\section*{Problem 1 - 161221}

Let \(R\) be a rectangle with area \(A\), side lengths \(a, b\), and diagonal length \(d\). Furthermore, let \(a\) be the arithmetic mean of \(b\) and \(d\).

Determine \(a, b\), and \(d\) for this rectangle in terms of \(A\).
}

The following relationships then apply

\[
A=a \cdot b \quad(1) \quad ; \quad d=\sqrt{a^{2}+b^{2}} \quad(2) \quad ; \quad a=\frac{b+d}{2}
\]

Rearranging (1) and (2) and substituting into (3) yields

\[
\begin{aligned}
a & =\frac{A+\sqrt{a^{4}+f^{2}}}{2 a} \\
\left(2 a^{2}-A\right)^{2} & =a^{4}+A^{2} \\
a^{2}\left(3 a^{2}-4 A\right) & =0
\end{aligned}
\]

with the solutions \(a_{1}=\frac{2 \sqrt{3}}{3} \sqrt{A}\) and \(a_{2}=0\), where \(a_{2}\) is discarded.

Substituting backwards into (1), (2) yields \(b=\frac{\sqrt{3}}{2} \sqrt{A}\) and \(d=\frac{5 \sqrt{3}}{6} \sqrt{A}\). A check in (3) confirms the correctness of the solution.