Olympiad problems

1991 All Soviet Union Mathematical Olympiad, 544

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?

1983 Tournament Of Towns, (046) 3

Tags: quadrilateral construction , sidelengths , geometry

Construct a quadrilateral given its side lengths and the length of the segment joining the midpoints of its diagonals. (IZ Titovich)

1950 Moscow Mathematical Olympiad, 176

Tags: sidelengths , geometric inequality , angle , inequalities

Let $a, b, c$ be the lengths of the sides of a triangle and $A, B, C$, the opposite angles. Prove that $$Aa + Bb + Cc \ge \frac{Ab + Ac + Ba + Bc + Ca + Cb}{2}$$

2010 Swedish Mathematical Competition, 5

Tags: sidelengths , angle , geometry

Consider the number of triangles where the side lengths $a,b,c$ satisfy $(a + b + c) (a + b -c) = 2b^2$. Determine the angles in the triangle for which the angle opposite to the side with the length $a$ is as big as possible.

1998 Belarus Team Selection Test, 3

Tags: geometry , sidelengths , angle , combinatorics

For any given triangle $A_0B_0C_0$ consider a sequence of triangles constructed as follows: a new triangle $A_1B_1C_1$ (if any) has its sides (in cm) that equal to the angles of $A_0B_0C_0$ (in radians). Then for $\vartriangle A_1B_1C_1$ consider a new triangle $A_2B_2C_2$ (if any) constructed in the similar พay, i.e., $\vartriangle A_2B_2C_2$ has its sides (in cm) that equal to the angles of $A_1B_1C_1$ (in radians), and so on. Determine for which initial triangles $A_0B_0C_0$ the sequence never terminates.

2022 German National Olympiad, 3

Tags: incircle , circumcircle , touching circles , sidelengths , geometry

Let $M$ and $N$ be the midpoints of segments $BC$ and $AC$ of a triangle $ABC$, respectively. Let $Q$ be a point on the line through $N$ parallel to $BC$ such that $Q$ and $C$ are on opposite sides of $AB$ and $\vert QN\vert \cdot \vert BC\vert=\vert AB\vert \cdot \vert AC\vert$. Suppose that the circumcircle of triangle $AQN$ intersects the segment $MN$ a second time in a point $T \ne N$. Prove that there is a circle through points $T$ and $N$ touching both the side $BC$ and the incircle of triangle $ABC$.

2011 Abels Math Contest (Norwegian MO), 2a

Tags: diagonal , geometry , perpendicular , sidelengths

In the quadrilateral $ABCD$ the side $AB$ has length $7, BC$ length $14, CD$ length $26$, and $DA$ length $23$. Show that the diagonals are perpendicular. You may assume that the quadrilateral is convex (all internal angles are less than $180^o$).

2010 Bundeswettbewerb Mathematik, 1

Tags: geometric inequality , inequalities , algebra , sidelengths

Let $a, b, c$ be the side lengths of an non-degenerate triangle with $a \le b \le c$. With $t (a, b, c)$ denote the minimum of the quotients $\frac{b}{a}$ and $\frac{c}{b}$ . Find all values that $t (a, b, c)$ can take.

2022/2023 Tournament of Towns, P2

Tags: geometry , sidelengths

Medians $BK{}$ and $CN{}$ of triangle $ABC$ intersect at $M{}.$ Consider quadrilateral $ANMK$ and find the maximum possible number of its sides having length 1. [i]Egor Bakaev[/i]

2016 Finnish National High School Mathematics Comp, 1

Tags: geometry , algebra , sidelengths

Which triangles satisfy the equation $\frac{c^2-a^2}{b}+\frac{b^2-c^2}{a}=b-a$ when $a, b$ and $c$ are sides of a triangle?

2005 Sharygin Geometry Olympiad, 12

Tags: construction , sidelengths , geometry

Construct a quadrangle along the given sides $a, b, c$, and $d$ and the distance $I$ between the midpoints of its diagonals.

2009 Oral Moscow Geometry Olympiad, 1

Tags: geometry , sidelengths , diagonal

Are there two such quadrangles that the sides of the first are less than the corresponding sides of the second, and the corresponding diagonals are larger? (Arseniy Akopyan)

2018 Finnish National High School Mathematics Comp, 2

Tags: geometry , angle bisector , sidelengths

The sides of triangle $ABC$ are $a = | BC |, b = | CA |$ and $c = | AB |$. Points $D, E$ and $F$ are the points on the sides $BC, CA$ and $AB$ such that $AD, BE$ and $CF$ are the angle bisectors of the triangle $ABC$. Determine the lengths of the segments $AD, BE$, and $CF$ in terms of $a, b$, and $c$.

2019 Durer Math Competition Finals, 2

Tags: number theory , right triangle , geometry , sidelengths , prime , integer

Prove that if a triangle has integral side lengths and its circumradius is a prime number then the triangle is right-angled.

2002 Junior Balkan Team Selection Tests - Romania, 3

Tags: geometry , area , max , midpoint , sidelengths

Let $ABC$ be a triangle and $a = BC, b = CA$ and $c = AB$ be the lengths of its sides. Points $D$ and $E$ lie in the same halfplane determined by $BC$ as $A$. Suppose that $DB = c, CE = b$ and that the area of $DECB$ is maximal. Let $F$ be the midpoint of $DE$ and let $FB = x$. Prove that $FC = x$ and $4x^3 = (a^2+b^2 + c^2)x + abc$.