MHB What Are the Possible Values of (ab+cd)/(ad+bc)?

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    2017
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The discussion focuses on finding the possible values of the expression (ab + cd) / (ad + bc) given the conditions a^2 - ad + d^2 = b^2 + bc + c^2 and a^2 + b^2 = c^2 + d^2, where a, b, c, and d are positive real numbers. Participants explore various mathematical approaches and techniques to derive the solution. The correct solution was provided by Opalg, highlighting the importance of understanding the relationships between the variables. The problem encourages engagement with advanced algebraic concepts and fosters collaborative problem-solving. Overall, the thread emphasizes the analytical exploration of mathematical expressions under specific constraints.
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Here is this week's POTW:

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Given positive real numbers $a,\,b,\,c,\,d$ satisfy the equalities

$a^2-ad+d^2=b^2+bc+c^2$ and $a^2+b^2=c^2+d^2$,

find all possible values of the expression $\dfrac{ab+cd}{ad+bc}$.

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Congratulations to Opalg for his correct solution:), which you can find below:
Everything in this problem is homogeneous of degree 2, so we can multiply all four numbers $a,b,c,d$ by the same positive constant and the problem will be unchanged. Therefore, choosing a suitable constant, we can assume that $a^2+b^2 = c^2 + d^2 = 1.$ Then there must exist $\theta,\ \phi$ in $(0,\pi/2)$ such that $$a = \cos\theta, \quad b = \sin\theta, \quad c = \cos\phi,
\quad d = \sin\phi.$$ The equation $a^2 - ad + d^2 = b^2 + bc + c^2$ becomes $$\cos^2\theta - \cos\theta\sin\phi + \sin^2\phi = \sin^2\theta + \sin\theta \cos\phi + \cos^2\phi,$$ so that $$\sin\theta \cos\phi + \cos\theta\sin\phi = \cos^2\theta - \sin^2\theta - \cos^2\phi + \sin^2\phi,$$ $$\sin(\theta + \phi) = \cos(2\theta) - \cos(2\phi) = 2\sin(\theta + \phi)\sin(\theta - \phi).$$ Therefore $\sin(\theta - \phi) = \frac12$, which implies that $\cos(\theta - \phi) = \frac{\sqrt3}2.$

Next, $$\begin{aligned} \frac{ab+cd}{ad+bc} &= \frac{\cos\theta\sin\theta + \cos\phi\sin\phi}{\cos\theta\sin\phi + \sin\theta\cos\phi} \\ &= \frac{\sin(2\theta) + \sin(2\phi)}{2\sin(\theta + \phi)} \\ &= \frac{2\sin(\theta + \phi)\cos(\theta - \phi)}{2\sin(\theta + \phi)} \\ &= \cos(\theta - \phi) = \frac{\sqrt3}2. \end{aligned}$$

Therefore $$\frac{ab+cd}{ad+bc} = \frac{\sqrt3}2.$$ So that is the only possible value of the expression.
 
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