Olympiad problems

2013 Oral Moscow Geometry Olympiad, 5

In triangle $ABC, \angle C= 60^o, \angle A= 45^o$. Let $M$ be the midpoint of $BC, H$ be the orthocenter of triangle $ABC$. Prove that line $MH$ passes through the midpoint of arc $AB$ of the circumcircle of triangle $ABC$.

2019 Peru EGMO TST, 2

Tags: orthocenter , midpoints arc , concyclic , arc midpoint , geometry , circumcircle

Let $\Gamma$ be the circle of an acute triangle $ABC$ and let $H$ be its orthocenter. The circle $\omega$ with diameter $AH$ cuts $\Gamma$ at point $D$ ($D\ne A$). Let $M$ be the midpoint of the smaller arc $BC$ of $\Gamma$ . Let $N$ be the midpoint of the largest arc $BC$ of the circumcircle of the triangle $BHC$. Prove that there is a circle that passes through the points $D, H, M$ and $N$.

2016 Hanoi Open Mathematics Competitions, 13

Tags: orthocenter , midpoint , equal segments , geometry

Let $H$ be orthocenter of the triangle $ABC$. Let $d_1, d_2$ be lines perpendicular to each-another at $H$. The line $d_1$ intersects $AB, AC$ at $D, E$ and the line d_2 intersects $B C$ at $F$. Prove that $H$ is the midpoint of segment $DE$ if and only if $F$ is the midpoint of segment $BC$.

2009 Brazil Team Selection Test, 1

Tags: right triangle , cyclic , pentagon , geometry , orthocenter

Let $A, B, C, D, E$ points in circle of radius r, in that order, such that $AC = BD = CE = r$. The points $H_1, H_2, H_3$ are the orthocenters of the triangles $ACD$, $BCD$ and $BCE$, respectively. Prove that $H_1H_2H_3$ is a right triangle .

2019 Romania Team Selection Test, 3

Tags: orthocenter , reflection , geometry , concurrency

Let $AD, BE$, and $CF$ denote the altitudes of triangle $\vartriangle ABC$. Points $E'$ and $F'$ are the reflections of $E$ and $F$ over $AD$, respectively. The lines $BF'$ and $CE'$ intersect at $X$, while the lines $BE'$ and $CF'$ intersect at the point $Y$. Prove that if $H$ is the orthocenter of $\vartriangle ABC$, then the lines $AX, YH$, and $BC$ are concurrent.

2010 Sharygin Geometry Olympiad, 6

Tags: orthocenter , right triangle , incircle , geometry

The incircle of triangle $ABC$ touches its sides in points $A', B',C'$ . It is known that the orthocenters of triangles $ABC$ and $A' B'C'$ coincide. Is triangle $ABC$ regular?

1993 Tournament Of Towns, (358) 1

Tags: orthocenter , geometry

Let $M$ be a point on the side $AB$ of triangle $ABC$. The length $AB = c$ and $\angle CMA=\phi$ are given. Find the distance between the orthocentres (intersection points of altitudes) of the triangles $AMC$ and $BMC$. (IF Sharygin)

2009 Ukraine Team Selection Test, 5

Tags: orthocenter , right triangle , circumcircle , circumcenter , geometry

Let $A,B,C,D,E$ be consecutive points on a circle with center $O$ such that $AC=BD=CE=DO$. Let $H_1,H_2,H_3$ be the orthocenters triangles $ACD,BCD,BCE$ respectively. Prove that the triangle $H_1H_2H_3$ is right.

Indonesia Regional MO OSP SMA - geometry, 2017.3

Tags: symmetric , orthocenter , circumcircle , geometry

Given triangle $ABC$, the three altitudes intersect at point $H$. Determine all points $X$ on the side $BC$ so that the symmetric of $H$ wrt point $X$ lies on the circumcircle of triangle $ABC$.

2002 India IMO Training Camp, 4

Tags: orthocenter , triangle , concurrency , circumcircle , geometry

Let $O$ be the circumcenter and $H$ the orthocenter of an acute triangle $ABC$. Show that there exist points $D$, $E$, and $F$ on sides $BC$, $CA$, and $AB$ respectively such that \[ OD + DH = OE + EH = OF + FH\] and the lines $AD$, $BE$, and $CF$ are concurrent.

2007 Sharygin Geometry Olympiad, 4

Tags: locus , orthocenter , locus problems , geometry

Determine the locus of orthocenters of triangles, given the midpoint of a side and the feet of the altitudes drawn on two other sides.

2018 Saudi Arabia IMO TST, 1

Tags: midpoint , orthocenter , circumcenter , incenter , geometry

Let $ABC$ be an acute, non isosceles triangle with $M, N, P$ are midpoints of $BC, CA, AB$, respectively. Denote $d_1$ as the line passes through $M$ and perpendicular to the angle bisector of $\angle BAC$, similarly define for $d_2, d_3$. Suppose that $d_2 \cap d_3 = D$, $d_3 \cap d_1 = E,$ $d_1 \cap d_2 = F$. Let $I, H$ be the incenter and orthocenter of triangle $ABC$. Prove that the circumcenter of triangle $DEF$ is the midpoint of segment $IH$.

