Olympiad problems

2010 Gheorghe Vranceanu, 1

Let $ A_1,B_1,C_1 $ be the middlepoints of the sides of a triangle $ ABC $ and let $ A_2,B_2,C_2 $ be on the middle of the paths $ CAB,ABC,BCA, $ respectively. Prove that $ A_1A_2,B_1B_2,C_1C_2 $ are concurrent.

2004 Spain Mathematical Olympiad, Problem 5

Tags: geometry , median , inscribed circle

Demonstrate that the condition necessary so that, in triangle ${ABC}$, the median from ${B}$ is divided into three equal parts by the inscribed circumference of a circle is: ${A/5 = B/10 = C/13}$.

Swiss NMO - geometry, 2005.1

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

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.

2018 Hanoi Open Mathematics Competitions, 5

Tags: geometry , median , angle bisector , altitude

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. $5$

2005 BAMO, 2

Tags: geometry , equal , median , isosceles

Prove that if two medians in a triangle are equal in length, then the triangle is isosceles. (Note: A median in a triangle is a segment which connects a vertex of the triangle to the midpoint of the opposite side of the triangle.)

2009 Sharygin Geometry Olympiad, 4

Tags: geometry , angle bisector , median

Given is $\triangle ABC$ such that $\angle A = 57^o, \angle B = 61^o$ and $\angle C = 62^o$. Which segment is longer: the angle bisector through $A$ or the median through $B$? (N.Beluhov)

2015 Junior Regional Olympiad - FBH, 3

Tags: geometry , median

Let $AD$ be a median of $ABC$ and $S$ its midpoint. Let $E$ be a intersection point of $AB$ and $CS$. Prove that $BE=2AE$

2014 Austria Beginners' Competition, 4

Tags: median , geometry , perpendicular

Consider a triangle $ABC$. The midpoints of the sides $BC, CA$, and $AB$ are denoted by $D, E$, and $F$, respectively. Assume that the median $AD$ is perpendicular to the median $BE$ and that their lengths are given by $AD = 18$ and $BE = 13.5$. Compute the length of the third median $CF$. (K. Czakler, Vienna)

2011 Sharygin Geometry Olympiad, 19

Tags: geometry , median , altitude , bisector , equal segments

Does there exist a nonisosceles triangle such that the altitude from one vertex, the bisectrix from the second one and the median from the third one are equal?

1985 Tournament Of Towns, (080) T1

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

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 .

2005 Sharygin Geometry Olympiad, 9.5

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

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 $135^o$

2012 Oral Moscow Geometry Olympiad, 6

Tags: geometry , construction , orthocenter , median , angle bisector

Restore the triangle with a compass and a ruler given the intersection point of altitudes and the feet of the median and angle bisectors drawn to one side. (No research required.)

2006 BAMO, 3

Tags: centroid , median , geometry

In triangle $ABC$, choose point $A_1$ on side $BC$, point $B_1$ on side $CA$, and point $C_1$ on side $AB$ in such a way that the three segments $AA_1, BB_1$, and $CC_1$ intersect in one point $P$. Prove that $P$ is the centroid of triangle $ABC$ if and only if $P$ is the centroid of triangle $A_1B_1C_1$. Note: A median in a triangle is a segment connecting a vertex of the triangle with the midpoint of the opposite side. The centroid of a triangle is the intersection point of the three medians of the triangle. The centroid of a triangle is also known by the names ”center of mass” and ”medicenter” of the triangle.

2014 Saudi Arabia GMO TST, 1

Tags: median , concyclic , geometry , 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, 13

Tags: geometry , right triangle , median

In a triangle $ABC$, right angled at $A$, the altitude through $A$ and the internal bisector of $\angle A$ have lengths $3$ and $4$, respectively. Find the length of the median through $A$.

