Olympiad problems

2016 Czech-Polish-Slovak Junior Match, 1

Let $ABC$ be a right-angled triangle with hypotenuse $AB$. Denote by $D$ the foot of the altitude from $C$. Let $Q, R$, and $P$ be the midpoints of the segments $AD, BD$, and $CD$, respectively. Prove that $\angle AP B + \angle QCR = 180^o$. Czech Republic

2006 Sharygin Geometry Olympiad, 7

Tags: geometry , square , Angle Chasing , congruent triangles

The point $E$ is taken inside the square $ABCD$, the point $F$ is taken outside, so that the triangles $ABE$ and $BCF$ are congruent . Find the angles of the triangle $ABE$, if it is known that$EF$ is equal to the side of the square, and the angle $BFD$ is right.

2019 Peru EGMO TST, 6

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.

2007 Estonia Team Selection Test, 4

Tags: Angle Chasing , angles , 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 $\angle EKF.$

2019 Yasinsky Geometry Olympiad, p4

Tags: geometry , cyclic quadrilateral , Angle Chasing , angles

Find the angles of the cyclic quadrilateral if you know that each of its diagonals is a bisector of one angle and a trisector of the opposite one (the trisector of the angle is one of the two rays that lie in the interior of the angle and divide it into three equal parts). (Vyacheslav Yasinsky)

2022 Iran MO (3rd Round), 1

Tags: geometry , Angle Chasing , Spiral Similarity , circumcircle , geometric transformation , reflection , angle bisector

Triangle $ABC$ is assumed. The point $T$ is the second intersection of the symmedian of vertex $A$ with the circumcircle of the triangle $ABC$ and the point $D \neq A$ lies on the line $AC$ such that $BA=BD$. The line that at $D$ tangents to the circumcircle of the triangle $ADT$, intersects the circumcircle of the triangle $DCT$ for the second time at $K$. Prove that $\angle BKC = 90^{\circ}$(The symmedian of the vertex $A$, is the reflection of the median of the vertex $A$ through the angle bisector of this vertex).

2017 All-Russian Olympiad, 8

Tags: geometry , incenter , Angle Chasing , incircle

Given a convex quadrilateral $ABCD$. We denote $I_A,I_B, I_C$ and $I_D$ centers of $\omega_A, \omega_B,\omega_C $and $\omega_D$,inscribed In the triangles $DAB, ABC, BCD$ and $CDA$, respectively.It turned out that $\angle BI_AA + \angle I_CI_AI_D = 180^\circ$. Prove that $\angle BI_BA + \angle I_CI_BI_D = 180^{\circ}$. (A. Kuznetsov)

2021 IMO Shortlist, G4

Tags: ISL 2021 , IMO Shortlist , geometry , cyclic quadrilateral , concurrency , Angle Chasing , imo shortlist g4

Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.

2017 Ukrainian Geometry Olympiad, 2

Tags: geometry , Angle Chasing , perpendicularity

On the side $AC$ of a triangle $ABC$, let a $K$ be a point such that $AK = 2KC$ and $\angle ABK = 2 \angle KBC$. Let $F$ be the midpoint of $AC$, $L$ be the projection of $A$ on $BK$. Prove that $FL \bot BC$.

2010 Sharygin Geometry Olympiad, 6

Tags: geometry , angles , square , midpoints , Angle Chasing , geometry solved , Sharygin Geometry Olympiad

Let $E, F$ be the midpoints of sides $BC, CD$ of square $ABCD$. Lines $AE$ and $BF$ meet at point $P$. Prove that $\angle PDA = \angle AED$.

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 $ABC$.

2011 IMO Shortlist, 5

Tags: geometry , incenter , circumcircle , geometric transformation , homothety , IMO Shortlist , Angle Chasing

Let $ABC$ be a triangle with incentre $I$ and circumcircle $\omega$. Let $D$ and $E$ be the second intersection points of $\omega$ with $AI$ and $BI$, respectively. The chord $DE$ meets $AC$ at a point $F$, and $BC$ at a point $G$. Let $P$ be the intersection point of the line through $F$ parallel to $AD$ and the line through $G$ parallel to $BE$. Suppose that the tangents to $\omega$ at $A$ and $B$ meet at a point $K$. Prove that the three lines $AE,BD$ and $KP$ are either parallel or concurrent. [i]Proposed by Irena Majcen and Kris Stopar, Slovenia[/i]

