Olympiad problems

Geometry Mathley 2011-12, 1.2

Let $ABC$ be an acute triangle with its altitudes $BE,CF$. $M$ is the midpoint of $BC$. $N$ is the intersection of $AM$ and $EF. X$ is the projection of $N$ on $BC$. $Y,Z$ are respectively the projections of $X$ onto $AB,AC$. Prove that $N$ is the orthocenter of triangle $AYZ$. Nguyễn Minh Hà

IV Soros Olympiad 1997 - 98 (Russia), 9.3

Tags: orthocenter , angle , ratio , geometry

What is angle $B$ of triangle$ ABC$, if it is known that the altitudes drawn from $A$ and $C$ intersect inside the triangle and one of them is divided by of intersection point into equal parts, and the other one in the ratio of $2: 1$, counting from the vertex?

2015 Romania Team Selection Test, 1

Tags: orthocenter , isogonal conjugate , circumcenter , geometry

Let $ABC$ be a triangle, let $O$ be its circumcenter, let $A'$ be the orthogonal projection of $A$ on the line $BC$, and let $X$ be a point on the open ray $AA'$ emanating from $A$. The internal bisectrix of the angle $BAC$ meets the circumcircle of $ABC$ again at $D$. Let $M$ be the midpoint of the segment $DX$. The line through $O$ and parallel to the line $AD$ meets the line $DX$ at $N$. Prove that the angles $BAM$ and $CAN$ are equal.

Kyiv City MO Juniors Round2 2010+ geometry, 2012.9.4

Tags: circumcircle , geometry , angle , parallel , orthocenter , equal segments

In an acute-angled triangle $ABC$, the point $O$ is the center of the circumcircle, and the point $H$ is the orthocenter. It is known that the lines $OH$ and $BC$ are parallel, and $BC = 4OH $. Find the value of the smallest angle of triangle $ ABC $. (Black Maxim)

2019 Saudi Arabia Pre-TST + Training Tests, 2.3

Tags: circumcircle , geometry , euler circle , circumcenter , orthocenter

Let $ABC$ be an acute, non isosceles triangle with $O,H$ are circumcenter and orthocenter, respectively. Prove that the nine-point circles of $AHO,BHO,CHO$ has two common points.

2023 Bulgaria JBMO TST, 4

Tags: orthocenter , shortlist , geometry

Given is an acute angled triangle $ABC$ with orthocenter $H$ and circumcircle $k$. Let $\omega$ be the circle with diameter $AH$ and $P$ be the point of intersection of $\omega$ and $k$ other than $A$. Assume that $BP$ and $CP$ intersect $\omega$ for the second time at points $Q$ and $R$, respectively. If $D$ is the foot of the altitude from $A$ to $BC$ and $S$ is the point of the intersection of $\omega$ and $QD$, prove that $HR = HS$.

2019 Brazil Team Selection Test, 1

Tags: orthocenter , circumcircle , equilateral triangle , collinear , geometry

Let $ABC$ be an acute triangle, with $\angle A > 60^\circ$, and let $H$ be it's orthocenter. Let $M$ and $N$ be points on $AB$ and $AC$, respectively, such that $\angle HMB = \angle HNC = 60^\circ$. Also, let $O$ be the circuncenter of $HMN$ and $D$ be a point on the semiplane determined by $BC$ that contains $A$ in such a way that $DBC$ is equilateral. Prove that $H$, $O$ and $D$ are collinear.

1996 IMO Shortlist, 3

Tags: geometry , circumcircle , symmetry , orthocenter , trigonometry

Let $O$ be the circumcenter and $H$ the orthocenter of an acute-angled triangle $ABC$ such that $BC>CA$. Let $F$ be the foot of the altitude $CH$ of triangle $ABC$. The perpendicular to the line $OF$ at the point $F$ intersects the line $AC$ at $P$. Prove that $\measuredangle FHP=\measuredangle BAC$.

