Olympiad problems

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]

1999 Estonia National Olympiad, 4

Tags: altitude , vector , median , geometry

For the given triangle $ABC$, prove that a point $X$ on the side $AB$ satisfies the condition $\overrightarrow{XA} \cdot\overrightarrow{XB} +\overrightarrow{XC} \cdot \overrightarrow{XC} = \overrightarrow{CA} \cdot \overrightarrow{CB} $, iff $X$ is the basepoint of the altitude or median of the triangle $ABC$.

2008 Czech and Slovak Olympiad III A, 3

Tags: geometric inequality , triangle , median , inequalities

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.

2006 Sharygin Geometry Olympiad, 12

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

In the triangle $ABC$, the bisector of angle $A$ is equal to the half-sum of the height and median drawn from vertex $A$. Prove that if $\angle A$ is obtuse, then $AB = AC$.

2019 India PRMO, 29

Tags: median , angle bisector , geometry

In a triangle $ABC$, the median $AD$ (with $D$ on $BC$) and the angle bisector $BE$ (with $E$ on $AC$) are perpedicular to each other. If $AD = 7$ and $BE = 9$, find the integer nearest to the area of triangle $ABC$.

2012 Czech And Slovak Olympiad IIIA, 2

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

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

2019 BAMO, E/3

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

In triangle $\vartriangle ABC$, we have marked points $A_1$ on side $BC, B_1$ on side $AC$, and $C_1$ on side $AB$ so that $AA_1$ is an altitude, $BB_1$ is a median, and $CC_1$ is an angle bisector. It is known that $\vartriangle A_1B_1C_1$ is equilateral. Prove that $\vartriangle ABC$ is equilateral too. (Note: A median connects a vertex of a triangle with the midpoint of the opposite side. Thus, for median $BB_1$ we know that $B_1$ is the midpoint of side $AC$ in $\vartriangle ABC$.)

1969 IMO Longlists, 10

Tags: trigonometry , geometry , trigonometric equation , 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 $\frac{\sin(\phi+\Psi)}{\sin(\phi-\Psi)}= 5.$

2013 Sharygin Geometry Olympiad, 5

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

The altitude $AA'$, the median $BB'$, and the angle bisector $CC'$ of a triangle $ABC$ are concurrent at point $K$. Given that $A'K = B'K$, prove that $C'K = A'K$.

1994 Tournament Of Towns, (437) 3

Tags: geometry , median , angle

The median $AD$ of triangle $ABC$ intersects its inscribed circle (with center $O$) at the points $X$ and $Y$. Find the angle $XOY$ if $AC = AB + AD$. (A Fedotov)

2024 Yasinsky Geometry Olympiad, 5

Tags: geometry , angle bisector , median , parallelogram

Let $ AL $ be the bisector of triangle $ ABC $, $ O $ the center of its circumcircle, and $ D $ and $ E $ the midpoints of $ BL $ and $ CL $, respectively. Points $ P $ and $ Q $ are chosen on segments $ AD $ and $ AE $ such that $ APLQ $ is a parallelogram. Prove that $ PQ \perp AO $. [i]Proposed by Mykhailo Plotnikov[/i]

2022 Yasinsky Geometry Olympiad, 1

Tags: geometry , median , angle

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)

1959 IMO, 4

Tags: geometry , trigonometry , construction , median

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.

1967 All Soviet Union Mathematical Olympiad, 084

Tags: geometry , altitude , median , equilateral

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.

2015 Romania National Olympiad, 3

Tags: geometry , median , area

Let be a point $ P $ in the interior of a triangle $ ABC. $ The lines $ AP,BP,CP $ meet $ BC,AC, $ respectively, $ AB $ at $ A_1,B_1, $ respectively, $ C_1. $ If $$ \mathcal{A}_{PBA_1} +\mathcal{A}_{PCB_1} +\mathcal{A}_{PAC_1} =\frac{1}{2}\mathcal{A}_{ABC} , $$ show that $ P $ lies on a median of $ ABC. $ $ \mathcal{A} $ [i]denotes area.[/i]

1992 Denmark MO - Mohr Contest, 4

Tags: median , geometry , geometric inequality

Let $a, b$ and $c$ denote the side lengths and $m_a, m_b$ and $m_c$ of the median's lengths in an arbitrary triangle. Show that $$\frac34 < \frac{m_a + m_b + m_c}{a + b + c}<1$$ Also show that there is no narrower range that for each triangle that contains the fraction $$\frac{m_a + m_b + m_c}{a + b + c}$$

1999 All-Russian Olympiad Regional Round, 8.3

Tags: median , geometry , inscribed , ratio

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 $ABC$.

1991 All Soviet Union Mathematical Olympiad, 544

Tags: median , sidelengths , geometry

Does there exist a triangle in which two sides are integer multiples of the median to that side? Does there exist a triangle in which every side is an integer multiple of the median to that side?

2020 AMC 10, 11

Tags: median

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

Kyiv City MO Juniors Round2 2010+ geometry, 2014.7.4

Tags: ratio , median , geometry , angle

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 $AB: BM$.

2004 All-Russian Olympiad Regional Round, 8.3

Tags: geometry , equilateral , median

In an acute triangle, the distance from the midpoint of any side to the opposite vertex is equal to the sum of the distances from it to sides of the triangle. Prove that this triangle is equilateral.

1970 All Soviet Union Mathematical Olympiad, 135

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

The angle bisector $[AD]$, the median $[BM]$ and the height $[CH]$ of the acute-angled triangle $ABC$ intersect in one point. Prove that the $\angle BAC> 45^o$.

1967 IMO Shortlist, 4

Tags: geometry , geometric inequality , construction , median , triangle

Suppose medians $m_a$ and $m_b$ of a triangle are orthogonal. Prove that: a.) Using medians of that triangle it is possible to construct a rectangular triangle. b.) The following inequality: \[5(a^2+b^2-c^2) \geq 8ab,\] is valid, where $a,b$ and $c$ are side length of the given triangle.

2017 Canada National Olympiad, 3

Tags: combinatorics , average , median , set

Define $S_n$ as the set ${1,2,\cdots,n}$. A non-empty subset $T_n$ of $S_n$ is called $balanced$ if the average of the elements of $T_n$ is equal to the median of $T_n$. Prove that, for all $n$, the number of balanced subsets $T_n$ is odd.

2017 Yasinsky Geometry Olympiad, 6

Tags: geometry , angle bisector , ratio , median

In the triangle $ABC$ , the angle bisector $AD$ divides the side $BC$ into the ratio $BD: DC = 2: 1$. In what ratio, does the median $CE$ divide this bisector?