2021 JBMO Shortlist, G5

Tags: Junior , Balkan , shortlist , 2021 , geometry , Angle Chasing

Let $ABC$ be an acute scalene triangle with circumcircle $\omega$. Let $P$ and $Q$ be interior points of the sides $AB$ and $AC$, respectively, such that $PQ$ is parallel to $BC$. Let $L$ be a point on $\omega$ such that $AL$ is parallel to $BC$. The segments $BQ$ and $CP$ intersect at $S$. The line $LS$ intersects $\omega$ at $K$. Prove that $\angle BKP = \angle CKQ$. Proposed by [i]Ervin Macić, Bosnia and Herzegovina[/i]

2018 Yasinsky Geometry Olympiad, 6

Tags: geometry , Angle Chasing , midpoints , 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).

2018 Romania Team Selection Tests, 1

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 $ABC$.

2022 Switzerland Team Selection Test, 9

Tags: ISL 2021 , IMO Shortlist , geometry , cyclic quadrilateral , concurrency , Angle Chasing , imo shortlist g4

Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.

2008 USAMO, 2

Tags: geometry , circumcircle , geometric transformation , homothety , geometry solved , symmedian , Angle Chasing

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.

2020 Yasinsky Geometry Olympiad, 1

Tags: geometry , angles , Angle Chasing , right triangle , circumcircle

Given a right triangle $ABC$, the point $M$ is the midpoint of the hypotenuse $AB$. A circle is circumscribed around the triangle $BCM$, which intersects the segment $AC$ at a point $Q$ other than $C$. It turned out that the segment $QA$ is twice as large as the side $BC$. Find the acute angles of triangle $ABC$. (Mykola Moroz)

2017 Ukrainian Geometry Olympiad, 1

Tags: geometry , midpoints , equal angles , Angle Chasing

In the triangle $ABC$, ${{A}_{1}}$ and ${{C}_{1}} $ are the midpoints of sides $BC $ and $AB$ respectively. Point $P$ lies inside the triangle. Let $\angle BP {{C}_{1}} = \angle PCA$. Prove that $\angle BP {{A}_{1}} = \angle PAC $.

2022 Saudi Arabia IMO TST, 2

Tags: ISL 2021 , IMO Shortlist , geometry , cyclic quadrilateral , concurrency , Angle Chasing , imo shortlist g4

Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.

2024 Auckland Mathematical Olympiad, 4

Tags: Angle Chasing , geometry

The altitude $AH$ and the bisector $CL$ of triangle $ABC$ intersect at point $O$. Find the angle $BAC$, if it is known that the difference between angle $COH$ and half of angle $ABC$ is $46$.

2019 Hong Kong TST, 2

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.

2019 USEMO, 1

Tags: geometry , cyclic quadrilateral , USEMO , Angle Chasing

Let $ABCD$ be a cyclic quadrilateral. A circle centered at $O$ passes through $B$ and $D$ and meets lines $BA$ and $BC$ again at points $E$ and $F$ (distinct from $A,B,C$). Let $H$ denote the orthocenter of triangle $DEF.$ Prove that if lines $AC,$ $DO,$ $EF$ are concurrent, then triangle $ABC$ and $EHF$ are similar. [i]Robin Son[/i]

2007 Sharygin Geometry Olympiad, 9

Tags: geometry , Angle Chasing , convex quadrilateral , Angle Chase

Suppose two convex quadrangles are such that the sides of each of them lie on the perpendicular bisectors of the sides of the other one. Determine their angles,

2017 Philippine MO, 4

Tags: geometry , Spiral Similarity , Angle Chasing , Locus problems , incenter , circumcircle

Circles $\mathcal{C}_1$ and $\mathcal{C}_2$ with centers at $C_1$ and $C_2$ respectively, intersect at two points $A$ and $B$. Points $P$ and $Q$ are varying points on $\mathcal{C}_1$ and $\mathcal{C}_2$, respectively, such that $P$, $Q$ and $B$ are collinear and $B$ is always between $P$ and $Q$. Let lines $PC_1$ and $QC_2$ intersect at $R$, let $I$ be the incenter of $\Delta PQR$, and let $S$ be the circumcenter of $\Delta PIQ$. Show that as $P$ and $Q$ vary, $S$ traces the arc of a circle whose center is concyclic with $A$, $C_1$ and $C_2$.