Olympiad problems

2013 Romania National Olympiad, 1

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 $DM = BF$

2024 Yasinsky Geometry Olympiad, 1

Tags: geometry , median

Let $BE$ and $CF$ be the medians of an acute triangle $ABC.$ On the line $BC,$ points $K \ne B$ and $L \ne C$ are chosen such that $BE = EK$ and $CF = FL.$ Prove that $AK = AL.$ [i]Proposed by Heorhii Zhilinskyi[/i]

2016 IMO Shortlist, G2

Tags: projective geometry , mixtilinear incircle , median , geometry

Let $ABC$ be a triangle with circumcircle $\Gamma$ and incenter $I$ and let $M$ be the midpoint of $\overline{BC}$. The points $D$, $E$, $F$ are selected on sides $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ such that $\overline{ID} \perp \overline{BC}$, $\overline{IE}\perp \overline{AI}$, and $\overline{IF}\perp \overline{AI}$. Suppose that the circumcircle of $\triangle AEF$ intersects $\Gamma$ at a point $X$ other than $A$. Prove that lines $XD$ and $AM$ meet on $\Gamma$. [i]Proposed by Evan Chen, Taiwan[/i]

1994 Swedish Mathematical Competition, 2

Tags: geometry , geometric inequalities , median

In the triangle $ABC$, the medians from $B$ and $C$ are perpendicular. Show that $\cot B + \cot C \ge \frac23$.

2008 Czech and Slovak Olympiad III A, 3

Tags: geometric inequality , inequalities , triangle , median

Find the greatest value of $p$ and the smallest value of $q$ such that for any triangle in the plane, the inequality \[p<\frac{a+m}{b+n}<q\] holds, where $a,b$ are it's two sides and $m,n$ their corresponding medians.

2011 Sharygin Geometry Olympiad, 17

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

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?

2004 All-Russian Olympiad Regional Round, 9.2

Tags: geometry , median , 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.

1968 All Soviet Union Mathematical Olympiad, 106

Tags: incircle , geometry , median

Medians divide the triangle onto $6$ smaller ones. $4$ of the circles inscribed in those small ones are equal. Prove that the triangle is equilateral.