE$ was drawn. Find the values of the angles of the triangle $ABC$, if it is known that the bisectors of the angles $ABD$ and $AED$ intersect on the line $AD$. (Fedak Ivan)

2012 Dutch IMO TST, 4

Tags: right angle , angle chasing , geometry , equal angles

Let $\vartriangle ABC$ be a triangle. The angle bisector of $\angle CAB$ intersects$ BC$ at $L$. On the interior of line segments $AC$ and $AB$, two points, $M$ and $N$, respectively, are chosen in such a way that the lines $AL, BM$ and $CN$ are concurrent, and such that $\angle AMN = \angle ALB$. Prove that $\angle NML = 90^o$.

2019 Macedonia National Olympiad, 3

Tags: symmetry , angle chasing , miquel point , geometry

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.

2018 Danube Mathematical Competition, 2

Tags: angle chasing , perpendicularity , geometry , perpendicular

Let $ABC$ be a triangle such that in its interior there exists a point $D$ with $\angle DAC = \angle DCA = 30^o$ and $ \angle DBA = 60^o$. Denote $E$ the midpoint of the segment $BC$, and take $F$ on the segment $AC$ so that $AF = 2FC$. Prove that $DE \perp EF$.

1996 Bosnia and Herzegovina Team Selection Test, 3

Tags: parallelogram , angle chasing , geometry

Let $M$ be a point inside quadrilateral $ABCD$ such that $ABMD$ is parallelogram. If $\angle CBM = \angle CDM$ prove that $\angle ACD = \angle BCM$

2021 All-Russian Olympiad, 1

Tags: geometry , angle chasing

On the side $BC$ of the parallelogram $ABCD$, points $E$ and $F$ are given ($E$ lies between $B$ and $F$) and the diagonals $AC, BD$ meet at $O$. If it's known that $AE, DF$ are tangent to the circumcircle of $\triangle AOD$, prove that they're tangent to the circumcircle of $\triangle EOF$ as well.

2017 Yasinsky Geometry Olympiad, 5

Tags: angle , geometry , angle chasing , diameter

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.

Kharkiv City MO Seniors - geometry, 2014.10.4

Tags: angle chasing , square , angle , equal angles , geometry

Let $ABCD$ be a square. The points $N$ and $P$ are chosen on the sides $AB$ and $AD$ respectively, such that $NP=NC$. The point $Q$ on the segment $AN$ is such that that $\angle QPN=\angle NCB$. Prove that $\angle BCQ=\frac{1}{2}\angle AQP$.

2017 Swedish Mathematical Competition, 4

Tags: angle chasing , angle , angle bisector , geometry

Let $D$ be the foot of the altitude towards $BC$ in the triangle $ABC$. Let $E$ be the intersection of $AB$ with the bisector of angle $\angle C$. Assume that the angle $\angle AEC = 45^o$ . Determine the angle $\angle EDB$.

2014 Taiwan TST Round 3, 4

Tags: circumcircle , isosceles triangle , angle chasing , inversion , geometry

Let $ABC$ be a triangle with $\angle B > \angle C$. Let $P$ and $Q$ be two different points on line $AC$ such that $\angle PBA = \angle QBA = \angle ACB $ and $A$ is located between $P$ and $C$. Suppose that there exists an interior point $D$ of segment $BQ$ for which $PD=PB$. Let the ray $AD$ intersect the circle $ABC$ at $R \neq A$. Prove that $QB = QR$.

2021 Argentina National Olympiad, 3

Tags: angle chasing , geometry

Let $ABCD$ be a quadrilateral inscribed in a circle such that $\angle ABC=60^{\circ}.$ a) Prove that if $BC=CD$ then $AB= CD+DA.$ b) Is it true that if $AB= CD+DA$ then $BC=CD$?

