Olympiad problems

1997 Tournament Of Towns, (557) 2

Let $a$ and $b$ be two sides of a triangle. How should the third side $c$ be chosen so that the points of contact of the incircle and the excircle with side $c$ divide that side into three equal segments? (The excircle corresponding to the side $c$ is the circle which is tangent to the side $c$ and to the extensions of the sides $a$ and $b$.) (Folklore)

1957 Polish MO Finals, 5

Tags: trisection , construction , geometry

Given a line $ m $ and a segment $ AB $ parallel to it. Divide the segment $ AB $ into three equal parts using only a ruler, i.e. drawing only the lines.

2013 Cuba MO, 2

Tags: geometry , trisection

Two equal isosceles triangles $ABC$ and $ADB$, with $C$ and $D$ located in different halfplanes with respect to the line $AB$, share the base $AB$. The midpoints of $AC$ and $BC$ are denoted by $E$ and $F$ respectively. Show that $DE$ and $DF$ divide $AB$ into three equal parts length.

Durer Math Competition CD Finals - geometry, 2008.C1

Tags: parallelogram , trisection , geometry , isosceles

Given the parallelogram $ABCD$. The trisection points of side $AB$ are: $H_1, H_2$, ($AH_1 = H_1H_2 =H_2B$). The trisection points of the side $DC$ are $G_1, G_2$, ($DG_1 = G_1G_2 = G_2C$), and $AD = 1, AC = 2$. Prove that triangle $AH_2G_1$ is isosceles.

2010 Flanders Math Olympiad, 2

Tags: parallelogram , trisection , ratio , geometry

A parallelogram with an angle of $60^o$ has $a$ as the longest side and a shortest side $b$. Let's take the perpendiculars down from the vertices of the obtuse angles to the longest diagonal, then it is divided into three equal parts. Determine the ratio $\frac{a}{b}$.

1950 Poland - Second Round, 5

Tags: geometry , construction , trisection , concentric

Given two concentric circles and a point $A$. Through point $A$, draw a secant such that its segment contained by the larger circle is divided by the smaller circle into three equal parts.

Durer Math Competition CD Finals - geometry, 2018.C3

Tags: perpendicular , trisection , geometry

Points $A, B, C, D$ are located in the plane as follows: sections $AB$ and $CD$ are perpendicular to each other and are of equal length, moreover, D is just the trisection point of segment $AB$ closer to $A$. The perpendicular from point $D$ on segment $BC$ intersects it at $E$. Let the trisection point of segment $DE$ closer to $E$ be $H$. Prove that segments $CH$ and the sections $AE$ are perpendicular to each other.

Ukrainian From Tasks to Tasks - geometry, 2013.9

Tags: geometry , rhombus , trisection

The perpendicular bisectors of the sides $AB$ and $CD$ of the rhombus $ABCD$ are drawn. It turned out that they divided the diagonal $AC$ into three equal parts. Find the altitude of the rhombus if $AB = 1$.

2004 Paraguay Mathematical Olympiad, 4

Tags: geometry , square , trisection

In a square $ABCD$, $E$ is the midpoint of $BC$ and $F$ is the midpoint of $CD$. Prove that $AF$ and $AE$ divide the diagonal $BD$ in three equal segments.