Olympiad problems

1991 Chile National Olympiad, 6

Given a triangle with $ \triangle ABC $, with: $ \angle C = 36^o$ and $ \angle A = \angle B $. Consider the points $ D $ on $ BC $, $ E $ on $ AD $, $ F $ on $ BE $, $ G $ on $ DF $ and $ H $ on $ EG $, so that the rays $ AD, BE, DF, EG, FH $ bisect the angles $ A, B, D, E, F $ respectively. It is known that $ FH = 1 $. Calculate $ AC$.

2009 Greece Junior Math Olympiad, 2

Tags: angle chasing , equilateral

From vertex $A$ of an equilateral triangle $ABC$, a ray $Ax$ intersects $BC$ at point $D$. Let $E$ be a point on $Ax$ such that $BA =BE$. Calculate $\angle AEC$.

KoMaL A Problems 2024/2025, A. 885

Tags: incenter , angle chasing , geometry

Let triangle $ABC$ be a given acute scalene triangle with altitudes $BE$ and $CF$. Let $D$ be the point where the incircle of $\,\triangle ABC$ touches side $BC$. The circumcircle of $\triangle BDE$ meets line $AB$ again at point $K$, the circumcircle of $\triangle CDF$ meets line $AC$ again at point $L$. The circumcircle of $\triangle BDE$ and $\triangle CDF$ meet line $KL$ again at $X$ and $Y$, respectively. Prove that the incenter of $\triangle DXY$ lies on the incircle of $\,\triangle ABC$. [i]Proposed by Luu Dong, Vietnam[/i]

2019 Peru EGMO TST, 6

Tags: geometry , angle chasing , miquel point , symmetry

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 India PRMO, 29

Tags: angle chasing , geometry , incenter

Let $D$ be an interior point of the side $BC$ of a triangle $ABC$. Let $I_1$ and $I_2$ be the incentres of triangles $ABD$ and $ACD$ respectively. Let $AI_1$ and $AI_2$ meet $BC$ in $E$ and $F$ respectively. If $\angle BI_1E = 60^o$, what is the measure of $\angle CI_2F$ in degrees?

2023 Bangladesh Mathematical Olympiad, P9

Tags: geometry , angle chasing , inversion , cyclic quadrilateral , radical axis , parallel

Let $\Delta ABC$ be an acute angled triangle. $D$ is a point on side $BC$ such that $AD$ bisects angle $\angle BAC$. A line $l$ is tangent to the circumcircles of triangles $ADB$ and $ADC$ at point $K$ and $L$, respectively. Let $M$, $N$ and $P$ be its midpoints of $BD$, $DC$ and $KL$, respectively. Prove that $l$ is tangent to the circumcircle of $\Delta MNP$.

2012 Sharygin Geometry Olympiad, 5

Tags: angle chasing , geometry , right triangle , isosceles

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)

2018 Hanoi Open Mathematics Competitions, 8

Tags: angle chasing , rhombus , angle , geometry

Let $ABCD$ be rhombus, with $\angle ABC = 80^o$: Let $E$ be midpoint of $BC$ and $F$ be perpendicular projection of $A$ onto $DE$. Find the measure of $\angle DFC$ in degree.

2014 India IMO Training Camp, 3

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

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 $QB = QR$.

2019 Silk Road, 1

Tags: angle chasing , altitude , geometry , equal angles

The altitudes of the acute-angled non-isosceles triangle $ ABC $ intersect at the point $ H $. On the segment $ C_1H $, where $ CC_1 $ is the altitude of the triangle, the point $ K $ is marked. Points $ L $ and $ M $ are the feet of perpendiculars from point $ K $ on straight lines $ AC $ and $ BC $, respectively. The lines $ AM $ and $ BL $ intersect at $ N $. Prove that $ \angle ANK = \angle HNL $.

