Found problems: 4
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?
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.
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.