ÖMO 2004

Solve the following system of equations in real numbers: $\begin{cases} a^2 = \cfrac{\sqrt{bc}\sqrt[3]{bcd}}{(b+c)(b+c+d)} b^2 =\cfrac{\sqrt{cd}\sqrt[3]{cda}}{(c+d)(c+d+a)} c^2 =\cfrac{\sqrt{da}\sqrt[3]{dab}}{(d+a)(d+a+b)} d^2 =\cfrac{\sqrt{ab}\sqrt[3]{abc}}{(a+b)(a+b+c)} \end{cases}$

Over the sides of an equilateral triangle with area $1$ are triangles with the opposite angle $60^{\circ}$ to each side drawn outside of the triangle. The new corners are $P$ , $Q$ and $R$ . (and the new triangles $APB$ , $BQC$ and $ARC$ ) 1)What is the highest possible area of the triangle $PQR$ ? 2)What is the highest possible area of the triangle whose vertexes are the midpoints of the inscribed circles of the triangles $APB$ , $BQC$ and $ARC$ ?