Olympiad problems

2017 Yasinsky Geometry Olympiad, 4

Median $AM$ and the angle bisector $CD$ of a right triangle $ABC$ ($\angle B=90^o$) intersect at the point $O$. Find the area of the triangle $ABC$ if $CO=9, OD=5$.

Kyiv City MO Juniors 2003+ geometry, 2017.9.51

Tags: median , geometry , tangential quadrilateral , isosceles

In the triangle $ABC$, the medians $BB_1$ and $CC_1$, which intersect at the point $M$, are drawn. Prove that a circle can be inscribed in the quadrilateral $AC_1MB_1$ if and only if $AB = AC$.

Kyiv City MO Juniors Round2 2010+ geometry, 2012.7.3

Tags: geometry , median , equal segments

In the triangle $ABC $ the median $BD$ is drawn, which is divided into three equal parts by the points $E $ and $F$ ($BE = EF = FD$). It is known that $AD = AF$ and $AB = 1$. Find the length of the segment $CE$.

2009 Belarus Team Selection Test, 1

Tags: geometry , circumcenter , angle bisector , median

In a triangle $ABC, AM$ is a median, $BK$ is a bisectrix, $L=AM\cap BK$. It is known that $BC=a, AB=c, a>c$. Given that the circumcenter of triangle $ABC$ lies on the line $CL$, find $AC$ I. Voronovich

2020 AMC 12/AHSME, 8

Tags: median

What is the median of the following list of $4040$ numbers$?$ $$1, 2, 3, ..., 2020, 1^2, 2^2, 3^2, ..., 2020^2$$ $\textbf{(A) } 1974.5 \qquad \textbf{(B) } 1975.5 \qquad \textbf{(C) } 1976.5 \qquad \textbf{(D) } 1977.5 \qquad \textbf{(E) } 1978.5$

2013 Romania National Olympiad, 1

Tags: geometry , angle bisector , median , parallel

In the triangle $ABC$, the angle - bisector $AD$ ($D \in BC$) and the median $BE$ ($E \in AC$) intersect at point $P$. Lines $AB$ and $CP$ intesect at point $F$. The parallel through $B$ to $CF$ intersects $DF$ at point $M$. Prove that $DM = BF$

1995 All-Russian Olympiad Regional Round, 10.3

Tags: tangent , geometry , median

In an acute-angled triangle $ABC$, the circle $S$ with the altitude $BK$ as the diameter intersects $AB$ at $E$ and $BC$ at $F$. Prove that the tangents to $S$ at $E$ and $F$ meet on the median from $B$.

2019 Romanian Master of Mathematics Shortlist, G1

Tags: geometry , circumcircle , circumcenter , median

Let $BM$ be a median in an acute-angled triangle $ABC$. A point $K$ is chosen on the line through $C$ tangent to the circumcircle of $\vartriangle BMC$ so that $\angle KBC = 90^\circ$. The segments $AK$ and $BM$ meet at $J$. Prove that the circumcenter of $\triangle BJK$ lies on the line $AC$. Aleksandr Kuznetsov, Russia

2013 Oral Moscow Geometry Olympiad, 1

Tags: geometry , angle bisector , perpendicular , median

In triangle $ABC$ the angle bisector $AK$ is perpendicular on the median is $CL$. Prove that in the triangle $BKL$ also one of angle bisectors are perpendicular to one of the medians.

2006 Thailand Mathematical Olympiad, 3

Tags: median , geometry

The three medians of a triangle has lengths $3, 4, 5$. What is the length of the shortest side of this triangle?

2015 European Mathematical Cup, 3

Tags: circles , median , geometry

Circles $k_1$ and $k_2$ intersect in points $A$ and $B$, such that $k_1$ passes through the center $O$ of the circle $k_2$. The line $p$ intersects $k_1$ in points $K$ and $O$ and $k_2$ in points $L$ and $M$, such that the point $L$ is between $K$ and $O$. The point $P$ is orthogonal projection of the point $L$ to the line $AB$. Prove that the line $KP$ is parallel to the $M-$median of the triangle $ABM$. [i]Matko Ljulj[/i]

1969 IMO Shortlist, 10

Tags: trigonometric equation , trigonometry , geometry , median

$(BUL 4)$ Let $M$ be the point inside the right-angled triangle $ABC (\angle C = 90^{\circ})$ such that $\angle MAB = \angle MBC = \angle MCA =\phi.$ Let $\Psi$ be the acute angle between the medians of $AC$ and $BC.$ Prove that $\frac{\sin(\phi+\Psi)}{\sin(\phi-\Psi)}= 5.$

1989 Tournament Of Towns, (204) 2

Tags: geometry , inradius , median , incircle

In the triangle $ABC$ the median $AM$ is drawn. Is it possible that the radius of the circle inscribed in $\vartriangle ABM$ could be twice as large as the radius of the circle inscribed in $\vartriangle ACM$ ? ( D . Fomin , Leningrad)

2017 QEDMO 15th, 8

Tags: median , geometry , triangle inequality

Let $ABC$ be a triangle of area $1$ with medians $s_a, s_b,s_c$. Show that there is a triangle whose sides are the same length as $s_a, s_b$, and $s_c$, and determine its area.