2021 Science ON Juniors, 3

Tags: angle chasing , geometry , circles

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]

1997 Bosnia and Herzegovina Team Selection Test, 2

Tags: angle chasing , circumcircle , geometry , incenter , cyclic quadrilateral

In isosceles triangle $ABC$ with base side $AB$, on side $BC$ it is given point $M$. Let $O$ be a circumcenter and $S$ incenter of triangle $ABC$. Prove that $$ SM \mid \mid AC \Leftrightarrow OM \perp BS$$

2008 USAMO, 2

Tags: symmedian , angle chasing , homothety , geometry , geometric transformation , circumcircle

Let $ ABC$ be an acute, scalene triangle, and let $ M$, $ N$, and $ P$ be the midpoints of $ \overline{BC}$, $ \overline{CA}$, and $ \overline{AB}$, respectively. Let the perpendicular bisectors of $ \overline{AB}$ and $ \overline{AC}$ intersect ray $ AM$ in points $ D$ and $ E$ respectively, and let lines $ BD$ and $ CE$ intersect in point $ F$, inside of triangle $ ABC$. Prove that points $ A$, $ N$, $ F$, and $ P$ all lie on one circle.

2014 Brazil Team Selection Test, 3

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

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 $QB = QR$.

2010 Contests, 3

Tags: cyclic quadrilateral , angle chasing , geometry , circumcircle , radical axis

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 Yasinsky Geometry Olympiad, 6

Tags: angle chasing , midpoint , geometry , perpendicular

In the quadrilateral $ABCD$, the points $E, F$, and $K$ are midpoints of the $AB, BC, AD$ respectively. Known that $KE \perp AB, K F \perp BC$, and the angle $\angle ABC = 118^o$. Find $ \angle ACD$ (in degrees).

2014 Sharygin Geometry Olympiad, 1

Tags: angle chasing , geometry , square , isosceles

The vertices and the circumcenter of an isosceles triangle lie on four different sides of a square. Find the angles of this triangle. (I. Bogdanov, B. Frenkin)

2011 Oral Moscow Geometry Olympiad, 5

Tags: angle chasing , angle , perpendicular , geometry

In a convex quadrilateral $ABCD, AC\perp BD, \angle BCA = 10^o,\angle BDA = 20^o, \angle BAC = 40^o$. Find $\angle BDC$.

2019 Flanders Math Olympiad, 3

Tags: angle chasing , angle , geometry

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 $\angle CBP$.

1997 Canada National Olympiad, 4

Tags: angle chasing , geometry , parallelogram

The point $O$ is situated inside the parallelogram $ABCD$ such that $\angle AOB+\angle COD=180^{\circ}$. Prove that $\angle OBC=\angle ODC$.

2010 Sharygin Geometry Olympiad, 1

Tags: angle chasing , altitude , angle , angle bisector , geometry

For each vertex of triangle $ABC$, the angle between the altitude and the bisectrix from this vertex was found. It occurred that these angle in vertices $A$ and $B$ were equal. Furthermore the angle in vertex $C$ is greater than two remaining angles. Find angle $C$ of the triangle.

2008 Postal Coaching, 1

Tags: angle , angle chasing , right angle , arc midpoint , circumcircle , geometry , incenter

In triangle $ABC,\angle B > \angle C, T$ is the midpoint of arc $BAC$ of the circumcicle of $ABC$, and $I$ is the incentre of $ABC$. Let $E$ be point such that $\angle AEI = 90^0$ and $AE$ is parallel to $BC$. If $TE$ intersects the circumcircle of $ABC$ at $P(\ne T)$ and $\angle B = \angle IPB$, determine $\angle A$.

2013 Czech-Polish-Slovak Junior Match, 4

Tags: angle chasing , geometry , equal angles , circumcircle

Let $ABCD$ be a convex quadrilateral with $\angle DAB =\angle ABC =\angle BCD > 90^o$. The circle circumscribed around the triangle $ABC$ intersects the sides $AD$ and $CD$ at points $K$ and $L$, respectively, different from any vertex of the quadrilateral $ABCD$ . Segments $AL$ and $CK$ intersect at point $P$. Prove that $\angle ADB =\angle PDC$.

2007 Estonia Team Selection Test, 4

Tags: angle , angle chasing , square , geometry

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 $\angle EKF.$

2009 May Olympiad, 2

Tags: geometry , angle chasing , equilateral triangle , convex quadrilateral , isosceles

Let $ABCD$ be a convex quadrilateral such that the triangle $ABD$ is equilateral and the triangle $BCD$ is isosceles, with $\angle C = 90^o$. If $E$ is the midpoint of the side $AD$, determine the measure of the angle $\angle CED$.