10.9

Part of V Soros Olympiad 1998 - 99 (Russia)

Problems(2)

min network in equiangular hexagon (V Soros Olympiad 1998-99 Round 1 10.9)

Source:

Six cities are located at the vertices of a convex hexagon, all angles of which are equal. Three sides of this hexagon have length

a

, and the remaining three have length

b

a \le b

). It is necessary to connect these cities with a network of roads so that from each city you can drive to any other (possibly through other cities). Find the shortest length of such a road network.

geometrygeometric inequalityhexagon

cut a paper triangle of area 1 (V Soros Olympiad 1998-99 Round 3 10.9)

Source:

A triangle of area

1

is cut out of paper. Prove that it can be bent along a straight segment so that the area of the resulting figure is less than

s_0

s_0=\frac{\sqrt5-1}{2}

.
Note. The value

s_0

specified in the condition can be reduced (the smallest value of

s_0

is unknown to the authors of the problem). If you manage to do this (and justify it), write.

geometrygeometric inequalityareas