$$\begin{align} a+b+c+d &= 110 \\ \frac{a}{b} + \frac{c}{d} &= \frac{5}{4} \\ ad - bc &=-242 \\ (c-a)(b-d) &=- 605\end{align}$$
Solve for a,b,c,d
Origin of the question:
This crazy question came as a part of a practice sheet of questions which is required for foreign nationals to write if they want to attend a compulsory preparatory course to take university in Germany.
How an Ideal solution should look like:
I'm trying to figure out a clean way to solve this.
My thoughts
Sneaky Determinants?
The third equation looks like the determinant of the following matrix:
$$ \begin{bmatrix} a & b \\ c & d\end{bmatrix}$$
Following this, we can think of the second equation of the determinant of the following matrix:
$$ \frac{1}{\sqrt{bd}} \begin{bmatrix} a& -b \\ c & d \end{bmatrix}$$
Observing this maybe we can say something about the product? Or maybe something like the adjoint. This particular topic, it's been so long since I've done that I've forgotten any trick to apply.
Brute forcing?
I notice that through the 2,3 equation we can solve for ad and bc in terms of bd:
$$ \begin{align} ad &= \frac{\frac{5}{4} bd -242}{2} \\ bc &= \frac{\frac{5}{4} bd +242}{2} \end{align}$$
We can add equation these two new equations with the third one to get an expression having five out of the six possible product of variables taken two at a time:
$$ ab+cd+ad+bc = \frac{5}{2} bd +605$$
One issue that I think leads to the difficulty of solving this is that in the first equation only involves a product of single terms, while all others have product of two terms.
It all seems like a dead end. Any tips?