There is a one-to-one correspondence betwen ordered triples having properties (i) and (ii) and the set of functions \(f:A\rightarrow\left\{1,2,3,\ \left\{1,2\right\},\ \left\{1,3\right\},\ \left\{2,3\right\}\right\}\).  Since each value of \(f\) is determined independently, there are \(6^{\left|A\right|}=6^{10}\) such functions.  Thus, the answer is \(c=d=0,\ a=10,\ b=10\).
A2.  Let  \(R\) and \(T\) be fixed.   Denote by \(U\) the acute triangle formed by the top edge of and the two intersecting sides of \(T\).  Because it is similar to T\(U\) is an acute triangle, and \(U\) is fixed.  We will first find the maximum value of the ratio \(\frac{A\left(S\right)}{A\left(U\right)}\) for each \(U\).  Let \(V\) be the triangle formed by the top of \(S\) and the sides of \(U\).  Then