Olympiad problems

1995 Romania Team Selection Test, 3

The altitudes of a triangle have integer length and its inradius is a prime number. Find all possible values of the sides of the triangle.

2003 Estonia National Olympiad, 3

Tags: altitude , geometry , equal segments , parallelogram

In the acute-angled triangle $ABC$ all angles are greater than $45^o$. Let $AM$ and $BN$ be the heights of this triangle and let $X$ and $Y$ be the points on $MA$ and $NB$, respecively, such that $|MX| =|MB|$ and $|NY| =|NA|$. Prove that $MN$ and $XY$ are parallel.

2010 Sharygin Geometry Olympiad, 1

Tags: geometry , angle bisector , angle chasing , altitude , angle

For each vertex of triangle $ABC$, the angle between the altitude and the bisectrix from this vertex was found. It occurred that these angle in vertices $A$ and $B$ were equal. Furthermore the angle in vertex $C$ is greater than two remaining angles. Find angle $C$ of the triangle.

2014 Belarus Team Selection Test, 1

Tags: altitude , geometry , parallel

Let $AA_1, BB_1$ be the altitudes of an acute non-isosceles triangle $ABC$. Circumference of the triangles $ABC$ meets that of the triangle $A_1B_1C$ at point $N$ (different from $C$). Let $M$ be the midpoint of $AB$ and $K$ be the intersection point of $CN$ and $AB$. Prove that the line of centers the circumferences of the triangles $ABC$ and $KMC$ is parallel to the line $AB$. (I. Kachan)

2024 Austrian MO National Competition, 2

Tags: altitude , touching circles , geometry

Let $ABC$ be an acute triangle with $AB>AC$. Let $D,E,F$ denote the feet of its altitudes on $BC,AC$ and $AB$, respectively. Let $S$ denote the intersection of lines $EF$ and $BC$. Prove that the circumcircles $k_1$ and $k_2$ of the two triangles $AEF$ and $DES$ touch in $E$. [i](Karl Czakler)[/i]

2009 Hanoi Open Mathematics Competitions, 10

Tags: altitude , geometry

Let $ABC$ be an acute-angled triangle with $AB =4$ and $CD$ be the altitude through $C$ with $CD = 3$. Find the distance between the midpoints of $AD$ and $BC$

Indonesia MO Shortlist - geometry, g6

Tags: geometry , geometric inequality , altitude

Suppose the points $D, E, F$ lie on sides $BC, CA, AB$, respectively, so that $AD, BE, CF$ are the altitudes. Also, let $AD$ and $EF$ intersect at $P$. Prove that $$\frac{AP}{AD} \ge 1 - \frac{BC^2}{AB^2 + CA^2}$$

1994 Czech And Slovak Olympiad IIIA, 5

Tags: altitude , equal area , geometry , equilateral , triangle area

In an acute-angled triangle $ABC$, the altitudes $AA_1,BB_1,CC_1$ intersect at point $V$. If the triangles $AC_1V, BA_1V, CB_1V$ have the same area, does it follow that the triangle $ABC$ is equilateral?

1999 Estonia National Olympiad, 4

Tags: altitude , vector , median , geometry

For the given triangle $ABC$, prove that a point $X$ on the side $AB$ satisfies the condition $\overrightarrow{XA} \cdot\overrightarrow{XB} +\overrightarrow{XC} \cdot \overrightarrow{XC} = \overrightarrow{CA} \cdot \overrightarrow{CB} $, iff $X$ is the basepoint of the altitude or median of the triangle $ABC$.

2018 Silk Road, 1

Tags: altitude , geometry

In an acute-angled triangle $ABC$ on the sides $AB$, $BC$, $AC$ the points $H$, $L$, $K$ so that $CH \perp AB$, $HL \parallel AC$, $HK \parallel BC$. Let $P$ and $Q$ feet of altitudes of a triangle $HBL$, drawn from the vertices $H$ and $B$ respectively. Prove that the feet of the altitudes of the triangle $AKH$, drawn from the vertices $A$ and $H$ lie on the line $PQ$.

2012 Sharygin Geometry Olympiad, 1

Tags: altitude , geometry

The altitudes $AA_1$ and $BB_1$ of an acute-angled triangle ABC meet at point $O$. Let $A_1A_2$ and $B_1B_2$ be the altitudes of triangles $OBA_1$ and $OAB_1$ respectively. Prove that $A_2B_2$ is parallel to $AB$. (L.Steingarts)

2000 Saint Petersburg Mathematical Olympiad, 11.5

Tags: altitude , geometry , circumscribed quadrilateral

Let $AA_1$, $BB_1$, $CC_1$ be the altitudes of an acute angled triangle $ABC$. On the side $BC$ point $K$ is taken such that $\angle BB_1K=\angle A$. On the side $AB$ a point $M$ is taken such that $\angle BB_1M\angle C$. Let $L$ be the intersection of $BB_1$ and $A_1C_1$. Prove that the quadrilateral $B_1KLM$ is circumscribed. [I]Proposed by A. Khrabrov, D. Rostovski[/i]