2024 Indonesia MO, 3

Tags: geometry , circumcircle , orthocenter

The triangle $ABC$ has $O$ as its circumcenter, and $H$ as its orthocenter. The line $AH$ and $BH$ intersect the circumcircle of $ABC$ for the second time at points $D$ and $E$, respectively. Let $A'$ and $B'$ be the circumcenters of triangle $AHE$ and $BHD$ respectively. If $A', B', O, H$ are [b]not[/b] collinear, prove that $OH$ intersects the midpoint of segment $A'B'$.

2019 Junior Balkan Team Selection Tests - Romania, 3

Tags: orthocenter , isosceles , circumcircle , geometry

Let $d$ be the tangent at $B$ to the circumcircle of the acute scalene triangle $ABC$. Let $K$ be the orthogonal projection of the orthocenter, $H$, of triangle $ABC$ to the line $d$ and $L$ the midpoint of the side $AC$. Prove that the triangle $BKL$ is isosceles.

2018 Oral Moscow Geometry Olympiad, 5

Tags: geometry , locus , orthocenter , midpoint , fixed

The circle circumscribed about an acute triangle $ABC$ and the vertex $C$ are fixed. Orthocenter $H$ moves in a circle with center at point $C$. Find the locus of the midpoints of the segments connecting the feet of altitudes drawn from vertices $A$ and $B$.

2021 Israel TST, 3

Tags: orthocenter , common tangent , incircle , 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 $ACH$.

2017 Saudi Arabia BMO TST, 3

Tags: orthocenter , collinear , circumcircle , concurrency , geometry

Let $ABC$ be an acute triangle and $(O)$ be its circumcircle. Denote by $H$ its orthocenter and $I$ the midpoint of $BC$. The lines $BH, CH$ intersect $AC,AB$ at $E, F$ respectively. The circles $(IBF$) and $(ICE)$ meet again at $D$. a) Prove that $D, I,A$ are collinear and $HD, EF, BC$ are concurrent. b) Let $L$ be the foot of the angle bisector of $\angle BAC$ on the side $BC$. The circle $(ADL)$ intersects $(O)$ again at $K$ and intersects the line $BC$ at $S$ out of the side $BC$. Suppose that $AK,AS$ intersects the circles $(AEF)$ again at $G, T$ respectively. Prove that $TG = TD$.

2018 Bundeswettbewerb Mathematik, 3

Tags: parallel , orthocenter , circles , geometry , perpendicular

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.

2018 Peru MO (ONEM), 3

Tags: orthocenter , right angle , angle chasing , isosceles , geometry

Let $ABC$ be an acute triangle such that $BA = BC$. On the sides $BA$ and $BC$ points $D$ and $E$ are chosen respectively, such that $DE$ and $AC$ are parallel. Let $H$ be the orthocenter of the triangle $DBE$ and $M$ be the midpoint of $AE$. If $\angle HMC = 90^o$, determine the measure of angle $\angle ABC$.

Kyiv City MO Seniors 2003+ geometry, 2018.11.4.1

Tags: geometry , equal angles , orthocenter

In the quadrilateral $ABCD$, the diagonal $AC$ is the bisector $\angle BAD$ and $\angle ADC = \angle ACB$. The points $X, \, \, Y$ are the feet of the perpendiculars drawn from the point $A$ on the lines $BC, \, \, CD$, respectively. Prove that the orthocenter $\Delta AXY$ lies on the line $BD$.

2012 India Regional Mathematical Olympiad, 4

Tags: geometry , orthocenter , area of a triangle , area

$H$ is the orthocentre of an acuteangled triangle $ABC$. A point $E$ is taken on the line segment $CH$ such that $ABE$ is a rightangled triangle. Prove that the area of the triangle $ABE$ is the geometric mean of the areas of triangles $ABC$ and $ABH$.

2008 Balkan MO Shortlist, G4

Tags: perpendicular bisector , centroid , orthocenter , barycenter , geometry

A triangle $ABC$ is given with barycentre $G$ and circumcentre $O$. The perpendicular bisectors of $GA, GB$ meet at $C_1$,of $GB,GC$ meet at $A _1$, and $GC,GA$ meet at $B_1$. Prove that $O$ is the barycenter of the triangle $A_1B_1C_1$.

2019 Cono Sur Olympiad, 6

Tags: tangent , orthocenter , circumcircle , geometry

Let $ABC$ be an acute-angled triangle with $AB< AC$, and let $H$ be its orthocenter. The circumference with diameter $AH$ meets the circumscribed circumference of $ABC$ at $P\neq A$. The tangent to the circumscribed circumference of $ABC$ through $P$ intersects line $BC$ at $Q$. Show that $QP=QH$.

1988 Tournament Of Towns, (195) 2

Tags: orthocenter , circumradius , geometry , circumcircle

Let $N$ be the orthocentre of triangle $ABC$ (i .e. the point where the altitudes meet). Prove that the circumscribed circles of triangles $ABN, ACN$ and $BCN$ each have equal radius.

2021 Israel TST, 3

Tags: orthocenter , common tangent , incircle , 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 $ACH$.