Olympiad problems

Brazil L2 Finals (OBM) - geometry, 2023.2

Tags: circumcenter , isogonal lines , similar triangles , orthocenter , midpoint , cyclic quadrilateral , geometry

Consider a triangle $ABC$ with $AB < AC$ and let $H$ and $O$ be its orthocenter and circumcenter, respectively. A line starting from $B$ cuts the lines $AO$ and $AH$ at $M$ and $M'$ so that $M'$ is the midpoint of $BM$. Another line starting from $C$ cuts the lines $AH$ and $AO$ at $N$ and $N'$ so that $N'$ is the midpoint of $CN$. Prove that $M, M', N, N'$ are on the same circle.

2018 Sharygin Geometry Olympiad, 7

Tags: geometry , tangent , median , isogonal lines

Let $B_1,C_1$ be the midpoints of sides $AC,AB$ of a triangle $ABC$ respectively. The tangents to the circumcircle at $B$ and $C$ meet the rays $CC_1,BB_1$ at points $K$ and $L$ respectively. Prove that $\angle BAK = \angle CAL$.

2014 IFYM, Sozopol, 5

Tags: isogonal lines , geometry

Let $\Delta ABC$ be an acute triangle. Points $P,Q\in AB$ so that $P$ is between $A$ and $Q$. Let $H_1$ and $H_2$ be the feet of the perpendiculars from $A$ to $CP$ and $CQ$ respectively. Let $H_3$ and $H_4$ be the feet of the perpendiculars from $B$ to $CP$ and $CQ$ respectively. Let $H_3 H_4\cap BC=X$ and $H_1 H_2\cap AC=Y$, so that $X$ is after $B$ and $Y$ is after $A$. If $XY\parallel AB$, prove that $CP$ and $CQ$ are isogonal to $\Delta ABC$.

2001 Saint Petersburg Mathematical Olympiad, 9.5

Tags: geometry , midline , isogonal lines , sine chasing , angle bisector , isogonal conjugate

Points $A_1$, $B_1$, $C_1$ are midpoints of sides $BC$, $AC$, $AB$ of triangle $ABC$. On midlines $C_1B_1$ and $A_1B_1$ points $E$ and $F$ are chosen such that $BE$ is the angle bisector of $AEB_1$ and $BF$ is the angle bisector of $CFB_1$. Prove that bisectors of angles $ABC$ and $FBE$ coincide. [I]Proposed by F. Baharev[/i]

2021 Azerbaijan EGMO TST, 2

Tags: isogonal lines , geometry

Let $\omega$ be a circle with center $O,$ and let $A$ be a point with tangents $AP$ and $AQ$ to the circle. Denote by $K$ the intersection point of $AO$ and $PQ.$ $l_1$ and $l_2$ are two lines passing through $A$ that intersect $\omega.$ Call $B$ the intersection point of $l_1$ with $\omega,$ which is located nearer to $A$ on $l_1.$ Call $C$ the intersection point of $l_2$ with $\omega,$ which is located further to $A$ on $l_2.$ Prove that $\angle PAB = \angle QAC$ if and only if the points $B, K, C$ are on line.

2019 Philippine TST, 6

Tags: parallelogram , isogonal lines , angle , geometry

Let $D$ be an interior point of triangle $ABC$. Lines $BD$ and $CD$ intersect sides $AC$ and $AB$ at points $E$ and $F$, respectively. Points $X$ and $Y$ are on the plane such that $BFEX$ and $CEFY$ are parallelograms. Suppose lines $EY$ and $FX$ intersect at a point $T$ inside triangle $ABC$. Prove that points $B$, $C$, $E$, and $F$ are concyclic if and only if $\angle BAD = \angle CAT$.

2016 USA Team Selection Test, 2

Tags: reflection , mixtilinear incircle , isogonal lines , excircle , geometry

Let $ABC$ be a scalene triangle with circumcircle $\Omega$, and suppose the incircle of $ABC$ touches $BC$ at $D$. The angle bisector of $\angle A$ meets $BC$ and $\Omega$ at $E$ and $F$. The circumcircle of $\triangle DEF$ intersects the $A$-excircle at $S_1$, $S_2$, and $\Omega$ at $T \neq F$. Prove that line $AT$ passes through either $S_1$ or $S_2$. [i]Proposed by Evan Chen[/i]

2023 Brazil National Olympiad, 2

Tags: cyclic quadrilateral , geometry , orthocenter , circumcenter , isogonal lines , midpoint , similar triangles

Consider a triangle $ABC$ with $AB < AC$ and let $H$ and $O$ be its orthocenter and circumcenter, respectively. A line starting from $B$ cuts the lines $AO$ and $AH$ at $M$ and $M'$ so that $M'$ is the midpoint of $BM$. Another line starting from $C$ cuts the lines $AH$ and $AO$ at $N$ and $N'$ so that $N'$ is the midpoint of $CN$. Prove that $M, M', N, N'$ are on the same circle.