Found problems: 33
2010 IMAC Arhimede, 4
Let $M$ and $N$ be two points on different sides of the square $ABCD$. Suppose that segment $MN$ divides the square into two tangential polygons. If $R$ and $r$ are radii of the circles inscribed in these polygons ($R> r$), calculate the length of the segment $MN$ in terms of $R$ and $r$.
(Moldova)
2022 Belarusian National Olympiad, 10.2
A positive integer $n$ is given. On the segment $[0,n]$ of the real line $m$ distinct segments whose endpoints have integer coordinates are chosen. It turned out that it is impossible to choose some of thos segments such that their total length is $n$ and their union is $[0,n]$
Find the maximum possible value of $m$
2006 Sharygin Geometry Olympiad, 8
The segment $AB$ divides the square into two parts, in each of which a circle can be inscribed.
The radii of these circles are equal to $r_1$ and $r_2$ respectively, where $r_1> r_2$.
Find the length of $AB$.
2003 Austrian-Polish Competition, 3
$ABC$ is a triangle. Take $a = BC$ etc as usual.
Take points $T_1, T_2$ on the side $AB$ so that $AT_1 = T_1T_2 = T_2B$. Similarly, take points $T_3, T_4$ on the side BC so that $BT_3 = T_3T_4 = T_4C$, and points $T_5, T_6$ on the side $CA$ so that $CT_5 = T_5T_6 = T_6A$.
Show that if $a' = BT_5, b' = CT_1, c'=AT_3$, then there is a triangle $A'B'C'$ with sides $a', b', c'$ ($a' = B'C$' etc).
In the same way we take points $T_i'$ on the sides of $A'B'C' $ and put $a'' = B'T_6', b'' = C'T_2', c'' = A'T_4'$.
Show that there is a triangle $A'' B'' C'' $ with sides $a'' b'' , c''$ and that it is similar to $ABC$.
Find $a'' /a$.
1984 Tournament Of Towns, (072) 3
On a plane there is a finite set of $M$ points, no three of which are collinear . Some points are joined to others by line segments, with each point connected to no more than one line segment . If we have a pair of intersecting line segments $AB$ and $CD$ we decide to replace them with $AC$ and $BD$, which are opposite sides of quadrilateral $ABCD$. In the resulting system of segments we decide to perform a similar substitution, if possible, and so on . Is it possible that such substitutions can be carried out indefinitely?
(V.E. Kolosov)
1952 Moscow Mathematical Olympiad, 224-
You are given a segment $AB$. Find the locus of the vertices $C$ of acute-angled triangles $ABC$.
1982 Brazil National Olympiad, 5
Show how to construct a line segment length $(a^4 + b^4)^{1/4}$ given segments lengths $a$ and $b$.
2011 Greece JBMO TST, 2
On every side of a square $ABCD$, we consider three points different (to each other).
a) Find the number of line segments defined with endpoints those points , that do not lie on sides of the square.
b) If there are no three of the previous line segments passing through the same point, find how many of the intersection points of those segmens line in the interior of the square.