2023 Sharygin Geometry Olympiad, 9.8

Tags: geometry , angle chasing

Let $ABC$ be a triangle with $\angle A = 120^\circ$, $I$ be the incenter, and $M$ be the midpoint of $BC$. The line passing through $M$ and parallel to $AI$ meets the circle with diameter $BC$ at points $E$ and $F$ ($A$ and $E$ lie on the same semiplane with respect to $BC$). The line passing through $E$ and perpendicular to $FI$ meets $AB$ and $AC$ at points $P$ and $Q$ respectively. Find the value of $\angle PIQ$.

2018 Dutch BxMO TST, 4

Tags: angle chasing , angle , geometry , circumcircle

In a non-isosceles triangle $\vartriangle ABC$ we have $\angle BAC = 60^o$. Let $D$ be the intersection of the angular bisector of $\angle BAC$ with side $BC, O$ the centre of the circumcircle of $\vartriangle ABC$ and $E$ the intersection of $AO$ and $BC$. Prove that $\angle AED + \angle ADO = 90^o$.

2002 IMO, 2

Tags: perpendicular bisector , incenter , circumcircle , angle chasing , geometry

The circle $S$ has centre $O$, and $BC$ is a diameter of $S$. Let $A$ be a point of $S$ such that $\angle AOB<120{{}^\circ}$. Let $D$ be the midpoint of the arc $AB$ which does not contain $C$. The line through $O$ parallel to $DA$ meets the line $AC$ at $I$. The perpendicular bisector of $OA$ meets $S$ at $E$ and at $F$. Prove that $I$ is the incentre of the triangle $CEF.$

2012 Indonesia TST, 3

Tags: angle chasing , complex number , cyclic quadrilateral , geometry

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.

Kvant 2021, M2654

Tags: geometry , angle chasing

On the side $BC$ of the parallelogram $ABCD$, points $E$ and $F$ are given ($E$ lies between $B$ and $F$) and the diagonals $AC, BD$ meet at $O$. If it's known that $AE, DF$ are tangent to the circumcircle of $\triangle AOD$, prove that they're tangent to the circumcircle of $\triangle EOF$ as well.

2017 Ukrainian Geometry Olympiad, 2

Tags: angle chasing , trapezoid , equal angles , geometry

Point $M$ is the midpoint of the base $BC$ of trapezoid $ABCD$. On base $AD$, point $P$ is selected. Line $PM$ intersects line $DC$ at point $Q$, and the perpendicular from $P$ on the bases intersects line $BQ$ at point $K$. Prove that $\angle QBC = \angle KDA$.

2022 Belarusian National Olympiad, 10.6

Tags: angle chasing , geometry

Circles $\omega_1$ and $\omega_2$ intersect at $X$ and $Y$. Through point $Y$ two lines pass, one of which intersects $\omega_1$ and $\omega_2$ for the second time at $A$ and $B$, and the other at $C$ and $D$. Line $AD$ intersects for the second time circles $\omega_1$ and $\omega_2$ at $P$ and $Q$. It turned out that $YP=YQ$ Prove that the circumcircles of triangles $BCY$ and $PQY$ are tangent to each other.

2018 Pan-African Shortlist, G3

Tags: cyclic quadrilateral , angle chasing , geometry

Given a triangle $ABC$, let $D$ be the intersection of the line through $A$ perpendicular to $AB$, and the line through $B$ perpendicular to $BC$. Let $P$ be a point inside the triangle. Show that $DAPB$ is cyclic if and only if $\angle BAP = \angle CBP$.

2014 Flanders Math Olympiad, 3

Tags: angle , angle chasing , geometry , equal segments

Let $PQRS$ be a quadrilateral with $| P Q | = | QR | = | RS |$, $\angle Q= 110^o$ and $\angle R = 130^o$ . Determine $\angle P$ and $\angle S$ .

2016 Junior Regional Olympiad - FBH, 4

Tags: angle chasing , geometry

Let $C$ and $D$ be points inside angle $\angle AOB$ such that $5\angle COD = 4\angle AOC$ and $3\angle COD = 2\angle DOB$. If $\angle AOB = 105^{\circ}$, find $\angle COD$