Olympiad problems

1999 All-Russian Olympiad Regional Round, 8.3

On sides $BC$, $CA$, $AB$ of triangle $ABC$, points $A_1$, $B_1$, $C_1$ are chosen, respectively, so that the medians $A_1A_2$, $B_1B_2$, $C_1C_2$ of the triangle $A_1B_1C_1$ are respectively parallel to straight lines $AB$, $BC$, $CA$. Determine in what ratio points $A_1$, $B_1$, $C_1$ divide the sides of the triangle $ABC$.

2017 Brazil Team Selection Test, 3

Tags: mixtilinear incircle , median , projective geometry , 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]

2018 India PRMO, 10

Tags: perpendicular , median , geometry

In a triangle $ABC$, the median from $B$ to $CA$ is perpendicular to the median from $C$ to $AB$. If the median from $A$ to $BC$ is $30$, determine $\frac{BC^2 + CA^2 + AB^2}{100}$.

1998 Italy TST, 2

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

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 $ABC$.

2009 Oral Moscow Geometry Olympiad, 4

Tags: concurrency , diameter , median , geometry

Three circles are constructed on the medians of a triangle as on diameters. It is known that they intersect in pairs. Let $C_1$ be the intersection point of the circles built on the medians $AM_1$ and $BM_2$, which is more distant from the vertex $C$. Points $A_1$ and $B_1$ are defined similarly. Prove that the lines $AA_1, BB_1$ and $CC_1$ intersect at one point. (D. Tereshin)

2015 Belarus Team Selection Test, 2

Tags: circles , median , tangent , geometry

The medians $AM$ and $BN$ of a triangle $ABC$ are the diameters of the circles $\omega_1$ and $\omega_2$. If $\omega_1$ touches the altitude $CH$, prove that $\omega_2$ also touches $CH$. I. Gorodnin

2014 Saudi Arabia GMO TST, 1

Tags: geometry , median , concyclic , cyclic

Let $ABC$ be a triangle with $\angle A < \angle B \le \angle C$, $M$ and $N$ the midpoints of sides $CA$ and $AB$, respectively, and $P$ and $Q$ the projections of $B$ and $C$ on the medians $CN$ and $BM$, respectively. Prove that the quadrilateral $MNPQ$ is cyclic.

2018 India PRMO, 21

Tags: orthocenter , median , geometry

Let $\Delta ABC$ be an acute-angled triangle and let $H$ be its orthocentre. Let $G_1, G_2$ and $G_3$ be the centroids of the triangles $\Delta HBC , \Delta HCA$ and $\Delta HAB$ respectively. If the area of $\Delta G_1G_2G_3$ is $7$ units, what is the area of $\Delta ABC $?

2012 Czech And Slovak Olympiad IIIA, 2

Tags: geometry , median , area of a triangle , maximum

Find out the maximum possible area of the triangle $ABC$ whose medians have lengths satisfying inequalities $m_a \le 2, m_b \le 3, m_c \le 4$.

2008 Balkan MO Shortlist, G1

Tags: geometry , geometric inequality , median , exradius , excircle

In acute angled triangle $ABC$ we denote by $a,b,c$ the side lengths, by $m_a,m_b,m_c$ the median lengths and by $r_{b}c,r_{ca},r_{ab}$ the radii of the circles tangents to two sides and to circumscribed circle of the triangle, respectively. Prove that $$\frac{m_a^2}{r_{bc}}+\frac{m_b^2}{r_{ab}}+\frac{m_c^2}{r_{ab}} \ge \frac{27\sqrt3}{8}\sqrt[3]{abc}$$

2023 Sharygin Geometry Olympiad, 9.1

Tags: concyclic , median , 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.

1959 IMO Shortlist, 4

Tags: construction , geometry , median , trigonometry

Construct a right triangle with given hypotenuse $c$ such that the median drawn to the hypotenuse is the geometric mean of the two legs of the triangle.

