Olympiad problems

2019 Brazil Team Selection Test, 1

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 Polish MO Finals, 1

Tags: radical axis , angle chasing , geometry

An acute triangle $ABC$ in which $AB<AC$ is given. The bisector of $\angle BAC$ crosses $BC$ at $D$. Point $M$ is the midpoint of $BC$. Prove that the line though centers of circles escribed on triangles $ABC$ and $ADM$ is parallel to $AD$.

2019 Sharygin Geometry Olympiad, 6

Tags: angle chasing , circumcircle , geometry , angle bisector , median

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 $FDA$.

2018 Yasinsky Geometry Olympiad, 1

Tags: midpoint , altitude , angle chasing , equal angles , geometry

In the triangle $ABC$, $AD$ is altitude, $M$ is the midpoint of $BC$. It is known that $\angle BAD = \angle DAM = \angle MAC$. Find the values of the angles of the triangle $ABC$

2018 Taiwan TST Round 3, 4

Tags: angle chasing , projective geometry , geometry , similar triangles

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 $ABC$.

India EGMO 2023 TST, 6

Tags: angle chasing , isosceles triangle , geometry

Let $ABC$ be an isosceles triangle with $AB = AC$. Suppose $P,Q,R$ are points on segments $AC, AB, BC$ respectively such that $AP = QB$, $\angle PBC = 90^\circ - \angle BAC$ and $RP = RQ$. Let $O_1, O_2$ be the circumcenters of $\triangle APQ$ and $\triangle CRP$. Prove that $BR = O_1O_2$. [i]Proposed by Atul Shatavart Nadig[/i]

2017 Ukrainian Geometry Olympiad, 2

Tags: angle chasing , perpendicularity , geometry

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: midpoint , angle chasing , geometry , angle , square

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 Bosnia and Herzegovina Team Selection Test, 6

Tags: similar triangles , angle chasing , projective geometry , geometry

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 $ABC$.

EGMO 2017, 1

Tags: barycentric coordinates , angle chasing , quadrilateral , cyclic , geometry

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.

2025 Sharygin Geometry Olympiad, 1

Tags: geometry , 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

2006 Sharygin Geometry Olympiad, 7

Tags: congruent triangles , square , angle chasing , geometry

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.

2024 Bangladesh Mathematical Olympiad, P5

Tags: angle chasing , parallelogram , geometry , incenter

Consider $\triangle XPQ$ and $\triangle YPQ$ such that $X$ and $Y$ are on the opposite sides of $PQ$ and the circumradius of $\triangle XPQ$ and the circumradius of $\triangle YPQ$ are the same. $I$ and $J$ are the incenters of $\triangle XPQ$ and $\triangle YPQ$ respectively. Let $M$ be the midpoint of $PQ$. Suppose $I, M, J$ are collinear. Prove that $XPYQ$ is a parallelogram.

2000 Saint Petersburg Mathematical Olympiad, 9.2

Tags: angle chasing , geometry

Let $AA_1$ and $CC_1$ be altitudes of acute angled triangle $ABC$. A point $D$ is chosen on $AA_1$ such that $A_1D=C_1D$. Let $E$ be the midpoint of $AC$. Prove that points $A$, $C_1$, $D$, $E$ are concylic. [I]Proposed by S. Berlov[/i]

1987 Spain Mathematical Olympiad, 5

Tags: geometry , angle chasing

In a triangle $ABC, D$ lies on $AB, E$ lies on $AC$ and $ \angle ABE = 30^o, \angle EBC = 50^o, \angle ACD = 20^o$, $\angle DCB = 60^o$. Find $\angle EDC$.

2010 Sharygin Geometry Olympiad, 5

Tags: angle chasing , excircle , geometry , incircle

The incircle of a right-angled triangle $ABC$ ($\angle ABC =90^o$) touches $AB, BC, AC$ in points $C_1, A_1, B_1$, respectively. One of the excircles touches the side $BC$ in point $A_2$. Point $A_0$ is the circumcenter or triangle $A_1A_2B_1$, point $C_0$ is defined similarly. Find angle $A_0BC_0$.

1986 China Team Selection Test, 1

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

If $ABCD$ is a cyclic quadrilateral, then prove that the incenters of the triangles $ABC$, $BCD$, $CDA$, $DAB$ are the vertices of a rectangle.

2023 Bangladesh Mathematical Olympiad, P6

Tags: geometry , angle chasing , cyclic quadrilateral

Let $\triangle ABC$ be an acute angle triangle and $\omega$ be its circumcircle. Let $N$ be a point on arc $AC$ not containing $B$ and $S$ be a point on line $AB$. The line tangent to $\omega$ at $N$ intersects $BC$ at $T$, $NS$ intersects $\omega$ at $K$. Assume that $\angle NTC = \angle KSB$. Prove that $CK\parallel AN \parallel TS$.

2014 Harvard-MIT Mathematics Tournament, 8

Tags: angle chasing , radical axis , geometry , symmedian , circumcircle

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 $BX$.

2019 Thailand TST, 1

Tags: angle chasing , miquel point , symmetry , 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.

2023 pOMA, 6

Tags: angle chasing , geometry

Let $\Omega$ be a circle, and let $A$, $B$, $C$, $D$ and $K$ be distinct points on it, in that order, and such that lines $BC$ and $AD$ are parallel. Let $A'\neq A$ be a point on line $AK$ such that $BA=BA'$. Similarly, let $C'\neq C$ be a point on line $CK$ such that $DC=DC'$. Prove that segments $AC$ and $A'C'$ have the same length.

2020 Macedonia Additional BMO TST, 2

Tags: angle chasing , geometry , parallel

Let $ABCD$ be a convex quadrilateral. On the sides $AB$ and $CD$ there are interior points $K$ and $L$, respectively, such that $\angle BAL = \angle CDK$. Prove that the following statements are equivalent: $i) \angle BLA= \angle CKD$ $ii) AD \parallel BC $

2022 Bulgaria JBMO TST, 2

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

Let $ABC$ ($AB < AC$) be a triangle with circumcircle $k$. The tangent to $k$ at $A$ intersects the line $BC$ at $D$ and the point $E\neq A$ on $k$ is such that $DE$ is tangent to $k$. The point $X$ on line $BE$ is such that $B$ is between $E$ and $X$ and $DX = DA$ and the point $Y$ on the line $CX$ is such that $Y$ is between $C$ and $X$ and $DY = DA$. Prove that the lines $BC$ and $YE$ are perpendicular.

1998 North Macedonia National Olympiad, 1

Tags: pentagon , angle chasing , equal segments , geometry

Let $ABCDE$ be a convex pentagon with $AB = BC =CA$ and $CD = DE = EC$. Let $T$ be the centroid of $\vartriangle ABC$, and $N$ be the midpoint of $AE$. Compute $\angle NT D$

2012 India Regional Mathematical Olympiad, 7

Tags: angle chasing , circles , angle , geometry

On the extension of chord $AB$ of a circle centroid at $O$ a point $X$ is taken and tangents $XC$ and $XD$ to the circle are drawn from it with $C$ and $D$ lying on the circle, let $E$ be the midpoint of the line segment $CD$. If $\angle OEB = 140^o$ then determine with proof the magnitude of $\angle AOB$.