2005 Sharygin Geometry Olympiad, 5

Tags: locus , circumcenter , centroid , orthocenter , circumcircle , geometry

There are two parallel lines $p_1$ and $p_2$. Points $A$ and $B$ lie on $p_1$, and $C$ on $p_2$. We will move the segment $BC$ parallel to itself and consider all the triangles $AB'C '$ thus obtained. Find the locus of the points in these triangles: a) points of intersection of heights, b) the intersection points of the medians, c) the centers of the circumscribed circles.

2019 Latvia Baltic Way TST, 10

Tags: tangent circle , circumcircle , orthocenter , geometry

Let $\triangle ABC$ be an acute angled triangle with orthocenter $H$ and let $M$ be a midpoint of $BC$. Circle with diameter $AH$ is $\omega_1$ and circle with center $M$ is $\omega_2$. If $\omega_2$ is tangent to circumcircle of $\triangle ABC$, then prove that circles $\omega_1$ and $\omega_2$ are tangent to each other.

2011 Sharygin Geometry Olympiad, 6

Tags: incenter , orthocenter , incircle , geometry , altitude , contact triangle

Let $BB_1$ and $CC_1$ be the altitudes of acute-angled triangle $ABC$, and $A_0$ is the midpoint of $BC$. Lines $A_0B_1$ and $A_0C_1$ meet the line passing through $A$ and parallel to $BC$ in points $P$ and $Q$. Prove that the incenter of triangle $PA_0Q$ lies on the altitude of triangle $ABC$.

2005 Switzerland - Final Round, 8

Tags: orthocenter , collinear , geometry , circles

Let $ABC$ be an acute-angled triangle. $M ,N$ are any two points on the sides $AB , AC$ respectively. The circles with the diameters $BN$ and $CM$ intersect at points $P$ and $Q$. Show that the points $P, Q$ and the orthocenter of the triangle $ABC$ lie on a straight line.

1998 Argentina National Olympiad, 2

Tags: geometry , orthocenter , tangential quadrilateral , parallelogram

Let a quadrilateral $ABCD$ have an inscribed circle and let $K, L, M, N$ be the tangency points of the sides $AB, BC, CD$ and $DA$, respectively. Consider the orthocenters of each of the triangles $\vartriangle AKN, \vartriangle BLK, \vartriangle CML$ and $\vartriangle DNM$. Prove that these four points are the vertices of a parallelogram.

2024 Turkey MO (2nd Round), 2

Tags: orthocenter , geometry

Let $\triangle ABC$ be an acute triangle, where $H$ is the orthocenter and $D,E,F$ are the feet of the altitudes from $A,B,C$ respectively. A circle tangent to $(DEF)$ at $D$ intersects the line $EF$ at $P$ and $Q$. Let $R$ and $S$ be the second intersection points of the circumcircle of triangle $\triangle BHC$ with $PH$ and $QH$, respectively. Let $T$ be the point on the line $BC$ such that $AT\perp EF$. Prove that the points $R,S,D,T$ are concyclic.

1996 Argentina National Olympiad, 3

Tags: geometry , circumcircle , orthocenter , circumcenter , hexagon

The non-regular hexagon $ABCDEF$ is inscribed on a circle of center $O$ and $AB = CD = EF$. If diagonals $AC$ and $BD$ intersect at $M$, diagonals $CE$ and $DF$ intersect at $N$, and diagonals $AE$ and $BF$ intersect at $K$, show that the heights of triangle $MNK$ intersect at $O$.

2021 Yasinsky Geometry Olympiad, 4

Tags: geometry , orthocenter , angle bisector

In triangle $ABC$, the point $H$ is the orthocenter. A circle centered at point $H$ and with radius $AH$ intersects the lines $AB$ and $AC$ at points $E$ and $D$, respectively. The point $X$ is the symmetric of the point $A$ with respect to the line $BC$ . Prove that $XH$ is the bisector of the angle $DXE$. (Matthew of Kursk)

2023 Sharygin Geometry Olympiad, 9

Tags: geometry , orthocenter , midpoint of altitude

It is known that the reﬂection of the orthocenter of a triangle $ABC$ about its circumcenter lies on $BC$. Let $A_1$ be the foot of the altitude from $A$. Prove that $A_1$ lies on the circle passing through the midpoints of the altitudes of $ABC$.