2019 Ramnicean Hope, 2

Tags: geometry , circumcircle , vector geometry , median

Let $ P,Q,R $ be the intersections of the medians $ AD,BE,CF $ of a triangle $ ABC $ with its circumcircle, respectively. Show that $ ABC $ is equilateral if $ \overrightarrow{DP} +\overrightarrow{EQ} +\overrightarrow{FR} =0. $ [i]Dragoș Lăzărescu[/i]

2017 QEDMO 15th, 8

Tags: geometry , median , 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.

2011 Sharygin Geometry Olympiad, 14

Tags: geometry , altitude , median , ratio

In triangle $ABC$, the altitude and the median from vertex $A$ form (together with line $BC$) a triangle such that the bisectrix of angle $A$ is the median; the altitude and the median from vertex $B$ form (together with line AC) a triangle such that the bisectrix of angle $B$ is the bisectrix. Find the ratio of sides for triangle $ABC$.

1967 All Soviet Union Mathematical Olympiad, 084

Tags: altitude , median , equilateral , geometry

a) The maximal height $|AH|$ of the acute-angled triangle $ABC$ equals the median $|BM|$. Prove that the angle $ABC$ isn't greater than $60$ degrees. b) The height $|AH|$ of the acute-angled triangle $ABC$ equals the median $|BM|$ and bisectrix $|CD|$. Prove that the angle $ABC$ is equilateral.

2007 Sharygin Geometry Olympiad, 5

Tags: centroid , acute , median , geometry

Medians $AA'$ and $BB'$ of triangle $ABC$ meet at point $M$, and $\angle AMB = 120^o$. Prove that angles $AB'M$ and $BA'M$ are neither both acute nor both obtuse.

2017 Morocco TST-, 3

Tags: mixtilinear incircle , median , projective geometry , 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]

Kyiv City MO Juniors Round2 2010+ geometry, 2014.7.4

Tags: angle , ratio , median , geometry

The median $BM$ is drawn in the triangle $ABC$. It is known that $\angle ABM = 40 {} ^ \circ$ and $\angle CBM = 70 {} ^ \circ $ Find the ratio $AB: BM$.

2005 Switzerland - Final Round, 1

Tags: geometry , equilateral , centroid , concyclic , median , cyclic quadrilateral

Let $ABC$ be any triangle and $D, E, F$ the midpoints of $BC, CA, AB$. The medians $AD, BE$ and $CF$ intersect at point $S$. At least two of the quadrilaterals $AF SE, BDSF, CESD$ are cyclic. Show that the triangle $ABC$ is equilateral.

2019 Romanian Master of Mathematics Shortlist, G1

Tags: circumcenter , median , circumcircle , geometry

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

1999 Czech And Slovak Olympiad IIIA, 3

Tags: ratio , geometry , sum , median

Show that there exists a triangle $ABC$ such that $a \ne b$ and $a+t_a = b+t_b$, where $t_a,t_b$ are the medians corresponding to $a,b$, respectively. Also prove that there exists a number $k$ such that every such triangle satisfies $a+t_a = b+t_b = k(a+b)$. Finally, find all possible ratios $a : b$ in such triangles.

1985 Tournament Of Towns, (080) T1

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

A median , a bisector and an altitude of a certain triangle intersect at an inner point $O$ . The segment of the bisector from the vertex to $O$ is equal to the segment of the altitude from the vertex to $O$ . Prove that the triangle is equilateral .

2022 Yasinsky Geometry Olympiad, 1

Tags: geometry , angle , median

In the triangle $ABC$, the median $AM$ is extended to the intersection with the circumscribed circle at point $D$. It is known that $AB = 2AM$ and $AD = 4AM$. Find the angles of the triangle $ABC$. (Gryhoriy Filippovskyi)

2018 Hanoi Open Mathematics Competitions, 5

Tags: median , altitude , angle bisector , geometry

Let $ABC$ be an acute triangle with $AB = 3$ and $AC = 4$. Suppose that $AH,AO$ and $AM$ are the altitude, the bisector and the median derived from $A$, respectively. If $HO = 3 MO$, then the length of $BC$ is [img]https://cdn.artofproblemsolving.com/attachments/e/c/26cc00629f4c0ab27096b8bdc562c56ff01ce5.png[/img] A. $3$ B. $\frac72$ C. $4$ D. $\frac92$ E. $5$