Olympiad problems

2012 Austria Beginners' Competition, 4

A segment $AB$ is given. We erect the equilateral triangles $ABC$ and $ADB$ above and below $AB$. We denote the midpoints of $AC$ and $BC$ by $E$ and $F$ respectively. Prove that the straight lines $DE$ and $DF$ divide the segment $AB$ into three parts of equal length .

2015 Costa Rica - Final Round, G3

Tags: trisector , trisect , geometry , parallel

Let $\vartriangle A_1B_1C_1$ and $l_1, m_1, n_1$ be the trisectors closest to $A_1B_1$, $B_1C_1$, $C_1A_1$ of the angles $A_1, B_1, C_1$ respectively. Let $A_2 = l_1 \cap n_1$, $B_2 = m_1 \cap l_1$, $C_2 = n_1 \cap m_1$. So on we create triangles $\vartriangle A_nB_nC_n$ . If $\vartriangle A_1B_1C_1$ is equilateral prove that exists $n \in N$, such that all the sides of $\vartriangle A_nB_nC_n$ are parallel to the sides of $\vartriangle A_1B_1C_1$.

2022 Yasinsky Geometry Olympiad, 1

Tags: construction , trisect , geometry

An angle whose degree measure is equal to $108^o$ is given . Describe how with help compass and ruler can divide this angle into three equal parts. (Yukhim Rabinovych)

1966 Dutch Mathematical Olympiad, 1

Tags: geometry , trisector , trisect

A chord $AB$ is drawn in a circle, with center $M$ and radius $r$, that the two diameters which divide the largest arc $AB$ into three equal parts also divide the chord $AB$ into three equal parts. Express the length of that chord in terms of $r$.

1974 Spain Mathematical Olympiad, 6

Tags: construction , trisect , chord , geometry

Two chords are drawn in a circle of radius equal to unit, $AB$ and $AC$ of equal length. a) Describe how you can construct a third chord $DE$ that is divided into three equal parts by the intersections with $AB$ and $AC$. b) If $AB = AC =\sqrt2$, what are the lengths of the two segments that the chord $DE$ determines in $AB$?

2014 Belarus Team Selection Test, 3

Tags: equal segments , geometry , trisect , incircle

Point $L$ is marked on the side $AB$ of a triangle $ABC$. The incircle of the triangle $ABC$ meets the segment $CL$ at points $P$ and $Q$ .Is it possible that the equalities $CP = PQ = QL$ hold if $CL$ is a) the median? b) the bisector? c) the altitude? d) the segment joining vertex $C$ with the point $L$ of tangency of the excircle of the triangie $ABC$ with $AB$ ? (I. Gorodnin)