Found problems: 4
2011 Oral Moscow Geometry Olympiad, 6
One triangle lies inside another. Prove that at least one of the two smallest sides (out of six) is the side of the inner triangle.
2006 Sharygin Geometry Olympiad, 8.5
Is there a convex polygon with each side equal to some diagonal, and each diagonal equal to some side?
1982 Bundeswettbewerb Mathematik, 2
In a convex quadrilateral $ABCD$ sides $AB$ and $DC$ are both divided into $m$ equal parts by points $A, S_1 , S_2 , \ldots , S_{m-1} ,B$ and $D,T_1, T_2, \ldots , T_{m-1},C,$ respectively (in this order).
Similarly, sides $BC$ and $AD$ are divided into $n$ equal parts by points $B,U_1,U_2, \ldots, U_{n-1},C$ and $A,V_1,V_2, \ldots,V_{n-1}, D$. Prove that for $1 \leq i \leq m-1$ each of the segments $S_i T_i$ is divided by the segments $U_j V_j$ ($1\leq j \leq n-1$) into $n$ equal parts
1992 Chile National Olympiad, 5
In the $\triangle ABC $, points $ M, I, H $ are feet, respectively, of the median, bisector and height, drawn from $ A $. It is known that $ BC = 2 $, $ MI = 2-\sqrt {3} $ and $ AB > AC $.
a) Prove that $ I$ lies between $ M $ and $ H $.
b) Calculate $ AB ^ 2-AC ^ 2 $.
c) Determine $ \dfrac {AB} {AC} $.
d) Find the measure of all the sides and angles of the triangle.