Found problems: 200
2019 Saudi Arabia JBMO TST, 2
In triangle $ABC$ point $M$ is the midpoint of side $AB$, and point $D$ is the foot of altitude $CD$.
Prove that $\angle A = 2\angle B$ if and only if $AC = 2MD$
2023 Sharygin Geometry Olympiad, 16
Let $AH_A$ and $BH_B$ be the altitudes of a triangle $ABC$. The line $H_AH_B$ meets the circumcircle of $ABC$ at points $P$ and $Q$. Let $A'$ be the reflection of $A$ about $BC$, and $B'$ be the reflection of $B$ about $CA$. Prove that $A',B', P,Q$ are concyclic.
1995 North Macedonia National Olympiad, 2
Let $ a, $ $ b $, and $ c $ be sides in a triangle, a $ h_a, $ $ h_b $, and $ h_c $ are the corresponding altitudes.
Prove that $h ^ 2_a + h ^ 2_b + h ^ 2_c \leq \frac{3}{4} (a ^ 2 + b ^ 2 + c ^ 2). $ When is the equation valid?
2004 Switzerland Team Selection Test, 10
In an acute-angled triangle $ABC$ the altitudes $AU,BV,CW$ intersect at $H$.
Points $X,Y,Z$, different from $H$, are taken on segments $AU,BV$, and $CW$, respectively.
(a) Prove that if $X,Y,Z$ and $H$ lie on a circle, then the sum of the areas of triangles $ABZ, AYC, XBC$ equals the area of $ABC$.
(b) Prove the converse of (a).
2019 Silk Road, 1
The altitudes of the acute-angled non-isosceles triangle $ ABC $ intersect at the point $ H $. On the segment $ C_1H $, where $ CC_1 $ is the altitude of the triangle, the point $ K $ is marked. Points $ L $ and $ M $ are the feet of perpendiculars from point $ K $ on straight lines $ AC $ and $ BC $, respectively. The lines $ AM $ and $ BL $ intersect at $ N $. Prove that $ \angle ANK = \angle HNL $.
Kyiv City MO Seniors 2003+ geometry, 2018.10.4
In the acute-angled triangle $ABC$, the altitudes $BP$ and $CQ$ were drawn, and the point $T$ is the intersection point of the altitudes of $\Delta PAQ$. It turned out that $\angle CTB = 90 {} ^ \circ$. Find the measure of $\angle BAC$.
(Mikhail Plotnikov)
2014 Estonia Team Selection Test, 4
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 $|MP| = |MQ|$.
2018 Ukraine Team Selection Test, 9
Let $AA_1, BB_1, CC_1$ be the heights of triangle $ABC$ and $H$ be its orthocenter. Liune $\ell$ parallel to $AC$, intersects straight lines $AA_1$ and $CC_1$ at points $A_2$ and $C_2$, respectively. Suppose that point $B_1$ lies outside the circumscribed circle of triangle $A_2 HC_2$. Let $B_1P$ and $B_1T$ be tangent to of this circle. Prove that points $A_1, C_1, P$, and $T$ are cyclic.
2019 BAMO, E/3
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 $\vartriangle ABC$.)
2013 Sharygin Geometry Olympiad, 5
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 $C'K = A'K$.
2011 NZMOC Camp Selection Problems, 2
In triangle $ABC$, the altitude from $B$ is tangent to the circumcircle of $ABC$. Prove that the largest angle of the triangle is between $90^o$ and $135^o$. If the altitudes from both $B$ and from $C$ are tangent to the circumcircle, then what are the angles of the triangle?
2018 Yasinsky Geometry Olympiad, 3
Construct triangle $ABC$, given the altitude and the angle bisector both from $A$, if it is known for the sides of the triangle $ABC$ that $2BC = AB + AC$.
(Alexey Karlyuchenko)
2005 Sharygin Geometry Olympiad, 9.5
It is given that for no side of the triangle from the height drawn to it, the bisector and the median it is impossible to make a triangle. Prove that one of the angles of the triangle is greater than $135^o$
2010 Sharygin Geometry Olympiad, 1
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.
1972 IMO Shortlist, 5
Prove the following assertion: The four altitudes of a tetrahedron $ABCD$ intersect in a point if and only if
\[AB^2 + CD^2 = BC^2 + AD^2 = CA^2 + BD^2.\]
2024 Austrian MO National Competition, 2
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]
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.
2018 Yasinsky Geometry Olympiad, 1
In the triangle $ABC$, $AD$ is altitude, $M$ is the midpoint of $BC$. It is known that $\angle BAD = \angle DAM = \angle MAC$. Find the values of the angles of the triangle $ABC$
2019 Estonia Team Selection Test, 7
An acute-angled triangle $ABC$ has two altitudes $BE$ and $CF$. The circle with diameter $AC$ intersects the segment $BE$ at point $P$. A circle with diameter $AB$ intersects the segment $CF$ at point $Q$ and the extension of this altitude at point $Q'$. Prove that $\angle PQ'Q = \angle PQB$.
2003 Estonia National Olympiad, 3
Let $ABC$ be a triangle and $A_1, B_1, C_1$ points on $BC, CA, AB$, respectively, such that the lines $AA_1, BB_1, CC_1$ meet at a single point. It is known that $A, B_1, A_1, B$ are concyclic and $B, C_1, B_1, C$ are concyclic. Prove that
a) $C, A_1, C_1, A$ are concyclic,
b) $AA_1,, BB_1, CC_1$ are the heights of $ABC$.
2009 Puerto Rico Team Selection Test, 3
Show that if $ h_A, h_B,$ and $ h_C$ are the altitudes of $ \triangle ABC$, and $ r$ is the radius of the incircle, then $$ h_A + h_B + h_C \ge 9r$$
2012 Oral Moscow Geometry Olympiad, 3
$H$ is the intersection point of the heights $AA'$ and $BB'$ of the acute-angled triangle $ABC$. A straight line, perpendicular to $AB$, intersects these heights at points $D$ and $E$, and side $AB$ at point $P$. Prove that the orthocenter of the triangle $DEH$ lies on segment $CP$.
2022 Canadian Junior Mathematical Olympiad, 1
Let $\triangle{ABC}$ has circumcircle $\Gamma$, drop the perpendicular line from $A$ to $BC$ and meet $\Gamma$ at point $D$, similarly, altitude from $B$ to $AC$ meets $\Gamma$ at $E$. Prove that if $AB=DE, \angle{ACB}=60^{\circ}$
(sorry it is from my memory I can't remember the exact problem, but it means the same)
1956 Moscow Mathematical Olympiad, 329
Consider positive numbers $h, s_1, s_2$, and a spatial triangle $\vartriangle ABC$. How many ways are there to select a point $D$ so that the height of tetrahedron $ABCD$ drawn from $D$ equals $h$, and the areas of faces $ACD$ and $BCD$ equal $s_1$ and $s_2$, respectively?
2019 AIME Problems, 15
In acute triangle $ABC$ points $P$ and $Q$ are the feet of the perpendiculars from $C$ to $\overline{AB}$ and from $B$ to $\overline{AC}$, respectively. Line $PQ$ intersects the circumcircle of $\triangle ABC$ in two distinct points, $X$ and $Y$. Suppose $XP=10$, $PQ=25$, and $QY=15$. The value of $AB\cdot AC$ can be written in the form $m\sqrt n$ where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.