Olympiad problems

2018 Stars of Mathematics, 1

Let $ABC$ be a triangle, and let $\ell$ be the line through $A$ and perpendicular to the line $BC$. The reflection of $\ell$ in the line $AB$ crosses the line through $B$ and perpendicular to $AB$ at $P$. The reflection of $\ell$ in the line $AC$ crosses the line through $C$ and perpendicular to $AC$ at $Q$. Show that the line $PQ$ passes through the orthocenter of the triangle $ABC$. Flavian Georgescu

2016 Saudi Arabia IMO TST, 1

Tags: geometry , circumcircle , incircle , orthocenter , collinear

Let $ABC$ be a triangle whose incircle $(I)$ touches $BC, CA, AB$ at $D, E, F$, respectively. The line passing through $A$ and parallel to $BC$ cuts $DE, DF$ at $M, N$, respectively. The circumcircle of triangle $DMN$ cuts $(I)$ again at $L$. a) Let $K$ be the intersection of $N E$ and $M F$. Prove that $K$ is the orthocenter of the triangle $DMN$. b) Prove that $A, K, L$ are collinear.

2021 Oral Moscow Geometry Olympiad, 6

Tags: geometry , angle bisector , orthocenter

Point $M$ is a midpoint of side $BC$ of a triangle $ABC$ and $H$ is the orthocenter of $ABC$. $MH$ intersects the $A$-angle bisector at $Q$. Points $X$ and $Y$ are the projections of $Q$ on sides $AB$ and $AC$. Prove that $XY$ passes through $H$.

2024 Philippine Math Olympiad, P3

Tags: geometry , orthocenter

Given triangle $ABC$ with orthocenter $H$, the lines through $B$ and $C$ perpendicular to $AB$ and $AC$, respectively, intersect line $AH$ at $X$ and $Y$, respectively. The circle with diameter $XY$ intersects lines $BX$ and $CY$ a second time at $K$ and $L$, respectively. Prove that points $H, K$ and $L$ are collinear.

2011 Sharygin Geometry Olympiad, 1

Tags: geometry , orthocenter , centroid , tangent circle

In triangle $ABC$ the midpoints of sides $AC, BC$, vertex $C$ and the centroid lie on the same circle. Prove that this circle touches the circle passing through $A, B$ and the orthocenter of triangle $ABC$.

2023 USEMO, 4

Tags: orthocenter , power of a point , geometry

Let $ABC$ be an acute triangle with orthocenter $H$. Points $A_1$, $B_1$, $C_1$ are chosen in the interiors of sides $BC$, $CA$, $AB$, respectively, such that $\triangle A_1B_1C_1$ has orthocenter $H$. Define $A_2 = \overline{AH} \cap \overline{B_1C_1}$, $B_2 = \overline{BH} \cap \overline{C_1A_1}$, and $C_2 = \overline{CH} \cap \overline{A_1B_1}$. Prove that triangle $A_2B_2C_2$ has orthocenter $H$. [i]Ankan Bhattacharya[/i]

2018 Bundeswettbewerb Mathematik, 3

Tags: geometry , orthocenter , perpendicular , parallel , circles

Let $H$ be the orthocenter of the acute triangle $ABC$. Let $H_a$ be the foot of the perpendicular from $A$ to $BC$ and let the line through $H$ parallel to $BC$ intersect the circle with diameter $AH_a$ in the points $P_a$ and $Q_a$. Similarly, we define the points $P_b, Q_b$ and $P_c,Q_c$. Show that the six points $P_a,Q_a,P_b,Q_b,P_c,Q_c$ lie on a common circle.

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.

2015 Estonia Team Selection Test, 9

Tags: geometry , angle bisector , orthocenter , perpendicular

The orthocenter of an acute triangle $ABC$ is $H$. Let $K$ and $P$ be the midpoints of lines $BC$ and $AH$, respectively. The angle bisector drawn from the vertex $A$ of the triangle $ABC$ intersects with line $KP$ at $D$. Prove that $HD\perp AD$.

2010 Puerto Rico Team Selection Test, 4

Tags: concyclic , orthocenter , geometry

Let $ABC$ be an acute triangle such that $AB>BC>AC$. Let $D$ be a point different from $C$ on the segment $BC$, such that $AC=AD$. Let $H$ be the orthocenter of triangle $ABC$ and let $A_1$ and $B_1$ be the intersections of the heights from $A$ and $B$ to the opposite sides, respectively. Let $E$ be the intersection of the lines $A_1B_1$ and $DH$. Prove that $B$, $D$, $B_1$, $E$ are concyclic.

2017 Auckland Mathematical Olympiad, 5

Tags: geometry , orthocenter , angle

The altitudes of triangle $ABC$ intersect at a point $H$.Find $\angle ACB$ if it is known that $AB = CH$.

2016 Germany National Olympiad (4th Round), 3

Tags: geometry , circumcircle , excenter , parallel , orthocenter

Let $I_a$ be the $A$-excenter of a scalene triangle $ABC$. And let $M$ be the point symmetric to $I_a$ about line $BC$. Prove that line $AM$ is parallel to the line through the circumcenter and the orthocenter of triangle $I_aCB$.

