Olympiad problems

2006 Thailand Mathematical Olympiad, 3

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

median geometry

2012 Czech And Slovak Olympiad IIIA, 2

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

median geometry area of a triangle maximum inequalities

2016 Sharygin Geometry Olympiad, 1

An altitude $AH$ of triangle $ABC$ bisects a median $BM$. Prove that the medians of triangle $ABM$ are sidelengths of a right-angled triangle. by Yu.Blinkov

right triangle median geometry altitude law of cosines right triangles

2016 All-Russian Olympiad, 8

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]

circumcircle median geometry angle chasing barycentric coordinates cyclic quadrilaterals homothety modular arithmetic power of a point symmedians

1959 Poland - Second Round, 2

What relationship between the sides of a triangle makes it similar to the triangle formed by its medians?

median geometry similar

2016 AMC 10, 7

The mean, median, and mode of the $7$ data values $60, 100, x, 40, 50, 200, 90$ are all equal to $x$. What is the value of $x$? $\textbf{(A)}\ 50 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 75 \qquad\textbf{(D)}\ 90 \qquad\textbf{(E)}\ 100$

median arithmetic mean mode algebra

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 $DM = BF$

median geometry angle bisector parallel

1995 All-Russian Olympiad Regional Round, 10.3

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 $B$.

tangent median geometry