Olympiad problems

2015 Ukraine Team Selection Test, 6

Given an acute triangle $ABC, H$ is the foot of the altitude drawn from the point $A$ on the line $BC, P$ and $K \ne H$ are arbitrary points on the segments $AH$ and$ BC$ respectively. Segments $AC$ and $BP$ intersect at point $B_1$, lines $AB$ and $CP$ at point $C_1$. Let $X$ and $Y$ be the projections of point $H$ on the lines $KB_1$ and $KC_1$, respectively. Prove that points $A, P, X$ and $Y$ lie on one circle.

2006 Estonia Team Selection Test, 4

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

The side $AC$ of an acute triangle $ABC$ is the diameter of the circle $c_1$ and side $BC$ is the diameter of the circle $c_2$. Let $E$ be the foot of the altitude drawn from the vertex $B$ of the triangle and $F$ the foot of the altitude drawn from the vertex $A$. In addition, let $L$ and $N$ be the points of intersection of the line $BE$ with the circle $c_1$ (the point $L$ lies on the segment $BE$) and the points of intersection of $K$ and $M$ of line $AF$ with circle $c_2$ (point $K$ is in section $AF$). Prove that $K LM N$ is a cyclic quadrilateral.

2013 Saudi Arabia IMO TST, 2

Tags: altitude , geometry , concyclic

Let $ABC$ be an acute triangle, and let $AA_1, BB_1$, and $CC_1$ be its altitudes. Segments $AA_1$ and $B_1C_1$ meet at point $K$. The perpendicular bisector of segment $A_1K$ intersects sides $AB$ and $AC$ at $L$ and $M$, respectively. Prove that points $A,A_1, L$, and $M$ lie on a circle.

2014 Switzerland - Final Round, 8

Tags: concyclic , altitude , geometry

In the acute-angled triangle $ABC$, let $M$ be the midpoint of the atlitude $h_b$ through $B$ and $N$ be the midpoint of the height $h_c$ through $C$. Further let $P$ be the intersection of $AM$ and $h_c$ and $Q$ be the intersection of $AN$ and $h_b$. Show that $M, N, P$ and $Q$ lie on a circle.

1989 Chile National Olympiad, 5

Tags: geometry , rational , altitude

The lengths of the three sides of a $ \triangle ABC $ are rational. The altitude $ CD $ determines on the side $AB$ two segments $ AD $ and $ DB $. Prove that $ AD, DB $ are rational.

2009 Puerto Rico Team Selection Test, 3

Tags: altitude , geometry

On an arbitrary triangle $ ABC$ let $ E$ be a point on the height from $ A$. Prove that $ (AC)^2 - (CE)^2 = (AB)^2 - (EB)^2$.

2022 3rd Memorial "Aleksandar Blazhevski-Cane", P4

Tags: altitude , excircle , incircle , geometry , concurrency

Let $ABC$ be an acute triangle with incircle $\omega$, incenter $I$, and $A$-excircle $\omega_{a}$. Let $\omega$ and $\omega_{a}$ meet $BC$ at $X$ and $Y$, respectively. Let $Z$ be the intersection point of $AY$ and $\omega$ which is closer to $A$. The point $H$ is the foot of the altitude from $A$. Show that $HZ$, $IY$ and $AX$ are concurrent. [i]Proposed by Nikola Velov[/i]

1995 Singapore Team Selection Test, 2

Tags: altitude , geometry , concyclic , diameter

Let $ABC$ be an acute-angled triangle. Suppose that the altitude of $\vartriangle ABC$ at $B$ intersects the circle with diameter $AC$ at $P$ and $Q$, and the altitude at $C$ intersects the circle with diameter $AB$ at $M$ and $N$. Prove that $P, Q, M$ and $N$ lie on a circle.

2008 Postal Coaching, 2

Tags: altitude , sidelenghts , rational , geometry

Does there exist a triangle $ABC$ whose sides are rational numbers and $BC$ equals to the altitude from $A$?

1984 Tournament Of Towns, (061) O2

Tags: tetrahedron , geometry , 3d geometry , plane , altitude , parallel

Six altitudes are constructed from the three vertices of the base of a tetrahedron to the opposite sides of the three lateral faces. Prove that all three straight lines joining two base points of the altitudes in each lateral face are parallel to a certain plane. (IF Sharygin, Moscow)

1952 Moscow Mathematical Olympiad, 212

Tags: altitude , equilateral , orthocenter , ratio , geometry

Prove that if the orthocenter divides all heights of a triangle in the same proportion, the triangle is equilateral.

2018 Estonia Team Selection Test, 7

Tags: altitude , geometry , concurrency , circumcircle , equal segments

Let $AD$ be the altitude $ABC$ of an acute triangle. On the line $AD$ are chosen different points $E$ and $F$ so that $|DE |= |DF|$ and point $E$ is in the interior of triangle $ABC$. The circumcircle of triangle $BEF$ intersects $BC$ and $BA$ for second time at points $K$ and $M$ respectively. The circumcircle of the triangle $CEF$ intersects the $CB$ and $CA$ for the second time at points $L$ and $N$ respectively. Prove that the lines $AD, KM$ and $LN$ intersect at one point.