2009 Sharygin Geometry Olympiad, 3

Tags: altitude , geometry , projection

Let $AH_a $ and $BH_b$ be the altitudes of triangle $ABC$. Points $P$ and $Q$ are the projections of $H_a$ to $AB$ and $AC$. Prove that line $PQ $ bisects segment $H_aH_b$. (A.Akopjan, K.Savenkov)

2017 Sharygin Geometry Olympiad, 5

Tags: altitude , parallel , reflection , geometry

Let $BH_b, CH_c$ be altitudes of an acute-angled triangle $ABC$. The line $H_bH_c$ meets the circumcircle of $ABC$ at points $X$ and $Y$. Points $P,Q$ are the reflections of $X,Y$ about $AB,AC$ respectively. Prove that $PQ \parallel BC$. [i]Proposed by Pavel Kozhevnikov[/i]

2015 Oral Moscow Geometry Olympiad, 1

Tags: geometry , bisects , altitude

In triangle $ABC$, the altitude $AH$ passes through midpoint of the median $BM$. Prove that in the triangle $BMC$ also one of the altitudes passes through the midpoint of one of the medians.

2015 Israel National Olympiad, 2

Tags: altitude , geometry , area of a triangle

A triangle is given whose altitudes' lengths are $\frac{1}{5},\frac{1}{5},\frac{1}{8}$. Evaluate the triangle's area.

2016 Latvia Baltic Way TST, 15

Tags: midpoint , altitude , concyclic , geometry

Let $ABC$ be a triangle. Let its altitudes $AD$, $BE$ and $CF$ concur at $H$. Let $K, L$ and $M$ be the midpoints of $BC$, $CA$ and $AB$, respectively. Prove that, if $\angle BAC = 60^o$, then the midpoints of the segments $AH$, $DK$, $EL$, $FM$ are concyclic.

1987 Tournament Of Towns, (160) 4

Tags: inradius , altitude , perpendicular , geometry

From point $M$ in triangle $ABC$ perpendiculars are dropped to each altitude. It can be shown that each of the line segments of altitudes, measured between the vertex and the foot of the perpendicular drawn to it, are of equal length. Prove that these lengths are each equal to the diameter of the circle inscribed in the triangle.

2006 Sharygin Geometry Olympiad, 12

Tags: altitude , median , angle bisector , geometry , isosceles

In the triangle $ABC$, the bisector of angle $A$ is equal to the half-sum of the height and median drawn from vertex $A$. Prove that if $\angle A$ is obtuse, then $AB = AC$.

2014 Estonia Team Selection Test, 4

Tags: altitude , geometry , circumcircle , equal segments

In an acute triangle the feet of altitudes drawn from vertices $A$ and $B$ are $D$ and $E$, respectively. Let $M$ be the midpoint of side $AB$. Line $CM$ intersects the circumcircle of $CDE$ again in point $P$ and the circumcircle of $CAB$ again in point $Q$. Prove that $|MP| = |MQ|$.

2019 BAMO, E/3

Tags: geometry , angle bisector , median , altitude , equilateral

In triangle $\vartriangle ABC$, we have marked points $A_1$ on side $BC, B_1$ on side $AC$, and $C_1$ on side $AB$ so that $AA_1$ is an altitude, $BB_1$ is a median, and $CC_1$ is an angle bisector. It is known that $\vartriangle A_1B_1C_1$ is equilateral. Prove that $\vartriangle ABC$ is equilateral too. (Note: A median connects a vertex of a triangle with the midpoint of the opposite side. Thus, for median $BB_1$ we know that $B_1$ is the midpoint of side $AC$ in $\vartriangle ABC$.)

1986 Polish MO Finals, 6

Tags: geometry , concyclic , cyclic quadrilateral , perpendicular , altitude

$ABC$ is a triangle. The feet of the perpendiculars from $B$ and $C$ to the angle bisector at $A$ are $K, L$ respectively. $N$ is the midpoint of $BC$, and $AM$ is an altitude. Show that $K,L,N,M$ are concyclic.

1985 Bundeswettbewerb Mathematik, 2

Tags: projection , geometry , collinear , altitude

Prove that in every triangle for each of its altitudes: If you project the foof of one altitude on the other two altitudes and on the other two sides of the triangle, those four projections lie on the same line.

Mexican Quarantine Mathematical Olympiad, #4

Tags: altitude , orthocenter , circumcenter , geometry

Let $ABC$ be an acute triangle with orthocenter $H$. Let $A_1$, $B_1$ and $C_1$ be the feet of the altitudes of triangle $ABC$ opposite to vertices $A$, $B$, and $C$ respectively. Let $B_2$ and $C_2$ be the midpoints of $BB_1$ and $CC_1$, respectively. Let $O$ be the intersection of lines $BC_2$ and $CB_2$. Prove that $O$ is the circumcenter of triangle $ABC$ if and only if $H$ is the midpoint of $AA_1$. [i]Proposed by Dorlir Ahmeti[/i]

2013 Sharygin Geometry Olympiad, 5

Tags: geometry , angle bisector , median , altitude , concurrency

The altitude $AA'$, the median $BB'$, and the angle bisector $CC'$ of a triangle $ABC$ are concurrent at point $K$. Given that $A'K = B'K$, prove that $C'K = A'K$.