1959 Czech and Slovak Olympiad III A, 1

Tags: median , construction , triangle

Construct a triangle $ABC$ with the right angle at vertex $C$ given lengths of its medians $m_a$, $m_b$. Discuss conditions of solvability.

2000 Belarus Team Selection Test, 5.1

Tags: median , geometric inequalities , inequalities , angle bisector

Let $AM$ and $AL$ be the median and bisector of a triangle $ABC$ ($M,L \in BC$). If $BC = a, AM = m_a, AL = l_a$, prove the inequalities: (a) $a\tan \frac{a}{2} \le 2m_a \le a \cot \frac{a}{2} $ if $a < \frac{\pi}{2}$ and $a\tan \frac{a}{2} \ge 2m_a \ge a \cot \frac{a}{2} $ if $a > \frac{\pi}{2}$ (b) $2l_a \le a\cot \frac{a}{2} $.

1996 Israel National Olympiad, 3

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

The angles of an acute triangle $ABC$ at $\alpha , \beta, \gamma$. Let $AD$ be a height, $CF$ a median, and $BE$ the bisector of $\angle B$. Show that $AD,CF$ and $BE$ are concurrent if and only if $\cos \gamma \tan\beta = \sin \alpha$ .

2004 All-Russian Olympiad Regional Round, 9.2

Tags: median , geometry , isosceles

In triangle $ABC$, medians $AA'$, $BB'$, $CC'$ are extended until they intersect with the circumcircle at points $A_0$, $B_0$, $C_0$, respectively. It is known that the intersection point M of the medians of triangle $ABC$ divides the segment $AA_0$ in half. Prove that the triangle $A_0B_0C_0$ is isosceles.

2011 Sharygin Geometry Olympiad, 17

Tags: geometry , median , bisector , compare , geometric inequality

a) Does there exist a triangle in which the shortest median is longer that the longest bisectrix? b) Does there exist a triangle in which the shortest bisectrix is longer that the longest altitude?

2000 Belarus Team Selection Test, 3.1

Tags: median , geometry

In a triangle $ABC$, let $a = BC, b = AC$ and let $m_a,m_b$ be the corresponding medians. Find all real numbers $k$ for which the equality $m_a+ka = m_b +kb$ implies that $a = b$.

2023 Sharygin Geometry Olympiad, 9.1

Tags: median , concyclic , geometry

The ratio of the median $AM$ of a triangle $ABC$ to the side $BC$ equals $\sqrt{3}:2$. The points on the sides of $ABC$ dividing these side into $3$ equal parts are marked. Prove that some $4$ of these $6$ points are concyclic.

2000 Argentina National Olympiad, 2

Tags: geometry , angle bisector , median , parallel , perpendicular

Given a triangle $ABC$ with side $AB$ greater than $BC$, let $M$ be the midpoint of $AC$ and $L$ be the point at which the bisector of angle $\angle B$ intersects side $AC$. The line parallel to $AB$, which intersects the bisector $BL$ at $D$, is drawn by $M$, and the line parallel to the side $BC$ that intersects the median $BM$ at $E$ is drawn by $L$. Show that $ED$ is perpendicular to $BL$.

1996 Bundeswettbewerb Mathematik, 3

Tags: combinatorial geometry , geometry , median

Four lines are given in a plane so that any three of them determine a triangle. One of these lines is parallel to a median in the triangle determined by the other three lines. Prove that each of the other three lines also has this property.

1998 Italy TST, 2

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

In a triangle $ABC$, points $H,M,L$ are the feet of the altitude from $C$, the median from $A$, and the angle bisector from $B$, respectively. Show that if triangle $HML$ is equilateral, then so is triangle $ABC$.

2006 Sharygin Geometry Olympiad, 10.4

Tags: median , circumcircle , concurrency , geometry

Lines containing the medians of the triangle $ABC$ intersect its circumscribed circle for a second time at the points $A_1, B_1, C_1$. The straight lines passing through $A,B,C$ parallel to opposite sides intersect it at points $A_2, B_2, C_2$. Prove that lines $A_1A_2,B_1B_2,C_1C_2$ intersect at one point.