Mathley 2014-15, 3

Tags: projection , circumcircle , orthocenter , game strategy

A point $P$ is interior to the triangle $ABC$ such that $AP \perp BC$. Let $E, F$ be the projections of $CA, AB$. Suppose that the tangents at $E, F$ of the circumcircle of triangle $AEF$ meets at a point on $BC$. Prove that $P$ is the orthocenter of triangle $ABC$. Do Thanh Son, High School of Natural Sciences, National University, Hanoi

2019 Bulgaria National Olympiad, 2

Tags: circumcenter , geometry , orthocenter , circumcircle , perpendicular bisector

Let $ABC$ be an acute triangle with orthocenter $H$ and circumcenter $O.$ Let the intersection points of the perpendicular bisector of $CH$ with $AC$ and $BC$ be $X$ and $Y$ respectively. Lines $XO$ and $YO$ cut $AB$ at $P$ and $Q$ respectively. If $XP+YQ=AB+XY,$ determine $\measuredangle OHC.$

2008 Oral Moscow Geometry Olympiad, 5

Tags: midpoint , geometry , orthocenter , construction

Reconstruct an acute-angled triangle given the orthocenter and midpoints of two sides. (A. Zaslavsky)

2013 Sharygin Geometry Olympiad, 6

Tags: geometry , orthocenter , circles

A line $\ell$ passes through the vertex $B$ of a regular triangle $ABC$. A circle $\omega_a$ centered at $I_a$ is tangent to $BC$ at point $A_1$, and is also tangent to the lines $\ell$ and $AC$. A circle $\omega_c$ centered at $I_c$ is tangent to $BA$ at point $C_1$, and is also tangent to the lines $\ell$ and $AC$. Prove that the orthocenter of triangle $A_1BC_1$ lies on the line $I_aI_c$.

2019 Switzerland Team Selection Test, 1

Tags: geometry , orthocenter , similarity

Let $ABC$ be a triangle and $D, E, F$ be the foots of altitudes drawn from $A,B,C$ respectively. Let $H$ be the orthocenter of $ABC$. Lines $EF$ and $AD$ intersect at $G$. Let $K$ the point on circumcircle of $ABC$ such that $AK$ is a diameter of this circle. $AK$ cuts $BC$ in $M$. Prove that $GM$ and $HK$ are parallel.

2016 IMO Shortlist, G8

Tags: geometry , incenter , orthocenter

Let $A_1, B_1$ and $C_1$ be points on sides $BC$, $CA$ and $AB$ of an acute triangle $ABC$ respectively, such that $AA_1$, $BB_1$ and $CC_1$ are the internal angle bisectors of triangle $ABC$. Let $I$ be the incentre of triangle $ABC$, and $H$ be the orthocentre of triangle $A_1B_1C_1$. Show that $$AH + BH + CH \geq AI + BI + CI.$$

2021 Israel TST, 3

Tags: incircle , orthocenter , common tangent , geometry

Let $ABC$ be an acute triangle with orthocenter $H$. Prove that there is a line $l$ which is parallel to $BC$ and tangent to the incircles of $ABH$ and $ACH$.

2014 Bulgaria National Olympiad, 1

Tags: trigonometry , geometry , orthocenter , locus

Let $k$ be a given circle and $A$ is a fixed point outside $k$. $BC$ is a diameter of $k$. Find the locus of the orthocentre of $\triangle ABC$ when $BC$ varies. [i]Proposed by T. Vitanov, E. Kolev[/i]

2021 Mexico National Olympiad, 4

Tags: geometry , orthocenter , euler line , circumcircle , tangent circle

Let $ABC$ be an acutangle scalene triangle with $\angle BAC = 60^{\circ}$ and orthocenter $H$. Let $\omega_b$ be the circumference passing through $H$ and tangent to $AB$ at $B$, and $\omega_c$ the circumference passing through $H$ and tangent to $AC$ at $C$. [list] [*] Prove that $\omega_b$ and $\omega_c$ only have $H$ as common point. [*] Prove that the line passing through $H$ and the circumcenter $O$ of triangle $ABC$ is a common tangent to $\omega_b$ and $\omega_c$. [/list] [i]Note:[/i] The orthocenter of a triangle is the intersection point of the three altitudes, whereas the circumcenter of a triangle is the center of the circumference passing through it's three vertices.

2014 Danube Mathematical Competition, 3

Tags: midpoint , geometry , right angle , orthocenter

Let $ABC$ be a triangle with $\angle A<90^o, AB \ne AC$. Denote $H$ the orthocenter of triangle $ABC$, $N$ the midpoint of segment $[AH]$, $M$ the midpoint of segment $[BC]$ and $D$ the intersection point of the angle bisector of $\angle BAC$ with the segment $[MN]$. Prove that $<ADH=90^o$

2012 Sharygin Geometry Olympiad, 6

Tags: geometry , circumcircle , orthocenter

Let $\omega$ be the circumcircle of triangle $ABC$. A point $B_1$ is chosen on the prolongation of side $AB$ beyond point B so that $AB_1 = AC$. The angle bisector of $\angle BAC$ meets $\omega$ again at point $W$. Prove that the orthocenter of triangle $AWB_1$ lies on $\omega$ . (A.Tumanyan)

2015 EGMO, 6

Tags: geometry , circumcircle , orthocenter , centroid , angle

Let $H$ be the orthocentre and $G$ be the centroid of acute-angled triangle $ABC$ with $AB\ne AC$. The line $AG$ intersects the circumcircle of $ABC$ at $A$ and $P$. Let $P'$ be the reflection of $P$ in the line $BC$. Prove that $\angle CAB = 60$ if and only if $HG = GP'$

2018 District Olympiad, 2

Tags: geometry , centroid , orthocenter

Consider a right-angled triangle $ABC$, $\angle A = 90^{\circ}$ and points $D$ and $E$ on the leg $AB$ such that $\angle ACD \equiv \angle DCE \equiv \angle ECB$. Prove that if $3\overrightarrow{AD} = 2\overrightarrow{DE}$ and $\overrightarrow{CD} + \overrightarrow{CE} = 2\overrightarrow{CM}$ then $\overrightarrow{AB} = 4\overrightarrow{AM}$.