2007 Balkan MO Shortlist, G4

Tags: geometric inequality , geometry , altitude

Points $M,N$ and $P$ on the sides $BC, CA$ and $AB$ of $\vartriangle ABC$ are such that $\vartriangle MNP$ is acute. Denote by $h$ and $H$ the lengths of the shortest altitude of $\vartriangle ABC$ and the longest altitude of $\vartriangle MNP$. Prove that $h \le 2H$.

Mexican Quarantine Mathematical Olympiad, #4

Tags: geometry , altitude , circumcenter , orthocenter

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]

2011 German National Olympiad, 3

Tags: geometry , altitude , circumcircle , orthic triangle

Let $ABC$ be an acute triangle and $D$ the foot of the altitude from $A$ onto $BC$. A semicircle with diameter $BC$ intersects segments $AB,AC$ and $AD$ in the points $F,E$ resp. $X$. The circumcircles of the triangles $DEX$ and $DXF$ intersect $BC$ in $L$ resp. $N$ other than $D$. Prove $BN=LC$.

2016 Saint Petersburg Mathematical Olympiad, 3

Tags: tetrahedron , sphere , geometry , concurrency , altitude , 3d geometry

In a tetrahedron, the midpoints of all the edges lie on the same sphere. Prove that it's altitudes intersect at one point.

2021-IMOC, G1

Tags: geometry , altitude , antipode , cyclic quadrilateral

Let $\overline{BE}$ and $\overline{CF}$ be altitudes of triangle $ABC$, and let $D$ be the antipodal point of $A$ on the circumcircle of $ABC$. The lines $\overleftrightarrow{DE}$ and $\overleftrightarrow{DF}$ intersect $\odot(ABC)$ again at $Y$ and $Z$, respectively. Show that $\overleftrightarrow{YZ}$, $\overleftrightarrow{EF}$ and $\overleftrightarrow{BC}$ intersect at a point.

Kyiv City MO 1984-93 - geometry, 1991.10.2

Tags: geometry , altitude , orthocenter

In an acute-angled triangle $ABC$ on the sides $AB$, $BC$, $AC$, the points $C_1$, $A_1$, and $B_1$ are marked such that the segments $AA_1$, $BB_1$, $CC_1$ intersect at some point $O$ and the angles $AA_1C$, $BB_1A$, $CC_1B$ are equal. Prove that $AA_1$, $BB_1$, and $CC_1$ are the altitudes of the triangle.

Swiss NMO - geometry, 2014.8

Tags: altitude , concyclic , geometry

In the acute-angled triangle $ABC$, let $M$ be the midpoint of the atlitude $h_b$ through $B$ and $N$ be the midpoint of the height $h_c$ through $C$. Further let $P$ be the intersection of $AM$ and $h_c$ and $Q$ be the intersection of $AN$ and $h_b$. Show that $M, N, P$ and $Q$ lie on a circle.

2013 Saudi Arabia GMO TST, 4

Tags: geometry , perpendicular , altitude , circles

In acute triangle $ABC$, points $D$ and $E$ are the feet of the perpendiculars from $A$ to $BC$ and $B$ to $CA$, respectively. Segment $AD$ is a diameter of circle $\omega$. Circle $\omega$ intersects sides $AC$ and $AB$ at $F$ and $G$ (other than $A$), respectively. Segment $BE$ intersects segments $GD$ and $GF$ at $X$ and $Y$ respectively. Ray $DY$ intersects side $AB$ at $Z$. Prove that lines $XZ$ and $BC$ are perpendicular

Durer Math Competition CD 1st Round - geometry, 2015.D4

Tags: geometry , circumcircle , altitude

The altitude of the acute triangle $ABC$ drawn from $A$ , intersects the side $BC$ at $A_1$ and the circumscribed circle at $A_2$ (different from $A$). Similarly, we get the points $B_1$, $B_2$, $C_1$, $C_2$. Prove that $$\frac{AA_2}{AA_1}+\frac{BB_2}{BB_1}+\frac{CC_2}{CC_1}= 4.$$

2007 Sharygin Geometry Olympiad, 1

Tags: construction , geometry , circumcircle , altitude , angle bisector

In an acute triangle $ABC$, altitudes at vertices $A$ and $B$ and bisector line at angle $C$ intersect the circumcircle again at points $A_1, B_1$ and $C_0$. Using the straightedge and compass, reconstruct the triangle by points $A_1, B_1$ and $C_0$.

1987 Tournament Of Towns, (160) 4

Tags: perpendicular , altitude , inradius , 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.

2011 Oral Moscow Geometry Olympiad, 4

Tags: trapezoid , perpendicular , midpoint , altitude , isosceles , geometry

In the trapezoid $ABCD, AB = BC = CD, CH$ is the altitude. Prove that the perpendicular from $H$ on $AC$ passes through the midpoint of $BD$.

2006 Belarusian National Olympiad, 7

Tags: concurrency , altitude , geometry

Let $AH_A, BH_B, CH_C$ be altitudes and $BM$ be a median of the acute-angled triangle $ABC$ ($AB > BC$). Let $K$ be a point of intersection of $BM$ and $AH_A$, $T$ be a point on $BC$ such that $KT \parallel AC$, $H$ be the orthocenter of $ABC$. Prove that the lines passing through the pairs of the points $(H_c, H_A), (H, T)$ and $(A, C)$ are concurrent. (S. Arkhipov)