2017 Morocco TST-, 3

Tags: mixtilinear incircle , projective geometry , 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]

2006 Junior Tuymaada Olympiad, 7

Tags: geometry , circumcircle , median , geometric inequality

The median $ BM $ of a triangle $ ABC $ intersects the circumscribed circle at point $ K $. The circumcircle of the triangle $ KMC $ intersects the segment $ BC $ at point $ P $, and the circumcircle of $ AMK $ intersects the extension of $ BA $ at $ Q $. Prove that $ PQ> AC $.

2016 All-Russian Olympiad, 8

Tags: circumcircle , median , geometry

Medians $AM_A,BM_B,CM_C$ of triangle $ABC$ intersect at $M$.Let $\Omega_A$ be circumcircle of triangle passes through midpoint of $AM$ and tangent to $BC$ at $M_A$.Define $\Omega_B$ and $\Omega_C$ analogusly.Prove that $\Omega_A,\Omega_B$ and $\Omega_C$ intersect at one point.(A.Yakubov) [hide=P.S]sorry for my mistake in translation :blush: :whistling: .thank you jred for your help :coolspeak: [/hide]

Estonia Open Senior - geometry, 2012.1.3

Tags: right triangle , median , circumcenter , geometry

Let $ABC$ be a triangle with median AK. Let $O$ be the circumcenter of the triangle $ABK$. a) Prove that if $O$ lies on a midline of the triangle $ABC$, but does not coincide with its endpoints, then $ABC$ is a right triangle. b) Is the statement still true if $O$ can coincide with an endpoint of the midsegment?

2015 Sharygin Geometry Olympiad, 6

Tags: perpendicular bisector , perpendicular , median , geometry

Lines $b$ and $c$ passing through vertices $B$ and $C$ of triangle $ABC$ are perpendicular to sideline $BC$. The perpendicular bisectors to $AC$ and $AB$ meet $b$ and $c$ at points $P$ and $Q$ respectively. Prove that line $PQ$ is perpendicular to median $AM$ of triangle $ABC$. (D. Prokopenko)

2020-21 IOQM India, 10

Tags: statistics , median , mean , algebra

Five students take a test on which any integer score from $0$ to $100$ inclusive is possible. What is the largest possible difference between the median and the mean of the scores? [i](The median of a set of scores is the middlemost score when the data is arranged in increasing order. It is exactly the middle score when there are an odd number of scores and it is the average of the two middle scores when there are an even number of scores.)[/i]

1995 Denmark MO - Mohr Contest, 3

Tags: ratio , geometry , median

From the vertex $C$ in triangle $ABC$, draw a straight line that bisects the median from $A$. In what ratio does this line divide the segment $AB$? [img]https://1.bp.blogspot.com/-SxWIQ12DIvs/XzcJv5xoV0I/AAAAAAAAMY4/Ezfe8bd7W-Mfp2Qi4qE_gppbh9Fzvb4XwCLcBGAsYHQ/s0/1995%2BMohr%2Bp3.png[/img]

1983 Tournament Of Towns, (040) O2

Tags: vector , median , midpoint , geometry

On sides $AB, BC$ and $CA$ of triangle $ABC$ are located points $P, M$ and $K$, respectively, so that $AM, BK$ and $CP$ intersect in one point and the sum of the vectors $\overrightarrow{AM}, \overrightarrow{BK}$ and $\overrightarrow{CP}$ equals $ \overrightarrow{0}$. Prove that $K, M$ and $P$ are midpoints of the sides of triangle $ABC$ on which they are located.

Kyiv City MO Juniors 2003+ geometry, 2014.851

Tags: geometry , ratio , median , equal segments

On the side $AB$ of the triangle $ABC$ mark the point $K$. The segment $CK$ intersects the median $AM$ at the point $F$. It is known that $AK = AF$. Find the ratio $MF: BK$.

2019 Sharygin Geometry Olympiad, 6

Tags: angle bisector , median , circumcircle , angle chasing , geometry

Let $AK$ and $AT$ be the bisector and the median of an acute-angled triangle $ABC$ with $AC > AB$. The line $AT$ meets the circumcircle of $ABC$ at point $D$. Point $F$ is the reflection of $K$ about $T$. If the angles of $ABC$ are known, find the value of angle $FDA$.