2011 Ukraine Team Selection Test, 10

Tags: orthocenter , geometry , concurrency , equal angles

Let $ H $ be the point of intersection of the altitudes $ AP $ and $ CQ $ of the acute-angled triangle $ABC$. The points $ E $ and $ F $ are marked on the median $ BM $ such that $ \angle APE = \angle BAC $, $ \angle CQF = \angle BCA $, with point $ E $ lying inside the triangle $APB$ and point $ F $ is inside the triangle $CQB$. Prove that the lines $AE, CF$, and $BH$ intersect at one point.

2019 Abels Math Contest (Norwegian MO) Final, 4

Tags: similar , centroid , orthocenter , geometry

The diagonals of a convex quadrilateral $ABCD$ intersect at $E$. The triangles $ABE, BCE, CDE$ and $DAE$ have centroids $K,L,M$ and $N$, and orthocentres $Q,R,S$ and $T$. Show that the quadrilaterals $QRST$ and $LMNK$ are similar.

2018 Saudi Arabia GMO TST, 3

Tags: geometry , orthocenter , incenter , circumcenter

Let $I, O$ be the incenter, circumcenter of triangle $ABC$ and $A_1, B_1, C_1 $be arbitrary points on the segments $AI, BI, CI$ respectively. The perpendicular bisectors of $AA_1, BB_1, CC_1$ intersect each other at $X, Y$ and $Z$. Prove that the circumcenter of triangle $XYZ$ coincides with $O$ if and only if $I$ is the orthocenter of triangle $A_1B_1C_1$

2024 Austrian MO National Competition, 4

Tags: orthocenter , geometry , centroid , barycenter

Let $ABC$ be an obtuse triangle with orthocenter $H$ and centroid $S$. Let $D$, $E$ and $F$ be the midpoints of segments $BC$, $AC$, $AB$, respectively. Show that the circumcircle of triangle $ABC$, the circumcircle of triangle $DEF$ and the circle with diameter $HS$ have two distinct points in common. [i](Josef Greilhuber)[/i]

2005 Sharygin Geometry Olympiad, 10.6

Tags: concurrency , orthocenter , geometry , circles

Let $H$ be the orthocenter of triangle $ABC$, $X$ be an arbitrary point. A circle with a diameter of $XH$ intersects lines $AH, BH, CH$ at points $A_1, B_1, C_1$ for the second time, and lines $AX BX, CX$ at points $A_2, B_2, C_2$. Prove that lines A$_1A_2, B_1B_2, C_1C_2$ intersect at one point.

2017 Mexico National Olympiad, 3

Tags: parallel , orthocenter , geometry

Let $ABC$ be an acute triangle with orthocenter $H$. The circle through $B, H$, and $C$ intersects lines $AB$ and $AC$ at $D$ and $E$ respectively, and segment $DE$ intersects $HB$ and $HC$ at $P$ and $Q$ respectively. Two points $X$ and $Y$, both different from $A$, are located on lines $AP$ and $AQ$ respectively such that $X, H, A, B$ are concyclic and $Y, H, A, C$ are concyclic. Show that lines $XY$ and $BC$ are parallel.

2019 Balkan MO Shortlist, G4

Tags: orthocenter , geometry , projective

Given an acute triangle $ABC$, let $M$ be the midpoint of $BC$ and $H$ the orthocentre. Let $\Gamma$ be the circle with diameter $HM$, and let $X,Y$ be distinct points on $\Gamma$ such that $AX,AY$ are tangent to $\Gamma$. Prove that $BXYC$ is cyclic.

III Soros Olympiad 1996 - 97 (Russia), 9.6

Tags: geometry , orthocenter , incenter

In triangle $ABC$, angle $B$ is not right. The circle inscribed in $ABC$ touches $AB$ and $BC$ at points $C_1$ and $A_1$, and the feet of the altitudes drawn to the sides $AB$ and $BC$ are points $C_2$ and $A_2$. Prove that the intersection point of the altitudes of triangle $A_1BC_1$ is the center of the circle inscribed in triangle $A_2BC_2$.